Parity-violating two-pion exchange nucleon-nucleon interaction

We calculate in chiral perturbation theory the parity-violating two-pion exchange nucleon-nucleon potentials at leading one-loop order. At a distance of r = m−1 π ≃ 1.4 fm they amount to about ±16% of the parity-violating 1π-exchange potential. We evaluate also the parity-violating effects arising from 2π-exchange with excitation of virtual ∆(1232)-isobars. These come out to be relatively small in comparison to those from diagrams with only nucleon intermediate states. The reason for this opposite behavior to the parity-conserving case is the blocking of the dominant isoscalar central channel by CP-invariance. Furthermore, we calculate the T-matrix related to the iteration of the parity-violating 1π-exchange with the parity-conserving one. The analytical results presented in this work can be easily implemented into calculations of parity-violating nuclear observables. PACS: 12.20.Ds, 12.38.Bx, 21.30.Cb. Nuclear parity violation is an important tool to study the standard model of strong and electroweak interactions [1, 2]. In the two-nucleon system parity violation has been traditionally represented by parity-violating one-meson exchange where the strong and weak interactions are parametrized through parity-conserving and parity-violating meson-nucleon vertices. Due to CP-invariance, there exists no parity-violating coupling of neutral scalar or pseudoscalar mesons to nucleons (Barton’s theorem). Therefore, standard parametrizations of the parity-violating NN-potential involve the exchange of charged pions (π) as well as vector mesons (ρ and ω). The pertinent parity-violating meson-nucleon coupling constants h π,ρ,ω ∼ 10 have been calculated in quark models [3] and soliton models [4]. Due to a variety of uncertainties (e.g. quark wavefunctions, strong interaction enhancements, choice of particular soliton model) only “reasonable” ranges could be derived so far. For the parity-violating πNN -coupling constant hπ an experimental upper bound is known to be |hπ| < 1.43 · 10 [1]. An improved determination of this coupling constant is expected from a measurement of the photon asymmetry Aγ in the radiative capture of thermal neutrons ~np → dγ [5, 6, 7]. Recently, nuclear parity violation has been reformulated in the framework of effective field theory [8, 9]. As in the case of the parity-conserving NN-interaction one exploits the separation of scales. At low energies only the pions are kept as explicit degrees of freedom while all heavier particles are integrated out and their dynamical effects are subsumed in contact terms. At very low energies Ecm < 10MeV even the pion can be integrated out and then one is working with the pionless version of the effective field theory for nuclear parity violation. It has been shown in ref.[8] that at leading order O(q) there are in total five parity-violating low-energy constants associated with the respective contact operators linear in the nucleon momenta. One therefore requires a minimum of five independent, low-energy observables. For the effective field theory with dynamical pions there appear up to order O(q) three more parameters: the parity-violating πN -coupling constant hπ, a next-to-next-to-leading order correction to it, and a new electromagnetic operator [8]. This new electromagnetic operator is a specific feature of 1 the systematic effective field theory framework and entirely absent in the one-meson exchange phenomenology. Clearly, a lot of work is still necessary e.g. in order to pin down the parity-violating lowenergy constants and thus to reach a level where the effective field theory framework becomes predictive. In that situation it seems worthwhile to investigate separately the hierarchy of longrange pion-induced parity-violating NN forces. This is the purpose of the present short paper. We will compare directly the parity-violating 2π-exchange potentials at leading one-loop order with the parity-violating 1π-exchange. For the parity-conserving NN-potential the 2π-exchange with excitation of the low-lying spin-isospin-3/2 ∆(1232)-resonance plays a major role. The corresponding potentials exceed the ones from diagrams with only nucleon intermediate states typically by an order of magnitude [10, 11]. (Recently, is has been found that this feature remains when electromagnetic (one-photon exchange) corrections are included [12].) Therefore it is important to check whether a similar dominance of the ∆-induced processes holds for the parity-violating 2π-exchange interaction. We find that this is not the case. The basic reason for this opposite behavior is CP-invariance which forbids parity-violating effect in the isoscalar central channel. As a further possibly sizeable one-loop contribution we calculate the T-matrix related to the iteration of the parity-violating 1π-exchange with the parity-conserving one. Let us begin with recalling the parity-violating pion-nucleon vertex. It has the form: Lpv = hπ √ 2 N̄(~π × ~τ )N , (1) where N denotes a nucleon Dirac-spinor and hπ ∼ 10 is the weak πNN -coupling constant. In the center-of-mass frame the T-matrix of parity-violating 1π-exchange is readily computed as: T (1π) pv = −i gAhπ 2 √ 2fπ (~τ1 × ~τ2) (~σ1 + ~σ2) · ~q mπ + q 2 +O(M N ) . (2) Here, gA ≃ 1.3 is the nucleon axial vector coupling constant and fπ = 92.4MeV denotes the pion decay constant. ~σ1,2 and ~τ1,2 are the usual spinand isospin-operators of the two nucleons and ~q stands for the momentum transfer. Note that relativistic corrections start at order M N (with MN = 939MeV the nucleon mass) and therefore are negligibly small (typically less than 1%). Fourier transformation of the static term in Eq.(2) leads to a parity-violating NN potential in coordinate space: Vpv(~r ) = (~τ1 × ~τ2) (~σ1 + ~σ2) · r̂ U(r) + (τ 3 1 + τ 3 2 )(~σ1 × ~σ2) · r̂ W (r) , (3) with the radial dependence given by the derivative of a Yukawa function: U(r) = − gAhπ 8 √ 2πfπ eπ r2 (1 +mπr) . (4) The second term in Eq.(3) proportional to the cross product ~σ1 × ~σ2 of spin-operators has also been introduced because the associated “vector-type” potentialW (r) does receive contributions from parity-violating 2π-exchange. Next, we come to the parity-violating two-pion exchange. Some representative diagrams with leading order vertices are shown in Fig. 1. These are to be supplemented by further diagrams with the parity-violating vertex at a different position and/or both nucleon lines interchanged. We are interested only in the nonpolynomial or finite-range parts of these one-loop diagrams (disregarding the zero-range δ(~r )-terms from the polynomial pieces). For that purpose it is sufficient to calculate their spectral functions or imaginary parts using the Cutkosky cutting rule. The pertinent two-body phase space integral is most conveniently performed 2 in the ππ center-of-mass frame where it becomes proportional to a simple angular integral: