## July 30, 2024

### The Zinn-Justin Equation

A note from my QFT class. Finally, I understand what Batalin-Vilkovisky anti-fields are for.

The Ward-Takahashi Identities are central to understanding the renormalization of QED. They are an (infinite tower of) constraints satisfied by the vertex functions in the 1PI generating functional $\Gamma(A_\mu,\psi,\tilde\psi,b,c,\chi)$. They are simply derived by demanding that the BRST variations

(1)$\begin{split} \delta_{\text{BRST}} b&= -\frac{1}{\xi}(\partial\cdot A-\xi^{1/2}\chi)\\ \delta_{\text{BRST}} A_\mu&= \partial_\mu c\\ \delta_{\text{BRST}} \chi &= \xi^{-1/2} \partial^\mu\partial_\mu c\\ \delta_{\text{BRST}} \psi &= i e c\psi\\ \delta_{\text{BRST}} \tilde{\psi} &= -i e c\tilde{\psi}\\ \delta_{\text{BRST}} c &= 0 \end{split}$

annihilate $\Gamma$: $\delta_{\text{BRST}}\Gamma=0$ (Here, by a slight abuse of notation, I’m using the same symbol to denote the sources in the 1PI generating functional and the corresponding renormalized fields in the renormalized action $\mathcal{L}= -\frac{Z_A}{4}F_{\mu\nu}F^{\mu\nu} + Z_\psi \left(i\psi^\dagger \overline{\sigma}\cdot(\partial-i e A)\psi+ i\tilde{\psi}^\dagger \overline{\sigma}\cdot(\partial+i e A)\tilde{\psi} -Z_m m(\psi\tilde{\psi}+\psi^\dagger\tilde{\psi}^\dagger) \right) +\mathcal{L}_{\text{GF}}+\mathcal{L}_{\text{gh}}$ where $\begin{split} \mathcal{L}_{\text{GF}}+\mathcal{L}_{\text{gh}}&= \delta_{\text{BRST}}\frac{1}{2}\left(b(\partial\cdot A+\xi^{1/2}\chi)\right)\\ &=-\frac{1}{2\xi} (\partial\cdot A)^2+ \frac{1}{2}\chi^2 - b\partial^\mu\partial_\mu c \end{split}$ They both transform under BRST by (1).)

The situation in nonabelian gauge theories is more cloudy. Unlike in QED, $\mathcal{N}\coloneqq Z_g Z_A^{1/2}\neq 1$. Hence the BRST transformations need to be renormalized. Let $\tilde{D}_\mu = \partial_\mu -i g\mathcal{N}A_\mu$ be the renormalized covariant derivative and $\tilde{F}_{\mu\nu} = \frac{i}{g\mathcal{N}}[\tilde{D}_\mu,\tilde{D}_\nu]= \partial_\mu A_\nu-\partial_\nu A_\mu -i g\mathcal{N}[A_\mu,A_\nu]$ the renormalized field strength. The renormalized BRST transformations

(1)$\begin{split} \delta_{\text{BRST}} b&= -\frac{1}{\xi}(\partial\cdot A-\xi^{1/2}\chi)\\ \delta_{\text{BRST}} A_\mu&= Z_{\text{gh}}\tilde{D}_\mu c = Z_{\text{gh}} (\partial_\mu c -i g\mathcal{N}[A_\mu,c])\\ \delta_{\text{BRST}} \chi &= \xi^{-1/2}Z_{\text{gh}} \partial^\mu\tilde{D}_\mu c\\ \delta_{\text{BRST}} c &= \frac{i g}{2}Z_{\text{gh}} \mathcal{N}\{c,c\} \end{split}$

explicitly involve both $\mathcal{N}$ and the ghost wave-function renormalization, $Z_{\text{gh}}$ and are corrected order-by-order in perturbation theory. Hence the relations which follow from $\delta_{\text{BRST}} \Gamma=0$ (called the Slavnov-Taylor Identities) are also corrected order-by-order in perturbation theory.

This is … awkward. The vertex functions are finite quantities. And yet the relations (naively) involve these infinite renormalization constants (which, moreover, are power-series in $g$).

But if we step up to the full-blown Batalin-Vilkovisky formalism, we can do better. Let’s introduce a new commuting adjoint-valued scalar field $\Phi$ with ghost number $-2$ and an anti-commuting adjoint-valued vector-field $S_\mu$ with ghost number $-1$ and posit that they transform trivially under BRST: $\begin{split} \delta_{\text{BRST}} \Phi&=0\\ \delta_{\text{BRST}} S_\mu&=0 \end{split}$ The renormalized Yang-Mills Lagrangian1 is

(2)$\mathcal{L}= -\frac{Z_A}{2} Tr\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu} +\mathcal{L}_{\text{GF}}+\mathcal{L}_{\text{gh}}+\mathcal{L}_{\text{AF}}$

where $\begin{split} \mathcal{L}_{\text{GF}}+\mathcal{L}_{\text{gh}}&= \delta_{\text{BRST}}Tr\left(b(\partial\cdot A+\xi^{1/2}\chi)\right)\\ &=-\frac{1}{\xi} Tr(\partial\cdot A)^2+ Tr\chi^2 -2Z_{\text{gh}}Tr b\partial\cdot\tilde{D} c \end{split}$ and $\mathcal{L}_{\text{AF}} =Z_{\text{gh}}Tr(S^\mu\tilde{D}_\mu c) +\frac{i g}{2} Z_{\text{gh}}\mathcal{N} Tr(\Phi\{c,c\})$ $\mathcal{L}_{\text{AF}}$ is explicitly BRST-invariant because what appears multiplying the anti-fields $S^\mu$ and $\Phi$ are BRST variations (respectively of $A_\mu$ and $c$). These were the “troublesome” BRST variations where the RHS of (1) were nonlinear in the fields (and hence subject to renormalization).

Now we can replace the “ugly” equation $\delta_{\text{BRST}}\Gamma=0$, which has explicit factors of $\mathcal{N}$ and $Z_{\text{gh}}$ and is corrected order-by-order, with

(3)$\frac{\delta\Gamma}{\delta A^a_\mu}\frac{\delta\Gamma}{\delta S_a^\mu} + \frac{\delta \Gamma}{\delta c^a}\frac{\delta \Gamma}{\delta \Phi_a} - \xi^{-1/2} \chi_a\frac{\delta\Gamma}{\delta b_a} = 0$

which is an exact (all-orders) relation among finite quantites. The price we pay is that the Zinn-Justin equation (3) is quadratic, rather than linear, in $\Gamma$.

1 The trace is normalized such that $Tr(t_a t_b) = \frac{1}{2}\delta_{a b}$.

Posted by distler at July 30, 2024 1:56 PM

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## 1 Comment & 0 Trackbacks

### Re: The Zinn-Justin Equation

Hello, Prof. Distler. I just read this post:
https://golem.ph.utexas.edu/~distler/blog/archives/002943.html#more

What would you say is a good undergrad particle physics book and a graduate particle physics book that you would use for such classes you’d be teaching. I suppose you’d be most likely using your own notes, but I’m curious about what you’d recommend an advanced undergraduate should look at or what a graduate student should look at. Thanks!

Posted by: Rich on August 2, 2024 4:49 PM | Permalink | Reply to this

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