Renormalization of the spectral action for the Yang-Mills system

We establish renormalizability of the full spectral action for the Yang-Mills system on a flat 4-dimensional background manifold. Interpreting the spectral action as a higher-derivative gauge theory, we find that it behaves unexpectedly well as far as renormalization is concerned. Namely, a power counting argument implies that the spectral action is superrenormalizable. From BRST-invariance of the one-loop effective action, we conclude that it is actually renormalizable as a gauge theory.


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This spectral action has firm roots in the noncommutative geometrical description of the Yang-Mills system, we refer to [7] for more details. For our purposes, it suffices to know that D A is a Dirac operator with coefficients in a SU(N )-vector bundle equipped with a connection A. That is, locally we have with ∇ µ the spin connection on a Riemannian spin manifold M . For simplicity, we take M to be flat (i.e. vanishing Riemann curvature tensor) and 4-dimensional. Furthermore, we will assume that f is a Laplace transform: f (x) = t>0 e −tx 2 g(t)dt, even though this assumption could be avoided by using spectral densities instead ( [14] and also [23, section 8.4

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Proposition 1 ([4]). In the above notation, there is an asymptotic expansion (as Λ → ∞): Recall that the Seeley-De Witt coefficients a m (x, D 2 A ) are gauge invariant polynomials in the fields A µ . Indeed, the Weitzenböck formula gives Consequently, a Theorem by Gilkey [16,Theorem 4.8.16] shows that (in this case) a m are polynomial gauge invariants in F µν and its covariant derivatives. The order ord of a m is m, where we set on generators: Consequently, the curvature F µν has order 2, and F µ 1 µ 2 ;µ 3 ···µ k has order k. For example, a 4 (x, D 2 A ) is proportional to Tr F µν F µν and more generally: for some constants c k and the Laplacian ∆ A = −(∂ µ − iA µ ) 2 (see also [1] and references therein). The remainder is of third and higher order in F , plus its covariant derivatives, adding up to an order equal to 4 + 2k.
Remark 2. It is the term a 4 that gives rise to the Yang-Mills action functional, the higherorder terms are usually ignored (being proportional to an inverse power of the 'cut-off ' Λ).
More recently, also the higher-order terms, or even the full spectral action were studied in specific cases in [6,17] and from a more general point of view in [22].

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We assume that the first term is the usual (free part of the) Yang-Mills action, that is, we adjust the positive function f so that f 0 c 0 = 1/4. For the other coefficients, we have the following neat expression. 1 Lemma 4. The coefficients f −2k are related to the 2k'th derivatives of f at zero: Proof. With f (x) = e −tx 2 g(t)dt we derive for its derivatives: We end this section by introducing a formal expansion (that starts with 1) so that we can write more concisely This form motivates the interpretation of S 0 [A] (and of S[A]) as a higher-derivative gauge theory. As we will see below, this indeed regularizes the theory in such a way that S[A] defines a superrenormalizable field theory.

Gauge fixing in the YM-system
We add a gauge-fixing term of the following higher-derivative form: We derive the propagator by inverting the non-degenerate quadratic form given by

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which for the moment is a formal expansion in Λ. We will come back to it in more detail in the next section.
As usual, the above gauge fixing requires a Jacobian, conveniently described by a Faddeev-Popov ghost Lagrangian: Here C, C are the Faddeev-Popov ghost fields and their propagator is .
is invariant under the BRSTtransformations: On the other hand, which modulo vanishing boundary terms is minus the previous expression.
Note that s 2 = 0, which can be cured by standard homological methods: introduce an auxiliary (aka Nakanishi-Lautrup) field h so that C and h form a contractible pair in BRST-cohomology. In other words, we replace the above transformation in (3.3) on C by sC = −h and sh = 0. If we replace S gf + S gh by sΨ with Ψ an arbitrary gauge fixing fermion, it follows from gauge invariance of S and nilpotency of s that S + sΨ is BRST-invariant. The above special form of S gf + S gh can be recovered by choosing Ψ = Tr ϕ Λ (∆)(C) 1 2 ξh + ∂ µ A µ .

Renormalization of the spectral action for the YM-system
As said, we consider the spectral action for the Yang-Mills system as a higher-derivative field theory. This means that we will use the higher derivatives of F µν that appear in the asymptotic expansion as natural regulators of the theory, similar to [20,21] (see also [15, section 4.4]). However, note that the regularizing terms are already present in the spectral action S[A] and need not be introduced as such. Let us consider the expansion (2.1) up to order n (which we assume to be at least 8), i.e. we set f 4−m = 0 for all m > n. Also, assume a gauge fixing of the form (3.1) and (3.2). Then, we easily derive from the structure of ϕ Λ (p 2 ) that the propagators of both the gauge field and the ghost field behave as |p| −n+2 as |p| → ∞. Indeed, in this case: Moreover, the weights of the interaction in terms of powers of momenta is given by the table (figure 1). We will use v k to indicate the number of gauge interaction vertices of valence k, and with v the number of ghost-gauge vertices.
Let us now find an expression for the superficial degree of divergence ω of a graph consisting of I internal gauge edges, I internal ghost edges, v k valence k gauge vertices and v ghost-gauge vertices. In 4 dimensions, we find at loop order L:

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Proof. We use the relations where E and E are the number of external gauge and ghost legs, respectively. Indeed, these formulas count the number of half (gauge/ghost) edges in a graph in two ways: from the number of edges and from the valences of the vertices. We use them to substitute for 2I and 2 I in the above expression for ω so as to obtain from which the result follows at once from Euler's formula As a consequence, ω < 0 if L ≥ 2 (provided n ≥ 8): all Feynman graphs are finite at loop order greater than 1. If L = 1, then there are finitely many graphs which are divergent, namely those for which E + E ≤ 4. We conclude that the spectral action for the Yang-Mills system is superrenormalizable.
Of course, the spectral action being a gauge theory, there is more to renormalizability than just power counting: we have to establish gauge invariance of the counterterms. We already know that the counterterms needed to render the perturbative quantization of the spectral action finite are of order 4 or less in the fields and arise only from one-loop graphs.
The key property of the effective action at one loop is that it is BRST-invariant: In particular, assuming a regularization compatible with gauge invariance, the divergent part Γ 1,∞ is BRST-invariant. Results from [2, 10-13] on BRST-cohomology for Yang-Mills type theories ascertain that the only BRST-closed functional of order 4 or less in the fields is represented by δZ F µν F µν for some constant δZ. The counterterm Γ 1,∞ can thus be added to S and absorbed by a redefinition of the fields and coupling constant: Equivalently, one could leave A and g invariant, and redefine f 0 → f 0 + δZ, leaving all other coefficients f 4−m invariant. Intriguingly, renormalization of the spectral action for the YM-system can thus be accomplished merely by shifting the function f in such a way that f (0) → f (0) + δZ, whilst leaving all its higher derivatives at 0 invariant.
Remark 8. The above form for Γ 1,∞ can actually be established by an explicit computation in dimensional regularization following [18,19]. We intend to present the full details elsewhere.