View source for Pion-pion scattering: the rho resonance

== <math>\pi\pi\rightarrow\rho</math> scattering ==

Craig Pelissier defended his proposal to measure the mass and width of the <math>\rho</math> resonance.
You can find his proposal [[Media:Craigs-proposal.pdf|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 <math>a = 0.0997991 {\rm fm}</math>
* pion mass <math>m_\pi = 330 {\rm MeV}</math>
* lattice size <math>N_x=N_y=N_z = 18, 20, 22, 24, 26</math>  (20 is the lattice size with the smallest difference between the first excited state and the resonance).
* <math>N_t = 48</math> 


===Cubic Box in lab frame===

{| class="wikitable" 
|-
|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
|-
|<math>m_\pi\times L</math> || 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 <math>\rho</math>.


===COM===

* lattice spacing <math>a = 0.0997991 fm</math>
* pion mass <math>m_\pi = 330  MeV</math>
* 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).
* <math>N_t = 48</math>
* <math>N_x=N_y= 24</math>

====COM Elongated Box    <math>24^2 \times N_z \times N_t</math>====


{|class = "wikitable"
|-
|<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>.

{| 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
|}

The bold numbers indicate the relevant sizes to measure <math>\rho</math>..

----

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>
* <math>n_{OR} = 8</math>
* <math>n_{skip} = 180</math>