BPHZ

Prerequisites: Renormalization. Full path (23 pages)

Why is it interesting?
Intuitive
Concrete
Abstract

Why is it interesting?

The Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization scheme is an conceptually extremely interesting way to get rid of the infinities that arise in Canonical Quantum Field Theory. In contrast to other popular scheme, the BPHZ method to get rid of infinites involves no UV cutoff. This is interesting, for example if one considers the famous hierarchy problem. Usually it is argued that a small mass for the Higgs particle is unnatural, because it gets loop corrections that are proportional to the cutoff $\Lambda$ squared

$m_H^2 = m_0^2 + c \Lambda^2 + \ldots$

For a large cutoff scale like, for example, the Planck scale, one would therefore suspect a large Higgs mass, unless these corrections somehow cancel with the bare Higgs mass parameter $m_0^2$ . In the BPHZ scheme there is no cutoff and therefore no cancellation against $\Lambda^2$ . Heavy particles coupled to the Higgs still shift its mass squared by an amount proportional to their own mass squared, so this does not by itself remove the hierarchy problem:

It has been shown 5) that a different renormalization scheme (BPHZ) needs no fine tuning of the bare paramters. FINE-TUNING PROBLEM AND THE RENORMALIZATION GROUP by Wetterich

Though it was now obvious that QED was the correct theory to describe electromagnetic interactions, the renormalization procedure itself, allowing the extraction of finite results from initial infinite quantities, had remained a matter of some concern for theorists: the meaning of the renormalization ‘recipe’ and, thus, of the bare parameters remained obscure. Much effort was devoted to try to overcome this initial conceptual weakness of the theory. Several types of solutions were proposed:

(i) The problem came from the use of an unjustified perturbative expansion and a correct summation of the expansion in powers of α would solve it. Somewhat related, in spirit, was the development of the so-called axiomatic QFT, which tried to derive rigorous, non-perturbative, results from the general principles on which QFT was based.

(ii) The principle of QFT had to be modified: only renormalized perturbation theory was meaningful. The initial bare theory with its bare parameters had no physical meaning. This line of thought led to the BPHZ (Bogoliubov, Parasiuk, Hepp, Zimmerman) formalism and, finally, to the work of Epstein and Glaser, where the problem of divergences in position space (instead of momentum space) was reduced to a mathematical problem of a correct definition of singular products of distributions. The corresponding efforts much clarified the principles of perturbation theory, but disguised the problem of divergences in such a way that it seemed never having existed in the first place.

(iii) Finally, the cut-off had a physical meaning and was generated by additional interactions, non-describable by QFT. In the 1960s some physicists thought that strong interactions could play this role (the cut-off then being provided by the range of nuclear forces). Renormalizable theories could then be thought as theories some- what insensitive to this additional unknown short-distance structure, a property that obviously required some better understanding.

This latter point of view is in some sense close to our modern understanding, even though the cut-off is no longer provided by strong interactions. "Phase Transitions and Renormalization Group" by Zinn-Justin

Intuitive

A loop diagram in quantum field theory is an integral over the momentum running around the loop. For some diagrams this integral diverges because the integrand falls off too slowly at large loop momentum. The usual cure is to first make the integral finite by hand, for example by cutting it off at some large momentum, then absorb the dependence on that cutoff into the parameters of the theory, and at the end remove the cutoff.

BPHZ does the subtraction before integrating. The large-momentum part of a divergent integrand depends on the external momenta of the diagram in a very simple way: up to terms that fall off fast enough, it is a polynomial in them. BPHZ subtracts from the integrand the first few terms of its Taylor expansion in the external momenta. The subtracted integrand then decays fast enough and the integral is finite. No cutoff and no modified dimension of space-time ever appears.

The subtracted polynomial terms are exactly what an extra local term in the Lagrangian (a counterterm) would produce, so the procedure is equivalent to ordinary renormalization with counterterms. The complication is that a diagram can contain smaller divergent diagrams inside it, nested or overlapping. Subtracting only the overall divergence then fails. The forest formula of Zimmermann gives the rule for which combinations of subdiagrams have to be subtracted, so that every divergence is removed exactly once.

Concrete

A good explanation of the BPHZ renormalization scheme can be found at page 53ff and page 133ff in Collins "Renormalization".

Another good explanation can be found at page 626 and 640 in Duncan "The Conceptual Framework of Quantum Field Theory".

Abstract

Let $\Gamma$ be a Feynman graph with Euclidean momentum-space integrand $I_\Gamma(p,k)$ , where $p$ are the external and $k$ the loop momenta, and assume all propagators are massive. A renormalization part is a one-particle-irreducible subgraph $\gamma\subseteq\Gamma$ with superficial degree of divergence $\omega(\gamma)\geq 0$ . Two subgraphs are non-overlapping if they are disjoint or one contains the other. A forest $F$ is a set of pairwise non-overlapping renormalization parts; the empty set is a forest, and $\Gamma$ itself belongs to forests if $\omega(\Gamma)\geq0$ . For each $\gamma$ let $T_\gamma$ be the operator that replaces the part of the integrand belonging to $\gamma$ by its Taylor polynomial of degree $\omega(\gamma)$ in the external momenta of $\gamma$ , expanded around zero. Zimmermann's forest formula defines the renormalized integrand

$R_\Gamma = \sum_{F} \prod_{\gamma\in F}\left(-T_\gamma\right) I_\Gamma ,$

where in each product the operator of a subgraph acts before the operator of any graph containing it. Equivalently, Bogoliubov's recursion defines $\bar R_\Gamma$ by subtracting all proper divergent subgraphs, each already renormalized, and sets $R_\Gamma = (1-T_\Gamma)\bar R_\Gamma$ .

Theorem (BPHZ). For every graph of a theory with massive propagators, $\int dk\, R_\Gamma$ converges absolutely in Euclidean space. In Minkowski space, with the propagator denominators $p^2 - m^2 + i\epsilon$ , the limit $\epsilon\to 0$ exists as a tempered distribution in the external momenta. Moreover the sum over graphs of $R_\Gamma$ equals (formally, or with any intermediate regularization) the unsubtracted Feynman expansion of a Lagrangian $\mathcal{L}+\Delta\mathcal{L}$ , where $\Delta\mathcal{L}$ is a formal power series of local counterterms whose form is fixed by the degrees $\omega(\gamma)$ .

Bogoliubov and Parasiuk formulated the subtraction procedure, Hepp gave a complete proof of convergence by decomposing the integration domain into sectors, and Zimmermann solved the recursion by the forest formula and proved absolute convergence directly. Taylor expansion at zero external momenta fixes the renormalization conditions at that point; for massless fields this produces infrared divergences, and the method has to be modified (the BPHZL scheme of Lowenstein and Zimmermann). Subtracting to a higher degree than $\omega(\gamma)$ (oversubtraction) is also consistent and defines Zimmermann's normal products of composite operators, which are used to derive Ward identities and the operator product expansion within perturbation theory.

In the Hopf algebra language of Connes and Kreimer (see renormalization), the forest formula is the twisted antipode: with $\phi$ the unrenormalized Feynman rules and $T$ the subtraction operator, the counterterm map is defined recursively by $S^\phi_T(\Gamma) = -T\left[\phi(\Gamma) + \sum_{\gamma} S^\phi_T(\gamma)\,\phi(\Gamma/\gamma)\right]$ , the sum running over proper non-empty unions of divergent subgraphs, and the renormalized value is $S^\phi_T * \phi$ .

BPHZ

Why is it interesting?

Intuitive

Concrete

Abstract

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