Renormalizable nc QFT

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1 Faculty of Physics, University of Vienna (joint work with Raimar Wulkenhaar, arxiv: )

2 Introduction Improve QFT use NCG and RG flow Scalar field on Moyal space, modified action 2004 Taming the Landau ghost M Disertori, R Gurau, J Magnen, V Rivasseau, = Paris group, 2006 used Ward identity and Schwinger-Dyson equ for singular part Progress in solving the model Ward identity 2 pt Schwinger-Dyson equation Integral equation for ren 2 pt fct Perturbative solution 4 pt Schwinger-Dyson equ Conclusions

3 Introduction Project Minkowski-Euclidean QFT. Require: Covariance, spectrum condition, locality, vacuum,... Algebraic, constructive, perturbative RG flow approaches regularization - renormalization UV, IR, zero convergence radius Landau ghost, trivial Higgs model? add Gravity or deform Space-Time many approaches limited wedge localization merge general relativity with quantum field theory through noncommutative geometry Manifold: nc algebra, differential calculus, projective modules,...

4 Φ 4 Θ φ 4 on [x µ, x ν ] = iθ µν, IR/UV mixing: nonrenenormalizable Theorem: Action Modified action S = ( 1 ) d 4 x 2 µφ µ φ+ Ω2 2 ( x µφ) ( x µ φ)+ µ2 0 2 φ φ+λφ φ φ φ (x) for x µ := (θ 1 ) µν x ν is perturbativly renormalizable to all orders in λ, cyclic order of momenta leads to ribbon graphs 3 proofs: Polchinski, multiscale analysis (Paris) Action has Langmann-Szabo position-momentum duality S [ φ;µ 0,λ,Ω ] Ω 2 S [ φ; µ 0 Ω, ] λ, 1 Ω 2 Ω Curvature of cutoff-algebra explains potential term (Buric-Wohlgenannt)

5 β function β λ = λ2 2 phys (1 Ωphys ) 48π 2 (1+Ω 2 + O(λ3 phys )3 phys ) flow bounded, L. ghost killed! Due to wave fct. renormalization Ω = 1 betafunction vanishes up to all loops! Paris group 0.8 Ω 2 [Λ] 1 (λ[λ] diverges in comm. case) λ[λ] π λ R Λ 2 R = 1 θ λ R = Ω R = ln Λ 2 θ perturbation theory remains valid at all scales! non-perturbative construction of the model seems possible!

6 Matrix model Action in matrix base at Ω = 1 Action functionals for bare mass µ bare Wave function renormalisation φ Z 1 2φ. Fix θ = 4, φ mn = φ nm real: S = m,n N 2 Λ 1 2 φ mnh mn φ nm + V(φ), H mn = Z ( µ 2 bare + m + n ), V(φ) = Z 2 λ 4 m,n,k,l N 2 Λ Λ is cut-off. µ bare, Z divergent No infinite renormalisation of coupling constant m, n,... belong to N 2, m := m 1 + m 2. φ mn φ nk φ kl φ lm,

7 Ward identity inner automorphism φ UφU of M Λ, infinitesimally φ mn φ mn + i k N 2 Λ(B mk φ kn φ mk B kn ) not a symmetry of the action, but invariance of measure Dφ = m,n N 2 Λ dφ mn gives 0 = δw = 1 iδb ab Z = 1 Dφ Z n ( Dφ δs + δ ( ) ) tr(φj) e S+tr(φJ) iδb ab iδb ab ( ) (H nb H an)φ bn φ na + (φ bn J na J bn φ na) e S+tr(φJ) where W[J] = ln Z[J] generates connected functions J nm trick φ mn { ( ( δ 0 = n (H nb H an ) 2 δj nb δj an + ( exp V ( ) ) δ δj e 1 2 p,q JpqH 1 δ J na } pq J qp δj nb J bn c )) δ δj an

8 Interpretation The insertion of a special vertex Vab ins := (H an H nb )φ bn φ na n into an external face of a ribbon graph is the same as the difference between the exchanges of external sources J nb J na and J an J bn The dots stand for the remaining face indices. Z( a b )G ins [ab]... = G b... G a...

9 SD equation 2 vertex is Z 2 λ, connected two-point function is G ab : first graph equals Z 2 λ q G aq open p-face in Σ R and compare with insertion into connected two-point function; insert either into 1P reducible line or into 1PI function:

10 Amputate upper G ab two-point function, sum over p, multiply by vertex Z 2 λ, obtain: Σ R ab : Σ R ab = Z 2 λ p (G ab ) 1 G ins [ap]b = Zλ p (G ab ) 1 G bp G ba p a. case a = b = 0 and Z = 1 treated. Use G 1 ab = H ab Γ ab and Tab L = Z 2 λ q G aq gives for 2 point function: Z 2 λ q G aq Zλ p (G ab ) 1 G bp G ba p a = H ab G 1 ab. Symmetry Γ ab = Γ ba is not manifest!

11 Renormalize express SD equation in terms of the 1PI function Γ ab perform renormalisation for the 1PI part Γ ab = Z 2 λ p ( 1 H bp Γ bp + 1 H ap Γ ap 1 (Γ bp Γ ab ) ). H bp Γ bp Z( p a ) Taylor expand: Γ ren is derivative in a 1, a 2, b 1, b 2. Implies Γ ab = Zµ 2 bare µ2 ren + (Z 1)( a + b ) + Γren ab, Γ ren 00 = 0 ( Γren ) 00 = 0,... the resulting equation is G 1 ab = a + b + µ2 ren Γren ab. (Z 1)( a + b ) + Γ ren ab Z 2 + Z p + µ 2 Γren 0p = Λ 0 ( Z p d p b + p + µ 2 Γ ren + bp Z b + p + µ 2 Γ ren bp Γ ren bp Γren ab ( p a ) + Z 2 a + p + µ 2 Γ ren ap Z p + µ 2 Γ ren 0p Γ ren ) 0p, p

12 change variables,... a =: µ 2 α 1 α, b =: β µ2 1 β, p =: ρ µ2 1 ρ, p d p = ρ dρ µ4 (1 ρ) 3 Γ ab =: µ 2 Γ αβ (1 α)(1 β), Λ =: ξ µ2 1 ξ...get expression for ren.constant,..take cutoff limit: Theorem The renormalised planar two-point function G αβ of self-dual noncommutative φ 4 4-theory (with continuous indices) satisfies the integral equation ( 1 α ( G αβ = 1 + λ Mβ L β βy ) + 1 β ( Mα L α αy ) 1 αβ 1 αβ + 1 β ( Gαβ ) (Mα ) α(1 β) ( ) 1 L α + αn α0 Lβ + N αβ N α0 1 αβ G 0α 1 αβ ) (1 α)(1 β) + (G αβ 1)Y, 1 αβ

13 expansion Integral equation for Γ ab is non-perturbatively defined. Resisted an exact treatment. We look for an iterative solution G αβ = n=0 λn G (n) αβ. This involves iterated integrals labelled by rooted trees. Up to O(λ 3 ) we need 1 α I α := dx = ln(1 α), 0 1 αx 1 α I x I α := dx 0 1 αx = Li 2(α) + 1 ( ) 2 ln(1 α) 2 1 α Ix Ix ( I α = dx 0 1 αx = 2 Li 3 α ) 1 α 1 α I x ( I α = dx 0 1 αx = 2Li 3 α ) 2Li 3 (α) ln(1 α)ζ(2) 1 α + ln(1 α)li 2 (α) ( ln(1 α) ) 3

14 In terms of I t and A := 1 α 1 αβ { } G αβ = 1 + λ A(I β β) + B(I α α), B := 1 β 1 αβ : + λ 2{ A ( ) ( βi β βi β αab (Iβ ) 2 ) 2βI β + I β + B ( ) ( αi α αi α βba (Iα) 2 ) 2αI α + I α + AB ( (I α α) + (I β β) + (I α α)(i β β) + αβ(ζ(2) + 1) )} + λ 3{ AW β + αab ( ) 2 U β + I αi β + I α I β + αa BVβ + BW α + βba ( ) 2 U α + I β I α + I β I α + βb AVα + AB ( T β + T α I β (I α) 2 I α(i β ) 2 ) 6I αi β + AB 2( (1 α)(i α α) + 3I αi β + I β I α + I β (I 2) α) + BA 2( (1 β)(i β β) + 3I αi β + I α I β + I α(i β ) 2)} + O(λ 4 ) T β := βi β βi β + (I β β), Similar for U β, V β, W β Remark: I β β β = 1 0 dx βx 1 βx

15 Observations Polylogarithms and multiple zeta values appear in singular part of individual graphs of e.g. φ 4 -theory (Broadhurst-Kreimer) We encounter them for regular part of all graphs together Conjecture G αβ takes values in a polynom ring with generators A, B, α, β, {I t }, where t is a rooted tree with root label α or β at order n the degree of A, B is n, the degree of α, β is n, the number of vertices in the forest is n. If true: There are n! forests of rooted trees with n vertices at order n estimate: G (n) αβ n!(c αβ) n may lead to Borel summability?.

16 Schwinger-Dyson equ 4 pt fct Follow a-face, there is a vertex at which ab-line starts: 1 First graph: Index c and a are opposite It equals Z 2 λg ab G bc G ins [ac]d 2 Second graph: Summation index p and a are opposite. We open the p-face to get an insertion. This is not into full connected four-point function, which would contain an ab-line not present in the graph.

17 second graph equals ( ) = Z 2 λ G ab G[ap]bcd ins Gins [ap]b G abcd 1PI four-point function Γ ren abcd = Zλ { G 1 p ad G 1 cd a c + p G ( pb Gdp )} Γ ren pbcd a p G Γren abcd ad

18 Theorem The renormalised planar 1PI four-point function Γ αβγδ of self-dual noncommutative φ 4 4-theory satisfies Γ αβγδ = λ (1 (1 α)(1 γδ)(g αδ G γδ ) 1 + ρ dρ (1 β)(1 αδ)g βρg δρ Γ ρβγδ Γ ) αβγδ G γδ (1 δ)(α γ) 0 (1 βρ)(1 δρ) ρ α ( 1 G αδ +λ (M β L β Y)G αδ + dρ G αδg βρ (1 β) 1 0 (1 δρ)(1 βρ) + ρ dρ (1 β)(1 αδ)g βρ (G ρδ G αδ ) ) 0 (1 βρ)(1 δρ) (ρ α) Corollary Γ αβγδ = 0 is not a solution! We have a non-trivial (interacting) QFT in four dimensions!

19 Conclusions Studied model at Ω = 1 RG flows save Used Ward identity and Schwinger-Dyson equation ren. 2 point fct fulfills nonlinear integral equ ren. 4 pt fct linear inhom. integral equ perturbative solution: Γ αβγδ = λ λ 2( (1 γ)(i α α) (1 α)(i γ γ) α γ + (1 δ)(i β β) (1 β)(i δ δ) β δ ) + O(λ 3 ) is nontrivial and cyclic in the four indices nontrivial Φ 4 model?