Pion-pion scattering: the rho resonance

\(\pi\pi\rightarrow\rho\) 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:

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RESULTS

• Dispersion relations

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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 \(a = 0.0997991 {\rm fm}\)
• pion mass \(m_\pi = 330 {\rm MeV}\)
• 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 bold numbers indicate the relevant sizes to measure \(\rho\).

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

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