The electric dipole form factor of the nucleon in chiral perturbation theory to sub-leading order

Abstract

The electric dipole form factor (EDFF) of the nucleon stemming from the QCD θ term and from the quark color-electric dipole moments is calculated in chiral perturbation theory to sub-leading order. This is the lowest order in which the isoscalar EDFF receives a calculable, non-analytic contribution from the pion cloud. In the case of theθ term, the expected lower bound on the deuteron electric dipole moment is |d d | > ∼ 1.4·10 −4θ e fm. The momentum dependence of the isovector EDFF is proportional to a non-derivative time-reversal-violating pion-nucleon coupling, and the scale for momentum variation -appearing, in particular, in the radius of the form factor-is the pion mass.The electric dipole form factor (EDFF) completely specifies the parity (P ) and timereversal (T ) -violating coupling of a spin 1/2 particle to a single photon [1,2]. At zero momentum, it reduces to the electric dipole moment (EDM), and its radius provides a contribution to the Schiff moment (SM) of a bound state containing the particle [3]. The full momentum dependence of the form factor can be used in lattice simulations to extract the EDM by extrapolation from a finite-momentum calculation [4] (in addition to the required extrapolations in quark masses and volume [5]).There has been some recent interest [1,2,6,7,8,9] in the nucleon EDFF motivated by prospects of experiments that aim to improve the current bound on the neutron EDM, |d n | < 2.9 · 10 −13 e fm [10], by nearly two orders of magnitude [11], and to constrain the proton and deuteron EDMs at similar levels [12]. We would like to understand the implications of a possible signal in these measurements to the sources of T violation at the quark level, which include, in order of increasing dimension, the QCDθ term, the quark color-EDM (qCEDM) and EDM, the gluon color-EDM, etc. [13,14]. Unfortunately, as with other low-energy observables, both the EDM and the SM of hadrons and nuclei are difficult to calculate directly in QCD. However, long-range contributions from pions can, to some extent, be calculated using the low-energy effective field theory of QCD, chiral perturbation theory (ChPT) [15,16,17]. ChPT affords a systematic expansion of lowenergy observables in powers of Q/M QCD , where Q represents low-energy scales such as external momenta and the pion mass m π , and M QCD ∼ 1 GeV denotes the characteristic QCD scale. (For introductions, see for example Refs. [18,19].)In Refs. [1,9] the nucleon EDFF stemming from T -violation sources of effective dimension up to 6 was considered in ChPT to the lowest order where momentum dependence appears. It was argued [9] that the nucleon EDFF partially reflects the source of T violation at the quark level. The various sources differ in particular in the expectation for the behavior of the isoscalar EDFF. Forθ and qCEDM, the isoscalar momentum dependence appears only at NLO. The nucleon EDFF fromθ was calculated at LO in Ref.[1], generalizing to finite momenta earlier calculations of the EDM [20, 21] and SM [3]. At this order, the momentum depe...

Nuclear electric dipole moments in chiral effective field theory

“…The only other relevant EDM contribution up to NLO (i.e. contributions suppressed by a factor of ∼ M π /m N ) is the isoscalar sum of the single-nucleon EDMs [15,16,56,57]. In this combination, the large isovector loop contribution to the single-nucleon EDM cancels [58].…”

Electric dipole moments of light nuclei from chiral effective field theory

“…The dimension-six operators in Eq. (8) have different transformation properties still [27,26,23,28,34]. The isoscalar and isovector qEDM and qCEDM transform as the fourth and third components of two other SO(4) vectors.…”

“…Using these interactions, the nucleon EDM has been calculated in χPT up to NLO for all sources of dimension up to six [25,26,23].…”

“…where γ E ≃ 0.577 is the Euler-Mascheroni constant, the isoscalar and isovector EDMs can be expressed at NLO respectively as [25,26,23] …”

“…Thus the non-analytic terms serve as lower-bound estimates for the size of d p,n . We then expect, forθ [11] and qCEDM [23]…”