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May be this is trivial but I need to understand why the renormalization of conserved current is not necessary ? As for example, in this paper, they demand (2$^{nd}$ paragraph of the 2$^{nd}$ column in page no. 3157) that the renormalization of the operator $$ Q_7 = \frac{e^2}{4\pi}\,\left[ \bar{s}\,\gamma^\mu (1-\gamma_5)d\right]\left[\bar{e}\,\gamma_\mu \,e \right]$$ is not needed because I quote from the reference [14] of the above mentioned paper:

"[14] Although the hadronic part of $Q_7$ is a composite operator involving two quark fields at the same point, it does not require renormalization since it is a partially conserved current. Thus, its matrix elements are not $\mu$ dependent."

Yet, in another paper (where one of the authors is M. B. Wise, the author of the first article I cited), they say otherwise, I quote from the abstract:

"...It is commonly asserted that the electromagnetic current is conserved and therefore is not renormalized. Within QED we show (a) that this statement is false..."

For now I need to understand why do we assert that conserved currents don't need renormalization. References to articles and books are most welcome.

Qmechanic
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quanta
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1 Answers1

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This is a long comment, in which I summarise Collins's points in @HansMoleman's source (part of the book's Sec 6.6), but I'm not an expert on this topic.

Given a basic current $j^\mu_b$, both $[j^\mu_b]$ and the unrenormalised Lagrangian's conserved current $j^\mu$ add minimal subtraction counterterms to $j^\mu_b$. Since $\varepsilon^\mu:=j^\mu-[j^\mu_b]$ is the difference of two solutions of the Ward identity, $\partial_\mu\langle 0|T\varepsilon^\mu (x)X|0\rangle=0$. The question is whether $\partial_\mu\varepsilon^\mu=0$ holds off-shell. If this were only true on-shell, $\partial_\mu\langle 0|T\varepsilon^\mu (x)X|0\rangle$ would be a non-zero term of the same mass dimension as $j^\mu_b$ (or $j^\mu$, or $[j^\mu_b]$), so one has to consider which terms of this dimension the theory can construct. Examples include:

  • $\partial_\nu F^{\mu\nu}$;
  • in a $4D$ non-Abelian theory, $\varepsilon_{\kappa\lambda\mu\nu}\partial^\kappa (A_a^\lambda A_b^\nu)$ (which, in view of the Levi-Civita symbol, indicates a chiral symmetry, and Collins doesn't treat it fully until Chapter 12);
  • and, because we can extend this analysis to $2$-dimensional "currents", $(\partial_\mu\partial_\nu-g_{\mu\nu}\square)\phi^2$ is a counterterm in $T_{\mu\nu}$.

In some cases no such terms can be constructed, so $\partial_\mu\varepsilon^\mu=0$. Collins lists cases where the result can fail:

  • Space-time symmetries - for these he recommends the sources Callan, Coleman & Jackiw (1970), Freedman, Muzinich & Weinberg (1974), Collins (1976), Brown & Collins (1980) and Joglekar (1976);
  • Non-conserved currents with breaking term of dimension $\ge$ that of the Lagrangian density;
  • Non-linear transformations;
  • Gauge theories, if a suitable generalisation fails (again, this is treated in full in his Chapter 12).

I can't find a discussion of $Q_7$ in the book but, considering that he spends the rest of Chapter 6 studying a scalar field's mass renormalization, you should usually expect renormalization to involve a lot of calculation, sometimes revealing that "nothing changes". (For example, a suitable generalisation of the above might succeed in a gauge theory.) In such cases, there can be an effort to summarise why nothing changes, but these can be confusing if the source leaves it there, rather than letting it serve as a preface to a full demonstration.

J.G.
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