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Bounds on Microscopic Physics from P and T

arXiv:hep-ph/9205233v1 26 May 1992UTTG-06-92Bounds on Microscopic Physics from P and TViolation in Atoms and Molecules∗Willy Fischler, Sonia Paban†, and Scott Thomas‡Theory GroupDepartment of PhysicsUniversity of TexasAustin, Texas 78712AbstractAtomic and molecular electric dipole moments are calculated within the mini-mal supersymmetric standard model. Present experiments already provide strongbounds on the combination of phases responsible for the dipole moments of theneutron and closed shell atoms.

For a supersymmetry breaking scale of 100 GeV,these phases must be smaller than ∼10−2.∗Research supported in part by the Robert A. Welch Foundation and NSF Grant PHY9009850.†paban@utaphy.bitnet‡thomas@utaphy.bitnet

The experimental bounds on electric dipole moments (edm’s) of the neutron,atoms, and molecules, are reaching a level of precision that makes them usefulprobes of P and T violation that may originate at scales beyond the standardmodel. Aside from the QCD vacuum angle, the only source of T (or CP) violationin the standard model is the phase in the quark mass matrix.

This generates edm’sorders of magnitude beyond experimental reach [1]. Extensions of the standardmodel however generally have CP violating phases that produce edm’s of the orderof, or larger than, present experimental bounds.

A finite ¯θQCD that saturates thebound from the neutron edm [2], would also produce atomic edm’s of the sameorder.In this paper the main sources for atomic and molecular edm’s in theminimal supersymmetric extension of the standard model will be identified.In order to identify the microscopic sources of P and T violation responsiblefor the edm’s, it is instructive to present the analysis starting from the largestrelevant scale, namely the atomic scale. The next step is then to proceed downto the nuclear scale.

This will enable finally a discussion of the origin of P and Tviolation at the subnuclear level.At the atomic scale an atom or molecule is a composite system of electrons andnuclei. These constituents contribute to the edm via T and P odd electromagneticmoments and local non-electromagnetic interactions.

Some of the electromagneticmoments are suppressed. In the nonrelativistic limit, the contributions from theelectron and nuclear edm’s vanish.This is known in the literature as Schiff’stheorem [3].

The electron edm can contribute though in heavy atoms where theelectrons are relativistic [4, 5]. But in closed shell atoms and molecules with pairedelectron spins, the electron edm contributes only through hyperfine interactionswith the nucleus, and is therefore suppressed [6].

Schiff’s theorem does not applyto higher moments such as the nuclear magnetic quadrupole moment. In additionthere is another T and P odd moment known as Schiff’s moment, which is pro-portional to the offset of electric charge and dipole distributions of the nucleus.

Itleads to a local electromagnetic coupling of the electron to the nucleus [3].1

At the nuclear scale all of the contributions discussed above are contained inthe following HamiltonianH = 12de¯eσµνiγ5eF µν + 12dN ¯Nσµνiγ5NF µν + CenP S,SGF√2(¯eiγ5e)( ¯NN)+CenS,P SGF√2(¯ee)( ¯Niγ5N) + CenT,P TGF√2(¯eσµνe)( ¯Nσµνiγ5N)+QenV,P V (¯eγµe)( ¯N↔∂µ γ5N) + QenP V,V (¯e↔∂µ γ5e)( ¯NγµN)+ CnnS,P SGF√2( ¯NN)( ¯Niγ5N) + QnnV,P V ( ¯NγµN)( ¯N↔∂µ γ5N)(1)where N = (p, n) and isospin violation is ignored. The first and second terms arethe electron and nucleon edm’s respectively.

The remaining terms are local inter-actions among the electrons and nucleons. The nucleon edm and nucleon-nucleoncouplings contribute to the Schiffand magnetic quadrupole moments [7].

Theelectron-nucleon couplings of course contribute to the electron-nucleus coupling[8].Calculations of the contributions of the terms in H to the edm’s consid-ered below may be found in Ref.s [5,7-11] and are summarized in Table 1. Thecontributions from QenV,P V , QenP V,V , and QnnV,P V have not been considered in thesereferences although QenV,P V does contribute directly to the Schiffmoment.

Thesederivative operators have the same nonrelativistic limit as some of the nonderiva-tive operators. This allows a rough estimate of the corresponding contributions tothe edm’s.In this paper the microscopic origin of the operators in (1) will be calculatedwithin the context of the minimal supersymmetric extension of the standard model[12].

In addition to the KM like phases, this model has other CP violating phasesin the superpotential and soft supersymmetry breaking terms. The superpotentialcontributes a phase from the term |µ|eiφµ ˆH1 ˆH2.

The phases in the soft breakingterms arise from the gaugino masses, −12|m˜a|eiφ˜aλ˜aλ˜a + h.c., and the mass param-eters coupling left and right handed squarks (sleptons), −f ∗Lmf|Af|eiφAf fR + h.c.,where mf is the corresponding quark (lepton) mass and Af is defined bymfAf = mf AFhf+ µ∗v1v2! (2)2

AF is the scalar trilinear soft breaking coupling and hf the Yukawa coupling. Theimportance of these new phases is that P and T odd operators can be generatedirrespective of generational mixing.1The supersymmetric origin of the terms in the Hamiltonian (1) will now beconsidered.1.

The electron edm is a chirally violating operator of effective dimension 6. Itarises at the one loop level from electroweak gaugino exchange as shown in Fig.

1.The sum of all such diagrams contains several unknown masses and phases [13].In order to display the order of magnitude, only the photino contribution will beconsidered [14],de = −e α24πme|Ae|m3˜γsin(φAe −φ˜γ)f(x) + · · ·(3)where x = m2˜e/m2˜γ, f(x) is a loop function of order unity, f(1) = 1 [14], and + · · ·represents the contributions from the other electroweak gauginos. Numerically,for m˜e = m˜γ Eq.

(3) gives [14]de ≃−1 × 10−25(100 GeV)2m3˜γ/|Ae|sin(φAe −φ˜γ) e cm + · · ·(4)2. The lowest dimension operators which potentially contribute to the nucleonedm are the light quark electric and chromoelectric dipole moments [15, 16], andWeinberg’s 3-gluon operator [17, 18]O1 = 12 ¯qσµνiγ5qF µνO2 = 12 ¯qσµνiγ5TaqGµνaO3 = 13fabc ˜GaµνGµρb Gνcρwhere ˜Gaµν = 12ǫµνρσGρσaand Ta are the SU(3)C generators.

As for the electronedm the quark dipole moments are effectively dimension-6. The operators O1 andO2 are generated at the one loop level through gluino and electroweak gauginoexchange [16].

The sum of all such diagrams is again a function of several masses1If the hidden sector is Polonyi and the electroweak gaugino masses vanish at tree level, thesupersymmetric phases can be rotated away.3

and phases.To establish the order of magnitude only the gluino and photinoexchange will be consideredC1(µ) = eQq24π mq|Aq| αsm3˜λsin(φAq −φ˜λ)43f(y) + αQ2qm3˜γsin(φAq −φ˜γ)f(z) + · · ·!ζ1(5)C2(µ) = −gs24πmq|Aq| αsm3˜λsin(φAq −φ˜λ)(h(y) + 16f(y)) −αQ2qm3˜γsin(φAq −φ˜γ)f(z) + · · ·!ζ2(6)where Qq is the quark charge, y = m2˜q/m2˜λ, z = m2˜q/m2˜γ, f is the same functionthat appears in Eq. (3), and h(1) = 1 [16].

ζi are renormalization group correc-tions for the evolution of the operators from the supersymmetric scale, M, to thehadronic scale, µ. Using the known anomalous dimensions [19] with αs(M) = .1and αs(µ) = 4π/6 [17] gives ζ1 ≃1.6, and ζ2 ≃3.6.

Note that the photino con-tribution is down by ∼αα−1scompared with the gluino. Barring any cancellationamong the phases, and if all the mass parameters are of the same order, the gluinocontribution will dominate the quark dipole moments.The 3-gluon operator is generated at the two loop level.

The largest contribu-tions come from integrating out the chromoelectric dipole moments of quarks withmass mQ > ΛQCD [18, 19],C3(µ) = αs(mQ)8πC2(mQ)mQζ3(7)Keeping only the charm quark as the dominant contribution gives ζ3 ≃.38.An estimate of the edm requires an evaluation of the matrix elements of theseoperators on the nucleon. There is however no systematic approximation schemeto reliably calculate hadronic matrix elements of this type.

Here, some empiricalrules known as “naive dimensional analysis” that keep track of factors of 4π andmass scales [17, 20] will be used. This is the most uncertain aspect of the entireanalysis.

Even so, there is no physical reason to expect a substantial suppressionor enhancement as compared to estimates given by these rules. Using these rules,the nucleon edm isdN ≃C1(µ) + e4πC2(µ) + e4πΛχC3(µ)(8)4

where Λχ ≃4πfπ is the chiral symmetry breaking scale. Numerically, keeping onlythe gluino contributions with m˜q = m˜λ = m˜γ ≡˜m, and Au = Ad = Ac ≡Aq, Eq.

(8) then gives|dN| ≃(2.2 + 1.1 + .1) (100 GeV)2˜m3/|Aq|sin(φAq −φ˜λ) 10−23 e cm(9)where the the terms on the right side come from O1, O2, and O3 respectively.3. The nucleon-nucleon couplings are just the T odd component of the nuclearforce.

Following Weinberg’s analysis of nuclear forces [21], it is useful to think ofthese as arising from the T odd exchange of low lying mesons and direct contactinteractions. For illustrative purposes only the exchange of the lightest meson,the pion, will be considered; and an estimate of the T odd pion-nucleon coupling,¯gπNN ¯NπN, arising from the microscopic physics will be made.

The leading con-tribution will come from the light quark chromoelectric dipole moment. Using“naive dimensional analysis” the coupling is¯gπNN ≃gπNNΛχ4π C2(µ)(10)where gπNN is the usual pseudoscalar pion-nucleon coupling constant.

Integratingout the pion, the contribution to the nonderivative nucleon-nucleon coupling isCnnS,P S ∼¯gπNNgπNN1mπ2√2GF(11)Numerically, keeping only the gluino contribution with m˜λ = m˜q, Eqs. (6), (10),and (11) giveCnnS,P S ≃.7 (100 GeV)2m3˜q/|Aq|sin(φAq −φ˜λ)(12)The exchange of heavier mesons will also contribute to this coupling and thederivative nucleon-nucleon coupling.

This is not expected to substantially changethe estimates given below based solely on the nonderivative coupling from pionexchange.4. The electron-nucleon couplings in (1) arise from two classes of operators.

Thefirst operators involve an electromagnetic infrared enhancement and are suppressed5

by two powers of a heavy mass. The second class of operators are local at theatomic scale and suppressed by four powers of a heavy mass.The first class of operators arise from the effective nucleon-photon couplings¯NNFµν ˜F µν and ¯Niγ5NFµνF µν.

The two photons couple to electrons to producean effective electron-nucleon interaction.Integrating out the photons gives aninfrared enhancement, cutoffby the electron mass. One way in which such aneffective interaction arises is through the pion pole as shown in Fig.

2, wherethe CP violation occurs in the pion-nucleon coupling.This contributes to theelectron-nucleon pseudoscalar-scalar couplingCenP S,S ∼gπee¯gπNN1m2π√2GF(13)where ¯gπNN is given in (10), gπee is the effective pion-electron coupling computedin Ref. [22]gπee ∼−3α22π2mefπlnΛχme(14)and the approximation that the electron is on shell has been used.

Note that theeffective electron-nucleon interaction is suppressed by only two powers of a heavymass (from the light quark chromoelectric dipole moment). A similar diagram withthe CP violation in the vertex π0FµνF µν, contributes to the scalar-pseudoscalarcoupling.The edm’s in Table 1.are however somewhat less sensitive to thiscoupling.The nucleon-photon couplings can also arise from the following microscopicoperators which are not suppressed by a light quark massO4 = GaµνGµνa Fρσ ˜F ρσO5 = Gaµν ˜Gµνa FρσF ρσThese operators arise at the two loop level.

The dominant contribution to O4comes from integrating out the electric dipole moment of a quark with mass mQ >ΛQCD, as shown in Fig. 3.

This involves an infrared enhancement, cutoffof bythe quark mass. The coefficient is therefore suppressed by only two powers of a6

heavy mass. Similar operators have been considered in Ref.

[23]. Here,C4 ≃egs2QQ256π2mQ3C1(mQ)(15)Using “naive dimensional analysis” thenCenP S,S ≃−6απ meΛχ lnmQme √2GFC4(16)A similar discussion applies to O5 with the chromoelectric dipole moment of heavyquarks.

All these operators scale like (mQM)−2.It should be noted that the infrared enhancement from the photon-electronloop comes from momenta ∼me. Treating the resulting interactions as local issomewhat dubious for heavy atoms for which pe ∼Zαme.

This is not expected tosignificantly alter the estimates of the edm’s given below. Among the first class ofoperators, the estimate in Eq.

(13) turns out to be numerically most important,CenP S,S ∼2 × 10−8(100 GeV)2m3˜q/|Aq|sin(φAq −φ˜λ)(17)The second class of local operators that lead to electron-nucleon couplings in-cludeO6 = ¯eiγ5e¯qqO7 = ¯ee¯qiγ5qO8 = ¯eσµνe¯qσµνiγ5qO9 = ¯eiγ5eGaµνGµνaO10 = ¯eeGaµν ˜GµνaO11 = ¯eγµe¯q↔∂µ γ5qO12 = ¯e↔∂µ γ5e¯qγµqBecause of the chiral properties, all these operators are effectively dimension 8.The operators O6, O7, and O8 are generated, after Fierz reordering, by box dia-grams2 of the type shown in Fig. 4.

Neglecting running corrections, the coefficientsare then related by C6 = C7 = 12C8 ≡C. A typical contribution is of orderC ∼α4πGFmemq|µ|sin2β m3˜γsin(φµ −φ˜γ)(18)where tan β = v2/v1.

With all the supersymmetric mass parameters of order M,this scale like GFM−2.2Tree level Higgs exchange is CP conserving in the minimal supersymmetric standard modeland will not contribute to the electron-nucleon couplings.7

The chiral suppression of a light quark mass is avoided in the operators O9and O10. These can arise from the box diagrams of Fig.

4 with the light quarksreplaced by heavy quarks, Q. The heavy quarks are then integrated out as shownin Fig.

5. In the limit M >> mQ,C9 =XQ−23αs4πC6(mQ)mQC10 =XQ−αs4πC7(mQ)mQ(19)There are no dimension-8 operators without a light quark mass suppression thatgive a tensor-pseudotensor Lorentz structure.

There is however an effective dimension-10 operator, O13 = 13¯eσµνeidabc ˜GaρσGρσb Gµνc , which is generated analogously to O9and O10, but with at least three gluons on the heavy quark loop. With the rules of“naive dimensional analysis” this operator is not much suppressed compared withO9 and O10.

It is therefore included in the estimate below. Using “naive dimen-sional analysis” the contributions of O6 through O10, and O13, to the nonderivativeelectron-nucleon couplings areCenP S,S =√2GFC6(µ) −XQ23αs4πΛχmQC6(mQ)CenS,P S =√2GFC7(µ) −XQαs4πΛχmQC7(mQ)CenT,P T =√2GFC8(µ) −XQ124 gs4π3 ΛχmQ!3C8(mQ)(20)where again running corrections have been neglected.The operators O11 and O12 are generated by box diagrams of the type shownin Fig.

6a,C11 ∼α4πe2Q2qmq|Aq|m5˜γsin(φ˜γ −φAq)C12 ∼α4πe2Q2qme|Ae|m5˜γsin(φ˜γ −φAe)(21)8

There are also contributions that result from a weak interaction between the elec-tron(quark) and the weak-edm of the quark(electron) as shown in Fig. 6b.C11 = ±(1 −4|Qq|sin2θw)(1 −4sin2θw)GF√2C1(µ)eC12 = ±(1 −4|Qq|sin2θw)cotθwGF√2d(W )ee(22)where ± refers to up or down type quarks, d(W )eis the electron edm arising fromwino exchange, and C1(µ) is given above.

These operators scale like either GFM−2or M−4. Using “naive dimensional analysis” the contribution to the derivativeelectron-nucleon couplings in (1) areQenV,P V = C11QenP V,V = C12(23)All of the local operators turn out to be numerically somewhat less important thanthe estimate in Eq.

(13).The atomic and molecular edm’s can now be estimated as functions of themicroscopic parameters. In order to identify the most important effects at theatomic scale, the contributions from the electron edm, nucleon edm, nonderivativenucleon-nucleon coupling, and the largest electron-nucleon coupling (i.e.fromEq.

(13)) will be retained.Since an edm is proportional to spin, the leadingcontributions in open and closed shell atoms are different. For open shell atomswe consider 133Cs and 205Tl since good experimental bounds are available [24, 25].Putting together the results cataloged above with the results of the atomic andnuclear calculations from Table 1.,dCs ≃( −1.2 sin(φAe −φ˜γ) + 2 × 10−3sin(φAq −φ˜λ) + 1 × 10−1sin(φAq −φ˜λ)+ 1 × 10−3sin(φAq −φ˜λ) )100 GeVM210−23 e cm(24)dTl ≃( 6 sin(φAe −φ˜γ) + 2 × 10−4sin(φAq −φ˜λ) + 2 × 10−3sin(φAq −φ˜λ)−1 × 10−2sin(φAq −φ˜λ) )100 GeVM210−23 e cm(25)9

where, for simplicity of notation, all the supersymmetric mass parameters havebeen assumed to be of order M.The quantities on the right hand side ariserespectively from the electron edm, nucleon edm, nonderivative nucleon-nucleoncoupling, and pseudoscalar-scalar electron-nucleon coupling. In both cases, theelectron edm gives the dominant contribution since it is enhanced in heavy openshell atoms.

For 133Cs, with a nuclear quadrupole moment, the nucleon-nucleoncoupling is only a factor ∼10 less important than the electron edm (again assumingall the masses and phases are the same order). The present experimental boundsare [24, 25]3|dCs| < 7.2 × 10−24 e cm|dTl| < 6.6 × 10−24 e cmThe 205Tl result gives a bound on the phases contributing to the electron edm ofsin(φAe −φ˜γ)100 GeVM2< .1For closed shell atoms, good experimental bounds are available for 129Xe and199Hg.

Combining our results with those from Table 1.,dXe ≃( 8 × 10−3sin(φAe −φ˜γ) −7 × 10−2sin(φAq −φ˜λ) + 3 sin(φAq −φ˜λ)+ 2 × 10−4sin(φAq −φ˜λ) )100 GeVM210−26 e cm(26)dHg ≃( −1.2 × 10−2sin(φAe −φ˜γ) −3 × 10−2sin(φAq −φ˜λ) + 4 sin(φAq −φ˜λ)+ 2 × 10−4sin(φAq −φ˜λ) )100 GeVM210−25 e cm(27)The electron edm is here suppressed since the electron spins are paired.Theleading contribution comes from the nucleon-nucleon coupling. The present ex-perimental bounds are [26-29],|dXe| < 1.4 × 10−26 e cm|dHg| < 3 × 10−27 e cm (95% C.L.

)3Unless stated otherwise, the experimental bounds given here are the sum of the reportedmeasurement and experimental error. To date, all measurements are consistent with zero.10

The 199Hgresult gives a bound on the phases contributing to the light quarkchromoelectric dipole moment,sin(φAq −φ˜λ)100 GeVM2< .008It should be noted that, due to the uncertainties in the hadronic matrix elementsand nuclear calculations [7], this bound is much more uncertain than that fromthe electron edm given above.At present, the best bound on a molecular edm is for 205TlF. Again combiningour results with Table 1.,dTlF ≃( −8 × 10−2sin(φAe −φ˜γ) −1.5 × 10−1sin(φAq −φ˜λ) + sin(φAq −φ˜λ)−1 × 10−3sin(φAq −φ˜λ) )100 GeVM210−22 e cm(28)The leading contribution again comes from the nucleon-nucleon coupling since theelectron spins are paired.

The current experimental bound [10, 30]|dTlF| < 4.6 × 10−23 e cmdoes not yet yield a substantial bound on the phases if M ≃100 GeV.Because of the different structure of open and closed shell atoms, the exper-iments bound different microscopic CP violating parameters. Open shell atomsare sensitive to phases in the weak gaugino sector contributing to the electronedm.

Closed shell atoms and molecules with paired electron spin get the leadingcontribution from nuclear effects. This provides a stringent bound on the gluino-squark phases contributing to the light quark chromoelectric dipole moment.

Forcomparison, the neutron edm receives contributions from both the light quarkelectric and chromoelectric dipole moments [16]. The experimental bound [2] of|dn| < 8 × 10−26 e cm, with the estimate (9), givessin(φAq −φ˜λ)100 GeVM2< .003This is of the same order as the bound from 199Hg.

Taken together, the experimen-tal bounds provide information on different combinations of CP violating phases.11

Other sources of CP violation would give rise to different patterns of edm’s [31].For example, a nonzero ¯θQCD would contribute to the edm’s mainly through the Todd pion-nucleon coupling [7, 32]. This would give a definite relation among theedm’s of open and closed shell atoms and the neutron.The prospects for improving the present measurements are encouraging.

Theultimate sensitivity of the current 199Hgexperiment is expected to reach thelevel of 3 × 10−28 e cm [29].New techniques may allow further improvementin sensitivity for open shell atoms [33]. An unexplored area where the phasescontributing to the light quark chromoelectric dipole moment could be measuredis in light atoms.

In this case the electrons are nonrelativistic and the primarycontributions are nuclear. In summary, the experimental study of atomic edm’sprovides a promising probe for exploring physics beyond the standard model.We would like to thank D. Heinzen and V. Kaplunovsky for useful discussions.We would also like to thank N. Fortson for providing us with the unpublishedbound on 199Hg.

This research was supported in part by the Robert A. WelchFoundation and NSF Grant PHY 9009850.12

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.133Cs205Tl129Xe199Hg205TlFd/de120−600−.8 × 10−31.2 × 10−280d/dN7 × 10−47 × 10−52 × 10−5∼10−4.5d(e cm)/CnnS,P S2 × 10−243 × 10−265 × 10−266 × 10−252 × 10−22d(e cm)/CenP S,S7 × 10−19−5 × 10−189 × 10−231 × 10−21−6 × 10−18d(e cm)/CenT,P T9 × 10−215 × 10−215 × 10−21−6 × 10−201 × 10−16d(e cm)/CenS,P S−1 × 10−233 × 10−23−4 × 10−19Table 1. Contributions to electric dipole moments from the Hamiltonian (1)..16

Figure CaptionsFig. 1 A typical contribution to de from photino exchange.Fig.

2 The contribution to an effective electron-nucleon coupling from pion ex-change with the T odd pion-nucleon coupling.Fig. 3 The contribution to the operator GaµνGµνa Fρσ ˜F ρσ from the electric dipolemoment of a heavy quark.

Other diagrams related by gauge invariance arenot shown.Fig. 4 A typical contribution to the electron-quark operators.Fig.

5 The contribution to the electron-gluon operators from the electron-heavyquark operators. Other diagrams related by gauge invariance are not shown.Fig.

6 Typical contributions to the operator ¯e↔∂µ γ5e¯qγµq; (6a) from a box dia-gram, (6b) from Z-exchange and the electron electroweak dipole moment.17

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