We classify the parity-and time-reversal-violating operators involving quark and gluon fields that have effective dimension six: the quark electric dipole moment, the quark and gluon chromo-electric dipole moments, and four four-quark operators. We construct the effective chiral Lagrangian with hadronic and electromagnetic interactions that originate from them, which serves as the basis for calculations of low-energy observables. The form of the effective interactions depends on the chiral properties of these operators. We develop a power-counting scheme and calculate within this scheme, as an example, the parity-and time-reversal-violating pion-nucleon form factor. We also discuss the electric dipole moments of the nucleon and light nuclei.The Standard Model (SM) of particle physics contains, in its minimal version, two sources of time-reversal (T ), or, equivalently, CP violation. In the electroweak sector, the phase in the quark mixing matrix [1] is associated with the Jarlskog parameter J CP 3 × 10 −5 [2]. The strong sector contains the QCD vacuum angleθ [3], but the experimental upper limit on the neutron electric dipole moment (EDM) shows thatθ is unnaturally small,θ < ∼ 10 −10 [4]. CP violation within the SM is believed to be insufficient for a successful baryogenesis scenario [5], and therefore new sources of CP violation are expected in order to explain the cosmological matter-antimatter asymmetry. This is not a surprise since the SM is likely but the dimensionfour part of an effective field theory (EFT) that contains higher-dimensional operators, some of which will violate CP .Powerful probes of such CP violation beyond the SM are EDMs of nucleons, nuclei, atoms, and molecules [6,7], which violate both parity and time reversal (/ P/ T ). Since the SM predictions from the quark mixing matrix [8] are orders of magnitude away from current experimental limits, a finite EDM in upcoming experiments would be an unambiguous sign of new physics. In addition to impressive improvements [9] on the time-honored EDM experiments with neutrons and neutral atoms, in particular 199 Hg, which have resulted in very precise limits [10,11], novel ideas exist for the measurement of EDMs of charged particles, such as the proton, deuteron and helion, in storage rings [12]. An important question that comes up is whether, when future experiments measure nonzero EDMs, we will be able to pinpoint the microscopic source of P and T violation.EDMs of strongly interacting particles arise from the higher-dimensional / P/ T operators at the quark-gluon level. These non-renormalizable operators might have their origin in a renormalizable theory at a higher-energy scale, such as, for example, supersymmetric (SUSY) extensions of the SM. At the SM scale, the most important higher-dimensional / P/ T operators should be those of dimension six, as we are not concerned here with CP violation in the dimension-five leptonic operator [13] that gives rise to neutrino masses and mixings. From symmetry considerations it is found [14,15,16,17] that the...

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.

We set up the framework for the calculation of electric dipole moments (EDMs) of light nuclei using the systematic expansion provided by chiral effective field theory (EFT). We take into account parity (P ) and time-reversal (T ) violation which, at the quark-gluon level, originates from the QCD vacuum angle and dimension-six operators capturing physics beyond the Standard Model. We argue that EDMs of light nuclei can be expressed in terms of six low-energy constants that appear in the P -and T -violating nuclear potential and electric current. As examples, we calculate the EDMs of the deuteron, the triton, and 3 He in leading order in the EFT expansion.

We present a master formula describing the neutrinoless-double-beta decay (0νββ) rate induced by lepton-number-violating (LNV) operators up to dimension nine in the Standard Model Effective Field Theory. We provide an end-to-end framework connecting the possibly very high LNV scale to the nuclear scale, through a chain of effective field theories. Starting at the electroweak scale, we integrate out the heavy Standard Model degrees of freedom and we match to an SU (3) c ⊗ U (1) em effective theory. After evolving the resulting effective Lagrangian to the QCD scale, we use chiral perturbation theory to derive the lepton-number-violating chiral Lagrangian. The chiral Lagrangian is used to derive the twonucleon 0νββ transition operators to leading order in the chiral power counting. Based on renormalization arguments we show that in various cases short-range two-nucleon operators need to be enhanced to leading order. We show that all required nuclear matrix elements can be taken from existing calculations. Our final result is a master formula that describes the 0νββ rate in terms of phase-space factors, nuclear matrix elements, hadronic low-energy constants, QCD evolution factors, and high-energy LNV Wilson coefficients, including all the interference terms. Our master formula can be easily matched to any model where LNV originates at energy scales above the electroweak scale. As an explicit example, we match our formula to the minimal left-right-symmetric model in which contributions of operators of different dimension compete, and we discuss the resulting phenomenology.

We calculate the electric-dipole and magnetic-quadrupole form factors of the deuteron that arise as a low-energy manifestation of parity and time-reversal violation in quark-gluon interactions. We consider the QCD vacuum angle and the dimension-six operators that originate from physics beyond the standard model: the quark electric and chromoelectric dipole moments and the gluon chromoelectric dipole moment. Within the framework of two-flavor chiral perturbation theory, we show that in combination with the nucleon electric dipole moment, the deuteron moments would allow an identification of the dominant source(s) of symmetry violation.

Within the framework of chiral effective field theory, we discuss the leading contributions to the neutrinoless double-beta decay transition operator induced by light Majorana neutrinos. Based on renormalization arguments in both dimensional regularization with minimal subtraction and a coordinate-space cutoff scheme, we show the need to introduce a leading-order short-range operator, missing in all current calculations. We discuss strategies to determine the finite part of the short-range coupling by matching to lattice QCD or by relating it via chiral symmetry to isospin-breaking observables in the two-nucleon sector. Finally, we speculate on the impact of this new contribution on nuclear matrix elements of relevance to experiment.

We analyze neutrinoless double beta decay (0νββ) within the framework of the Standard Model Effective Field Theory. Apart from the dimension-five Weinberg operator, the first contributions appear at dimension seven. We classify the operators and evolve them to the electroweak scale, where we match them to effective dimension-six, -seven, and -nine operators. In the next step, after renormalization group evolution to the QCD scale, we construct the chiral Lagrangian arising from these operators. We develop a power-counting scheme and derive the two-nucleon 0νββ currents up to leading order in the power counting for each lepton-number-violating operator. We argue that the leadingorder contribution to the decay rate depends on a relatively small number of nuclear matrix elements. We test our power counting by comparing nuclear matrix elements obtained by various methods and by different groups. We find that the power counting works well for nuclear matrix elements calculated from a specific method, while, as in the case of light Majorana neutrino exchange, the overall magnitude of the matrix elements can differ by factors of two to three between methods. We calculate the constraints that can be set on dimension-seven lepton-number-violating operators from 0νββ experiments and study the interplay between dimension-five and -seven operators, discussing how dimension-seven contributions affect the interpretation of 0νββ in terms of the effective Majorana mass m ββ .

The process at the heart of neutrinoless double-beta decay, nn → pp e − e − induced by a light Majorana neutrino, is investigated in pionless and chiral effective field theory. We show in various regularization schemes the need to introduce a short-range lepton-numberviolating operator at leading order, confirming earlier findings. We demonstrate that such a short-range operator is only needed in spin-singlet S-wave transitions, while leading-order transitions involving higher partial waves depend solely on long-range currents. Calculations are extended to include next-to-leading corrections in perturbation theory, where to this order no additional undetermined parameters appear. We establish a connection based on chiral symmetry between neutrinoless double-beta decay and nuclear charge-independence breaking induced by electromagnetism. Data on the latter confirm the need for a leadingorder short-range operator, but do not allow for a full determination of the corresponding lepton-number-violating coupling. Using a crude estimate of this coupling, we perform ab initio calculations of the matrix elements for neutrinoless double-beta decay for 6 He and 12 Be. We speculate on the phenomenological impact of the leading short-range operator on the basis of these results.