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All-order bounds for correlation functions of gauge-invariant operators in Yang-Mills theory. (English) Zbl 1356.81167
A rigorous self-contained proof that Euclidean Yang-Mills theories are perturbatively renoramlisable, is given. The proof is consisted by analytic part (Chapter V) and algebraic part (Chapter VI). To be self-contained, necessary tools used in the proof are explained in Chapter II, the classical gauge theory, BRST transformations, and cohomologies in Chapter III, the flow equation framework in Chapter VI. Trees, definition and properties of weighted trees, which are heavily used, are introduced in Chapter V.
The analytic part gives bounds uniform in the UV cutoff (denoted by $$\Lambda_0$$) from the connected amputated correlation functiosn (CACs) with an arbitrary number of basic fields (Chapter III, IV and V). They are derived using renormalization flow equation (Chapter III (75); \begin{aligned} \partial_\Lambda L^{\Lambda,\Lambda_0}+\partial_\Lambda I^{\Lambda,\Lambda_0}=&\frac{\hslash}{2}\big\langle\frac{\delta}{\delta\phi_K},(\partial_\Lambda C^{\Lambda,\Lambda_0}_{KL}\ast \frac{\delta}{\delta\phi_L}\big\rangle L^{\Lambda,\Lambda_0}- \\ &-\frac{1}{2}\big\langle\frac{\delta}{\delta\phi_k}L^{\Lambda,\Lambda_0},(\partial_\Lambda C^{\Lambda,\Lambda_0}_{KL})\ast \frac{\delta}{\delta\phi_l}L^{\Lambda,\Lambda_0}\big\rangle,\end{aligned} where $$\Lambda$$ is the IR cutoff, $$L^{\Lambda_0}$$ is the interacting part of the action with modification from UV cutoff, and $$I^{\Lambda,\Lambda_0}=-\hslash\log Z^{\Lambda,\Lambda_0}(0)$$) approach to quantum field theory [V. F. Müller, Rev. Math. Phys. 15, No. 5, 491–558 (2003; Zbl 1081.81543); C. Kopper, Prog. Math. 251, 161–174 (2007; Zbl 1166.81350)]. The gauge structure is treated in the algebraic part (Chapters II and VI). The gauge fixed theory is set up with appropriate ghost fields and auxiliary fields in such a way that the action remains BRST invariant [N. Nakanishi, “Covariant quantization of the electromagnetic field in Landau gauge”, Prog. Theor. Phys. 35, No. 6, 1111–1116 (1966; doi:10.1143/PTP.35.1111)]. At the quantum level, BRST invariance of the theory is expressed by certain Ward identities connecting different correlation functions [J. C. Ward, Phys. Rev., II. Ser. 78, 182 (1950; Zbl 0041.33012); Y. Takahashi, Nuovo Cimento, X. Ser. 6, 371–375 (1957; Zbl 0078.20202)]. The renormalization schema adopted in the analytic part will violate these identities. But suitable “anomalous” versions of these identities are derived, which states the theory is gauge invariant in quantum level.
Precisely, main results are summarized as follows: Let $$\langle \mathcal{O}_{a_1}(x_1),\ldots, \mathcal{O}_{As}(x_s)\rangle_c$$ be the correlation functions of an arbitrary massless Euclidean quantum field theory that is superficially renormalisable in the sense that the classical Lagrangian without source only contains of positive or zero (mass) dimensions ($$A_i$$ is the gauge potential in Yang-Mills theory). Then up to an arbitrary but fixed perturbation order, the connected correlation functions with composite operator insertions are tempered distributions. Smeared with test functions, they satisfy the bound $\langle \mathcal{O}_{A_1}(x_1),\ldots,\mathcal{O}_{A_s}(x_s)\rangle_C\leq C\prod_{i=1}^x\|f_i\|_{D+1}$ (Chapter I A. Corollary 2). Here $$D=\sum_{i=1}^s[\mathcal{O}_{A\i}]$$, $$[\cdot ]$$ is the engineering dimension, and $$\|\cdot\|_k$$ is the Schwartz norm. Authors say this bound is weak, and introduce the notion of weighted tree (Chapter IV, A), sharp bound of $$|\langle \mathcal{O}_{A_i}(s_1),\ldots,\mathcal{O}_{A_s}(x_s)\rangle_c|$$ is obtained by using the weight of trees (Chapter I,A. Theorem 3).
While these bounds are still valid, at intermediate steps of the construction gauge invariance violated, and must be restore afterwards by an appropriate finite change in the renormalisation conditions implicit in the definition of the correlation functions. Gauge invariance in the quantum field theory is expressed by a set of Ward identities. Using Batalin-Vikovisky (BV) formalism [S. Weinberg, The quantum theory of fields. Vol. 2. Modern applications. Cambridge: Cambridge University Press (2005; Zbl 1069.00007)] for an arbitrary massless, superficially renormalisable gauge theory and up to an arbitrary, but fixed perturbation order, existence of renormalisation conditions, such that the CACs with insertion fulfill the Ward identity \begin{aligned} &\sum_{i=1}^s\langle\mathcal{O}_{A_i}(x_1),\ldots,\mathcal{O}_{A_s}(x_s)\rangle_c \\ =&\hslash\sum_{1\leq k\leq l\leq s}\langle\mathcal{O}_{A_1}(x_1),\ldots(\mathcal{O}_{A_k}(x_k),\mathcal{O}_{A_l}8x_l))_h,\ldots,\mathcal{O}_{A_s}(x_s)\rangle_c,\end{aligned} if $$H^{1,4}_{\mathrm{E}(4)}(\hat{\mathbf{s}}|\mathrm{d})=0$$ (Chapter I. Theorem 4). Here $$H^{1,4}_{\mathrm{E}(4)}(\hat{\mathbf{s}}|\mathrm{d})$$ means the equivariant classical cohomology of $$\hat{\mathbf{s}}$$, the classical Slavnov-Taylor differential, at form degree 4 and ghost number 1 (details are explained in Chapter II). It is remarked these Ward identities are valid for bosonic operators, while for fermionic operators, additional minus sign appear.
Physically, Theorem 4 is the statement that the theory is gauge invariant in the quantum level. For Yang-Mills theory based on a semisimple Lie group, $$H^{1,4}(\hat{\mathbf{s}}|\mathrm{d})$$ is generted by the gauge anomaly [G. Barnich et al., Commun. Math. Phys. 174, No. 1, 57–91 (1995; Zbl 0844.53059); ibid. 174, No. 1, 93–116 (1995; Zbl 0844.53060)]]. As a consequence, when $$\mathrm{E}(4)\cong \mathrm{O}(4)\rtimes \mathbb{R}^4$$, $$H^{1,4}_{\mathrm{E}(4)}(\hat{\mathbf{s}}|\mathrm{d})$$ vanishes. Therefore the Euclidean Yang-Mills theories are perturbatively renormalisable.
Proofs of statements of analytic parts based on detailed expression of bounds of UV cut off via the analysis by weighted trees. Results are somewhat more general than Corollary 2 and Theorem 3 (Chapter V. Propositions 11 and 12). Several inequalities used in Chapter V (and Chapter VI) are proved in the appendix. They may used as hard exercises in advanced Calculus. The proof of Theorem 4 is based on the analysis of anomalies; gauge anomaly and anomalies governing the local gauge invariance of renormalised composite operators inside a correlation functions and the local gauge invariance of contact type terms of renormalised composite operators inside a correlation functions (Chapter VI. Propositions 17 and 18).
In Chapter VII, Discussion, the authors state that proofs of this paper can be extended to other gauge fields which have a BV-extended action linear in antifields, provided one can remove the anomaly for the functionals without insertions. This question is decided by the relevant equivariant cohomology of the corresponding classical BV differential $$\hat{\mathbf{s}}$$ at dimension 4 and ghost number 1. The authors also say in a next step, to extend the results on the Operator Product Expansion for scalar fields to gauge theories is desirable [S. Hollands and C. Kopper, Commun. Math. Phys. 313, No. 1, 257–290 (2012; Zbl 1332.81119); J. Holland et al., ibid. 342, No. 2, 385–440 (2016; Zbl 1337.81092)].

##### MSC:
 81T13 Yang-Mills and other gauge theories in quantum field theory 81T08 Constructive quantum field theory 81T15 Perturbative methods of renormalization applied to problems in quantum field theory 81T17 Renormalization group methods applied to problems in quantum field theory 81T50 Anomalies in quantum field theory
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