Probing exotic phenomena at the interface of nuclear and particle physics with the electric dipole moments of diamagnetic atoms: A unique window to hadronic and semileptonic CP violation
Abstract
The current status of electric dipole moments of diamagnetic atoms which involves the synergy between atomic experiments and three different theoretical areas – particle, nuclear and atomic is reviewed. Various models of particle physics that predict CP violation, which is necessary for the existence of such electric dipole moments, are presented. These include the standard model of particle physics and various extensions of it. Effective hadron level combined charge conjugation (C) and parity (P) symmetry violating interactions are derived taking into consideration different ways in which a nucleon interacts with other nucleons as well as with electrons. Nuclear structure calculations of the CPodd nuclear Schiff moment are discussed using the shell model and other theoretical approaches. Results of the calculations of atomic electric dipole moments due to the interaction of the nuclear Schiff moment with the electrons and the P and timereversal (T) symmetry violating tensorpseudotensor electronnucleus are elucidated using different relativistic manybody theories. The principles of the measurement of the electric dipole moments of diamagnetic atoms are outlined. Upper limits for the nuclear Schiff moment and tensorpseudotensor coupling constant are obtained combining the results of atomic experiments and relativistic manybody theories. The coefficients for the different sources of CP violation have been estimated at the elementary particle level for all the diamagnetic atoms of current experimental interest and their implications for physics beyond the standard model is discussed. Possible improvements of the current results of the measurements as well as quantum chromodynamics, nuclear and atomic calculations are suggested.
pacs:
11.30.ErCP invariance and 14.20.DhProtons and neutrons and 24.80.+yNuclear tests of fundamental interactions and symmetries and 31.15.veElectron correlation calculations for atoms and ions: ground state1 Introduction
The important predictions of the standard model (SM) of particle physics SM1 (); SM2 (); SM3 () have been verified largely due to the remarkable advances in accelerator technology higgsatlas (); higgscms (). A number of ingenious high energy experiments are currently underway to search for new phenomena beyond the SM. Many of these experiments are being performed using the Large Hadron Collider (LHC) at the TeV scale. A complementary approach to search for new physics beyond the SM is characterized by nonaccelerator low energy precision tests of fundamental physics. It involves measuring observables and comparing the experimental results with the predictions of the SM. This is an indirect approach to new physics beyond the SM, but the observation of rare or forbidden phenomena is an indubitable proof of the existence of a new theory. Although conclusions reached by such an approach may in some case not be as specific in identifying the underlying fundamental theory as in the direct high energy physics approach, its sensitivity to new physics may well exceed the energy of collider experiments.
The combined charge conjugation (C) and parity (P) symmetry (CP) violation is considered to have relevance to the huge discrepancy from the SM prediction which is observed in the matterantimatter asymmetry of the Universe sakharov (), and is currently an issue of primary importance in elementary particle physics khriplovich (); bigibook (); roberts (). CP violation has been studied in various physical systems, but has so far been observed only in the christenson () and mesons abe (); aubert (); aaij (); alvarez (), in which cases the experiments are in agreement with predictions of the SM. In the SM, it arises from the complex phase of the CabibboKobayashiMaskawa (CKM) matrix KM (); Jarlskog (). It is well known that this phase cannot generate excess of matter over antimatter in the early Universe farrar (); huet (); dine (). It is therefore imperative to find one or several new sources of CP violation beyond the SM. A variety of studies on CP violation including experiments to observe the electric dipole moments (EDMs) of different systems have lent themselves to searches for new physics beyond the SM purcell (); hgedm1987 (); rosenberry (); regan (); baker (); griffith (); hudson (); baron (); bishof (); graner (); Pendlebury ().
A nondegenerate physical system can possess a permanent EDM due to violations of P and timereversal (T) symmetries ramsey1 (); fortson (). T violation implies the CP violation as a consequence of the CPT theorem luders (). An atom could possess an EDM due to the possible existence of (i) the electron EDM () (ii) P and T violating (P,Todd) electronnucleus interactions and (iii) the hadronic CP violation. EDMs of open shell (paramagnetic) atoms arise primarily due and the P,Todd electronnucleus scalarpseudoscalar (SPS interaction, but the dominant contributions to the EDMs of closedshell (or diamagnetic) atoms come from the hadronic CP violation and the electronnucleus tensorpseudotensor (TPT) interaction. Atomic EDMs are sensitive to new physics beyond the standard model (BSM) and can probe CP violating interactions corresponding to mass scales of tens of TeV or larger bernreuther (); barr (); pospelovreview (); ramsey (). The results of atomic EDM experiments and theory currently constrain various extensions of the SM. Experiments are underway to improve the limits of EDMs in paramagnetic (openshell) weiss (); heinzen (); harada () and diamagnetic (closedshell) atoms furukawaxe (); inouexe (); rand (); tardiff (); fierlinger (); schmidt (); yoshimi (). Their results in combination with state of the art theoretical calculations can extract various CP violating coupling constants at the elementary particle level via the hadronic, nuclear and atomic theories khriplovich (); pospelovreview (); heedmreview (); ginges (); Fukuyama (); dzubareview (); engel ().
It is necessary at this stage to emphasize the importance of the study of EDMs of the diamagnetic atoms. Many low energy observables used in the precision tests of fundamental physics, including EDMs of the paramagnetic atoms, are sensitive to limited sectors (e.g. leptonic, hadronic, Higgs, etc) of a particular particle physics model. However, the EDMs of diamagnetic atoms arise from new physics in multiple sectors of a variety of extensions of the SM, since the hadronic sector opens up many possible scenarios for CP violation at the elementary level (quark EDMs, quark chromoEDMs, gluon chromoEDMs, quarkquark (qq) interactions, etc.). This means that one experimental constraint cannot in principle determine the unknown coupling constants of the models. Unraveling new physics beyond the SM in the context of EDMs of diamagnetic atoms is equivalent to finding the values for the couplings of new interactions that are solutions of a set of coupled equations obtained from experiments on atomic EDMs. The number of systems for EDM experiments must be at least equal to the number of coupling constants in order to uniquely determine those constants; assuming that uncertainties associated in all the results are of similar order. It is therefore desirable to perform EDM experiments on a number of different diamagnetic atoms.
The experimental limit on the EDM of mercury atom (Hg) has improved several times since the first measurement in 1987 hgedm1987 (), and it is currently the lowest limit reported for the EDM of any system ( cm) graner (). Improvements are expected in the EDM measurements of other diamagnetic systems such as the Xe and Ra in the near future. However, since the EDMs of the diamagnetic atoms depend on many fundamental sectors, considerable theoretical effort has to be put in relating these EDMs to new physics beyond the SM (see Fig. 1). In particular, the atomic and nuclear level manybody physics as well as the nonperturbative effects of quantum chromodynamics (QCD) contribute to the theoretical uncertainties in the determination of their sensitivity to fundamental theories. Recent advances in the atomic and nuclear manybody as well as QCD calculations using numerical methods have reduced these uncertainties, but further progress is necessary in this direction.
The focus of this review article is the recent advances in the EDMs of the diamagnetic atoms which arise predominantly from the nuclear Schiff moment (NSM) schiff () and CP violating electronnuclear interaction. The former arises from CP violating nucleonnucleon (NN) interactions and EDMs of nucleons, which in turn originate from CP violating quark level. The latter is fundamentally due to the CP violating electronquark (eq) interactions. We shall summarize our current understanding of physics beyond the SM that has been obtained by combining the results of experiment as well as atomic theory, nuclear theory and QCD relevant in the evaluation of the EDMs of diamagnetic atoms. The theoretical uncertainty in the determination of these EDMs is the combined uncertainties resulting from the calculations in these three different theories. It is therefore important to identify the large sources of errors in extracting the CP violating couplings at the particle physics level from the EDM experimental data.
The article is organized in the following manner: Sec. 2 covers CP violations at the particle physics level that are suitable for the kind of atomic EDM that is considered in this review. The derivation of hadron level effective CPodd interactions are then presented in Sec. 3. Sec. 4 deals with the NSM and the nuclear structure issues involved in its calculation. Different features of relativistic manybody theories which are necessary to calculate the EDMs of diamagnetic atoms are presented in Sec. 5.1. An introduction to the principles of the measurement of EDMs of diamagnetic atoms and the current status of the search for EDMs of these atoms are given in Sec. 6. We summarize the effect of CPodd interactions at the particle physics level on the EDMs of diamagnetic atoms in Sec. 7, and analyze the candidates for BSM physics which can be constrained. Finally, our concluding remarks regarding the search for the EDMs of diamagnetic atoms are made in Sec. 8.
2 Sources of P and CP violations in particle physics
In this section, we describe the physics of CP violation at the level of elementary particle physics. First, we present the relevant CP violating operators, and then show that the SM contribution to them is small. We then briefly review several motivated candidates of new theories beyond SM. We also introduce the PecceiQuinn mechanism which is almost mandatory to resolve the problem of too large QCD term. We finally see the procedure to renormalize the CPodd operators from the elementary physics to the hadronic scale, to pass on to the hadron level analysis.
2.1 CP violating operators after integration of heavy degrees of freedom
After integrating out heavy new physics particles of BSM, the Higgs boson and massive electroweak gauge bosons (and eventually the top quark), we are left with an infinite number of operators which form the quark and gluon level effective interactions. As the coupling constants of those interactions are suppressed by the power of the energy scale of new physics, operators with the lowest mass dimension are important in the physics of strong interaction. Here we list the CP violating operators generated at the elementary level up to mass dimension six, which are relevant in the physics of the EDM of atoms:

term:
(1) 
Fermion EDM:
(2) where denotes the electron or the quark and also it follows .

quark chromoEDM:
(3) where is the field operator of the quark .

Weinberg operator:
(4) where is the (3) structure constant of the Lie algebra.

P, CPodd or equivalently P,Todd 4quark interactions:
(5) where the color indices and were explicitly written when the color contraction is not taken in the same fermion bilinear.

P, CPodd or equivalently P,Todd eq interactions:
(6)
where superscripts SP, PS, and T denote the scalarpseudoscalar (SPS), pseudoscalarscalar (PSS), and TPT eq interactions, respectively.
We must note that these effective interactions are defined at some energy scale. In perturbative evaluations, they are usually given at the energy scale where the new particle BSM is integrated out (typically at the TeV scale).
2.2 The SM contribution
Let us start with the SM contribution to the elementary level CP violation SM1 (); SM2 (); SM3 (). Apart from the strong term, CPviolation comes from the KobayashiMaskawa phase KM () in the form of Jarlskog invariant Jarlskog (). The standard form of CabibboKobayashiMaskawa (CKM) matrix is given by
and the Jarlskog invariant is
(7) 
Here implies an imaginary part and and . This combination of CKM matrix elements is the minimal requirement to generate CP violation.
The CP violation in the SM therefore requires at least two boson exchanges. For the quark EDM and the chromoEDM, the twoloop level contribution is also known to vanish due to the GIM mechanism GIM (); donoghuesmedm (); shabalin1 (); shabalin2 (), and the leading order one is given by at the threeloop level czarnecki () (see Fig. 3). Their effect on the nucleon EDM is around cm, much smaller than the present experimental limit of that of the neutron ( cm) baker (); Pendlebury ().
The EDM of the electron is also generated by the CP phase of the CKM matrix. This effect starts from the fourloop level, and its value is cm pospelovsmelectronedm (); booth (); pospelovsmatomicedm (). We must note that the effect of the CP phase of the neutrino mixing matrix is negligible due to the small neutrino mass. If the neutrinos are Majorana fermions the effect of additional CP phases can generate the electron EDM from the twoloop level, and a larger value will be allowed for Archambault (); hemfv1 (); hemfv2 (); novales ().
Purely gluonic CPodd processes such as the term or the Weinberg operator are also known to be very small. The term generated by the CKM phase is ellissmtheta (); khriplovichtheta (), which yields a nucleon EDM of cm. The Weinberg operator gives an even smaller nucleon EDM, of order cm smweinbergop ().
In the strongly interacting sector, the most widely accepted leading hadronic CP violation due to the CP phase of the CKM matrix is generated by the long distance effect. The long distance contribution of the CKM phase arises from the interference between the tree level strangeness violating boson exchange process and the penguin diagram (see Fig. 2), which forms the Jarlskog invariant (7). From a naive dimensional analysis, the nucleon and nuclear EDMs are estimated as cm, which is larger than the contribution from the short distance processes (quark EDM, chromoEDM, Weinberg operator, etc). Previous calculations of the nucleon EDM are in good agreement with this estimations ellissmedm (); nanopoulossmedm (); Deshpande (); gavelasmedm (); smneutronedmkhriplovich (); eeg (); hamzaouismedm (); smneutronedmmckellar (); mannel (); seng ().
The CP violating effects in the SM exhibit an EDM well smaller than the experimental detectability, and a large room is left for the discovery of new source of CP violation BSM.
2.3 Sources of CP violation from BSM physics
In many scenarios of BSM, large EDMs are predicted, because of higher order contributions that can arise at the one or twoloop levels. These contributions are overwhelmingly exceed over the loop suppressed SM contribution. In Fig. 4, we present the typical lowest order CP violating processes of BSM contributing to the EDMs at the elementary level. In this subsection, we would like to elaborate several such well motivated candidates of BSM which can generate EDMs.
2.3.1 Higgs doublet models
The Higgs boson was recently discovered higgsatlas (); higgscms (), but the detailed Higgs potential is still unknown. There are currently many wellmotivated extensions of the Higgs sector BSM. The most wellknown one is the twoHiggs doublet model (2HDM), and extensive studies have been performed barrzee (); weinbergop (); dicus (); chang2hdm1 (); leigh (); chang2hdm2 (); mahanta (); kao2hdm (); barger (); bowserchao (); buras2hdm (); brod (); sinoue2hdm (); cheung2hdm (); jung2hdmedm (); tabe (); bian (); chenedm (); eeghiggs ().
As the Higgs boson has a small coupling with light fermions, the oneloop level fermion EDM and the CPodd fourfermion interactions are suppressed in 2HDM barrmasiero (); enint (). The leading contribution to the elementary level CP violation contributing to the EDM is the twoloop level BarrZee type diagram barrzee () [Fig. 4 (b)], enhanced by the large Yukawa coupling of the top quark of the inner loop. The BarrZee type diagram contribution to the EDM of SM fermion can be written as barrzee ()
(8) 
where
(9)  
(10) 
and and are the Yukawa couplings relating the lightest Higgs boson ( GeV) with the fermion and the top quark, respectively. The first (second) term of Eq. (8) is generated by the vacuum expectation value (), where is the Higgs doublet interacting with the uptype quark () or the downtype fermion (, for the electron or downtype quarks). These vacuum expectation values strongly depend on the Higgs potential. Those couplings are obtained from the diagonalization of the Higgs doublets.
In diamagnetic atoms, the most important CP violating process is the quark chromoEDM:
(11) 
With , we have cm. We must note that the Weinberg operator (4) is also generated in the 2HDM [see Fig. 4 (d)] weinbergop (); dicus (). Its typical value is GeV, with GeV. We will see in later sections that this contribution is subleading for the nucleon EDM.
2.3.2 Supersymmetric (SUSY) models
As the next attractive model for BSM physics, we have the minimal supersymmetric standard model (MSSM) haber (); gunionmssm (); martinmssm (). The MSSM contains several phenomenological interactions which generically possess CP phases. In the most simplified parametrization, the Higgs bilinear term
(12) 
from the superpotential, which is required to give mass to higgsinos, and the supersymmetry breaking sfermion trilinear interactions
(13)  
are CP violating. Here the dot denotes the inner product. For the sfermion trilinear interactions, we often assume a flavor diagonal one, with a common CP phase . This assumption is due to the strong constraints on flavor changing neutral current from phenomenology ramsey (); gabbiani ().
Under this MSSM Lagrangian, the fermion EDM appears at the oneloop level barrmasiero (); ellismssmedm (); buchmullermssmedm (); polchinski (); aguilamssmedm (); nanopoulosmssmedm (); duganmssmedm (); nathmssmedm (); kizukurimssmedm1 (); kizukurimssmedm2 (); inuimssmedm (); mssmreloaded (); bingli () [see Fig. 4 (a)]. The electron EDM and the quark EDM, in the simplified parametrization of MSSM where masses of all the supersymmetric particles as well as are given by , are given by pospelovreview ()
(14)  
(15) 
respectively, with is the electric charge of the quark , and , and are the couplings of the and gauge theories and QCD, respectively. The quark chromoEDM is similarly given by
(16) 
where is the ratio between the vacuum expectation values of the uptype and downtype Higgs fields. As for the Higgs doublet models, we also see here a dependence on . By assuming and , the MSSM contribution to the EDMs of the fermions and the chromoEDMs of quarks at the scale TeV [] become cm, cm and cm^{1}^{1}1Unfortunately such models, except for the few predictive ones, have so many undetermined parameters and we should be careful about under what assumptions such and such predictions have been made. On this point one can refer to Ref. FA () for more detailed clarification..
To conceive natural scenarios in MSSM, it is often assumed that the first and the second generations have no sfermion trilinear interactions. In such a case, the leading order CP violation are given by the twoloop level effect, namely the BarrZee type diagrams [Fig. 4 (b)] west (); kadoyoshi (); chang (); pilaftsis1 (); chang2 (); mssmrainbow1 (); demir (); feng1 (); feng2 (); lihiggshiggsino (); mssmrainbow2 (); carena (); nakai () and the Weinberg operator [Fig. 4 (d)] daimssm (); arnowitt1 (); arnowitt2 (). We must note that the BarrZee type diagram and the fourfermion interaction are enhanced when is large demir (); fischler (); falk (); lebedevmssm (); pilaftsishiggsmssm (). Global analyses with constrained supersymmetric parameters by the Grand unification theory (GUT) strongly constrain CP phases mssmunify1 (); mssmunify2 (); mssmunify3 (); mssmunify4 (); mssmunify5 (); mssmunify6 ().
Another natural supersymmetric scenario is the split SUSY model splitsusy1 (); splitsusy2 (), relying on the GUT. In this case, the sfermions are much heavier than the gauginos, and oneloop level diagrams, which must contain sfermions, are suppressed. The BarrZee type diagram with chargino inner loop therefore becomes dominant splitsusy3 (); splitsusy4 (); dhuria (); sarellis ().
The SUSY model can be extended with additional interactions, with several motivations. The first possibility is to take into account additional soft supersymmetry breaking terms, in particular the flavor violating ones which are not forbidden by any symmetries or by other experimental constraints. The flavor nondiagonal soft breaking terms can generically have CP phases. This extension was motivated by the deviation of the CP violating decay grossman (); barbieri () suggested by Belle experiment belle (). The effects of those flavor violating terms on the EDM are however large, and it was found that the EDM experimental data can strongly constrain their CP phases hisanoshimizu1 (); hisanoshimizu2 (); endomssmflavor (); chomssmflavor (); hisanonagai1 (); hisanonagai2 (); hisanonagai3 (); Altmannshofer ().
Another possible way to extend the MSSM is to add new interactions in the superpotential. The scenario on these lines is the nexttominimal supersymmetric standard model (NMSSM) which considers an additional scalar superfield in the Higgs sector nmssm (). This model can dynamically generate the term (12) and circumvent the problem of term. It is also motivated by the difficulty to explain the appearance of the light Higgs boson in the simple parametrization of the MSSM. In the NMSSM, the EDMs of fermions do not become large nmssmedm (). If we further enlarge the superpotential by adding new local gauged terms (BLMSSM) blmssm1 (); blmssm2 (), the fermion EDMs can become large, and the CP phases will be strongly constrained by the current experimental data blmssmedm (). The EDM is even more enhanced if we also allow the Rparity violation, where baryon and lepton numbers are not conserved rpvreview1 (); rpvreview2 (); rpvreview3 (); rpvreview4 (). If we neglect the oneloop level fermion EDM which is only generated in the presence of soft breaking bilinear Rparity violating interaction keum1 (); keum2 (); choirpv (); chiourpv (), the leading CP violation processes are the BarrZee type diagram godbole (); abelrpv (); changrpv (); rpvedm1 (); rpvedm2 (); rpvedm3 (); rpvedm4 () and the CPodd fourfermion interaction rpvedm1 (); rpv4f1 (); faessler1 (); faessler2 (). The majority of CP phases of the Rparity violating couplings are strongly constrained by the current EDM experimental data.
Obviously, the SUSY extensions allow larger observable EDMs as the number of parameters increases. This fact does not depend on whether we have extended the superpotential or the soft supersymmetry breaking interaction. The supersymmetric SM is an excellent example of new physics which contributes to the EDM of composite systems through various elementary level CPodd operators. Current EDM experimental data strongly constrain the CP phases of models with large degree of freedom. In the analysis of theories and models which have a large parameter space, it was often assumed that only a small numbers of couplings are active, and the effect of the others were neglected. We however have to note that cancellations may occur among supersymmetric CP phases susycancel1 (); susycancel2 (); susycancel3 (); susycancel4 (); susycancel5 (); susycancel6 (); susycancel7 (); susycancel8 (); susycancel9 (); susycancel10 (); susycancel11 (); susycancel12 (); susycancel13 (); yamanakabook (); linearprogramming (). In that case, still large CP phases may be allowed, and they may be relevant in the ongoing EDM experiments.
2.3.3 LeftRight symmetric models
The LeftRight symmetric models contain an additional gauge theory which couples to the righthanded fermions of SM leftright1 (); leftright2 (); leftright3 (). An gauge group is assumed to be spontaneously broken at some high energy scale, and gives the SM as an effective theory below it. Phenomenologically, a mixing of boson with a heavier boson is possible. The mass of additional weak gauge boson is constrained by LHC experiment, and the current lower bound is a few TeV LRLHC1 (); CMSW'1 (); ATLASW' (); CMSW'2 (); CMSW'3 ().
In low energy effective theory, we obtain a 4quark interaction with the structure :
(17)  
where the coupling constant scales as . The terms in the last line are the color octet fourquark interaction, with the generator of the group. If has a CP phase, the EDM is induced in hadronic systems LRedm1 (); LRedm2 (); LRedm3 (); LRzhang1 (); LRzhang2 (); LRxu (); eft6dim (); LRLHC2 (); dekens (). It is important to note that the above fourquark interaction breaks both the chiral and isospin symmetries eft6dim (). This property is useful in estimating the leading CPodd hadron level effective interaction generated by it sengisovector (). Moreover, the effective interaction (17) is generated at the scale , where the boson is integrated out.
2.3.4 Models with vectorlike fermion
The vectorlike fermions are spin particles which have the same gauge charges for their left and righthanded components Aguilar (). They are not constrained by the analysis of the Higgs boson in collider experiments, as it was for extensions with extra generations of chiral fermions djouadi (). This class of models are attractive since those particles are often relevant in extensions of SM with composite sectors dimopouloscomposite (); kaplancomposite (); peterson () or extradimensions contino (); hewette6 ().
As a model independent feature, the vectorlike fermions may mix with SM fermions, but those processes are strongly constrained by the flavor changing neutral current Aguilar (); vectorlikefcnc1 (); vectorlikefcnc2 (); vectorlikefcnc3 (); vectorlikefcnc4 (); vectorlikefcnc5 (); vectorlikefcnc6 (); vectorlikefcnc7 (); vectorlikefcnc8 (); vectorlikefcnc9 (); vectorlikefcnc10 (); vectorlikefcnc11 (). Regarding more model dependent aspects, additional dynamically generated bosons may accompany vectorlike fermions, such as the Higgs bosons, KaluzaKlein particles, or higher energy resonances, and their interactions with SM fermions may generate EDM at the oneloop level. This process is also strongly constrained by phenomenology vectorlikefcnc8 (); liaockm (); Iltan (); chang5d (); agashe (); nishiwaki (); Kalinowski (). Under those constraints, the vectorlike fermions may appear in the intermediate states connected only by the exchange of gauge bosons fan (). The leading CP violating process is therefore the Weinberg operator choi ().
The contribution of the Weinberg operator in the vectorlike fermion models can be written as
(18) 
with . Here we have assumed a boson which couples to vectorlike fermions with mass (). In the limiting case , we have mssmreloaded (). In technicolor theories, an effective interaction is generated by a similar mechanism appelquist1 ().
2.3.5 Leptoquark models
The leptoquarks are bosons which couple to both leptons and baryons, and often appears in scenarios with GUT. Those which violate the baryon number are strongly constrained by the proton decay, but those which conserve lepton and baryon numbers are allowed up to the constraints from the LHC experiments leptoquarkcms1 (); leptoquarkcms2 (); leptoquarkcms3 (); leptoquarkcms4 (); leptoquarkatlas1 (); leptoquarkatlas2 (), and their interaction can be probed using low energy precision tests davidson (). The simplest interaction of the scalar leptoquark is given as
(19) 
where is the leptoquark field, and the indices denote the flavor.
If the couplings and have relative CP phases, the EDM will be induced in atomic systems. The leading CP violation is given by the oneloop level fermion EDM barrmasiero (); geng () and the CPodd eq interaction [see Eq. (6)] enint (); heeNedm (); herczegleptoquark (). For the atomic system, the latter is especially important, since it contributes to the tree level. The Leptoquark model is one of the rare models which contribute to the TPT CPodd eN interaction [the term with in Eq. (6)].
2.4 Renormalization group evolution (RGE)
In the usual discussion of particle physics, the effect of BSM physics is calculated at some high energy scale, much higher than that of the strong interaction MeV. On the other hand, their matching with the hadronic effective interaction is done at the hadron scale, we must evolve the Wilson coefficients of elementary level interactions down to the hadronic scale. In this Subsec., we first present the RGE of purely hadronic CPodd operators, and then that of CPodd eq interactions, which do not mix with each other.
2.4.1 RGE of strong CPodd operators
The effective CPodd Lagrangian and their Wilson coefficients are given as
(20)  
with
(21)  
(22)  
(23)  
(24)  
(25)  
(26)  
(27)  
(28)  
(29) 
where the color indices and were explicitly written when the color contraction is not taken in the same fermion bilinear. The summation of the quark for the above operators must be taken for the relevant flavor at the renormalization scale chosen (e.g. for GeV).
We note that the above CPodd operators are defined in a scale where the Higgs boson, massive electroweak gauge bosons and the top quark are integrated out. Their Wilson coefficients therefore also involve the BSM CPodd effect related with those particles at a higher energy scale through the RGE brod (); dekens1 (); brod2 (); cirigliano2 (); cirigliano3 (). In this section, we do not treat them explicitly, but consider their effects through the renormalized Wilson coefficients at the electroweak scale as the initial condition.
The evolution of the Wilson coefficients is dictated by the renormalization group equation, which mixes the CPodd operators when the scale is changed. It is given by the following differential equation
(30) 
The anomalous dimension matrix is given by
(31) 
with the renormalization matrix. By integrating (30) with the initial condition at the scale of new physics , we have
(32) 
where
(33) 
with the strong coupling , and the coupling ordered product operator . The anomalous dimension matrix and the beta function are expanded in terms of the QCD coupling as
(34)  
(35) 
Let us see the leading logarithmic order contribution. The leading order coefficient of the beta function is with the color number . The anomalous dimension matrix , depending on , is expressed in terms of submatrices as tensorrenormalization1 (); braaten1 (); boyd (); braaten2 (); dineweinbergop (); tensorrenormalization2 (); degrassi (); yang (); dekens1 ()
(36) 
where is the null matrix with arbitrary dimension, and
(37)  
(38)  
(39)  
(40)  
(41) 
where , , , , , and .
Let us show the results for three explicit cases with the initial condition TeV. For the quark EDM, there is no mixing with other operators. If only the quark EDM is dominant at the initial scale, we have
(42) 
for GeV. The running of the quark mass is
(43) 
We have used the quark masses , GeV, and GeV as input pdg ().
If the quark chromoEDM is dominant at , the Wilson coefficients at the hadronic scale mixes with the quark EDM:
(44)  
(45) 
Note that the flavor of the quark is conserved during the running in the leading logarithmic order. It is also to be noted that the running of the quark EDM in Eq. (42) and the chromoEDM in Eq. (45) are additionally affected by the integration of the Higgs boson, heavy electroweak gauge bosons and the top quark, if those particles have CP violating interactions in the BSM physics dekens ().
In the case where only the Weinberg operator is present at , we have
(46)  
(47)  
(48) 
Here the Wilson coefficients and are generated for all relevant quark flavors (). It is also important to note that is sizably suppressed after the running. By comparing Eqs. (45) and (48), we see that the chromoEDM becomes large at the hadronic scale, even if the Wilson coefficients of the Weinberg operator and the chromoEDM are of the same order of magnitude. This is the case for 2HDM, where the contribution from BarrZee type diagrams are the most important.
We also show the evolution of the fourquark operator of the Leftright symmetric model [see Sec. 2.3.3]. The CPodd fourquark coupling of Eq. (17), renormalized at the electroweak scale , is evolved down to the hadronic scale as dekens1 ()
(49)  
Although we obtain several other Wilson coefficients at the hadronic scale, here we focus on , , and , since their corresponding operators are the components of the operator , which is suggested to be the leading contribution of the isovector pionnucleon interaction (see Sec. 3.3). We also note again that the running of the Wilson coefficient begins at the electroweak scale , since the boson has to be integrated out to generate the fourquark operator in Leftright symmetric model. At the scale above , the coupling of the righthanded boson with quarks does not run. In running from to , the leftright fourquark operator mixes with several other fourquark operators, but it is interesting to note that it does not mix with the quark EDM, the quark chromoEDM, and the Weinberg operator.
In the case where several CPodd processes are simultaneously relevant at the TeV scale, the RGE of them down to the hadronic scale is just given by the linear combination of Wilson coefficients seen above. This is because the RGE is calculated only in QCD and the effect of CPodd interactions on the running is negligible.
Finally, let us also briefly present the running of SM contribution, although we do not discuss the detail. The SM contribution at the electroweak scale is expressed by ten fourquark operators buras2 (). The nexttonexttoleading logarithmic order evolution of the SM contribution enhances one of the penguin operator (see Fig. 2) by a factor of about 40 when the scale is varied from to GeV buras2 (); buras1 (); smnuclearedm (). This effect is nontrivial and enhances the SM contribution to the nucleon level CPodd processes from the naive estimation.
Note that the RGE of this subsection is calculated in the perturbative framework, and systematics due to nonperturabative effects may be important at the hadronic scale GeV.
2.4.2 RGE of CPodd eq interaction
We now present the QCD RGE of the CPodd eq interactions. The change of the Wilson coefficients of the CPodd eq interactions depends on the Lorentz structure of the quark bilinears. For the SPS and PSS type ones [terms with and of Eq. (6), respectively], the renormalization is the same as that of the quark mass. We therefore have
(50)  
with TeV. Here we also show the ratio for , for which we have less theoretical uncertainty due to the nonperturbative effect of QCD. This renormalization point is often used in the lattice QCD calculations of nucleon matrix elements.
For the TPT CPodd eq interaction [the term with of Eq. (6)], the renormalization is the same as that of the quark EDM. The renormalization group evolution is then
(51)  
The SPS and SPS type P,CPodd eq interactions with heavy quarks are integrated out at scale below the quark masses, but their effects remain relevant through the P,CPodd electrongluon (eg) interaction. The P,CPodd eg interaction is defined as