Difference between revisions of "Pion-pion scattering: the rho resonance"

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Line 36: Line 36:
  
  
−
**COM**
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===COM===
  
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* lattice spacing $a = 0.0997991 fm$
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* lattice spacing <math>a = 0.0997991 fm</math>
−
* pion mass $m_\pi = 330  MeV$
+
* pion mass <math>m_\pi = 330  MeV</math>
−
* elongated direction $N_z = 40, 42, 44, 46, 48, 50 ,52, 54, 56$ (48 is the lattice size set such that the first excited state equals the resonance).
+
* elongated direction <math>N_z = 40, 42, 44, 46, 48, 50 ,52, 54, 56</math> (48 is the lattice size set such that the first excited state equals the resonance).
−
* $N_t = 48$
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* <math>N_t = 48</math>
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* $N_x=N_y= 24$
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* <math>N_x=N_y= 24</math>
  
−
***COM Elongated Box    $24^2 \times N_z \times N_t$***
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====COM Elongated Box    <math>24^2 \times N_z \times N_t</math>====
  
  
−
$m_\pi \times L_s$ | $L_s$ \[fm\] | $a$\[fm\] | $L_e$\[fm\] | $m_\pi$\[MeV\] | $m_\rho$ \[MeV\]  
+
{|class = "wikitable"
−
:-:|:-:|:-:|:-:|:-:|:-:
+
|-
−
4.00572 | 2.39518 | 0.0997991 | 4.79036 | 330 | 838.769
+
|<math>m_\pi \times L_s</math> || <math>L_s</math> [fm] || <math>a</math>[fm] || <math>L_e</math>[fm] || <math>m_\pi</math>[MeV] || <math>m_\rho</math> [MeV]  
 +
|-
 +
|4.00572 || 2.39518 || 0.0997991 || 4.79036 || 330 || 838.769
 +
|}
  
  
 +
E = the energy of the first excited state <math>p = \frac{2\pi}{L_e}</math>.
  
−
E = the energy of the first excited state $p = \frac{2\pi}{L_e}$.
+
{| class = "wikitable"
 +
|<math>N_z</math> || 40 || '''42''' || '''44''' || '''46''' || '''48''' || '''50''' || '''52''' || '''54''' || '''56''' || 58
 +
|-
 +
|<math>L_e</math> [fm] || 3.99196 || 4.19156 || 4.39116 || 4.59076 || 4.79036 || 4.98996 || 5.18955 || 5.38915 || 5.58875 || 5.78835
 +
|-
 +
|E [MeV] || 906.325 || 886.314 || 868.598 || 852.841 || 838.769 || 826.152 || 814.799 || 804.55 || 795.266 || 786.834
 +
|}
  
−
$N_z$ | $40$ | \[42 | 44 | 46 | 48 | 50 | 52 | 54 | 56\] | $58$
+
The brackets indicate the relevant sizes to measure <math>\rho</math>.
−
-|-|-|-|-|-|-|-|-|-|-
 
−
$L_e$\[fm\] | 3.99196 | 4.19156 | 4.39116 | 4.59076 | 4.79036 | 4.98996 | 5.18955 | 5.38915 | 5.58875 | 5.78835
 
−
E \[MeV\] | 906.325 | 886.314 | 868.598 | 852.841 | 838.769 | 826.152 | 814.799 | 804.55 | 795.266 | 786.834
 
  
−
The brackets indicate the relevant sizes to measure $\rho$.
+
----
  
−
***
+
To generate a lattice spacing with the right value we set <math>\beta</math> using the results quoted in [[Lattice spacing]]. We find the following parameters:
−
 
+
* <math>\beta=5.961</math>
−
To generate a lattice spacing with the right value we set $\beta$ using the results quoted in [[Lattice spacing]]. We find the following parameters:
+
* <math>n_{OR} = 8</math>
−
* $\beta=5.961$
+
* <math>n_{skip} = 180</math>
−
* $n_{OR} = 8$
 
−
* $n_{skip} = 180$
 

Revision as of 11:57, 16 December 2009

\(\pi\pi\rightarrow\rho\) scattering

Craig Pelissier defended his proposal to measure the mass and width of the \(\rho\) resonance. You can find his proposal here.

We will generate two sets of lattices:

  • symmetric lattices with cubic symmetry ( _LAB_ )
  • elongated lattices ( _COM_ )

Here is the e-mail from Craig regarding the choice of lattice sizes:

LAB

  • lattice spacing \(a = 0.0997991 {\rm fm}\)
  • pion mass \(m_\pi = 330 {\rm MeV}\)
  • lattice size \(N_x=N_y=N_z = 18, 20, 22, 24, 26\) (20 is the lattice size with the smallest difference between the first excited state and the resonance).
  • \(N_t = 48\)


Cubic Box in lab frame

N 10 12 14 16 18 20 22 24 26 28
L [fm] 0.997991 1.19759 1.39719 1.59679 1.79638 1.99598 2.19558 2.39518 2.59478 2.79438
E [MeV] 1032.54 966.921 917.955 880.12 850.118 825.846 805.893 789.273 775.277 763.377
\(m_\pi\times L\) 1.66905 2.00286 2.33667 2.6704 3.0042 3.338 3.6719 4.0057 4.3395 4.67334

The bold numbers indicate the relevant sizes to measure \(\rho\).


COM

  • lattice spacing \(a = 0.0997991 fm\)
  • pion mass \(m_\pi = 330 MeV\)
  • elongated direction \(N_z = 40, 42, 44, 46, 48, 50 ,52, 54, 56\) (48 is the lattice size set such that the first excited state equals the resonance).
  • \(N_t = 48\)
  • \(N_x=N_y= 24\)

COM Elongated Box \(24^2 \times N_z \times N_t\)

\(m_\pi \times L_s\) \(L_s\) [fm] \(a\)[fm] \(L_e\)[fm] \(m_\pi\)[MeV] \(m_\rho\) [MeV]
4.00572 2.39518 0.0997991 4.79036 330 838.769


E = the energy of the first excited state \(p = \frac{2\pi}{L_e}\).

\(N_z\) 40 42 44 46 48 50 52 54 56 58
\(L_e\) [fm] 3.99196 4.19156 4.39116 4.59076 4.79036 4.98996 5.18955 5.38915 5.58875 5.78835
E [MeV] 906.325 886.314 868.598 852.841 838.769 826.152 814.799 804.55 795.266 786.834

The brackets indicate the relevant sizes to measure \(\rho\).

To generate a lattice spacing with the right value we set \(\beta\) using the results quoted in Lattice spacing. We find the following parameters:

  • \(\beta=5.961\)
  • \(n_{OR} = 8\)
  • \(n_{skip} = 180\)
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