On the Localization of a Renormalizable Translation Invariant U(1) NCGM

Transcription

1 On the Localization of a Renormalizable Translation Invariant U(1) NCGM Institute for Theoretical Physics, Vienna University of Technology in collaboration with: D. Blaschke, A. Rofner, M. Schweda May 17, 2009

2 Prerequisites and Assumptions: simple deformation, non-commuting space-time coordinates on R 4 θ : [ˆx µ, ˆx ν ] = iθ µν, with θ µν = θ , and θ R

3 Prerequisites and Assumptions: simple deformation, non-commuting space-time coordinates on R 4 θ : [ˆx µ, ˆx ν ] = iθ µν, with θ µν = θ , and θ R definition of the Groenewold-Moyal -product: f(x) g(x) = e i 2 θµν x µ y ν f(x)g(y) x=y g(x) f(x)

4 Prerequisites and Assumptions: simple deformation, non-commuting space-time coordinates on R 4 θ : [ˆx µ, ˆx ν ] = iθ µν, with θ µν = θ , and θ R definition of the Groenewold-Moyal -product: f(x) g(x) = e i 2 θµν x µ y ν f(x)g(y) x=y g(x) f(x) invariance under cyclic permutations of the integral d 4 x f(x) g(x) h(x) = d 4 x h(x) f(x) g(x) = d 4 xf(x) g(x) = d 4 xf(x)g(x)

5 Scalar approach Gauge model Localization by auxiliary fields A first scalar approach Naïve implementation of scalar Klein Gordon theory in R 4 θ : S[φ] = d 4 x 1 2 µφ µ φ m2 2 φ 2 λ 4! φ 4 leads to a propagator 1 G φ (k) = k 2 + m 2 1 R. Gurau, J. Magnen, V. Rivasseau and A. Tanasa, A translation-invariant renormalizable non-commutative scalar model, Commun. Math. Phys.287 (2009) , [arxiv: ]

6 Scalar approach Gauge model Localization by auxiliary fields A first scalar approach Naïve implementation of scalar Klein Gordon theory in R 4 θ : S[φ] = d 4 x 1 2 µφ µ φ m2 2 φ 2 λ 4! φ 4 leads to a propagator 1 G φ (k) = k 2 + m 2 k p c1 p + c 2 2 m 2 ln p 2 + F (p) UV/IR mixing 1 R. Gurau, J. Magnen, V. Rivasseau and A. Tanasa, A translation-invariant renormalizable non-commutative scalar model, Commun. Math. Phys.287 (2009) , [arxiv: ]

7 Scalar approach Gauge model Localization by auxiliary fields A first scalar approach Naïve implementation of scalar Klein Gordon theory in R 4 θ : S[φ] = d 4 x 1 2 µφ µ φ m2 2 φ 2 λ 4! φ 4 φ a2 φ leads to a propagator G φ (k) = 1 k 2 + m 2 + a2 k 2 k p c1 p + c 2 2 m 2 ln p 2 + F (p) UV/IR mixing 1 1 R. Gurau, J. Magnen, V. Rivasseau and A. Tanasa, A translation-invariant renormalizable non-commutative scalar model, Commun. Math. Phys.287 (2009) , [arxiv: ]

8 Scalar approach Gauge model Localization by auxiliary fields A first scalar approach Naïve implementation of scalar Klein Gordon theory in R 4 θ : S[φ] = d 4 x 1 2 µφ µ φ m2 2 φ 2 λ 4! φ 4 φ a2 φ leads to a propagator G φ (k) = 1 k 2 + m 2 + a2 k 2 k p c1 p + c 2 2 m 2 ln p 2 + F (p) UV/IR mixing...still the same result Q: Where are the improvements? 1 1 R. Gurau, J. Magnen, V. Rivasseau and A. Tanasa, A translation-invariant renormalizable non-commutative scalar model, Commun. Math. Phys.287 (2009) , [arxiv: ]

9 Scalar approach Gauge model Localization by auxiliary fields IR Damping Q: Where are the improvements? A: The new propagator damps in higher loop insertions... 2 D. N. Blaschke, F. Gieres, E. Kronberger, T. Reis, M. Schweda and R. I. P. Sedmik, Quantum Corrections for Translation-Invariant Renormalizable Non-Commutative Φ 4 Theory, JHEP 11 (2008) 074, [arxiv: ]

10 Scalar approach Gauge model Localization by auxiliary fields IR Damping Q: Where are the improvements? A: The new propagator damps in higher loop insertions... p q 1 q 2 q 3 p c η=±1 e iηkθp d 4 k (k 2 ) n [ ] k 2 + m 2 n+1 }{{} k k 0 c { 1 (k 2 n ) naïve model 2 D. N. Blaschke, F. Gieres, E. Kronberger, T. Reis, M. Schweda and R. I. P. Sedmik, Quantum Corrections for Translation-Invariant Renormalizable Non-Commutative Φ 4 Theory, JHEP 11 (2008) 074, [arxiv: ]

11 Scalar approach Gauge model Localization by auxiliary fields IR Damping Q: Where are the improvements? A: The new propagator damps in higher loop insertions... p q 1 q 2 q 3 p c η=±1 d 4 k e iηkθp (k 2 ) n [ ] k 2 + m 2 + a2 n+1 k }{{ 2 } k k 0 c { 1 (k 2 ) n naïve model k 2 a 2(n+1) 1 p 2 model 2 D. N. Blaschke, F. Gieres, E. Kronberger, T. Reis, M. Schweda and R. I. P. Sedmik, Quantum Corrections for Translation-Invariant Renormalizable Non-Commutative Φ 4 Theory, JHEP 11 (2008) 074, [arxiv: ]

12 Scalar approach Gauge model Localization by auxiliary fields IR Damping Q: Where are the improvements? A: The new propagator damps in higher loop insertions... p q 1 q 2 q 3 p c η=±1 d 4 k e iηkθp (k 2 ) n [ ] k 2 + m 2 + a2 n+1 k }{{ 2 } k k 0 c { 1 (k 2 ) n naïve model k 2 a 2(n+1) 1 p 2 model This result is independent of the order! 2 2 D. N. Blaschke, F. Gieres, E. Kronberger, T. Reis, M. Schweda and R. I. P. Sedmik, Quantum Corrections for Translation-Invariant Renormalizable Non-Commutative Φ 4 Theory, JHEP 11 (2008) 074, [arxiv: ]

13 Scalar approach Gauge model Localization by auxiliary fields How to gauge this model? Ansatz: d 4 xφ(x) 1 φ(x) d 4 x 1 4 F µν 1 D 2 D 2 F µν, U(1) θ : δ ε A µ = µ ε + ig [A µ, ε], F µν = µ A ν ν A µ ig [A µ, A ν ], D µ µ ig [A µ, ] and D µ = θ µν D ν. 3 D. N. Blaschke, F. Gieres, E. Kronberger, M. Schweda and M. Wohlgenannt, Translation-invariant models for non-commutative gauge fields, J. Phys. A41 (2008) , [arxiv: ]

14 Scalar approach Gauge model Localization by auxiliary fields How to gauge this model? Ansatz: d 4 xφ(x) 1 φ(x) d 4 x 1 4 F µν 1 D 2 D 2 F µν, U(1) θ : δ ε A µ = µ ε + ig [A µ, ε], F µν = µ A ν ν A µ ig [A µ, A ν ], D µ µ ig [A µ, ] and D µ = θ µν D ν. Expression 1 D 2 F Y transforms BRST covariantly (sy = ig [c, Y ]) 3 D. N. Blaschke, F. Gieres, E. Kronberger, M. Schweda and M. Wohlgenannt, Translation-invariant models for non-commutative gauge fields, J. Phys. A41 (2008) , [arxiv: ]

15 Scalar approach Gauge model Localization by auxiliary fields How to gauge this model? Ansatz: d 4 xφ(x) 1 φ(x) d 4 x 1 4 F µν 1 D 2 D 2 F µν, U(1) θ : δ ε A µ = µ ε + ig [A µ, ε], F µν = µ A ν ν A µ ig [A µ, A ν ], D µ µ ig [A µ, ] and D µ = θ µν D ν. Expression 1 D 2 F Y transforms BRST covariantly (sy = ig [c, Y ]) Y is of inherently nonlocal nature 3 D. N. Blaschke, F. Gieres, E. Kronberger, M. Schweda and M. Wohlgenannt, Translation-invariant models for non-commutative gauge fields, J. Phys. A41 (2008) , [arxiv: ]

16 Scalar approach Gauge model Localization by auxiliary fields How to gauge this model? Ansatz: d 4 xφ(x) 1 φ(x) d 4 x 1 4 F µν 1 D 2 D 2 F µν, U(1) θ : δ ε A µ = µ ε + ig [A µ, ε], F µν = µ A ν ν A µ ig [A µ, A ν ], D µ µ ig [A µ, ] and D µ = θ µν D ν. Expression 1 D 2 F Y transforms BRST covariantly (sy = ig [c, Y ]) Y is of inherently nonlocal nature Formal power series in the gauge field A µ (by recursion): F = D 2 1 D 2 F D2 Y = Y ig µ [A µ, Y ] ig [A µ, µ Y ] +..., Y = 1 F f(y ) Problem: Infinite series infinite number of gauge boson vertices! 3 3 D. N. Blaschke, F. Gieres, E. Kronberger, M. Schweda and M. Wohlgenannt, Translation-invariant models for non-commutative gauge fields, J. Phys. A41 (2008) , [arxiv: ]

17 Scalar approach Gauge model Localization by auxiliary fields Alternative way Rewrite d 4 x 1 4 F µν a 2 D 2 D F 2 µν d 4 x [ab µν F µν B µν D ] 2 D 2 B µν 4 D. N. Blaschke, A. Rofner, M. Schweda and R. I. P. Sedmik, One-Loop Calculations for a Translation Invariant Non-Commutative Gauge Model, [arxiv: ]

18 Scalar approach Gauge model Localization by auxiliary fields Alternative way Rewrite d 4 x 1 4 F µν a 2 D 2 D F 2 µν d 4 x [ab µν F µν B µν D ] 2 D 2 B µν New field B µν = B νµ localizes the action 4 D. N. Blaschke, A. Rofner, M. Schweda and R. I. P. Sedmik, One-Loop Calculations for a Translation Invariant Non-Commutative Gauge Model, [arxiv: ]

19 Scalar approach Gauge model Localization by auxiliary fields Alternative way Rewrite d 4 x 1 4 F µν a 2 D 2 D F 2 µν d 4 x [ab µν F µν B µν D ] 2 D 2 B µν New field B µν = B νµ localizes the action equivalence by introducing the tree level equation of motion δs (2) inv δb ρσ = af ρσ 2 D 2 D 2 B ρσ = 0 4 D. N. Blaschke, A. Rofner, M. Schweda and R. I. P. Sedmik, One-Loop Calculations for a Translation Invariant Non-Commutative Gauge Model, [arxiv: ]

20 Scalar approach Gauge model Localization by auxiliary fields Alternative way Rewrite d 4 x 1 4 F µν a 2 D 2 D F 2 µν d 4 x [ab µν F µν B µν D ] 2 D 2 B µν New field B µν = B νµ localizes the action equivalence by introducing the tree level equation of motion δs (2) inv δb ρσ = af ρσ 2 D 2 D 2 B ρσ = 0 B µν has its own dynamics: G BB ρσ,τɛ(k) = 1 [ 4k 2 k 2 δ ρτ δ σɛ δ ρɛ δ στ + a 2 k σk τ δ ρɛ + k ρ k ɛ δ στ k σ k ɛ δ ρτ k ρ k τ δ σɛ k 2 k ( 2 k 2 + a 2 Q: How can B µν be interpreted? k 2 ) ]. 4 D. N. Blaschke, A. Rofner, M. Schweda and R. I. P. Sedmik, One-Loop Calculations for a Translation Invariant Non-Commutative Gauge Model, [arxiv: ]

21 Scalar approach Gauge model Localization by auxiliary fields Alternative way Rewrite d 4 x 1 4 F µν a 2 D 2 D F 2 µν d 4 x [ab µν F µν B µν D ] 2 D 2 B µν New field B µν = B νµ localizes the action equivalence by introducing the tree level equation of motion δs (2) inv δb ρσ = af ρσ 2 D 2 D 2 B ρσ = 0 B µν has its own dynamics: G BB ρσ,τɛ(k) = 1 [ 4k 2 k 2 δ ρτ δ σɛ δ ρɛ δ στ + a 2 k σk τ δ ρɛ + k ρ k ɛ δ στ k σ k ɛ δ ρτ k ρ k τ δ σɛ k 2 k ( 2 k 2 + a 2 k 2 ) Q: How can B µν be interpreted? A: New degrees of freedom! 4 4 D. N. Blaschke, A. Rofner, M. Schweda and R. I. P. Sedmik, One-Loop Calculations for a Translation Invariant Non-Commutative Gauge Model, [arxiv: ] ].

22 Scalar approach Gauge model Localization by auxiliary fields Auxiliary fields and BRST doublets I Idea (Vilar et al. 5 ): Make the auxiliary field complex and assign appropriate ghosts to it 5 L. C. Q. Vilar, O. S. Ventura, D. G. Tedesco and V. E. R. Lemes, Renormalizable Noncommutative U(1) Gauge Theory Without IR/UV Mixing, [arxiv: ]

23 Scalar approach Gauge model Localization by auxiliary fields Auxiliary fields and BRST doublets I Idea (Vilar et al. 5 ): Make the auxiliary field complex and assign appropriate ghosts to it Fields: B µν R 4 θ {B µν, Bµν }, ghosts: {ξ µν, ξ µν } BRST doublets(quartets): s ξ µν = B µν ig { c, ξ µν }, s Bµν = ig [ c, Bµν ] sb µν = ξ µν ig [c, B µν ], sξ µν = ig [ c, ξ µν ]. Note: already known from QED, where s c = b, sb = 0. 5 L. C. Q. Vilar, O. S. Ventura, D. G. Tedesco and V. E. R. Lemes, Renormalizable Noncommutative U(1) Gauge Theory Without IR/UV Mixing, [arxiv: ]

24 Scalar approach Gauge model Localization by auxiliary fields Auxiliary fields and BRST doublets I Idea (Vilar et al. 5 ): Make the auxiliary field complex and assign appropriate ghosts to it Fields: B µν R 4 θ {B µν, Bµν }, ghosts: {ξ µν, ξ µν } BRST doublets(quartets): s ξ µν = B µν ig { c, ξ µν }, s Bµν = ig [ c, Bµν ] sb µν = ξ µν ig [c, B µν ], sξ µν = ig [ c, ξ µν ]. Note: already known from QED, where s c = b, sb = 0. Relevant part of the action (including an additional doublet {χ µν, χ µν } with ghosts {ψ µν, ψ µν }): d 4 x 1 4 F µν γ 2 (D 2 ) 2 F µν [ d 4 x s ψµν D 2 B µν + ξ µν D 2 χ µν + γ 2 ψµν χ µν] i 2 γb µνf µν + i 2 γ B µν F µν 5 L. C. Q. Vilar, O. S. Ventura, D. G. Tedesco and V. E. R. Lemes, Renormalizable Noncommutative U(1) Gauge Theory Without IR/UV Mixing, [arxiv: ]

25 Scalar approach Gauge model Localization by auxiliary fields Auxiliary fields and BRST doublets II Targets achieved: Locality

26 Scalar approach Gauge model Localization by auxiliary fields Auxiliary fields and BRST doublets II Targets achieved: Locality Translation invariance

27 Scalar approach Gauge model Localization by auxiliary fields Auxiliary fields and BRST doublets II Targets achieved: Locality Translation invariance BRST invariance (almost).the only remaining physical field (not BRST exact) is A µ no additional dof.

28 Scalar approach Gauge model Localization by auxiliary fields Auxiliary fields and BRST doublets II Targets achieved: Locality Translation invariance BRST invariance (almost).the only remaining physical field (not BRST exact) is A µ no additional dof.? Renormalizability? Algebraic renormalization (UV)+ explicit computation of coefficients Explicit renormalization in the IR?

29 Scalar approach Gauge model Localization by auxiliary fields Auxiliary fields and BRST doublets III Drawback: complicated! 12 fields, 4 BRST sources, 14 Slavnov sources (yielding 9 propagators and 13 vertices)

30 Scalar approach Gauge model Localization by auxiliary fields Auxiliary fields and BRST doublets III Drawback: complicated! 12 fields, 4 BRST sources, 14 Slavnov sources (yielding 9 propagators and 13 vertices) Just a glimpse: { Σ= R d 4 x 1 4 Fµν F µν +ib µa µ + c µ D µc+ χ µν D 2 B µν + B µν D 2 χ µν +γ 2 χ µν χ µν + J µναβ {B µν,f αβ }+J µναβ { B µν,f αβ } ψ µν D 2 ξ µν ξ µν D 2 ψ µν γ 2 ψµν ψ µν Q µναβ {ξ µν,f αβ } Ω µ D µ c i 2 L g{c,c} iū µν g{c,ξ µν}+u µν ( B µν ig{c, ξ µν}) + v µν (ξ µν ig[c,b µν]) iv µν g[c, B µν] i P µν g{c,ψ µν}+p µν ( χ µν ig{c, ψ µν}) + R µν (ψ µν ig[c,χ µν]) ir µν g[c, χ µν] + M µναβ (J µναβ ig{c,q µναβ })+M µναβ ( J µναβ ig{c, Q µναβ }) i N µναβ g[c,j µναβ ] in µναβ g[c, J µναβ ] },

31 Proposition of a new model Thinking... Q: Can t we simplify this model in any way?

32 Proposition of a new model Thinking... Q: Can t we simplify this model in any way? A: Yes, we can. Reduce the quartet to a doublet!

33 Construction of the model I Localization by fields {B µν, Bµν }, with ghosts {ψ µν, ψ µν } d 4 x 1 4 F µν λ 2 µ 2 D 2 D F 2 µν [ λ S loc = d 4 x 2 ( Bµν + B µν ) F µν µ 2 Bµν D 2 D2 B µν + µ 2 ψµν D 2 D2 ψ µν ], Landau gauge fixing S φπ = d 4 x (b µ A µ c µ D µ c) (Note: All products are products, even if not denoted explicitely!)

34 Construction of the model II BRST transformations with doublet structures: sa µ = D µ c, sc = igcc, s c = b, sb = 0, sf µν = ig [c, F µν ], s ψ µν = B µν + ig { c, ψ } µν, s Bµν = ig [ c, B ] µν, sb µν = ψ µν + ig [c, B µν ], sψ µν = ig {c, ψ µν }. S loc = [ ( )] λ λ d 4 x 2 B µνf µν + s 2 ψ µν F µν µ 2 ψµν D 2 D2 B µν, The action is almost BRST invariant (up to a soft breaking ).

35 Construction of the model II BRST transformations with doublet structures: sa µ = D µ c, sc = igcc, s c = b, sb = 0, sf µν = ig [c, F µν ], s ψ µν = B µν + ig { c, ψ } µν, s Bµν = ig [ c, B ] µν, sb µν = ψ µν + ig [c, B µν ], sψ µν = ig {c, ψ µν }. S loc = [ ( )] λ λ d 4 x 2 B µνf µν + s 2 ψ µν F µν µ 2 ψµν D 2 D2 B µν, The action is almost BRST invariant (up to a soft breaking ). Ready for algebraic renormalization?

36 Breaking term Non-BRST-invariant term in the action: ss break = s d 4 x λ 2 B µνf µν = d 4 x λ 2 ψ µνf µν 6 L. Baulieu and S. P. Sorella, Soft breaking of BRST invariance for introducing non-perturbative infrared effects in a local and renormalizable way, Physics Letters B 671 (2009) 481, [arxiv: ]

37 Breaking term Non-BRST-invariant term in the action: ss break = s d 4 x λ 2 B µνf µν = d 4 x λ 2 ψ µνf µν has to be eliminated to enable UV renormalization. 6 introduction of pairs of BRST sources {Q µναβ, Q µναβ } and {J µναβ, J µναβ }, s Q µναβ = J µναβ + ig { c, Q µναβ }, s Jµναβ = ig [ c, J µναβ ], sq µναβ = J µναβ + ig {c, Q µναβ }, sj µναβ = ig [c, J µναβ ], S break s d 4 x Q µναβ B µν F αβ = d 4 x J µναβ B µν F αβ Q µναβ ψ µν F αβ 6 L. Baulieu and S. P. Sorella, Soft breaking of BRST invariance for introducing non-perturbative infrared effects in a local and renormalizable way, Physics Letters B 671 (2009) 481, [arxiv: ]

38 Soft breaking mechanism But: Breaking term implements the actual IR damping of the 1 p 2 model. 7 D. Zwanziger, Local and Renormalizable Action from the Gribov Horizon, Nucl. Phys. B323 (1989) , D. Zwanziger, Renormalizability of the critical limit of lattice gauge theory by BRS invariance, Nucl. Phys. B399 (1993)

39 Soft breaking mechanism But: Breaking term implements the actual IR damping of the 1 p model. 2 Trick by Zwanziger 7 : Assign physical values in the IR, recovering the breaking: Q µναβ phys = 0, Jµναβ phys = λ 4 (δ µαδ νβ δ µβ δ να ), Q µναβ phys = 0, J µναβ phys = λ 4 (δ µαδ νβ δ µβ δ να ). IR S break = λ 2 B µνf µν UV S break = s d 4 x Q µναβ B µν F αβ 7 D. Zwanziger, Local and Renormalizable Action from the Gribov Horizon, Nucl. Phys. B323 (1989) , D. Zwanziger, Renormalizability of the critical limit of lattice gauge theory by BRS invariance, Nucl. Phys. B399 (1993)

40 Complete action Z S inv = d 4 x 1 4 FµνF µν, Z Z S φπ = d 4 x s ( c µ A µ) = d 4 x (b µ A µ c µ D µc), Z S new = d 4 x s J ψµν µναβ F αβ µ 2 ψµνd 2 D e 2 B µν Z = d 4 x J Bµν µναβ F αβ µ 2 BµνD 2 D e 2 B µν + µ 2 ψµνd 2 D e 2 ψ µν, Z S break = d 4 x s Qµναβ B µν F αβ Z = d 4 x Jµναβ B µν F αβ Q µναβ ψ µν F αβ, Z S ext = d 4 x Ω A µ D µ c + ig Ω c cc + Ω B µν (ψ µν + ig [c, B µν ]) + ig Ω B µν ˆc, Bµν +ig Ω ψ µν {c, ψ µν } + Ω ψ ` Bµν µν + ig c, ψ µν + Ω Q µναβ J µναβ + ig nc, Q µναβo h +ig Ω J µναβ c, J µναβi + Ω Q µναβ J µναβ + ig nc, Q µναβo h + ig Ω J µναβ c, J µναβi.

41 Propagators G cc (k) = 1 k 2, a λ µ, Gµν,ρσ(k) ψψ (δµρδνσ δµσδνρ) = 2µ 2 k 2 k, 2 G AA µν (k) = 1 k 2 + a 2 k 2 δ µν kµkν k 2 «, G AB µ,ρσ(k) = ia (k ρδ µσ k σδ µρ) 2µ k 2 k = G A B µ,ρσ (k) = Gρσ,µ(k) BA, 2 k 2 + a 2 k Gµν,ρσ(k) BB = 1 2µ 2 k 2 k 4δ µρδ νσ δ µσδ νρ a 2 k µk ρδ νσ + k νk σδ µρ k µk σδ νρ k νk ρδ µσ 2 2k 2 k 5, 2 k 2 + a 2 k G BB µν,ρσ(k) = a 2 4k 2 k 2 kµkρδνσ 4 + kνkσδµρ kµkσδνρ kνkρδµσ µ 2 k 2 k 2 k 2 + a 2 k 2 5 B B = Gµν,ρσ(k),

42 Power counting Superficial degree of (UV) divergence in 4 dimensions: d γ = 4 E A E c/ c 2E B 2E B 2E ψ ψ 2E θ, and with E λ = E B + E B + E ψ/ ψ + E θ, d γ = 4 E A E c/ c 2E λ, E x counts the number of external legs of field x, E λ and E θ count the overall powers of λ and θ in the graph.

43 One loop computations Tadpoles with one external boson vanish, as d 4 k sin k p 2 0 p 0 p k

44 One loop computations Tadpoles with one external boson vanish, as d 4 k sin k p 2 0 p 0 p k Vacuum Polarization Leading terms for p 0 are 2g2 p µ p ν π 2 ( p 2 ) + finite. 2 Transversality as expected 18 graphs, 7 thereof are divergent

45 One loop computations Tadpoles with one external boson vanish, as d 4 k sin k p 2 0 p 0 p k Vacuum Polarization Leading terms for p 0 are 2g2 p µ p ν π 2 ( p 2 ) + finite. 2 Transversality as expected Sectors B µν and ψ µν cancel as expected 18 graphs, 7 thereof are divergent

46 Conclusion What has been achieved?

47 Conclusion What has been achieved? Non-commutative U(1) θ gauge theory

48 Conclusion What has been achieved? Non-commutative U(1) θ gauge theory Translation invariant and local

49 Conclusion What has been achieved? Non-commutative U(1) θ gauge theory Translation invariant and local Potentially renormalizable

50 Outlook What are the next steps?

51 Outlook What are the next steps? Next step: Completition of the algebraic renormalization procedure in the UV

52 Outlook What are the next steps? Next step: Completition of the algebraic renormalization procedure in the UV IR renormalization (as was achieved for the scalar model)

53 Outlook What are the next steps? Next step: Completition of the algebraic renormalization procedure in the UV IR renormalization (as was achieved for the scalar model) Transition to Minkowski? - we ll try

54 Thank you Thank you for your attention! Thanks to my collaborators: Daniel Blaschke, Arnold Rofner, Manfred Schweda

55 Equivalence of local and nonlocal terms Z = = = = { ( D( ψψ BBA) exp D( BBA) det 4 ( µ 2 D 2 D2 ) exp D( BBA) ( ) { det 4 µ 2 D 2 D2 exp ( B µν λ 1 2µ 2 DA det 4 4( ) { D 2 D2 exp d 4 x 1 4 F µνf µν + S loc )} D 2 D 2 F µν { [ 1 d 4 x 4 F µνf µν + λ ( Bµν + 2 B ) µν F µν ]} µ 2 Bµν D 2 D2 B µν d 4 x [ 1 d 4 x ) µ 2 D 2 D2 4 F µνf µν + λ2 4µ 2 F 1 µν D 2 D F µν ( )]} 2 B µν [ 1 4 F µνf µν + λ2 4µ 2 F µν λ 1 2µ 2 D 2 D F µν 2 1 D 2 D 2 F µν ]}.

56 Properties of fields, ghosts and sources Table: Properties of fields and sources. Field A µ c c B µν Bµν ψ µν ψµν J αβµν Jαβµν Q αβµν Qαβµν g Mass dim Statistics b f f b b f f b b f f Source Ω A µ Ω c b Ω B µν Ω B µν Ω ψ µν Ω ψ µν Ω J αβµν Ω J αβµν Ω Q αβµν Ω Q αβµν g Mass dim Statistics f b b f f b b f f b b

57 1-loop vacuum polarization graphs i j k l m n o p q r s t u v w x y z aa ab