Physics, in its quest to unravel the mysteries of the universe, often involves controversy, with seemingly established ideas needing to be reevaluated. One such controversy concerns the “principle of maximal conformality” in QCD, the quantum field theory of quarks and gluons, some of the universe’s fundamental components. Generally viewed as a viable approach to solving a significant theoretical challenge, this “principle” undergoes searching criticism in a recent study.

Professor Paul Stevenson from Rice University, in a study published in Physics Letters B, argues that the idea fails to resolve the issue of renormalization scheme dependence, as it purports to do. He emphasizes the role of RG invariance, a fundamental property of quantum field theories whereby physical predictions do not depend on the specific way that the renormalized coupling constant is defined (the “renormalization scheme”). Delving deeper into the nuances, he explains, “For the exact theory, this symmetry is trivial, quite banal; it is just the fact that one is allowed to make an algebraic substitution, a change of variables. The problem arises with the approximation because we don’t have an exact result, only the first few terms of its perturbation series – and truncating the series spoils the symmetry.”

Professor Stevenson proposed his own method to resolve the problem in a well-known, but controversial, paper of 1981. It is based on what he called the “principle of minimal sensitivity”: the approximate result, though not exactly invariant, is locally invariant around an optimum scheme choice where any small scheme change produces almost no change in the result. He recently completed a book (available open access online) explaining his method in detail.

His new paper, he says, is not about defending that principle – “that is a separate argument” – but about explaining what is, and what is not, invariant. He shows that a key quantity in the “maximal conformality” approach is not invariant, so that the method gives results that still depend on the initial, arbitrary scheme choice. “It just does not work,” he says. He also highlights some widespread misconceptions in the scientific community about the nature of scheme dependency. He emphasizes that it is not just a matter of identifying the right energy scale for the coupling constant. Indeed, the scale alone is not meaningful; what matters is the ratio of this scale to a parameter Λ that is itself scheme dependent, but in a simple and definite manner. He shows that there are specific quantities, one at each order, that are calculable invariants. He stresses that “RG invariance is a symmetry, and any viable method for resolving the scheme-dependence problem should be formulatable in terms of the invariants of that symmetry.”

Professor Stevenson’s critical examination of “maximal conformality” is a reminder that diligent scrutiny and reassessment in scientific research is always required. Controversy and debate are needed if our theoretical models are to be as robust and precise as possible, so that we can advance our understanding of the quantum world.

ArXiv links to see further discussions:


Paul Stevenson, ‘Maximal conformality’ does not work, Physics Letters B, 2023. DOI:


Paul Stevenson received his BA from Cambridge University in 1976 and his Ph.D. from Imperial College, London in 1979. His thesis was on QCD jets in e+e- and leptoproduction. After postdoctoral work at the University of Wisconsin-Madison, and at CERN, he moved to Rice University in Houston in 1984. At Rice he regularly taught courses in quantum field theory, quantum mechanics, and classical mechanics and was twice awarded one of the university’s prestigious George R. Brown teaching awards. He took early retirement in 2015 and now lives in southwest England. Besides his work on renormalization, he has worked on the Gaussian Effective Potential,  λφ4  scalar field theory, and the possibility of hydrodynamic excitations of the symmetry-broken vacuum. He also wrote (with Duck and Sudarshan) an influential early paper on “weak measurements” in quantum mechanics.