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Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Bhabha scatteringâ Status of Theory Predictions â
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
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
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. BhabhaâThe Scattering of Positrons by Electrons with Exchange on Diracâs Theory of the Positronâ[1] Proc. Roy. Soc. A154 (1936) 195
Homi J. Bhabha (1909 â 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 Indiaâs 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 â Bhabha published [1] â the first calculation of the the cross section of electron-positron scattering.1945 â He established the Tata Institute of Fundamental Research in Bombay . . .1948 â . . . 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.
2/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Bhabhaâs article
Here is the first page of [1]H. BhabhaThe Scattering of Positrons byElectrons with Exchange onDiracâs Theory of the PositronProc. Roy. Soc., A154, p. 195(1936)
3/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Bhabhaâs formula I
⢠|Ms +Mt |2
⢠simple process, me = 0
⢠strong forward peak⟠1/t2
⢠But: ¾-pair etc. productionadvantageous ?
dĎ0
2ĎdĎ=Îą2
s
( s
t+ 1 +
t
s
)2
Beam energy E , scattering angle Ď,and s, t:
s = 4E2
t =s
2(1â cosĎ)
4/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Outline
Earlier presentations at FCAL meetings:[2] 2005 Tel Aviv Forward Bhabha Scattering â 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.
5/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Electroweak Born diagrams
The Born diagramsDiagram 1; topology 4s
Îł
e
e
e
e
Diagram 2; topology 4s
Z
e
e
e
e
Diagram 3; topology 4s
H
e
e
e
e
Diagram 4; topology 4s
Ď
e
e
e
e
Diagram 5; topology 4t
Îł
e
e
e
e
Diagram 6; topology 4t
Z
e
e
e
e
Diagram 7; topology 4t
H
e
e
e
e
Diagram 8; topology 4t
Ď
e
e
e
e
s = M2Z ,500,3000 GeV2 Ďmin = 26,39,44 mrad
tmin = s2(1â cosĎmin) = 1.5,0.2,1.5 GeV2
m2e
t ⤠2Ă 10â6 safely neglect me
m2Âľ
t ⤠6Ă 10â2 care about heavier massesHiggs diagrams: play no role [even at meson factories]Z diagrams: contribute at GigaZ and ILC, see later.
6/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Electroweak Born cross-sections
The Bhabha Born cross-section:
dĎew
dΊ=
Îą2
4s(Ts + Tst + Tt ) ,
with Z axial coupling a = 1 and vector coupling v = 1â 4s2w ,
Ts = (1 + cos2θ)
[1 + 2ReĎ(s)
(v2)
+ |Ď(s)|2(
1 + v2)2]
+ 2 cos θ[2ReĎ(s) + |Ď(s)|2
(4v2)],
Tst = â2(1 + cos θ)2
(1â cos θ)
1 + [Ď(t) + ReĎ(s)]
(1 + v2
)+ Ď(t)ReĎ(s)
[(1 + v2)2 + 4v2
],
Tt = 2(1 + cos θ)2
(1â cos θ)2
1 + 2Ď(t)
(1 + v2
)+ Ď(t)2
[(1 + v2)2 + 4v2
]+
8
(1â cos θ)2
[1â Ď(t)
(1â v2
)]2.
We choose the following conventions:
Ď(s) =GFâ
2
M2Z
8ĎÎą
s
s â M2Z + iMZ ÎZ
,
Ď(t) =GFâ
2
M2Z
8ĎÎą
t
t â M2Z
, t < 0.
7/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Input quantities
Beware:Among the quantities
Îą, GF , s2w , MZ , ÎZ , MW
there are only three independent, and ÎZ is predicted by thetheory.
The phrasing effective Born cross-section means here that weuse, in the Born estimates the following five input variables:
Îą = 1/137.036,s2
w = 0.23,MZ = 91.1876Âą 0.0021 GeV,
ÎZ = 2.4952Âą 0.0023 GeV,
GF = (1.16637Âą 0.00001)Ă 10â5 GeVâ2.
8/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Born ew/Born QED at large and small angles [7]
Bhabha scatteringÎŁew
ÎŁQED
cos Î=-0.1
cos Î=+0.1
0 500 1000 1500 2000 2500 30000
5
10
15
20
25
!!!!s
Bhabha scattering
ÎŁew
ÎŁQED
cos Î=0.99
cos Î=0.999
cos Î=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
âs.
9/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Born ew/Born QED at different energies; large angles [7]
Bhabha scattering
ÎŁew
ÎŁQED
cos Î=-0.1
cos Î=+0.1
20 40 60 80 1000
5
10
15
20
25
!!!!s
Bhabha scattering
ÎŁew
ÎŁQED
cos Î=-0.1
cos Î=+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).
10/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
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/Ëmoenig//bhabha_ilc.pdf
⢠ILC: e+eâ â W +Wâ, f f with O(106) eventsâ 10â3
⢠ILC: e+eâ â e+eâ, a probe for New Physics with O(105) events/yearâ 10â3
⢠GigaZ: e+eâ â Z â had, l+lâ, the latter with O(108) eventsâ 10â4
Conclude: will need:
âL/L â 2Ă 10â4
Luminosity L â 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:
(Îą/Ď) = 2Ă 10â3 NLO â 1-loop + real
(Îą/Ď)2 = 0.6Ă 10â5 NNLO â 2-loop + real
Additional factors of the type log(
sme
)n, n = 1, 2 [, 3] at two loops
11/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
804 electroweak 1-loop Bhabha diagrams
The calculation of electroweak 1-loop corrections plus real QEDbremsstrahlung is fully controlled and automatized.QED bremsstrahlungâ MC programs: there are other experts,not me.
Diagram 1; topology 4s7a
XÎł
e
e
e
e
Diagram 2; topology 4s7a
XZ
e
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e
Diagram 3; topology 4s7a
XH
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Diagram 4; topology 4s7a
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Diagram 5; topology 4t7a
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Îł
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Diagram 6; topology 4t7a
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Diagram 7; topology 4t7a
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Diagram 8; topology 4t7a
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Diagram 9; topology 4t6a
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Diagram 10; topology 4t6a
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Diagram 11; topology 4t6a
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Diagram 12; topology 4t6a
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Diagram 13; topology 4s6a
XÎł
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Diagram 14; topology 4s6a
XZ
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Diagram 15; topology 4s6a
XH
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Diagram 16; topology 4s6a
XĎ
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Diagram 17; topology 4s5a
WÎł Îł
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Diagram 18; topology 4s5a
ĎÎł Îł
e
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e
Diagram 19; topology 4s5a
XÎł Îł
e
e
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Diagram 20; topology 4s5a
WÎł Z
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Diagram 21; topology 4s5a
ĎÎł Z
e
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Diagram 22; topology 4s5a
XÎł Z
e
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Diagram 23; topology 4s5a
WZ Îł
e
e
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e
Diagram 24; topology 4s5a
ĎZ Îł
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e
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Diagram 25; topology 4s5a
XZ Îł
e
e
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Diagram 26; topology 4s5a
WZ Z
e
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Diagram 27; topology 4s5a
HZ Z
e
e
e
e
Diagram 28; topology 4s5a
ĎZ Z
e
e
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Diagram 29; topology 4s5a
ĎZ Z
e
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e
Diagram 30; topology 4s5a
XZ Z
e
e
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e
Diagram 31; topology 4s5a
ZH H
e
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Diagram 32; topology 4s5a
WH H
e
e
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e
Diagram 33; topology 4s5a
HH H
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Diagram 34; topology 4s5a
ĎH H
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Diagram 35; topology 4s5a
ĎH H
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Diagram 36; topology 4s5a
XH H
e
e
e
e
Diagram 37; topology 4s5a
ZĎ Ď
e
e
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e
Diagram 38; topology 4s5a
WĎ Ď
e
e
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e
Diagram 39; topology 4s5a
HĎ Ď
e
e
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e
Diagram 40; topology 4s5a
ĎĎ Ď
e
e
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e
Diagram 41; topology 4s5a
ĎĎ Ď
e
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Diagram 42; topology 4s5a
XĎ Ď
e
e
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e
Diagram 43; topology 4t5a
W
Îł
Îł
e
e
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e
Diagram 44; topology 4t5a
Ď
Îł
Îł
e
e
e
e
Diagram 45; topology 4t5a
X
Îł
Îł
e
e
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e
Diagram 46; topology 4t5a
W
Îł
Z
e
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Diagram 47; topology 4t5a
Ď
Îł
Z
e
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e
Diagram 48; topology 4t5a
X
Îł
Z
e
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Diagram 49; topology 4t5a
W
Z
Îł
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Diagram 50; topology 4t5a
Ď
Z
Îł
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Diagram 51; topology 4t5a
X
Z
Îł
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Diagram 52; topology 4t5a
W
Z
Z
e
e
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Diagram 53; topology 4t5a
H
Z
Z
e
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e
e
Diagram 54; topology 4t5a
Ď
Z
Z
e
e
e
e
Diagram 55; topology 4t5a
Ď
Z
Z
e
e
e
e
Diagram 56; topology 4t5a
X
Z
Z
e
e
e
e
Diagram 57; topology 4t5a
Z
H
H
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Diagram 58; topology 4t5a
W
H
H
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Diagram 59; topology 4t5a
H
H
H
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Diagram 60; topology 4t5a
Ď
H
H
e
e
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e
Diagram 61; topology 4t5a
Ď
H
H
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Diagram 62; topology 4t5a
X
H
H
e
e
e
e
Diagram 63; topology 4t5a
Z
Ď
Ď
e
e
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e
Diagram 64; topology 4t5a
W
Ď
Ď
e
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Diagram 65; topology 4t5a
H
Ď
Ď
e
e
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e
Diagram 66; topology 4t5a
Ď
Ď
Ď
e
e
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e
Diagram 67; topology 4t5a
Ď
Ď
Ď
e
e
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e
Diagram 68; topology 4t5a
X
Ď
Ď
e
e
e
e
Diagram 69; topology 4s5b
W
W
Îł Îł
e
e
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e
Diagram 70; topology 4s5b
W
Ď
Îł Îł
e
e
e
e
Diagram 71; topology 4s5b
W
Ď
Îł Îł
e
e
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e
Diagram 72; topology 4s5b
ĎĎ
Îł Îł
e
e
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e
Diagram 73; topology 4s5b
e
e
Îł Îł
e
e
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e
Diagram 74; topology 4s5b
¾¾
Îł Îł
e
e
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Diagram 75; topology 4s5b
ĎĎ
Îł Îł
e
e
e
e
Diagram 76; topology 4s5b
u
u
Îł Îł
e
e
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e
Diagram 77; topology 4s5b
c
c
Îł Îł
e
e
e
e
Diagram 78; topology 4s5b
t
t
Îł Îł
e
e
e
e
Diagram 79; topology 4s5b
d
d
Îł Îł
e
e
e
e
Diagram 80; topology 4s5b
s
s
Îł Îł
e
e
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e
Diagram 81; topology 4s5b
b
b
Îł Îł
e
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e
Diagram 82; topology 4s5b
Ρ W-
Ρ W-
Îł Îł
e
e
e
e
Diagram 83; topology 4s5b
Ρ W+
Ρ W+
Îł Îł
e
e
e
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Diagram 84; topology 4s5b
W
W
Îł Z
e
e
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e
Diagram 85; topology 4s5b
W
Ď
Îł Z
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Diagram 86; topology 4s5b
W
Ď
Îł Z
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Diagram 87; topology 4s5b
ĎĎ
Îł Z
e
e
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Diagram 88; topology 4s5b
e
e
Îł Z
e
e
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Diagram 89; topology 4s5b
¾¾
Îł Z
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Diagram 90; topology 4s5b
ĎĎ
Îł Z
e
e
e
e
Diagram 91; topology 4s5b
u
u
Îł Z
e
e
e
e
Diagram 92; topology 4s5b
c
c
Îł Z
e
e
e
e
Diagram 93; topology 4s5b
t
t
Îł Z
e
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Diagram 94; topology 4s5b
d
d
Îł Z
e
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Diagram 95; topology 4s5b
s
s
Îł Z
e
e
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Diagram 96; topology 4s5b
b
b
Îł Z
e
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Diagram 97; topology 4s5b
Ρ W-
Ρ W-
Îł Z
e
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Diagram 98; topology 4s5b
Ρ W+
Ρ W+
Îł Z
e
e
e
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Diagram 99; topology 4s5b
W
W
Îł H
e
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Diagram 100; topology 4s5b
W
Ď
Îł H
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Diagram 101; topology 4s5b
W
Ď
Îł H
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Diagram 102; topology 4s5b
ĎĎ
Îł H
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Diagram 103; topology 4s5b
e
e
Îł H
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Diagram 104; topology 4s5b
¾¾
Îł H
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Diagram 105; topology 4s5b
ĎĎ
Îł H
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Diagram 106; topology 4s5b
u
u
Îł H
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Diagram 107; topology 4s5b
c
c
Îł H
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Diagram 108; topology 4s5b
t
t
Îł H
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Diagram 109; topology 4s5b
d
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Îł H
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Diagram 110; topology 4s5b
s
s
Îł H
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Diagram 111; topology 4s5b
b
b
Îł H
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Diagram 112; topology 4s5b
Ρ W-
Ρ W-
Îł H
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Diagram 113; topology 4s5b
Ρ W+
Ρ W+
Îł H
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Diagram 114; topology 4s5b
W
Ď
Îł Ď
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Diagram 115; topology 4s5b
W
Ď
Îł Ď
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Diagram 116; topology 4s5b
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Îł Ď
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Diagram 117; topology 4s5b
¾¾
Îł Ď
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Diagram 118; topology 4s5b
ĎĎ
Îł Ď
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Diagram 119; topology 4s5b
u
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Îł Ď
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Diagram 120; topology 4s5b
c
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Îł Ď
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Diagram 121; topology 4s5b
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Îł Ď
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Diagram 122; topology 4s5b
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Îł Ď
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Diagram 123; topology 4s5b
s
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Îł Ď
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Diagram 124; topology 4s5b
b
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Îł Ď
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Diagram 125; topology 4s5b
Ρ W-
Ρ W-
Îł Ď
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Diagram 126; topology 4s5b
Ρ W+
Ρ W+
Îł Ď
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Diagram 127; topology 4s5b
W
W
Z Îł
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Diagram 128; topology 4s5b
W
Ď
Z Îł
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Diagram 129; topology 4s5b
W
Ď
Z Îł
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Diagram 130; topology 4s5b
ĎĎ
Z Îł
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Diagram 131; topology 4s5b
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Z Îł
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Diagram 132; topology 4s5b
¾¾
Z Îł
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Diagram 133; topology 4s5b
ĎĎ
Z Îł
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Diagram 134; topology 4s5b
u
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Z Îł
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Diagram 135; topology 4s5b
c
c
Z Îł
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e
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e
Diagram 136; topology 4s5b
t
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Z Îł
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Diagram 137; topology 4s5b
d
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Z Îł
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Diagram 138; topology 4s5b
s
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Z Îł
e
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Diagram 139; topology 4s5b
b
b
Z Îł
e
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e
Diagram 140; topology 4s5b
Ρ W-
Ρ W-
Z Îł
e
e
e
e
Diagram 141; topology 4s5b
Ρ W+
Ρ W+
Z Îł
e
e
e
e
Diagram 142; topology 4s5b
H
Z
Z Z
e
e
e
e
Diagram 143; topology 4s5b
W
W
Z Z
e
e
e
e
Diagram 144; topology 4s5b
W
Ď
Z Z
e
e
e
e
Diagram 145; topology 4s5b
W
Ď
Z Z
e
e
e
e
Diagram 146; topology 4s5b
H
Ď
Z Z
e
e
e
e
Diagram 147; topology 4s5b
ĎĎ
Z Z
e
e
e
e
Diagram 148; topology 4s5b
νe
νe
Z Z
e
e
e
e
Diagram 149; topology 4s5b
ν¾
ν¾
Z Z
e
e
e
e
Diagram 150; topology 4s5b
ν Ď
ν Ď
Z Z
e
e
e
e
Diagram 151; topology 4s5b
e
e
Z Z
e
e
e
e
Diagram 152; topology 4s5b
¾¾
Z Z
e
e
e
e
Diagram 153; topology 4s5b
ĎĎ
Z Z
e
e
e
e
Diagram 154; topology 4s5b
u
u
Z Z
e
e
e
e
Diagram 155; topology 4s5b
c
c
Z Z
e
e
e
e
Diagram 156; topology 4s5b
t
t
Z Z
e
e
e
e
Diagram 157; topology 4s5b
d
d
Z Z
e
e
e
e
Diagram 158; topology 4s5b
s
s
Z Z
e
e
e
e
Diagram 159; topology 4s5b
b
b
Z Z
e
e
e
e
Diagram 160; topology 4s5b
Ρ W-
Ρ W-
Z Z
e
e
e
e
Diagram 161; topology 4s5b
Ρ W+
Ρ W+
Z Z
e
e
e
e
Diagram 162; topology 4s5b
W
W
Z H
e
e
e
e
Diagram 163; topology 4s5b
W
Ď
Z H
e
e
e
e
Diagram 164; topology 4s5b
W
Ď
Z H
e
e
e
e
Diagram 165; topology 4s5b
ĎĎ
Z H
e
e
e
e
Diagram 166; topology 4s5b
e
e
Z H
e
e
e
e
Diagram 167; topology 4s5b
¾¾
Z H
e
e
e
e
Diagram 168; topology 4s5b
ĎĎ
Z H
e
e
e
e
Diagram 169; topology 4s5b
u
u
Z H
e
e
e
e
Diagram 170; topology 4s5b
c
c
Z H
e
e
e
e
Diagram 171; topology 4s5b
t
t
Z H
e
e
e
e
Diagram 172; topology 4s5b
d
d
Z H
e
e
e
e
Diagram 173; topology 4s5b
s
s
Z H
e
e
e
e
Diagram 174; topology 4s5b
b
b
Z H
e
e
e
e
Diagram 175; topology 4s5b
Ρ W-
Ρ W-
Z H
e
e
e
e
Diagram 176; topology 4s5b
Ρ W+
Ρ W+
Z H
e
e
e
e
Diagram 177; topology 4s5b
H
Z
Z Ď
e
e
e
e
Diagram 178; topology 4s5b
W
Ď
Z Ď
e
e
e
e
Diagram 179; topology 4s5b
W
Ď
Z Ď
e
e
e
e
Diagram 180; topology 4s5b
H
Ď
Z Ď
e
e
e
e
Diagram 181; topology 4s5b
e
e
Z Ď
e
e
e
e
Diagram 182; topology 4s5b
¾¾
Z Ď
e
e
e
e
Diagram 183; topology 4s5b
ĎĎ
Z Ď
e
e
e
e
Diagram 184; topology 4s5b
u
u
Z Ď
e
e
e
e
Diagram 185; topology 4s5b
c
c
Z Ď
e
e
e
e
Diagram 186; topology 4s5b
t
t
Z Ď
e
e
e
e
Diagram 187; topology 4s5b
d
d
Z Ď
e
e
e
e
Diagram 188; topology 4s5b
s
s
Z Ď
e
e
e
e
Diagram 189; topology 4s5b
b
b
Z Ď
e
e
e
e
Diagram 190; topology 4s5b
Ρ W-
Ρ W-
Z Ď
e
e
e
e
Diagram 191; topology 4s5b
Ρ W+
Ρ W+
Z Ď
e
e
e
e
Diagram 192; topology 4s5b
W
W
H Îł
e
e
e
e
Diagram 193; topology 4s5b
W
Ď
H Îł
e
e
e
e
Diagram 194; topology 4s5b
W
Ď
H Îł
e
e
e
e
Diagram 195; topology 4s5b
ĎĎ
H Îł
e
e
e
e
Diagram 196; topology 4s5b
e
e
H Îł
e
e
e
e
Diagram 197; topology 4s5b
¾¾
H Îł
e
e
e
e
Diagram 198; topology 4s5b
ĎĎ
H Îł
e
e
e
e
Diagram 199; topology 4s5b
u
u
H Îł
e
e
e
e
Diagram 200; topology 4s5b
c
c
H Îł
e
e
e
e
Diagram 201; topology 4s5b
t
t
H Îł
e
e
e
e
Diagram 202; topology 4s5b
d
d
H Îł
e
e
e
e
Diagram 203; topology 4s5b
s
s
H Îł
e
e
e
e
Diagram 204; topology 4s5b
b
b
H Îł
e
e
e
e
Diagram 205; topology 4s5b
Ρ W-
Ρ W-
H Îł
e
e
e
e
Diagram 206; topology 4s5b
Ρ W+
Ρ W+
H Îł
e
e
e
e
Diagram 207; topology 4s5b
W
W
H Z
e
e
e
e
Diagram 208; topology 4s5b
W
Ď
H Z
e
e
e
e
Diagram 209; topology 4s5b
W
Ď
H Z
e
e
e
e
Diagram 210; topology 4s5b
ĎĎ
H Z
e
e
e
e
Diagram 211; topology 4s5b
e
e
H Z
e
e
e
e
Diagram 212; topology 4s5b
¾¾
H Z
e
e
e
e
Diagram 213; topology 4s5b
ĎĎ
H Z
e
e
e
e
Diagram 214; topology 4s5b
u
u
H Z
e
e
e
e
Diagram 215; topology 4s5b
c
c
H Z
e
e
e
e
Diagram 216; topology 4s5b
t
t
H Z
e
e
e
e
Diagram 217; topology 4s5b
d
d
H Z
e
e
e
e
Diagram 218; topology 4s5b
s
s
H Z
e
e
e
e
Diagram 219; topology 4s5b
b
b
H Z
e
e
e
e
Diagram 220; topology 4s5b
Ρ W-
Ρ W-
H Z
e
e
e
e
Diagram 221; topology 4s5b
Ρ W+
Ρ W+
H Z
e
e
e
e
Diagram 222; topology 4s5b
Z
Z
H H
e
e
e
e
Diagram 223; topology 4s5b
Z
Ď
H H
e
e
e
e
Diagram 224; topology 4s5b
W
W
H H
e
e
e
e
Diagram 225; topology 4s5b
W
Ď
H H
e
e
e
e
Diagram 226; topology 4s5b
W
Ď
H H
e
e
e
e
Diagram 227; topology 4s5b
H
H
H H
e
e
e
e
Diagram 228; topology 4s5b
ĎĎ
H H
e
e
e
e
Diagram 229; topology 4s5b
ĎĎ
H H
e
e
e
e
Diagram 230; topology 4s5b
e
e
H H
e
e
e
e
Diagram 231; topology 4s5b
¾¾
H H
e
e
e
e
Diagram 232; topology 4s5b
ĎĎ
H H
e
e
e
e
Diagram 233; topology 4s5b
u
u
H H
e
e
e
e
Diagram 234; topology 4s5b
c
c
H H
e
e
e
e
Diagram 235; topology 4s5b
t
t
H H
e
e
e
e
Diagram 236; topology 4s5b
d
d
H H
e
e
e
e
Diagram 237; topology 4s5b
s
s
H H
e
e
e
e
Diagram 238; topology 4s5b
b
b
H H
e
e
e
e
Diagram 239; topology 4s5b
Ρ Z
Ρ Z
H H
e
e
e
e
Diagram 240; topology 4s5b
Ρ W-
Ρ W-
H H
e
e
e
e
Diagram 241; topology 4s5b
Ρ W+
Ρ W+
H H
e
e
e
e
Diagram 242; topology 4s5b
W
Ď
H Ď
e
e
e
e
Diagram 243; topology 4s5b
W
Ď
H Ď
e
e
e
e
Diagram 244; topology 4s5b
e
e
H Ď
e
e
e
e
Diagram 245; topology 4s5b
¾¾
H Ď
e
e
e
e
Diagram 246; topology 4s5b
ĎĎ
H Ď
e
e
e
e
Diagram 247; topology 4s5b
u
u
H Ď
e
e
e
e
Diagram 248; topology 4s5b
c
c
H Ď
e
e
e
e
Diagram 249; topology 4s5b
t
t
H Ď
e
e
e
e
Diagram 250; topology 4s5b
d
d
H Ď
e
e
e
e
Diagram 251; topology 4s5b
s
s
H Ď
e
e
e
e
Diagram 252; topology 4s5b
b
b
H Ď
e
e
e
e
Diagram 253; topology 4s5b
Ρ W-
Ρ W-
H Ď
e
e
e
e
Diagram 254; topology 4s5b
Ρ W+
Ρ W+
H Ď
e
e
e
e
Diagram 255; topology 4s5b
W
Ď
Ď Îł
e
e
e
e
Diagram 256; topology 4s5b
W
Ď
Ď Îł
e
e
e
e
Diagram 257; topology 4s5b
e
e
Ď Îł
e
e
e
e
Diagram 258; topology 4s5b
¾¾
Ď Îł
e
e
e
e
Diagram 259; topology 4s5b
ĎĎ
Ď Îł
e
e
e
e
Diagram 260; topology 4s5b
u
u
Ď Îł
e
e
e
e
Diagram 261; topology 4s5b
c
c
Ď Îł
e
e
e
e
Diagram 262; topology 4s5b
t
t
Ď Îł
e
e
e
e
Diagram 263; topology 4s5b
d
d
Ď Îł
e
e
e
e
Diagram 264; topology 4s5b
s
s
Ď Îł
e
e
e
e
Diagram 265; topology 4s5b
b
b
Ď Îł
e
e
e
e
Diagram 266; topology 4s5b
Ρ W-
Ρ W-
Ď Îł
e
e
e
e
Diagram 267; topology 4s5b
Ρ W+
Ρ W+
Ď Îł
e
e
e
e
Diagram 268; topology 4s5b
H
Z
Ď Z
e
e
e
e
Diagram 269; topology 4s5b
W
Ď
Ď Z
e
e
e
e
Diagram 270; topology 4s5b
W
Ď
Ď Z
e
e
e
e
Diagram 271; topology 4s5b
H
Ď
Ď Z
e
e
e
e
Diagram 272; topology 4s5b
e
e
Ď Z
e
e
e
e
Diagram 273; topology 4s5b
¾¾
Ď Z
e
e
e
e
Diagram 274; topology 4s5b
ĎĎ
Ď Z
e
e
e
e
Diagram 275; topology 4s5b
u
u
Ď Z
e
e
e
e
Diagram 276; topology 4s5b
c
c
Ď Z
e
e
e
e
Diagram 277; topology 4s5b
t
t
Ď Z
e
e
e
e
Diagram 278; topology 4s5b
d
d
Ď Z
e
e
e
e
Diagram 279; topology 4s5b
s
s
Ď Z
e
e
e
e
Diagram 280; topology 4s5b
b
b
Ď Z
e
e
e
e
Diagram 281; topology 4s5b
Ρ W-
Ρ W-
Ď Z
e
e
e
e
Diagram 282; topology 4s5b
Ρ W+
Ρ W+
Ď Z
e
e
e
e
Diagram 283; topology 4s5b
W
Ď
Ď H
e
e
e
e
Diagram 284; topology 4s5b
W
Ď
Ď H
e
e
e
e
Diagram 285; topology 4s5b
e
e
Ď H
e
e
e
e
Diagram 286; topology 4s5b
¾¾
Ď H
e
e
e
e
Diagram 287; topology 4s5b
ĎĎ
Ď H
e
e
e
e
Diagram 288; topology 4s5b
u
u
Ď H
e
e
e
e
Diagram 289; topology 4s5b
c
c
Ď H
e
e
e
e
Diagram 290; topology 4s5b
t
t
Ď H
e
e
e
e
Diagram 291; topology 4s5b
d
d
Ď H
e
e
e
e
Diagram 292; topology 4s5b
s
s
Ď H
e
e
e
e
Diagram 293; topology 4s5b
b
b
Ď H
e
e
e
e
Diagram 294; topology 4s5b
Ρ W-
Ρ W-
Ď H
e
e
e
e
Diagram 295; topology 4s5b
Ρ W+
Ρ W+
Ď H
e
e
e
e
Diagram 296; topology 4s5b
H
Z
Ď Ď
e
e
e
e
Diagram 297; topology 4s5b
W
Ď
Ď Ď
e
e
e
e
Diagram 298; topology 4s5b
W
Ď
Ď Ď
e
e
e
e
Diagram 299; topology 4s5b
H
Ď
Ď Ď
e
e
e
e
Diagram 300; topology 4s5b
e
e
Ď Ď
e
e
e
e
Diagram 301; topology 4s5b
¾¾
Ď Ď
e
e
e
e
Diagram 302; topology 4s5b
ĎĎ
Ď Ď
e
e
e
e
Diagram 303; topology 4s5b
u
u
Ď Ď
e
e
e
e
Diagram 304; topology 4s5b
c
c
Ď Ď
e
e
e
e
Diagram 305; topology 4s5b
t
t
Ď Ď
e
e
e
e
Diagram 306; topology 4s5b
d
d
Ď Ď
e
e
e
e
Diagram 307; topology 4s5b
s
s
Ď Ď
e
e
e
e
Diagram 308; topology 4s5b
b
b
Ď Ď
e
e
e
e
Diagram 309; topology 4s5b
Ρ W-
Ρ W-
Ď Ď
e
e
e
e
Diagram 310; topology 4s5b
Ρ W+
Ρ W+
Ď Ď
e
e
e
e
Diagram 311; topology 4t5b
WW
Îł
Îł
e
e
e
e
Diagram 312; topology 4t5b
WĎ
Îł
Îł
e
e
e
e
Diagram 313; topology 4t5b
WĎ
Îł
Îł
e
e
e
e
Diagram 314; topology 4t5b
ĎĎ
Îł
Îł
e
e
e
e
Diagram 315; topology 4t5b
ee
Îł
Îł
e
e
e
e
Diagram 316; topology 4t5b
¾¾
Îł
Îł
e
e
e
e
Diagram 317; topology 4t5b
ĎĎ
Îł
Îł
e
e
e
e
Diagram 318; topology 4t5b
uu
Îł
Îł
e
e
e
e
Diagram 319; topology 4t5b
cc
Îł
Îł
e
e
e
e
Diagram 320; topology 4t5b
tt
Îł
Îł
e
e
e
e
Diagram 321; topology 4t5b
dd
Îł
Îł
e
e
e
e
Diagram 322; topology 4t5b
ss
Îł
Îł
e
e
e
e
Diagram 323; topology 4t5b
bb
Îł
Îł
e
e
e
e
Diagram 324; topology 4t5b
Ρ W-Ρ W
-
Îł
Îł
e
e
e
e
Diagram 325; topology 4t5b
Ρ W+Ρ W
+
Îł
Îł
e
e
e
e
Diagram 326; topology 4t5b
WW
Îł
Z
e
e
e
e
Diagram 327; topology 4t5b
WĎ
Îł
Z
e
e
e
e
Diagram 328; topology 4t5b
WĎ
Îł
Z
e
e
e
e
Diagram 329; topology 4t5b
ĎĎ
Îł
Z
e
e
e
e
Diagram 330; topology 4t5b
ee
Îł
Z
e
e
e
e
Diagram 331; topology 4t5b
¾¾
Îł
Z
e
e
e
e
Diagram 332; topology 4t5b
ĎĎ
Îł
Z
e
e
e
e
Diagram 333; topology 4t5b
uu
Îł
Z
e
e
e
e
Diagram 334; topology 4t5b
cc
Îł
Z
e
e
e
e
Diagram 335; topology 4t5b
tt
Îł
Z
e
e
e
e
Diagram 336; topology 4t5b
dd
Îł
Z
e
e
e
e
Diagram 337; topology 4t5b
ss
Îł
Z
e
e
e
e
Diagram 338; topology 4t5b
bb
Îł
Z
e
e
e
e
Diagram 339; topology 4t5b
Ρ W-Ρ W
-
Îł
Z
e
e
e
e
Diagram 340; topology 4t5b
Ρ W+Ρ W
+
Îł
Z
e
e
e
e
Diagram 341; topology 4t5b
WW
Îł
H
e
e
e
e
Diagram 342; topology 4t5b
WĎ
Îł
H
e
e
e
e
Diagram 343; topology 4t5b
WĎ
Îł
H
e
e
e
e
Diagram 344; topology 4t5b
ĎĎ
Îł
H
e
e
e
e
Diagram 345; topology 4t5b
ee
Îł
H
e
e
e
e
Diagram 346; topology 4t5b
¾¾
Îł
H
e
e
e
e
Diagram 347; topology 4t5b
ĎĎ
Îł
H
e
e
e
e
Diagram 348; topology 4t5b
uu
Îł
H
e
e
e
e
Diagram 349; topology 4t5b
cc
Îł
H
e
e
e
e
Diagram 350; topology 4t5b
tt
Îł
H
e
e
e
e
Diagram 351; topology 4t5b
dd
Îł
H
e
e
e
e
Diagram 352; topology 4t5b
ss
Îł
H
e
e
e
e
Diagram 353; topology 4t5b
bb
Îł
H
e
e
e
e
Diagram 354; topology 4t5b
Ρ W-Ρ W
-
Îł
H
e
e
e
e
Diagram 355; topology 4t5b
Ρ W+Ρ W
+
Îł
H
e
e
e
e
Diagram 356; topology 4t5b
WĎ
Îł
Ď
e
e
e
e
Diagram 357; topology 4t5b
WĎ
Îł
Ď
e
e
e
e
Diagram 358; topology 4t5b
ee
Îł
Ď
e
e
e
e
Diagram 359; topology 4t5b
¾¾
Îł
Ď
e
e
e
e
Diagram 360; topology 4t5b
ĎĎ
Îł
Ď
e
e
e
e
Diagram 361; topology 4t5b
uu
Îł
Ď
e
e
e
e
Diagram 362; topology 4t5b
cc
Îł
Ď
e
e
e
e
Diagram 363; topology 4t5b
tt
Îł
Ď
e
e
e
e
Diagram 364; topology 4t5b
dd
Îł
Ď
e
e
e
e
Diagram 365; topology 4t5b
ss
Îł
Ď
e
e
e
e
Diagram 366; topology 4t5b
bb
Îł
Ď
e
e
e
e
Diagram 367; topology 4t5b
Ρ W-Ρ W
-
Îł
Ď
e
e
e
e
Diagram 368; topology 4t5b
Ρ W+Ρ W
+
Îł
Ď
e
e
e
e
Diagram 369; topology 4t5b
WW
Z
Îł
e
e
e
e
Diagram 370; topology 4t5b
WĎ
Z
Îł
e
e
e
e
Diagram 371; topology 4t5b
WĎ
Z
Îł
e
e
e
e
Diagram 372; topology 4t5b
ĎĎ
Z
Îł
e
e
e
e
Diagram 373; topology 4t5b
ee
Z
Îł
e
e
e
e
Diagram 374; topology 4t5b
¾¾
Z
Îł
e
e
e
e
Diagram 375; topology 4t5b
ĎĎ
Z
Îł
e
e
e
e
Diagram 376; topology 4t5b
uu
Z
Îł
e
e
e
e
Diagram 377; topology 4t5b
cc
Z
Îł
e
e
e
e
Diagram 378; topology 4t5b
tt
Z
Îł
e
e
e
e
Diagram 379; topology 4t5b
dd
Z
Îł
e
e
e
e
Diagram 380; topology 4t5b
ss
Z
Îł
e
e
e
e
Diagram 381; topology 4t5b
bb
Z
Îł
e
e
e
e
Diagram 382; topology 4t5b
Ρ W-Ρ W
-
Z
Îł
e
e
e
e
Diagram 383; topology 4t5b
Ρ W+Ρ W
+
Z
Îł
e
e
e
e
Diagram 384; topology 4t5b
HZ
Z
Z
e
e
e
e
Diagram 385; topology 4t5b
WW
Z
Z
e
e
e
e
Diagram 386; topology 4t5b
WĎ
Z
Z
e
e
e
e
Diagram 387; topology 4t5b
WĎ
Z
Z
e
e
e
e
Diagram 388; topology 4t5b
HĎ
Z
Z
e
e
e
e
Diagram 389; topology 4t5b
ĎĎ
Z
Z
e
e
e
e
Diagram 390; topology 4t5b
νeνe
Z
Z
e
e
e
e
Diagram 391; topology 4t5b
ν¾ν¾
Z
Z
e
e
e
e
Diagram 392; topology 4t5b
ν Ďν Ď
Z
Z
e
e
e
e
Diagram 393; topology 4t5b
ee
Z
Z
e
e
e
e
Diagram 394; topology 4t5b
¾¾
Z
Z
e
e
e
e
Diagram 395; topology 4t5b
ĎĎ
Z
Z
e
e
e
e
Diagram 396; topology 4t5b
uu
Z
Z
e
e
e
e
Diagram 397; topology 4t5b
cc
Z
Z
e
e
e
e
Diagram 398; topology 4t5b
tt
Z
Z
e
e
e
e
Diagram 399; topology 4t5b
dd
Z
Z
e
e
e
e
Diagram 400; topology 4t5b
ss
Z
Z
e
e
e
e
Diagram 401; topology 4t5b
bb
Z
Z
e
e
e
e
Diagram 402; topology 4t5b
Ρ W-Ρ W
-
Z
Z
e
e
e
e
Diagram 403; topology 4t5b
Ρ W+Ρ W
+
Z
Z
e
e
e
e
Diagram 404; topology 4t5b
WW
Z
H
e
e
e
e
Diagram 405; topology 4t5b
WĎ
Z
H
e
e
e
e
Diagram 406; topology 4t5b
WĎ
Z
H
e
e
e
e
Diagram 407; topology 4t5b
ĎĎ
Z
H
e
e
e
e
Diagram 408; topology 4t5b
ee
Z
H
e
e
e
e
Diagram 409; topology 4t5b
¾¾
Z
H
e
e
e
e
Diagram 410; topology 4t5b
ĎĎ
Z
H
e
e
e
e
Diagram 411; topology 4t5b
uu
Z
H
e
e
e
e
Diagram 412; topology 4t5b
cc
Z
H
e
e
e
e
Diagram 413; topology 4t5b
tt
Z
H
e
e
e
e
Diagram 414; topology 4t5b
dd
Z
H
e
e
e
e
Diagram 415; topology 4t5b
ss
Z
H
e
e
e
e
Diagram 416; topology 4t5b
bb
Z
H
e
e
e
e
Diagram 417; topology 4t5b
Ρ W-Ρ W
-
Z
H
e
e
e
e
Diagram 418; topology 4t5b
Ρ W+Ρ W
+
Z
H
e
e
e
e
Diagram 419; topology 4t5b
HZ
Z
Ď
e
e
e
e
Diagram 420; topology 4t5b
WĎ
Z
Ď
e
e
e
e
Diagram 421; topology 4t5b
WĎ
Z
Ď
e
e
e
e
Diagram 422; topology 4t5b
HĎ
Z
Ď
e
e
e
e
Diagram 423; topology 4t5b
ee
Z
Ď
e
e
e
e
Diagram 424; topology 4t5b
¾¾
Z
Ď
e
e
e
e
Diagram 425; topology 4t5b
ĎĎ
Z
Ď
e
e
e
e
Diagram 426; topology 4t5b
uu
Z
Ď
e
e
e
e
Diagram 427; topology 4t5b
cc
Z
Ď
e
e
e
e
Diagram 428; topology 4t5b
tt
Z
Ď
e
e
e
e
Diagram 429; topology 4t5b
dd
Z
Ď
e
e
e
e
Diagram 430; topology 4t5b
ss
Z
Ď
e
e
e
e
Diagram 431; topology 4t5b
bb
Z
Ď
e
e
e
e
Diagram 432; topology 4t5b
Ρ W-Ρ W
-
Z
Ď
e
e
e
e
Diagram 433; topology 4t5b
Ρ W+Ρ W
+
Z
Ď
e
e
e
e
Diagram 434; topology 4t5b
WW
H
Îł
e
e
e
e
Diagram 435; topology 4t5b
WĎ
H
Îł
e
e
e
e
Diagram 436; topology 4t5b
WĎ
H
Îł
e
e
e
e
Diagram 437; topology 4t5b
ĎĎ
H
Îł
e
e
e
e
Diagram 438; topology 4t5b
ee
H
Îł
e
e
e
e
Diagram 439; topology 4t5b
¾¾
H
Îł
e
e
e
e
Diagram 440; topology 4t5b
ĎĎ
H
Îł
e
e
e
e
Diagram 441; topology 4t5b
uu
H
Îł
e
e
e
e
Diagram 442; topology 4t5b
cc
H
Îł
e
e
e
e
Diagram 443; topology 4t5b
tt
H
Îł
e
e
e
e
Diagram 444; topology 4t5b
dd
H
Îł
e
e
e
e
Diagram 445; topology 4t5b
ss
H
Îł
e
e
e
e
Diagram 446; topology 4t5b
bb
H
Îł
e
e
e
e
Diagram 447; topology 4t5b
Ρ W-Ρ W
-
H
Îł
e
e
e
e
Diagram 448; topology 4t5b
Ρ W+Ρ W
+
H
Îł
e
e
e
e
Diagram 449; topology 4t5b
WW
H
Z
e
e
e
e
Diagram 450; topology 4t5b
WĎ
H
Z
e
e
e
e
Diagram 451; topology 4t5b
WĎ
H
Z
e
e
e
e
Diagram 452; topology 4t5b
ĎĎ
H
Z
e
e
e
e
Diagram 453; topology 4t5b
ee
H
Z
e
e
e
e
Diagram 454; topology 4t5b
¾¾
H
Z
e
e
e
e
Diagram 455; topology 4t5b
ĎĎ
H
Z
e
e
e
e
Diagram 456; topology 4t5b
uu
H
Z
e
e
e
e
Diagram 457; topology 4t5b
cc
H
Z
e
e
e
e
Diagram 458; topology 4t5b
tt
H
Z
e
e
e
e
Diagram 459; topology 4t5b
dd
H
Z
e
e
e
e
Diagram 460; topology 4t5b
ss
H
Z
e
e
e
e
Diagram 461; topology 4t5b
bb
H
Z
e
e
e
e
Diagram 462; topology 4t5b
Ρ W-Ρ W
-
H
Z
e
e
e
e
Diagram 463; topology 4t5b
Ρ W+Ρ W
+
H
Z
e
e
e
e
Diagram 464; topology 4t5b
ZZ
H
H
e
e
e
e
Diagram 465; topology 4t5b
ZĎ
H
H
e
e
e
e
Diagram 466; topology 4t5b
WW
H
H
e
e
e
e
Diagram 467; topology 4t5b
WĎ
H
H
e
e
e
e
Diagram 468; topology 4t5b
WĎ
H
H
e
e
e
e
Diagram 469; topology 4t5b
HH
H
H
e
e
e
e
Diagram 470; topology 4t5b
ĎĎ
H
H
e
e
e
e
Diagram 471; topology 4t5b
ĎĎ
H
H
e
e
e
e
Diagram 472; topology 4t5b
ee
H
H
e
e
e
e
Diagram 473; topology 4t5b
¾¾
H
H
e
e
e
e
Diagram 474; topology 4t5b
ĎĎ
H
H
e
e
e
e
Diagram 475; topology 4t5b
uu
H
H
e
e
e
e
Diagram 476; topology 4t5b
cc
H
H
e
e
e
e
Diagram 477; topology 4t5b
tt
H
H
e
e
e
e
Diagram 478; topology 4t5b
dd
H
H
e
e
e
e
Diagram 479; topology 4t5b
ss
H
H
e
e
e
e
Diagram 480; topology 4t5b
bb
H
H
e
e
e
e
Diagram 481; topology 4t5b
Ρ ZΡ Z
H
H
e
e
e
e
Diagram 482; topology 4t5b
Ρ W-Ρ W
-
H
H
e
e
e
e
Diagram 483; topology 4t5b
Ρ W+Ρ W
+
H
H
e
e
e
e
Diagram 484; topology 4t5b
WĎ
H
Ď
e
e
e
e
Diagram 485; topology 4t5b
WĎ
H
Ď
e
e
e
e
Diagram 486; topology 4t5b
ee
H
Ď
e
e
e
e
Diagram 487; topology 4t5b
¾¾
H
Ď
e
e
e
e
Diagram 488; topology 4t5b
ĎĎ
H
Ď
e
e
e
e
Diagram 489; topology 4t5b
uu
H
Ď
e
e
e
e
Diagram 490; topology 4t5b
cc
H
Ď
e
e
e
e
Diagram 491; topology 4t5b
tt
H
Ď
e
e
e
e
Diagram 492; topology 4t5b
dd
H
Ď
e
e
e
e
Diagram 493; topology 4t5b
ss
H
Ď
e
e
e
e
Diagram 494; topology 4t5b
bb
H
Ď
e
e
e
e
Diagram 495; topology 4t5b
Ρ W-Ρ W
-
H
Ď
e
e
e
e
Diagram 496; topology 4t5b
Ρ W+Ρ W
+
H
Ď
e
e
e
e
Diagram 497; topology 4t5b
WĎ
Ď
Îł
e
e
e
e
Diagram 498; topology 4t5b
WĎ
Ď
Îł
e
e
e
e
Diagram 499; topology 4t5b
ee
Ď
Îł
e
e
e
e
Diagram 500; topology 4t5b
¾¾
Ď
Îł
e
e
e
e
Diagram 501; topology 4t5b
ĎĎ
Ď
Îł
e
e
e
e
Diagram 502; topology 4t5b
uu
Ď
Îł
e
e
e
e
Diagram 503; topology 4t5b
cc
Ď
Îł
e
e
e
e
Diagram 504; topology 4t5b
tt
Ď
Îł
e
e
e
e
Diagram 505; topology 4t5b
dd
Ď
Îł
e
e
e
e
Diagram 506; topology 4t5b
ss
Ď
Îł
e
e
e
e
Diagram 507; topology 4t5b
bb
Ď
Îł
e
e
e
e
Diagram 508; topology 4t5b
Ρ W-Ρ W
-
Ď
Îł
e
e
e
e
Diagram 509; topology 4t5b
Ρ W+Ρ W
+
Ď
Îł
e
e
e
e
Diagram 510; topology 4t5b
HZ
Ď
Z
e
e
e
e
Diagram 511; topology 4t5b
WĎ
Ď
Z
e
e
e
e
Diagram 512; topology 4t5b
WĎ
Ď
Z
e
e
e
e
Diagram 513; topology 4t5b
HĎ
Ď
Z
e
e
e
e
Diagram 514; topology 4t5b
ee
Ď
Z
e
e
e
e
Diagram 515; topology 4t5b
¾¾
Ď
Z
e
e
e
e
Diagram 516; topology 4t5b
ĎĎ
Ď
Z
e
e
e
e
Diagram 517; topology 4t5b
uu
Ď
Z
e
e
e
e
Diagram 518; topology 4t5b
cc
Ď
Z
e
e
e
e
Diagram 519; topology 4t5b
tt
Ď
Z
e
e
e
e
Diagram 520; topology 4t5b
dd
Ď
Z
e
e
e
e
Diagram 521; topology 4t5b
ss
Ď
Z
e
e
e
e
Diagram 522; topology 4t5b
bb
Ď
Z
e
e
e
e
Diagram 523; topology 4t5b
Ρ W-Ρ W
-
Ď
Z
e
e
e
e
Diagram 524; topology 4t5b
Ρ W+Ρ W
+
Ď
Z
e
e
e
e
Diagram 525; topology 4t5b
WĎ
Ď
H
e
e
e
e
Diagram 526; topology 4t5b
WĎ
Ď
H
e
e
e
e
Diagram 527; topology 4t5b
ee
Ď
H
e
e
e
e
Diagram 528; topology 4t5b
¾¾
Ď
H
e
e
e
e
Diagram 529; topology 4t5b
ĎĎ
Ď
H
e
e
e
e
Diagram 530; topology 4t5b
uu
Ď
H
e
e
e
e
Diagram 531; topology 4t5b
cc
Ď
H
e
e
e
e
Diagram 532; topology 4t5b
tt
Ď
H
e
e
e
e
Diagram 533; topology 4t5b
dd
Ď
H
e
e
e
e
Diagram 534; topology 4t5b
ss
Ď
H
e
e
e
e
Diagram 535; topology 4t5b
bb
Ď
H
e
e
e
e
Diagram 536; topology 4t5b
Ρ W-Ρ W
-
Ď
H
e
e
e
e
Diagram 537; topology 4t5b
Ρ W+Ρ W
+
Ď
H
e
e
e
e
Diagram 538; topology 4t5b
HZ
Ď
Ď
e
e
e
e
Diagram 539; topology 4t5b
WĎ
Ď
Ď
e
e
e
e
Diagram 540; topology 4t5b
WĎ
Ď
Ď
e
e
e
e
Diagram 541; topology 4t5b
HĎ
Ď
Ď
e
e
e
e
Diagram 542; topology 4t5b
ee
Ď
Ď
e
e
e
e
Diagram 543; topology 4t5b
¾¾
Ď
Ď
e
e
e
e
Diagram 544; topology 4t5b
ĎĎ
Ď
Ď
e
e
e
e
Diagram 545; topology 4t5b
uu
Ď
Ď
e
e
e
e
Diagram 546; topology 4t5b
cc
Ď
Ď
e
e
e
e
Diagram 547; topology 4t5b
tt
Ď
Ď
e
e
e
e
Diagram 548; topology 4t5b
dd
Ď
Ď
e
e
e
e
Diagram 549; topology 4t5b
ss
Ď
Ď
e
e
e
e
Diagram 550; topology 4t5b
bb
Ď
Ď
e
e
e
e
Diagram 551; topology 4t5b
Ρ W-Ρ W
-
Ď
Ď
e
e
e
e
Diagram 552; topology 4t5b
Ρ W+Ρ W
+
Ď
Ď
e
e
e
e
Diagram 553; topology 4s2c
e
eÎł
Îł
e
e
e
e
Diagram 554; topology 4s2c
e
eZ
Îł
e
e
e
e
Diagram 555; topology 4s2c
e
eH
Îł
e
e
e
e
Diagram 556; topology 4s2c
e
eĎ
Îł
e
e
e
e
Diagram 557; topology 4s2c
W
Wνe
Îł
e
e
e
e
Diagram 558; topology 4s2c
ĎW
νe
Îł
e
e
e
e
Diagram 559; topology 4s2c
W
Ďνe
Îł
e
e
e
e
Diagram 560; topology 4s2c
ĎĎ
νe
Îł
e
e
e
e
Diagram 561; topology 4s2c
e
eÎł
Z
e
e
e
e
Diagram 562; topology 4s2c
e
eZ
Z
e
e
e
e
Diagram 563; topology 4s2c
νe
νeW
Z
e
e
e
e
Diagram 564; topology 4s2c
e
eH
Z
e
e
e
e
Diagram 565; topology 4s2c
e
eĎ
Z
e
e
e
e
Diagram 566; topology 4s2c
νe
νeĎ
Z
e
e
e
e
Diagram 567; topology 4s2c
W
Wνe
Z
e
e
e
e
Diagram 568; topology 4s2c
ĎW
νe
Z
e
e
e
e
Diagram 569; topology 4s2c
W
Ďνe
Z
e
e
e
e
Diagram 570; topology 4s2c
ĎĎ
νe
Z
e
e
e
e
Diagram 571; topology 4s2c
H
Ze
Z
e
e
e
e
Diagram 572; topology 4s2c
Z
He
Z
e
e
e
e
Diagram 573; topology 4s2c
ĎH
eZ
e
e
e
e
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H
Ďe
Z
e
e
e
e
Diagram 575; topology 4s2c
e
eÎł
H
e
e
e
e
Diagram 576; topology 4s2c
e
eZ
H
e
e
e
e
Diagram 577; topology 4s2c
e
eH
H
e
e
e
e
Diagram 578; topology 4s2c
e
eĎ
H
e
e
e
e
Diagram 579; topology 4s2c
W
Wνe
H
e
e
e
e
Diagram 580; topology 4s2c
ĎW
νe
H
e
e
e
e
Diagram 581; topology 4s2c
W
Ďνe
H
e
e
e
e
Diagram 582; topology 4s2c
ĎĎ
νe
H
e
e
e
e
Diagram 583; topology 4s2c
Z
Ze
H
e
e
e
e
Diagram 584; topology 4s2c
ĎZ
eH
e
e
e
e
Diagram 585; topology 4s2c
H
He
H
e
e
e
e
Diagram 586; topology 4s2c
Z
Ďe
H
e
e
e
e
Diagram 587; topology 4s2c
ĎĎ
eH
e
e
e
e
Diagram 588; topology 4s2c
e
eÎł
Ď
e
e
e
e
Diagram 589; topology 4s2c
e
eZ
Ď
e
e
e
e
Diagram 590; topology 4s2c
e
eH
Ď
e
e
e
e
Diagram 591; topology 4s2c
e
eĎ
Ď
e
e
e
e
Diagram 592; topology 4s2c
ĎW
νe
Ď
e
e
e
e
Diagram 593; topology 4s2c
W
Ďνe
Ď
e
e
e
e
Diagram 594; topology 4s2c
H
Ze
Ď
e
e
e
e
Diagram 595; topology 4s2c
Z
He
Ď
e
e
e
e
Diagram 596; topology 4s2c
ĎH
eĎ
e
e
e
e
Diagram 597; topology 4s2c
H
Ďe
Ď
e
e
e
e
Diagram 598; topology 4t2c
e e
Îł
Îłe
e
e
e
Diagram 599; topology 4t2c
e e
Z
Îłe
e
e
e
Diagram 600; topology 4t2c
e e
H
Îłe
e
e
e
Diagram 601; topology 4t2c
e e
Ď
Îłe
e
e
e
Diagram 602; topology 4t2c
W W
νe
Îłe
e
e
e
Diagram 603; topology 4t2c
W Ď
νe
Îłe
e
e
e
Diagram 604; topology 4t2c
Ď W
νe
Îłe
e
e
e
Diagram 605; topology 4t2c
Ď Ď
νe
Îłe
e
e
e
Diagram 606; topology 4t2c
e e
Îł
Ze
e
e
e
Diagram 607; topology 4t2c
e e
Z
Ze
e
e
e
Diagram 608; topology 4t2c
νe νe
W
Ze
e
e
e
Diagram 609; topology 4t2c
e e
H
Ze
e
e
e
Diagram 610; topology 4t2c
e e
Ď
Ze
e
e
e
Diagram 611; topology 4t2c
νe νe
Ď
Ze
e
e
e
Diagram 612; topology 4t2c
W W
νe
Ze
e
e
e
Diagram 613; topology 4t2c
W Ď
νe
Ze
e
e
e
Diagram 614; topology 4t2c
Ď W
νe
Ze
e
e
e
Diagram 615; topology 4t2c
Ď Ď
νe
Ze
e
e
e
Diagram 616; topology 4t2c
Z H
e
Ze
e
e
e
Diagram 617; topology 4t2c
H Z
e
Ze
e
e
e
Diagram 618; topology 4t2c
H Ď
e
Ze
e
e
e
Diagram 619; topology 4t2c
Ď H
e
Ze
e
e
e
Diagram 620; topology 4t2c
e e
Îł
He
e
e
e
Diagram 621; topology 4t2c
e e
Z
He
e
e
e
Diagram 622; topology 4t2c
e e
H
He
e
e
e
Diagram 623; topology 4t2c
e e
Ď
He
e
e
e
Diagram 624; topology 4t2c
W W
νe
He
e
e
e
Diagram 625; topology 4t2c
W Ď
νe
He
e
e
e
Diagram 626; topology 4t2c
Ď W
νe
He
e
e
e
Diagram 627; topology 4t2c
Ď Ď
νe
He
e
e
e
Diagram 628; topology 4t2c
Z Z
e
He
e
e
e
Diagram 629; topology 4t2c
Z Ď
e
He
e
e
e
Diagram 630; topology 4t2c
H H
e
He
e
e
e
Diagram 631; topology 4t2c
Ď Z
e
He
e
e
e
Diagram 632; topology 4t2c
Ď Ď
e
He
e
e
e
Diagram 633; topology 4t2c
e e
Îł
Ďe
e
e
e
Diagram 634; topology 4t2c
e e
Z
Ďe
e
e
e
Diagram 635; topology 4t2c
e e
H
Ďe
e
e
e
Diagram 636; topology 4t2c
e e
Ď
Ďe
e
e
e
Diagram 637; topology 4t2c
W Ď
νe
Ďe
e
e
e
Diagram 638; topology 4t2c
Ď W
νe
Ďe
e
e
e
Diagram 639; topology 4t2c
Z H
e
Ďe
e
e
e
Diagram 640; topology 4t2c
H Z
e
Ďe
e
e
e
Diagram 641; topology 4t2c
H Ď
e
Ďe
e
e
e
Diagram 642; topology 4t2c
Ď H
e
Ďe
e
e
e
Diagram 643; topology 4t1c
ee
Îł
Îł
e
e
e
e
Diagram 644; topology 4t1c
ee
Z
Îł
e
e
e
e
Diagram 645; topology 4t1c
ee
H
Îł
e
e
e
e
Diagram 646; topology 4t1c
ee
Ď
Îł
e
e
e
e
Diagram 647; topology 4t1c
WW
νe
Îł
e
e
e
e
Diagram 648; topology 4t1c
ĎW
νe
Îł
e
e
e
e
Diagram 649; topology 4t1c
WĎ
νe
Îł
e
e
e
e
Diagram 650; topology 4t1c
ĎĎ
νe
Îł
e
e
e
e
Diagram 651; topology 4t1c
ee
Îł
Z
e
e
e
e
Diagram 652; topology 4t1c
ee
Z
Z
e
e
e
e
Diagram 653; topology 4t1c
νeνe
W
Z
e
e
e
e
Diagram 654; topology 4t1c
ee
H
Z
e
e
e
e
Diagram 655; topology 4t1c
ee
Ď
Z
e
e
e
e
Diagram 656; topology 4t1c
νeνe
Ď
Z
e
e
e
e
Diagram 657; topology 4t1c
WW
νe
Z
e
e
e
e
Diagram 658; topology 4t1c
ĎW
νe
Z
e
e
e
e
Diagram 659; topology 4t1c
WĎ
νe
Z
e
e
e
e
Diagram 660; topology 4t1c
ĎĎ
νe
Z
e
e
e
e
Diagram 661; topology 4t1c
HZ
e
Z
e
e
e
e
Diagram 662; topology 4t1c
ZH
e
Z
e
e
e
e
Diagram 663; topology 4t1c
ĎH
e
Z
e
e
e
e
Diagram 664; topology 4t1c
HĎ
e
Z
e
e
e
e
Diagram 665; topology 4t1c
ee
Îł
H
e
e
e
e
Diagram 666; topology 4t1c
ee
Z
H
e
e
e
e
Diagram 667; topology 4t1c
ee
H
H
e
e
e
e
Diagram 668; topology 4t1c
ee
Ď
H
e
e
e
e
Diagram 669; topology 4t1c
WW
νe
H
e
e
e
e
Diagram 670; topology 4t1c
ĎW
νe
H
e
e
e
e
Diagram 671; topology 4t1c
WĎ
νe
H
e
e
e
e
Diagram 672; topology 4t1c
ĎĎ
νe
H
e
e
e
e
Diagram 673; topology 4t1c
ZZ
e
H
e
e
e
e
Diagram 674; topology 4t1c
ĎZ
e
H
e
e
e
e
Diagram 675; topology 4t1c
HH
e
H
e
e
e
e
Diagram 676; topology 4t1c
ZĎ
e
H
e
e
e
e
Diagram 677; topology 4t1c
ĎĎ
e
H
e
e
e
e
Diagram 678; topology 4t1c
ee
Îł
Ď
e
e
e
e
Diagram 679; topology 4t1c
ee
Z
Ď
e
e
e
e
Diagram 680; topology 4t1c
ee
H
Ď
e
e
e
e
Diagram 681; topology 4t1c
ee
Ď
Ď
e
e
e
e
Diagram 682; topology 4t1c
ĎW
νe
Ď
e
e
e
e
Diagram 683; topology 4t1c
WĎ
νe
Ď
e
e
e
e
Diagram 684; topology 4t1c
HZ
e
Ď
e
e
e
e
Diagram 685; topology 4t1c
ZH
e
Ď
e
e
e
e
Diagram 686; topology 4t1c
ĎH
e
Ď
e
e
e
e
Diagram 687; topology 4t1c
HĎ
e
Ď
e
e
e
e
Diagram 688; topology 4s1c
e
e
γγ
e
e
e
e
Diagram 689; topology 4s1c
e
e
ZÎł
e
e
e
e
Diagram 690; topology 4s1c
e
e
HÎł
e
e
e
e
Diagram 691; topology 4s1c
e
e
ĎÎł
e
e
e
e
Diagram 692; topology 4s1c
W
W
νe
Îł
e
e
e
e
Diagram 693; topology 4s1c
W
Ď
νe
Îł
e
e
e
e
Diagram 694; topology 4s1c
Ď
W
νe
Îł
e
e
e
e
Diagram 695; topology 4s1c
Ď
Ď
νe
Îł
e
e
e
e
Diagram 696; topology 4s1c
e
e
ÎłZ
e
e
e
e
Diagram 697; topology 4s1c
e
e
ZZ
e
e
e
e
Diagram 698; topology 4s1c
νe
νe
WZ
e
e
e
e
Diagram 699; topology 4s1c
e
e
HZ
e
e
e
e
Diagram 700; topology 4s1c
e
e
ĎZ
e
e
e
e
Diagram 701; topology 4s1c
νe
νe
ĎZ
e
e
e
e
Diagram 702; topology 4s1c
W
W
νe
Z
e
e
e
e
Diagram 703; topology 4s1c
W
Ď
νe
Z
e
e
e
e
Diagram 704; topology 4s1c
Ď
W
νe
Z
e
e
e
e
Diagram 705; topology 4s1c
Ď
Ď
νe
Z
e
e
e
e
Diagram 706; topology 4s1c
Z
H
eZ
e
e
e
e
Diagram 707; topology 4s1c
H
Z
eZ
e
e
e
e
Diagram 708; topology 4s1c
H
Ď
eZ
e
e
e
e
Diagram 709; topology 4s1c
Ď
H
eZ
e
e
e
e
Diagram 710; topology 4s1c
e
e
ÎłH
e
e
e
e
Diagram 711; topology 4s1c
e
e
ZH
e
e
e
e
Diagram 712; topology 4s1c
e
e
HH
e
e
e
e
Diagram 713; topology 4s1c
e
e
ĎH
e
e
e
e
Diagram 714; topology 4s1c
W
W
νe
H
e
e
e
e
Diagram 715; topology 4s1c
W
Ď
νe
H
e
e
e
e
Diagram 716; topology 4s1c
Ď
W
νe
H
e
e
e
e
Diagram 717; topology 4s1c
Ď
Ď
νe
H
e
e
e
e
Diagram 718; topology 4s1c
Z
Z
eH
e
e
e
e
Diagram 719; topology 4s1c
Z
Ď
eH
e
e
e
e
Diagram 720; topology 4s1c
H
H
eH
e
e
e
e
Diagram 721; topology 4s1c
Ď
Z
eH
e
e
e
e
Diagram 722; topology 4s1c
Ď
Ď
eH
e
e
e
e
Diagram 723; topology 4s1c
e
e
ÎłĎ
e
e
e
e
Diagram 724; topology 4s1c
e
e
ZĎ
e
e
e
e
Diagram 725; topology 4s1c
e
e
HĎ
e
e
e
e
Diagram 726; topology 4s1c
e
e
ĎĎ
e
e
e
e
Diagram 727; topology 4s1c
W
Ď
νe
Ď
e
e
e
e
Diagram 728; topology 4s1c
Ď
W
νe
Ď
e
e
e
e
Diagram 729; topology 4s1c
Z
H
eĎ
e
e
e
e
Diagram 730; topology 4s1c
H
Z
eĎ
e
e
e
e
Diagram 731; topology 4s1c
H
Ď
eĎ
e
e
e
e
Diagram 732; topology 4s1c
Ď
H
eĎ
e
e
e
e
Diagram 733; topology 4x1d
e
Îł
e
Îł
e
e
e
e
Diagram 734; topology 4x1d
e
Z
e
Îł
e
e
e
e
Diagram 735; topology 4x1d
e
H
e
Îł
e
e
e
e
Diagram 736; topology 4x1d
e
Ď
e
Îł
e
e
e
e
Diagram 737; topology 4x1d
e
Îł
e
Z
e
e
e
e
Diagram 738; topology 4x1d
e
Z
e
Z
e
e
e
e
Diagram 739; topology 4x1d
e
H
e
Z
e
e
e
e
Diagram 740; topology 4x1d
e
Ď
e
Z
e
e
e
e
Diagram 741; topology 4x1d
νe
W
νe
W
e
e
e
e
Diagram 742; topology 4x1d
νe
Ď
νe
W
e
e
e
e
Diagram 743; topology 4x1d
e
Îł
e
H
e
e
e
e
Diagram 744; topology 4x1d
e
Z
e
H
e
e
e
e
Diagram 745; topology 4x1d
e
H
e
H
e
e
e
e
Diagram 746; topology 4x1d
e
Ď
e
H
e
e
e
e
Diagram 747; topology 4x1d
e
Îł
e
Ď
e
e
e
e
Diagram 748; topology 4x1d
e
Z
e
Ď
e
e
e
e
Diagram 749; topology 4x1d
e
H
e
Ď
e
e
e
e
Diagram 750; topology 4x1d
e
Ď
e
Ď
e
e
e
e
Diagram 751; topology 4x1d
νe
W
νe
Ď
e
e
e
e
Diagram 752; topology 4x1d
νe
Ď
νe
Ď
e
e
e
e
Diagram 753; topology 4x1d
W
νe
W
νe
e
e
e
e
Diagram 754; topology 4x1d
W
νe
Ď
νe
e
e
e
e
Diagram 755; topology 4x1d
Ď
νe
W
νe
e
e
e
e
Diagram 756; topology 4x1d
Ď
νe
Ď
νe
e
e
e
e
Diagram 757; topology 4x1d
Îł
e
Îł
e
e
e
e
e
Diagram 758; topology 4x1d
Îł
e
Z
e
e
e
e
e
Diagram 759; topology 4x1d
Îł
e
H
e
e
e
e
e
Diagram 760; topology 4x1d
Îł
e
Ď
e
e
e
e
e
Diagram 761; topology 4x1d
Z
e
Îł
e
e
e
e
e
Diagram 762; topology 4x1d
Z
e
Z
e
e
e
e
e
Diagram 763; topology 4x1d
Z
e
H
e
e
e
e
e
Diagram 764; topology 4x1d
Z
e
Ď
e
e
e
e
e
Diagram 765; topology 4x1d
H
e
Îł
e
e
e
e
e
Diagram 766; topology 4x1d
H
e
Z
e
e
e
e
e
Diagram 767; topology 4x1d
H
e
H
e
e
e
e
e
Diagram 768; topology 4x1d
H
e
Ď
e
e
e
e
e
Diagram 769; topology 4x1d
Ď
e
Îł
e
e
e
e
e
Diagram 770; topology 4x1d
Ď
e
Z
e
e
e
e
e
Diagram 771; topology 4x1d
Ď
e
H
e
e
e
e
e
Diagram 772; topology 4x1d
Ď
e
Ď
e
e
e
e
e
Diagram 773; topology 4x2d
Îł
e
Îł
e
e
e
e
e
Diagram 774; topology 4x2d
Îł
e
Z
e
e
e
e
e
Diagram 775; topology 4x2d
Îł
e
H
e
e
e
e
e
Diagram 776; topology 4x2d
Îł
e
Ď
e
e
e
e
e
Diagram 777; topology 4x2d
Z
e
Îł
e
e
e
e
e
Diagram 778; topology 4x2d
Z
e
Z
e
e
e
e
e
Diagram 779; topology 4x2d
Z
e
H
e
e
e
e
e
Diagram 780; topology 4x2d
Z
e
Ď
e
e
e
e
e
Diagram 781; topology 4x2d
H
e
Îł
e
e
e
e
e
Diagram 782; topology 4x2d
H
e
Z
e
e
e
e
e
Diagram 783; topology 4x2d
H
e
H
e
e
e
e
e
Diagram 784; topology 4x2d
H
e
Ď
e
e
e
e
e
Diagram 785; topology 4x2d
Ď
e
Îł
e
e
e
e
e
Diagram 786; topology 4x2d
Ď
e
Z
e
e
e
e
e
Diagram 787; topology 4x2d
Ď
e
H
e
e
e
e
e
Diagram 788; topology 4x2d
Ď
e
Ď
e
e
e
e
e
Diagram 789; topology 4x3d
Îł
e
Îł
ee
e
e
e
Diagram 790; topology 4x3d
Z
e
Îł
ee
e
e
e
Diagram 791; topology 4x3d
H
e
Îł
ee
e
e
e
Diagram 792; topology 4x3d
Ď
e
Îł
ee
e
e
e
Diagram 793; topology 4x3d
Îł
e
Z
ee
e
e
e
Diagram 794; topology 4x3d
Z
e
Z
ee
e
e
e
Diagram 795; topology 4x3d
H
e
Z
ee
e
e
e
Diagram 796; topology 4x3d
Ď
e
Z
ee
e
e
e
Diagram 797; topology 4x3d
Îł
e
H
ee
e
e
e
Diagram 798; topology 4x3d
Z
e
H
ee
e
e
e
Diagram 799; topology 4x3d
H
e
H
ee
e
e
e
Diagram 800; topology 4x3d
Ď
e
H
ee
e
e
e
Diagram 801; topology 4x3d
Îł
e
Ď
ee
e
e
e
Diagram 802; topology 4x3d
Z
e
Ď
ee
e
e
e
Diagram 803; topology 4x3d
H
e
Ď
ee
e
e
e
Diagram 804; topology 4x3d
Ď
e
Ď
ee
e
e
e
12/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
1991 â 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/
13/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
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
14/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Bhabha scattering atâ
s = 500 GeV â EWSM versus QED[10] Fleischer, Lorca, Riemann, LCWS Paris 2004, Automatized calculation of2-fermion production with DIANA and aITALC
e+eâ â e+eââ
s = 500 GeV
cos θ[
dĎd cos θ
]Born
/pb[
dĎd cos θ
]B+1-loop
/pb Model
â.9000 0.216 998 0.144 359 EWSMâ.9000 0.523 873 0.387 798 QEDâ.5000 0.261 360 0.181 086 EWSMâ.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 Îąem has been switched off here
15/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Bhabha scattering atâ
s = MZ GeV
[11] Fleischer Gluza Lorca Riemann EPJC 2006, First order radiative corrections toBhabha scattering in d dimensions
cos θ Born EWSM O(Îą) EWSMâ0.9 0.12201 ¡ 104 0.11767 ¡ 104
â0.7 0.10099 ¡ 104 0.95012 ¡ 103
â0.5 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 atâ
s = mZ . Borncontribution and the O(Îą) correction are shown; the maximum soft-photon energy isâ
s/10.
16/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Bhabha scattering atâ
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(Îą) EWSM O(Îą) QED Nf = 92.691 â0.9 2.16999 ¡ 10â1 1.93445 ¡ 10â1 â10.85% 4.69800 ¡ 10â1 116.50%2.346 â0.7 2.30098 ¡ 10â1 2.08843 ¡ 10â1 â9.24% 5.03879 ¡ 10â1 118.98%2.094 â0.5 2.61360 ¡ 10â1 2.38707 ¡ 10â1 â8.67% 5.66238 ¡ 10â1 116.65%1.571 0 5.98142 ¡ 10â1 5.46677 ¡ 10â1 â8.60% 1.09322 ¡ 100 82.77%1.047 +0.5 4.21273 ¡ 100 3.81301 ¡ 100 â9.49% 5.13530 ¡ 100 21.90%0.795 +0.7 1.58240 ¡ 101 1.43357 ¡ 101 â9.41% 1.64548 ¡ 101 3.99%0.451 +0.9 1.89160 ¡ 102 1.72928 ¡ 102 â8.58% 1.76464 ¡ 102 â6.71%0.142 +0.99 2.06556 ¡ 104 1.90607 ¡ 104 â7.72% 1.91774 ¡ 104 â7.16%0.045 +0.999 2.08236 ¡ 106 1.91624 ¡ 106 â7.98% 1.92546 ¡ 106 â7.53%0.014 +0.9999 2.08429 ¡ 108 1.91402 ¡ 108 â8.17% 1.92270 ¡ 108 â7.75%
Differential cross-sections in pbarn for Bhabha scattering atâ
s = 500. Borncontribution, the O(Îą) correction, and also a QED prediction are shown; the maximumsoft-photon energy is
âs/10.
17/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Bhabha scattering atâ
s = 500 GeV
[12] Gluza Lorca Riemann ACAT 2003, KEK Tsukuba, Automated use of DIANA fortwo-fermion production at colliders
Table: Cross-sections for eâe+ â eâe+ (Îł) atâ
s = 500 GeV with aphoton energy cut Emax
Îłsoft=â
s/10.
cos θ[
dĎd cos θ
]O(Îą3)
(pb) Group
â0.9 0.19344 50785 26863 6 aITALCâ0.9 0.19344 50785 26862 2 FA/FC/LTâ0.9 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
18/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Overview
The evaluation of the NNLO corrections is nearly finished.The present MC-programs contain âthe most important of the NNLO correctionsâalready â whatever this means.What it means is not in all cases evident.Why?
(I) The leading NNLLO [next-to-next-toâleading-Log] corrections are proportional to
Îą2
Ď2log2(t/m2
e),Îą2
Ď2log(t/m2
e)
But we seem to need also the constant terms [in m2e/s], without lop-factors
Îą2
Ď2log2(t/m2
e),Îą2
Ď2
(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 â not
New theoretical techniques are needed to finalize the theoretical predictions.
19/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
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, Âľ, Ď, quarks
Add the realcorrections
⢠real 2-photon emissionâ JG
⢠electron pair emissionâ JG
20/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added
Recent progress for the Nf = 1 massive predictions
Massive next-to-next-to-leadingâLogarithmic 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
21/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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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,â PRLTransform the massless 2-loop results of Bern, Dixon, Ghinculov (2002) with InfraRed(IR) regulation by D = 4â 2Îľ into the on-mass-shell renormalization with me â 0 andIR regulation by Îť = mÎł 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).
22/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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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:
eâ
e+ e+
eâe+
eâ
Îł
Îł
Îł
Îł
eâ eâ
eâ
e+
e+ e+
eâ eâ
eâ
e+
e+e+
Îł
Îł
Figure: Samples of the 36 diagrams contributing toe+eâ â e+eâ(e+eâ).
23/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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Not done for NNLO Nf = 1: loop-by-loop contributions from real photonemission
The one-loop diagrams for real photon emission:
e+ + eâ â e+ + eâ + Îł at QED 1-loop
In preparation: Gluza, Riemann, Yundin and Czyz et al.Show here the Âľ-pair sub-sample of diagrams:
Diagram 47; topology u34_
e γγ¾
Âľ
e
eÂľ
Âľ
Îł
Diagram 48; topology u35_
e
Îł
γ¾
Âľ
e
e Âľ
Âľ
Îł
Diagram 49; topology u36_
Îłe
Îł
e
Âľ
e
e Âľ
Âľ
Îł
Diagram 50; topology u37_
Îłe
Îłe
Âľ
e
e Âľ
Âľ
Îł
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+eâ â Âľ+ÂľâÎł
Note: Need 5-point tensor reduction [21] Fleischer, Riemann + C++ by Yundin
24/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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What remains: the virtual and real Nf = 2 corrections I
Show only the non-factorizing, truly complicated diagrams and the real corrections
Îł
eâ
e+e+
eâ
Îł Îł
Îł
eâ
e+e+
eâ
Îł
eâ
e+
eâ
e+
Îł
Îł
Îł
Figure: (a)â(c) are sample two-loop diagrams
25/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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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)
26/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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Backup: virtual Nf = 2 corrections II
mf
me
me
SE3l2M1m
B5l2M2md B5l2M2m
V4l2M2md
SE3l2M1md
V4l2M1m
V4l2M2m
V4l2M1md
27/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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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 â > [23] APP B38 (2007)and Bonciani,Ferroglia,Penin 0710.4775 (2007)
⢠m2e << m2
hadrons, s, t : ACGR 0711.3847 â > [6] PRL 100 (2008)and Kuehn et al. 0807.1284 (2008)
28/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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Backup: virtual Nf = 2 corrections I
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 ââ 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
29/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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Backup: virtual Nf = 2 corrections I
Dispersion Integrals
g¾νq2 + i δ
âg¾ι
q2 + i δ
(q2 gιβ â qÎą qβ
)Î had(q2)
gβνq2 + i δ
,
the once-subtracted dispersion integral
Î had(q2) = âq2
Ď
⍠â4M2
Ď
dzz
Im Î had(z)
q2 â z + i δ.
Finally, one relates Im Î had to the hadronic cross-section ratio Rhad,
ImÎ had(z) = âÎą
3Rhad(z) = â
Îą
3Ďe+eââhadrons(z)
(4ĎÎą2)/(3z),
For heavy fermion insertions, we have instead of Rhad(z):
Rf (z) = Q2f Cf (1 + 2m2
f /z)â
1â 4m2f /z,
Replacing the Î had(q2) in a vertex or in box diagram by the z-dispersion integral and
exchanging theâŤ
d4k with theâŤ
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
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Backup: virtual Nf = 2 corrections I
The kernel functions for the dispersion integrals
âÎą(x) = âÎą(5)had(x) + Î e(x) +
âf =Âľ,Ď,t
Î f (x)
âÎą(5)had(x) =
Îą
Ď
x3
⍠â4m2
Ď
dzR(5)
had(z)
z1
x â z + iδ
V2(x) = V2e(x) + V2rest(x)
V2rest(x) =
⍠â4M2
dzR(z)
zKV (x + iδ; z)
KV (x ; z) =13
â
78â
z2 x
+( 3
4+
z2 x
)ln(â
xz
)â
12
(1 +
zx
)2 [Îś2 â Li2
(1 +
xz
) ]
Bi (x , y) =
⍠â4M2
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.
31/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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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].
32/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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The real Âľ pair corrections
Under study: Worek, Gluza, TR et al.
eâ
e+
e+
eâ
Îł
Îł
Âľ+
Âľâ
eâ
e+ Âľ+
Âľâ
e+
eâ
Îł
Îł
eâ eâ
e+e+
Îł
Îł
Âľ+
Âľâ
eâ
e+ e+
eâ
Âľ+
Âľâ
Îł
Îł
Figure: Samples of the 12 diagrams contributing toe+eâ â e+eâÂľ+Âľâ.
33/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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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+
eâ
eâ
Ď+
Ďâ eâ
eâe=
e+
Ď+
Ďâ
γγ
e+ e+
Ď+
Ďâ
eâeâ
Îł
eâ eâ
e+ e+
Ď+
ĎâÎł
eâ eâ
e+ e+
Ď+
ĎâÎł
Îł
Sample diagrams with real pion pair emission, taken from [26].
34/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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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 . . .
35/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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References I
H. Bhabha, The Scattering of Positrons by Electrons with Exchange on Diracâs Theory of the Positron, Proc.Roy. Soc. A154 (1936) 195.
T. Riemann, Forward Bhabha scattering â theoretical problemstalk given at the ILC FCAL Collaboration Meeting, Sep. 2005, Tel-Aviv University, Israel,http://www-zeuthen.desy.de/Ëriemann/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/Ëriemann/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/Ëriemann/Talks/zeuthen-FCAL-2009-06.pdf.
S. Jadach, Theoretical error of luminosity cross section at LEP, talk presented at Mini-WorkshopâElectroweak Physics Data and the Higgs Massâ, 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(Îą2) corrections toBhabha scattering, Phys. Rev. D78 (2008) 085019, [0807.4691].
36/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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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, âBhabha scattering at the ILC.âTalk 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) 35â52, [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) 289â292, [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) 33â38, [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].
37/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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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) 121â179, [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) 419â427, [0909.1750].
K. Kajda, T. Sabonis, and V. Yundin, Qed pentagon contributions to e+eâ â Âľ+ÂľâÎł, 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) 26â51, [0704.2400v2].
M. Czakon, J. Gluza, and T. Riemann, On the massive two-loop corrections to bhabha scattering, Acta Phys.Polon. B36 (2005) 3319â3326, [hep-ph/0511187].
H. Burkhardt, âNew numerical analysis of the hadronic vacuum polarization.â
38/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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References IV
âFortran program rintpl.f.âPrivate 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+eâ â e+eâĎ+Ďâ and the pion form factor measurementsvia the radiative return method, Phys. Lett. B634 (2006) 493â497, [hep-ph/0601169].
K. Hagiwara, A. D. Martin, D. Nomura, and T. Teubner, Predictions for g â 2 of the muon and Îąqed(m2z ),
Phys. Rev. D69 (2004) 093003, [hep-ph/0312250].
K. Hagiwara, A. D. Martin, D. Nomura, and T. Teubner, Improved predictions for g â 2 of the muon and
Îąqed(m2z ), Phys. Lett. B649 (2007) 173â179, [hep-ph/0611102].
39/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010
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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.
40/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010