1. JW 2011
Donji Milanovac, Auguts 28, 2011
noncommutative standard model
Michael Wohlgenannt
– Typeset by FoilTEX – 1

2. in collaboration with
X. Calmet, B. Jurˇo, P.Schupp and Julius Wess (hep-ph/0111115)
c
lateron also with
B. Meli´, K. Passek-Kummeriˇki and J. Trampeti´ (hep-ph/0502249, hep-
c c c
ph/0503064)
– Typeset by FoilTEX – 2

3. straight forward way to deform a field theory
• take a commutative action
• insert ⋆,
i µν x y
f ⋆g =e 2 θ ∂µ ∂ν f (x)g(y) ,
y→x
ˆ
replace fields by NC fields, φ → φ
ˆ ˆ
• φ = φ[φ] given by SW-maps
• expand everything in terms of θµν
– Typeset by FoilTEX – 3

4. S= d4 x ¯(i) / (i)
ψL iDψL + ¯(i) / (i)
ψR iDψR
i i
1 1 L 1 S Sµν
− ′
fµν f µν − Fµν F Lµν − Fµν F
4g 2g 2gS
+ (Dµφ)†Dµφ − µ2φ†φ − λ(φ†φ)2
¯ (j) (i) (j)
− (W ij LLφ eR + W †ij eR φ†LL )
ij
– Typeset by FoilTEX – 4

5. Problems:
• gauge group SU (3) ⊗ SU (2) ⊗ U (1)
• charge quantisation
• Yukawa couplings and Higgs
• kinetic terms for the gauge bosons
– Typeset by FoilTEX – 5

6. SU (3) ⊗ SU (2) ⊗ U (1)
g gS
Vµ = g AµY + Bµaσ + Gµaλa ,
′ a
2 2
ˆα[V ] = Vα + 1 θµν {Vν , ∂µVα} + 1 θµν {Fµα, Vν } + O(θ2)
V (1)
4 4
ˆ
charge quantisation: solved by the SW-map A[A]
Yukawa couplings
ˆ ˆ
φ = φ[φ, A, A′]
¯
ˆ ˆ
LL ⋆ ρL(φ) ⋆ eR
ˆ
Y = 1/2 − 1/2 + 1 −1
– Typeset by FoilTEX – 6

7. kinetic terms for gauge bosons:
4 1 ˆµν ⋆ Fµν + 1 tr2Fµν ⋆ Fµν + 1 tr3Fµν ⋆ Fµν
ˆ ˆ ˆ ˆ ˆ
− d x tr1F
2g ′ 2g 2gS
1 1 0
minimal model: use Y = 2 in tr1
0 −1
non-minimal model
1 ˆ ˆ
− d4 x κρTr ρ(Fµν ) ⋆ ρ(F µν )
2 ρ
– Typeset by FoilTEX – 7

8. Summary
• same particle content as commutative SM
• new interactions, processes forbidden in the commutative SM ⇒ limits
on θ
• in minimal model no triple photon interaction
– Typeset by FoilTEX – 8

9. some further literature
• Behr, Deshpande, Duplancic, Schupp, Trampetic, J. Wess, Z → γγ,
hep-ph/0202121
• Aschieri, Jurˇo, Schupp and J. Wess, NC GUTs, (hep-th/0205214)
c
• Schupp and Trampetic, NCSM and forbidden decays, hep-ph/0405163
• Ohl, Reuter, Testing NCSM, hep-ph/0406098
• Brandt, Martin and Ruiz-Ruiz, anomaly freedom, hep-th/0307292
– Typeset by FoilTEX – 9