As summarized by Wikipedia, different sources use different choices for the (pion) decay constant. This means that the numerical value can vary between $$ \sqrt 2\ f_\pi \quad\leftrightarrow\quad f_\pi \quad\leftrightarrow\quad \frac{1}{\sqrt 2}\ f_\pi $$

I suppose this is connected to the normalization of the state $|\pi(p)\rangle$ in the defining equation $$\tag{1} \langle 0|\,j_{A}^{\,\mu\,a}(x)|\pi^b(p)\rangle = \text{i} p^\mu f_\pi \text{e}^{-\text{i}p\cdot x}\delta^{ab},$$ where we normalize the pion state $$\tag{2} \langle \pi(p)|\pi(q)\rangle = \mathcal N \delta^{(3)}(\mathbf{p}-\mathbf{q})$$ usually with $$\mathcal N = (2\pi)^3\, 2\, p^0 = (2\pi)^3\, 2\, \sqrt{\mathbf p^2+m^2}.$$

(How) is the normalization of the pion state $|\pi\rangle$ connected to the numeric value of the decay constant?


Nowadays, one always uses (2) to normalize a pion component. And one always uses $$\tag{1} \langle 0|\,j_{A}^{\,\mu\,a}(x)|\pi^b(p)\rangle = \text{i} p^\mu f_\pi \text{e}^{-\text{i}p\cdot x}\delta^{ab},$$ to define $f_\pi\sim 93$MeV. (cf M Schwarz, Peskin & Schroeder, etc..., including Donoghue, Golowich & Holstein, which should be your vademecum in any and all such questions.) The mnemonic of this, the mainstream POV, is $j_A^\mu \sim f_\pi\partial^\mu \pi+...$

However, experimentally, (PDG, Li & Cheng,...) one looks at, e.g. charged pion decay, where $\pi^-=(\pi^1-i\pi^2)/\sqrt{2}$, and $ j_A^{\mu ~\bar ud}= j_A^{\mu ~1}+i j_A^{\mu ~2} $; to the effect that $$ \langle 0|\,j_{A}^{\,\mu\,\bar u d}(x)|\pi^-\rangle = \text{i} p^\mu \sqrt{2}f_\pi \text{e}^{-\text{i}p\cdot x} ,$$ which is the 130MeV expression; and you are warned they absorb the square root of 2 in the definition! At any moment, you make sure you appreciate what is actually being written in the analog of (1), and how the axial hadronic currents are normalized. The normalization of a pion component, however, is normally fixed.

  • Edit note. Indeed, the peculiar normalization of Weinberg vI (10.2.15) metastasizes to Weinberg vII (19.4.24) & (19.4.26), and, if you wished to compare to (1) here, as he notes in his footnote, $F_\pi=2f_\pi=186$MeV in our above discussion. A mere change of language, but perfectly consistent. Indeed, if you were (quixotically) inclined to do phenomenology with Weinberg (famous but not celebrated) normalizations, you'd have to build up a conversion table for all of such expressions. I have to reassure you, however, that the mainstream has shifted quite a bit since that book, and defined by the first group sampled.
  • $\begingroup$ I see. How does the value of $184$ MeV in Weinberg‘s book for into this explanation? $\endgroup$ – Stephan Oct 31 '18 at 0:38
  • $\begingroup$ I think eq. (10.2.15) on p.434 is relevant $\endgroup$ – Stephan Oct 31 '18 at 3:06

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