What kind of New Physics, if any, we expect to discover at the LHC? I will
try to address this formidable question by re-formulating it as follows: is the
breaking of the electroweak symmetry strong or weak ?Comment: Talk given at IFAE 2009, Bari, Italy, 15-17 April 200

Contino, Roberto

English

arXiv

CONTINO, ROBERTO

Archivio della ricerca- Università di Roma La Sapienza

New Physics at the LHC: Strong versus Weak symmetry breaking

What kind of New Physics, if any, we expect to discover at the LHC? I will try to address this formidable question by re-formulating it as follows: is the breaking of the electroweak symmetry strong or wea

Contino Roberto

Archivio istituzionale della Ricerca - Scuola Normale Superiore

What kind of New Physics, if any, we expect to discover at the LHC? I will try to address this formidable question by re-formulating it as follows: is the breaking of the electroweak symmetry strong or weak

Contino, R.

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2009-10427-3Colloquia: IFAE 2009IL NUOVO CIMENTO Vol. 32 C, N. 3-4 Maggio-Agosto 2009New Physics at the LHC: Strong vs. weak symmetry breakingR. ContinoDipartimento di Fisica, Universita` di Roma “La Sapienza” - Rome, ItalyINFN, Sezione di Roma - I-00185 Rome, Italy(ricevuto il 19 Settembre 2009; pubblicato online il 14 Ottobre 2009)Summary. — What kind of New Physics, if any, we expect to discover at theLHC? I will try to address this formidable question by re-formulating it as follows:is the breaking of the electroweak symmetry strong or weak?PACS 12.60.Cn – Extensions of electroweak gauge sector.PACS 12.60.Rc – Composite models.PACS 12.39.Fe – Chiral Lagrangians.1. – The need for an electroweak symmetry breaking sectorThe entire physics discovered so far in high-energy experiments can be described andcompactly summarized by the LagrangianL = L0 + Lmass,(1)L0 = −14WaμνWa μν − 14BμνBμν − 14GμνGμν +3∑j=1(Ψ¯(j)L i DΨ(j)L + Ψ¯(j)R i DΨ(j)R),Lmass = M2W W+μ W−μ +12M2Z ZμZμ−∑i,j(u¯(i)L Muiju(j)R + d¯(i)L Mdijd(j)R + e¯(i)L Meije(j)R + ν¯(i)L Mνijν(j)R)+ h.c.,where Ψ is a collective index for the Standard Model (SM) fermions and i, j are generationindices. A remarkable property that has emerged, and that is most clearly exhibited bythe Lagrangian L, is that while all the fundamental interactions among the particles aresymmetric under a local SU(2)L×U(1)Y invariance, the observed mass spectrum is not.In other words, the electroweak symmetry is hidden, i.e. spontaneously broken by thevacuum. In mathematical terms, the spontaneous breaking can be made more explicitby introducing as propagating degrees of freedom the Nambu-Goldstone bosons χa thatc© Societa` Italiana di Fisica 1112 R. CONTINOcorrespond to the longitudinal polarizations of the W and Z bosons (for simplicity, fromhere on I will omit the lepton terms and concentrate on the quark sector):Σ(x) = exp[iσaχa(x)/v], DμΣ = ∂μΣ− ig2σa2W aμΣ + ig1Σσ32Bμ,(2)Lmass = v24Tr[(DμΣ)†(DμΣ)]− v√2∑i,j(u¯(i)L d(i)L)Σ(λuij u(j)Rλdij d(j)R)+ h.c.The local SU(2)L × U(1)Y invariance is now manifest in the Lagrangian (2) with Σtransforming as(3) Σ→ UL(x)ΣU†Y (x), UL(x)=exp [iαaL(x)σa/2] UY (x)=exp[iαY (x)σ3/2].In the unitary gauge 〈Σ〉 = 1, the chiral Lagrangian (2) reproduces the mass term ofeq. (1) with(4) ρ ≡ M2WM2Z cos2 θW= 1,which is consistent with the experimental measured value to quite good accuracy. Theabove relation follows as the consequence of a larger global SU(2)L×SU(2)R approximateinvariance of (2), Σ→ UL ΣU†R, which is spontaneously broken to the diagonal subgroupSU(2)c by 〈Σ〉 = 1, and explicitly broken by g1 and λuij = λdij . In the limit of vanishingg1 the “custodial” SU(2)c implies MW = MZ , which is replaced by eq. (4) at tree levelfor arbitrary g1. Further corrections proportional to g1 and λu−λd arise at the one-looplevel and are small. In fact, the success of the tree-level prediction ρ = 1 a posteriorijustifies the omission in the chiral Lagrangian (2) of the additional term(5) c3 v2 Tr [Σ†DμΣT 3]2that is invariant under the local SU(2)L×U(1)Y but explicitly breaks the global SU(2)L×SU(2)R. In other terms, the coefficient c3 of such extra operator is experimentallyconstrained to be very small.The chiral formulation (2) makes the limit of our current mathematical descriptionmost transparent: There is a violation of perturbative unitarity in the scattering χχ →χχ at energies E  MW , which is ultimately linked to the non-renormalizability of thechiral Lagrangian. More specifically, the scattering amplitude grows with E2,A(χaχb → χcχd) = A(s) δabδcd + A(t) δacδbd + A(u) δadδbc,(6)A(s) =sv2[1 + O(M2Ws)]due to the derivative interaction among four Goldstones that comes from expanding thekinetic term of Σ in eq. (2). Intuitively, the χ’s are the degrees of freedom that are eatenin the unitary gauge to form the longitudinal polarizations of W and Z. The scatteringof four Goldstones thus corresponds to the physical scattering of four longitudinal vectorNEW PHYSICS AT THE LHC: STRONG vs. WEAK SYMMETRY BREAKING 13bosons: VLVL → VLVL, with VL = WL, ZL. Such correspondence is made formal by theEquivalence Theorem, which states that the amplitude for the emission or absorption ofa Goldstone field χ becomes equal at large energies to the amplitude for the emissionor absorption of a longitudinally polarized vector boson. As a consequence, the physicalscattering VLVL → VLVL violates perturbative unitarity at large energies E MW , andthe leading energy behavior of its cross-section is captured by that of the easier processχχ → χχ. The merit of the chiral formulation is that of isolating the problem to thesector of the Lagrangian which leads to the mass terms for the vector bosons and thefermions. In other terms:We need some new symmetry breaking dynamics that acts as an ultraviolet completionof the electroweak chiral Lagrangian (2) and restores unitarity.As required by the experimental evidence, such new dynamics must be (approxi-mately) custodially symmetric, so as to prevent large corrections to the ρ parameter.The question then arises: is this dynamics weak or strong?2. – Strong vs. weak symmetry breakingThe most economical example of electroweak symmetry breaking (EWSB) sector isthat of just one new scalar field h(x), singlet under SU(2)L×SU(2)R, in addition to theGoldstones χ. Assuming that h is coupled to the SM gauge fields and fermions only viaweak gauging and (proto)-Yukawa couplings, the most general EWSB Lagrangian hasthree free parameters a, b, c(1) at the quadratic order in h [1]LEWSB = 12(∂μh)2 + V (h) +v24Tr[(DμΣ)†(DμΣ)](1 + 2ahv+ bh2v2+ . . .)(7)−mψ ψ¯LΣ(1 + chv+ · · ·)ψR + h.c.,where V (h) is some potential, including a mass term, for h. Each of these parameterscontrols the unitarization of a different sector of the theory: For a = 1 the exchange ofthe scalar unitarizes the χχ → χχ (VLVL → VLVL) scattering (in the following, dashedand solid lines denote, respectively, the fields χ and h, whereas solid lines with an arrowdenote fermions):A(s) 	 sv2(1− a2);For b = a2 also the inelastic channel χχ→ hh (VLVL → hh) respects unitarity.(1) In general c will be a matrix in flavor space, but in the following we will assume for simplicitlyjust one fermion generation. The extension to three families is straightforward.14 R. CONTINOA(χaχb → hh) 	 δab sv2(b−a2);Finally, for ac = 1 the χχ→ ψψ¯ (VLVL → ψψ¯) scattering is unitarized.A(χaχb → ψψ¯) 	 δab√mψsv2(1−ac).Only for a = b = c = 1 the EWSB sector is weakly interacting (provided the scalar h islight), as for example a = 1 implies a strong V V → V V scattering with violation of per-turbative unitarity at energies√s ≈ 4πv/√1− a2, and similarly for the other channels.The point a = b = c = 1 in fact defines what I would call the “Higgs model”: LEWSB(with vanishing higher-order terms in h) can be rewritten in terms of the SU(2)L doublet(8) H(x) =1√2eiσaχa(x)/v(0v + h(x))and gets the usual form of the Standard Model Higgs Lagrangian. In other words, χaand h together form a linear representation of SU(2)L × SU(2)R. In terms of the Higgsdoublet H(x), the custodial invariance of the Lagrangian appears like an accidentalsymmetry: at the renormalizable level, all the (SU(2)L × U(1)Y )-invariant operatorsare functions of H†H =∑i ω2i , where ωi are the four real components parametrizingthe complex doublet H. An SO(4) ∼ SU(2)L × SU(2)R invariance follows under whichthese components are rotated, broken to SO(3) ∼ SU(2)c in the vacuum 〈H†H〉 = v2.The unitarity of the Higgs model can be traced back to its renormalizability, which isnow evident from the Lagrangian written in terms of H.The weakly interacting Higgs model has two main virtues: it is theoretically attractivebecause of its calculability, and it is insofar phenomenologically successful, passing inparticular all the LEP and SLD electroweak precision tests. Both calculability (whichstems from perturbativity) and the success in passing the precision tests follow from theHiggs boson h being light. It is however well known that an elementary light scalar, suchas h, is unstable under radiative corrections, hence highly unnatural in the absence ofsome symmetry protection. It is quite possible, on the other hand, that a light Higgs-likescalar arises as a bound state of a new strong dynamics: its being composite would solvethe SM Higgs hierarchy problem, while its being light would still be required to pass theelectroweak tests. The Lagrangian (7) with generic a, b, c gives a general parametrizationof such composite Higgs theories where all the other resonances have been integrated out.Away from the unitary point a = b = c = 1 the exchange of the light composite Higgs hfails to completely unitarize the theory, which eventually becomes strongly interacting athigh energies. Similarly to pion-pion scattering in QCD, unitarity is ultimately reinforcedat the strong dynamics scale through the exchange of the other (spin-1) resonances.Insofar we have tacitly assumed that these latter are heavier than the compositeHiggs. This is in fact required (unless some non-trivial symmetry protection mechanismNEW PHYSICS AT THE LHC: STRONG vs. WEAK SYMMETRY BREAKING 15v vW 3 BρFig. 1. – (Colour on-line) Upper left: contribution to the S parameter from the exchange ofheavy spin-1 resonances. Lower left: one-loop Higgs contribution to S and T . Right: correctionto 1,3 as a consequence of the modified Higgs coupling to the SM vectors: Δ1 = αT =−c1(1 − a2) log(Λ2/m2h), Δ3 = α/(4 sin2θW )S = −c3(1 − a2) log(Λ2/m2h). The black fat dotshows the SM prediction (a = 1) for mh = 120GeV. The smaller blue dots show how 1,3 aremodified by diminishing a from 1 to 0 in steps of 0.1 (for Λ = 2.5TeV). The 68%, 90%, 99%CL contours have been obtained by setting 2 and b to their SM value.is at work) to avoid large corrections to the precision observables, for example to thePeskin-Takeuchi S parameter, see fig. 1. As first pointed out by Georgi and Kaplan inthe eighties [2], the composite Higgs boson can be naturally lighter than the other reso-nances if it emerges as the (pseudo-)Goldstone boson of an enlarged global symmetry ofthe strong dynamics. For example, if the strong sector has an SO(5) global symmetryspontaneously broken down to SO(4) ∼ SU(2)L × SU(2)R, this implies four real Gold-stones trasforming as a fundamental of SO(4), or equivalently as a complex doublet Hof SU(2)L [3]. The couplings of the SM fermion and gauge fields to the strong sector willin general break explicitly its global symmetry, thus generating a Higgs potential at theone-loop level. By naive dimensional analysis, the expected mass scale of the other reso-nances is mρ ∼ 4πf , where f is the σ-model scale associated with the composite Higgs.This latter gets instead a much lighter mass mh ∼ gSMv at one-loop, with gSM beingsome SM coupling, thus implying a parametric hierarchy mh  mρ. In this context,the electroweak scale v is dynamically determined and will not in general coincide withf , differently from Technicolor theories where no separation of scales exists. The ratioξ = v2/f2 sets the size of the parametric suppression in all corrections to the precisionobservables, as f →∞ (ξ → 0) with fixed v is a decoupling limit where the Higgs stayslight and all the other resonances become infinitely heavy. As a matter of fact, v  0.3fis enough to largely suppress any correction from the heavy resonances.3. – The Higgs boson: elementary or composite?It is at this point clear that the discovery of a light Higgs boson alone will notbe sufficient to rule out the possibility of a strong electroweak symmetry breaking.Experimentally, one should measure the parameters a, b, c as precisely as possible and16 R. CONTINOlook for deviations from the unitary point a = b = c = 1. In general these parame-ters are independent of each other, although they will be related in specific compositeHiggs models as functions of ξ. Reference [4] also showed that the behavior of a andb at small ξ is universal whenever the light Higgs boson is part of a composite SU(2)Ldoublet.A first determination of a and c will come from a precise measurement of the couplingsof the Higgs to the SM fermions and vectors. This will require disentangling possiblemodifications of both the Higgs production cross-sections and decay rates. Preliminarystudies have shown that the LHC should be eventually able to extract the individualHiggs couplings with a ∼ 20% precision [5], though much will depend on the value ofits mass. This would imply a sensitivity on (1 − a) up to 0.1–0.2 [4]. As stressed bythe authors of ref. [6], the parameter a is already constrained by the LEP precisiondata: modifying the Higgs coupling to the SM vectors changes the infrared one-loopcontribution to 1,3 (1 = SM1 + αT , 3 = SM3 + α/(4 sin2θW )S) by an amount Δ1,3 =−c1,3(1 − a2) log(Λ2/m2h), where c1,3 are one-loop numerical coefficients and Λ denotesthe scale at which the other resonances set on and unitarity is ultimately restored in V Vscattering. For example, assuming no additional corrections to the precision observablesand setting mh = 120GeV, Λ = 2.5TeV, one obtains a  0.85 at 99% CL, see fig. 1.Measuring the Higgs couplings will give important clues on its nature and on the roleit plays in the EWSB mechanism. A “direct” probe of the strength of the symmetrybreaking dynamics will however come only from a precise study of the V V scattering.A smoking gun of strong electroweak symmetry breaking would be discovering a lightHiggs and at the same time finding an excess of events in V V → V V scattering comparedto the SM expectation: this would be the sign that the energy-growing behavior of thescattering cross-section of longitudinal W and Z’s is not saturated at a low scale by theHiggs exchange. As a matter of fact, it turns out that the onset of the strong scatteringis delayed to higher energies due to a large “pollution” from the scattering of transversepolarizations, (see for example [1]). This requires defining the signal as the differencebetween the observed number of events and the one predicted in the SM for a referencevalue of the Higgs mass [7], since the contribution from the transverse scattering isnot expected to change significantly in the presence of new physics: σ(VLVL signal) =σ(V V observed) − σ(SM). The pioneering preliminary study of ref. [7] focussed on the“golden” purely leptonic decay modes and showed that perhaps the best channel toobserve a strong scattering signal is W±W± →W±W± (all W ’s with the same charge),see table I. More recent analyses have appeared in the literature [8], also considering theTable I. – Event rates at the LHC (√s = 14TeV) with 100 fb−1 for the scattering V V → V Vin the purely leptonic channels (with both vectors decaying to leptons). From ref. [7]. Signal isdefined as the difference between the total number of events predicted for a = 0 and that predictedin the SM (a = 1) with mh = 100GeV. The “SM” column reports this latter number, while“Background” refers to the sum of additional reducible backgrounds. See ref. [7] for details onthe cuts and the simulation.Signal a = 0 SM BackgroundZZ (4l) 1.5 9 0.7W+W− 5.8 27 12W±Z 3.2 1.2 4.9W±W± 13 5.6 3.7NEW PHYSICS AT THE LHC: STRONG vs. WEAK SYMMETRY BREAKING 17Table II. – Event rates at the LHC (√s = 14TeV) with 300 fb−1 for the scattering V V → hhin three-lepton final states, with the Higgs decaying to two W ’s. From ref. [1]. The signalsignificance reported in the last column is computed using a Poisson statistics and expressed inunits of standard deviations. See ref. [1] for details on the cuts and the simulation.Events SignificanceSignal (a2 − b) = 1.0 7.4 3.7Signal (a2 − b) = 0.8 5.0 2.6Signal (a2 − b) = 0.5 2.3 1.2Background 1.5 –semileptonic decay modes, though none of them can yet be considered fully complete. Itis however quite clear that the study of V V scattering is extremely challenging due tothe small signal cross-section and the numerous SM backgrounds which can fake it.Another smoking gun of composite Higgs models and strong symmetry breaking wouldbe the observation of the V V → hh scattering [4], which in the SM has an extremelysmall cross-section. The importance of this channel comes from the fact that it is theonly process giving information on the parameter b, which is not constrained by thescattering V V → V V or the precision tests and cannot be determined by measuring thesingle Higgs couplings. Compared to V V → V V , this process is a more “pure” probeof the strong dynamics, since there is no pollution from transverse modes (the Higgsis the fourth Goldstone together with the W and Z longitudinal polarizations). Anexplorative analysis [1] has shown that the best chances of discovery are probably in thethree-lepton final states, with the Higgs decaying to W+W−: pp → hhjj → 4Wjj →l+l−l± ET jjjj, see table II. A more realistic study with full detector simulation willhowever be needed to confirm these result and establish the ultimate LHC sensitivity onthe parameter b.∗ ∗ ∗It is a pleasure to acknowledge the collaboration with C. Grojean, M. Moretti,F. Piccinini and R. Rattazzi which led to much of the material covered by this talk. Ialso thank the organizers of IFAE 2009 for inviting me to this interesting and stimulatingconference.REFERENCES[1] Contino R., Grojean C., Moretti M., Piccinini F. and Rattazzi R., to appear.[2] Georgi H. and Kaplan D. B., Phys. Lett. B, 136 (1984) 183.[3] Agashe K., Contino R. and Pomarol A., Nucl. Phys. B, 719 (2005) 165 [arXiv:hep-ph/0412089].[4] Giudice G. F., Grojean C., Pomarol A. and Rattazzi R., JHEP, 0706 (2007) 045[arXiv:hep-ph/0703164].[5] Du¨hrssen M., ATL-PHYS-2003-030, available from http://cdsweb.cern.ch; Du¨hrssenM., Heinemeyer S., Logan H., Rainwater D., Weiglein G. and Zeppenfeld D., Phys.Rev. D, 70 (2004) 113009 [arXiv:hep-ph/0406323].[6] Barbieri R., Bellazzini B., Rychkov V. S. and Varagnolo A., Phys. Rev. D, 76 (2007)115008 [arXiv:0706.0432 [hep-ph]].18 R. CONTINO[7] Bagger J. et al., Phys. Rev. D, 49 (1994) 1246 [arXiv:hep-ph/9306256]; 52 (1995) 3878[arXiv:hep-ph/9504426].[8] See for example: Butterworth J. M., Cox B. E. and Forshaw J. R., Phys. Rev. D, 65(2002) 096014 [arXiv:hep-ph/0201098]; Ballestrero A., Bevilacqua G. and Maina E.,arXiv:0812.5084 [hep-ph]; Aad G. et al. (The ATLAS Collaboration), arXiv:0901.0512[hep-ex].