10 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","D54"),
  "kappa" = c(0.139426500000, 0.138752850000, 0.137972174000),
  "mul_sim" = c(0.00072, 0.0006, 0.00054),
  "musig_sim" = c(0.1315052, 0.107146000965, 0.087911000000),#0.077706999882
  "mudel_sim" = c(0.1246864, 0.106585999855, 0.086224000000),#0.074646999976
  "ms_sim" = c(0,0,0),
  "mc_sim" = c(0,0,0),
  "Zp_Zs" = c(0.79018,  0.82308, 0.85095),
  "mul" = c(0.0006669, 0.0005864, 0.0004934),
  "musig" = c(0,0,0),
  "mudel" = c(0,0,0),
  "ms" = c(0.018267,  0.016053, 0.013559),
  "mc" = c(0.23134,   0.19849, 0.16474)
)

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] 0.230029899552 0.194874805726 0.161283312800

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

en	kappa	mul_sim	musig_sim	mudel_sim	ms_sim	mc_sim
B64	0.1394265	0.00072	0.1315052	0.1246864	0.03298050	0.2300299
C80	0.1387529	0.00060	0.1071460	0.1065860	0.01941720	0.1948748
D54	0.1379722	0.00054	0.0879110	0.0862240	0.01453869	0.1612833

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

Zp_Zs	mul	musig	mudel	ms	mc
0.79018	0.0006669	0.1248035	0.13482561	0.018267	0.23134
0.82308	0.0005864	0.1072715	0.11082580	0.016053	0.19849
0.85095	0.0004934	0.0891495	0.08883072	0.013559	0.16474

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	0.1315052	0.0006669	0.13216025
C80	0.1387529	0.00060	0.1071460	0.0005864	0.10895360
D54	0.1379722	0.00054	0.0879110	0.0004934	0.08963934

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	0.1315052	0.1309269	0.1239545
C80	0.1387529	0.00060	0.1071460	0.1241452	0.1272392
D54	0.1379722	0.00054	0.0879110	0.1217060	0.1259384

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

2*kap*mub =  0.03650935611963307 0.03445101120121027 0.0335840782922007

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

2*kap*epsb =  0.03456509229315717 0.03530960365515591 0.03475199451439068

cat("denominator\n")

denominator

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

2*kap*mub =  0.0366706195356 0.029733625999993 0.024258543577028

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

2*kap*epsb =  0.0347691766992 0.02957822249996168 0.023793025461952