Electroweak constraints on new physics
aa r X i v : . [ h e p - ph ] M a y Fortschritte der Physik, 10 September 2018
Electroweak constraints on new physics
F. del Aguila a ∗ and J. de Blas b ∗∗ a CAFPE and Depto. de F´ısica Te´orica y del Cosmos, Universidad de Granada, E-18071 Granada, Spain b Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USAReceived XXXX, revised XXXX, accepted XXXXPublished online XXXX
Key words
Effective Lagrangian, electroweak precision data constraints, Higgs limits, extended models.
Subject classification
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The time for establishing the mechanism of the Standard Model (SM) symmetry breaking seems to havearrived with the large hadron collider (LHC) era. Thus, it is widely believed that the SM is a low energyeffective theory and that new physics must exist near the TeV scale which makes natural the observedvalues of the gauge boson and fermion masses. However, the four scenarios with or without the SM Higgsand/or new physics observed at LHC are still possible. Although it shall be paradoxical that electroweakprecision data (EWPD) are in agreement with the SM predictions at the few per mille level [1,2], implyingthat the new physics scale is relatively large, but new resonances other than the SM Higgs are detectedat the LHC [3]. EWPD also disfavor a SM Higgs mass much larger than its present direct limit and thatno new physics is found up to a few TeV [4] (see below). At any rate, EWPD and direct searches arecomplementary and whatever physics LHC reveals, it shall fulfill the indirect constraints.Physics could be unexpected but we shall assume that we will be finally left with the SM plus somenew particles with masses above the electroweak scale. Such a scenario can be described for energiesbelow a few hundreds of GeV by an effective Lagrangian with the SM fields and gauge symmetry. Thenew physics being encoded in the operators of dimension d > . In the following we update the limitson these operators, assuming universality and taking one at a time. In general, only those contributing toobservables showing (small) deviations from the SM predictions are not suppressed at the per cent level.We then explain how to accommodate a large Higgs mass, which is the only SM parameter still unknown.What can be done invoking new heavy neutrinos and/or vector bosons. Finally, we comment on how tofulfill the EWPD constraints and still allow for new resonances at the LHC reach. Let us write the effective Lagrangian with the SM fields and symmetries L eff = ∞ X d =4 d − L d = L + 1Λ L + 1Λ L + . . . , L d = X i α di O di , (cid:2) O di (cid:3) = d, (1) ∗ E-mail: [email protected] ∗∗ E-mail: [email protected]
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F. del Aguila and J. de Blas: Electroweak constraints
Operator Z pole W data Low Energy LEP 2 Global fitcoefficient C.L. limits [TeV − ]LLLL α (1) ll Λ - - [ − . , . − . , . − . , . α (3) ll Λ [ − . , . − . , . − . , . − . , . − . , . α (1) lq Λ - - [ − . , . . , . − . , . α (3) lq Λ - [ − . , . − . , . . , . − . , . RRRR α ee Λ - - [ − . , . − . , . − . , . α eu Λ - - [ − . , . − . , − . − . , . α ed Λ - - [ − . , . . , . − . , . LRRL α le Λ - - [ − . , . − . , . − . , . α lu Λ - - [ − . , . . , . − . , . α ld Λ - - [ − . , . − . , − . − . , . α qe Λ - - [ − . , . . , . − . , . SVF α (1) φl Λ [ − . , . - [ − . , . − . , . − . , . α (1) φq Λ [ − . , . - [ − . , . − . , . − . , . α (1) φe Λ [ − . , . - [ − . , . − . , . − . , . α (1) φu Λ [ − . , . - [ − . , . − . , . − . , . α (1) φd Λ [ − . , . - [ − . , . − . , . − . , . α (3) φl Λ [ − . , . − . , . − . , . − . , . − . , . α (3) φq Λ [ − . , . − . , . − . , . (cid:2) · − , . (cid:3) [ − . , . α φud Λ - [ − . , . − . , . - [ − . , . Oblique α (3) φ Λ [ − . , . − . , − . − . , . − . , . − . , . α WB Λ [ − . , . (cid:2) − . , − · − (cid:3) [ − . , . − . , . − . , . Table 1
C.L. limits on ( confidence interval of) the dimension six operator coefficients entering in EWPD.The limits are obtained from a fit considering only one operator at a time and for each data set. Limits are in unitsof TeV − . The different columns show the results for different fits depending on the observables included. Thesecond column ( W data) also includes the constraints from CKM universality. When a given operator contributes to aphysical process from which any of the SM inputs is derived, it indirectly corrects the predictions for all electroweakobservables (e.g., the operator O (3) ll which modifies the prediction for the muon decay constant G µ ). where Λ is the (unknown) cutoff scale up to which the effective Lagrangian description is valid, and each L d contains all the local operators of canonical mass dimension d allowed by the symmetries. ( L onlycontains one operator [5, 6], which violates lepton number and can be neglected because it is proportionalto the very tiny neutrino masses and then plays no rˆole in our analysis [7].) The operators of dimensionsix O i are classified in [6] (see for a non-redundant set [8]). In the table we update the limits on theircoefficients α i / Λ . The data included in the fit are described in [4, 9–12], but updated to their more recentvalues. We separate them in four sets and collect the corresponding bounds in the first four columns. Theglobal fit to all data is gathered in the last column. We assume universality and the fits are performedadding one operator at a time to the SM.As can be observed, some of the most significant departures ( ∼ σ or larger) from the SM predictionscan be eased with few of these operators. This translates into asymmetric intervals. For instance, the excess Copyright line will be provided by the publisher dp header will be provided by the publisher 3 χ M H [GeV] a) SM N e W B χ M H [GeV] b) SM N e W B Fig. 1 a)
From top to bottom, minimum χ as a function of the Higgs mass for the SM fit, and the fits includingbesides a heavy neutrino singlet N e coupled to the first lepton family, a vector triplet of hyperchage one W , and aneutral vector singlet B (see [4, 9, 11] for conventions). b) The same but assuming that the SM Higgs is found to havea mass M H = 130 ±
10 GeV or M H = 250 ±
10 GeV (blue bands). of the hadronic cross section observed at LEP 2 can be explained by four-fermion operators involvingelectrons and quarks, like O (1 , lq , etcetera. Parity violation in Møller scattering can be improved by O (1) ll or O ee , for example. On the other hand, the relatively large value of the W mass can be accounted by O (3) φ and O W B . While the large forward-backward bottom asymmetry results in an asymmetric α (1) φd interval,although we assume universality. At any rate, the size and asymmetry of the intervals get reduced when alldata are considered. In the previous fits to dimension six operators the SM parameters are fixed to their best value in the fit tothe SM alone, except for the Higgs mass M H which is left free. This, in general, prefers to be next to itsdirect lower limit of 114 GeV [1, 2, 13]. In the left figure we show the χ dependence on M H in the fitto the SM alone with all SM parameters free (upper black solid line). As it is apparent, if the SM Higgsis found to be relatively heavy, further physics has to cancel its one-loop contributions to the differentelectroweak precision observables, in order to restore the excellent agreement with the data. In particular,it has to balance the negative quantum correction to the ρ = M W /M Z cos θ W parameter [14]. This canbe done at tree level increasing the numerator or decreasing the denominator, yielding in both cases therequired positive contribution. The former can be effectively achieved reducing the SM contribution to theFermi constant G µ by mixing the electron neutrino with a sterile heavy neutrino N e [9, 15], and the lattermixing the Z boson with heavier extra vector bosons [11, 16]. In the left figure we show the effect ofboth possibilities. The second upper line (blue dashed) corresponds to the heavy neutrino addition, whichcan not completely account for a heavy Higgs but improves the global fit. Whereas there are two gaugeboson additions balancing the heavy Higgs corrections to EWPD, named B and W in [11], respectively(bottom green dotted-dashed and second bottom red solid lines in the figure). In these three fits the onlySM parameters left free, besides the Higgs mass, are the strong coupling constant and the top mass. Thelarge χ values on the ordinate reminds the large number (212) of data included in the fits. Finally, in theright figure we plot the same as in the left one but replacing the present large collider bounds [13] by twoguesses of the Higgs mass eventually measured at CERN [17]. Note that in the operator basis chosen here corrections to G µ can be encoded either in O (3) φl or O (1 , ll . While, direct correctionsto the ρ parameter are accounted by O (3) φ , allowing for large M H values for negative α (3) φ (see Table 1). Copyright line will be provided by the publisher
F. del Aguila and J. de Blas: Electroweak constraints
The previous fits make apparent the paradox that the SM describes physics up to the LEP 2 energy ( ∼
209 GeV) with a precision in general below the per cent level, but we still expect that LHC will discoverfurther resonances near the TeV [3]. If so, a model dependent pattern of cancellations must arise resultingin small contributions to electroweak precision observables. The corresponding discussion for the case ofextra gauge bosons is presented in [11, 12]. Examples with cancellations based on custodial symmetriescan be found in [18] for extra quarks or in [19] for extra leptons. However, in these models flavor plays anessential rˆole because the new fermions mainly mix with the third family, as may be in Nature.
Acknowledgements
We are grateful to the Corfu Institute 2010 organizers for their habitual kind hospitality, and toM. P´erez-Victoria for collaboration in the work reviewed here. This work has been partially supported by MICINN(FPA2006-05294 and FPA2010-17915) and by Junta de Andaluc´ıa (FQM 101, FQM 3048 and FQM 6552). The workof J.B. has been supported in part by the U.S. National Science Foundation under Grant PHY-0905283-ARRA.
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