Perturbative Renormalisations of Noether currents for N=1 Super Yang-Mills theory for Wilson, tree-level Symanzik and Iwasaki gauge actions with stout smeared links

This dataset provides symbolic expressions for the $\bf{\overline{\rm{MS}}}$ renormalization factors (Z factors) of various operators for multiple gauge actions, namely the Wilson, tree-level Symanzik, and Iwasaki actions. The gluino action employs stout-smeared links, the full version of lattice the action is presented here, in Eq. (12). The results are distributed as a Wolfram language / Mathematica .m file (ZfactorsNoetherCurrentsSYM.m), which contains expressions for ZSS*, ZST*, ZSA1*, ZSB1*, ZSB2*, ZSC1*, ZSC2*, ZSC3*, ZSC4*, ZSC5*, ZSC6*, ZSC7*, ZSC8*, and ZJ* for each of the three actions. The chiral current is renormalized multiplicatively, whereas the renormalized supercurrent $S_\mu^R$ mixes with four classes of operators. The renormalization and mixing coefficients are defined as follows: $S_\mu^R = Z_{SS} S_\mu^B + Z_{ST} T_\mu^B + Z_{SA1} {\cal O}_{A1}^B + \sum_{i=1}^{2} Z_{SBi} {\cal O}_{Bi}^B + \sum_{i=1}^{8} Z_{SCi} {\cal O}_{Ci}^B$ $J_{5 \mu}^R = Z_J J_{5 \mu}^B $ where the superscript $B$ ($R$) denotes bare (renormalized) operators, and $Z_{X(Y)}$ are the corresponding renormalization/mixing coefficients. The coefficients $Z_{X(Y)}$ depend on the regularization, $\mathrm{Reg. = Lattice}$, and the renormalization scheme, $\mathrm{Sch. = {\overline{\rm{MS}}}}$, employed, and are therefore written as $Z^{\mathrm{Reg.},\mathrm{Sch.}}_{X(Y)}$. Operator classes: Class G: Gauge invariant $S_\mu = -\frac{1}{2} \, \mathrm{Tr} \left( u_{\rho\sigma} \, [\gamma_\rho, \gamma_\sigma] \, \gamma_\mu \lambda \right)$ (Supercurrent) $T_\mu = 2\, \mathrm{Tr} \left( u_{\mu\nu} \, \gamma_\nu \, \lambda \right)$ Class A: BRST invariant ${\mathcal{O}}_{A1} = \frac{1}{\alpha} \, \mathrm{Tr} \left( (\partial_\nu u_\nu)\, \gamma_\mu \lambda \right) - i g\, \mathrm{Tr} \left( [c, \bar{c}]\, \gamma_\mu \lambda \right)$ Class B: Vanishing by E.O.M. ${\mathcal{O}}_{B1} = \mathrm{Tr} \left( u_\mu\, \not{D} \lambda \right)$ ${\mathcal{O}}_{B2}= \mathrm{Tr} \left( \not{u}\, \gamma_\mu\, \not{D} \lambda \right)$ Class C: All other operators that may mix with $S_\mu$ ${\mathcal{O}}_{C1}= \mathrm{Tr} \left( u_\mu\, \lambda \right)$ ${\mathcal{O}}_{C2} = \mathrm{Tr} \left( \not{u}\, \gamma_\mu\, \lambda \right)$ ${\mathcal{O}}_{C3}= \mathrm{Tr} \left( \not{u}\, \partial_\mu \lambda \right)$ ${\mathcal{O}}_{C4}= \mathrm{Tr} \left( (\partial_\mu \not{u})\, \lambda \right)$ ${\mathcal{O}}_{C5}= \mathrm{Tr} \left( (\partial_\nu u_\nu)\, \gamma_\mu \lambda \right)$ ${\mathcal{O}}_{C6}= \mathrm{Tr} \left( u_\nu\, \gamma_\mu\, \partial_\nu \lambda \right)$ ${\mathcal{O}}_{C7} = i g\, \mathrm{Tr} \left( [u_\rho, u_\sigma]\, [\gamma_\rho, \gamma_\sigma]\, \gamma_\mu \lambda \right)$ ${\mathcal{O}_{C8}} = i g\, \mathrm{Tr} \left( [u_\mu, u_\nu]\, \gamma_\nu \lambda \right)$ $J_{5 \mu} = \mathrm{Tr} \left( \bar \lambda \gamma_5 \gamma_\mu \lambda \right)$ (Chiral current) The file includes expressions for: ZSSWILSON, ZSTWILSON, ZSA1WILSON, ZSB1WILSON, ZSB2WILSON, ZSC1WILSON, ZSC2WILSON, ZSC3WILSON, ZSC4WILSON, ZSC5WILSON, ZSC6WILSON, ZSC7WILSON, ZSC8WILSON ZSSTLS, ZSTTLS, ZSA1TLS, ZSB1TLS, ZSB2TLS, ZSC1TLS, ZSC2TLS, ZSC3TLS, ZSC4TLS, ZSC5TLS, ZSC6TLS, ZSC7TLS, ZSC8TLS ZSSIWASAKI, ZSTIWASAKI, ZSA1IWASAKI, ZSB1IWASAKI, ZSB2IWASAKI, ZSC1IWASAKI, ZSC2IWASAKI, ZSC3IWASAKI, ZSC4IWASAKI, ZSC5IWASAKI, ZSC6IWASAKI, ZSC7IWASAKI, ZSC8IWASAKI ZJWILSON, ZJTLS, ZJIWASAKI Variables / symbols used in the file: g: coupling constant alatt: lattice spacing Nc: number of colors beta: $\beta = 1 - \alpha$, where $\alpha$ is the gauge parameter csw: clover parameter wA: stout parameter stemming from the action wO: stout parameter stemming from the supercurrent operator mubar: MSbar renormalization scale pisq16: $16 \pi^2$ Usage (Mathematica / Wolfram Language): Get["ZfactorsNoetherCurrentsSYM.m"]; Then evaluate the desired symbol and substitute parameter values as needed. For example: Get["ZfactorsNoetherCurrentsSYM.m"];params = {Nc -> 3, beta -> 0, csw -> 1, r -> 1, wA -> 0,wO->0.15, pisq16 -> 16 Pi^2, alatt*mubar -> 1};ZSSWILSON /. params After these substitutions, the resulting expression depends only on the coupling g.