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Technicolor and the First Muon Collider 1 Kenneth Lane 8 Department of Physics, Boston University, 590 Commonwealth Ave, Boston, MA 02215 9 9 1 n Abstract. a J Themotivationsforstudyingdynamicalscenariosofelectroweakandflavorsymme- 1 try breakingarereviewedandthe latestideas,especiallytopcolor-assistedtechnicolor, 2 are summarized. Technicolor’s observable low-energy signatures are discussed. The superb energy resolution of the First Muon Collider may make it possible to resolve 1 theextraordinarilynarrowtechnihadronsthatoccurinsuchmodels—π0,ρ0,ω —and v T T T produce them at very large rates compared to other colliders. 5 8 3 1 0 I OVERVIEW OF TECHNICOLOR 8 9 / Technicolor—the strong interaction of fermions and gauge bosons at the scale h p Λ 1TeV—describes the breakdown of electroweak symmetry to electromag- TC - netism∼without elementary scalar bosons [1]. In its simplest form, technicolor is a p e scaled-up version of QCD, with massless technifermions whose chiral symmetry is h spontaneously broken at Λ . If left and right-handed technifermions are assigned : TC v to weak SU(2) doublets and singlets, respectively, then M = cosθ M = 1gF , i W W Z 2 π X where F = 246GeV is the technipion decay constant, 2 analogous to f = 93MeV π π r for the ordinary pion. a The principal signals in hadron and lepton collider experiments of “classical” technicolor were discussed long ago [2,3]. In the minimal technicolor model, with just one technifermion doublet, the only prominent collider signals are the enhancements in longitudinally-polarized weak boson production. These are the s-channel color-singlet technirho resonances near 1.5–2 TeV: ρ0 W+W− and ρ± W±Z0. The (α2) cross sections of these processes are quTite→smaLll atLsuch mTa→sses. LThLis O and the difficulty of reconstructing weak-boson pairs with reasonable efficiency make observing these enhancements a challenge. 1) Talk presented at the Workshop on Physics at the First Muon Collider and at the Front End of a Muon Collider 2) The only technipions in minimal technicolor are the massless Goldstone bosons that become, via the Higgs mechanism, the longitudinal components W± and Z0 of the weak gauge bosons. L L Nonminimal technicolor models are much more accessible because they have a rich spectrum of lower mass technirho vector mesons and technipion states into whichtheymaydecay.3 IfthereareN doubletsoftechnifermions, alltransforming D according to the same complex representation of the technicolor gauge group, there will be 4N2 1 technipions whose decay constant is D − F π F = . (1) T √N D Three of these are the longitudinal weak bosons; the remaining 4N2 4 await D − discovery. In the standard model and its extensions, the masses of quarks and leptons are produced by their Yukawa couplings to the Higgs bosons—couplings of arbi- trary magnitude and phase that are put in by hand. This option is not available in technicolor because there are no elementary scalars. Instead, quark and lepton chiral symmetries must be broken explicitly by gauge interactions alone. The most economical way to do this is to employ extended technicolor, a gauge group containing flavor, color and technicolor as subgroups [4–6]. Quarks, leptons and technifermions are unified into a few large representations of ETC. The ETC gauge symmetry is broken at high energy to technicolor color. Then quark and lepton ⊗ hard masses arise from their coupling (with strength g ) to technifermions via ETC ETC gauge bosons of generic mass M : ETC g2 m (M ) m (M ) ETC T¯T , (2) q ETC ≃ ℓ ETC ≃ M2 h iETC ETC where T¯T and m (M ) are the technifermion condensate and quark and ETC q,ℓ ETC h i lepton masses renormalized at the scale M . ETC If technicolor is like QCD, with a running coupling α rapidly becoming small TC above Λ 1TeV, then T¯T T¯T Λ3 . To obtain quark masses of a fewTCGe∼V, M /g h < i3E0TTCe≃V his reiqTuCire≃d. TTChis is excluded: Extended ETC ETC technicolor boson exchanges a∼lso generate four-quark interactions which, typically, include ∆S = 2 and ∆B = 2 operators. For these not to be in conflict with K0-K¯0 a|nd |B0-B¯0 mix|ing p|arameters, M /g must exceed several hundred d d ETC ETC TeV [5]. This implies quark and lepton masses no larger than a few MeV, and technipion masses no more than a few GeV—a phenomenological disaster. Because of this conflict between constraints on flavor-changing neutral currents and the magnitude of ETC-generated quark, lepton and technipion masses, classical technicolor was superseded over a decade ago by “walking” technicolor [7]. Here, the strong technicolor coupling α runs very slowly—walks—for a large TC range of momenta, possibly all the way up to the ETC scale of several hun- ¯ ¯ dred TeV. The slowly-running coupling enhances TT / TT by almost a ETC TC h i h i 3) The technipions of nonminimal technicolor include the longitudinal weak bosons as well as additionalGoldstonebosonsassociatedwithspontaneoustechnifermionchiralsymmetrybreaking. Thelattermustanddoacquiremass—fromtheextendedtechnicolorinteractionsdiscussedbelow. factor of M /Λ . This, in turn, allows quark and lepton masses as large ETC TC as a few GeV and M > 100GeV to be generated from ETC interactions at πT M = (100TeV). ∼ ETC O Walking technicolor requires a large number of technifermions in order that α TC runs slowly. These fermions may belong to many copies of the fundamental representation of the technicolor gauge group, to a few higher dimensional representations, or to both. 4 In many respects, walking technicolor models are very different from QCD with a few fundamental SU(3) representations. One example of this is that integrals of weak-current spectral functions and their moments converge much more slowly than they do in QCD. Consequently, simple dominance of the spectral integrals by a few resonances cannot be correct. This and other calculational tools based on naive scaling from QCD and on large-N arguments are suspect [10]. Thus, it is TC not yet possible to predict with confidence the influence of technicolor degrees of freedom on precisely-measured electroweak quantities—the S,T,U parameters to name the most discussed example [11]. The large mass of the top quark [12] motivated another major development in technicolor. Theorists have concluded that ETC models cannot explain the top quark’s large mass without running afoul of experimental constraints from the ρ ¯ parameter and the Z bb decay rate [13]. This state of affairs has led to the → proposal of “topcolor-assisted technicolor” (TC2) [14]. In TC2, as in top-condensate models of electroweak symmetry breaking [15], almost all of the top quark mass arises from a new strong “topcolor” interaction [16]. To maintain electroweak symmetry between (left-handed) top and bottom quarks and yet not generate m m , the topcolor gauge group under which (t,b) trans- b t ≃ formis usually taken to be a strongly-coupled SU(3) U(1). The U(1) provides the ⊗ difference that causes only top quarks to condense. Then, in order that topcolor interactions be natural—i.e., that their energy scale not be far above m —without t introducing large weak isospin violation, it is necessary that electroweak symmetry breaking remain due mostly to technicolor interactions [14]. Earlysteps inthe development of theTC2 scenario have been taken intwo recent papers [17]. The breaking of topcolor SU(3) U(1) near the electroweak scale gives ⊗ rise to a massive color octet of V colorons and a color-singlet Z′. The SU(3) may 8 be broken by some of the same technifermion condensates that break electroweak SU(2) U(1), so that the colorons (which are expected to be broad) have mass ⊗ near 500 GeV. However, in order that the strong topcolor U(1) interaction not contaminate the ordinary Z couplings to fermions, it and the weaker U(1) acting on light fermions must be broken down to their diagonal subgroup, ordinary weak hypercharge, in the vicinity of 2 TeV. This suggests that the Z′ mass is in the range 4) Thelastpossibilityinspired“multiscaletechnicolor”modelscontainingbothfundamentaland higher representations, and having an unusual phenomenology [8]. In multiscale models, there typically aretwo widely separatedscalesof electroweaksymmetry breaking,with the upper scale set by the weak decay constant, F =246GeV. Multiscale models in which the entire top quark π mass is generated by ETC interactions are excluded by such processes as b sγ [9]. → 1–3 TeV, out of reach of all but the highest energy colliders. As I discussed in my talk at the FMC workshop, the Z′ is so heavy that it may require a multi-TeV Big Muon Collider to find and study it. This subject deserves further study. In TC2 models, ETC interactions are still needed to generate the light and bottom quark masses, contribute a few GeV to m , 5 and give mass to the technipions. t The scale of ETC interactions still must be hundreds of TeV to suppress flavor- changing neutral currents and, so, the technicolor coupling still must walk. Thus, even though the phenomenology of TC2 is still in its infancy, it is expected to share general features with multiscale technicolor: many technifermion doublets bound into many technihadron states, some at relatively low masses, some carrying ordinary color and some not. The lightest technihadrons may have masses in the range 100–300 GeV and should be accessible at the Tevatron collider in Run III if not Run II. All of them are easily produced and detected at the LHC at moderate luminosities. If technihadrons exist, they will be discovered at hadron colliders before the First Muon Collider (FMC) is built. As we shall see, this is a good thing for the FMC: Several of the lightest technihadrons are very narrow and can be produced in the s-channel of µ+µ− annihilations. In the narrow-band FMC, it would be exceedingly difficult to find them by a standard scan procedure without a good idea of where to look. II TECHNICOLOR AT THE FMC A Technihadron Decay Rates I assume that the technicolor gauge group is SU(N ) and take N = 4 in TC TC calculations. Its gauge coupling must walk and I assume this is achieved by a large number of isodoublets of technifermions transforming according to thefundamental representation of SU(N ). I consider the phenomenology of only the lightest TC color-singlet technihadrons and assume that the constraint from the S-parameter on their spectrum still allows the lightest ones to be considered in isolation for a limited range of √s, the µ+µ− center-of-mass energy, about their masses. These technihadrons carry isospin I = 1 and 0 and consist of a single isotriplet and isosinglet of vectors, ρ0, ρ± and ω , and pseudoscalars π0, π±, and π0′. The T T T T T T latter are in addition to the longitudinal weak bosons, W± and Z0—those linear L L combinations of technipions that couple to the electroweak gauge currents. I adopt TC2 as a guide for guessing phenomenological generalities. In TC2 there is no need for large technifermion isospin splitting associated with the top-bottom mass difference. This implies that the lightest ρ and ω are approximately degenerate. T T The lightest charged and neutral technipions also should have roughly the same mass, but there may be appreciable π0–π0′ mixing. If that happens, the lightest T T 5) MasslessGoldstone“top-pions”arisefromtop-quarkcondensation. ThisETCcontributionto m is needed to give them a mass in the range of 150–250GeV. t neutral technipions are really U¯U and D¯D bound states. Finally, for purposes of discussing signals at the FMC, we take the lightest technihadron masses to be M = M 200GeV; M 100GeV. (3) ρT ∼ ωT ∼ πT ∼ The decays of technipions are induced mainly by ETC interactions which couple them to quarks and leptons. These couplings are Higgs-like, and so technipions are expected to decay into the heaviest fermion pairs allowed. Because only a few GeV of the top-quark’s mass is generated by ETC, there is no great preference for π T to decay to top quarks nor for top quarks to decay into them. Furthermore, the isosinglet component of neutral technipions may decay into a pair of gluons if its constituent technifermions are colored. Thus, the predominant decay modes of the light technipions are assumed to be π0 ¯bb, c¯c, τ+τ− πT0′→ gg, ¯bb, c¯c, τ+τ− (4) πT+ →c¯b, cs¯, τ+ν . T → τ To estimate branching ratios we use the following decay rates (for later use in the technihadron production cross sections, we quote the energy-dependent width [3,18]): 6 1 Γ(πT → f¯′f) = 16πF2 Nf pf Cf2(mf +mf′)2 T Γ(π0′ gg) = 1 α2 C N2 s32 . (5) T → 128π3F2 S πT TC T Here, C is an ETC-model dependent factor of order one except that TC2 suggests f C < m /m ; N isthenumberofcolorsoffermionf;p isthefermionmomentum; t b t f f α| |is∼the QCD coupling evaluated at M ; and C is a Clebsch of order one. For S πT πT M = 110GeV, F = F /3 = 82GeV, m = 4.2GeV, N = 4, α = 0.1, C = 1 πT T π b TC S b for π0 and π0′, and C = 4/3: T T πT Γ(π0 ¯bb) = Γ(π0′ ¯bb) = 35MeV T → T → (6) Γ(π0′ gg) = 10MeV. T → If technicolor were like QCD, we would expect the main decay modes of the lightest technivector mesons to be ρ0 π+π− and ω π+π−π0 with the T → T T T → T T T technihadrons all composed of the same technifermions. However, the large ratio ¯ ¯ TT / TT occurring in walking technicolor significantly enhances techni- ETC TC h i h i pion masses compared to technivector masses. Thus, ρ π π decay channels T T T may well be closed. If this happens, then ρ0 decays to W→+W− or W±π∓ and ω T L L L T T to γπ0 or Z0π0. [8,19,20,6] T T 6) The amplitude is taken to be (πT f¯′(p1)f(p2))=Cf(mf +mf′)/FT u¯(p2)γ5v(p1). M → We parameterize this for ρ decays by adopting a simple model of two isotriplets T of technipions which are mixtures of W±, Z0 and mass-eigenstate technipions π±, L L T π0. The lighter isotripletρ isassumed todecay dominantly into pairsof themixed T T state of isotriplets Π = sinχ W +cosχ π , where T L T | i | i | i sinχ = F /F . (7) T π Then, the energy-dependent decay rate for ρ0 π+π− (where π may be W , T → A B A,B L Z , or π ) is given by L T 2α 2 p3 Γ(ρ0 π+π−) = ρTCAB AB , (8) T → A B 3 s where p is the technipion momentum and α is obtained by naive scaling from AB ρT the QCD coupling for ρ ππ: → 3 α = 2.91 . (9) ρT N (cid:18) TC(cid:19) The parameter 2 is given by CAB sin4χ for W+W− L L 2 = 2sin2χ cos2χ for W+π− +W−π+ (10) CAB  cos4χ for π+Lπ−T L T T T Note that the ρ can bevery narrow. For √s = M = 210GeV, M = 110GeV, T ρT πT and sinχ = 1, we have Γ(ρ0 π+π−) = 680MeV, 80% of which is W±π∓. 3 AB T → A B L T We shall also need the decay rates of the ρ to fermion-antifermion states. The P T energy-dependent widths are N α2 p (s+2m2) Γ(ρ0 f¯f ) = f i i A0(s). (11) T → i i 3α s i ρT Here, α is the fine-structure constant, p is the momentum and m the mass of i i fermion f , and the factors A0 are given by i i 0 2 2 A (s) = (s) + (s) , i |AiL | |AiR | 2cos2θ s W (s) = Q + ζ , (12) Aiλ i sin22θW iλ s−MZ2 +i√sΓZ! 2 2 ζ = T Q sin θ , ζ = Q sin θ . iL 3i i W iR i W − − ¯ For M = 210GeV and other parameters as above, the ff partial decay widths ρT are: Γ(ρ0 u¯ u ) = 5.8MeV, Γ(ρ0 d¯d ) = 4.1MeV Γ(ρT0 → ν¯iνi) = 0.9MeV, Γ(ρT0 → ℓ+iℓi−) = 2.6MeV. (13) T → i i T → i i For the ω , phase space considerations suggest we consider only its γπ0 and T T fermionic decay modes. The energy dependent widths are: αp3 0 Γ(ω γπ ) = , T → T 3M2 T N α2 p (s+2m2) Γ(ω f¯f ) = f i i B0(s). (14) T → i i 3α s i ρT The mass parameter M in the ω γπ0 rate is unknown a priori; naive scaling from the QCD decay, ωT γπ0, sTug→gestsTit is several 100 GeV. The factor B0 is → i given by 0 2 2 B (s) = (s) + (s) , i |BiL | |BiR | 4sin2θ s W (s) = Q ζ (Q +Q ). (15) Biλ " i − sin22θW iλ s−MZ2 +i√sΓZ!# U D Here, Q and Q = Q 1 are the electric charges of the ω ’s constituent techni- U D U T − fermions. For M = 210GeV and M = 110GeV, and choosing M = 100GeV ωT πT T and Q = Q +1 = 4, the ω partial widths are: U D 3 T Γ(ω γπ0) = 115MeV T → T ¯ Γ(ω u¯ u ) = 6.8MeV, Γ(ω d d ) = 2.6MeV (16) T i i T i i Γ(ω → ν¯ν ) = 1.7MeV, Γ(ω → ℓ+ℓ−) = 5.9MeV. T i i T i i → → The beam momentum spread of the First Muon Collider has been quoted to be as narrow as σ /p = 3 10−5 at √s = 100GeV and 10−3 at √s = 200GeV. These p × correspond to beam energy spreads of σ = 5MeV at 100 GeV and 300 MeV at E 200 GeV. The resolution at 100 GeV is less than the expected π0, π0′ widths. At T T 200 GeV it is sufficient to resolve the ρ0, but not the ω , for the parameters we T T used. It is very desirable, therefore, that the 200 GeV FMC’s energy spread be about factor of 10 smaller. Since each of these technihadrons can be produced as an s-channel resonance in µ+µ− annihilation, it would then be possible to sit on the peak at √s = M. As we see next, the peak cross sections are enormous, 2–3 orders of magnitude larger than can be achieved at a hadron collider and even at a linear e+e− collider because of the latter’s inherent beam energy spread. B Technihadron Production Rates Like the standard Higgs boson, neutral technipions are expected to couple to µ+µ− with a strength proportional to m . Compared to the Higgs, however, this µ coupling is enhanced by a factor of F /F = 1/sinχ. This makes the resolution of π T the FMC well-matched to the π0 width. Thus, the FMC is a technipion factory, T overwhelming the rate at any other collider. Once a neutral technipion has been FIGURE 1. Theoretical (unsmeared) cross sections for µ+µ− π0′ ¯bb (dashed), gg → T → (dot-dashed) and total (solid) for M = 110GeV and other parameters defined in the text. πT The solid horizontal lines are the backgrounds from γ, Z0 ¯bb (lower) and Z0 q¯q (upper). → → Note the energy scale. found in ρ or ω decays at a hadron collider, it should be relatively easy in the T T FMCto locatetheprecise positionof theresonance andsit onit. The cross sections ¯ for ff and gg production are isotropic; near the resonance, they are given by dσ(µ+µ− π0 orπ0′ f¯f) N C C m m 2 s → T T → = f µ f µ f , dz 2π FT2 ! (s−Mπ2T)2 +sΓ2πT (17) dσ(µ+µ− π0′ gg) C C m α N 2 s2 → T → = πT µ µ S TC . dz 32π3 FT2 ! (s−Mπ2T)2 +sΓ2πT Here, z = cosθ where θ is the center-of-mass production angle. The π0′ production cross sections and the Z0 backgrounds are shown in Fig. 1 T for M = 110GeV and other parameters as above (C = C = 1, C = 4/3, πT µ f πT F = 82GeV, α = 0.1, N = 4). The peak signal rates approach 1nb. The T S TC ¯bb dijet rates are much larger than the Z0 ¯bb backgrounds, while the gg rate is comparable to Z0 q¯q. Details of these an→d the other calculations in this section, → including the effects of the finite beam energy resolution, will appear in Ref. [21]. See Ref. [22] for another example of neutral scalars that may be produced in µ+µ− annihilation. FIGURE 2. Theoretical (unsmeared) cross sections for µ+µ− ρ0,ω e+e− for input → T T → masses M =210GeV and M =212.5GeV and other parameters as defined in the text. ρT ωT The cross sections for technipion production via the decay of technirho and tech- niomega s-channel resonances are calculated using vector meson (γ, Z0) dominance [3,8,19,20]. They are given by: dσ(µ+µ− ρ0 π π ) πα2p3 A0(s) 2 (1 z2) → T → A B = AB µ CAB − , dz s12 (s−Mρ2T)2 +sΓ2ρT dσ(µ+µ− → ωT → γπT0) = πα3s12p3 Bµ0(s)(1+z2) , (18) dz 3α M2 (s M2 )2 +sΓ2 ρT T − ωT ωT where A0 and B0 were defined in Eqs. 12 and 15, respectively. For M = M = µ µ ρT ωT 210GeV, M = 110GeV, and other parameters as above, the total peak cross πT sections are [21]: σ(µ+µ− ρ0 π π ) = 1.1nb AB → T → A B (19) P σ(µ+µ− ω γπ0) = 8.9nb. → T → T The technirho rate is 20% W+W− and 80% W±π∓. T Finally,itisreasonabletoexpect asmallnonzeroisospinsplittingbetweenρ0 and T ω . This would appear as a dramatic interference in the µ+µ− f¯f cross section T → provided the FMC energy resolution is good enough in the ρ –ω region. The cross T T section is most accurately calculated by using the full γ–Z0–ρ –ω propagator T T matrix, ∆(s). With 2 = M2 i√sΓ (s) for V = Z0,ρ ,ω , this matrix is the MV V − V T T inverse of s 0 sf sf 0 s 2 −sfγρT −sfγωT ∆−1(s) =  sf −sfMZ s− Zρ2T − 0ZωT  . (20)  −−sfγγωρTT −−sfZZωρTT −0MρT s−M2ωT    Here, α α f = , f = (Q +Q ) γρT α γωT α U D s ρT s ρT α cos2θ α sin2θ W W f = , f = (Q +Q ). (21) ZρT α sin2θ ZωT − α sin2θ U D s ρT W s ρT W Then, the cross section is given in terms of matrix elements of ∆ by dσ(µ+µ− ρ0, ω f¯f ) N πα2 → T T → i i = f 2 + 2 (1+z)2 iLL iRR dz 8s |D | |D | (cid:26)(cid:16) (cid:17) 2 2 2 + + (1 z) , (22) iLR iRL |D | |D | − (cid:27) (cid:16) (cid:17) where 4 Diλλ′(s) = s QiQµ∆γγ(s)+ sin22θ ζiλζµλ′∆ZZ(s) (cid:20) W 2 + ζiλQµ∆Zγ(s)+Qiζµλ′∆γZ(s) . (23) sin2θ W (cid:18) (cid:19)(cid:21) Figure 2 shows the theoretical ρ0–ω interference effect in µ+µ− e+e− for T T → input masses M = 210GeV and M = 212.5GeV and other parameters as ρT ωT above. The propagator ∆ shifts the nominal positions of the resonance peaks by (α/α ). Thetheoreticalpeakcrosssectionsare5.0pbat210.7GeVand320pbat O ρT 214.0GeV. This demonstrates the importance of precise resolution in the 200GeV FMC. III CONCLUSIONS Modern technicolor models predict narrow neutral technihadrons, π , ρ and T T ω . These states would appear as spectacular resonances in a µ+µ− collider with T √s = 100–200GeV and energy resolution σ /E < 10−4. This is a very strong E physics motivation for building the First Muon Col∼lider. I thank other members of the First Muon Collider Workshop Strong Dynamics Subgroup for valuable interactions, especially Paul Mackenzie and Chris Hill for stressing the importance of a narrow π0 at a muon collider, and Pushpa Bhat, T