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*, 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: $S_\mu^R = Z_{SS} S_\mu^B + Z_{ST} T_\mu^B + Z_{SA1} {\cal O}_{A1}^B + \sum_{i=1}^{2} Z_{SBi} {\cal O}_{Bi}^B + \sum_{i=1}^{9} Z_{SCi} {\cal O}_{Ci}^B$ $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 $Z^{\mathrm{Reg.},\mathrm{Sch.}}_{X(Y)}$. Before defining the operators and their classes, we note that although the mixing pattern formally involves the operators ${\mathcal O}_{Ci}$ with $i=1,\ldots,9$, the present dataset contains results only for $i=1,\ldots,6$. Results for ${\mathcal O}_{C7}$, ${\mathcal O}_{C8}$, and ${\mathcal O}_{C9}$ will be presented at a later stage. 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 $S_\mu$ ${\mathcal{O}}_{C1}= \mathrm{Tr} \left( u_\mu\, \lambda \right)$ ${\mathcal{O}}_{C2} = \mathrm{Tr} \left( \not{u}\, \gamma_\mu\, \lambda \right)$ ${\mathcal{O}}_{C3}= \mathrm{Tr} \left( \not{u}\, \partial_\mu \lambda \right)$ ${\mathcal{O}}_{C4}= \mathrm{Tr} \left( (\partial_\mu \not{u})\, \lambda \right)$ ${\mathcal{O}}_{C5}= \mathrm{Tr} \left( (\partial_\nu u_\nu)\, \gamma_\mu \lambda \right)$ ${\mathcal{O}}_{C6}= \mathrm{Tr} \left( u_\nu\, \gamma_\mu\, \partial_\nu \lambda \right)$ $J_{5 \mu} = \mathrm{Tr} \left( \bar \lambda \gamma_5 \gamma_\mu \lambda \right)$ (Chiral current) The file includes expressions for: ZSSWILSON, ZSTWILSON, ZSA1WILSON, ZSB1WILSON, ZSB2WILSON, ZSC1WILSON, ZSC2WILSON, ZSC3WILSON, ZSC4WILSON, ZSC5WILSON, ZSC6WILSON ZSSTLS, ZSTTLS, ZSA1TLS, ZSB1TLS, ZSB2TLS, ZSC1TLS, ZSC2TLS, ZSC3TLS, ZSC4TLS, ZSC5TLS, ZSC6TLS ZSSIWASAKI, ZSTIWASAKI, ZSA1IWASAKI, ZSB1IWASAKI, ZSB2IWASAKI, ZSC1IWASAKI, ZSC2IWASAKI, ZSC3IWASAKI, ZSC4IWASAKI, ZSC5IWASAKI, ZSC6IWASAKI 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.