Talk:Beta function (physics)

it might be useful to add other important examples... Jpod2 22:59, 16 August 2006 (UTC)[reply]

Ginzburg-Landau theory

One could add the beta-function of the Ginzburg-Landau theory as another example.

Fabian.biebl (talk) 12:02, 19 August 2011 (UTC)[reply]

A Ginzburg-Landau theory is just a Wick-rotated scalar field theory in d+1 dimensions. Indeed, you can find this beta function written down in that article, in four dimensions, and somewhat misplaced. Please, note that this beta function just applies in the limit of a small self-interaction. In the opposite limit this takes the form

\beta =d\lambda

being d the dimension of the space-time and

\lambda

the coupling of the self-interaction term. You should check this and relative references at landau pole and the corresponding discussion, also this material being somewhat misplaced and not well-linked each other.--Pra1998 (talk) 12:57, 19 August 2011 (UTC)[reply]

QED

are you sure, this is correct?

$\beta (e)={\frac {e^{3}}{12\pi ^{2}}}~,$ $\beta (\alpha )={\frac {2\alpha ^{2}}{3\pi }}~,$

if I substitue α=e²/4π, I get (even if dropping ε, h and c): $\beta (e)={\frac {e^{4}}{24\pi ^{3}}}~,$

is the rest of the formulas likewise corrupted or is there some other magic in it? Ra-raisch (talk) 15:46, 2 August 2017 (UTC)[reply]

"Magic"? You just naively plugged in, as though β were a mere function, and not a gradient, as defined in the lede of the article!? Please read carefully. Consider

2e~de/(4\pi )

on the l.h.side instead. Yes, one is sure. Cuzkatzimhut (talk) 16:20, 2 August 2017 (UTC)[reply]

thx, I thought of that, but β(e) looks just like a mere function. It's hard to outline the difference. I then use

\beta _{e}

instead. Ra-raisch (talk) 16:53, 2 August 2017 (UTC)[reply]

Yes, but it is understood, once the operator was unambiguously defined in the lede that its eigenvalues (in the rest of the article) are functions of the argument indicated. This is so central to the renormalization group methodology that I am not sure this is the article to emphasize it. Cuzkatzimhut (talk) 18:40, 2 August 2017 (UTC)[reply]

QCD

This formula for the one-loop beta function of the strong coupling constant,

\beta (g)=-\left(11-{\frac {n_{s}}{3}}-{\frac {2n_{f}}{3}}\right){\frac {g^{3}}{16\pi ^{2}}}~,

and the one below that are not correct, no? There should be no contribution from the Higgs boson, i.e. the term with $n_{s}$ should be zero, since the Higgs does not couple strongly and hence, it cannot contribute to the beta function of the strong coupling constant at one-loop. If you take the general formula of Politzer, Gross and Wilczek, this is reflected by the fact that the Dynkin index $T(R_{s})$ is zero, since the Higgs boson in QCD is a singlet under charge transformations. So the correct formulae should be

\beta (g)=-\left(11-{\frac {2n_{f}}{3}}\right){\frac {g^{3}}{16\pi ^{2}}}~,

and

\beta (\alpha _{s})=-\left(11-{\frac {2n_{f}}{3}}\right){\frac {\alpha _{s}^{2}}{2\pi }}~,

— Preceding unsigned comment added by 129.13.37.94 (talk) 10:03, 18 January 2019 (UTC)[reply]

Fixed misleading name of notional colored scalars. This is a general expression to be used for theory analysis/speculation: Treating it as engineering handbook data is misguided. Cuzkatzimhut (talk) 14:49, 18 January 2019 (UTC)[reply]