Exploring Non-Supersymmetric New Physics in Polarized Møller Scattering
aa r X i v : . [ h e p - ph ] M a r Exploring Non-Supersymmetric New Physics in Polarized MøllerScattering
We-Fu Chang ∗ Department of Physics, National Tsing Hua University, Hsin Chu 300, Taiwan
John N. Ng † Theory Group, TRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., Canada
Jackson M. S. Wu ‡ Center for Research and Education in Fundamental Physics,Institute for Theoretical Physics, University of Bern,Sidlerstrasse 5, 3012 Bern, Switzerland (Dated: October 22, 2018)
Abstract
We study in an effective operator approach how the effects of new physics from various scenariosthat contain an extra Z ′ neutral gauge boson or doubly charged scalars, can affect and thusbe tested by the precision polarized Møller scattering experiments. We give Wilson coefficientsfor various classes of generic models, and we deduce constraints on the parameter space of therelevant coupling constants or mixing angles from the results of the SLAC E158 experiment whereapplicable. We give also constraints projected from the upcoming 1 ppb JLAB experiment. In thescenario where the extra Z ′ is light ( M Z ′ ≪ M W ), we obtain further constraints on the parameterspace using the BNL g − Z ′ neutral gauge boson. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION Polarized electron scattering has a long and illustrious history that began with the cel-ebrated SLAC polarized e D scattering [1], which was instrumental in putting the SandardModel (SM) on a firm experimental ground. The latest experiment of this class is the polar-ized Møller scattering. As only the electrons are involved, theory calculations at quantumloop levels can be performed free of hadronic uncertainties making it an ideal process forprecision SM testing; any deviation from expectations will also be a clear signal of newphysics beyond the SM.A while ago, an experiment of this type has been successfully completed [2], where parity-violating measurements have been made from the scattering of longitudinally polarized elec-trons – either left-handed (LH) or right-handed (RH) – on an unpolarized electron target.At a momentum transfer of Q = 0 .
026 GeV , the left-right asymmetry A LR ≡ dσ L − dσ R dσ L + dσ R , (1)has been measured to an accuracy of approximately 20 ppb, and the weak angle determinedto be sin θ W = 0 . ± . . ) ± . . ), with its running established at the6 σ level. In the SM, A LR arises from the interference between the electromagnetic and theweak neutral currents, and is given by A SM LR = G µ Q √ πα − y y + (1 − y ) h − θ MSW i , Q = ys , s = 2 m e E beam , (2)where G µ = 1 . × − GeV − , α − = 137 . m e have been dropped when appropriate. The electroweak corrections matching thisexperimental accuracy and more is given in Ref. [3]. The value of the weak angle at Q = 0is found to be sin θ MSW = 0 . ± . A LR to an accuracy of 1 ppb. Since thetheoretical uncertainty can in principle be calculated to this accuracy, a successful measure-ment will be sensitive to the new physics at the several TeV scale, which is also the physicslandscape to be explored by the LHC.Also recently, an anomalous abundance of positron flux in cosmic radiation has beenreported by the PAMELA collaboration [5]. This confirms the excess previously observed2y the HEAT [6] and AMS [7] experiments. The ATIC [8] and PPB-BETS [9] balloonexperiments have also recorded similar excess. One possible origin is from a hidden sectorand low scale dark matter annihilation [10]. The hidden sector scale is O (GeV), and so isat the opposite end of the new heavy physics. The characteristic of this class of model isthat possible couplings to SM matter fields must be extremely weak. A very simple modelwas constructed previously to study this possibility [11], and Møller scattering was found togive very stringent constraints.Motivated by all these encouraging developments, we examine in detail how an im-proved A LR measurement could shed light on new physics. We focus here on classes ofnon-superymmetric new physics and how they might manifest themselves in precision po-larized Møller scattering. It is well known that A LR can be used to probe the existenceof doubly charged scalars and extra Z ′ bosons. The doubly charged scalars are interestingin their own right, and they also arise in Type II seesaw models for neutrino masses [12].For extra Z ′ bosons, the literature is particularly rich with well motivated models. Theseinclude models of grand unified theory (GUT) based on the E or SO (10) gauge group [13],hidden sector Z ′ models [11, 14], and left-right symmetric [15] and warped extra-dimensionalRandall-Sundrum (RS) [16] models based on the gauge group SU (2) L × SU (2) R × U (1) B − L .The effects of a new particle such as the doubly charged scalar of the extra Z ′ boson canbe encoded in an effective operator approach, which for A LR involve only two dimension sixoperators: O = ¯ eγ µ ˆ Le ¯ eγ µ ˆ Le , O = ¯ eγ µ ˆ Re ¯ eγ µ ˆ Re , ˆ L = 12 (1 − γ ) , ˆ R = 12 (1 + γ ) . (3)In the next section, we describe this effective operator approach.The rest of the paper is organized as follows. In Sec. III, we discuss the classes of newphysics models. Besides updating relevant known results, we also present a new calculationof the contributions in the Minimal Custodial RS (MCRS) model with bulk symmetry SU (2) L × SU (2) R × U (1) B − L , and we give comparisons with the usual four dimensionalleft-right symmetric models. Sec. IV contains our conclusions. The details of the MCRSmodel and the specifics of the fermion localization are collected into Appendix. A.3 I. EFFECTIVE OPERATORSA. High Scale New Physics
If the new physics is at a scale higher than the Fermi scale v ≃
247 GeV, an effectiveoperator approach is powerful and avoids much model dependence. Denoting the scale ofnew physics by Λ and assuming that the SM gauge group holds between Λ and v , the effectiveLagrangian with dimension six four-lepton operators added is given by L = L SM + L , − L = c LL Λ ( ¯ L a γ µ L a )( ¯ L b γ µ L b ) + c LR Λ ( ¯ L a γ µ L a )( e R γ µ e R ) + c RR Λ ( e R γ µ e R )( e R γ µ e R )+ d LL Λ ( ¯ L a γ µ L b )( ¯ L b γ µ L a ) + h.c. , (4)where a, b are SU (2) indices, and L = ( ν, e ) TL . The subscripts L and R denote the usualchiral projection. We have included only terms relevant for the calculation of A LR , i.e. thefirst generation leptons. Note that for scattering of a polarized beam on a polarized target,there is also a scalar operator that contributes: L s = c S Le R e R L + h.c. (5)A full analysis of dimension six leptonic operators can be found in Ref. [17].The Wilson coefficients c and d are given by specific models at Λ. One has to do arenormalization group (RG) running analysis to determine their values at the relevant lowenergy scale where the experiments are performed. The RG equations for these coefficientscan be found in Ref. [17]. Numerically, we find that the RG running effects are not significant.They give about a 4% effect running between Λ (typically a few TeV) and v , and hence canbe neglected in the first approximation. Below v the running is governed by the well known β QED funtions. Including them leads to a change of about 10% in the values of the coefficientsbetween Λ and √ s ≃ . A LR is found by calculating the diagrams depicted in Fig. 1 The shadedblob schematically includes all new physics effects such as Z - Z ′ mixing and virtual exchanges4 , Zee ee ee ee FIG. 1: Feynman diagrams contributing to A LR . The shaded blob contains all new physics effects. of new particles. For Λ ≫ s ≫ m e valid for the 12 GeV experiment, one has A LR = A SM LR + δA LR = 4 G µ s √ πα y (1 − y )1 + y + (1 − y ) (cid:20)(cid:18) − sin θ MSW (cid:19) + c ′ LL − c RR √ G µ Λ (cid:21) , (6)where y = − ts , c ′ LL = c LL + d LL , and δA LR denotes the deviation of the asymmetry from theSM prediction arising from new physics. The one-loop SM radiative corrections are encodedin the precise value of sin θ W evaluated in the MS scheme. The effective operators, O , ,can arise from numerous new physics models, and those coming from the tree-level exchangeof virtual particles are the most important. We thus examine models with at least an extraneutral gauge boson, Z ′ , or doubly charged scalars below in Sec. III, where we give Wilsoncoefficients in various models. Of course there could be extra charged gauged bosons thatalso contribute in specific models, but these typically come in at the loop level and so areless important.We remark here that the new physics effects in Møller scattering can also come in throughthe Zee coupling being modified. This is particularly evident in cases when the heavy statesof the new physics also have very small couplings to electrons so that the virtual exchangeprocess is suppressed both by the propagator effect and the small coupling. Example ofthis can be found in the extra hidden sector/shadow U (1) and RS models which we discussbelow in Sec. III D and III E. B. Low Scale New Phyiscs
For the class of models where new physics arise at a scale Λ . v , two effects are at work.The first is due to the exchange of the new low mass particle. The propagator effect has to5e compensated by its small couplings to the electrons of different chiralities. The secondis due to the modification of the SM Z coupling to the electrons. Well known examples aremodels with extra Z ′ that can mix with the Z . Details are given below in Sec. III. III. MODEL CONSIDERATIONS
In this section, we discuss how new physics can be probed by polarized Møller scatteringin various non-supersymmetric extensions of the SM. We begin with models that containextra neutral Z ′ bosons. A comprehensive review on the physics of Z ′ can be found inRef. [18].Irrespective of the ultimate origin of the extra neutral gauge boson, its interactions canbe described by the gauge group, SU (2) L × U (1) Y × U (1) ′ , at a scale not too high compareto v [19]. Denoting the extra U (1) ′ gauge field by X µ , its interaction with the SM mattercontent is given by L ′ = − X µν X µν − ǫ B µν X µν + g ′ z f ¯ f γ µ f X µ − M X X µ X µ + | D µ H | , (7)with the covariant derivative given by D µ = ∂ µ − ig L T a W aµ − i g Y B µ − i g ′ z H X µ , (8)where g L , g Y , and g ′ are the gauge couplings of the SU (2) L , U (1) Y , and U (1) ′ respectively, f a species label that runs over all SM fermions, and z f ( z H ) the SM fermion (Higgs) chargeunder the U (1) ′ . Other notations are standard. The paramter ǫ characterizes the kineticmixing between B µ , the hypercharge gauge field, and X µ , and is expected in general to beof order 10 − to 10 − [11, 20].The kinetic energy terms for the gauge bosons can be recast into canonical form througha GL (2) transformation XB = c ǫ − s ǫ X ′ B ′ , s ǫ = ǫ √ − ǫ , c ǫ = 1 √ − ǫ . (9)In most cases involving high scale new physics, the kinetic mixing is negligible. However,this is not true in general for low scale cases. The mass term in Eq. (7) can arise from ahidden sector scalar interactions that breaks the U (1) ′ gauge symmetry. The details are not6mportant for this study. After the spontaneous breaking of the electroweak symmetry, thismass term will lead to a mass mixing among W , B , and X fields which depend on z H .For models with z H = 0 but a non-vanishing kinetic mixing between the neutral gaugebosons, the transformation between the weak and mass basis is given by B ′ W X ′ = c W − s W s W c W
00 0 1 c η − s η s η c η γZZ ′ , (10)where s W ( c W ) denotes sin θ W (cos θ W ), and similarly for the angle η . The first rotation isthe standard one that give rise to the would be SM Z , which we label as Z , while the seconddiagonalizes the mixing between Z and X ′ that produces the two neutral gauge eigenstates, Z and Z ′ . This mixing angle is given bytan 2 η = 2 s W s ǫ x + s W s ǫ − , x = c W M X M W , (11)and the masses of Z and Z ′ are given by M ∓ = M W c W (cid:26) x + 1 + s W s ǫ ∓ q ( x − s W s ǫ ) + 4 s W s ǫ (cid:27) . (12)where M − ( M + ) denotes the lighter (heavier) of the two states. For high scale new physics, M W ≪ M X . Since s ǫ . − , the mixing angle η is very small as η < ǫ when s ǫ is small and x >
1. For M W ≫ M X , ǫ and η can be of the same order. The relative sign between themdepends on the relative magnitude between x and s ǫ .Models in which ǫ is vanishing but z H is not are more typical of high scale new physicsscenarios. Since there is now no kinetic mixing, the Z mixes with the X . The mass matrixof this system can be written as M Z − X = M Z ∆ ∆ M X , (13)with the mixing angle and the masses of the resulting Z and Z ′ given by [18, 19]tan 2 θ = 2∆ M Z − M X , (14)and M Z, Z ′ = 12 h M Z + M X ∓ q ( M Z − M X ) + 4∆ i . (15)7ote that both ∆ and M X are model dependent. For example, if the SM Higgs is chargedunder both U (1) factors, ∆ = − z H v g ′ p g + g Y and M X = v g ′ z H + M where M isthe mass from higher scale.In either case, the Wilson coefficients can now be calculate from ǫ , g ′ , z f , z H , and Λ ina given model. For quick reference, we summarize here the results from below the Wilsoncoefficients for high scale new physics in Table I. New Physics c ′ LL c RR E GUT πα c W (cid:16) c β + q s β (cid:17) πα c W (cid:16) c β − q s β (cid:17) U (1) R × U (1) B − L g Y g R − g Y
14 (2 g Y − g R ) g R − g Y U (1) B − L g ′ g ′ U (1) X g ′ z L g ′ z e P ±± −| y ee | / c ′ LL and c RR for various generic high scale new physicsmodels containing an extra Z ′ neutral gauge boson or doubly charged scalars ( P ±± ). A. GUT Models
The appearance of extra U (1) gauge groups is exemplified by the E GUT model. Oneway they can arise is through the symmetry breaking chain E → SO (10) × U (1) ψ → SU (5) × U (1) χ × U (1) ψ . (16)The U (1) factors here are not anomalous. There are also no kinetic mixing, and the massmixings between the gauge bosons of the extra U (1) factors and the SM Z are phenomeno-logically found to be negligible. The X boson above is now a linear combination of the ψ and χ gauge bosons X µ ( β ) = B µψ sin β + B µχ cos β , (17)where B µψ, χ are gauge fields of the U (1) ψ, χ respectively. The X µ is taken to be the lighterof the two neutral linear combinations. From the fermion charges given in Ref. [18, 21], we8nd c ′ LL − c RR = 4 πα c W c β c β + r s β ! , (18)where c β ( s β ) denotes cos β (sin β ).Defining δ LR ≡ ( A LR − A SMLR ) /A SMLR = δA LR /A SMLR , we plot in Fig. 2 the resulting δ LR dueto GUT new physics. We plot also the constraints from the Tevatron new physics searches, E158 - - PSfrag replacements δ LR s η s ǫ M Z ′ (GeV) s β g R g ′ z L g ′ z e | y ee | Z ′ I Z ′ η Z LR FIG. 2: The ratio, δ LR , as a function of s β . The labelled solid lines are the SLAC E158 upperlimit, δ LR = 0 . Z ′ I and Z ′ η , Λ = 729 ,
891 GeV. Theother solid, dashed, and dotted lines denote cases where Λ = 1 , , | δ LR | < . the SLAC E158 experiment, and that projected from the the JLAB 1 ppb measurement. Weuse the Tevatron limits on the Z ′ I and Z ′ η masses to illustrate the current collider constraintson the new physics scale Λ for the GUT extra neutral gauge bosons . The 95% confidencelevel (CL) lower mass limit for Z ′ I ( Z ′ η ) is M Z ′ I >
729 GeV ( M Z ′ η >
891 GeV), representingthe lowest (highest) exclusion limit in the CDF search [22]. The SLAC E158 measurementof the left-right asymmetry [2] gives A LR = 131 ±
14 (stat) ±
10 (syst) ppb . (19) As can be seen from Eq. (14), with ∆ ∼ O (100) GeV and M X & O (1) TeV expected, the Z − X massmixing is small with sin θ . O (0 . A LR due to the mass mixing and the concurrent modification in the gauge-fermion couplings, whichare proportional to (sin θ ) . In our notation, Z ′ I and Z ′ η are just X with tan β = p / − p / A SMLR = 1 . × − for Q = 0 .
026 GeV and y = 0 .
6, one can infer from this that − . < δ LR < .
122 at 95% CL . Lastly, as a benchmark assume that the total 1 σ errorof the new JLAB measurement of A LR to be 1 ppb with a central value given by the SMvalue, A SMLR , one expects in such a case | δ LR | < . B. Left-Right Symmetric Models
For left-right symmetric models, the gauge group is taken to be SU (2) L × SU (2) R × U (1) B − L . The SU (2) R is broken down to a U (1) R by a triplet Higgs field at a scale greaterthan a few TeV, leaving SU (2) L × U (1) R × U (1) B − L . In this case, the U (1) R × U (1) B − L isequivalent to the U (1) Y × U (1) ′ [19]. The charges z f and z H can be determined by enforcinganomaly cancellation conditions (see below). We find c ′ LL − c RR = 14 (cid:0) g Y − g R (cid:1) , (20)where g R is the gauge coupling of SU (2) R . Interestingly, note that even if g R = g L , δ LR willnot vanish. We plot in Fig. 3 the resulting δ LR for left-right symmetric models, as well asthe relevant SLAC E158 constraint and that projected from the JLAB 1 ppb measurement.As a comparion, we plot also the curve that arises from the Z LR
95% CL lower mass limit, M Z LR >
630 GeV, obtained from the combination of electroweak data, direct Tevatron andindirect LEP II searches [18]. We see that current experiments do offer, if only barely, someconstraints on the TeV scale new physics here compared to the GUT case. But for usefulones, the precision of the upcoming JLAB experiment is still much needed. C. SU (2) L × U (1) Y × U (1) X Models
There are interesting bottom up models where the extra U (1) factors are not immediatelyrelated to the remnant of some broken down non-abelian groups. The prime example is the SU (2) L × U (1) Y × U (1) B − L model. The assumption is that the SM is charged under U (1) X , We have followed the more common practice here in taking the 1 σ total error as the statistical andsystematic uncertainties added in quadrature. The 95% CL corresponds to an error interval of ± . σ . - - - PSfrag replacements δ LR s η s ǫ M Z ′ (GeV) s β g R g ′ z L g ′ z e | y ee | Z ′ I Z ′ η Z LR FIG. 3: The ratio, δ LR , as a function of g R . The labelled solid lines are the SLAC E158 upperlimit, δ LR = 0 . Z LR lower mass limit, Λ = 630 GeV. The other solid,dashed, and dotted lines denote cases where Λ = 1 , , | δ LR | < . and since this group is known to be anomalous, one has to demand anomaly cancellations forchiral fermions under the full SU (3) C × SU (2) L × U (1) Y × U (1) X . As neutrinos are massive,we introduce one sterile neutrino, N R , per family and denote its charge by z N . Assumingthese charges are family independent, we have seven independent charges z f , z N , and z H .The anomaly conditions arise from[ SU (3)] U (1) X , [ SU (2)] U (1) X , [ U (1) Y ] U (1) X , [ U (1) X ] . (21)The mix gravitational and U (1) X anomaly give the same condition as [ U (1) X ] and is re-dundant. The Yukawa terms that give quark and lepton masses yielded one additionalcondition. Thus there are two independent charges, which we choose to be z L and z e . Theother charges are z Q = − z L , z u = 23 z L − z e , z d = − z L + z e , z N = 2 z L − z e , z H = z L − z e . (22)For the interesting case where X = B − L , we have the familiar lepton and baryonnumber assignments, and z H = 0. More importantly, if the extra Z ′ arises directly from thebreaking of a gauged U (1) B − L , Møller scattering will yield only the SM result, since in thatcase z L = z e ; this leads immediately to c ′ LL − c RR = 0. In general, z L = z e , and we find c ′ LL − c RR = g ′ ( z L − z e ) , (23)11here g ′ is the gauge coupling constant of the U (1) X . Note that in this analysis the massesof the sterile neutrinos do not come into play. Hence the details of the mechanism forgenerating active neutrinos masses will not change our results. We plot in Fig. 4 contoursof constant δ LR as a function of g ′ z L and g ′ z e for Λ = 1 TeV. For other values of Λ, thecontour plot is simply rescaled by a factor of (Λ / - - - - - - - - - - - PSfrag replacements δ LR s η s ǫ M Z ′ (GeV) s β g R g ′ z L g ′ z e | y ee | Z ′ I Z ′ η Z LR FIG. 4: Contours of constant δ LR for Λ = 1 TeV. Note the contours for the SLAC E158 lowerand upper limits, δ LR = − .
337 and δ LR = 0 . δ LR = ± . D. Hidden Sector/Shadow Z ′ Models
Another class of models with an extra U (1) factor, which has received renewed attentionrecently due to the observation of excess cosmic positron flux, is one such that there are nodirect couplings of the extra gauge field to SM fermions or Higgs fields. The extra U (1) hid is a gauge symmetry of a hidden sector with its own scalar sector. The hidden sector andthe SM communicate through hidden scalars couplings with the SM Higgs [11] and kineticmixing [23]. In this class of models z H = 0, and the fermion couplings to Z and Z ′ are given12y [11] iγ µ g L c W (cid:26)h c η Q LZ ( f ) − s η s W s ǫ Y Lf i ˆ L + ( L ↔ R ) (cid:27) ( Z µ ¯ f f ) , (24) − iγ µ g L c W (cid:26)h s η Q LZ ( f ) + 12 c η s W s ǫ Y Lf i ˆ L + ( L ↔ R ) (cid:27) ( Z ′ µ ¯ f f ) , (25)where Q L, RZ ( f ) = T ( f L, R ) − s W Q f with Q f = T f + Y f /
2. We see that the neutral currentcouplings are not only rotated as indicated by the c η factor, but they also contain an extrapiece proportional to the fermion hypercharge due to the U (1)- U (1) X mixing. Hence weneed to re-examine the electroweak precision data using the full couplings as well as takinginto account the effects due to virtual Z ′ exchanges.As mentioned above, there are two interesting limits with very different phenomenologiesdepending on whether M X is greater or less than M W , which correspond to high or low scalenew physics. M X ≫ M W From Eq. (11) we can see that η ≪ ǫ and the mixing can be ignored. The most stringentconstraints come from electroweak precision measurements at the Z -pole. As for Møllerscattering, the exchange of the Z ′ gives negligible effect because its large mass and verysmall coupling to the electrons (see Eq. (25)). The dominant effect comes from the modified Ze ¯ e coupling, and we find δ LR ≡ δA LR A SM LR ≃ − s η − s η s W s ǫ − s W − c η s η s W s ǫ s W + 11 − s W = − (cid:0) s η + 15 . s η s ǫ + 31 . c η s η s ǫ (cid:1) , (26)where the Thomson limit value of the weak angle, s W = 0 . s W for all numerical presentations below. We point outhere that this result is independent of M ′ . Also if s η and s ǫ have the same sign, δ LR isnegative. M X ≪ M W In this case we have a light Z ′ whose mass is essentially M X (see Eq. (12)), and themixing angle η can be large compare to ǫ . Since the Z ′ is light, δ LR is dominated by the13 - PSfrag replacements δ LR s η s ǫ M Z ′ ( G e V ) s β g R g ′ z L g ′ z e | y ee | Z ′ I Z ′ η Z LR FIG. 5: Constraint from the light Z ′ contribution to A LR . The contours are that for the SLAC E158lower limit (solid), δ LR = − . δ LR = − . δA LR > photon- Z ′ interference term: δ LR ≈ δ Z ′ LR ≃ u Le ) − ( u Re ) − s W M Z M Z ′ = (cid:0) s η − . c η s ǫ + 31 . c η s η s ǫ (cid:1) M Z M Z ′ , (27)where u L, Re = − (cid:2) s η Q L,RZ ( e ) + c η s W s ǫ Y e (cid:3) . From the SLAC E158 measurement of A LR , wehave at 95% CL s η − . c η s ǫ + 31 . c η s η s ǫ ∈ [ − . , . × − (cid:18) M Z ′ (cid:19) . (28)If we use the projected JLAB limits from above, the constraint interval on the RHS is changeto [ − . , .
16] in the same unit. Now from Eqs. (11) and (12), s ǫ can be written in termsof M Z ′ and s η as s ǫ = − s η s W c η (cid:18) − c W M Z ′ M W (cid:19) . (29)Using this, we plot in Figs 5 the allowed ( s η , M Z ′ ) region as given by Eq. (28). Note that forthe RHS of Eq. (28) greater than zero, there is no real solution, and so no allowed parameterspace. 14he mixing parameters η and ǫ are also constrained by low energy observables such asthe anomalous magnetic moment of the muon a µ . The new physics contribution from a lightextra Z ′ to a µ is found to be [11] δa µ = G µ m µ √ π M Z M Z ′ (cid:2) s η (4 s W − s W − − s W − s W s η c η s ǫ + s W c η s ǫ (cid:3) = − . × − (cid:0) s η − . c η s ǫ − . c η s η s ǫ (cid:1) M Z M Z ′ = − . × − s η (cid:26) M W − c W M Z ′ M W s W h M W ( s W − .
19) + 0 . M Z ′ i(cid:27) M Z M Z ′ , (30)where we have used Eq. (29) in the last line. We see that for M Z ′ ≪ M W , the curly bracketis positive definite. Thus a δa µ arising from a light Z ′ is always negative. This means thatif confirmed, the current BNL result of a 3 . σ deviation from SM expectation [24]: a exp µ − a SM µ = (2 . ± . × − , (31)cannot be due to the new physics arising from the light Z ′ alone. E. The MCRS Model
Recently, the warped five-dimensional (5D) RS model endowed with a bulk gauge group SU (3) c × SU (2) L × SU (2) R × U (1) X [25] – which we call the MCRS model – has beenactively studied as a framework for flavor physics [26, 27, 28]. The gauge hierarchy problemis solved when the warp factor is taken to be kπr c ≈
37 [16]. This sets the scale for thenew physics to m KK ≈ .The bulk SU (2) R factor protects the ρ parameter in the effective 4D theory from excessivecorrections coming from KK excitations, and is broken by orbifold boundary conditions onthe UV boundary (brane); this is a 5D generalization of the case studied in Sec. III B. Abrief summary of the model is given in Appendix A, which sets our notations.In the MCRS model, δ LR arise due to the interactions of KK excitations of the neutralgauge bosons and the charged leptons. Contributions come either from direct virtual ex-changes of neutral gauge KK modes, or through the Ze ¯ e coupling been modified due tomixings of SM modes with KK modes. The latter involve both fermions and gauge bosons, Note this is at the higher end of the LHC discovery limits. Z ′ case discussed above. More details of the vertex correctiondue to KK mixings can be found in Appendix A.A generic feature of the extra dimensional models is that gauge- f ¯ f couplings are notflavor diagonal in the mass eigenbasis. To calculate the contributions to δ LR in the masseigenbasis requires the knowledge of both the LH and RH rotation matrices, which aredetermined by the mass matrix of the charged leptons and their localization in the extradimension (see Ref. [26]). In Table II, we list the contributions pertinent to the case ofpolarized Møller scattering . We give results for five typical charged lepton configurations Config. Direct KK exchange Ze ¯ e correction TotalLL (10 − ) RR (10 − ) δ dirLR (10 − ) LL (10 − ) RR (10 − ) δ Ze ¯ eLR (10 − ) δ LR (10 − )I 1.98574 1.51504 8.09458 -1.07684 -7.98263 -2.45877 -1.64931II 1.98956 1.51484 8.11154 -1.07827 -7.98167 -2.47224 -1.66109III 1.99197 1.51050 8.12409 -1.07920 -7.96023 -2.49935 -1.68694IV 1.99188 1.51065 8.12365 -1.07916 -7.96172 -2.49767 -1.68530V 1.99049 1.51483 8.11566 -1.07858 -7.98162 -2.47504 -1.66347TABLE II: New physics contributions to the polarized Møller scattering in the MCRS model forfive typical charged lepton configurations that give rise to experimentally observed charged leptonmasses, and are admissible under current LFV constraints. that lead to charged lepton mass matrices compatible with the current lepton flavor violation(LFV) constraints, and give rise to experimentally observed charged lepton masses. Detailsof the charged lepton configurations and the particular realization of the admissible chargedlepton mass matrices used to calculate the new physics contributions in Table II are givenin Appendix B. Note that the direct KK LL (RR) contribution corresponds to the Wilsoncoefficient c ′ LL ( c RR ) scaled by a factor (2 √ G F ) − .As can be seen from Table II, the new physics contribution to A LR from direct exchangesof virtual KK gauge modes are subdominant compare to that from the shift in the SM Ze ¯ e We do not displayed the tiny imaginary parts, which is . O (10 − ), since the real part of the contributiondominates in the tree-level A LR . andcouplings of the electron to gauge KK modes are small. Note that the LL-type contributionis always greater that of the RR-type. This is a consequence of the fact that the LH charge isgreater than the RH charge, i.e. | Q LZ ( e ) | > | Q RZ ( e ) | . Note also the new physics contributionsare quite stable across the configurations, indicating their robustness with respect to thevariations in the charged lepton mass matrix. The MCRS values of δ LR at m KK ≈ | δ LR | < . F. Doubly Charged Scalars
Doubly charged scalars P ±± are motivated by Type II seesaw neutrino mass generationmechanisms. They can be either SU (2) L singlets or triplets, and they carry two unitsof lepton number. The contribution of P ±± to Møller scattering depends on only twoparameters, viz. its mass, M P , and its coupling to electrons, y ee . For example, if there isonly coupling to RH electrons, the effective contact interaction is | y ee | M P ( e cR e R )(¯ e R e cR ) = | y ee | M P (¯ eγ µ ˆ Re )(¯ eγ µ ˆ Re ) , (32)leading to δ LR = | y ee | √ − s W ) G µ M P . (33)For models where the coupling is to LH electrons, δ LR is similarly given with the RR Wilsoncoefficient changed to that of the LL type. The important difference is that δ LR is negativehere. We plot the δ LR arising from doubly charged scalars coupling only to RH electronsin Fig. 6, as well as the relevant constraints from the SLAC E158 and the projected JLAB1 ppb measurement. IV. CONCLUSIONS
We have examined in detail the implications of a very high precision polarized Møllerscattering experiment on new physics beyond the SM, which is well known to be sensitive toextra Z ′ and doubly charged scalars in the few TeV mass range. We have shown how such As required by electroweak precision measurements and very stringent quark sector flavor constraints, m KK &
158 JLAB 1.000.500.20 0.300.15 0.700.0010.0050.0100.0500.1000.5001.000
PSfrag replacements δ LR s η s ǫ M Z ′ (GeV) s β g R g ′ z L g ′ z e | y ee | Z ′ I Z ′ η Z LR FIG. 6: The ratio, δ LR , as a function of | y ee | . The solid, dashed, and dotted lines denote cases whereΛ = 1 , , δ LR = 0 . δ LR = 0 . a measurement can be particularly useful if such a state is found at the LHC. If a doublycharged scalar were discovered at the LHC, a measurement of the δ LR will determine itsYukawa couplings to electrons, and the sign of δ LR will determine whether it couples to LHor RH electrons (negative for the former, positive for the latter). Such scalars also carrytwo units of lepton number, hence measurements of δ LR will in turn have impact on thepredicted rates of neutrinoless double beta decays of nuclei [12], thus providing invaluableinformation on the origin of neutrino masses. Similarly, if an extra Z ′ were discovered,measuring δ LR will shed light on its origin. We have given detailed calculations for variousclasses of models including the E GUT model, left-right symmetric model, and two examplesof non-anomalous U (1).For high scale new physics, we found if the new particle is of the more exotic type suchas a hidden/shadow Z ′ or the pure KK Z ′ in warped extra-dimension RS scenarios, δ LR measures the change in the effective Ze ¯ e couplings as compare to the SM, and it is notsensitive to the mass of these new gauge bosons. In the scenario where the new physics is ofa very low scale ( ≪ v ), and a Z ′ with a small mass exists that couples very weakly to SMfermions, a δ LR measurement gives even more valuable information not available from highenergy hadron collider experiments. We have shown that the SLAC E158 experiment canalready tightly constrains the mixings of this type of Z ′ with the SM Z . The smallness of themixings allowed conforms with string-theoretic expectations, as well as phenomenological18eterminations [11]. On the other hand, the positive deviation from the SM expectation inthe measured value of muon g − Z ′ sinceit gives a negative contribution. To the best of our knowledge there is no cleaner or bettermeasurements on couplings of a hidden/shadow Z ′ to electrons. These are valuable inputsto dark matter model building employing a hidden or shadow sector.It is clear that an extra Z ′ or doubly charged scalars with masses in the 1 to 2 TeVrange can be found at the LHC relatively easily. We have shown that δ LR can providecross checks on these measurements as well as independent information on the couplings toelectrons of different chiralities. If the masses are & δ LR measurement. The current reach of newphysics of a 10 ppb measurement achieved at SLAC is around 1 to 2 TeV. Given the samecouplings, a 1% measurement will probe new physics at a scale Λ ∼ − Acknowledgments
The research of W.F.C. is supported by the Taiwan NSC under Grant No. 96-2112-M-007-020-MY3. The research of J.N.N. is partially supported by the Natural Scienceand Engineering Council of Canada. The research of J.M.S.Wu is supported in part bythe Innovations und Kooperationsprojekt C-13 of the Schweizerische UniversitaetskonferenzSUK/CRUS.
APPENDIX A: RS MODEL WITH SU (2) L × SU (2) R × U (1) B − L BULK SYMMETRY
We describe briefly in this appendix the basic set-up of the MCRS model to establishnotations relevant for studying flavor physics. A more detailed description can be found in,e.g. Ref. [25].The MCRS mode is formulated on a slice of
AdS space specified by the metric ds = G AB dx A dx B = e − σ ( φ ) η µν dx µ dx ν − r c dφ , (A1)19here σ ( φ ) = kr c | φ | , η µν = diag(1 , − , − , − k is the AdS curvature, and − π ≤ φ ≤ π .The theory is compactified on an S / ( Z × Z ′ ) orbifold, with r c the radius of the compactifiedfifth dimension, and the orbifold fixed points at φ = 0 and φ = π correspond to the UV(Planck) and IR (TeV) branes respectively. To solve the hierarchy problem, one takes kπr c ≈
37. The warped down scale is defined to be ˜ k = ke − kπr c . Note that ˜ k sets the scaleof the first KK gauge boson mass, m (1) gauge ≈ . k , which determines the scale of the newKK physics.The MCRS model has a bulk gauge group SU (3) c × SU (2) L × SU (2) R × U (1) X underwhich the IR brane-localized Higgs field and transforms as (1 , , . The SM fermions areembedded into SU (2) L × SU (2) R × U (1) X via the five-dimensional (5D) bulk Dirac spinors Q i = u iL [+ , +] d iL [+ , +] , U i = u iR [+ , +]˜ d iR [ − , +] , D i = ˜ u iR [ − , +] d iR [+ , +] ,L i = ν iL [+ , +] e iL [+ , +] , E i = ˜ ν iR [ − , +] e iR [ − , +] , i = 1 , , , (A2)where Q i transforms as (2 , / , U i and D i as (1 , / , L i as (2 , − / , and E i as (1 , − / .The parity assignment ± denote the boundary conditions applied to the spinors on the[UV , IR] brane, with + ( − ) being the Neumann (Dirichlet) boundary conditions. Onlyfields with the [+,+] parity contain zero-modes that do not vanish on the brane. Thesesurvive in the low energy spectrum of the 4D effective theory, and are identified as the SMfields. Note that since we are only interested in the charged leptons in this work, we needonly the RH charged leptons, and it is not necessary to have the doubling in the leptonsector as in the quark sector.A given 5D bulk fermion field, Ψ, can be KK expanded asΨ L,R ( x, φ ) = e σ/ √ r c π ∞ X n =0 ψ ( n ) L,R ( x ) f nL,R ( φ ) , (A3)where subscripts L and R label the chirality, and the KK modes f nL,R are normalized accord-ing to 1 π Z π dφ f n⋆L,R ( φ ) f mL,R ( φ ) = δ mn . (A4)The KK-mode profiles are obtained from solving the equations of motion. For zero-modes,we have f L,R ( φ, c L,R ) = s kr c π (1 ∓ c L,R ) e kr c π (1 ∓ c L,R ) − e (1 / ∓ c L,R ) kr c φ , (A5)20here the c-parameter is determined by the bulk Dirac mass parameter, m = c k , and theupper (lower) sign applies to the LH (RH) label. Depending on the orbifold parity of thefermion, one of the chiralities is projected out.After spontaneous symmetry breaking, the Yukawa interactions localized on the IR branelead to mass terms for the fermions on the IR brane S Yuk = Z d x v W kr c π h Q ( x, π ) λ u U ( x, π ) + Q ( x, π ) λ d D ( x, π ) + L ( x, π ) λ e E ( x, π ) i + h . c . , (A6)where v W = 174 GeV is the VEV acquired by the Higgs field, and λ u, d, e are (complex)dimensionless 5D Yukawa coupling matrices. For zero-modes, which are identified as theSM fermions, this gives the mass matrices for the SM fermions in the 4D effective theory( M RSf ) ij = v W kr c π λ f ,ij f L ( π, c Lf i ) f R ( π, c Rf j ) ≡ v W kr c π λ f ,ij F L ( c Lf i ) F R ( c Rf j ) , (A7)where f labels the fermion species. The mass matrices are diagonalized by a bi-unitarytransformation ( U fL ) † M RSf U fR = m f m f
00 0 m f , (A8)where m fi are the masses of the SM quarks and leptons. The mass eigenbasis is then definedby ψ ′ = U † ψ , and the CKM matrix given by V CKM = ( U uL ) † U dL .Because of KK interactions, the couplings of the Z to fermions are shifted from their SMvalues. These shifts are not universal in general, which leads to flavor off-diagonal couplingswhen the fermions are rotated from the weak eigenbasis to the mass eigenbasis: L FCNC ⊃ g L c W Z µ ( Q LZ ( f ′ ) X a,b ˆ κ Lab ¯ f ′ aL γ µ f ′ bL + Q RZ ( f ′ ) X a,b ˆ κ Rab ¯ f ′ aR γ µ f ′ bR ) . (A9)The mass eigenbasis is defined by f ′ = U † f , and the flavour off-diagonal couplings are givenby ˆ κ L,Rab = X i,j ( U † L,R ) ai κ L,Rij ( U L,R ) jb , (A10)where κ ij = diag( κ , κ , κ ) are the coupling shifts due to KK interactions in the weakeigenbasis. Note that there would be no flavor violations if κ is proportional to the identitymatrix. 21he coupling shifts κ ij arises from two sources: mixing between the SM Z and neutral KKgauge bosons, and that between the SM fermions and their KK excitations. These mixingprocesses are depicted in Fig. 7. In the gauge mixing diagram, X can be either the SM Z (cid:1) X KK Zf ¯ f < H > < H > (cid:1) f KKα f KKα Zf i f j < H > < H > FIG. 7: Effective Zf ¯ f coupling due to gauge and fermion KK mixings or the Z ′ that arises from the SU (2) R , while in the fermion mixing diagram, i , j , and α aregeneration indices. Note that the Yukawa interaction mixes LH (RH) SM fermions with theRH (LH) KK fermions. One can see from the diagrams in Fig. 7 that κ ij depend on theoverlap of fermion wavefunctions, which are controlled by how the fermions are localized inthe extra dimension. Details of the calculation and the full expressions of the contributionsdue to gauge and fermion KK mixings can be found in Ref. [26]. APPENDIX B: TYPICAL ADMISSIBLE CHARGED LEPTON MASS MATRI-CES
Parameterizing the complex 5D Yukawa couplings as λ ,ij = ρ ij e iφ ij , admissible massmatrices of the forms given by Eq. (A7) are found with ρ ij and φ ij randomly generated in theintervals (0 ,
2) and [ − π, π ) respectively. In the following, for each of the five charged leptonconfigurations determined by the c-parameter sets c L and c E , we list for a typical solutionthat passes all current LFV constraints the mass eigenvalues, the magnitude of the complexmass matrix, | M | , of the solution, and the magnitude and phase of the corresponding 5DYukawa coupling matrix . All numerical values are given to four significant figures (exceptfor the mass eigenvalues, which required more accuracy to distinguish m τ ). The masseigenvalues agree with the charged masses at 1 TeV found in Ref. [29] to within one standard We need not list arg M because it is the same as φ , as can be seen from Eq. (A7). • Configuration I: c L = { . , . , . } , c E = {− . , − . , − . }{ m e , m µ , m τ } = { . · − , . , . } (B1) | M | = . . . . . . . . . (B2) ρ = . . . . . . . . . , φ = . − . − . − .
509 2 . − . − .
294 1 .
547 0 (B3) • Configuration II: c L = { . , . , . } , c E = {− . , − . , − . }{ m e , m µ , m τ } = { . · − , . , . } (B4) | M | = . . . . . . . . . (B5) ρ = . . . . . . . . . , φ = − . − .
623 3 . − . − . − . − .
274 2 .
565 0 (B6) • Configuration III: c L = { . , . , . } , c E = {− . , − . , − . }{ m e , m µ , m τ } = { . · − , . , . } (B7) | M | = . . . . . . . . . (B8)23 = .
375 0 . . . .
014 0 . . . . , φ = − . − .
449 0 . − . − . − . .
814 2 .
844 0 (B9) • Configuration IV: c L = { . , . , . } , c E = {− . , − . , − . }{ m e , m µ , m τ } = { . · − , . , . } (B10) | M | = . . . . . . . . . (B11) ρ = .
556 0 . . .
009 1 .
108 0 . . . . , φ = .
701 1 . − . − . − .
094 2 . − .
407 0 . (B12) • Configuration V: c L = { . , . , . } , c E = {− . , − . , − . }{ m e , m µ , m τ } = { . · − , . , . } (B13) | M | = . . . . . . . . . (B14) ρ = .
268 0 . . . . . . . . , φ = . − .
288 1 . . − . − . − .
560 2 .
667 0 (B15) [1] C. Prescott et al. , Phys. Lett.
B77 , 347 (1978).[2] P. L. Anthony et al. [SLAC E158 Collaboration], Phys. Rev. Lett. , 081601 (2005).[3] A. Czarnecki and W. J. Marciano, Phys. Rev. D53 , 1066 (1996).[4] J. Erler and M. J. Ramsey-Musolf, Phys. Rev. D , 073003 (2005).[5] O. Adriani et al. [PAMELA Collabration], arXiv: 0810.4995 [astro-ph].
6] S.W. Barwick et al. [HEAT Collaboration], Astrophys. J.
L191 (1997).[7] M. Aguilar et al. [AMS-01 Collaboration], Phys. Lett.
B646
145 (2007).[8] J. Chang et al. [ATIC Collaboration], Nature
362 (2008).[9] S. Torii et al. [PPB-BETS Collaboration], arXiv: 0809.0760 [astro-ph].[10] N. Arkani-Hamed and N. Weiner, arXiv: 0810.0714 [hep-ph].[11] W. F. Chang, J. N. Ng and J. M. S. Wu, Phys. Rev.
D74
022 (2007).[13] For a review, see J. Hewett and T. Rizzo, Phys. Rep. , 193 (1989).[14] D. Feldman, Z. Liu and P. Nath, Phys. Rev. Lett. Unification and Supersymmetry
Springer-Verlag (1996).[16] L. Randall and R. Sundrum, Phys. Rev. Lett , 3370 (1999).[17] W. F. Chang and J. N. Ng, JHEP , 091 (2005).[18] A. Leike, Phys. Rep.
143 (1999), P. Langacker, arXiv: 0801.1345 [hep-ph].[19] T. Appelquist, B. A. Dobrescu and A. R. Hopper, Phys. Rev. D , 035012 (2003).[20] K. R. Dienes, C. Kolda, and J. March-Russell, Nucl. Phys. B492 , 104 (1997).[21] W. J. Marciano, Nucl. Phys. Proc. Suppl. , 5 (1989).[22] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. , 171802 (2007).[23] B. Holdom, Phy. Lett. B166 , 196 (1991), J. Kumar and J. D. Wells, Phy. Rev.
D74 , 115017(2006)[24] G. Bennet et al. [E821 Muon g − D 73 , 050 (2003)[26] W. F. Chang, J. N. Ng and J. M. S. Wu, Phys. Rev. D , 096003 (2008); arXiv: 0809.1390[hep-ph].[27] C. Csaki, A. Falkowski and A. Weiler, JHEP , 049 (2008).[28] S. Casagrande, F. Goertz, U. Haisch, M. Neubert and T. Pfoh, arXiv: 0807.4937 [hep-ph].[29] Z. Z. Xing, H. Zhang and S. Zhou, Phys. Rev. D , 113016 (2008)., 113016 (2008).