JW 2011

Donji Milanovac, Auguts 28, 2011

noncommutative standard model

Michael Wohlgenannt

– Typeset by FoilTEX – 1

in collaboration with

X. Calmet, B. Jurco, P.Schupp and Julius Wess (hep-ph/0111115)

lateron also with

B. Melic, K. Passek-Kummericki and J. Trampetic (hep-ph/0502249, hep-

ph/0503064)

– Typeset by FoilTEX – 2

straight forward way to deform a field theory

• take a commutative action

• insert ⋆,

f ⋆ g = ei2θµν∂x

µ∂yνf(x)g(y)

∣

∣

∣

y→x,

replace fields by NC fields, φ→ φ

• φ = φ[φ] given by SW-maps

• expand everything in terms of θµν

– Typeset by FoilTEX – 3

S =

∫

d4x

(

∑

i

ψ(i)L i /Dψ

(i)L +

∑

i

ψ(i)R i /Dψ

(i)R

−1

4g′fµνf

µν −1

2gFL

µνFLµν −

1

2gS

FSµνF

Sµν

+ (Dµφ)†Dµφ− µ2φ†φ− λ(φ†φ)2

−∑

ij

(W ijLLφ e(j)R +W †ije

(i)R φ†L

(j)L )

)

– Typeset by FoilTEX – 4

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

SU(3) ⊗ SU(2) ⊗ U(1)

Vµ = g′AµY +g

2Bµaσ

a +gS

2Gµaλ

a ,

Vα[V ] = Vα +1

4θµν{Vν, ∂µVα} +

1

4θµν{Fµα, Vν} + O(θ2) (1)

charge quantisation: solved by the SW-map A[A]

Yukawa couplingsφ = φ[φ,A,A′]

¯LL ⋆ ρL(φ) ⋆ eR

Y = 1/2 − 1/2 + 1 − 1

– Typeset by FoilTEX – 6

kinetic terms for gauge bosons:

−

∫

d4x

(

1

2g′tr1Fµν ⋆ Fµν +

1

2gtr2Fµν ⋆ Fµν +

1

2gS

tr3Fµν ⋆ Fµν

)

minimal model: use Y = 12

(

1 00 −1

)

in tr1

non-minimal model

−1

2

∫

d4x∑

ρ

κρTr(

ρ(Fµν) ⋆ ρ(Fµν))

– Typeset by FoilTEX – 7

Summary

• same particle content as commutative SM

• new interactions, processes forbidden in the commutative SM ⇒ limitson θ

• in minimal model no triple photon interaction

– Typeset by FoilTEX – 8

some further literature

• Behr, Deshpande, Duplancic, Schupp, Trampetic, J. Wess, Z → γγ,hep-ph/0202121

• Aschieri, Jurco, Schupp and J. Wess, NC GUTs, (hep-th/0205214)

• 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