Chapter 5 Renormalized couplings
we define the Fourier transform of the fiels as ˜ϕ(pμ)=∑xμeipμxμϕ(xμ) and the correlations functions
χi2=⟨˜ϕi(0)˜ϕ†i(0)⟩χi2(p∗)=⟨˜ϕi(p∗)˜ϕ†i(p∗)⟩χi4=⟨˜ϕi(0)˜ϕi(0)˜ϕ†i(0)˜ϕ†i(0)⟩χμR4=⟨Re{˜ϕ1(0)˜ϕ1(0)˜ϕ†0(0)˜ϕ†0(0)}⟩χgR4=⟨Re{˜ϕ1(0)†˜ϕ(0)˜ϕ0(0)˜ϕ0(0)}⟩ with p∗=(2π/L,0,0,0) (in x direction). We use the above correlators we impose the renormalization condition following Weisz and Wolff (2011) Z−1iχi2=1m2iRVZ−1iχi2(p∗)=1V1m2iR+p2∗ Note that miR is not the pole mass. Inverting the above renormalization condition we get Z−1i=(1χ2(p∗)−1χ2(0))1p2∗V.
with these quantities we define the renormalized quantities
m2iR=Zi/(χi2V)λiR=χi4−2χi2χi2(χi2)2Z2i(χi2)2μR=χμR4−χ02χ12(χ02χ12)Z0Z1χ02χ12gR=χgR4χ12(χ02)3√Z1Z32
=20
T=18
L<- c(-1.267, -1.26)
m0s<- c(-0.55, -1.267)
m1s<- 0.2*c(0:20)
gs<- 0
l0<- 0
l1<- 0
mu
<-gs*0
zeros
<- data.frame( "g"=zeros, "m02"=zeros, "m12"=zeros,
df"m_0R2"=zeros,"m_0R2_err"=zeros,
"m_1R2"=zeros,"m_1R2_err"=zeros,
"lambda0R"=zeros,"lambda0R_err"=zeros,
"lambda1R"=zeros, "lambda1R_err"=zeros,
"muR"=zeros, "muR_err"=zeros,
"gR"=zeros, "gR_err"=zeros
)
<- function(v, block=1){
create_jackif (block>1){
<-length(v)
N<- as.integer(N/block)
bins<-rep(0,bins-1)
vbfor (i in c(1:bins-1)){
<- mean(v[c(1:N/bins)+i *N/bins ])
vb[i]
}<- vb
v
}<-length(v)
N<-mean(v)
m<- (N*m-v)/(N-1)
jackreturn(jack)
}<- function(jack){
error_jack<-length(jack)
N<-mean(jack)
m<- sqrt( (N-1)*mean((jack-m)^2) )
sigma
return(sigma)
}
<- T*L*L*L
V=0
countfor(g in gs){
for(m0 in m0s){
for(m1 in m1s){
<-paste("../couplings_R/data")
dir=sprintf("%s/E_T%d_L%d_msq0%.6f_msq1%.6f_l0%.6f_l1%.6f_mu%.6f_g%.6f_rep0",dir,T,L,m0,m1,l0,l1,mu,g)
fileif(file.exists(file)){
<- read.table(file)
dfi# the code does the FT as 1/V \sum_x
# here we remove the factor 1/V for each filed
<- create_jack(dfi[,4], block = 100)*V^2
chi20<- create_jack(dfi[,5], block = 100)*V^2
chi21<- create_jack(dfi[,6], block = 100)*V^2
chi20p<- create_jack(dfi[,7], block = 100)*V^2
chi21p
<- create_jack(dfi[,8], block = 100)*V^4
chi40<- create_jack(dfi[,9], block = 100)*V^4
chi41<- create_jack(dfi[,10], block = 100)* V^4
chi4mu<- create_jack(dfi[,11], block = 100)* V^4
chi4g
<-(2*pi *V/18)* (chi20*chi20p)/(chi20-chi20p)
Z0<-(2*pi*V/18)* (chi21*chi21p)/(chi21-chi21p)
Z1<-Z0/(chi20*V)
m0r2<-Z1/(chi21*V)
m1r2<-(chi40-2*chi20*chi20)*Z0^2/(chi20^4)
l0r<-(chi41-2*chi21*chi21)*Z1^2/(chi21^4)
l1r<-(chi4mu-chi20*chi21)*(Z1 *Z0)/((chi20^2) * chi21^2)
mur<-(chi4g)*sqrt(Z1 *Z0^3)/((chi20^3) *chi21)
gr# dfi<- read.table(file)
# k0<- 1/(8+m0)
# k1<- 1/(8+m1)
# gl<- g*(4. * sqrt(k0 * k1 * k1 * k1))
+1,]<- list(g,m0,m1,
df[countmean(m0r2), error_jack(m0r2),
mean(m1r2), error_jack(m1r2),
mean(l0r), error_jack(l0r),
mean(l1r), error_jack(l1r),
mean(mur), error_jack(mur),
mean(gr), error_jack(gr)
)=count+1
count
}#masses
}} #g }
detach("package:Rose", unload=TRUE)
library("Rose")
<- myggplot(color=TRUE, shape=T, fill=FALSE)
gg
<- gg+ geom_point(data=df,
ggaes(x=g, y=m_0R2, color=paste("mR0",df$m02,df$m12),
shape=paste("mR0",df$m02,df$m12)))
<- gg + geom_errorbar(data=df,
gg aes(x=g, ymin=m_0R2-m_0R2_err, ymax=m_0R2+m_0R2_err ,
color=paste("mR0",df$m02,df$m12),
shape=paste("mR0",df$m02,df$m12)))
## Warning: Ignoring unknown aesthetics: shape
<- gg+ geom_point(data=df,
ggaes(x=g, y=m_1R2, color=paste("mR1",df$m02,df$m12),
shape=paste("mR1",df$m02,df$m12) ))
<- gg + geom_errorbar(data=df,
gg aes(x=g, ymin=m_1R2-m_1R2_err, ymax=m_1R2+m_1R2_err ,
color=paste("mR1",df$m02,df$m12),
shape=paste("mR1",df$m02,df$m12)))
## Warning: Ignoring unknown aesthetics: shape
myplotly(gg,"","g" , to_print = FALSE)
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
library(Rose)
library(ggplot2)
library(plotly)
<- myggplot(color=TRUE, shape=T, fill=FALSE)
gg<- gg+ geom_point(data=df,
ggaes(x=g, y=lambda0R, color=paste("lambdaR0",df$m02,df$m12) ))
<- gg + geom_errorbar(data=df,
gg aes(x=g, ymin=lambda0R-lambda0R_err,
ymax=lambda0R+lambda0R_err , color=paste("lambdaR0",df$m02,df$m12)
)) <- gg+ geom_point(data=df,
ggaes(x=g, y=lambda1R, color=paste("lambdaR1",df$m02,df$m12)))
<- gg + geom_errorbar(data=df,
gg aes(x=g, ymin=lambda1R-lambda1R_err,
ymax=lambda1R+lambda1R_err ,
color=paste("lambdaR1",df$m02,df$m12)))
myplotly(gg,"","g" , to_print = FALSE)
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
library(Rose)
library(ggplot2)
library(plotly)
<- myggplot(color=TRUE, shape=T, fill=FALSE)
gg<- gg+ geom_point(data=df,
ggaes(x=g, y=gR,
color=paste(df$m02,df$m12) ) )
<- gg + geom_errorbar(data=df,
gg aes(x=g, ymin=gR-gR_err, ymax=gR+gR_err ,
color=paste(df$m02,df$m12) ) )
myplotly(gg, to_print = FALSE)
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
## Warning: Use of `df$m02` is discouraged. Use `m02` instead.
## Warning: Use of `df$m12` is discouraged. Use `m12` instead.
References
Weisz, Peter, and Ulli Wolff. 2011. “Triviality of Theory: Small Volume Expansion and New Data.” Nuclear Physics B 846 (2): 316–37. https://arxiv.org/pdf/1012.0404.pdf.