We construct the effective chiral Lagrangian involving hadronic and electromagnetic interactions originating from the QCDθ term. We impose vacuum alignment at both quark and hadronic levels, including field redefinitions to eliminate pion tadpoles. We show that leading time-reversalviolating (TV) hadronic interactions are related to isospin-violating interactions that can in principle be determined from charge-symmetry-breaking experiments. We discuss the complications that arise from TV electromagnetic interactions. Some implications of the expected sizes of various pion-nucleon TV interactions are presented, and the pion-nucleon form factor is used as an example.

The electric dipole form factor of the nucleon stemming from the QCDθ term is calculated in chiral perturbation theory in leading order. To this order, the form factor originates from the pion cloud. Its momentum dependence 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 factoris the pion mass.

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...