Delta mix calculation

Ensemble Details

• Lattice Size: 32³ × 64 \( a = 0.085\rm fm\)

• Number of Configurations: 46 each with 4 different source positions (x, y, z, t). Total of 194 configurations.
  • Position 1: (16, 16, 16, 10)
  • Position 2: (0, 0, 0, 26)
  • Position 3: (16, 16, 16, 42)
  • Position 4: (0, 0, 0, 58)

• Masses

DWF Mass\[ma = 0.004 \] , \(m_\pi = \) 292 MeV

Overlap Masses

No.	ma	est \(m_\pi (\mbox{MeV})\)
1	0.00069	100
2	0.00148	140
3	0.00253	180
4	0.00349	210
5	0.0046	240
6	0.00585	270
7	0.00677	290
8	0.00765	307
9	0.00885	330
10	0.0112	364
11	0.0129	390
12	0.0152	419
13	0.018	450
14	0.024	510
15	0.03	570
16	0.036	630

Propagator Calculation Details

using rbc_conf_3264_m0.004_0.03_002190_hyp as a typical example

• Boundary Conditions: Periodic
• Location on Kraken: email Mike Lujan: mlujan@gwmail.gwu.edu
• Eigensystems
  • # Hwilson: 200 available. Timing: 1.4 hours including I/O time.
  • # Overlap: 400 available. Timing: 10.4 hours including I/O time.
• Inversions
  • CGM error: 10^-8
  • Eigenvectors used: #Hwilson=85, #Overlap=150.
  • Overlap Precision: 10^-9 --> Poly order 215.
  • Total # of CGM iterations: 3081 (257 average iterations per inversion).
  • Timing: 2.0 hours including I/O time.

Fitting Analysis

Method 1

In analyzing \( \Delta \rm mix \) we needed to take into account cross correlations between configurations and between masses. To successfully take into account the cross correlations, we bin jackknife the configurations. Within every jackknife sample we compute \( a^4 \Delta \rm mix \) according to

\( a^4\Delta_{\mbox{mix}}(m_v) =(m_{vs}a)^2 - \frac{1}{2}((m_{vv}a)^2 + (m_{ss}a)^2) \)

where \( m_v \) is the valence quark mass, \( m_{vv} \), \( m_{vs} \), \( m_{ss} \) are the overlap, mix and domain wall pion mass respectively. There were 12 values of valence masses \(m_v \), and one value for the sea mass, yielding 12 separate \(m_{vv} \) and \( m_{vs} \) masses and one \(m_{ss} \) mass. In each jackknife loop we fitted a single Cosh to extract the ground state of the system.

Fit details for extracting the pion masses.

Action	Fitting Range	\( \chi^2 \)
DWF	[11 - 17]	-
Overlap \( m_1 \rightarrow m_4 \)	[4, 11]	-
Overlap \( m_5 \rightarrow m_10 \)	[5, 14]	-
Overlap \( m_10 \rightarrow m_16 \)	[10,17]
Mix	[8,15]	-

For each jackknife bin we computed ∆mix (12 values in all). From this we computed cross correlations among the configurations and masses by constructing the following covariance matrix

\( C_{ij} = \frac{(n-1)}{n} \sum_{k=1}^n (\Delta_{\rm mix}(m_{k,i}) - \bar{\Delta}_{\rm mix}(m_i)) (\Delta_{\rm mix}(m_{k,j}) - \bar{\Delta}_{\rm mix}(m_j)). \)

The covariance matrix has an extra \( (n-1)^2 \) factor due to the jackknife sampling. We minimized \( \chi^2 \) in the usual fashion using a constant function to determine the value of ∆mix. We find

\( \mathbf{a^4 \Delta_{\rm mix} = 0.00079(20)} \)

Method 2

The second method of fitting used weighted averages as another means to determine \( \Delta_{\rm mix} \). For each bin we computed a weighted average of \( \Delta_{\rm mix} \) among the masses.

\( a^4\Delta_{\rm mix} = \frac{ \sum_i a^4 \Delta_{\rm mix}(m_i) / \sigma^2(m_i)}{ \sum_i 1 / \sigma^2 (m_i) } \)

where \( \sigma^2 (m_i) \) is determined by the following error propagation formula

\( \sigma^2 = 4 m_{vs}^2 \delta m_{vs}^2 + m_{vv}^2 \delta m_{vv}^2 + m_{ss}^2 \delta m_{ss}^2 \)

where the \( \delta 's \) are the errors resulting from the jackknife fits to the individual pion masses. This was done on each bin sample. To then compute \( a^2 \Delta_{\rm mix} \) for the system we averaged over all bins and computed the standard error from the binned samples. For this method we find

\( \mathbf{a^4 \Delta_{\rm mix} = 0.00090(26) }\)

Plots

Efective Mass Plots

Overlap(Red). DWF(Green), Mix(Blue)

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