This is a unitary setup \(\kappa_{sea}=\kappa_{valence}\)
P5P5 = 0.2358(36) \(\chi^2/dof=\) 883.81
V0P5 = 0.01329(25) \(\chi^2/dof=\) 8479.2
A0P5 = -0.00475(12) \(\chi^2/dof=\) 10.826
V0P5_insP = -0.313(22) \(\chi^2/dof=\) 5.3516
\(M_{PS}\)= 167.3(2.1) MeV
M_{PS} = 0.07698(95) \(\chi^2/dof=\) 0.43318
Here we compute the variation of \(M_PS\) respect to the bare mass \(m\)
DeltaM = -2.09(14) \(\chi^2/dof=\) 0.37252
DeltaMt = 18(41) \(\chi^2/dof=\) 0.33376
mpcac = -0.001079(78) \(\chi^2/dof=\) 0.79115
deltam = 0.0120(14) \(\chi^2/dof=\) 2.1906
deltam_sub = 0.00713(29) \(\chi^2/dof=\) 0.38428
Expanding in taylor series in \(m\) the correlator and observing that \(\partial_m \langle O\rangle= \langle O P_5\rangle\) (there is a factor two in the correlator that in the code is called \(\langle V_0 P_5 P_5\rangle\) since insertion of \(P_5\) is done in only one quark line)
\[ \langle V_0 P_5\rangle({m_{cr}})= \langle V_0 P_5\rangle({m})+i\delta m \langle V_0 P_5 P_5\rangle({m}) \] since \(\langle V_0 P_5\rangle({m_{cr}})=0\) then
\[ \delta m=\frac{ \langle V_0 P_5\rangle }{i\langle V_0 P_5 P_5\rangle} \] using the fact that the numerator is purely immaginary and the denominator real we have \[ \delta m=\frac{Im( \langle V_0 P_5\rangle )}{Re(\langle V_0 P_5 P_5\rangle)} \]
form the relation \(\kappa=1/(2m+8)\) we have
\[ \kappa_{cr}=\frac{\kappa}{(1+2\,\delta m\,\kappa) }. \] Since the correlator \(\langle V_0 P_5 P_5\rangle\) is not flat in time but linear when we are far from maximal twist, we can subtract the linear part \[ \langle V_0 P_5 \rangle=B(e^{-Mt}-e^{-M(T-t)})\\ \partial_m \langle V_0 P_5 \rangle=i\langle V_0 P_5 P_5\rangle =(\partial_mB)(e^{-Mt}-e^{-M(T-t)}) -(\partial_m M)B(te^{-Mt}-(T-t)e^{-M(T-t)}) \] with \(\partial_m M\) computed in the section before
## Current $kappa=0.140064$## $kappa_{cr}$= 0.139595(54)## $kappa_{cr,sub}$= 0.139785(11)This is not a unitary setup \(\kappa_{sea}=0.140064\)
\(M_{PS}\)= 143.3(1.6) MeV
M_{PS} = 0.06596(72) \(\chi^2/dof=\) 0.25425
DeltaM = 1.07(13) \(\chi^2/dof=\) 0.23577
DeltaMt = 39(20) \(\chi^2/dof=\) 0.031808
mpcac = 0.000431(44) \(\chi^2/dof=\) 0.90198
deltam = -0.00225(18) \(\chi^2/dof=\) 0.55268
deltam_sub = -0.00212(15) \(\chi^2/dof=\) 0.47876
## Current $kappa=0.139779$## $kappa_{cr}$= 0.1398671(71)## $kappa_{cr,sub}$= 0.1398617(58)This is not a unitary setup \(\kappa_{sea}=0.140064\)
M_{PS} = 0.06428(68) \(\chi^2/dof=\) 0.41313
\(M_{PS}\)= 139.7(1.5) MeV
DeltaM = 0.05(11) \(\chi^2/dof=\) 0.25218
DeltaMt = 1.4(4.9) \(\chi^2/dof=\) 2.2876
mpcac = 2.7(53.4)e-6 \(\chi^2/dof=\) 0.74192
deltam = -0.00024(17) \(\chi^2/dof=\) 0.36804
deltam_sub = -0.00024(17) \(\chi^2/dof=\) 0.36914
## Current $kappa=0.1398563$## $kappa_{cr}$= 0.1398656(65)## $kappa_{cr,sub}$= 0.1398656(65)