9 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", "C80"),
  "kappa" = c(0.139426500000, 0.138752850000),
  "mul_sim" = c(0.00072, 0.0006),
  "musig_sim" = c(0.1246864, 0.106585999855),
  "mudel_sim" = c(0.1315052, 0.107146000965),
  "ms_sim" = c(0,0),
  "mc_sim" = c(0,0),
  "Zp_Zs" = c(0.79018,  0.82308),
  "mul" = c(0.0006669, 0.0005864),
  "musig" = c(0,0),
  "mudel" = c(0,0),
  "ms" = c(0.018267,  0.016053),
  "mc" = c(0.23134,   0.19849)
)

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
)
print(df$mc_sim,digits = 12)

[1] 0.228599178936 0.194775730329

kable(df[, c(1:7)], digits = 20)

en	kappa	mul_sim	musig_sim	mudel_sim	ms_sim	mc_sim
B64	0.1394265	0.00072	0.1246864	0.1315052	0.02077362	0.2285992
C80	0.1387529	0.00060	0.1065860	0.1071460	0.01839627	0.1947757

kable(df[, c(8:13)], digits = 20)

Zp_Zs	mul	musig	mudel	ms	mc
0.79018	0.0006669	0.1248035	0.1348256	0.018267	0.23134
0.82308	0.0005864	0.1072715	0.1108258	0.016053	0.19849

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
C80	0.1387529	0.00060	0.1065860	0.0005864	0.1084431

library(dplyr)
library(knitr)
# keeping ms fixed and change mc form
# mc_sim =0.2285992
mc_targ =0.22887328 # 0.1 of the iso-sim step
df1 <- mutate(df,
  musig = (ms_sim + mc_targ) / 2,
  mudel = (mc_targ - ms_sim) / Zp_Zs / 2
)
kable(df1[, c(1, 2, 3, 4,  10,11)], digits = 16)

en	kappa	mul_sim	musig_sim	musig	mudel
B64	0.1394265	0.00072	0.1246864	0.1248235	0.1316786
C80	0.1387529	0.00060	0.1065860	0.1236348	0.1278594

options(digits=16)
cat("2*kap*mub = ", df1$kappa*2*df1$musig,"\n")

2*kap*mub =  0.0348073936511998 0.03430935469479172

cat("2*kap*epsb = ", df1$kappa*2*df1$mudel,"\n")

2*kap*epsb =  0.03671898440436382 0.0354817090476069

cat("denominator\n")

denominator

cat("2*kap*mub = ", df1$kappa*2*df1$musig_sim,"\n")

2*kap*mub =  0.0347691766992 0.02957822249996168

cat("2*kap*epsb = ", df1$kappa*2*df1$mudel_sim,"\n")

2*kap*epsb =  0.0366706195356 0.029733625999993