# Parity-Violating Electron Scattering as a Probe of Supersymmetry

###### Abstract

We compute the one-loop supersymmetric (SUSY) contributions to the weak charges of the electron () and proton () using the Minimal Supersymmetric Standard Model (MSSM). These vector couplings of the -boson to fermions will be determined in two fixed-target, parity-violating electron scattering experiments. The SUSY loop contributions to and can be substantial, leading to several percent corrections to the Standard Model values for these quantities. We show that the relative signs of the SUSY loop effects on and are correlated and positive over nearly all of the MSSM parameter space, whereas inclusion of R-parity nonconserving interactions can lead to opposite sign relative shifts in the weak charges. Thus, a comparison of and measurements could help distinguish between different SUSY scenarios.

###### pacs:

14.20.Dh, 11.55.-m, 11.55.Fv[

]

The search for physics beyond the Standard Model (SM) of electroweak and strong interactions is a primary objective for particle and nuclear physics. Historically, parity-violating (PV) interactions have played an important role in elucidating the structure of the electroweak interaction. In the 1970’s, PV deep inelastic scattering (DIS) measurements performed at the Stanford Linear Accelerator Center (SLAC) confirmed the SM prediction for the structure of weak neutral current interactions[1]. These results were consistent with a value for the weak mixing angle given by , implying a tiny (electron)(quark) neutral current interaction. Subsequent PV measurements – performed at both very low scales using atoms as well as at the -pole in annihilation – have been remarkably consistent with the results of the SLAC DIS measurement[1].

More recently, the results of cesium atomic parity-violation (APV) [2] and deep inelastic - (-) nucleus scattering[3] have been interpreted as determinations of the scale-dependence of . The SM predicts how this parameter will evolve from its precisely measured value at the -pole. The cesium APV and neutrino DIS measurements imply and deviations, respectively, from the predicted evolution of (defined in the scheme). If conventional atomic or hadron structure effects are ultimately unable to account for these discrepancies, the results of these precision measurements would point to new physics.

In light of this situation, two new measurements involving polarized electron scattering have taken on added interest: PV Möller () scattering at SLAC[4] and elastic, PV scattering at the Jefferson Lab (JLab)[5]. In the absence of new physics, both measurements could be used to determine at the same scale ( ) – falling between the scales relevant to the APV and neutrino DIS measurements – with comparable precision in each case. Any significant deviation from the SM prediction for at this scale would provide striking evidence for new physics, particularly if both measurements report a deviation. On the other hand, agreement would imply that the most likely explanations for the cesium APV and neutrino DIS results are atomic and hadron structure effects within the SM.

In this Letter, we analyze the prospective implications of the parity-violating electron scattering (PVES) measurements for supersymmetry (SUSY). Although no supersymmetric particle has yet been discovered, there exists strong motivation for believing that SUSY is a component of the “new” Standard Model. For example, the existence of low-energy SUSY is a prediction of many string theories; it offers a solution to the hierarchy problem, providing a mechanism for maintaining the stability of the electroweak scale against large radiative corrections; and it results in coupling unification close to the Planck scale. In light of such arguments, it is clearly of interest to determine what insight about SUSY the new PVES measurements might provide.

In the simplest version of SUSY – the Minimal Supersymmetric Standard Model (MSSM)[6] – low-energy precision observables experience SUSY only via tiny loop effects involving virtual supersymmetric particles. In the MSSM, the requirement of baryon minus lepton number () conservation leads to conservation of the R-parity quantum number, , where denotes spin. Every SM particle has while the corresponding superpartner, whose spin differs by unit, has . Conservation of implies that every vertex has an even number of superpartners. Consequently, for processes like , all superpartners must live in loops. Such loops generate corrections – relative to the SM amplitude – of order , where denotes a SM particle mass and is a superpartner mass. Thus, low-energy experiments must probe an observable with a precision of few tenths of a percent or better in order to discern SUSY loop effects. Low-energy charged current experiments have already reached such levels of precision, and the corresponding implications of these experiments for the MSSM have been discussed elsewhere[7].

The leading-order SM contribution to the PV and asymmetries is governed by the tree-level vector coupling of the -boson to these fermions – the so-called “weak charge”. At tree-level in the SM this coupling is suppressed: . One-loop SM electroweak radiative corrections further reduce this tiny number, leading to the predictions [8, 9] and [9]. The fortuitous suppression of these couplings in the SM renders them more transparent to the possible effects of new physics. Consequently, experimental precision of order a few percent, rather than a few tenths of a percent, is needed to probe SUSY loop corrections. As we show below, the possible magnitude of these effects may be as large as the proposed experimental error bars for the and measurements (8% and 4%, respectively [4, 5]). Moreover, the relative sign of the effect in both cases is correlated – and positive – over nearly all available SUSY parameter space. To our knowledge, this correlation is specific to the MSSM, making it a potential low-energy signature of this new physics scenario.

The content of the MSSM has been described in detail elsewhere[6], so we review only a few features here. The particle spectrum consists of the SM particles and the corresponding superpartners: spin-0 sfermions (), spin- gluinos (), and spin- mixtures of Higgsinos and electroweak gauginos, the neutralinos () and charginos (). In addition, the Higgs sector of the MSSM contains two doublets (“up”- and “down”-types, respectively), whose vacuum expectations and are parameterized in terms of and . Together with the SU(2) and U(1) couplings and respectively, is determined from , , and the Fermi constant extracted from the muon lifetime, , while remains a free parameter. The MSSM also introduces a coupling between the two Higgs doublets characterized by the dimensionful parameter .

Degeneracy between SM particles and their superpartners is lifted by the SUSY-breaking Lagrangian, which depends in general on 105 additional parameters. These include the SUSY-breaking Higgs mass parameters; the gaugino masses ; the left- (right-)handed sfermion mass parameters (); and terms which mix and into mass eigenstates . One expects the magnitude of the SUSY-breaking parameters to lie somewhere between the weak scale and TeV. Significantly larger values can reintroduce the hierarchy problem.

Theoretical models for SUSY-breaking mediation provide relations among this large set of parameters, generally resulting in only a few independent parameters at the SUSY-breaking or GUT scale[10]. According to the model-independent analysis of Ref. [7], however, the superpartner spectrum implied by these models conflicts with the combined constraints of low-energy charged current data, , and the muon anomalous magnetic moment unless one allows for nonconservation of . Here, we adopt a similar model-independent approach and do not impose any specific relations among SUSY-breaking parameters. To our knowledge, no other model-independent analysis of MSSM corrections to neutral current observables has appeared in the literature, nor have the complete set of corrections to low-energy PV observables been computed previously (see, e.g. [11]).

MSSM loop corrections to elementary - amplitudes consist of several topologies. Contributions to gauge-boson self energies can be expressed entirely in terms of the oblique parameters , , and in the limit that . Since present collider limits allow for fairly light superpartners, however, we do not work in this limit. Consequently, the corrections arising from the photon self-energy () and - mixing tensor () contain a residual -dependence not embodied by the oblique parameters. Vertex corrections, external leg corrections, and box graphs are process-dependent and cannot be parameterized in any general way. Note that one must consider corrections to both neutral current and charged current amplitudes, since the former are normalized to and since is calculated in the MSSM using , , and radiative corrections to these quantities. We evaluate these corrections using the modified dimensional reduction renormalization scheme () [12].

Including these corrections, the general structure for an elementary amplitude is

(1) |

with

(2) |

Here, and , with and denoting a universal set of corrections to PV amplitudes and containing the effects of process-specific vertex, external leg, and box graph effects. It is useful to express the SUSY contributions to and in terms of oblique parameters as well as a residual -dependence of and :

(3) | |||||

Here, the hat denotes quantities renormalized in the scheme; are the SUSY vertex, external leg, and box graph corrections to the -decay amplitude; () is the sine (cosine) of the weak mixing angle in the scheme defined at the scale ; and is the SUSY contribution to the difference between the fine structure constant and the electromagnetic coupling renormalized . Note that superpartner loops give .

In order to evaluate the potential size of SUSY corrections, we generated set of different combinations of SUSY-breaking parameters, chosen randomly from flat distribution in mass parameters and . The former were bounded below by present collider limits and bounded above by 1000 GeV, corresponding to the (TeV) naturalness limit. We also restricted to lie in the range as required by perturbativity of third generation quark Yukawa couplings, and we allowed for left-right mixing among sfermions. In order to avoid unacceptably large flavor-changing neutral currents, we have also assumed no intergenerational sfermion mixing.

For each combination of parameters, we evaluate superpartner masses and mixing angles, which we then use as inputs for computing the radiative corrections. We also separately evaluate the corresponding contributions to the oblique parameters. The latter are tightly constrained from precision electroweak data. We rule out any parameter combination leading to values of and lying outside the present 95% confidence limit contour for these quantities. We note that this procedure is not entirely self-consistent, since we have not evaluated non-universal MSSM corrections to other precision electroweak observables before extracting oblique parameter constraints. As noted in Ref. [11], where MSSM corrections to -pole observables were evaluated using different models for SUSY-breaking mediation, non-universal effects can be as large as oblique corrections. Nevertheless, we expect our procedure to yield a reasonable estimate of the oblique parameter constraints. Since and do not dominate the low-energy SUSY corrections (see below), our results depend only gently on the precise allowed ranges for these parameters.

For the case of charged current observables, gluino loops can generate substantial corrections when the masses and mixing angles for - and -type squarks are not identical[7]. In contrast, gluinos decouple entirely from the one-loop MSSM corrections to semi-leptonic neutral current PV observables. Moreover, MSSM Higgs contributions to vertex, external leg, and box graph corrections are negligible due to the small, first- and second-generation Yukawa couplings. The light Higgs contribution to gauge boson propagators has already been included via the oblique parameters, while the effects of other MSSM Higgs bosons are sufficiently small to be neglected[13].

In Fig. 1, we plot the shift in the weak charge of the proton, , versus the corresponding shift in the electron’s weak charge, , normalized to the respective SM values. The corrections can be as large as () and () – roughly the size of the proposed experimental errors. Generally speaking, the magnitudes of decrease as SUSY mass parameters are increased. The largest effects occur when at least one superpartner is relatively light. An exception occurs in the presence of significant mass splitting between sfermions, which may lead to sizable contributions. However, such weak isospin-breaking effects also increase the magnitude of , so their impact is bounded by oblique parameter constraints. This consideration has been implemented in arriving at Fig. 1. We also observe that the presence or absence of sfermion left-right mixing affects the distribution of points, but does not significantly change the range of possible corrections. For the situation of no left-right mixing, the points are more strongly clustered near the origin. Thus, while corrections of the order of several percent are possible in either case, large effects are more likely in the presence of left-right mixing.

The shifts are dominated by . Non-universal corrections involving vertex corrections and wavefunction renormalization experience significant cancellations, while box graphs are numerically suppressed. We find that is nearly always negative, corresponding to a reduction in the effective for the PVES experiments. Within itself, contributions from the various terms in Eq. (Parity-Violating Electron Scattering as a Probe of Supersymmetry) have comparable importance, with some degree of cancellation occurring between the effects of and . Thus, the oblique parameter approximation gives a rather poor description of the MSSM effects on the weak charges.

As evident from Fig. 1, the relative sign of the corrections to both and is nearly always the same and nearly always positive. Since () in the SM, SUSY loop corrections give (). This correlation is significant, since the effects of other new physics scenarios can display different signatures. For example, for the general class of neutral gauge bosons (with mass 1000 GeV), the effects on and also correlate, but can have either sign in this case[14, 9]. Leptoquark interactions, in contrast, would not lead to discernible effects in but could induce sizable shifts in [14].

As a corollary, we also note that the relative importance of SUSY loop corrections to the weak charge of heavy nuclei probed with APV is suppressed. The shift in the nuclear weak charge is given by . Since the sign of due to superpartner loops is nearly always the same, and since and in the SM, a strong cancellation between and occurs in heavy nuclei. This cancellation implies that the magnitude of is generally less than about 0.2% for cesium and is equally likely to have either sign. Since the presently quoted uncertainty for the cesium nuclear weak charge is about 0.6%, the present deviation from the SM prediction does not substantially constrain the SUSY parameter space.

While agreement between experimental values for and the SM predictions would not produce significant new constraints on the MSSM, a deviation of or more could help distinguish between the MSSM and other new physics scenarios. For example, a deviation for , along with the cesium APV result, would be consistent with an boson but fairly difficult to accomodate in the MSSM. To illustrate, we plot in Fig. 2 the impact of such a result on the MSSM parameter space. Here, we have assumed no L-R mixing, set , and taken GeV. The plot indicates the allowed left-handed slepton and squark masses ( and ), assuming for simplicity a common mass for all generations. Present collider searches rule out the dark shaded region, while charged current data exclude the light shaded area. The hatched region would be ruled out (at 95% C.L.) by a large, positive , yielding an upper bound GeV. Should future collider limits exceed this value, then a would be the favored explanation of of large cesium APV and deviations.

Alternately, one may relax the assumption of conservation. Doing so leads to new interactions of the type and with coupling strengths and (the subscripts refer to generation number). These interactions give rise to new tree-level exchange of sfermions between SM fermions. As discussed in Ref. [15], charged current data, , and the results of cesium APV can be accommodated under this scenario if and are nonzero. The interior of the truncated ellipse in Fig. 1 shows the possible corrections to and , given the constraints from other electroweak data. We observe that the prospective effects of non-conservation are quite distinct from SUSY loops. The shifts and have opposite signs over most of the allowed region, in contrast to the situation for SUSY loop effects or bosons. Thus, a comparison of results for the two PVES experiments could help determine whether this extension of the MSSM is to be favored over other new physics scenarios.

We thank R. Carlini, B. Filippone, S. Heinemeyer, R.D. McKeown, and M. Wise for useful comments. This work was supported in part by the U.S. Department of Energy and National Science Foundation.

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