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My question is a historical one: when was the phrase "beta function", as it pertains to the renormalization-group equations, used in physics? I am talking about this beta function:

$$\beta_g\equiv \frac{\partial g}{\partial \log \mu}$$

In fact some of the early seminal papers on this subject (as it pertains to quantum field theory) only use the phrase "renormalization-group equations" [Gell-Mann, Low; 1954] [Callan; 1970], so I am led to believe the terminology was adopted much later.

I am interested because I believe the other "beta function", i.e. the Euler integral of the first kind, is actually quite commonly used in physics (especially in QFT, when calculating physics beta-functions!), so I find it a little surprising that the slightly conflicting terminology was adopted.

I understand that questions on terminology and/or etymology are usually off-topic here, but I fear this question may be too technical to ask on the History of Science and Mathematics stack exchange site.

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  • $\begingroup$ You say the early papers use a different phrase. Do they use a $\beta$ as in your equation? $\endgroup$ – innisfree Oct 19 '18 at 21:32
  • $\begingroup$ @innisfree Yes, but not explicitly using the symbol $\beta$. They relate bare and renormalized quantities via counterterms defined by power series in the couplings with divergent coefficients, and discuss the dependence of those counterterms on the artificially introduced renormalization scale, but never do they explicitly write $\beta=\ldots$. $\endgroup$ – Arturo don Juan Oct 19 '18 at 23:18
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    $\begingroup$ Also note the History of Science and Mathematics Stack Exchange has been created. $\endgroup$ – Nihar Karve Feb 23 at 8:56
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I've seen references specifically to the "Callan-Symanzik beta function", so I don't think it was much later. (Curt Callan and Kurt Symanzik independently discovered their equation in 1970.) Although the notation of Callan (1970) is unclear to me, Symanzik (1970) implicitly defined $\beta(g)$ [Eq. (I.13)] Then, Symanzik (1971) clearly defined $\beta(g)$ as a derivative [Eq. (I.13b)] and discussed some of its properties. Symanzik simply named different coefficients $\alpha,\beta,\gamma$, so the notation itself is rather natural.

The earliest published instances I'm aware of where the function was referred to by the phrase (as opposed to "$\beta(g)$", "the function $\beta$", or similar) are from a couple years later. Politzer (1973) stated that the notation was already common "the coefficient function in the Callan-Symanzik equations commonly called $\beta(g)$...", and proceeded to explicitly write "$\beta$ functions". Later the same year, Weinberg (1973) did use "beta function" in text form. I cannot rule out someone else using the phrase earlier, but it nevertheless appears to be during the early 70s.

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