PHYSICAL REVIEW D 68, 035008 ͑2003͒

Probing supersymmetry with parity-violating electron scattering

A. Kurylov California Institute of Technology, Pasadena, California 91125, USA

M. J. Ramsey-Musolf California Institute of Technology, Pasadena, California 91125, USA, Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA, and Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195, USA

S. Su California Institute of Technology, Pasadena, California 91125, USA ͑Received 13 March 2003; published 15 August 2003͒ ͑ ͒ e We compute the one-loop supersymmetric SUSY contributions to the weak charges of the electron (QW), p Cs ͑ ͒ proton (QW), and cesium nucleus (QW ) in the minimal supersymmetric standard model MSSM . Such contributions can generate several percent corrections to the corresponding standard model values. The mag- e p nitudes of the SUSY loop corrections to QW and QW are correlated over nearly all of the MSSM parameter Cs space and result in an increase in the magnitudes of these weak charges. In contrast, the effects on QW are considerably smaller and are equally likely to increase or decrease its magnitude. Allowing for R-parity e p violation can lead to opposite sign relative shifts in QW and QW , normalized to the corresponding standard p e model values. A comparison of QW and QW measurements could help distinguish between different SUSY scenarios.

DOI: 10.1103/PhysRevD.68.035008 PACS number͑s͒: 12.60.Jv, 11.30.Er, 12.15.Lk, 13.60.Hb

I. INTRODUCTION ultimately unable to account for the NuTeV ‘‘anomaly,’’ the results of this precision measurement would point to new The search for physics beyond the standard model ͑SM͒ physics. of electroweak and strong interactions is a primary objective In light of this situation, two new measurements involving for particle and nuclear physics. Historically, parity-violating polarized electron scattering have taken on added interest: ͑PV͒ interactions have played an important role in elucidat- PV Mo¨ller ͑ee͒ scattering at SLAC ͓4͔ and elastic, PV ep ing the structure of the electroweak interaction. In the 1970s, scattering at the Jefferson Lab ͑JLab͓͒5͔. In the absence of PV deep inelastic scattering ͑DIS͒ measurements performed new physics, both measurements could be used to determine 2␪ ͉ 2͉Ϸ 2 at the Stanford Linear Accelerator Center ͑SLAC͒ confirmed sin W at the same scale ͓ q 0.03 (GeV/c) ͔—falling be- the SM prediction for the structure of weak neutral current tween the scales relevant to the APV and neutrino DIS 2 interactions ͓1͔. These results were consistent with a value measurements—with comparable precision in each case Ϫ 2 ⌬ 2␪ Ϸ ϫ 4 for the weak mixing angle given by sin ␪ Ϸ1/4, implying a ( sin W 7 10 ). Any significant deviation from the SM W 2␪ tiny V͑electron͒ϫA͑quark͒ neutral current interaction. Sub- prediction for sin W at this scale would provide striking evi- sequent PV measurements—performed at both very low dence for new physics, particularly if both measurements scales using atoms as well as at the Z pole in eϩeϪ report a deviation. On the other hand, agreement would im- annihilation—have been remarkably consistent with the re- ply that the most likely explanation for the neutrino DIS sults of the SLAC DIS measurement ͓1͔. result involves hadron structure effects within the SM. In this paper, we analyze the prospective implications of More recently, the results of cesium atomic parity- the parity-violating electron scattering ͑PVES͒ measure- ͑ ͓͒ ͔ ␯ ¯␯ violation APV 2 and deep inelastic -( -) nucleus scat- ments for supersymmetry ͑SUSY͒. Although no supersym- tering ͓3͔ have been interpreted as determinations of the metric particle has yet been discovered, there exists strong 2␪ scale dependence of sin W . The SM predicts how this quan- theoretical motivation for believing that SUSY is a compo- tity should depend on the momentum transfer squared (q2) nent of the ‘‘new’’ standard model. For example, the exis- of a given process.1 The cesium APV result appears to be tence of low-energy SUSY is a prediction of many string consistent with the SM prediction for q2Ϸ0, whereas the theories; it offers a solution to the hierarchy problem; and it neutrino DIS measurement implies a ϩ3␴ deviation at ͉q2͉ results in coupling unification close to the Planck scale. In ϳ20 (GeV/c)2. If conventional hadron structure effects are addition, if R parity is conserved ͑see below͒, SUSY pro- vides an excellent candidate for cold dark matter, the lightest neutralino ͑see Ref. ͓7͔ for a review͒. The existence of such 1The weak mixing angle and its q2 evolution are renormalization scheme-dependent. Here, we use the modified minimal subtraction (MS) scheme for the SM and the dimensional reduction (DR) 2In practice, the PV ep experiment will actually provide a value 2␪ 2ϭ ͓ ͔ scheme for supersymmetric extensions of the SM. for sin W(q 0), as discussed in Ref. 6 .

0556-2821/2003/68͑3͒/035008͑14͒/$20.0068 035008-1 ©2003 The American Physical Society KURYLOV, RAMSEY-MUSOLF, AND SU PHYSICAL REVIEW D 68, 035008 ͑2003͒ dark matter is required in most cosmological models. In light measurements for all phenomenologically acceptable choices of such arguments, it is clearly of interest to determine what of the MSSM parameters, even if such choices lie outside the insight about SUSY, if any, the new PVES measurements purview of standard SUSY-breaking models. In doing so, we might provide. follow the spirit of Ref. ͓11͔, where a similar analysis of In the simplest version of SUSY—the minimal supersym- SUSY loop effects in ␯ (¯␯)-nucleus scattering was per- metric standard model ͑MSSM͒ with conserved R parity formed. ͓8͔—low-energy precision observables experience SUSY In the case of PV electron scattering, we find that the only via tiny loop effects involving virtual supersymmetric magnitude of SUSY loop effects could be as large as the e p particles. The requirement of baryon minus lepton number proposed experimental uncertainties for the QW and QW (BϪL) conservation leads to conservation of the R-parity measurements ͑8% and 4%, respectively ͓4,5͔͒. Moreover, ϭ Ϫ 2Sϩ3(BϪL) ͑ quantum number, PR ( 1) , where S denotes the relative sign of the effect compared to the SM predic- ϭϩ tion͒ in both cases is correlated—and positive—over nearly spin. Every SM particle has PR 1 while the correspond- ϭ all available SUSY parameter space. To our knowledge, such ing superpartner, whose spin differs by 1/2 unit, has PR Ϫ correlation is specific to the MSSM ͑with R-parity conserva- 1. Conservation of PR implies that every vertex has an ͒ even number of superpartners. Consequently, for processes tion , making it a potential low-energy signature of this new → → physics scenario. We also find that the SUSY loop effects on like ee ee and ep ep, all superpartners must live in Cs loops, which generate corrections—relative to the SM QW , the weak charge of the cesium atom measured in APV, 2 Ϫ3 are much less pronounced. Thus, the present agreement be- amplitude—of order (␣/␲)(M/M˜ ) ϳ10 ͑where M de- Cs tween the experimental value for QW and the SM prediction notes a SM particle mass and M˜ is a superpartner mass͒. does not preclude the presence of relatively large effects in Generally speaking, then, low-energy experiments must the PV electron scattering asymmetries. probe an observable with a precision of few tenths of a per- We also investigate a scenario where PR is not conserved. cent or better in order to discern SUSY loop effects. Low- We find that, in contrast to the PR-conserving SUSY, the energy charged current experiments have already reached relative sign of the effect ͑compared to the SM prediction͒ is e p such levels of precision, and the corresponding implications always negative for QW and can have either sign for QW , of these experiments for the MSSM have been discussed with the positive sign being somewhat more likely than the elsewhere ͓9͔. negative sign. The potential magnitude of the effects are con- In the case of PV ee and elastic ep scattering, the preci- siderably larger than those generated by SUSY loops. In sion needed to probe SUSY loop effects is roughly an order e p principle, then, a comparison of QW and QW can potentially of magnitude less stringent, owing to a fortuitous suppres- establish whether or not R parity is violated within a SUSY sion of the SM PV asymmetries, ALR . At leading order in extension of the SM. Having an answer to this question 2 ϫ q , the A(e) V( f ) contributions to ALR are governed by would have consequences reaching beyond the realm of ac- f QW , the ‘‘weak charge’’ of the target fermion, f. The weak celerator physics. For instance, if PR is violated in PVES, charge of a particle f is defined as the strength of the effec- then the lepton number is not conserved, thereby implying tive A(e)ϫV( f ) interaction: that neutrinos have Majorana masses and making neutrino- less double beta decay possible ͑see, e.g. Ref. ͓12͔͒. R-parity G␮ L ef ϭϪ f ¯␥ ␥ ¯␥␮ ͑ ͒ violation also renders the lightest supersymmetric particle EFF QWe ␮ 5ef f . 1 2ͱ2 unstable, thus eliminating SUSY dark matter, which has sig- nificant implications for cosmology ͓7͔. At tree level in the SM the weak charges of both the electron Our discussion of these points is organized as follows. p ϭϪ e ϭ Ϫ 2␪ After briefly reviewing the minimal supersymmetric standard and the proton are suppressed: QW QW 1 4 sin W Ϸ0.1. One-loop SM electroweak radiative corrections fur- model in Sec. II, we discuss the structure of the one-loop e radiative corrections to Qe,p in Sec. III and the tree-level ther reduce this tiny number, leading to the predictions QW W ϭϪ ͓ ͔ p ϭ ͓ ͔ տ PR-violating contributions in Sec. IV. The analysis of the 0.0449 6,10 and QW 0.0716 6 . The factor of 10 suppression of these couplings in the SM renders them more prospective implications of the parity-violating electron scat- transparent to the possible effects of new physics. Conse- tering measurements for supersymmetry is presented in Sec. quently, experimental precision of order a few percent, rather V. We conclude in Sec. VI. Appendix A lists the counterterms ͑ ͒ than a few tenths of a percent, is needed to probe SUSY loop for the effective PVES Lagrangian of Eq. 1 necessary for corrections. ͑Theoretical uncertainties associated with QCD renormalization of the one-loop radiative corrections to the e,p ͓ ͔ ͒ weak charges. In Appendix B we explicitly prove that gluino corrections to QW are considerably smaller 6,10 . p e,p loops do not contribute to QW . In Appendix C we give com- In analyzing these SUSY loop contributions to QW ,we carry out a model-independent treatment, avoiding the plete expressions for all process-dependent one-loop SUSY choice of a specific mechanism for SUSY-breaking media- corrections to the ep and ee scattering; expressions for process-independent contributions are given in the appen- tion. While most analyses of precision electroweak observ- ͓ ͔ ables have been performed using one or more widely used dixes of Ref. 11 . models for SUSY-breaking mediation, the generic features of II. MSSM PARAMETERS the superpartner spectrum implied by such models may not be consistent with precision data ͓9͔. Consequently, we wish The content of the MSSM has been described in detail to determine the possible impact of SUSY on the two PVES elsewhere ͓8͔, so we review only a few features here. The

035008-2 PROBING SUPERSYMMETRY WITH PARITY-VIOLATING . . . PHYSICAL REVIEW D 68, 035008 ͑2003͒

particle spectrum consists of the SM particles and the corre- sponding superpartners: spin-0 sfermions (˜f , which include sneutrinos ˜␯, charged sleptons ˜l , and up- and down-type squarks ˜u and ˜d), spin-1/2 gluinos (˜g), spin-1/2 mixtures of ˜ 0 ˜ neutral Higgsinos (H1Ϫ2), the B-ino (B), and the neutral ˜ 3 ␹0 W-ino (W ), collectively called neutralinos ( 1Ϫ4), and spin-1/2 mixtures of charged Higgsinos (H˜ Ϯ) and charged ˜ Ϯ ␹Ϯ W-inos (W ), collectively called charginos ( 1,2). In addi- tion, the Higgs sector of the MSSM contains two doublets ͑up and down types, which give mass to the up- and down- ͒ FIG. 1. Types of radiative corrections to parity-violating elec- type fermions, respectively , whose vacuum expectations vu ͑ ͒ ͑ ͒ ␥ ͑ ͒ ϭͱ 2ϩ 2 ␤ tron scattering: a Z boson self-energy, b Z- mixing, c electron and vd are parametrized in terms of v vu vd and tan anapole moment contributions, ͑d͒ vertex corrections, and ͑e͒ box ϭ ͑ ͒ ͑ ͒ vu /vd . Together with the SU 2 L and U 1 Y couplings g graphs. External leg corrections are not explicitly shown. ␣ and gЈ, respectively, v is determined from , M Z , and G␮ , the Fermi constant extracted from the muon lifetime, while III. RADIATIVE CORRECTIONS TO Qf tan ␤ remains a free parameter. The MSSM also introduces a W coupling between the two Higgs doublets characterized by With higher-order corrections included, the weak charge the dimensionful parameter ␮. The complete set Feynman of a fermion f can be written as rules for the MSSM, which take into account SUSY breaking ͓ ͔ f ϭ␳ ͓ f Ϫ ␬ 2␪ ͔ϩ␭ ͑ ͒ and particle mixing, are given in Ref. 13 . QW PV 2I3 4Q f PVsin W f , 2 Degeneracy between SM particles and their superpartners is lifted by the SUSY-breaking Lagrangian, which depends in where I f and Q are, respectively, the weak isospin and the general on 105 additional parameters. These include the 3 f electric charge of the fermion f. The quantities ␳ and ␬ SUSY-breaking Higgs mass parameters, the electroweak PV PV ͑ ͒ are universal in that they do not depend on the fermion f gaugino masses M 1,2 , the gluino mass M˜g , the left- right- ␭ 2 2 under consideration. The correction f , on the other hand, handed sfermion mass parameters M˜ (M˜ ), and left-right f L f R does depend on the fermion species. At tree level, one has 2 ˜ ˜ ␳ ϭ1ϭ␬ and ␭ ϭ0, while at one-loop order, these pa- mixing terms M˜ which mix f L and f R into mass eigen- PV PV f f LR rameters are ˜ states f 1,2 . In our analysis, we take the sfermion mass ma- trices to be diagonal in flavor space to avoid large flavor- ␳ ϭ1ϩ␦␳SMϩ␦␳SUSY, changing neutral currents. We also set all CP-violating PV phases to zero. One expects the magnitude of the SUSY- ␬ ϭ ϩ␦␬SMϩ␦␬SUSY breaking parameters to lie somewhere between the weak PV 1 , scale and ϳ1 TeV. Significantly larger values can reintro- duce the hierarchy problem. ␭ ϭ␭SMϩ␭SUSY ͑ ͒ f f f , 3 Theoretical models for SUSY-breaking mediation provide relations among this large set of soft SUSY-breaking param- where the SUSY contributions to ␳ , ␬ , and ␭ are de- eters, generally resulting in only a few independent param- PV PV f ͓ ͔ noted in the above equation by the corresponding super- eters at the SUSY-breaking or GUT scale 14 . Evolution of script. In general, the corrections ␦␳, ␦␬, etc. depend on q2, the soft parameters down to the weak scale introduces flavor 2 ␬ and in particular, the q dependence of PV defines the scale- and species dependence into the superpartner spectrum due 2␪eff 2 dependence of the weak mixing angle: sin W (q ) to the presence of Yukawa and gauge couplings in the renor- ϭ␬ 2 2␪ 2␪ ͑ ͒ PV(q )sin W , with sin W being evaluated at some refer- malization group RG equations. According to the model- 2 ͑ 2ϭ 2 independent analysis of Ref. ͓9͔, however, generic features ence scale q0 usually q0 M Z). 2␪ ␬ 2 of this spectrum implied by typical SUSY-breaking models The precise definitions of sin W , PV(q ), etc. depend and RG evolution may conflict with the combined con- on one’s choice of renormalization scheme. We evaluate the straints of low-energy charged current data, M , and the SUSY contributions using the modified dimensional reduc- W ͓ ͔ muon anomalous magnetic moment unless one allows for tion renormalization scheme (DR) 16 and denote all quan- non-conservation of PR . In light of this situation, we adopt tities evaluated in this scheme by a hat. In DR, all momenta here a similar model-independent approach and do not im- are extended to dϭ4Ϫ2⑀ dimensions, while the Dirac alge- pose any specific relations among SUSY-breaking param- bra remains four dimensional as required by SUSY invari- eters. To our knowledge, no other model-independent analy- ance. The relevant classes of Feynman diagrams are shown sis of MSSM corrections to PV observables has appeared in in Fig. 1. Note that all gauge boson self-energies contribute ␳ ␬ the literature, nor have the complete set of corrections to only to PV and PV while all non-universal box diagrams as low-energy PV observables been computed previously ͑see, well as vertex and external leg corrections are combined in ␭ e.g. Ref. ͓15͔ for a study within minimal supergravity and f . The counterterms for the effective PVES Lagrangian in gauge mediated models of SUSY breaking͒. Eq. ͑1͒ in the DR scheme are given in Appendix A.

035008-3 KURYLOV, RAMSEY-MUSOLF, AND SU PHYSICAL REVIEW D 68, 035008 ͑2003͒

The Z boson self-energy contribution ͑Fig. 1a͒ simply res- cales the leading order amplitude. Its effect is naturally com- ␦ ˆ ͑ ͒ ␳ bined with the counterterm G␮ from Eq. A2 into PV :

␦Gˆ ␮ ⌸ˆ ͑0͒ ⌸ˆ ͑0͒ ⌸ˆ ͑0͒ ␳ ϭ ϩ ϩ ZZ ϭ Ϫ WW ϩ ZZ Ϫ␦ˆ ␮ PV 1 1 VB , G␮ 2 2 2 M Z M W M Z ͑4͒

where the Z boson self-energy is evaluated at q2ϭ0. This is FIG. 2. Electron anapole moment contributions to the parity- an appropriate approximation in this case because the mo- violating electron-fermion scattering amplitude. mentum transfer in ee and ep scattering will be much smaller than the masses of the particles that appear in the loop graphs. The error is of the order ͉q2͉/M 2 ϳ10Ϫ6, which is In computing the SUSY corrections to the weak charges one Z 2 ␮ must decide which value for sˆ 2 to use. Since ␦sˆ has negligible. The quantity ␦ˆ denotes the sum of electroweak SUSY VB ␬ ˆ 2 vertex, external leg, and box graph corrections to the muon already been absorbed into PV one must determine s from decay amplitude, which must be subtracted when the neutral Eq. ͑6͒ using the SM radiative corrections only. The corre- ␣ ͓ ͔ current ͑NC͒ amplitudes are normalized to G␮ . sponding value extracted using only , G␮ , and M Z is 12 : ͑ ͒ ͑ ͒ ␬ ͑ ͒ The graphs Fig. 1 b ,1 c contribute to PV in Eq. 2 . The expression is sˆ 2ϭ0.23120Ϯ0.00018. ͑8͒ ˆ ⌸ˆ 2͒ ␦ˆ 2 c Z␥͑q s ␬ ϭ1ϩ ϩ4cˆ 2Fe ͑q2͒ϩ , ͑5͒ In order to incorporate constraints from existing precision PV ˆ 2 A ˆ 2 s q s data ͑see Sec. V͒, it is useful to introduce the oblique param- ͓ ͔ e 2 eters S, T, and U 19 : where FA(q ) is the parity-violating electron-photon form factor, which—at q2ϭ0—is known as the anapole moment ͓ ͑ ͔͒ 3 ˆ 2 ˆ 2 ˆ 2Ϫˆ 2 of the electron see Eq. 12 . It should be noted that in the 4s c 2 c s 2 Sϭ Reͭ ⌸ˆ ͑0͒Ϫ⌸ˆ ͑M ͒ϩ ͓⌸ˆ ␥͑M ͒ MSSM one has ⌸ˆ SUSY(q2)ϳq2, so there is no singularity at ␣ˆ 2 ZZ ZZ Z ˆ ˆ Z Z Z␥ M Z cs q2ϭ0 in the above equation ͓17͔. The SM contribution con- New tains a singularity that is canceled by a corresponding singu- 2 Ϫ⌸ˆ ␥͑0͔͒ϩ⌸ˆ ␥␥͑M ͒ͮ , larity in the anapole moment contribution ͑Fig. 1c͒. Since in Z Z the following we consider only the new physics contribu- tions, this issue is irrelevant ͑a complete treatment of the SM New contributions is given in Refs. ͓6,10,18͔͒. The shift ␦sˆ 2 in 1 2sˆ 2 Tϭ ͭ cˆ ͩ ⌸ˆ ͑0͒ϩ ⌸ˆ ␥͑0͒ͪ Ϫ⌸ˆ ͑0͒ͮ , ˆ 2ϭ Ϫ ˆ 2ϵ 2␪ˆ 2 2 ZZ Z WW s 1 c sin W(MZ) arises from its definition in terms of ␣ˆ M cˆ ␣ W , G␮ , and M Z :

␲␣ ˆ 2 ⌸ˆ ͑ ͒Ϫ⌸ˆ ͑ 2 ͒ ⌸ˆ ͑ 2 ͒Ϫ⌸ˆ ͑ ͒ 4s WW 0 WW M W ZZ M Z ZZ 0 sˆ 2cˆ 2ϭ , Uϭ ͭ ϩcˆ 2 ͱ 2͑ Ϫ⌬ˆ ͒ ␣ˆ 2 2 2G␮M Z 1 r M W M Z ⌸ˆ ͑ 2 ͒Ϫ⌸ˆ ͑ ͒ ⌸ˆ ͑ 2 ͒ New ˆ ⌸ˆ ͑ ͒ ⌸ˆ ͑M 2 ͒ Z␥ M Z Z␥ 0 ␥␥ M Z s Z␥ 0 ZZ Z ϩ2cˆ sˆ ϩsˆ 2 ͮ , ͑9͒ ⌬rˆϭ⌸ˆ ␥␥Ј ͑0͒ϩ2 Ϫ 2 2 ˆ 2 2 M Z M Z c M Z M Z ⌸ˆ ͑ ͒ WW 0 ␮ ϩ ϩ␦ˆ , ͑6͒ where the superscript ‘‘New’’ indicates that only the new 2 VB physics contributions to the self-energies are included. Con- M W tributions to gauge-boson self-energies can be expressed en- 2 2 SM SUSY where ⌸ˆ Ј (q )ϵ⌸ˆ ␥␥ /q . Writing ⌬rˆϭ⌬rˆ ϩ⌬rˆ one tirely in terms of the oblique parameters S, T, and U in the ␥␥ ӷ has limit that M NEW M Z . However, since present collider lim- its allow for fairly light superpartners, we do not work in this 2 limit. Consequently, the corrections arising from the photon ␦sˆ cˆ 2 SUSY SUSY ⌸ ␥ ⌸ ϭ ⌬rˆ . ͑7͒ self-energy ( ␥␥) and -Z mixing tensor ( Z␥) contain a sˆ 2 cˆ 2Ϫsˆ 2 residual q2 dependence not embodied by the oblique param- ␳ ␬ eters. Expressing PV and PV in terms of S, T, and U we obtain 3 ␬ ͑ ͒ Note that our definition of PV , Eq. 5 , includes the anapole form factor of the electron Fe (q2), which may be absent in other A ␦␳SUSYϭ␣ˆ Ϫ␦ˆ ␮ definitions appearing in literature ͑see, e.g. Ref. ͓6͔͒. T VB ,

035008-4 PROBING SUPERSYMMETRY WITH PARITY-VIOLATING . . . PHYSICAL REVIEW D 68, 035008 ͑2003͒

FIG. 3. MSSM radiative corrections to the electron neutral current vertex. Radiative correc- tions to the down quark vertex are obtained by ͑ ˜ ͒ replacing charged leptons sleptons, Li with down type quarks ͑squarks͒ and sneutrinos with up type squarks.

ˆ 2 ␣ˆ ˆ ⌸ˆ ͑ 2͒ for all fermions—are given in Ref. ͓11͔. In addition, some c ␮ c Z␥ q ␦␬SUSYϭͩ ͪͩ SϪ␣ˆ Tϩ␦ˆ ͪ ϩ ͫ simplifications occur in the analysis for PVES that do not 2 2 2 2 VB 2 cˆ Ϫsˆ 4sˆ cˆ sˆ q arise in general. In the case of charged current observables, for example, gluino loops can generate substantial correc- 2 SUSY 2 2 ⌸ˆ ␥͑M ͒ cˆ ⌸ˆ ␥␥͑M ͒ ͓ ͔ Ϫ Z Z ͬ ϩͩ ͪͫ Ϫ Z tions 9,11 . In contrast, gluinos decouple entirely from the 2 ˆ 2Ϫˆ 2 2 one-loop MSSM corrections to semi-leptonic neutral current M Z c s M Z PV observables. The proof of this statement is given in Ap- ⌬␣ˆ SUSY pendix B. In addition, the MSSM Higgs contributions to ver- ϩ ͬ ϩ4cˆ 2Fe ͑q2͒SUSY, ͑10͒ ␣ A tex, external leg, and box graph corrections are negligible due to the small, first- and second-generation Yukawa cou- plings. The light Higgs contribution to gauge boson propa- ⌬␣ˆ where is the SUSY contribution to the difference be- gators has already been included via the oblique parameters, tween the fine structure constant and the electromagnetic while the effects of other MSSM Higgs bosons are suffi- ␮ϭ ⌬␣ˆ ϭ ␣ˆ Ϫ␣ SUSY coupling renormalized at M Z : ͓ (M Z) ͔ . ciently small to be neglected ͓20͔. Therefore, we do not dis- As noted above, we take q2→0 in our analysis. cuss gluino and Higgs contributions in the following. The non-universal contribution to the weak charge is de- Anapole moment corrections, corresponding to Fig. 1(c). termined by the sum of the renormalized vertex corrections In the presence of parity-violating interactions, higher-order ˆ f ͑ ͓͒ ͑ ͔͒ ␦ˆ ef contributions can generate the photon-fermion coupling of VV,A in Fig. 1 d see Eq. A6 and the box graphs Box in Fig. 1͑e͓͒see Eq. ͑C15͔͒: the form ͑see, e.g. Ref. ͓21͔͒: f ͑ 2͒ ␭ˆ ϭ f ˆ e ϩ e ˆ f ϩ␦ˆ ef ͑ ͒ FA q ␮ ␮ f gVVA gAVV Box , 11 MPV ϭϪ ¯͑ 2␥ Ϫ ͒␥ ␧ ͑ ͒ i ␥Ϫ f ie 2 f q q” q 5 f ␮ , 12 M Z f ͑ ͒ where gV,A are given in Eq. A3 . Finally, we note that by vector current conservation, ␦Q p W where f is a fermion spinor, ␧␮ is the photon polarization, can be computed directly from the shifts in the up- and and F f (q2) is the anapole moment form factor. down-quark weak charges: ␦Q p ϭ2␦Qu ϩ␦Qd . The analo- A W W W The quantity F f (q2) is, in general, gauge dependent. This gous relation in the SM is modified by non-perturbative A dependence cancels after the anapole moment contribution is strong interactions in the Z␥ box graph ͓6͔. The latter arise combined with other one-loop corrections to the given scat- because the loop contains a massless particle, rendering the tering process ͓21͔. The Feynman diagrams that contribute to corresponding loop integral sensitive to both low and high Fe (q2)SUSY are shown in Fig. 2, and the analytical expres- momentum scales. In contrast, the SUSY radiative correc- A tions are dominated by large loop momenta, and non- sions are presented in Appendix C. perturbative QCD corrections are suppressed by Vertex corrections, corresponding to Fig. 1(d). The rel- (⌳ /M )2Ӷ1. evant diagrams are shown in Figs. 3 and 4. The diagrams in QCD SUSY Fig. 3 cover ee, ed, and eu scattering when the radiative correction is for the projectile side. When the radiative cor- One-loop SUSY Feynman diagrams rection is to be applied to the target side, the diagrams in Fig. Here, we present the SUSY one-loop diagrams that are 3 can also be used for ee and ed scattering. In this case f particular to PVES. Such diagrams correspond to the generic ϭe is the projectile. To obtain the corrections to the down corrections shown in Figs. 1͑c͒,1͑d͒, and 1͑e͒. Contributions quark vertex, the electron can simply be replaced with the corresponding to Figs. 1͑a͒ and 1͑b͒ are universal, and the down quark. The diagrams in Fig. 4 show the radiative cor- relevant diagrams—together with the external leg corrections rections to the target side when the incoming electron inter-

FIG. 4. MSSM radiative corrections to the up ˜ ˜ quark neutral current vertex. Here, Ui(Di) de- notes up͑down͒-type squark.

035008-5 KURYLOV, RAMSEY-MUSOLF, AND SU PHYSICAL REVIEW D 68, 035008 ͑2003͒

FIG. 5. MSSM box graphs that contribute to the electron-electron scattering amplitude. Here, Ј ϭ pi(pi ), i 1,2, is the momentum of the initial ͑final͒ state fermion. Radiative corrections to the electron-down quark scattering are trivially ob- tained by replacing the target with the down quark.

acts with the up quark inside the proton. The explicit expres- from Eq. ͑14͒ of the parity-violating contact four-electron sions for the vertex corrections can be found in Appendix C. interaction. It is straightforward to show that the superpoten- Box corrections, corresponding to Fig. 1(e). These graphs tial in Eq. ͑13͒ can only produce parity-conserving contact ␦ˆ ef ͑ ͒ interactions between identical leptons. generate Box in Eq. 11 . The relevant diagrams are shown in Fig. 5 and Fig. 6. The explicit expressions are given in Contributions from PR-violating interactions to low en- Appendix C. ergy observables can be parametrized in terms of the follow- ing quantities: f IV. R-PARITY VIOLATING CONTRIBUTIONS TO QW ͉␭ ͉2 ijk ⌬ ͑˜f ͒ϭ у0, ͑15͒ When R parity is not conserved, new tree-level contribu- ijk ͱ 2 e,p 4 2G␮M˜f tions to QW appear. The latter are generated by the (B Ϫ L)-violating superpotential: with a similar definition for the primed quantities. In terms of ⌬ ijk , etc., one obtains for the relative shifts in the weak 1 1 ͓ ͔ W ϭ ␭ L L ¯E ϩ␭Ј L Q D¯ ϩ ␭Љ U¯ D¯ D¯ charges 22 : RPV 2 ijk i j k ijk i j k 2 ijk i j k ␦Qe ϩ␮Ј ͑ ͒ W 4 k i LiHu , 13 ϷϪͫ1ϩͩ ͪ ␭ ͬ⌬ ͑˜e ͒, e 1Ϫ4 sin2␪ x 12k R QW W where Li and Qi denote lepton and quark SU(2)L doublet superfields, E , U , and D are singlet superfields and the ␦ p i i i QW 2 ␭ , etc. are a priori unknown couplings. In order to avoid Ϸͩ ͪ ͓Ϫ2␭ ⌬ ͑˜ek ͒ϩ2⌬Ј ͑˜dk ͒ ijk p 1Ϫ4 sin2␪ x 12k R 11k R unacceptably large contributions to the proton decay rate, we QW W set the ⌬Bϭ0 couplings ␭Љ to zero. For simplicity, we also ” ijk Ϫ⌬Ј ͑˜qk ͔͒Ϫ⌬ ͑˜ek ͒, neglect the last term in Eq. ͑13͒. The purely leptonic terms 1k1 L 12k R ␭ ( 12k) contribute to the electron scattering amplitudes via the sˆ 2͑1Ϫsˆ 2͒ 1 normalization of NC amplitudes to G␮ and through the defi- ␭ ϭ Ϸ ͑ ͒ x 0.35. 16 nition of sˆ 2 ͓22͔. The remaining semileptonic, ⌬LϭϮ1 in- 1Ϫ2sˆ 2 1Ϫ⌬rˆ SM teractions (␭Ј ) give direct contributions to the eq scattering ijk ⌬ amplitudes. The latter may be obtained computing the Feyn- As discussed in Sec. V the quantities ijk , etc. are con- man amplitudes in Figs. 7͑b͒,7͑c͒ and preforming a Fierz strained from other precision data. Since they are non- ͑ ͒ e reordering. In this manner one obtains the following effec- negative, Eq. 16 indicates that the relative shift in QW is tive four-fermion Lagrangian: negative semidefinite. On the other hand, the relative shift in p QW can have either sign depending on the relative magni- ͉␭Ј ͉2 ͉␭Ј ͉2 tudes of ⌬ , ⌬Ј , and ⌬Ј . L EFFϭϪ 1k1 ¯ ␥␮ ¯ ␥ ϩ 11k ¯ ␥␮ ¯ ␥ 12k 11k 1k1 RPV 2 dR dReL ␮eL 2 uL uLeL ␮eL 2M˜qk 2M˜dk L R V. ANALYSIS OF THE SUSY CONTRIBUTIONS TO THE WEAK CHARGES ͉␭ ͉2 Ϫ 12k ͓¯␯ ␥␮␮ ¯ ␥ ␯ ϩ ͔ ͑ ͒ 2 ␮L LeL ␮ eL H.c. , 14 In order to evaluate the potential size of SUSY loop cor- 2M˜ek R rections, a set of about 3000 different combinations of SUSY-breaking parameters was generated, chosen randomly ͉ 2͉Ӷ 2 where we have taken q M˜f and have retained only the from a flat distribution in the soft SUSY mass parameters terms relevant for the PVES scattering. Note the absence ͑independent for each generation͒ and in ln tan ␤. The

FIG. 6. MSSM box graphs that contribute to the electron-up quark scattering. The meaning of momentum labels is the same as in Fig. 5.

035008-6 PROBING SUPERSYMMETRY WITH PARITY-VIOLATING . . . PHYSICAL REVIEW D 68, 035008 ͑2003͒

FIG. 7. Tree-level PR-violating contributions to the muon decay ͓plot ͑a͔͒, the eu scat- tering amplitude ͓plot ͑b͔͒, and the ed scattering amplitude ͓plot ͑c͔͒. ⌬ The quantities ijk , etc., are de- fined in Eq. ͑15͒.

͒ ϳ p ϳ former were bounded below by present collider limits and conserved can be as large as 4% (QW) and 8% bounded above by 1000 GeV, corresponding to the O(TeV) e (QW)—roughly the size of the proposed experimental errors naturalness limit. Also, tan ␤ was restricted to lie in the range for the two PVES measurements. Generally speaking, the Ͻ ␤Ͻ 1.4 tan 60. These limits follow from the requirement magnitudes of ␦Qe,p slowly increase with tan ␤ and decrease that the third generation quark Yukawa couplings remain per- W ͑ ͒ ͑ ͒ as SUSY mass parameters are increased. The largest effects turbative small up to the grand unified theory GUT scale. occur when at least one superpartner is relatively light. An Left-right mixing among sfermions was allowed. In order to exception occurs in the presence of significant mass splitting avoid unacceptably large flavor-changing neutral currents, no between sfermions, which may lead to sizable contributions. intergenerational sfermion mixing was permitted. The ranges However, such weak isospin-breaking effects also increase over which the soft SUSY breaking parameters and tan ␤ were scanned are shown in Table I. the magnitude of T, so their impact is bounded by oblique For each combination of parameters, we computed super- parameter constraints. This consideration has been imple- partner masses and mixing angles, which we then used as mented in arriving at Fig. 8. inputs for computing the radiative corrections. Only the pa- The effects of sfermion left-right mixing were studied rameters generating SUSY contributions to the muon anoma- separately. We observe that the presence or the absence of lous magnetic moment consistent with the latest results ͓23͔ the mixing affects the distribution of points, but does not were considered. We also separately evaluated the corre- significantly change the range of possible corrections. For sponding contributions to the oblique parameters. The latter the situation of no left-right mixing, the points are more are tightly constrained from precision electroweak data. We strongly clustered near the origin. Thus, while corrections of rule out any parameter combination leading to values of S the order of several percent are possible in either case, large and T lying outside the present 95% confidence limit contour effects are more likely in the presence of left-right mixing. ␦ e,p ␦␬SUSY for these quantities. We note that this procedure is not en- The shifts QW are dominated by . This feature is e tirely self-consistent, since we have not evaluated non- illustrated for QW in Fig. 9 where the soft SUSY breaking universal MSSM corrections to other precision electroweak parameters are chosen such that the total SUSY correction to observables before extracting oblique parameter constraints. e p QW is about 4%, about a half of its maximum value. For QW As noted in Ref. ͓15͔, where MSSM corrections to Z-pole the situation is similar. We observe that non-universal correc- observables were evaluated using different models for tions involving vertex corrections and wave function renor- SUSY-breaking mediation, non-universal effects can be as malization experience significant cancellations. In addition, large as oblique corrections. Nevertheless, we expect our procedure to yield a reasonable estimate of the oblique pa- rameter constraints. Since S and T do not dominate the low- energy SUSY corrections ͑see below͒, our results depend only gently on the precise allowed ranges for these param- eters. In Fig. 8 we plot the shift in the weak charge of the ␦ p ϭ ␦ u ϩ␦ d proton, QW 2 QW QW , versus the corresponding shift ␦ e in the electron’s weak charge, QW , normalized to the re- ͑ spective SM values. The corrections in the MSSM with PR

TABLE I. Ranges of SUSY parameters scanned. Here, M˜ de- ͉␮͉ notes any of , M 1,2 , or the diagonal sfermion mass parameters i ␮ M˜ . The parameter and M 1,2 can take either sign. The genera- f L,R tion index i runs from 1 to 3.

Parameter Min Max FIG. 8. Relative shifts in electron and proton weak charges due ␤ tan 1.4 60 to SUSY effects. Dots indicate SUSY loop corrections for ϳ3000 M˜ 50 GeV 1000 GeV randomly generated SUSY-breaking parameters. The interior of the 2 i Ϫ106 GeV2 106 GeV2 truncated elliptical region gives possible shifts due to P noncon- (M˜f )LR R serving SUSY interactions ͑95% confidence͒.

035008-7 KURYLOV, RAMSEY-MUSOLF, AND SU PHYSICAL REVIEW D 68, 035008 ͑2003͒

␦ e e FIG. 10. Contributions to QW/QW from various corrections to SUSY SUSY ␦ e e ␦␬ ͓see Eq. ͑10͔͒: total from ␦␬ ͑solid line͒, S parameter FIG. 9. Various contributions to QW/QW : total from SUSY ␮ SUSY SUSY ͑ ͒ ͑ ͒ ␦ˆ ͑ ͒ ␥ loops ͑solid line͒, from ␦␳ ͑dashed line͒, from ␦␬ ͑dash- dotted line , T parameter dashed line , VB dash-dotted line , Z- ͑ ͒ dotted line͒, from the vertex corrections ͑dotted line͒, and from the mixing and the photon self-energy open circles , and the electron ͑ ͒ box graphs ͑open circles͒. The x axis gives the lepton superpartner anapole moment crosses . The soft SUSY parameters are the same mass, chosen to be the same for both left- and right-handed first and as in Fig. 9. ␤ϭ second generation sleptons. For this graph, tan 10, the gaugino e soft mass parameters are 2M ϭM ϭ␮ϭ200 GeV, and masses of not lead to discernible effects in QW but could induce sizable 1 2 p ͓ ͔ the third generation sleptons and of all squarks are 1000 GeV. For shifts in QW 6,24 . ␦ e e As a corollary, we also find that the relative importance of this case, QW/QW is about half of its maximum possible value. The total relative correction is clearly dominated by ␦␬SUSY. SUSY loop corrections to the weak charge of heavy nuclei probed with APV is suppressed. The shift in the nuclear ␦ ϭ ϩ ␦ u ϩ corrections to Qe,p due to shifts in the ␳ parameter are weak charge is given by QW(Z,N) (2Z N) QW (2N W PV ϩ ␦ d ␦ f f Ϫ ˆ 2 Z) QW . Since the sign of QW/QW due to superpartner suppressed by 1 4s . u Ͼ d We find that ␦␬SUSY is nearly always negative, corre- loops is nearly always the same, and since QW 0 and QW Ͻ ␦ u ␦ d 2␪eff 2 0 in the SM, a strong cancellation between QW and QW sponding to a reduction in the value of sin W (q ) ϭ␬ 2 2␪ occurs in heavy nuclei. This cancellation implies that the PV(q )sin W for the parity-violating electron scattering ␦ experiments ͓see Eq. ͑2͔͒. In this case, the degree of cancel- magnitude of QW(Z,N)/QW(Z,N) is generally less than 3 ͑ ͒ about 0.2% for cesium and is equally likely to have either lation between 2I f and Q f terms in Eq. 2 is reduced, yield- f sign. Since the presently quoted uncertainty for the cesium ing an increased magnitude of QW . Since this effect is iden- ͓ ͔ e p ␦␬ nuclear weak charge is about 0.6% 25 , cesium APV does tical for both QW and QW , the dominant effect of not substantially constrain the SUSY parameter space. produces a linear correlation between the two weak charges. Equally as important, the present agreement of QCs with the Some scatter around this line arises from non-universal ef- W SM prediction does not preclude significant shifts in Qe,p ␭ˆ ͑ ͒ W fects in f see Fig. 8 . arising from SUSY. The situation is rather different, for ex- ␦␬SUSY As illustrated in Fig. 10, within itself, contribu- ample, in the E ZЈ scenario, where sizable shifts in Qe,p tions from the various terms in Eq. ͑10͒ have comparable 6 W would also imply observable deviations of QCs from the SM importance, with some degree of cancellation occurring be- W prediction. tween the effects of S and T. Thus, the oblique parameter The prospective ‘‘diagnostic power’’ of the two PVES approximation gives a rather poor description of the MSSM measurements is further increased when one relaxes the as- ␦ˆ ␮ effects on the weak charges. In particular the quantity VB in sumption of P conservation. Doing so leads to the tree-level ͑ ͒ ␦␬SUSY R Eq. 10 makes a significant contribution to . corrections to the weak charges shown in Eq. ͑16͒. The quan- As evident from Fig. 8, the relative sign of the corrections tities ⌬ , etc. in Eqs. ͑15͒ and ͑16͒ are constrained from the p e ijk to both QW and QW—normalized to the corresponding SM existing precision data ͓22͔. A summary of the existing values—is nearly always the same and nearly always posi- constraints—including the latest theoretical inputs into the p Ͼ e Ͻ Cs tive. Since QW 0(QW 0) in the SM, SUSY loop correc- extraction of Q from experiment ͓25͔—is given in Table II ␦ p Ͼ ␦ e Ͻ W tions give QW 0(QW 0). This correlation is signifi- of Ref. ͓11͔, which we partially reproduce here in Table II. cant, since the effects of other new physics scenarios can We list the PR-violating contribution to four relevant preci- display different signatures. For example, for the general sion observables: superallowed nuclear ␤ decay that con- ͉ ͉ ͓ ͔ class of theories based on E6 gauge group, with neutral strains Vud 26 , atomic PV measurements of the cesium Շ p Cs ͓ ͔ ␮ ␲ ͓ ͔ gauge bosons having mass 1000 GeV, the effects on QW weak charge QW 2 , the e/ ratio Re/␮ in l2 decays 27 , e ␦ e,p e,p and QW also correlate, but QW /QW can have either sign in and a comparison of the Fermi constant G␮ with the appro- ͓ ͔ ␣ 2␪ ͓ ͔ this case 6,24 . In contrast, leptoquark interactions would priate combination of , M Z , and sin W 28 . The values of

035008-8 PROBING SUPERSYMMETRY WITH PARITY-VIOLATING . . . PHYSICAL REVIEW D 68, 035008 ͑2003͒

␦͉ ͉2 ͉ ͉2 ␦ Cs Cs ␦ ␦ ␦ p p TABLE II. PR-violating contributions to Vud / Vud , QW /QW , Re/␮ , G␮ /G␮ , QW/QW , and ␦ e e ⌬Ј ⌬ QW/QW . Columns give the coefficients of the various corrections from ijk and 12k to the different ͑ p,e quantities. The last column gives the experimentally measured value of the corresponding quantity for QW , only the proposed experimental uncertainties are shown͒.

Quantity ⌬Ј ˜ k ⌬Ј ˜ k ⌬ ˜ k ⌬Ј ˜ k Value 11k(dR) 1k1(qL) 12k(eR) 21k(dR) ␦͉ ͉2 ͉ ͉2 Ϫ Ϫ Ϯ Vud / Vud 20 200.0029 0.0014 ␦ Cs Cs Ϫ Ϫ Ϯ QW /QW 4.82 5.41 0.05 0 0.0040 0.0066 ␦ Ϫ Ϫ Ϯ Re/␮ 200 2 0.0042 0.0033 ␦G␮ /G␮ 0 0 1 0 0.00025Ϯ0.001875 ␦ p p Ϫ Ϫ Ϯ QW/QW 55.9 27.9 18.7 0 0.040 ␦ e e Ϫ Ϯ QW/QW 0029.8 0 0.089 the experimental constraints on those quantities are given in SUSY-breaking mediation, as we have undertaken a model- the last column. We also list the PR-violating contributions to independent analysis in this study. We also find that the im- ␦Q p /Q p and ␦Qe /Qe , along with proposed experimental pact of SUSY radiative corrections on the cesium weak W W W W Cs uncertainties. charge are quite small, so that the present agreement of QW ␦ p p with the SM does not rule out potentially observable effects The 95% C.L. region allowed by this fit in the QW/QW ␦ e e in PVES. vs QW/QW plane is shown by the closed curve in Fig. 8. e p ⌬ ˜ ⌬Ј ˜ у ͓ In contrast, the effects on QW and QW induced by new Note that the sign requirements ijk( f ), ijk( f ) 0 see Eq. tree-level, P violating SUSY interactions display a different ͑ ͔͒ R 15 truncate the initially elliptical curve to the shape shown behavior. Given the constraints from other precision elec- in the figure. We observe that the prospective effects of PR troweak observables, such as the Fermi constant, first row non-conservation are quite distinct from SUSY loops. The Cs CKM unitarity, and QW , one would expect PR violation to value of ␦Qe /Qe is never positive in contrast to the situa- e W W cause a decrease in the size of QW . On the other hand, the tion for SUSY loop effects, whereas ␦Q p /Q p can have ei- p W W magnitude of QW can change either way, with an increase ther sign. Note, however, that the area enclosed by the curve being more likely. Moreover, the size of the P violating ␦ p p у R corresponding to QW/QW 0 is larger than the area corre- corrections could be even larger than those induced by SUSY ␦ p p Ͻ ␦ p p p sponding to QW/QW 0, implying that QW/QW is more loops, particularly in the case of QW . Should measurements likely to be positive. In addition, the magnitude of the of the weak charges be consistent with this signature of PR PR-violating effects can be roughly twice as large as the violation, they could have important implications for the na- e,p ture of cold dark matter ͑it would not be supersymmetric͒ possible magnitude of SUSY loop effects for both QW . Thus, a comparison of results for the two parity-violating and the nature of neutrinos ͑they would be Majorana fermi- ͒ electron scattering experiments could help determine ons . whether this extension of the MSSM is to be favored over From either standpoint, should the PVES measurements other new physics scenarios ͑see also Ref. ͓6͔͒. deviate significantly from the SM predictions, one may be able to draw interesting conclusions about the character of SUSY. But what if both measurements turn out to be consis- VI. CONCLUSIONS tent with the SM? In this case, one would add further con- straints to the possibility of PR violation, but only marginally A new generation of precise, PVES experiments are constrain the MSSM parameter space based on possible loop p e poised to probe a variety of scenarios for physics beyond the effects. In the latter case, the impact on both QW and QW is ͓ ͔ ␦ 2␪eff 2 SUSY SM 4–6,10 . The sensitivity of these measurements to new dominated by sin W (q ) . Although the projected, physics is enhanced because the SM values for the electron combined statistics of the two measurements would make and proton weak charges are suppressed and because theo- them more sensitive to SUSY radiative corrections than ei- retical uncertainties in the SM predictions are sufficiently ther measurement would be independently, additional preci- small ͓6,10͔. Here, we have studied the ability of these mea- sion would be advantageous. In this respect, a possible future e surements to shed new light on supersymmetric extensions measurement of QW with a factor of two better precision of the SM. We have observed that in a PR-conserving ver- than anticipated at SLAC would significantly enhance the sion of the MSSM, the effects of SUSY loop corrections to ability of PVES to shed new light on SUSY.4 the electron and proton weak charges are highly correlated and have the same relative sign ͑positive͒ compared to the ACKNOWLEDGMENTS SM prediction over nearly all the available MSSM parameter We thank R. Carlini, J. Erler, R. D. McKeown, and M. space. This correlation arises because the corrections are Wise for useful discussions. This work is supported in part 2␪eff 2 dominated by the SUSY loop contributions to sin W (q )—a result that would not have been obvious in the absence of an explicit calculation. Moreover, the appearance of this corre- 4We thank D. Mack, P. Reimer, and P. Souder for sharing with us lation does not result from the adoption of any model for the possibility of such a future measurement at the Jefferson Lab.

035008-9 KURYLOV, RAMSEY-MUSOLF, AND SU PHYSICAL REVIEW D 68, 035008 ͑2003͒ under U.S. Department of Energy contract No. DE-FG03- ␦Aˆ ϭϪge ␦Zˆ e ϩge ␦Zˆ e , 02ER41215 ͑A.K. and M.J.R.-M.͒, No. DE-FG03- e V A A V 00ER41132 ͑M.J.R.-M.͒, and No. DE-FG03-92-ER-40701 ␦Vˆ ϭg f ␦Zˆ f Ϫg f ␦Zˆ f , ͑S.S.͒. A.K. and M.J.R.-M. are supported by the National f V V A A Science Foundation under grant No. PHY00-71856. 1 ␦Zˆ e, f ϭ ͑␦Zˆ e, f Ϫ␦Zˆ e, f ͒, A 2 L R APPENDIX A: COUNTERTERMS FOR THE EFFECTIVE PVES LAGRANGIAN 1 ␦Zˆ e, f ϭ ͑␦Zˆ e, f ϩ␦Zˆ e, f ͒, ͑A4͒ The ‘‘bare’’ effective Lagrangian for the forward angle V 2 L R PVES scattering has the form: ␦ ˆ e, f ␦ ˆ e, f 0 where ZL and ZR are the field strength renormalization G␮ L ϭϪ f 0 ␮0ϫ 0 constants for left- and right-handed fermions, respectively. ef QW Ae V␮ f , 2ͱ2 One can write the one-loop correction to the NC vertex as Ϫ͑ ͒␦ ˆ f ϭϪ͑ ͒¯␥ ͑ ˆ f ϩ␥ ˆ f ͒ ͑ ͒ 0 2 ϩ␦ ˆ ig/4c V␮ ig/4c f ␮ GV 5GA f , A5 G␮ g0 G␮ G␮ ϭ ϭ , ͱ 8͑M 0 ͒2 ͱ where only the contributions that are not suppressed by pow- 2 W 2 ͱ͉ 2͉ ers of either the momentum transfer ( q /M SUSY) or the ˆ e Q f 0ϭ2I f Ϫ4Q s2ϭ2I f Ϫ4Q ͑sˆ 2ϩ␦sˆ 2͒, fermion mass (m f /M SUSY) are shown. The quantities GA W 3 f 0 3 f ˆ f and GV represent rescaling of the vertices by the one-loop ␦ ˆ radiative corrections. They must be combined with the ap- ␮ ␮ Ae A 0ϭ͑¯e␥␮␥ e͒ ϵA ͩ 1ϩ ͪ , propriate counterterms from Eq. ͑A4͒ to obtain the renormal- e 5 0 e f QW ized corrections:

␦ ˆ ˆ e ϭ ˆ e ϩ␦ ˆ ϭ ˆ e Ϫ e ␦ ˆ e ϩ e ␦ ˆ e 0 V f VA GA Ae GA gV ZA gA ZV , V ϭ͑¯f ␥␮ f ͒ ϵV␮ ͩ 1ϩ ͪ , ͑A1͒ ␮ f 0 f Q f W ˆ f ϭ ˆ f ϩ␦ ˆ ϭ ˆ f ϩ f ␦ ˆ f Ϫ f ␦ ˆ f ͑ ͒ VV GV V f GV gV ZV gA ZA . A6 where the bare quantities are indexed by ‘‘0.’’ Unless other- wise indicated, all higher-order contributions include both APPENDIX B: DECOUPLING OF GLUINOS FROM THE f the SM and the SUSY pieces. The quantity 1/QW in the WEAK CHARGE OF QUARKS parentheses in the last two lines of the above equation is explicitly factored out to make the definitions in Eq. ͑A4͒ It is sufficient to demonstrate the decoupling for one of ͑ ͒ below more convenient. the quark flavors e.g. the up quark since for other flavors the proof is identical. Consider the renormalized vector neu- The counterterm ␦Gˆ ␮ is entirely determined by the muon tral current vertex for the up quark ͓see Eq. ͑A6͔͒: lifetime. It can be taken from Eq. ͑62͒ in Ref. ͓17͔: Vˆ u ϭGˆ u ϩgu ␦Zˆ u Ϫgu ␦Zˆ u , ␦Gˆ ␮ ⌸ˆ ͑0͒ V V V V A A ϭϪ WW Ϫ␦ˆ ␮ ͑ ͒ VB , A2 G␮ 2 u u M W ␦Zˆ ϩ␦Zˆ ␦ ˆ u ϭ L R ZV , ⌸ˆ 2 ␦ˆ ␮ 2 where WW(q ) is the W boson self-energy and VB is the sum of the vertex and box corrections to the muon decay ␦ ˆ u Ϫ␦ ˆ u ZL ZR amplitude. ␦Zˆ u ϭ . ͑B1͒ We use the following convention for the Z-fermion inter- A 2 action: ˆ u The gluino contributions to GV are given by the graph shown g f f ␮ in Fig. 4͑b͒, with the neutralino replaced by the gluino. By V ϭϪ ¯f ␥␮͑g ϩg ␥ ͒fZ , Zf 4c V A 5 using Eq. ͑C8͒ below with the appropriate coupling constants it is straightforward to show that g f ϭ2I f Ϫ4Q s2, V 3 f ␣ 4 S 4 Gˆ u ͑Gluino͒ϭϪ ͚ ͩ ͚ Z*IiZIjϪ sˆ 2␦ ͪ ͓g*ujgui f ϭϪ f ͑ ͒ V ␲ U U ij GL GL gA 2I3 . A3 3 4 i, j I 3 uj ui ϩg* g ͔V ͑M˜ ,m˜ ,m˜ ͒, ͑B2͒ In this convention, the counterterms for the vector and the GR GR 2 g Ui U j axial vector currents can be read off from Ref. ͓17͔:5 ␣ where S is the strong coupling constant, M˜g is the gluino ͑ ͒ mass, V2(M,m1 ,m2) is defined in Eq. C1 , the couplings 5 ͓ ͔ f Ij ͓ ͔ Note that Ref. 17 has the opposite sign convention for gA . ZU are defined in the appendixes of Ref. 11 , and

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ui ϭϪͱ 1i ␣ gGL 2ZU* , u 4 S ␦Zˆ ͑Gluino͒ϭ ͓V ͑M˜ ,m˜ ,m˜ ͒ V 3 4␲ 2 g U1 U1 gui ϭͱ2Z*4i . ͑B3͒ GR U ϩV ͑M˜ ,m˜ ,m˜ ͔͒, 2 g U4 U4 In this work, no flavor mixing in the squark sector is al- 1i 4i ␣ lowed. Therefore, Z ,Z Þ0 only if iϭ1,4. Since Z is 4 S U U U ␦Zˆ u ͑Gluino͒ϭ ͑1Ϫ2͉Z14͉2͒ unitary we find A 3 4␲ U ϫ ͒ uj ui ϩ uj ui ϭ ͑ 1 j 1iϩ 4 j 4i͒ϭ ␦ ͓V2͑M˜g ,mU˜ ,mU˜ gGL* gGL gGR* gGR 2 ZU ZU* ZU ZU* 2 ij 1 1 ͑B4͒ ϪV ͑M˜ ,m˜ ,m˜ ͔͒. ͑B9͒ 2 g U4 U4 for i, jϭ1,4. Finally, After substitution of Eqs. ͑B5͒ and ͑B9͒ into Eq. ͑B1͒ the 4 ␣ gluino corrections to the vector neutral current vertex of the ˆ u ͑ ͒ϭϪ S ͩ ͉ 1i͉2 GV Gluino ͚ ZU up quark cancel exactly. Therefore, gluino loops do not 3 2␲ iϭ1,4 renormalize the weak charge of the up quark. 4 2 Ϫ s ͪ V ͑M˜ ,m˜ ,m˜ ͒ 3 2 g Ui Ui APPENDIX C: COMPLETE EXPRESSIONS FOR FEYNMAN DIAGRAMS 4 ␣ gu ϭϪ S ͭ V͓ ͑ ͒ In this appendix we list analytical expressions for all V M˜ ,m˜ ,m˜ 3 2␲ 2 2 g U1 U1 SUSY one-loop vertex and box Feynman diagrams that con- u tribute to PV electron scattering. The complete expressions gA ͑ ͒ ϩV ͑M˜ ,m˜ ,m˜ ͔͒Ϫ ͑1 for remaining diagrams see Sec. III as well as the Feynman 2 g U4 U4 2 rules are given in the appendixes of Ref. ͓11͔. We use the 14 2 capitalized letters I and J to denote the family index for Ϫ2͉Z ͉ ͓͒V ͑M˜ ,m˜ ,m˜ ͒ U 2 g U1 U1 quarks and leptons (I,Jϭ1,...,3), small letters i and j to denote the index for squarks and sleptons (i, jϭ1,...,6ex- Ϫ ͑ ͔͒ͮ ͑ ͒ cept for sneutrino, when i, jϭ1,...,3),and small letters p V2 M˜g ,mU˜ ,mU˜ , B5 4 4 and n to denote the index for the neutralinos (p,n ϭ1,...,4)andcharginos (p,nϭ1,2). where the closure property Eq. ͑B4͒ was used together with

8 1. Vertex corrections gu ϭ2͑IuϪ2Qusˆ 2͒ϭ1Ϫ sˆ 2, V 3 3 The Feynman diagrams are shown in Figs. 3 and 4. Let us start with the corrections to the eϪeϪZ vertex. The loop gu ϭϪ2IuϭϪ1. ͑B6͒ integral functions V1(m1 ,m2 ,m3) and V2(m1 ,m2 ,m3) are A 3 defined as

On the other hand, the gluino-induced wave function renor- 1 1 y ͑ ͒ϭ ͵ ͵ malization constants of the up quark have the form V1 M,m1 ,m2 dx dy ͒ , 0 0 D3͑M,m1 ,m2 4 ␣ u S ui 2 ui 2 ␦Zˆ ͑Gluino͒ϭ ͚ ͉͑g ͉ ϩ͉g ͉ ͒F ͑m˜ ,M˜,0͒, 1 1 V ␲ GL GR 1 Ui g 3 8 i ͑ ͒ϭ ͵ ͵ ͓ ͑ ͒ ␮2͔ V2 M,m1 ,m2 dx dyy ln D3 M,m1 ,m2 / , 0 0 ␣ 4 S ␦ ˆ u ͑ ͒ϭ ͉͑ ui ͉2Ϫ͉ ui ͉2͒ ͑ ͒ 2 2 ZA Gluino ͚ gGL gGR F1 mU˜ ,M˜g,0 , D ͑M,m ,m ͒ϭ͑1Ϫy͒M 2ϩy͓͑1Ϫx͒m ϩxm ͔, ͑C1͒ 3 8␲ i i 3 1 2 1 2 ͑B7͒ where ␮ is the renormalization scale. Explicitly where F1(m1 ,m2 ,m3) is given by m2 2 1 1 m1ln 2 ͑ ͒ϭ ͵ ͕͓ 2ϩ͑ Ϫ ͒ 2Ϫ ͑ M F1 m1 ,m2 ,m3 x ln xm1 1 x m2 x 1 V ͑M,m ,m ͒ϭ 0 1 1 2 ͑ 2Ϫ 2͒͑ 2Ϫ 2͒ M m1 m2 m1 Ϫ ͒ 2͔ ␮2͖ ͑ ͒ x m3 / . B8 m2 2 2 m2ln 2 ͑ ͒ M Note that according to Eq. C1 , F1(m1 ,m2,0) ϩ , ϵ ͑ ͒ ͑ ͒ ͑M 2Ϫm2͒͑m2Ϫm2͒ V2(m2 ,m1 ,m1). Using Eqs. B3 and B4 we find 2 1 2

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1 ␣ ͑ ͒ϭ 2Ϫ ␦Vˆ u(a)ϭ ͚ ͩ ͚ Z*IiZIj ͫ ␮ V2 M,m1 ,m2 2lnM 3 D D 4 2␲ i, j,p I 4 2 2 2m1 m1 2 ujp uip ϩ Ϫ sˆ ␦ ͪ V ͑m␹ϩ,m˜ ,m˜ ͒¯u␥␮͓g* g Pˆ ln ij 2 D D L L L ͑ 2Ϫ 2͒͑ 2Ϫ 2͒ 2 3 p i j M m1 m2 m1 M 2m4 m2 ϩg*ujpguipPˆ ͔u, ͑C7͒ ϩ 2 2 R R R ͑ 2Ϫ 2͒͑ 2Ϫ 2͒ ln 2 M m2 m1 m2 M ␣ u(b) Ii Ij 2 ␦Vˆ ϭϪ ͚ ͩ ͚ Z* Z Ϫ ␮ ͬ ͑ ͒ ␮ U U 2ln . C2 2␲ i, j,p I 4 2 ujp uip ͓ ˆ ϭ Ϫ␥ ˆ ϭ ϩ␥ Ϫ sˆ ␦ ͪ V ͑m␹0,m˜ ,m˜ ͒¯u␥␮͓g* g Pˆ We have P (1 )/2, P (1 )/2] ij 2 U U 0L 0L L L 5 R 5 3 p i j ␣ e(a) Ii Ij ϩ ujp uip ˆ ͔ ͑ ͒ ␦Vˆ ϭϪ ͚ ͩ ͚ Z* Z g0*R g0R PR u, C8 ␮ ␯ ␯ 2␲ i, j,p I

2 ejp eip ␣ Ϫ2Q␯sˆ ␦ ͪ V ͑m␹ϩ,m˜␯ ,m˜␯ ͒¯e␥␮͓g* g Pˆ u(c) LЈ uin uip ij 2 L L L ␦ ˆ ϭϪ ¯ ␥ ͕͓ ˆ p i j V␮ ͚ u ␮ OnpgL* gL PL 2␲ i,p,n ϩ ejp eip ˆ ͔ ͑ ͒ Ј gR* gR PR e. C3 ϩ R uin uip ˆ ϩ ϩ ϩ ϩ͒ O g* g PR͔2m␹ m␹ V1͑mD˜ ,m␹ ,m␹ np R R p n i p n ij Note that Q␯ϭ0 and Z␯ is a unitary 3ϫ3 matrix. Therefore, Ј Ј Ϫ͓OR g*uinguipPˆ ϩOL g*uinguipPˆ ͔ ͚ Ii IjϪ ˆ 2␦ ϭ␦ np L L L np R R R one identically has IZ␯* Z␯ 2Q␯s ij ij . The explicit ϫ ϩ ϩ ϩ͒ ͑ ͒ form is kept so that the down quark neutral current vertex ͓1 2V2͑mD˜ ,m␹ ,m␹ ͔͖u, C9 → ͑ i p n may be easily obtained by the replacement e d with ZL → ␯→ ZD) and u. ␣ ␦ ˆ u(d)ϭϪ ¯ ␥ ͕͓ LЉ *uin uip ˆ ␣ V␮ ␲ ͚ u ␮ Onpg0L g0L PL ␦Vˆ e(b)ϭ ͚ ͩ ͚ Z*IiZIj 2 p,n,i ␮ L L 2␲ i, j,p I ϩ RЉ uin uip ˆ ͒ O g* g PR͔2m␹0m␹0V1͑mU˜ ,m␹0,m␹0 np 0R 0R p n i p n 2 ejp eip ϩ2Q sˆ ␦ ͪ V ͑m␹0,m˜ ,m˜ ͒¯e␥␮͓g* g Pˆ e ij 2 L L 0L 0L L p i j Ϫ͓ RЉ uin uip ˆ ϩ LЉ uin uip ˆ ͔ Onpg0*L g0L PL Onpg0*R g0R PR ϩ ejp eip ˆ ͑ ͒ ϫ ϩ ͒ ͑ ͒ g* g PR͔e, C4 ͓1 2V2͑mU˜ ,m␹0,m␹0 ͔͖u. C10 0R 0R i p n ␣ ␦ ˆ e(c)ϭ ¯␥ ͕͓ RЈ ein eip ˆ V␮ ͚ e ␮ O pngL* gL PL 2. Anapole moment corrections 2␲ i,p,n ͓ ͔ Ј Using formulas in the Appendix of Ref. 11 we find for ϩ L ein eip ˆ ϩ ϩ ϩ ϩ͒ e O g* g PR͔2m␹ m␹ V1͑m˜␯ ,m␹ ,m␹ pn R R p n i p n FA(0) of the electron: Ϫ͓ LЈ ein eip ˆ ϩ RЈ ein eip ˆ ͔ Fe ͑0͒ϭF(a)͑0͒ϩF(b)͑0͒, O pngL* gL PL O pngR* gR PR A A A

ϫ ϩ ϩ ϩ͒ ͑ ͒ ͓1 2V2͑m˜␯ ,m␹ ,m␹ ͔͖e, C5 i p n ␣M 2 (a)͑ ͒ϭϪ Z ͉ eip͉2 FA 0 ͚ gL ␣ 48␲ i,p ␦ ˆ e(d)ϭϪ ¯␥ ͕͓ LЉ *ein eip ˆ V␮ ␲ ͚ e ␮ Onpg0L g0L PL 2 p,n,i 1 x2͑xϪ3͒dx ϫ ͵ ͑ ͒ 2 2 , C11 ϩ RЉ ein eip ˆ ͒ 0 ͑1Ϫx͒m ϩxm ϩ O g* g PR͔2m␹0m␹0V1͑m˜L ,m␹0,m␹0 ˜␯ ␹ np 0R 0R p n i p n i p Љ Љ Ϫ͓OR g*eingeipPˆ ϩOL g*eingeipPˆ ͔ np 0L 0L L np 0R 0R R ␣ 2 M Z ϫ ϩ ͒ ͑ ͒ (b)͑ ͒ϭ ͉͑ eip͉2Ϫ͉ eip͉2͒ ͓1 2V2͑m˜L ,m␹0,m␹0 ͔͖e. C6 FA 0 ͚ g0L g0R i p n 48␲ i,p

The vector and axial vector pieces can be readily read off 1 x3dx ϫ ͵ ͑ ͒ from the above formulas. The radiative corrections to the up 2 2 . C12 0 ͑ Ϫ ͒ ϩ 1 x m␹0 xm˜L quark neutral current vertex are as follows ͑see Fig. 4͒: p i

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3. Box graphs ͒ B1͑M 1 ,M 2 ,m1 ,m2 Let us introduce the following notation: 1 1 1 z͑1Ϫz͒ ϭ ͵ dx͵ dy͵ dz , ϭ¯͑ Ј͒␥ ͑ Ϫ␥ ͒ ͑ ͒ 0 0 0 D ͑M ,M ,m ,m ͒ L fi f pi ␮ 1 5 f pi , 4 1 2 1 2

ϭ¯͑ Ј͒␥ ͑ ϩ␥ ͒ ͑ ͒ ͑ ͒ ͑ ͒ R fi f pi ␮ 1 5 f pi . C13 B2 M 1 ,M 2 ,m1 ,m2 As explicitly shown below, each box graph has the following 1 1 1 z͑1Ϫz͒ ϭ ͵ dx͵ dy͵ dz , structure: 2͑ ͒ 0 0 0 D4 M 1 ,M 2 ,m1 ,m2

G␮ ef ϭ ͑ ef ϫ ϩ ef ϫ ϩ ef ϫ ͑ ͒ M Box i A L f 2 Le1 B R f 2 Re1 C R f 2 Le1 D4 M 1 ,M 2 ,m1 ,m2 ͱ2 ϭ ͓͑ Ϫ ͒ 2ϩ 2͔ϩ͑ Ϫ ͓͒͑ 2ϩ͑ Ϫ ͒ 2͔ z 1 x M 1 xM2 1 z ym1 1 y m2 . ϩDefL ϫR ͒. ͑C14͒ f 2 e1 ͑C19͒ To study the effects of the parity-violating electron scattering we need to pick only the term that has axial vector current on Explicitly, the projectile side e1 and the vector current on the target side m2 f 2. Therefore, it is easily seen that the box diagram contri- 4 1 ͓ ͑ ͔͒ m1ln 2 bution to C1 f is see Eq. 11 M ͑ ͒ϭ 2 B1 M 1 ,M 2 ,m1 ,m2 ͑ 2Ϫ 2͒͑ 2Ϫ 2͒͑ 2Ϫ 2͒ ␦ˆ ef ϭϪ ͑Ϫ efϩ efϪ efϩ ef͒ ͑ ͒ 2 m1 M 1 m1 m2 m1 M 2 Box 2 A B C D C15 2 m2 in the above notation. The explicit expressions for the ee box m4ln 2 M 2 graphs in Fig. 5 are given below: ϩ 2 ͑ 2Ϫ 2͒͑ 2Ϫ 2͒͑ 2Ϫ 2͒ 2 m2 m1 m2 M 1 m2 M 2 ␣ 2 ˆ 2 G␮ M Ws ␦M ee(a)ϭi ͚ ͕͓g*ejngejpg*eingeipL ϫL M 2 Box ͱ ␲ 0L 0L 0L 0L e2 e1 4 1 2 4 n,p,i, j M 1ln 2 M 2 ϩ , ϩ *ejn ejp *ein eip ͑ 2Ϫ 2͒͑ 2Ϫ 2͒͑ 2Ϫ 2͒ g0R g0R g0R g0R Re2 2 M 1 m1 M 1 m2 M 1 M 2 ϫ Re1͔m␹0m␹0B2͑m˜L ,m˜L ,m␹0,m␹0͒ p n i j p n 2 M 2 m2ln ϩ͓g*ejngejpg*eingeipR ϫL 1 m2 0R 0R 0L 0L e2 e1 ͑ ͒ϭ 1 B2 M 1 ,M 2 ,m1 ,m2 ͑ 2Ϫ 2͒͑ 2Ϫ 2͒͑ 2Ϫ 2͒ ϩ ejn ejp ein eip m1 M 1 m1 m2 m1 M 2 g0*L g0L g0*R g0R Le2 2 ϫ ͔ ͑ ͖͒ ͑ ͒ M 2 Re1 B1 m˜L ,m˜L ,m␹0,m␹0 , C16 m2ln i j p n 2 m2 ϩ 2 ␣ 2 ˆ 2 ͑m2Ϫm2͒͑m2ϪM 2͒͑m2ϪM 2͒ G␮ M Ws 2 1 2 1 2 2 ␦M ee(b)ϭϪi ͚ ͕͓g*ejngejpgeing*eipL Box ␲ 0L 0L 0L 0L e2 2 ͱ2 4 n,p,i, j M 2 M 2ln 1 M 2 ϫL ϩg*ejngejpgeing*eipR ϩ 1 e1 0R 0R 0R 0R e2 ͑ 2Ϫ 2͒͑ 2Ϫ 2͒͑ 2Ϫ 2͒ . M 1 m1 M 1 m2 M 1 M 2 ϫ Re1͔B1͑m˜L ,m˜L ,m␹0,m␹0͒ i j p n ͑C20͒ ϩ͓g*ejngejpgeing*eipR ϫL 0R 0R 0L 0L e2 e1 The box graphs for the electron-down quark scattering are ϩ ejn ejp ein eip easily obtained from the above expressions by replacing the g0*L g0L g0R g0*R Le2 target electron e2 with the down quark. Also, all quantities ϫ ͑ ͒ Re1͔m␹0m␹0B2͑m˜L ,m˜L ,m␹0,m␹0͖͒, C17 that have the running index j re to be replaced with the cor- i j p n p n responding quantities for the first generation down squarks: ejn→ djn ␣ 2 ˆ 2 g0L g0L , etc. G␮ M Ws ␦M ee(c)ϭϪi ͚ g*eipgeingejpg*ejnL The box graphs for the electron-up quark scattering are Box ␲ L L L L e2 ͱ2 4 n,p,i, j shown in Fig. 6. The graphs 6͑a͒ and 6͑b͒ are easily obtained from the corresponding graphs for the ee scattering by re- ϫ ͑ ϩ ϩ͒ ͑ ͒ Le1B1 m˜␯ ,m˜␯ ,m␹ ,m␹ . C18 placing all quantities that have the running index i with the i j p n corresponding quantities for the first up generation squarks: ein→ uin ͑ ͒ In all the above formulas the following functions are used: g0L g0L , etc. The result for the last graph 6 c is

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␣ 2 ˆ 2 G␮ M Ws ␦ eu(c)ϭ ujn ujp ein eip ϩ ϩ ͑ ϩ ϩ͒ ͑ ͒ M Box i ͚ gL* gL gL* gL Lu2Le1m␹ m␹ B2 m˜␯ ,mD˜ ,m␹ ,m␹ . C21 ␲ p n i j p n ͱ2 4 n,p,i, j

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