Method for Determining G_Ep/G_Mp

The differential cross section for e-p scattering can be written:

$$\sigma(E,\theta) = \sigma_0(E,\theta)(G_E^2 + \epsilon^{-1}G_M^2Q^2\kappa)$$

where E is the incident electron energy, $\theta$ the electron scattering angle, $\sigma_0$ the Mott scattering cross section, $\epsilon$ is the virtual photon polarization parameter, and $\kappa = (\frac{\mu_p}{2M_p})^2$ = 2.212. G_E and G_M are, of course, functions of Q² alone.

For a given E there is a one-to-one correspondence between $\theta$ and Q² and the cross section can be written:

$$\sigma(E,Q^2) = \sigma_0G_M^2(\rho^2 + \epsilon^{-1}Q^2\kappa)$$

where $\rho = \frac{G_E}{G_M}$.

For two energies, E_A and E_B, at the same Q² the ratio of the cross sections is:

$$\frac{\sigma(E_A,Q^2)}{\sigma(E_B,Q^2)} = K \frac{\rho^2 + \epsilon_A^{-1}Q^2\kappa}{\rho^2 + \epsilon_B^{-1}Q^2\kappa}$$

where K is a kinematic factor.

If measurements at each energy are made simultaneously at two values of Q², Q₁² and Q₂², then there are two experimentally determined ratios:

$$R_A = \frac{\sigma(E_A, Q_1^2)}{\sigma(E_A,Q_2^2)} = K_A\fra... ...n_{A1}^{-1}Q_1^2\kappa}{\rho_2^2+\epsilon_{A2}^{-1}Q_2^2\kappa}$$

and

$$R_B = \frac{\sigma(E_B, Q_1^2)}{\sigma(E_B,Q_2^2)} = K_B\fra... ...n_{B1}^{-1}Q_1^2\kappa}{\rho_2^2+\epsilon_{B2}^{-1}Q_2^2\kappa}$$

and 2 ratios of physical interest, corresponding to the values of G_E/G_M at Q₁² and Q₂²:

$$R_1 = \frac{\sigma(E_A, Q_1^2)}{\sigma(E_B,Q_1^2)} = K_1\fra... ...n_{A1}^{-1}Q_1^2\kappa}{\rho_1^2+\epsilon_{B1}^{-1}Q_1^2\kappa}$$

and

$$R_2 = \frac{\sigma(E_A, Q_2^2)}{\sigma(E_B, Q_2^2)} = K_2\fr... ...n_{A2}^{-1}Q_2^2\kappa}{\rho_2^2+\epsilon_{B2}^{-1}Q_2^2\kappa}$$

note that $\frac{R_A}{R_B} = \frac{R_1}{R_2}$, or $R_1 = R_2\frac{R_A}{R_B}$.

The idea of the proposed measurements is to pick one value of Q² (Q₁²) where the recently reported $\frac{G_E}{G_M}$ from the polarization transfer experiment [4] is very different from unity, and another Q² (Q₂²) where $\frac{G_E}{G_M}$ must be close to its low-energy value of unity (with G_M in units of $\mu_p$) and to pick kinematics such that $\epsilon_1$ covers a wide range while $\epsilon_2$ does not change a great deal. If, then, R_A and R_B are accurately measured and R₂ can be accurately calculated then R₁ is accurately determined. R₁ is a function of only $\rho_1$(=$\frac{G_E}{G_M}$(Q²₁)) and known quantities. We propose to do this at each of 3 values of Q₁², 1.45, 3.20 and 4.90 GeV², with a common Q₂², 0.5 GeV². These points are shown in Figure 3.

  

Figure 3: $\epsilon$ as a function of Q² at the proposed electron energies. The solid circles show the points at which data will be taken.

The proposed measurement is similar to the conventional Rosenbluth separation technique, but it has two major differences that give significant advantages. With one spectrometer, we perform a conventional Rosenbluth separation, but detect the protons, rather than the electrons. This gives a much larger range in $\epsilon$ by allowing us to measure at kinematics where the electron is at very small and very large angles. Detecting the proton leads to a reduced cross section dependence on the kinematics (beam energy and scattering angle) and reduces several systematic uncertainties when comparing the forward and backward angle measurements. While we make the primary measurement with one arm, we make a simultaneous measurement at low Q² where G_E/G_M is well known and where the $\epsilon$ range is very small. This will allow us to use the second arm as a luminosity monitor, removing the uncertainties due to beam charge and target density fluctuations. The major sources of uncertainty in the SLAC measurements [1] were uncertainty in the scattering kinematics, the total charge, and the target density. Because we measure the protons, we are less sensitive to knowledge of the scattering kinematics, and because we use the low Q² measurement as a luminosity monitor, we are insensitive to the measured charge and target thickness.