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.本数据集提供了威尔逊（Wilson）、树级Symanzik（tree-level Symanzik）与岩崎（Iwasaki）三种规范作用量下，各类算符的$overline{ ext{MS}}$（改进型最小减除）重整化因子（Z因子）符号表达式。胶微子（gluino）作用量采用粗涂抹（stout-smeared）格链，完整的格点作用量形式见式(12)。 结果以Wolfram语言/Mathematica的.m格式文件`ZfactorsNoetherCurrentsSYM.m`分发，该文件包含三种规范作用量各自对应的ZSS*、ZST*、ZSA1*、ZSB1*、ZSB2*、ZSC1*、ZSC2*、ZSC3*、ZSC4*、ZSC5*、ZSC6*以及ZJ*的符号表达式。 手征流（chiral current）为乘性重整化，而重整化超流（supercurrent）$S_mu^R$会与四类算符发生混合。重整化与混合系数的定义如下： $$S_mu^R = Z_{SS} S_mu^B + Z_{ST} T_mu^B + Z_{SA1} {cal O}_{A1}^B + sum_{i=1}^{2} Z_{SBi} {cal O}_{Bi}^B + sum_{i=1}^{9} Z_{SCi} {cal O}_{Ci}^B$$ $$J_{5 mu}^R = Z_J J_{5 mu}^B$$ 其中上标$B$（$R$）代表裸（重整化）算符，$Z_{X(Y)}$为对应的重整化/混合系数。系数$Z_{X(Y)}$依赖于所采用的正则化方案$mathrm{Reg. = Lattice}$（格点正则化）与重整化方案$mathrm{Sch. = overline{ ext{MS}}}$，因此可写为$Z^{mathrm{Reg.},mathrm{Sch.}}_{X(Y)}$。需说明的是，尽管形式上混合模式涉及$i=1,dots,9$的算符${mathcal O}_{Ci}$，但本数据集仅包含$i=1,dots,6$的结果，${mathcal O}_{C7}$、${mathcal O}_{C8}$与${mathcal O}_{C9}$的结果将在后续阶段发布。 ### 算符类别 1. **G类：规范不变算符（gauge invariant）** 超流表达式： $$S_mu = -frac{1}{2} , mathrm{Tr} left( u_{ hosigma} , [gamma_ ho, gamma_sigma] , gamma_mu lambda ight)$$ $$T_mu = 2, mathrm{Tr} left( u_{mu u} , gamma_ u , lambda ight)$$ 2. **A类：BRST不变算符（BRST invariant）** $$mathcal{O}_{A1} = frac{1}{alpha} , mathrm{Tr} left( (partial_ u u_ u), gamma_mu lambda ight) - i g, mathrm{Tr} left( [c, ar{c}], gamma_mu lambda ight)$$ 3. **B类：满足运动方程（Equations of Motion, E.O.M.）后消失的算符** $$O_{B1} = mathrm{Tr} left( u_mu, ot{D} lambda ight)$$ $$O_{B2}= mathrm{Tr} left( ot{u}, gamma_mu, ot{D} lambda ight)$$ 4. **C类：所有可与$S_mu$发生混合的其余算符** $$O_{C1}= mathrm{Tr} left( u_mu, lambda ight)$$ $$O_{C2} = mathrm{Tr} left( ot{u}, gamma_mu, lambda ight)$$ $$O_{C3}= mathrm{Tr} left( ot{u}, partial_mu lambda ight)$$ $$O_{C4}= mathrm{Tr} left( (partial_mu ot{u}), lambda ight)$$ $$O_{C5}= mathrm{Tr} left( (partial_ u u_ u), gamma_mu lambda ight)$$ $$O_{C6}= mathrm{Tr} left( u_ u, gamma_mu, partial_ u lambda ight)$$ 手征流表达式为： $$J_{5 mu} = mathrm{Tr} left( ar lambda gamma_5 gamma_mu lambda ight)$$ 该文件包含以下表达式： - 威尔逊规范作用量对应项：`ZSSWILSON`、`ZSTWILSON`、`ZSA1WILSON`、`ZSB1WILSON`、`ZSB2WILSON`、`ZSC1WILSON`、`ZSC2WILSON`、`ZSC3WILSON`、`ZSC4WILSON`、`ZSC5WILSON`、`ZSC6WILSON`、`ZJWILSON` - 树级Symanzik规范作用量对应项：`ZSSTLS`、`ZSTTLS`、`ZSA1TLS`、`ZSB1TLS`、`ZSB2TLS`、`ZSC1TLS`、`ZSC2TLS`、`ZSC3TLS`、`ZSC4TLS`、`ZSC5TLS`、`ZSC6TLS`、`ZJTLS` - 岩崎（Iwasaki）规范作用量对应项：`ZSSIWASAKI`、`ZSTIWASAKI`、`ZSA1IWASAKI`、`ZSB1IWASAKI`、`ZSB2IWASAKI`、`ZSC1IWASAKI`、`ZSC2IWASAKI`、`ZSC3IWASAKI`、`ZSC4IWASAKI`、`ZSC5IWASAKI`、`ZSC6IWASAKI`、`ZJIWASAKI` ### 文件中使用的变量与符号 - $g$：耦合常数（coupling constant） - $alatt$：格点间距（lattice spacing） - $N_c$：颜色数（number of colors） - $eta$：$eta = 1 - alpha$，其中$alpha$为规范参数（gauge parameter） - $csw$：clover参数（clover parameter） - $w_A$：源于作用量的粗涂抹参数（stout parameter） - $w_O$：源于超流算符的粗涂抹参数（stout parameter） - $ar{mu}$：$overline{ ext{MS}}$重整化标度（renormalization scale） - $pisq16$：$16pi^2$ ### 使用方法（Mathematica/Wolfram语言） mathematica Get["ZfactorsNoetherCurrentsSYM.m"]; 随后执行所需的符号表达式并按需代入参数值，示例如下： mathematica 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 完成参数代入后，所得表达式仅依赖于耦合常数$g$。