LEP1 vs. Future Colliders:
수식 1과 같은 두 가지 방법으로 기존의 효과적 람다 방식을 일반화할 수 있다.
기존의 효과적 람다 방식:
L = φ+σiφW i
µνBµν + 1
3!ǫijkW νj
µ W λkν W µiλ
장대게이지 군(context) 내에서는 다음과 같은 추가 적분을 가능하게 한다.
L = L 기존 + φ+σiφW i
µνB′µν
이 논문에서는 이 새로운 방법의 적용을 테스트하기 위해 특정 모델, 즉 SU(2)×U(1)×U(1)의 확장 게이지 군(context)를 가정했다. 이 확장 게이지 군(context)는 한 가닥 초격자 필드(extra gauge boson field)인 B′를 포함한다.
이 모델에서 우리는 W와 Z 사이의 교차 섭동을 계산했다. 결과는 다음과 같다:
dσ(e+e−→W +W −)/d cos θ = (s/4-M2
W)1/2|A|2
여기서 A는 t-채널 ν 교환 및 s-채널 γ, Z, Z′ 교환을 포함한다. 이 논문에서는 εWB′를 0에 설정하고 Z-Z′混合으로 인한 교차 섭동을 계산했다.
결과적으로 우리는 다음과 같은 결과를 얻었다:
∆sin2 θW|eff ≤ 0.001 (1σ)
이 값은 기존의 효과적 람다 방식에서 εWB로 설정할 때 얻을 수 있었던 값보다 크게 나타났다. 이것은 작은 Z-Z′ 혼합角 θ3으로 인해 발생하는 screening 효과 때문이라고 한다.
한편, 우리는 200 GeV와 260 GeV 에너지에 대해 WW 교차 섭동의 차분크로스를 계산했다. 결과는 다음과 같다:
dσ(e+e−→W +W −)/d cos θ = (s/4-M2
W)1/2|A|2
이 값은 기존의 효과적 람다 방식에서 εWB로 설정할 때 얻을 수 있었던 값보다 크게 나타났는데, 이는 작은 Z-Z′ 혼합각 θ3으로 인해 발생하는 screening 효과 때문이라고 한다.
따라서, 이 논문에서는 표준 모델(SM) 이외의 물리학을 연구하기 위한 새로운 방법을 제안했다. 이 방법은 기존의 효과적 람다 방식과 달리 Z-Z′ 혼합각 θ3으로 인한 교차 섭동을 계산할 수 있다.
영역에 대한 간단한 예를 통해, 우리는 새로운 방법이 표준 모델(SM)에서 벗어날 수 있는 지수를 찾았다. 결과는 다음과 같다:
∆sin2 θW|eff ≤ 0.001 (1σ)
이 값은 기존의 효과적 람다 방식에서 εWB로 설정할 때 얻을 수 있었던 값보다 크게 나타났는데, 이는 작은 Z-Z′ 혼합각 θ3으로 인해 발생하는 screening 효과 때문이라고 한다.
기존의 효과적 람다 방식을 일반화한 새로운 방법은 표준 모델(SM) 이외의 물리학을 연구하기 위한 유용한 도구가 될 수 있다.
LEP1 vs. Future Colliders:
arXiv:hep-ph/9207258v1 24 Jul 1992CERN–TH.6573/92ULB–TH–04/92hepth@xxx/9207258July 1992LEP1 vs. Future Colliders:Effective Operators And Extended Gauge GroupJ.-M. Fr`ere†, J.M. Moreno∗, M. Tytgat♭Service de Physique Th´eoriqueUniversit´e Libre de Bruxelles, Boulevard du TriompheCP 225, B-1050 Bruxelles, BelgiumandJ.
Orloff♯Theory Division, CERNCH-1211 Geneva 23, SwitzerlandABSTRACTIn an effective Lagrangian approach to physics beyond the Standard Model,it has been argued that imposing SU(2) × U(1) invariance severely restricts thediscovery potential of future colliders. We exhibit a possible way out in an extendedgauge group context.† Maˆıtre de Recherche FNRS.∗Supported in part by IISN.♭Aspirant FNRS.♯e-mail: orloff@dxcern.cern.ch
1. INTRODUCTIONIn studying departures from the Standard Model (SM), effective Lagrangians[1,2]prove a convenient tool, since they unify the various manifestations of a given con-tribution.
Usually an operator expansion is used and the various possible operatorsare ranked by increasing dimensionality. To any given order, only a finite numberof operators need be considered, and this is true irrespective of the nature of theunderlying SM extension (which only determines the coefficients).Gauge invariance imposed on an effective Lagrangian and precise measurementsat LEP1 have recently been shown[3] [4] [5] to severely restrict the possible impact ofnew physics on observations at forthcoming colliders (LEP2, ...).
The argument,with which we are in complete agreement, proceeds in two steps:• In the framework of gauge theories, extra terms induced by new physics inan effective Lagrangian must at least obey the SU(2)L × U(1) invariance. Inparticular, they must be expressible (above the scale of symmetry breakdown)in terms of [SU(2)L × U(1)]-invariant products of fields, including the SMscalar doublet(s).• These operators can be classified by increasing dimensionality.
If the newphysics scale is high enough, only the lowest-dimensional operators will con-tribute.Gauge invariance thus strongly limits the number of operators available fora given dimensionality; consequently, the typical effects expected at future col-liders, such as new 3-vector vertices, are intimately related to modifications oftypical LEP1 observables (widths, asymmetries, ...). When the analysis is lim-ited to dimension 6 operators, and apart from a few blind directions, the LEP1measurements[6] tend to give stronger bounds on the coefficients of these operatorsthan those expected from the direct observation of the 3-vector vertices.We found it interesting to explore some limitations of this approach.Ourgoal was to see how the above predictions would be affected when the implicit2
assumption of a very high mass scale for any new physics was relaxed.We have chosen to study a specific example, built upon the extended groupSU(2)×U(1)×U(1)′ [7]. This introduces essentially two new fields, namely an extragauge boson (B′µ) and an extra scalar singlet (χ) (an extra νR per generation is alsoneeded, so as to avoid anomalies, but it plays no role in the present discussion).The mass of the extra gauge boson will characterize the scale of this part of thenew physics.
If this scale lies far beyond the one accessible at future colliders, theextra degrees of freedom may be integrated out in an effective lagrangian approach,which leads, for dimension-6 operators, to the same analysis as Ref. [4].We have instead assumed the mass of the new gauge gauge boson to be rela-tively close to the LEP2 scale.One systematic way to treat this model would be to consider the higher ordersin a dimensional expansion.
This, however, quickly becomes intractable—in viewof the number of new operators involved—and unreliable, since doubts arise aboutthe convergence of the expansion (specially close to a resonance); it is then saferto bring in the full gauge group structure SU(2) × U(1) × U(1)′.In the spirit of Ref. [4], we continue to use an effective Lagrangian approach(limited to dimension-6 operators) to parametrize any new physics beyond theZ′ (i.e.
the heavy-mass eigenstate). The only difference lies in the fact that theeffective operators are now to be classified following their invariance under thegauge group SU(2) × U(1) × U(1)′ rather than SU(2) × U(1).At first sight, one might expect that extending the requested symmetry groupwould further restrict the choice of operators.
It turns out that this is more thancompensated for by the presence of the new degrees of freedom Z′ and χ.A typical example involves couplings of W bosons. In the approach of Ref.
[4],these can only arise from two dimension-6 operators, namely:OW B = φ+σiφW iµνBµν(1)3
OW = 13!ǫijkW νjµW λkνW µiλ . (2)In the present case, we can obviously add to this list the operator:OW B′ = φ+σiφW iµνB′µν.
(3)Such an operator is absent from the expansion of Ref.[4]. Nevertheless, ((3)) willgenerate anomalous couplings between the physical Z (now a mixture of W 3, B, B′)and W’s.One needs to formally integrate out B′ (and χ) to compare the two approaches.The situation is most easily exemplified in terms of graphs.In particular, forZW +W −coupling, we get contributions from :BB′φφ†φφ†W +W −Upon integrating out B′ and χ (the presence of the χ field is implicit in the B′mass term), we get:Bφφφ† φ†W +W −containing e.g.
the new operator (of dimension 8 in the fields)1M2B′φ+φ Bµν φ+τjφ W jµν. (4)On top of other dimension-8 operators, higher dimensionalities will of course appear4
through the low-momentum expansion of the B′ propagator.Let us now turn back to the full SU(2) × U(1) × U(1)′ and review briefly theparameters involved (we use g2, g1, g′1 for the respective gauge couplings and Y, Y ′for the hypercharges v =√2| ⟨φ⟩|, V = | ⟨χ⟩|). The field content and quantumnumber assignments are detailed in Table 1.We now include the additional operator (3) in the Lagrangian density:LW B′ = εv2.OW B′.
(5)We have followed Ref. [4] in the definition of ε.In addition to 3-gauge boson couplings of interest for future colliders, thisimplies (after symmetry breaking) corrections to the kinetic energy of neutral gaugebosons.
The Z-propagator is thus modified, with potential effects at LEP1. It ishowever easy to check that the propagator of the light Z physical state is notdirectly affected by the ε correction, but only by the combination εθ3, where θ3 isthe standard Z-Z′ mixing angle.
As a result, the departure from a pure SU(2) ×U(1) × U(1)′ is kept minimal once θ3 remains within its usual bounds[8].The same cannot be said of the e+e−→W +W −cross section. The effectiveoperator (3) manifests itself in two ways in this process:• The currents are modified by the redefinition of the boson fields (e.g.
the B′content of Z)• Direct contributions to the three boson vertices appear. In particular, theZ′ −W +W −(O(θ3)) vertex already present in the extended SU(2)×U(1)×U(1)′ now receives an O(ε) contribution.
That coupling is not suppressed byθ3 and thus becomes competitive with the other contributions, with which itinterferes. Its angular dependence makes it quite conspicuous.5
2. OVERVIEW OF THE CALCULATIONSThe differential cross-section for the process e+e−→W +W −(see Ref.
[9]) isdσ(e+e−→W +W −)d cos θ= s1/2(s/4 −M2W)1/2|A|2,(6)where A takes into account the contributions from the four usual diagrams corre-sponding to the t-channel ν exchange and the s-channel γ, Z, Z′ exchanges|A|2 = 18πXα,βaVα Spinα,β aVβ + aAα Spinα,β aAβ,(7)withaV (A)α=ν,γ,Z,Z′ = g224t,eγV (A)gγs,eZV (A)gZs −M2Z,eZ′V (A)gZ′s −M2Z′! (8)andSpinν,ν = utM4W−1 t24s2 + M4Ws2+t2sM2WSpinν,i = utM4W−1 κit4s −M2W t2s2 −M4Ws2+ (1 + κi)t2M2W−ts + M2WsSpini,j = utM4W−1 κiκj4−M2Ws(1 + κiκj)2+ 3M4Ws2+ (1 + κi)(1 + κj)s4M2W−1;i = γ, Z, Z′(9)describing the angular dependence.
The different couplings eiV (A), gi, κi are definedthrough the matrix S relating the physical neutral fields to the original gaugefields⋆:W 3µBµB′µ= S.AµZµZ′µ(10)⋆S is not unitary, as εWB′ induces a non-canonical kinetic term requiring a rescaling of thefields.6
by(gγ, gZ, gZ′) = (g2, 0, 0).Sgγ(1 + κγ), gZ(1 + κZ), gZ′(1 + κZ′)= (g2, 0, ε).SeγL(R), eZL(R), eZ′L(R)= (g2T 3eL(R), g1Y BeL(R), g′1Y B′eL(R)).SeV (A) = 12(eR + (−)eL).(11)3. DISCUSSION OF THE NUMERICAL RESULTSWe have not pursued a systematic search of the parameter space, but presenthere a simple example.
While the parameters are obviously chosen to make ourpoint clear, we have not attempted to maximize the effects to the extreme limitsallowed by current data. We have taken a smallish value for the new couplingstrength g′1 (λ = g′1/g2 = 0.1), which allows for a relatively small Z′-mass (MZ′ =300 GeV) without dangerous direct contributions.
In the same spirit, the Z-Z′mixing is kept small (θ3 ∼0.008) to control the indirect ones[8].As we mentioned above, all the effects of the additional operator OW B′ at LEP1are suppressed by a factor θ3ε. We do not present here complete fits to the LEP1data, but use instead the evolution of sin2 θW |eff as the dominant constraint†.Taking values of ε between −0.2 and 0.2, we find ∆sin2 θW |eff ≤0.001 ∼1σ[6].The fact that the values for ε are larger than those found for the similar parameterεW B[4] is just an illustration of the screening effect due to the small mixing angleθ3.We have plotted the differential cross-section at 200 GeV, which correspondsapproximately to the maximum of the WW cross-section.
We also give a plot for260 GeV, to show how the effect of ε is boosted by nudging the energy only a littlecloser to the pole of the Z′.† One can check that the same na¨ıve approach applied to the parameter εWB of Ref. [4]essentially reproduces their global fit.7
We first make out some qualitative remarks. With ε set to 0, the cross-sectionis indistinguishable from the SM one.
It is an interesting question to know whetherother channels might reveal the presence of the extended gauge group[10] indepen-dently of gauge boson couplings. We have only checked the most obvious channel—µ-pair production—and found that while the relative difference in cross-section wasindeed sizeable, the overall value of this cross-section was very small, which mightcause a problem with statistics.The presence of the anomalous gauge boson couplings controlled by ε reflectsin a modification of the cross-section.
This is not uniform, and a decrease in thebackward part is compensated for by an increase in the forward one. A detailedstudy of the best observables to detect the effect depends obviously on the detailedproperties of the detectors (angular resolution, ...) and falls beyond the scope ofthis paper.
If one wants to use a simple number to quantify the departures fromthe SM, inspection of the curves suggests resorting to A135 = σθ>135−σθ<135σtotal(theangle 135 is close to optimal at 200 GeV, and should be adapted as a function ofenergy) or to the deviation at the maximum of the cross-section.The numerical results are displayed in figs.1 and 2 and gathered in table2 for two values of the centre-of-mass energy (e.g.200 and 260 GeV). As canbe expected, the effects become more important when approaching the Z′ pole.We have not taken into account the radiative corrections for WW production(but included them for LEP1) as we do not expect them to qualitatively alterour conclusions.Indeed they were shown in the Standard Model to be almostθ-independent,[11] depleting at most the irrelevant small θ region by 15%.
Altoughthe presence of the Z′ will slightly change their behaviour, this can be consideredas a second-order correction for our discussion.As we mentioned above, we have not done an exhaustive exploration of theparameter space, since our purpose was rather to illustrate our proposition than togive a full account of the effects of this kind of operators in the context of extendedgauge groups.8
4. CONCLUSIONSThe model we have examined here shows one possible way to evade the limitsof Ref.
[4].It also gives some idea of the price to pay to achieve this goal.Itrequires both a relatively light Z′ and anomalous couplings of that Z′ (themselvesattributed to unspecified new physics) which are quite sizeable.What can we conclude from the above approach?• Despite strong constraints arising from the (high-luminosity) precise mea-surements at LEP1 and lower energies, the introduction of a larger gaugegroup, broken at a scale higher but still comparable with the SM, consider-ably increases the allowed freedom.• In particular, the anomalous couplings of an extra Z boson are not consid-erably restricted at the LEP1 level, and may lead to important departuresfrom SM expectations at energies reachable in the near future.The present observations do not detract from the importance of Ref. [4], butstress the point that e+e−colliders should be designed with enough flexibility tobe operated as discovery machines.AcknowledgementsWe wish to thank Alain Blondel, Andy Cohen, Alvaro de R´ujula, Bel´en Gavela,Gordy Kane, Olivier P`ene and Mariano Quir´os.9
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TABLE CAPTIONS1: Matter content and charges assignment of the extended model SU(2)×U(1)×U(1)′.2: A135-values for the SM and for SU(2) × U(1) × U(1)′ with ε = 0, ±0.2. A135is defined as follows:A135 = σθ>135 −σθ<135σtotal.11
Hypercharge ud!LucLdcL νe!LecLνcL φ1φ2!χY16−2313−1210120Y ′1515−35−35151−251Table 1Model√s = 200 GeV√s = 260 GeVSM−0.0640.297ε = 0−0.0660.292ε = +0.2−0.0610.290ε = −0.2−0.0890.180Table 212
FIGURE CAPTIONS1) Detail of the unpolarized differential cross-section for the process e+e−→W +W −at √s = 200 GeV: the solid line shows the SM, the dashed one isfor ε = 0.2 and the dash-dotted one for ε = −0.2; θ is the angle between e−and W +.2) Same as fig.1 at √s = 260 GeV.13
Figure 1SMε=+0.2ε=-0.2150160170180θe-W+θe-W+30354045dσ/dcosθ (pb)50556065707580dσ/dcosθ (pb)160165170175180θe-W+Figure 2SMε=+0.2ε=-0.214
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