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The 'LEP Paradox' View metadata, citation and similar papers at core.ac.uk brought to you by CORE hep-ph/0007265 IFUP–TH/2000–22 SNS-PH/00–12provided by CERN Document Server The `LEP paradox' Riccardo Barbieri Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy and INFN Alessandro Strumia Dipartimento di Fisica, Universit`a di Pisa and INFN, Pisa, Italia Abstract Is there a Higgs? Where is it? Is supersymmetry there? Where is it? By discussing these questions, we call attention to the `LEP paradox', which is how we see the naturalness problem of the Fermi scale after a decade of electroweak precision measurements, mostly done at LEP. Is it wise to spend time in reviewing a subject, which the chiral SU(3) SU(3) of strong interactions in the L ⊗ R can be summarized in one sentence: neither the Higgs pseudoscalar octet sector. nor supersymmetry have been found so far? Admit- The predictive power of such a non-linear Lagrangian tedly the question makes sense. For sure these topics is reduced with respect to the SM. In practice, at present, are crucial to the central problem of particle physics: the comparison can be made by considering 2 “(g 2)- − the ElectroWeak symmetry breaking. Our main moti- like” quantities, 1 and 3 [2], which include the EW vation here is, however, more specific. We want to bring radiative correction effects more sensitive to the Higgs the focus on what we like to call the ‘LEP paradox’. By sector. The experimentally determined 1 and 3 [1], 2 this we mean the way several years of (mostly) LEP re- mostly by ΓZ , MW /MZ and sin θW, are shown in fig. 1 sults [1] make us see the old and well known naturalness and compared with the SM prediction. All the radiative problem of the Fermi scale. corrections not included in 1 and 3, less sensitive to the The questions we address, in logical order, are: Higgs mass, are fixed at their SM values. The agreement with the SM for a Higgs mass below the triviality bound 1. Is there a Higgs? of about 600 GeV is remarkable and constitutes indirect 2. If yes, where is it? evidence for the existence of the Higgs boson. With a non linear Lagrangian, neither 1 and 3 3. Is there supersymmetry? can be computed. Some believe, however, that suit- able models of EW symmetry breaking may exist where 4. If yes, where is it? both 1 and 3 deviate from the SM values for mh = 3 (100 200) GeV by less than (1 2)10− , having there- All of these questions, as well as their answers, have to ÷ ÷ do with the ‘LEP paradox’ that was just mentioned. fore a chance of also reproducing the data without an explicit Higgs boson in the spectrum [3]. In the case where a reliable estimate can be made, technicolour 1 IsthereaHiggs? models with QCD-like dynamics, this is known not to happen [4]. Any decent theory of the EW interactions must contain the Goldstone bosons, two charged and one neutral, that provide the longitudinal degrees of freedom for the W 2 Where is the Higgs? and Z bosons. On top of them, the Standard Model has a neutral Higgs boson. Without the Higgs and without If one accepts the existence of a Higgs boson, the SM specifying what replaces it, one deals with a gauge La- Lagrangian becomes an unavoidable effective low LSM grangian with SU(2) U(1) non-linearly realized in the energy approximation of any sensible theory. A devia- L ⊗ Goldstone boson sector. This is in formal analogy with tion from it could occur for the need of describing new 1 10 3 Is supersymmetry there? The naturalness problem of the Fermi scale, caused by 8 the quadratic divergences in the Higgs mass, is with us since more than 20 years. We think that a Higgs mass in the (100 200) GeV range and, especially, a 3 6 ε ÷ lower bound on the scale of new physics of about 5 TeV turn the naturalness problem of the Fermi scale into a 1000 4 clear paradox. The loop with a top of 170 GeV gives a contribution to the Higgs mass 2 2 3 2 2 2 δm (top) = GFm k =(0.3 kmax) (1) h √2π2 t max 0 0 2 4 6 8 10 where k is the maximum momentum of the virtual 1000 ε max 1 top. The paradox arises if one thinks that 5 TeV is also a 2 lower bound on kmax, since in this case δmh(top) would Figure 1: Level curves at 68%, 90%, 99%, 99.9% CL of 2 { } exceed (1.5TeV), about 100 times the indirect value of 1 and 3 compared with the SM prediction for mh = 2 mh. We like to call this the “LEP paradox”, for obvious 100, 300, 600, 1000 GeV, from right to left. reasons. Supersymmetry offers a neat solution to this para- dox. A stop loop counteracts the top loop contribution 2 degrees of freedom with mass comparable or lower than to the Higgs mass, turning kmax of eq. (1) into the Fermi scale. Barring this possibility, the predic- tions of the SM — hence the indirect determination of 2 2 2 kmax kmax m~ ln (2) mh from the EW Precision Tests — could only be al- t m2 → t~ tered by the presence of operators (4+p) of dimension Oi 4+p 5 weighted by inverse powers of a cut-off scale, ≥ In this way a stop mass mt~ in the Fermi-scale range Λ, associated with some kind of new physics keeps the top-stop contribution to mh under control, while not undoing the success of the SM in passing the ci (4+p) eff (E<Λ) = SM + X . L L Λp Oi EWPT. This is a non trivial constraint for any possible i;p solution of the LEP paradox. The success of supersym- It is unavoidable that the respect gauge invariance. metric grand unification adds significant support to this Oi For the purposes of the following discussion, it is con- view [7]. servative that we restrict them to be flavour universal The contrary arguments to the supersymmetric so- and B, L, CP conserving. lution of the LEP paradox are of general character. One How does this modified Lagrangian compare with argument is that power divergences in field theory are the EWPT [5]? Table 1 gives a list of the (independent) not significant. This looks problematic to us: the top operators that affect the EWPT, together with the lower loop is there and something must be done with it. More limits that the same EWPT set on the corresponding relevant may be the observation that the cosmological Λ parameters. We take one operator at a time with constant poses another serious unsolved problem, also the dimensionless coefficients c =+1orc = 1and related to power divergences. i i − different values of the Higgs mass. The blanks in the Alternative physical pictures are proposed for solv- columns with mh = 300 or 800 GeV are there because ing the hierarchy problem (top-colour [8], extra dimen- no fit is possible, at 95% C.L., for whatever value of Λ. sions without supersymmetry [9], ...). As far as we A fit is possible, however, for m = (300 500) GeV with know, they all share a common problem: the lack of h ÷ suitable operators and with Λ inadefinedrange[6], as calculative techniques and/or of suitable conceptual de- shown in fig. 2. velopments do not allow to address the LEP paradox. For this reason one is cautious about saying that the Maybe the fundamental scale of these theories is low Higgs is between 100 and 200 GeV, as obtained in a pure and the agreement of the EWPT with the SM and a SM fit with Λ = . To fake a light Higgs, however, high cut-off is accidental. Alternatively, the separation ∞ a coincidence is needed. From table 1, a more likely between the Higgs mass and the scale of these theories conclusion seems that Λ is indeed bigger than about may be considerable. In this last case, unfortunately, the 5 TeV and the Higgs is light. related experimental signatures may become elusive. 2 Dimensions six mh = 100 GeV mh = 300 GeV mh = 800 GeV operators c = 1 c =+1 c = 1 c =+1 c = 1 c =+1 i − i i − i i − i a a WB =(H†τ H)Wµν Bµν 10 9.7 6.9—6.0— O 2 H = H†DµH 5.54.5 3.7—3.2— O |1 ¯ a | 2 LL = 2 (Lγµτ L) 8.15.9 6.3— —— O a ¯ a HL0 = i(H†Dµτ H)(Lγµτ L) 8.88.3 6.6— —— O a ¯ a HQ0 = i(H†Dµτ H)(Qγµτ Q) 6.66.9 ———— O ¯ HL = i(H†DµH)(LγµL) 7.68.9 ———— O ¯ HQ = i(H†DµH)(QγµQ) 5.73.5 —3.7 —— O ¯ HE = i(H†DµH)(EγµE) 8.87.2 —7.1 —— O ¯ = i(H†D H)(Uγ U) 2.43.3 ———— OHU µ µ = i(H†D H)(Dγ¯ D) 2.22.5 ———— OHD µ µ Table 1: 95% lower bounds on Λ/ TeV for the individual operators and different values of mh. c = 1 c = 1 c = 1 WB − H − LL − 1 1 1 0.3 0.3 0.3 higgs mass in TeV higgs mass in TeV higgs mass in TeV 0.1 0.1 0.1 1 3 10 30 100 1 3 10 30 100 1 3 10 30 100 Scale of new physics in TeV Scale of new physics in TeV Scale of new physics in TeV Figure 2: Level curves of ∆χ2 = 1, 2.7, 6.6, 10.8 that correspond to 68%, 90%, 99%, 99.9% CL for the first 3 { } { } operators in table 1 ( , ,and in the order) and c = 1.
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