# BPHZ

## 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

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

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 $m_0$. In the BPHZ scheme there is no cutoff and therefore no fine-tuning 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

## Layman

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

## Student

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".

## Researcher

The motto in this section is: the higher the level of abstraction, the better.
Common Question 1
Common Question 2

Example1
Example2: