An object is scale invariant if it is preserved by global rescaling of length. An object is conformally invariant if it is preserved by transformations that preserve angles between the tangents to smooth curves at their intersections, hence by local rescaling of length.
Yu Nakayama, Scale invariance vs conformal invariance [arXiv:1302.0884]
Martin Lüscher, Gerhard Mack: Global Conformal Invariance in Quantum Field Theory, Comm. Math. Phys. 41 3 (1975) 203-234 [doi:10.1007/BF01608988, jstor:cmp/1103898909, inspire:90687]
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