We review the use of the exact renormalization group for realization of symmetry in renormalizable field theories. The review consists of three parts. In part I ( § §2-4), we start with the perturbative construction of a renormalizable field theory as a solution of the exact renormalization group (ERG) differential equation. We show how to characterize renormalizability by an appropriate asymptotic behavior of the solution for a large momentum cutoff. Renormalized parameters are introduced to control the asymptotic behavior. In part II ( § §5-9), we introduce two formalisms to incorporate symmetry: one by imposing the Ward-Takahashi identity, and the other by imposing the generalized Ward-Takahashi identity via sources that generate symmetry transformations. We apply the two formalisms to concrete models such as QED, YM theories, and the Wess-Zumino model in four dimensions, and the O(N ) non-linear sigma model in two dimensions. We end this part with calculations of the abelian axial and chiral anomalies. In part III ( § §10 and 11), we overview the Batalin-Vilkovisky formalism adapted to the Wilson action of a bare theory with a UV cutoff. We provide a few appendices to give details and extensions that can be omitted for the understanding of the main text. The last appendix is a quick summary for the reader's convenience.

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We report preliminary results on the analysis of the three-body Υ( 10860) → B Bπ, Υ(10860) → [B B * + c.c.]π and Υ(10860) → B * B * π decays including an observation of the Υ(10860) → Z ± b (10610)π ∓ → [B B * + c.c.] ± π ∓ and Υ(10860) → Z ± b (10650)π ∓ → [B * B * ] ± π ∓ decays as intermediate channels. We measure branching fractions of the three-body decays to be B(Υ(10860) → [B B * + c.c.] ± π ∓ ) = (28.3 ± 2.9 ± 4.6) × 10 −3 and B(Υ(10860) → [B * B * ] ± π ∓ ) = (14.1 ± 1.9 ± 2.4) × 10 −3 and set 90% C.L. upper limit B(Υ(10860) → [B B] ± π ∓ ) < 4.0 × 10 −3 . We also report results on the amplitude analysis of the three-body Υ(10860) → Υ(nS)π + π − , n = 1, 2, 3 decays and the analysis of the internal structure of the three-body Υ(10860) → h b (mP )π + π − , m = 1, 2 decays. The results are based on a 121.4 fb −1 data sample collected with the Belle detector at a center-of-mass energy near the Υ(10860).

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We show, explicitly within perturbation theory, that the quantum master equation and the Wilsonian renormalization group flow equation can be combined such that for the continuum effective action, quantum BRST invariance is not broken by the presence of an effective ultraviolet cutoff $\Lambda$, despite the fact that the structure demands quantum corrections that naïvely break the gauge invariance, such as a mass term for a non-Abelian gauge field. Exploiting the derivative expansion, BRST cohomological methods fix the solution up to choice of renormalization conditions, without inputting the form of the classical, or bare, interactions. Legendre transformation results in an equivalent description in terms of solving the modified Slavnov–Taylor identities and the flow of the Legendre effective action under an infrared cutoff $\Lambda$ (i.e. effective average action). The flow generates a canonical transformation that automatically solves the Slavnov–Taylor identities for the wavefunction renormalization constants. We confirm this structure in detail at tree level and one loop. Under flow of $\Lambda$, the standard results are obtained for the beta function, anomalous dimension, and physical amplitudes, up to the choice of the renormalization scheme.