A Renormalizable 4-Dimensional Tensor Field Theory

Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme

“…Crucially, the overall Gurau degree is conserved under the coproduct [42] such that renormalizability is given even though ω sd (Γ) = ω(∂Γ) only up to ω g Γ . One can further show that ω g Γ ≥ ω g ∂Γ [4] such that the whole term in brackets is always greater or equal to zero and renormalizable theories are determined by the first two terms. Thus, a tensorial φ n d,r theory is super renormalizable if d r = 2ζ and just renormalizable if n = 2dr dr−2ζ .…”

“…Just-renormalizable quartic theories are those with d r = d(r − 1) = 4ζ. The tensorial equivalent to standard local φ 4 4 theory with quadratic propagator ζ = 1 are therefore theories with d r = 4, that is theories with (d, r) = (4, 2), (2,3) or (1,5). Let us her simplify explicit renormalization even further and chose ζ = 1/2.…”

“…Let us her simplify explicit renormalization even further and chose ζ = 1/2. The two just renormalizable theories are then tensorial φ 4 2,2 and φ 4 1,3 theory. The divergence degree in these two theories is…”

“…A classical example is BPHZ momentum subtraction as used for the general proof of renormalization [8,27,49]. In [47] I have shown that the Connes-Kreimer Hopf algebra describes renormalization not only in field theories with point-like interactions but also with full generality in combinatorially non-local field theories (cNLFT) such as non-commutative and matrix field theory [23,48,28], tensorial field theories [4,2] and group field theory [35,20,14].…”

“…In fact, exact solutions for matrix field theory have been found recently [38,21,12] and the perturbative results are mainly used as a check of consistency [22,28]. For other examples of cNLFT, renormalizability has been proven for various field theories with tensorial interactions [4,3,2,15,16,14] but no explicit results of amplitudes, let alone of full renormalized correlation functions, are known 1 . Using the Hopf-algebraic structure of renormalization I will give explicit amplitudes of perturbative tensorial field theory here for the first time.…”