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arXiv:hep-ph/0307293 v2 27 Jul 2003 ffrin ocsu oitoueanew large a theory of introduce relativistic to lack the us in the forces sea However, Dirac of the fermions. in surface of Fermi su- but pair elementary, the a not to is rather analogously condensate more The much perconductivity. EWSB the describe problem. hierarchy” “little the as to referred o ueprilsa E-I(see, LEP-II al- at getting search unsuccessful superparticles is the for after the fine-tuned be- somewhat has that ready it concern Recently, theory. a while perturbation divergence come in scalar logarithmic orders elementary a all an only to still receives su- is mass-squared using supersymme- boson its it of Higgs stabilize use the conventional to try, this possible the in is rendering mass- It persymmetry; divergence boson unstable. quadratic Higgs theory breaks the a model because receives The scale squared TeV boson. below Higgs the down spin- called elementary of particle condensation less the by accomplished is future and LHC, collider. , linear at decade electron-positron activity next experimental the and the intense this of of in nature target the the understanding is and condensate GeV electroweak 250 con- the of is of densate physics scale energy the The (EWSB). is breaking symmetry This short-ranged. action in hspitimdaeypssdffiut nmodel in difficulty poses immediately point This tion. o aeegh of Comp- wavelengths finite the ton finite is the superconductors in pair of length analog penetration Cooper The to with superconductors. in filled analogous condensates is condensate, Universe our Bose–Einstein that a implies mass finite Their ooso h electroweak the of bosons etto ntennaeingueter,teei lit- ex- is the there the with theory, that gauge agreed doubt non-abelian that tle the [1] in LEP-II pectation at vertices boson ehioo hois nteohrhn,atmtto attempt hand, other the on theories, Technicolor nteSadr oe fpril hsc,teEWSB the physics, particle of Model Standard the In rmtebatflmaueet ftetil gauge triple the of measurements beautiful the From sclual;Teeetoekpeiinosralsae( are observables precision electroweak The calculable; is ehioo etrwt ehiGMmcaimadhnethe hence “ ultravio and The a mechanism constraints; by and techni-GIM neutral-current generated problem with are alignment sector masses technicolor no and The is boson exist; There Higgs bosons elementary hierarchy; exa light certain the no with explaining is pro better there numerous much here, solving dynamics presented pro of walking I exchange understand difficulties. in to building bosons model scalar severe eliminate from suffers it while hoeia hsc ru,Lwec eklyNtoa Lab National Berkeley Lawrence Group, Physics Theoretical ehioo civseetoeksmer raig(EWSB breaking symmetry electroweak achieves Technicolor INTRODUCTION W W eateto hsc,Uiest fClfri,Berkeley California, of University Physics, of Department and and SU Z Z oos aigteinter- the making bosons, (2) oosaeide gauge indeed are bosons L e.g. × ehiooflSupersymmetry Technicolorful U 2) sometimes [2]), , (1) strong Y e + symmetry. e − Dtd uy9 2004) 9, July (Dated: interac- ioh Murayama Hitoshi rdcino ehisae ntesprofra window superconformal the in techni-states of production ” oo oesaea follows: as are models color refer- of for list review [3]. up-to-date excellent ences an an and to attempts readers model-building refer I phenomenolog- difficulties. technical other ical this numerous to been addition had strong there In difficulty, non-perturbative theories. on gauge rely of dynamics models the as building, oosfo h hoy h rueti htw have we that is argument The theory. spinless eliminate the to from bosons models: technicolor for motivation [7]. etc topseesaw, technicol dimensions, extra -assisted include Other directions above. possible listed and problems interactions anoma- phenomenological ETC the large In the solve are enhances therein). there that that references dimensions assumed and lous is [3] it also theories, (see these 6] “walking” 5, the [4, using theories especially dynamics, non-QCD-like h andffiute nteoiia C-ietechni- QCD-like original the in difficulties main The pr rmteES tef hr a enanother been has there itself, EWSB the from Apart using problems these solve to efforts been had There • • • nfruaeyntcalculable. not un)fortunately esrdaepeitdt eoto h allowed the of out be to ranges. predicted are measured observables electroweak measured precisely Certain ruled experimentally are out. which (PNGB) pseudo-Nambu-Goldstone bosons light successfully are state there ground not chosen, correct is the do tech- if which Even in EWSB. of the states achieve some ground theory degenerate nicolor are namely problem, there alignment that the have models instability.Most flavor- proton the from Some suffer TeV. while 10–100 even above models TeV, be ETC re- to few scale constraints ETC a the (FCNC) quire as current low neutral ETC the changing as implies mass is (ETC) scale top technicolor large extended The to an has sector. It by . augmented and be of of mechanism a masses provide generating not does itself Technicolor oet bno t eodr olto goal secondary its abandon to pose lm sn uesmer.I helps It supersymmetry. using blems e-opeernraial extended renormalizable let-complete rtr,Bree,C 42,USA 94720, CA Berkeley, oratory, h WBi ul yaia,hence dynamical, fully is EWSB the etri aefo flavor-changing- from safe is sector trsls ntepriua model particular the In results. ct olgtpseudo-Nambu-Goldstone light no na lgn n aua way, natural and elegant an in ) A970 S and USA 94720, CA , LBNL-53467 or, 2 not yet seen any spinless elementary bosons in Nature, quantum number assignment as the minimal technicolor and their masses suffer from the quadratic divergences model in the non-supersymmetric case. The dynamics which render the theory unstable and fine-tuned. I con- of technicolor treats two components of T , t+, and t− sider this a secondary goal less important than the pri- all equal, and hence there is an SU(4) global symmetry. mary goal of the elegant EWSB. If the secondary goal The model described here was studied by Luty, Terning, is abandoned, it is attractive to consider the possibility and Grant in [11]. that the technicolor model is supersymmetric. It neces- The low-energy effective theory of this supersymmetric sarily introduces spinless bosons into the theory, while it is known, and is described by the meson will not cause dangerous quadratic divergences. On the composites made of techniquarks [8]. It is known to lead other hand, non-perturbative dynamics of supersymmet- to the so-called quantum modified constraint: ric gauge theories are much better understood than the 4 non-supersymmetric counter parts which provides better M+M− − MSMs =Λ , (2) tools in model building (see [8] for a review). In fact, where the mesons are defined by I will make use of the superconformal dynamics which allows exact predictions on the anomalous dimensions. M± = (Tt±),MS = (TT ),Ms = (t+t−). (3) Moreover, the presence of spinless bosons will provide simple solutions to the alignment problem and the suc- The contraction of technicolor indices is understood cessful construction of the ETC sector as I present below. in each parentheses. At this point the model suf- Here are the relevant energy scales in the particular fers from the alignment problem just like in the non- model presented below: supersymmetric model. It is not clear how the desired ground state with expectation values in electroweak dou- • mSUSY ∼ 0.2 TeV. blet composites M+ and M− is chosen over that with • ΛTC ∼ 2 TeV. singlets MS and Ms that do not break the electroweak gauge group at all. It is often assumed that the higher • ΛETC ∼ 2 TeV. the remaining symmetry the lower the energy is; if so, The EWSB is solely due to the strong dynamics of tech- it is more likely that the ground state does not achieve nicolor gauge interaction, and the EWSB. (Here, Λ is the holormorphic dynamical scale, is considered a small perturbation. The main advantage which is different from the strong scale of technicolor the- of using supersymmetry is to have clear predictions on ory. We will come back to this point later.) the dynamics. Certain quantities are exactly calculable In this model, it is very simple to solve the alignment despite strong dynamics of the theory. Despite a rela- problem. I introduce two singlet fields S and s, and a tively low ETC scale, supersymmetry allows a renormal- superpotential izable implementation of the techni-GIM mechanism [16] W = S(TT )+ s(t t ). (4) to avoid the FCNC problems. + − Earlier attempts to make use of supersymmetry in the This superpotential is renormalizable and hence is ul- context of technicolor have assumed mSUSY > ΛTC , and traviolet complete. Once the technicolor interaction be- moreover were made before the dynamics of supersym- comes strong at the scale Λ, the superpotential turns into metric gauge theories were understood [7, 9]. The effect mass terms of meson composites together with the intro- of soft supersymmetry breaking on dynamics had been duced singlets of the order of Λ. The superpotential does worked out only relatively recently [10]. There is one im- not allow the MS = (TT ) and Ms = (t+t−) mesons to portant exception that used mSUSY < ΛTC and made acquire expectation values. This way, the desired ground use of supersymmetric dynamics by Luty, Terning, and state with the EWSB Grant [11] which, however, needed a light . 2 The present work makes use of these developments. hM+i = hM−i =Λ (5)

is uniquely chosen together with the D-term potential, TECHNICOLOR solving the alignment problem. By further making the soft masses for t+ and t− different in the ultraviolet, one The simplest choice of the technicolor group is can achieve different VEVs for M± as well (i.e., tan β 6= SU(2)TC with the following particle content of chiral su- 1). perfields (techniquarks) Note that the dynamics has the custodial SU(2) sym- metry and hence leads to ρ = 1 naturally. Out of the 1 1 T (2, 2, 0), t (2, 1, + ), t (2, 1, − ) (1) SU(4) ≃ SO(6) global symmetry, the superpotential + 2 − 2 breaks it to SO(4) ≃ SU(2)L×SU(2)R. U(1)Y is embed- where the quantum numbers are shown under the gauge ded into SU(2)R. Therefore, the custodial SU(2) symme- group SU(2)TC × SU(2)L × U(1)Y . This is the same try is broken only by the U(1)Y interaction and fermion 3 masses (see the next section), in complete analogy to the The meson composite fields Mˆ ± have different normal- minimal . ization from M± as will be determined shortly below. 2 2 2 The effect of soft supersymmetry breaking in this It gives mW ≃ g v with v ∼ ΛTC/4π. This result model had been worked out by Luty and Rattazzi [10]. is analogous to the result in non-supersymmetric QCD Out of five chiral superfields that survive the constraint, fπ ∼ Λ/(4π). Xˆ is a Lagrange multiplier field to enforce the five imaginary components acquire positive mass the quantum modified constraint. The last term is the 2 squared of O(mSUSY ) while the five real components desired ETC interaction responsible for the remain massless. The massless ones are the Nambu– mass. It leads to the mass term Goldstone bosons of the spontaneously broken symmetry 2 1 ΛTC SU(4)/Sp(2) = SO(6)/SO(5). Together with the terms QU (10) (4π)2 Λ from the superpotential, two full chiral multiplets become ETC massive, leaving only three real and imaginary compo- which implies nents left. The three massless imaginary components are 2 2 1 ΛTC v eaten by the as a consequence of the mf ∼ ≃ . (11) (4π)2 Λ Λ EWSB, while the real parts remain with supersymmetry- ETC ETC breaking scale. They are no Higgs bosons, however, as In order to reproduce the top quark mass, I need ΛETC ∼ the Higgs boson is completely eliminated by the con- v. Hence the presumed high ETC scale is brought down straint. They are rather the analog of H0 and H± in to the and it does not fulfill the goal. It two-doublet Higgs models while they lack scalar-scalar- is also important to note that the strong scale ΛTC ≃ 4πv vector vertices. is different from Seiberg’s holomorphic scale Λ ∼ v and ˆ Therefore the dynamical EWSB works successfully in the meson composites are related by M± ∼ 4πM±/Λ. this model. Another quantity that cannot be calculated rigorously is the contribution to the oblique electroweak parameters such as Peskin–Takeuchi S and T [15]. ETC

In the original QCD-like technicolor, the fermion WALKING masses are obtained through dimension-six four-fermion interactions giving As in the non-supersymmetric case, walking dynamics can be employed to enhance the ETC interaction to raise 1 Λ3 TC the ETC scale and suppress the possible FCNC effects. mf ∼ 2 2 . (6) (4π) ΛETC The main benefit of the supersymmetry is to allow me to The trouble is that one also expects flavor-changing neu- calculate the anomalous dimension factors exactly. 3 tral currents from the ETC boson exchange suppressed In supersymmetric SU(Nc) QCD for 2 Nc < Nf < 2 by the same power, 1/ΛETC. This causes the well-known 3Nc, the theory is in superconformal phase [8]. It has dilemma; a large enough fermion mass, especially that for a dual magnetic description in terms of the dual gauge the top quark, is incompatible with the apparent lack of group SU(Nf − Nc). The important point is that the flavor-changing neutral current processes. wave function factor for quark fields is The fermion masses are obtained through dimension- given exactly in the infrared as five superpotential terms µ (3Nc−Nf )/Nf Z = , (12) ij 1 Λ WETC = hu (Tt+)Qiuj (7)   ΛETC where µ (Λ) is the infrared (ultraviolet) scale. In partic- ij 1 ij 1 ular, the case N = 2 and N = 4 will be used in the +hd (Tt−)Qidj + hl (Tt−)Liej . (8) c f ΛETC ΛETC next section, giving In the supersymmetric case based on the model in the µ 1/2 previous section, the ETC scale cannot be raised high Z = . (13) [12]. The naive dimensional analysis is used to relate Λ various scales in the problem [13, 14], as the holormorphy Note that the suppressed wave function renormalization in supersymmetry is not powerful enough to constrain the factor Z at lower energies enhances the couplings of tech- K¨ahler potential. It suggests that the Lagrangian for the niquarks. meson composites is given by

1 4 † AN OVERKILL ETC MODEL L = d θMˆ Mˆ ± (4π)2 Z ± 2 ˆ ˆ ˆ 2 ˆ QU The model introduces four additional chiral multiplets + d θX(M+M− − ΛTC )+ΛTCM− . (9) Z ΛETC  with the same quantum numbers as the Higgs doublets 4

Hu and Hd in the Minimal Supersymmetric Standard can be obtained relative to the previous case. The techni- Model (MSSM). They are singlets under the technicolor color model reduces to that of the two doublets discussed gauge group. earlier by adding a mass term to the extra two doublets m3,4 which I take to be a common mass m. Because the ′ 1 ′ 1 Φ , Φ (1, 2, + ), Φ , Φ (1, 2, − ). (14) dynamics is already strong, ΛTC is expected to be at the u u 2 d d 2 physical mass of the extra doublets ΛTC ∼ m, leading to They have the superpotential the quantum modified constraint and hence the EWSB. The fermion mass is then enhanced to ′ ′ ′ ′ WETC = ΛETC (ΦuΦd +ΦuΦd)+Φu(Tt−)+Φd(Tt+) 1/2 ij ij ij 2 +h Qiuj Φu + h Qidj Φd + h LiejΦd. (15) v ΛETC u d l mf ∼ 4π , (20) ΛETC  ΛTC  The only flavor-violating couplings are the Yukawa cou- ij ij ij plings hu , hd , hl , and hence the model realizes the re- and in order to obtain mf ∼ v (top quark), I need quirement of the techni-GIM mechanism. By integrating Λ4 out the massive Φu and Φd, I find Λ ∼ Λ ∼ . (21) ETC TC (4π)2 eff 1 ij ij ij WETC = [(hd Qidj +hl Liej)(Tt−)+hu Qiuj (Tt+)]. ΛETC Therefore, this model eliminates Higgs doublets at (16) scale v and additional degrees of freedom such as Φu,d It has all the operators needed to generate fermion are pushed up to ΛTC. masses. This model does not lead to any phenomenolog- ically problematic flavor-changing effects from the ETC operators. As it stands now, combined with the SU(2) techni- color model with two flavors in the earlier section, the Phenomenology of the model presented is quite rich. ETC scale cannot be raised high, ΛETC ∼ v. Therefore Below TeV, the model looks very much like any super- this overkill model actually predicts Φu,d are light Higgs symmetric models except that there is no light Higgs bo- ± 0 doublets and ΛETC is nothing but the µ parameter [11]. son. There are analogs of heavy Higgses H and H but The rest of the discussion is how the superconformal the- not A0, even though they are not really Higgs bosons ory can be used to raise ΛETC and hence eliminate light and there is no vector-vector-scalar vertex. On the other Higgs from the spectrum. hand, the absence of light Higgs implies that the W W Let me take the same SU(2) technicolor model as be- scattering amplitude grows and is unitarized only above fore, but introduce two more flavors (four more doublets) TeV. It may even lead to some techni-resonances. If the to the model. The theory becomes strong at a scale Λ4 ETC sector is not an overkill, there may be small devia- (the subscript stands for four flavors) which is taken to tions from the Standard Model in K- and B-physics, and be much higher than the ETC scale. Below Λ4, the some -flavor violating signals. O(1) Yukawa interactions between the Φ doublets and The most striking prediction of this model is the pres- ′ ′ the techniquarks Φu(Tt−)+Φd(Tt+) are enhanced down ence of superconformal dynamics above ΛTC . In particu- + − to the mass ΛETC of Φ doublets as lar, the “e e ” cross section for producing techni-states can be predicted exactly [17] even though the S-matrix Λ 1/2 4 . (17) elements are ill-defined in conformal theories. ΛETC  It is important to note that the usual flavor-changing and CP problems in supersymmetry exist also in this However, a too-large Yukawa interaction is likely to upset framework. The flavor-changing problems need to be the delicate conformal dynamics. It is probably wise to solved by flavor-blind supersymmetry breaking mecha- restrict the growth of the Yukawa coupling to values less nisms such as gauge mediation [18], mediation than 4π, which I take as the maximum allowed value of [19] (supplemented by U(1) D-terms to make it viable the Yukawa coupling at the ETC scale. After integrating [20]), or gaugino mediation [21]. The hierarchy is sta- out the ETC doublets Φ , the effective interaction is u,d ble because of the technicolor rather than supersymme- further enhanced by an additional factor try, and it may be possible to push the supersymme- 1/2 try breaking scale to the technicolor scale or even be- ΛETC . (18) yond, while maintaining the natural hierarchy of the elec-  Λ  TC troweak scale. If so, the flavor-changing problems of su- In the end, an overall enhancement factor of persymmetry may be suppressed partly by decoupling, and also the little can be ameliorated. Λ 1/2 The overkill ETC model above should be considered 4π ETC (19)  ΛTC  only a toy model as it does not explain the origin of 5

flavor. It is conceivable that this framework can be com- techni-GIM mechanism and hence the sector is safe from bined with Froggatt–Nielsen mechnanism [22] at the ETC flavor-changing neutral currents effects. The electroweak scale. The effect of O(1) Yukawa couplings on supersym- precision observables are not calculable and I cannot as- metry breaking parameters can still be made immune sess the phenomenological constraints at the moment. from FCNC effects using the anomaly mediation. On the other hand the “e+e−” production of techni- sector can be calculable despite its walking dynamics. Given the model building is relatively simple, there may OPEN QUESTIONS well be models consistent with grand unification.

The result in this letter is a good start, but poses many I thank the organizers of European Physical Society, open questions, some theoretical, some phenomenologi- International Europhysics Conference on High Energy cal, and some aesthetical. Here is an incomplete list: Physics, July 17th–23rd, 2003, in Aachen, Germany, where this work was conceived and completed. I es- • Is there a way to reliably calculate the electroweak pecially thank Markus Luty, who pointed out a serious precision observables such as S and T parameters? mistake in the first version of the paper, and explained the naive dimensional analysis patiently to me. I also • Is there a way to reliably relate the techni-pion de- thank Alex Kagan and Ken Lane for comments. This cay constant to ΛTC ? work was supported in part by the Director, Office of Sci- ence, Office of High Energy and Nuclear Physics, of the • How far can the supersymmetry breaking scale be U.S. Department of Energy under Contract DE-AC03- pushed up? Because the hierarchy is explained by 76SF00098, and in part by the National Science Founda- technicolor rather than supersymmetry, it may be tion under grant PHY-00-98840. pushed up to ΛTC , ameliorating the “little hierar- chy” problem. If it is pushed even beyond ΛTC, at some point the theoretical control thanks to super- symmetry is lost, but maybe reasonable extrapola- tion on dynamics is possible. [1] See, for example, a talk by Renaud Bruneliere presented at European Physical Society, Inter- • Can one push up the ETC scale further? national Europhysics Conference on High Energy Physics, July 17th–23rd, 2003, in Aachen, Germany, • How is the coincidence understood that mSUSY , http://eps2003.physik.rwth-aachen.de ΛETC and ΛTC (determined by the masses of extra [2] R. Barbieri and A. Strumia, arXiv:hep-ph/0007265. doublets) are not very different? This is the analog [3] C. T. Hill and E. H. Simmons, Phys. Rept. 381, 235 of the µ-problem in the MSSM. (2003) [arXiv:hep-ph/0203079]. [4] B. Holdom, Phys. Rev. D 24, 1441 (1981). • The true ETC sector is supposed to explain the [5] K. Yamawaki, M. Bando and K. i. Matumoto, Phys. Rev. origin of flavor. Can a realistic model of flavor be Lett. 56, 1335 (1986). implemented within this framework? [6] T. W. Appelquist, D. Karabali and L. C. Wijewardhana, Phys. Rev. Lett. 57, 957 (1986). • Is there a grand-unifiable model? [7] See [3] and references therein. [8] K. A. Intriligator and N. Seiberg, Nucl. Phys. Proc. Suppl. 45BC, 1 (1996) [arXiv:hep-th/9509066]. [9] M. Dine, A. Kagan and S. Samuel, Phys. Lett. B 243, CONCLUSION 250 (1990). [10] M. A. Luty and R. Rattazzi, JHEP 9911, 001 (1999) I showed that the technicolor models with supersym- [arXiv:hep-th/9908085]. metry retain the beauty of the original technicolor model [11] M. A. Luty, J. Terning and A. K. Grant, Phys. Rev. D in explaining the EWSB dynamically with a natural ori- 63, 075001 (2001) [arXiv:hep-ph/0006224]. [12] I thank Markus Luty for patiently explaining this prob- gin for the hierarchy. It abandons one of the conventional lem he encountered in the work [11]. motivation for technicolor to eliminate spinless bosons [13] M. A. Luty, Phys. Rev. D 57, 1531 (1998) [arXiv:hep- from the theory. On the other hand it solves many of the ph/9706235]. problems that have plagued technicolor models without [14] A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. Lett. supersymmetry by walking dynamics under good theo- B 412, 301 (1997) [arXiv:hep-ph/9706275]. retical control. There is no alignment problem and no [15] M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 light pseudo-Nambu-Goldstone bosons exist. It is easy (1990). [16] L. Randall, Nucl. Phys. B 403, 122 (1993) [arXiv:hep- to generate fermion masses while suppressing the flavor- ph/9210231]. changing effects. In an overkill model presented in this [17] A. de Gouvea, A. Friedland and H. Murayama, Phys. letter, the fermion masses are generated by a ultraviolet- Rev. D 59, 105008 (1999) [arXiv:hep-th/9810020]. complete renormalizable extended technicolor sector with [18] M. Dine, A. E. Nelson, Y. Nir and Y. Shirman, Phys. 6

Rev. D 53, 2658 (1996) [arXiv:hep-ph/9507378]. [arXiv:hep-ph/0003081]. [19] L. Randall and R. Sundrum, Nucl. Phys. B 557, 79 [21] D. E. Kaplan, G. D. Kribs and M. Schmaltz, Phys. Rev. (1999) [arXiv:hep-th/9810155]; D 62, 035010 (2000) [arXiv:hep-ph/9911293]; G. F. Giudice, M. A. Luty, H. Murayama and R. Rat- Z. Chacko, M. A. Luty, A. E. Nelson and E. Ponton, tazzi, JHEP 9812, 027 (1998) [arXiv:hep-ph/9810442]. JHEP 0001, 003 (2000) [arXiv:hep-ph/9911323]. [20] N. Arkani-Hamed, D. E. Kaplan, H. Murayama and [22] C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B 147, Y. Nomura, JHEP 0102, 041 (2001) [arXiv:hep- 277 (1979). ph/0012103]; I. Jack and D. R. Jones, Phys. Lett. B 482, 167 (2000)