Transverse Beam Polarization as an Alternate View into New Physics at CLIC
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Transverse Beam Polarization as an Alternate Viewinto New Physics at CLIC ∗ †
Thomas G. Rizzo
SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, CA, 94025
Abstract In e + e − collisions, transverse beam polarization can be a useful tool in studying theproperties of particles associated with new physics beyond the Standard Model(SM).However, unlike in the case of measurements associated with longitudinal polarization,the formation of azimuthal asymmetries used to probe this physics in the case oftransverse polarization requires both e ± beams to be simultaneously polarized. Inthis paper we discuss the further use of transverse polarization as a probe of newphysics models at a high energy, √ s = 3 TeV version of CLIC. In particular, we show( i ) how measurements of the sign of these asymmetries is sufficient to discriminate theproduction of spin-0 supersymmetric states from the spin-1/2 Kaluza-Klein excitationsof Universal Extra Dimensions. Simultaneously, the contribution to this asymmetryarising from the potentially large SM W + W − background can be made negligiblysmall. We then show ( ii ) how measurements of such asymmetries and their associatedangular distributions on the peak of a new resonant Z ′ -like state can be used to extractprecision information on the Z ′ couplings to the SM fermions. ∗ Work supported in part by the Department of Energy, Contract DE-AC02-76SF00515 † e-mail:[email protected] Introduction and Background
With the LHC now running at √ s = 7 TeV and eventually running at 14 TeV, NewPhysics(NP) beyond the SM at the TeV scale is expected to show up sometime soon. Basedon theoretical prejudice this NP should assist in our understanding of the hierarchy prob-lem and may be intimately involved in the breaking of electroweak symmetry. Althoughthere are many speculations, it is not known what form this new physics might take withSupersymmetry[1], extra dimensions[2] and extended gauge theories[3] being among the po-tential candidates. Once the LHC discovers this physics the more difficult task of uncoveringthe underlying theory will still lay ahead. In all probability it is quite likely that the LHCmay not provide us with sufficient information to address this problem in full detail andmany in the community expect that the precision measurements available at a lepton col-lider, whose center of mass energy will depend on the precise mass scale of this NP, will benecessary to provide the complete answer to these questions.After the determination of its mass, the most elementary and important properties ofany new particle are its spin and the nature of its couplings to the familiar SM fields; varioustools will be necessary to obtain this information. The possibility of using transverse beampolarization in e + e − collisions to explore the details of many various NP scenarios has nowbecome rather well-established[4] over the last few years. However, unlike in the case of thephysics studies employing longitudinal beam polarization, to study NP with asymmetriesproduced in the transversely polarized case both e ± beams need to be polarized. The reasonfor this is that the corresponding asymmetry parameters, A , associated with azimuthalangular distributions are directly proportional to the product of these two polarizations, i.e. , A ∼ p T p T . ‡ In this paper we will explore two scenarios in which transverse polarization canbe used to obtain useful information about the properties of new particles in high energy e + e − collisions. To be specific, we will focus on very high energy collisions, √ s = 3 TeV,as are eventually envisioned at CLIC. We will assume that integrated luminosities in thefew ab − range are available and further assume that that the product of the magnitudes ofthe transverse polarizations | P T P T | = 0 .
5, not an unreasonable value given the estimatesfor possible positron polarization, for purposes of demonstration. Our philosophy in thepreliminary studies presented here is not to ignore the well-known techniques that employlongitudinal polarization to address these issues but to explore instead the capabilities oftransverse polarization to provide an alternative window into the same NP.The first scenario we consider is new particle spin discrimination. As is well-known,measurements at lepton colliders provide at least two relatively straightforward ways todetermine the spin of a particle which is pair produced in, e.g. , the s -channel. These tech-niques would then allow us to discriminate, e.g. , the production of SUSY particles from theKaluza-Klein(KK) excitations of Universal Extra Dimensions (UED)[5]. First, a threshold ‡ Here we also recall that transverse beam polarization is the ‘natural’ state of polarized beams at suchcolliders and that spin rotators are employed to produce the more conventionally studied case of longitudinalbeam polarization. ∼ β turn-on in the case of spin-0 sleptons and squarks, with β being the particle’s velocity, whereas the spin-1/2 KK excitations would instead have a ∼ β behavior. Second, far above the threshold region, the production of the spin-0 SUSY stateswould lead to a ∼ − cos θ angular distribution whereas the KK states, being vector-like,would instead yield a ∼ θ distribution. In a classic work[6], it has been explicitlyshown how such observables can be used at CLIC, running at √ s =3 TeV, to differentiate ∼
500 GeV smuons from the level-1 KK muons in UED. Although these two techniques arepowerful there could be situations where even observing these states associated with NPmight prove to be difficult[7, 8] so that it can be worthwhile having an additional tool athand to assist in spin discrimination. As we will demonstrate below, transverse polarizationcan provide such an additional tool. To be specific, the analysis presented below is based ona few simple observations: ( i ) The magnitudes of the transverse polarization asymmetries, A ,for scalars and vector-like fermions are large and are of opposite sign whereas A for W -bosonpair production, the largest SM background, is highly suppressed. ( ii ) The cos θ dependenceof A for both scalars and vector-like fermions behaves as ∼ − cos θ ; this also remains truein the cases of W -pair production to a very good approximation so that the signal shape isnot distorted by the background. ( iii ) Suitable angular cuts can be used to remove the bulkof this W -pair background.In the second scenario we consider the existence of a Z ′ -like state and ask the hypotheticalquestion: ‘What information about this state can be provided through the use of transversepolarization if it is employed instead of longitudinal polarization?’. As is very well-known,the existence of longitudinal polarization allows for the determination of the Left-Rightpolarization asymmetry, A LR , as well as the polarized Forward-Backward asymmetry forany final state fermion f , A polF B ( f ), from which coupling information can be extracted whencombined with unpolarized data such as partial widths. Here we will show that detailedmeasurements of the decay azimuthal angular distribution can be also employed to extractanalogously useful information on the couplings of this Z ′ state to the SM fermions. At CLIC luminosities ( ∼ ab − ) and energies ( ∼ W -pair production with both W ’s decaying to muons which provides the largestSM background after γγ initiated states are removed with the appropriate cuts. (In whatfollows it will be assumed for simplicity that both the smuons and the KK states will decaywith 100% branching fractions to muons plus missing energy.)Let us first consider the production of smuons; ignoring for the moment small terms2igure 1: Cross section for the production of left- and right-handed smuons (red and bluecurves, respectively) and the corresponding level-1 KK states in UED (green and magentacurves, respectively) as a function of their mass at a 3 TeV e + e − collider. The correspondingcross section for W -pair production followed by their subsequent decay to muons is alsoshown (solid line) for comparison purposes. 3roportional to ∼ Γ Z /M Z away from the Z -pole, the double differential cross section isgiven by dσdzdφ = α β s (1 − z ) [ F A − P T P T F B cos 2 φ ] , (1)where β is the speed of the smuon as usual, z = cos θ , and F A ( B ) = Q e Q f + 2 Q e Q f v e G V R + ( v e ± a e ) G V R , (2)with v ( a ) e being the vector(axial-vector) coupling of the electron to the Z , Q e,f being theelectric charges of the electron and final state particle such that G V = B ( T f − x w Q f ) , v e = B ( − / x w ) , a e = − B/ . (3)for any f in the final state. Here we have defined the familiar dimensionless quantities R = s ( s − m Z ) / [( s − m Z ) + Γ Z M Z ] , R = s / [( s − m Z ) + Γ Z M Z ] , (4)with B = [ √ G F M Z πα ( M Z ) ] / employed in the coupling definitions above. We note that the z − and φ − dependencies of this distribution are observed to factorize in a rather simple manner.Integration over z and normalizing to the total cross section yields the azimuthal distribution1 σ dσdφ = 12 π (1 + λ cos 2 φ ) , (5)where we have defined λ = − P T P T F B F A , (6)Further separate integration over the ‘odd’ and ‘even’ regions where cos 2 φ takes on oppositesigns yields the transverse polarization asymmetry A = R odd dσ R all dσ = 2 λπ . (7)If we had inverted the orders of the z and φ integrations we would observe that this azimuthalasymmetry itself has a very simple sin θ behavior: dAdz = 34 (1 − z ) A . (8)Now let us examine the corresponding results for the case of the level-1 KK muons;recalling that these are vector-like fermions we quickly obtain dσdzdφ = α β s (cid:16) F A [(1 + z ) + (1 − β )(1 − z )] + P T P T (1 − z ) β F B cos 2 φ (cid:17) , (9)4here all of the quantities are as defined above. Following the same procedure as in the caseof smuons we find that for particles with the same electroweak quantum numbers A KK = − β − β A ˜ µ , (10)with the asymmetry having same sin θ angular dependence as was found for smuons. Thusboth sets of particles yield asymmetries of comparable size but of opposite sign.In the case of W -pair production the analogous differential cross-section can be writtenas[9] dσdzdφ ∼ | M L ( z ) | + | M R ( z )) | + 2 P T P T Re ( M L ( z ) M R ( z ) ∗ ) cos 2 φ , (11)where M L,R are the corresponding, z -dependent, left- and right-handed helicity amplitudes.The resulting transverse polarization asymmetry is then found to be proportional to theratio of the integrals over z : A ∼ R dz Re ( M L M ∗ R ) R dz ( | M L ( z ) | + | M R ( z )) | ) . (12)Now as is well-known, in the case of W -pair production, since the W couples in a purelyleft-handed manner to the SM fermions we can (symbolically) observe that M L >> M R .This leads to a rather small value for the asymmetry since A ∼ M R /M L ; to obtain thecorresponding A ( z ) distribution in this case, since it is defined relative to the total crosssection, we simply omit the z integration in the numerator above. One then finds that A ( z )in the case of W -pair production also roughly behaves as 1 − z at CLIC energies.Fig. 2 shows the values of the transverse polarization asymmetry, A , for both smuonsand level-1 KK muons, as a function of their mass, as well as for W -pairs at a 3 TeVCLIC. Whereas both smuons and muon KK states are observed to have large transversepolarization asymmetries, that for W ’s is seen to be highly suppressed by comparison due tothe dominantly LH couplings of the W . The upper panel of Fig. 3 shows the correspondingshapes of the idealized azimuthal angular distributions for these same three states whereasin the lower panel of Fig. 3 we find their binned A ( z ) distributions. Since the W -pairasymmetry is so small in comparison to the two cases of interest it is difficult to see that italso has a roughly sin θ shape.Although the W -pair background has a very small asymmetry, it’s presence as a back-ground will end up diluting the asymmetry signal from either of the two NP sources makingspin discrimination more difficult. Ordinarily, when we are performing the identical parallelstudy employing longitudinal polarization, we can freely choose this polarization to be right-handed to remove a very large part of the W -pair induced background; we can’t do that hereso we need to resort to some other cut(s) to remove the W -pair contamination. We recall,however, that the z -distribution for W -pair production, which for √ s = 3 TeV is shownin the top panel of Fig. 4, is highly peaked in the forward direction. This means that the5igure 2: The transverse polarization asymmetry, A , for smuons and level-1 KK muons as afunction of their masses at a 3 TeV e + e − collider with the curves labeled as in the previousfigure.negatively charged muon from the W − will be correspondingly very forwardly peaked due tothe large boost. On the otherhand, negatively charged muons arising from either the decayof the smuons or KK states will be just as likely to go in either the forward or backwarddirections since the angular distributions shown above for the production of pairs of theseparticles are seen to be even functions of z . Thus removing events with negatively(positively)charged muons in the forward(backward) hemisphere will reduce the signal by only a factorof 2 while substantially reducing the W induced background. In order too see how large ofan effect this cut has on this background, the lower panel in Fig. 4 shows the W angularevent rate integrated over the range − ≤ z ≤ z cut . For z cut = 0, i.e. , performing the cutas described above and removing negative(positive) muons in the forward(backward) hemi-sphere, this is seen to reduce the background by more than a factor of ≃
60 which is morethan adequate for our purposes since the original W -pair cross section and the signal crosssections are roughly comparable.What will these azimuthal angular distributions for the NP look like at CLIC? In order tobe specific, we assume smuon/KK masses of 500 GeV and an integrated luminosity of 2 ab − ;the results for the event distributions are then shown in Fig. 5. The corresponding result forthe W -induced background, both before and after applying the angular cut discussed above,is shown in Fig. 6; here we see that the cut makes this background negligible. We note thatthe azimuthal distributions for the smuons and the KK states are quite easily distinguishable6igure 3: (Top) The idealized azimuthal angular distributions for 500 GeV smuons and level-1 KK muons at a 3 TeV e + e − collider. The results for W -pairs is also shown for comparison.The curves for smuons and muon KK states are interchanged in comparison to the previousfigures. (Bottom) Binned A ( z ) distributions for the same cases as described in the previousfigure. 7igure 4: (Top) Angular distribution for unpolarized e + e − → W + W − at √ s =3 TeV.(Bottom) The number of W induced muon events at this same energy requiring the negativemuon to lie in the range − ≤ z ≤ z cut . 8or these masses with the assumed polarizations and integrated luminosities.This preliminary analysis indicates that transverse polarization asymmetries may be auseful tool at CLIC to help to discriminate particle spins. A more detailed study includingfull SM backgrounds, ISR/beamstrahlung and detector effects should be performed to verifythese results. Z ′ Coupling Determinations
If a new Z ′ -like resonance is discovered at the LHC, we will want to know all of its properties,in particular its couplings to the SM fields, as well as the underlying theory which gave riseto it. Unfortunately, it is very likely that the LHC will be unable to perform this analysisin all generality, even if the Z ′ is relatively light, due to the lack of a sufficient number ofobservables and limited integrated luminosity[3, 10]. If this is indeed the case, then the datafrom an e + e − collider will be crucial, especially so if the Z ′ mass is within the √ s range ofthis collider so that we can then sit on the Z ′ pole. Here we imagine that such a state existswith a mass of 3 TeV (or less) and that we can sit on top of this resonance with CLIC.Ordinarily, with longitudinal beam polarization, the following observables are availableto perform coupling determinations for any final state f : Γ f , the partial widths, A F B ( f ),the Forward-Backward asymmetries, A polF B ( f ), the Polarized Forward-Backward asymmetriesand A LR , the Left-Right asymmetry. If longitudinal polarization is not available then wecan’t employ the last two observables and we need some ‘replacements’ from obtained bythe use of transverse polarization. To this end let us re-examine the normalized azimuthalangular distribution for massless fermions on top of the Z ′ resonance; the general form ofthis distribution is given by1Γ d Γ dφ ∼ P T P T ( λ cos 2 φ − τ f sin 2 φ ) , (13)where the parameter λ , apart from a different choice of normalization factor, was describedabove. On the Z ′ pole, we find that λ only depends upon the vector and axial vectorcouplings of the electron to the Z ′ and in that sense is completely analogous to A LR : λ = v ′ e − a ′ e v ′ e + a ′ e . (14)Here, and in what follows, a (un)primed coupling is one that corresponds to a coupling tothe ( Z ) Z ′ . The parameters τ f do not appear in the original expression for this distributionin the previous section since they originate from the absorptive part of the amplitude andare not significant away from resonances and were dropped in that analysis. On the Z ′ pole,however, we find (dropping terms that are subleading in M Z /M Z ′ << τ f = 2 Q e Q f a ′ e v ′ f + v e a ′ e ( v f v ′ f + a f a ′ f )( v ′ e + a ′ e )( v ′ f + a ′ f ) Γ Z ′ M Z ′ , (15)9igure 5: Azimuthal event distributions for smuons(top) and KK muons(bottom) assumingmasses of 500 GeV, √ s = 3 TeV and a luminosity of 2 ab − . The histogram color labels areas in the previous figures. 10igure 6: Same as the previous figure but now for the W -pair background both before(blackhistogram) and after(red histogram) the cut on z is applied.which are parametrically ‘small’ since the width-to-mass ratio for a typical Z ′ is likely to bein the range of a few percent. However, given the anticipated CLIC luminosity the eventsamples are expected to be huge as the peak cross section ( after ISR and beamstrahlungcorrections are accounted for) for the fermion f is expected to be σ f ≃ . × ( B e B f / . B e ( f ) are the electron and f branching fractions of the Z ′ . With this enormousamount of statistics, precision measurements of λ and the τ f should be rather straightforwardand it is likely that many of these and other observables on the Z ′ resonance will becomesystematics limited.How sensitive are these observables to changes in the Z ′ couplings and can they beused (in conjunction with the with the unpolarized observables) to obtain the parameters ofthe underlying theory? This, of course, requires a detailed study but we can get a strongindication of what may be possible by looking at a few examples. Fig. 7 shows the values ofthe observables λ and τ f = l,b,c for the well-known set of Z ′ originating in E models[3]. Notethat in such models, a single parameter, θ , controls all of the fermionic couplings to the Z ′ .From this figure we see that these transverse polarization observables are quite sensitive tothe value of this parameter.As a further example to probe the coupling sensitivity of these observables, we considera Z ′ from the Left-Right Symmetric Model[3] where the only free parameter is the ratio ofthe two SU (2) L,R gauge couplings, κ = g R /g L . Fig. 8 shows the values of λ and τ f = l,b,c forthis class of models. Again we see that transverse polarization observables are quite sensitive11igure 7: Values of the observables λ (top) and τ f = l,b,c (bottom), as represented by the red,green and blue curves, respectively, as a function of the parameter θ for Z ′ originating from E models. 12o the value of κ through the various fermion couplings. Clearly, these new observables doa respectable job at providing substitute coupling information to that obtainable from A LR and A polF B ( f ) when transverse polarization is available instead of longitudinal polarization.Figure 8: Same as the previous figure but now for the Left-Right Symmetric Model as afunction of the parameter κ = g R /g L .It is also interesting to consider what would happen if the new resonance were not aspin-1, Z ′ but were instead a spin-2 graviton KK excitation as in the original version of theRandall-Sundrum model[2, 11, 12]. In such a case the on-resonance double differential cross13ection for a massless fermion in the final state would take the generic form dσdzdφ ∼ (1 − z + 4 z )[1 − P T P T cos 2 φ ] + Γ M P T P T F ( z, v, a ) sin 2 φ , (16)where F is a rather complex function of the couplings and z , from which we learn severalthings. Most importantly, the z -dependence of the the unpolarized cross section and thatof the cos 2 φ part of the azimuthal distribution are seen to be identical which is somewhatreminiscent of the case of scalar particle pair production through s -channel spin-1 gaugeboson exchange as was discussed above. Secondly, we see that the value of the λ parameteris completely fixed, i.e. , it is universal and independent of the fermion flavor since grav-itational couplings are universal in this scenario. Thirdly, as was the case for the Z ′ , awidth-suppressed sin 2 φ term is again present (although we do not give its explicit form herevia the function F ) that depends upon the interference of the graviton with the usual SM γ and Z exchanges. Clearly this graviton KK resonance will be easily distinguishable from a Z ′ at CLIC. In this paper we have considered the further use of transverse polarization and the analogousazimuthal angular distributions as means to explore the properties of new states producedat the e + e − collider CLIC running at √ s = 3 TeV. Here we have shown that ( i ) transversepolarization asymmetries can be used as a discriminator of particle spin; in particular, thetwo possibilities of smuon or UED KK-fermion pair production can be easily distinguished.( ii ) Furthermore, we have shown that the general form of the azimuthal angular distribution,as measured on top of the pole of a new Z ′ -like state, can provide a powerful handle on thecouplings of the various SM fermions to this new state in a manner analogous to observablesemployed in the case of longitudinal beam polarization. In both of these scenarios, furtherwork will be necessary to more fully understand the power of these observables in a CLIC-like detector environment including the influence of the full SM and other new physicsbackgrounds and the effects of the significant CLIC machine-induced beamstrahlung. Acknowledgments
The author would like to thank J.L. Hewett, R.M. Godbole, G. Moortgat-Pick and S.Riemann for various discussions related to this work.
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