By Makeenko Y.M.

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65). In the quantum case, the scale invariance is broken by the (dimensional) cutoff a. The energy-momentum tensor is no longer traceless due to loop effects. e. 72) ¯ is a gauge-invariant functional of A, ψ and ψ. ¯ where F [A, ψ, ψ] For a renormalizable theory like QED, the RHS of Eq. 72) is proportional to the Gell-Mann—Low function B(e2 ) which is defined by a de2 = B e2 . 73) da A non-trivial property of a renormalizable theory is that the RHS in this formula is a function solely of e2 — the bare charge.

64 CHAPTER 1. 3. QUANTUM ANOMALIES FROM PATH INTEGRAL 65 The one-loop Gell-Mann–Low function can now be calculated using Eqs. 74), which reproduces the known result for QED. The higher-order corrections in e do not vanish for the scale anomaly. k − q1 q + q1 ❆ ❆ q❆ ❆❑ Remark on non-Abelian scale anomaly ✁ ✁ ❆ ✁ ❆r✁ ✁ ☛✁ q + k q r 1 1 a) Fig. 74) holds in the non-Abelian Yang–Mills theory as well if Fµν a is substituted by the non-Abelian field strength Fµν given by Eq. 63). The corresponding formula, reads q+k θµµ b) The diagrams which contribute to the scale anomaly in d = 4.

20) and using Eq. 21) which is obviously satisfied due to the Jacobi identity. We have thus proven the well-known fact that the Bianchi identity is explicitly satisfied in the second-order formalism, where Fµν is expressed via Aµ by virtue of Eq. 15). 19) are essential. The concept of the first- and second-order formalisms comes from the theory of gravity. dz µ Aµ (z) . 22) 0 dσ z˙ µ (σ)Aµ (z(σ)) . 23) Therefore, the path-ordered exponential in Eq. 22) is to be understood as1 τ [1 + dt z˙ µ (t)Aµ (z(t))] .