The leading order VAP arises from the imaginary part of the interference of one- and two-photon exchange amplitudes. Sub-leading contributions are generated by the nucleon magnetic moment and charge radius, as well as recoil corrections to the leading-order amplitude.
Introduction
Effective Field Theory
Let us now look at the effect of a mass term in the Lagrangian of Eq. Since we need kinetic terms in the Lagrangian construction, a covariant derivative in the chiral pion field is also required.
Vector Analyzing Power in Electron-Proton Scat- teringtering
Because the main QED contribution to An stems from ImMγγ, experimental measurements of the VAP provide an important test of theoretical calculations of Mγγ needed for the interpretation of other measurements. We also consider Anat's forward scatter angles and energies slightly higher than those of the SAMPLE experiment.
Parity-Violating Electron-Deuteron Scattering
This calculation was therefore made to better understand the magnitude of two-body effects in quasi-elastic deuteron electron scattering. They were shown to be insignificant in the asymmetry calculation, which neglected final-state interactions.
The Vector Analyzing Power in Elastic Electron-Proton Scattering
- Introduction
- General Considerations
- Two-Photon Exchange
- Results and Discussion
- Conclusions
When included in the loop plots of Figure 2.1, these interactions generate contributions to the ep amplitude My and Myy up to and including (p/M)2 relative to the leading term. The O(p/M)2 loop contributions arise from two O(p2)γ operators (for example, two insertions of the nucleon's magnetic moment operator, Figure 2.3d) or one O(p) and one O(p3) term (i.e. the proton charge radius). In the present case, where we are interested in backward angular scattering at non-zero q2, we would ideally like to use this formalism. the heavy baryon propagator does not allow direct copying of the t'Hooft-Passarino-Veltmann formulation.
On general grounds, the dependence on the regulator should be canceled out by a corresponding dependence on the bremsstrahlung contribution to the spin-dependent cross-section. Although it also contributes to the absorptive part of Myγ, the resulting terms do not contribute to the spin-dependent correlation of Eq. Including the magnetic moment, charge radius and retrace order in My along with the loop contributions in Eq. 2.10 leads to the following expression for the VAP:.
In Figure 2.7, the relative importance of the recoil, magnetic moment and charge radius contributions is also indicated. On the other hand, the use of the relativistic form of the operator, O7aeN, leads to the correlation µναβSµPνPαKβ which does not vanish for P =P.
Appendix A: Bremsstrahlung Computation
One can also ask how competitive the SAMPLE measurement is with other direct searches for new T-odd, P-even interactions. 12, 13], direct searches are most relevant in symmetry-breaking scenarios where parity is broken at or above the scale of the partitioning of T. Existing direct searches imply that αT <∼ few ×10−3, where αT is the ratio of a typical T-odd, P-even core matrix element to those of the remaining strong interaction. When translated into limits of generic, dimension seven operator coefficients C7 [under the normalization of Eq.
Given that conventional hadronic final state effects integrated into our calculations obviously imply a value of C7a with a magnitude of unit order, it seems unlikely that one will be able to circumvent the corresponding theoretical hadronic uncertainties if one needs to achieve the VAP a direct inquiry into new physics. On the other hand, low-energy studies of An could provide important information for the theoretical interpretation of other accurate electroweak observations. depends on ten different products of leptonic and hadronic tensors. Hacµν+Hadµν)(Lacµν +Ladµν) +Haaµν(Laaµν+Labµν+Lbbµν) + Lµνcc(Hµνcc +Hµνcd+Hµνdd). 2.20) where the sum is over all polarizations of the emitted photon.
If we evaluate the B functions for the kinematics involved here, we find that none of the integrals of Eq.
Appendix B: Local Operators
2.28) Since all the C's are real, we see that there is no contribution from dimension six terms.
Appendix C: Loop Integrals
First, the necessary framework for the calculation is drawn up. kinematics considered here, the passarino and veltman momentums and masses are: The Passarino-Veltman R functions are shown only for the Da integral:. and where the inverse of the momentum matrix X is:
Parity-Violating Electron-Deuteron Scattering
Introduction
The Parity-Violating Asymmetry
Note that, in the amplitude MZ, the Q2 dependence of the Z0 propagator has been ignored, since here we restricted ourselves to |Q2| m2Z. For this purpose, it is first convenient to separate the weak current jZ,σ into its vector j0,σ and axial vector j5,σ components and write accordingly:. The response functions can then be expressed as: 3.14) where mi is the mass of the target (assumed at rest in the laboratory), Ef is the energy of the final nuclear state (generally a scattering state), and in Eq.
Note that there is a sum over the final states and an average over the initial spin projection states of the target, as implied by the notation i. In the above expression for theRs, it has been assumed that the three-momentum transfer q is along the z axis , which defines the spin quantization axis of the core states.
Operators in Effective Field Theory
- Ordering of Terms
- Connection with Phenomenological Model
- One- and Two-Body Currents
The first corrections come in tree-level one-body flows from O(q/Λχ) magnetic corrections in the Lagrangian. Second corrections are of two types: (i) one-loop corrections and O(q2/Λ2χ) interactions in one-body flows shown in Figure 3.4, and (ii) tree-level two-body flows in Figure 3.5. Direct Z exchange between nucleons (PV in the deuteron initial state) is not included in the calculation.
At the next order in the expansion, we have the one-body contributions from Figur. Finally, Figure 3.6 shows the first order corrections to the two-body contribution, but these are not included in the calculation. EFT language corresponds to the inclusions of terms beyond those mentioned in the previous section.
The corresponding diagram is shown in Figure 3.11, which has a contribution proportional to Qµ, the transfer of four impulses. There is a contribution to the axial charge operator which, however, is not included in the asymmetry calculation.
Phenomenological Model
- Electromagnetic Operators
- Weak Operators
The main features of the two-body parts of the electromagnetic current operator are summarized below. While the main parts of the two-body currents are linked to the form of the two-nucleon interaction by the continuity equation, the main two-body charge operators are model-dependent and should be considered as relativistic corrections. Indeed, a consistent calculation of two-body charge effects in nuclei would require the inclusion of relativistic effects in both the interaction models and nuclear wave functions.
Nevertheless, there are quite clear indications of the importance of two-body charge operators due to the failure of the impulse approximation to predict the observable polarization of the deuteron tensor [66] and the charge form factors of three- and four-nucleon systems [65, 67]. The π- and ρ-meson exchange charge operators consist of isospin-dependent spin-spin and tensor interactions. that of Argonne v18 here), using the same recipe adopted for the corresponding current operators. However, it should be emphasized that for q≤2 fm−1 the contributions due to these two-body charge operators are very small compared to those from the one-body operator.
There are relativistic corrections toj5,1 as well as two-body contributions arising from π-, ρ-,ρπ and ∆ excitation exchange mechanisms [64]. For example, in weak proton-to-proton capture at KeV energies [70] (this process is induced by an axial weak charge-changing current) the two-body operators π, ρ, ρπ and ∆ increase the predicted one-body cross-section by 1.5% .
Calculation
ΛA is considered to be 1 GeV/c2, as obtained from the analysis of pion electroproduction data [68] and measurements of the reaction νµp→µ+n[69]. Such an estimate is also expected to hold in the quasi-elastic regime considered here. The δLST factor ensures the antisymmetry of the wave function, while the Clebsch-Gordan coefficients limit the sum over L and L .
The radial functions u(−)LL are obtained by solving the Schr¨odinger equation in the J ST channel, and behave asymptotically as:. 3.46) where SLJSTL is the S matrix in the J ST channel, and the Hankel functions are defined as (1,2)L (x)=jL(x)±inL(x),jlandnL being the spherical Bessel and Neumann functions respectively. In the quasi-elastic regime of interest here, these interaction effects are found to be negligible for Jmax>7. The response functions are written as (only Rγ,γL are given below for illustration):. where the contributions of the individual spin-isospin states are:.
The initial and final state wave functions are written as vectors in the spin-isospin space of the two nucleons for any given spatial configuration r.
Results and Conclusions
However, it is multiplied by the small factor (−1 + 4 sin2θW) = −0.074, and thus the asymmetry appears to be largely independent of nuclear structure details. The contribution of the axial flow to the asymmetry is in the range of 13% to 24% over the entire kinematic range considered. Since we performed the calculations with the SAMPLE kinematics, the above results can be used to account for two-body flow corrections in the analysis of the experimental data.
We can now use our results for the ratios of total to one-body contributions in the asymmetry and cross section to adjust for two-body effects in the experiment. Finally, we have presented a calculation of the asymmetry in quasi-elastic electron-deuteron scattering arising from Z0 exchange. Parity violation in the deuteron initial state or in the pion-nucleon angle (anapole moment) is not included.
Since we find that, when the cross-section is large at the quasi-elastic peak, the change in the asymmetry due to inter-body currents is negligible, we expect that these two-body corrections will cause a change in the analysis of the sample . percent-level experiment, too small to significantly affect the extraction of the nucleon's foreign and axial form factors. Right panels: ratios of one-plus-two-body contributions of only pion (dotted line) and full currents (solid line) to one-body contributions for the asymmetry|A|/|A1|(top) and cross section σ/σ1 (bottom).
Bibliography
