La “lista” de impactos ambientales potenciales de un proyecto minero .1 Proyectos mineros e impactos

+ +

(e)

+

· · ·

(f )

+ +

(g)

Figure 6.3: Contributions to the P - and T -violating 3N transition current:

(a)+(b) N²LO contribution (for a g^θ₀ vertex) , (c)+(d) N⁴LO contributions (for a g₀^θ vertex), (e)+(f) N⁴LO contributions, (g) N²LO contribution (for a g₀^θ vertex).

Diagrams (a)-(d) and (f) with g₁^θ vertices are further suppressed by one order in magnitude. the P - and T -violating pion-nucleon vertex is depicted by a black box. A P - and T -conserving pion-nucleon vertex is depicted by a solid dot. For each class of diagrams only one representative is shown.

6.2.3 Power counting of irreducible transition current op-erators

In order to compare the power counting of a 3N potential operator to the one of a 3N transition current operator, the former has to be multiplied by a factor of mN/M_π² accounting for an additional free three-nucleon propagator and a factor of e for the γN N coupling. The power counting scales of the P - and T -violating 3N transition currents from LO to N²LO are listed in tab. 6.1. Various diagram classes of irreducible transition current operators are displayed in fig. 6.3.

The diagram classes of fig. 6.3 (a) and (b) have the same order estimates since they only differ by a transposition of the P - and T -violating vertex and a P - and T -conserving vertex. The order estimate of the P - and T -conserving γN N vertex is given by emN/M_π², which yields the following order estimates for the diagram

classes fig. 6.3 (a) and (b): which is the order estimate of an N²LO diagram class. The power counting of the diagrams pictured in fig. 6.3 (c) and (d) with one Weinberg-Tomozawa vertex each is straight forward and yields N⁴LO:

Mπ which gives the following N⁴LO order estimate:

Mπ Finally, the P - and T -violating γππN N vertex in fig. 6.3 (f) originates from terms of the fouth-order pion-nucleon Lagrangian eq. (D.21). The order estimate of such a vertex is g^θ₀eMπ/(Fπm²_N). The order estimate of the diagram class depicted in which is the order estimate of an N⁴LO diagram class. The order estimate of the P - and T -conserving γππN N vertex in fig. 6.3 (g) is e/F_π². The order estimate of the entire diagram fig. 6.3 (g) is then given by:

e This means that there is no irreducible transition current operator to be con-sidered in this chapter. Other classes of diagrams not mentioned here are either N N diagrams which have been discussed in the previous chapter or are of irrel-evant subleading orders, i.e. of an order beyond NLO. The order estimates of all vertices mentioned in this section are also listed in tab. 5.2.

6.3 Numerical analysis technique

The numerical analysis technique of the leading nuclear contributions to the EDMs of ³He and ³H from P - and T -violating N N potential operators is ex-plained in this section. The technique described below is a generalization of the

6.3. NUMERICAL ANALYSIS TECHNIQUE 105

3He ³He ³He ³He

(a) (b)

Figure 6.4: Total LO P - and T -violating ³He transition current. The P - and T -violating pion-nucleon vertex is depicted by a black box. For each class of diagrams only one representative is shown. The shaded box in the center of diagram (b) denotes P - and T -conserving interactions in the intermediate state.

numerical analysis of the leading order N N contribution to the deuteron EDM presented in section 5.3 to the 3N system. The subsequent explanation of our numerical analysis technique is exactly the same for both 3N bound states. The form factor of ³He (or ³H) is given by the generalization of eq. (5.28) to the 3N system:

hψ³He|Ψi = hψ³He|O(~q)|ψ³Hei

+ hψ³He| VP //T G0+ VP //T G0V G
O(~q)|ψ³Hei + · · · , (6.16) where VP //T is either the P - and T -violating one-pion exchange N N potential operator induced by a general g0 or g1 vertex. Since the focus of this section is on the leading EDM contributions, the transition current operator O(~q) can be identified with the leading order P - and T -conserving γN N coupling in heavy baryon ChPT to all nucleon lines as in section 5.3. The P - and T -violating component of the³He form factor is then given by the second matrix element on the right-hand side of eq. (6.16).

The development of the parallel program designed to run on the supercom-puters JUROPA and JUQUEEN in order to numerically compute this matrix element involved a considerable programing effort. The architecture of the pro-gram requires all operators to be decomposed into their partial wave components.

The partial wave decomposition of VP //T for the considered N N potential opera-tors in the 3N system is provided by eqs. (F.26) and (F.27) and the partial wave decomposition ofO(~q)|ψ³Hei by eq. (F.28) in appendix F. The second matrix ele-ment on the right-hand side of eq. (6.16) has to be multiplied by a factor of two to account for the inverse time ordering. The completely symmetrized form for all potential and transition current operators is implied. The two matrix elements on the righthand side of eq. (6.16) correspond either to diagrams without P -and T -conserving interactions in the intermediate state as the one depicted in fig. 6.4 (a) or to diagrams with intermediate state interactions as the one depicted

in fig. 6.4 (b).

Whereas VP //T is a pure N N potential operator, the P - and T -conserving 3N potential V comprises N N and 3N interactions. Let V_2N^ij denote the N N -interaction of nucleons (i) and (j) for i, j = 1, 2, 3 and i 6= j. The 3N-interaction of nucleons can be decomposed into three parts with each of them being symmet-ric under an exchange of two nucleons:

V = V_2N¹²+ V_3N⁽¹⁾+ V_2N²³+ V_3N⁽²⁾+ V_2N¹³+ V_3N⁽³⁾, (6.17) where the 3N potentials V_3N⁽ⁱ⁾ are defined by

P23V_3N⁽¹⁾P23= V_3N⁽¹⁾, P13V_3N⁽²⁾P13= V_3N⁽²⁾, P12V_3N⁽³⁾P12 = V_3N⁽³⁾. (6.18) The nucleon transposition operator Pij, i6= j, transposes nucleon (i) and nucleon (j) in the 3N system. By the means of nucleon transposition operators, the potential V can be re-expressed solely in terms of V_2N⁽¹²⁾ and V_3N⁽³⁾:

V = V_2N¹² + V_3N⁽³⁾+ P12P23(V_2N¹²+ V_3N⁽³⁾) P23P12+ P13P23(V_2N¹² + V_3N⁽³⁾) P23P13. (6.19) Exploiting the fact that cyclic permutation operators commute with the full 3N propagator G and do not alter O(~q)|³Hei, one arrives at

P23P12GO(~q)|ψ³Hei = P²³P13GO(~q)|ψ³Hei = G O(~q)|ψ³Hei . (6.20) Inserting eq. (6.19) into eq. (6.16) the third term on the right-hand side becomes

V GO(~q)|ψ³Hi = (1 + P12P23+ P13P23)(V_2N¹² + V_3N⁽³⁾)GO(~q)|ψ³Hei

≡ (1 + P )(V2N¹² + V_3N⁽³⁾)GO(~q)|ψ³Hei , (6.21) where P is defined by P = P12P23+ P13P23such that1+P is the (cyclic) nucleon symmetrization operator. This allows us to define the Faddeev component |U⁽³⁾i by

V GO(~q) |ψ^3Hei ≡ (1 + P ) |U⁽³⁾i . (6.22) A concise introduction into Faddeev equations can be found in [125]. The Faddeev component |U⁽³⁾i obeys the following equation:

|U⁽³⁾i = (V2N¹² + V_3N⁽³⁾)GO(~q)|ψ³Hei

= (V_2N¹² + V_3N⁽³⁾)G0O(~q)|ψ³Hei

+(V_2N¹²+ V_3N⁽³⁾)G0(1 + P )(V2N¹² + V_3N⁽³⁾)GO(~q)|ψ³Hei

= (V_2N¹² + V_3N⁽³⁾)G0O(~q)|ψ³Hei + (V2N¹² + V_3N⁽³⁾)G0(1 + P )|U⁽³⁾i , (6.23)