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RADIATIVE CONTRIBUTION TO WEAK CHARGE

The weak charge of a particle f is defined as the strength of the effective A(e) V( f ) ef Gµ f × interaction: L = Q e¯γµ γ5e f¯γµ f . With higher-order corrections included, the EFF − 2√2 W f ρ f κ 2 θ λ weak charge can be written as QW = PV 2T3 4Q f PV sin W + f . The quantities h − i ρPV and κPV are universal, while the correction λ f , on the other hand, does depend on the species. At tree-level, one has ρPV = 1 = κPV and λ f = 0, while at one-loop ρ δρSM δρSUSY κ λ order PV = 1 + PV + PV , and similar formulae apply to PV and f . The counterterm δGˆ µ determined by life time and the Z0 boson self-energy are combined into ρPV (expressed in terms of the oblique parameters S, T [8]):

δGˆ µ Πˆ (0) Πˆ (0) Πˆ (0) µ µ ρ ZZ WW ZZ δˆ αˆ δˆ PV = 1 + + 2 = 1 2 + 2 VB = 1 + T VB. (1) Gµ MZ − MW MZ − − δˆ µ The quantity VB denotes the the electroweak vertex, external leg, and box graph correc- tions to the muon decay amplitude. e Z γ mixing and parity-violating electron- coupling F (0) contribute to κPV : − A Πˆ 2 δ 2 2 α cˆ γZ(q ) 2 e sˆnew cˆ ˆ ˆ µ κPV = 1 + + 4c ˆ F (0) + = 1 + S αˆ T + δ sˆ q2 A sˆ2 cˆ2 sˆ2 4s ˆ2cˆ2 − VB 2 2 2 2 − cˆ Πˆ Zγ(q ) Πˆ Zγ(M ) cˆ Πˆ γγ (M ) ∆αˆ + Z + Z + + 4c ˆ2Fe(q2) (2) sˆ q2 − M2 cˆ2 sˆ2 − M2 α A h Z i  − h Z i δ 2 2 2 α The shift sˆnew ins ˆ follows from the definition ofs ˆ in terms of , Gµ , and MZ[6]. The non-universal contribution λ f to the weak charge is determined by the sum of the the renormalized vertex corrections and the box graphs.

SUSY CORRECTION TO WEAK CHARGES

In order to evaluate the potential size of SUSY loop corrections, a set of about 3000 different combinations of SUSY-breaking parameters was generated. Fig. 1(a) shows the δ p δ u δ d shift in the weak charge of the proton, QW = 2 QW + QW , versus the corresponding shift in the electron’s weak charge, δQe , normalized to the respective SM values. The W p corrections in the MSSM (with R-parity conserved) can be as large as 4% (QW ) and e ∼ 8% (QW ) – roughly the size of the proposed experimental errors for the two PVES ∼ δ e,p δκSUSY measurements. The shifts QW are dominated by PV , which is nearly always −4 x 10 20

0.1 15 SM ) p W 10 | ν /(Q R 0 δ SUSY ) 5 p W (Q δ 0 −0.1 (b)

−0.2 −0.15 −0.1 e−0.05 e0 0.05 0.1 −1 0 1 2 δ (Q ) /(Q ) δ R −3 W SUSY W SM ν x 10 FIGURE 1. Plot(a) shows the relative shifts in electron and proton weak charges due to SUSY effects. Plot(b) shows the MSSM contribution to Rν and Rν¯ . Dots indicate MSSM loop corrections for 3000 randomly-generatedSUSY-breaking parameters. Interior of truncated elliptical region gives possible∼ shifts due to RPV SUSY interactions (95% confidence).

2 θ eff 2 κ 2 2 θ negative,corresponding to a reduction in the value of sin W (q ) = PV (q )sin W for e p the PVES experiments. Since this effect is identical for both QW and QW , the dominant effect of δκPV produces a linear correlation between the two weak charges. p As evident from Fig. 1 (a), the relative sign of the loop corrections to both QW and e QW is nearly always the same and positive. This correlation is significant, since the effects of other new physics scenarios can display different signatures. For example, for the general class of theories based on E6 gauge group, with neutral gauge bosons Z′ p e δ e,p e,p having mass < 1000 GeV, the effects on QW and QW also correlate, but QW /QW can have either sign in this case[9, 10]. In contrast, leptoquark interactions would not lead to e p discernible effects in QW but could induce sizable shifts in QW [9, 10]. As a corollary, we also find that SUSY loop corrections to the weak charge of cesium δ Cs Cs is suppressed: QW /QW < 0.2% and is equally likely to have either sign, which is smaller than the presently quoted uncertainty for the cesium nuclear weak charge of about 0.6% [11]. Therefore, the present agreement of QCs with the SM prediction e,p W does not preclude significant shifts in QW arising from SUSY. The situation is rather e,p different, for example, in the E6 Z′ scenario, where sizable shifts in QW would also Cs imply observable deviations of QW from the SM prediction. New tree-level SUSY contributions to the weak charges can be generated when the R parity in MSSM is not conserved. The effects of R-parity violating (RPV) contribution can be parametrized by positive, semi-definite, dimensionless quantities ∆i jk( f˜) and ∆ ˜ i′ jk( f )[12], which are constrained from the existing precision data [12]. The 95% CL δ p p δ e e region allowed in the QW /QW vs. QW /QW plane is shown by the closed curve in Fig. 1 (a). We observe that the prospective effects of RPV are quite distinct from SUSY δ e e loops. The value of QW /QW is never positive in contrast to the situation for SUSY loop δ p p effects, whereas QW /QW can have either sign. Thus, a comparison of the two PVES measurements could help determine which extension of the MSSM is to be favored over other new physics scenarios [10]. NUTEV MEASUREMENT

Recently, the NuTeV collaboration has performed a precise determination of the ratio Rν (Rν¯ ) of neutral and charged current deep-inelastic νµ (ν¯µ )-nucleus cross sections[2], which can be expressed in terms of the effective ν q hadronic couplings (geff )2: − L,R σ ν ν ν ν ( ( ¯ )N ( ¯ )X) eff 2 ( 1) eff 2 Rν ν¯ = → =(g ) + r − (g ) , (3) ( ) σ(ν(ν¯)N l (+)X) L R → − σCC σCC eff 2 where r = ν¯ N / νN . Comparing the SM predictions[13] for (gL,R) with the values exp SM obtained by the NuTeV Collaboration yields deviations δRν ν¯ = R R , δRν = ( ) ν(ν¯ ) − ν(ν¯ ) 0.0029 0.0015, δRν¯ = 0.0015 0.0026. − ± − ± The numerical results for SUSY contributions to Rν and Rν¯ via the correction to the eff 2 effective hadronic couplings (gL,R) are shown in Fig. 1 (b). For detailed analysis, see Ref. [7]. SUSY loop contributions to Rν and Rν¯ are smaller than the observed deviations. More significantly, the sign of the SUSY loop corrections is nearly always positive, in contrast to the sign of the NuTeV anomaly. Tree-level RPV contributions to Rν and Rν¯ are by and large positive. While small negative corrections are also possible, they are numerically too small to be interesting.

CONCLUSION

In summary, we have studied the SUSY corrections to the weak charge of the electron and proton, which could be measured at PV ee and ep scattering experiments. The cor- relation between these two quantities could be used to distinguish various new physics. We also examined the SUSY contributions to the NuTeV measurements and found that it is hard to explain the NuTeV anomaly in the framework of MSSM.

REFERENCES

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