Eric Vazquez
During this summer I was working in HERMES group with Norair Akopov
as supervisor.
As a first job, I analized the cross section (SIGMA) for elastic
electron scattering from a target of mass M, getting the necessary
information in (1). This cross section is obtained by the equation:
µ² cos²(ø/2) E'
SIGMA = ------------------- ---- (W2(q²) + 2 tan²(ø/2) W1(q²))
4 Eo² (sin²(ø/2))² Eo
Where:
Eo is the incident energy.
ø is the laboratory scattering angle.
Eo
E' = --------------------------
1 + (2 Eo / M) sin²(ø/2)
µ = Fine-structure constant
1/137.03604 aprox.
W1(q²) and W2(q²) are the structure functions which are defined under
latter equation for cross section scattering.
First case was an elastic electron-proton scattering; in this
situation:
M = Mp = 0.938256 GeV
Gep² + § Gmp²
W1(q²) = § Gmp² W2(q²) = ---------------
1 + §
Where:
Gmp and Gep are, respectively, the elastic electric and magnetic form
factors for the proton and:
§ = q² / 4 Mp²
(As is known, q is the four momentum of the exchanged virtual photon).
Second case was elastic electron scattering from deuterium, with this
kind of scattering is possible yield information about the charge and magnetic
moment distribution in the deuteron and the nucleon electromagnetic form
factors, as it will see. Now:
M = 1.87537 GeV
W1(q²) = Fd² (2/3) § Gs² W2(q²) = Fd² (Gp² + (2/3) § Gs²
Here:
Fd = F(q) = (1.580/q) (arctan(q/0.930) - 2 arctan(q/3.19) + arctan(q/5.45))
With q in 1/fm.
Gp and Gs are getting from the nucleon form factors.
Third case was quasielastic electron scattering from deuterium:
M = 0.938256 Gev
Ge'² + § Gm'²
W1(q²) = § Gm'² W2(q²) = -----------------
1 + §
With Gm' and Ge' the form factors for this case.
As a second job, I analized some distributions of Q² (the invariant
mass of the exchanged photon, which defines the resolution of the scattering
process), W² (missing mass, or effective mass of the unobserved particles
produced in the reaction) and Û (energy transfer of the virtual photon to the
nucleon).
Relativistic kinematics indicates that:
Û = Eo - E' W² = M² + 2 M Û - Q² Q² = -q² = 4 Eo E' sin²(ø/2)
I compared the generated distributions (Using Monte Carlo) with the
reconstructed distributions from the HERMES experiment.
I analized for the regions:
Elastic W² < 1
Resonance 1 < W² < 4
Deep Inelastic Scattering (DIS) W² > 4
With the next cuts:
Q² reconstructed > 1
0.04 < |arctan( sin(theta_r) * sin(phi_r) )| < 0.2
Û / Eo < 0.85
Being theta_r and phi_r, angles of HERMES experiment. For more
details see (2)
As a third and last job I checked some variations of the structure
function F2(x,Q²) for the proton and deuteron, getting all necessary in (3).
This quantity was parameterized by:
S. I. Bilen'kaya et al. Elastic region
A. Bodek et al. Resonance region
NMC Collaboration DIS region
I just work in the resonance region and in this case the structure
function is parameterized as follows:
Û W2(Û,Q²) = F2(x,Q²) = A(W,Q²) f(÷) x
Where:
f(÷) = ÷ (C3 (1 - (1/÷))³ + ... + C7 (1 - (1/÷)) )
x = Q² / 2 M Û
With: 2 M Û + a²
÷ = ------------
Q² + b²
And A(W,Q²) is a modulating function that contains 12 parameters
representing the masses, widths and amplitudes of the cross sections for
electroproduction of the four most prominent nucleon resonances and 8
parameters representing the W dependence of the non-resonant background
under these resonances.
The a², b², C3, C4, C5, C6 and C7 parameters are given below for the
proton and deuteron case; they are called the global fit parameters.
Proton Deuteron
a² = C1 1.642 ± 0.0110 1.512 ± 0.0090
b² = C2 0.376 ± 0.0050 0.351 ± 0.0040
C3 0.256 ± 0.0256 0.477 ± 0.0477
C4 2.178 ± 0.2178 2.160 ± 0.2160
C5 0.898 ± 0.0898 3.627 ± 0.3627
C6 -6.716 ± 0.6716 -10.470 ± 1.0470
C7 3.756 ± 0.3756 4.927 ± 0.4927
The variation of the structure function F2(x,Q²) was analized when
small diferences of the global fit parameters are taking into account
(values in the uncertainity range, of course), as well as, delta
variations, with delta defined as |F2(x,Q²) - F2'(x,Q²)| / F2(x,Q²).
Being F2'(x,Q²) the value of the structure function after the change of the
parameter. It is necessary to know that it was taken variations of
each parameter and then one of all parameters together.
ANALYSIS
The main goal is to study the Q²-dependence of the generalised
Gerasimov-Drell-Hearn (GDH) integral for the proton and principally it has
been studied systematics of the structure function F2(x,Q²) coming from
possible variations of parameters and the goal is agree the diferents
parametrizations of this function in the frontier of the three regions
(elastic, resonance and DIS). For this, it is important to know that 'the
limited W²-resolution of the HERMES experiment in the resonance region did
not allow the contributions of the individual nucleon resonances to be
distinguished and contaminations into the resonance region are needed to
take into account'(4).
REFERENCES
(1) S. Stein et al., Phys. Rev. D 12 (1975) 1884
(2) HERMES Collaboration DESY HERMES-95-02 July 1995
(3) A. Bodek et al., Phys. Rev. D 20 (1979) 1471
(4) HERMES Collaboration DESY 00-096 August 2000