6  ndg params

A matching between the ndg parameter and the OS can be defined as

\begin{cases} m_s = \mu_\sigma - \frac{Z_P}{Z_S}\mu_\delta \\ m_c = \mu_\sigma + \frac{Z_P}{Z_S}\mu_\delta \\ \end{cases} \quad\quad \begin{cases} \mu_\sigma =\frac{1}{2}( m_s + m_c) \quad\,\,\,\,\\ \mu_\sigma =\frac{1}{2}\frac{Z_S}{Z_P}( m_c - m_s) \\ \end{cases} \,. This matching scheme has the problem that \frac{Z_P}{Z_S} needs to be know very precise to estimate well the cancellation in the m_s.

An alternative scheme used in the gm2 strange and charm paper is \begin{cases} m_c = \mu_\sigma + \frac{Z_P}{Z_S}\mu_\delta \\ aM_K^{ndg}(\mu_\sigma,\mu_\delta) = aM_K^{OS}(m_s)= aM_K^{OS}(m_s^{ref}) +(m_s^{ref}-m_s)\frac{\partial M_K^{OS}}{\partial m_s}\Bigg|_{m_s} \\ \end{cases} where m_s^{ref} is some value close enough to m_s such higer order in the expansion are negligible. The derivative is splitted in sea and valence contribution \frac{\partial M_K^{OS}}{\partial m_s}=\frac{\partial^{val} M_K^{OS}}{\partial m_s} + \frac{\partial^{sea} M_K^{OS}}{\partial m_s} the derivative with respect to the valence is computed as finite difference and the derivative with respect to the sea is computed with the scalar insertions.

df <- data.frame(
  "en" = c("B64"),
  "kappa" = c(0.139426500000),
  "mul_sim" = c(0.00072),
  "musig_sim" = c(0.1246864),
  "mudel_sim" = c(0.1315052),
  "ms_sim" = c(0),
  "mc_sim" = c(0),
  "Zp_Zs" = c(0.79018),
  "mul" = c(0.0006669),
  "musig" = c(0),
  "mudel" = c(0),
  "ms" = c(0.018267),
  "mc" = c(0.23134)
)

library(dplyr)

Attaching package: 'dplyr'

The following objects are masked from 'package:stats':

    filter, lag

The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union

library(knitr)
# df$ms_sim<- df$musig_sim- df$
df <- mutate(df,
  ms_sim = musig_sim - Zp_Zs * mudel_sim,
  mc_sim = musig_sim + Zp_Zs * mudel_sim
)

df <- mutate(df,
  musig = (ms + mc) / 2,
  mudel = (mc - ms) / Zp_Zs / 2
)

kable(df[, c(1:7)])

en	kappa	mul_sim	musig_sim	mudel_sim	ms_sim	mc_sim
B64	0.1394265	0.00072	0.1246864	0.1315052	0.0207736	0.2285992

kable(df[, c(8:13)])

Zp_Zs	mul	musig	mudel	ms	mc
0.79018	0.0006669	0.1248035	0.1348256	0.018267	0.23134

now we cange only the charm keeping the strange at the simulation point

library(dplyr)
library(knitr)

df1 <- mutate(df,
  musig = (ms_sim + mc) / 2,
  mudel = (mc - ms_sim) / Zp_Zs / 2
)
  # musigma = 0.1260567
  # mudelta = 0.1332397
  # 2Kappamubar   = 0.0351512889651
  # 2KappaEpsBar  = 0.0371542900641
  # kappa = 0.1394265
  # 
  # #epsbar = 0.1394265
  # 2Kappamubar2  = 0.034769065158
  # 2KappaEpsBar2 = 0.036670563765
  # kappa2 = 0.1394265


kable(df1[, c(1, 2, 3, 4, 9, 10)], digits = 13)

en	kappa	mul_sim	musig_sim	mul	musig
B64	0.1394265	0.00072	0.1246864	0.0006669	0.1260568