Recently a new dynamical symmetry breaking model of electroweak interactions
was proposed based on interacting fermions. Two fermions of different SU(2)
representations form a symmetry breaking condensate and generate the lepton and
quark masses. The weak gauge bosons get their usual standard model masses from
a gauge invariant Lagrangian of a composite doublet scalar field. The new
fermion fields become massive by condensation. In this note the gap equations
are given in the linearized (mean field) approximation and the conditions for
symmetry breaking and mass generation are presented. Perturbative unitarity
constrains the self-couplings and the masses of the new fermions, a raw
spectrum is given.Comment: 10 pages, 4 figure

Agashe K

Cynolter G

E Lendvai

G Cynolter

Hill C T

Klevansky S P

Yamawaki K

Yao W-M

English

arXiv

Cynolter, G

Lendvai, E

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-ph/0607021v1  3 Jul 2006Gap Equations and Eletroweak SymmetryBreakingG. Cynolter and E. LendvaiTheoretial Physis Researh Group of Hungarian Aademy of Sienes,Eötvös University, Budapest, 1117 Pázmány Péter sétány 1/A, HungaryAbstratReently a new dynamial symmetry breaking model of eletroweakinterations was proposed based on interating fermions. Two fermionsof dierent SUL(2) representations form a symmetry breaking ondensateand generate the lepton and quark masses. The weak gauge bosons gettheir usual standard model masses from a gauge invariant Lagrangian ofa omposite doublet salar eld. The new fermion elds beome massiveby ondensation. In this note the gap equations are given in the linearized(mean eld) approximation and the onditions for symmetry breaking andmass generation are presented. Perturbative unitarity onstrains the self-ouplings and the masses of the new fermions, a raw spetrum is given.With the advent of the LHC the problem of eletroweak symmetry breaking be-omes more and more important. The LHC experiments are expeted to shedlight on the dynamis of the symmetry breaking. As there is no diret evidenefor elementary salar (Higgs) partiles, alternative, dynamial mehanisms havealso been investigated, like tehniolour, top ondensate or topolour models[1, 2, 3℄. These models were reborn in extra dimensional senarios like higgslessmodels [4℄. Reently a new dynamial symmetry breaking model was proposedbased on new fermions of dierent representations of the weak gauge group[5℄, in the model four-fermion interations generate the (symmetry breaking)ondensates. The ondensates generate masses for the new fermions and anauxiliary omposite salar eld is responsible for the weak gauge boson masses.The model is non-renormalizable and regulated by a four dimensional uto,we present the model after the introdution in more detail. In this work weextend the model to inlude more general ondensates and investigate the non-onventional struture of the gap equations in the modied model. We wereable to analyse the oupled gap equations and give the ondition of nding asymmetry breaking solution and fullling the onstraints of perturbative uni-tarity. The possible masses for various oupling onstants are presented and itis shown that the model an generate fermions with a mass of few hundred GeVfor utos in the TeV range.To start with we summarize the fermion ondensate model of eletroweakinterations [5℄. In the model to be outlined the Higgs setor is replaed with new1fermions with non-renormalizable four-fermion interations. Under SUL(2) ×UY (1) the new fermions are a neutral singlet ΨS, and a weak doublet ΨD =(Ψ+DΨ0D)with hyperharge 1. Ψ+D(Ψ0D)is a positively harged (neutral) eld.The new fermions have eetive four-fermion interations, valid up to somephysial uto, the ultraviolet ompletion of the model is not speied.The new Lagrangian with gauge invariant kineti terms and invariant 4-fermion interations of the new fermions is LΨ,LΨ = iΨDDµγµΨD + iΨS∂µγµΨS −m0DΨDΨD −m0SΨSΨS ++λ1(ΨDΨD)2+ λ2(ΨSΨS)2+ 2λ3(ΨDΨD) (ΨSΨS), (1)m0D,m0S are bare masses and Dµ is the ovariant derivativeDµ = ∂µ − ig2τ Aµ − ig′2Bµ, (2)where Aµ,Bµ and g, g′are the usual weak gauge boson elds and ouplings,respetively. Additional four-fermion ouplings are possible but those will notbasially hange the symmetry breaking and mass generation. We will show inwhat follows that for ouplings λi exeeding the ritial value the four-fermioninterations of (1) generate ondensates〈Ψ0DαΨ0Dβ〉0= a1δαβ , (3)〈Ψ+DαΨ+Dβ〉0= a+δαβ , (4)〈ΨSαΨSβ〉0= a2δαβ , (5)〈ΨSΨD〉0=〈(ΨSΨ+DΨSΨ0D)〉06= 0 (6)The formation of the harged ondensate (4) is more general then in [5℄, butit does not hange the original arguments about symmetry breaking. The non-diagonal ondensate in (6) spontaneously breaks SUL(2)× UY (1) to Uem(1) ofeletromagnetism. With the gauge transformations of ΨD the ondensate (6)an always be transformed into a real lower omponent,〈ΨSαΨ0Dβ〉0= a3δαβ,〈ΨSαΨ+Dβ〉0= 0, (7)where a3 is real. The omposite operator ΨSΨD resembles the standard salardoublet. Assuming invariant four-fermion interations for the new and knownfermions,Lf = gf(ΨfLΨfR) (ΨSΨD)+ gf(ΨfRΨfL) (ΨDΨS), (8)the ondensate (7) generates masses to the standard femions. In the linearized,mean eld approximation the eletron mass, for example, isme = −4gea3. (9)Up type quark masses an be generated via the harge onjugate eld Ψ̃D =iτ2 (ΨD)†.2The masses of the weak gauge bosons arise from the eetive interations ofthe auxiliary omposite Y = 1 salar doublet,Φ =(Φ+Φ0)= ΨSΨD. (10)Φ develops a gauge invariant kineti term in the low energy eetive desriptionLH = h (DµΦ)†(DµΦ) , (11)where Dµ is the usual ovariant derivative (2).The oupling onstant h sets the dimension of LH , [h] = −4 in mass dimen-sion, we assume h > 0. (11) is a non-renormalizable Lagrangian and it providesthe weak gauge boson masses and some of the interations of the new fermionswith the standard gauge bosons.The terms with Φ0 in LH an be written ash−1LH =g22W−µ W+µΦ0†Φ0 +g24 · cos2 θWZµZµΦ0†Φ0 + (12)+[∂µΦ0†∂µΦ0 − i2gcos θW(∂µΦ0†)Φ0Zµ +i2gcos θWΦ0†Zµ(∂µΦ0)]in terms of the usual vetor boson elds.In the linearized approximation in (12) we puthΦ0†Φ0 → h〈Φ0†Φ0〉0= h(16a23 − 4a1a2)=v22, (13)leading to the standard massesmW =gv2, mZ =gv2 cos θW. (14)v2 is, as usual,(√2GF)−1; v = 254 GeV . The tree masses naturally fulll theimportant relation ρtree = 1.One the ondensates (3-6) are formed, dynamial mass terms are generatedin the Lagrangian (1) beside the bare mass terms.Lψ → LlinΨ = −m+Ψ+DΨ+D −m1Ψ0DΨ0D −m2ΨSΨS −m3(Ψ0DΨS +ΨSΨ0D),(15)withm+ = m0D − 6λ1a+ − 8 (λ1a1 + λ3a2) = m1 + 2λ1 (a+ − a1) (16)m1 = m0D − 6λ1a1 − 8 (λ1a+ + λ3a2) , (17)m2 = m0S − 6λ2a2 − 8λ3 (a1 + a+) , (18)m3 = 2λ3a3. (19)If m3 = 0 (λ3 = 0 or a3 = 0) then (15) is diagonal, the original gauge eigenstatesare the physial elds, the eletroweak symmetry is not broken, λ3a3, the non-diagonal ondensate triggers the mixing and symmetry breaking. If m3 6= 0 (15)is diagonalized via unitary transformation to get physial mass eigenstates3ψ 0Dm 1 ψ 0Dψ0Dψ0Dψ 0Dλ 1 λ 3ψSψ+Dψ0Dψ 0Dψ 0Dλ 1ψ 0D= + +Figure 1. Feynman graphs for the gap equation (17). Similar graphsorresponding to (16,18) with exhanged legs and lines.ψD0 ψD0ψD0 ψSm 3 λ 3ψS ψS=Figure 2. Feynman graphs for the gap equation (19).Ψ1 = cΨ0D + sΨS,Ψ2 = −sΨ0D + cΨS , (20)where c = cosφ and s = sinφ, φ is the mixing angle. The masses of the physialfermions Ψ1, Ψ2 are2M1,2 = m1 +m2 ±m1 −m2cos 2φ. (21)The mixing angle is dened by2m3 = (m1 −m2) tan 2φ. (22)Again we see, one m3 = 0 the mixing angle vanishes (for m1 6= m2), M1 = m1and M2 = m2. The physial masses may be equal (M1 = M2) only if m1 = m2,the original neutral fermions are degenerate in mass and then the mixing angleis meaningless from the point of view of mass matrix diagonalization.It follows that the physial eigenstates themselves form ondensates sinec2〈Ψ1αΨ1β〉0+ s2〈Ψ2αΨ2β〉0= a1δαβ,s2〈Ψ1αΨ1β〉0+ c2〈Ψ2αΨ2β〉0= a2δαβ, (23)cs〈Ψ1αΨ1β〉0− cs〈Ψ2αΨ2β〉0= a3δαβ.There is no non-diagonal ondesate as Ψ1, Ψ2 are independent. Combining theequations of (23) one ndsa3 =12tan 2φ (a1 − a2) . (24)For a1 = a2, a3 6= 0 is not possible for cos 2φ 6= 0. As is seen, (24) is equivalentto〈Ψ1αΨ2β〉0= 0. Comparing (24) to (22) yieldsm1 −m2 = 2λ3 (a1 − a2) . (25)4Using the equations (16-19) we are lead to a onsisteny onditions(λ3 − λ1)(a1 +43a+)= (λ3 − λ2) a2, (26)λ1 6= λ2 goes with a1 + 43a+ 6= a2.The equations (16-19) an be formulated as gap equations in terms of thephysial elds expressing both the masses and the ondensates with Ψ1, Ψ2and Ψ+ ≡ Ψ+D. Assuming vanishing original masses, m0S = 0, m0D = 0, theomplete set of gap equations arec · s (M1 −M2) = 2λ3 c · s (I1 − I2) , (27)c2M1 + s2M2 = −λ1(6(c2I1 + s2I2)+ 8I+)− 8λ3(s2I1 + c2I2), (28)s2M1 + c2M2 = −6λ2(s2I1 + c2I2)− 8λ3(c2I1 + s2I2 + I+), (29)M+ = −λ1(8(c2I1 + s2I2)+ 6I+)− 8λ3(s2I1 + c2I2). (30)The main task of the present work is to explore the struture of the gapequations. There are four algebrai equations for four variables M1, M2, M+,c2 = cos2 φ. As in almost all approximation Ii ∼ Mi, (27-30) show gap equationharateristis, Mi = 0 is always a symmetri solution, whih is stable for small|λi|. Inreasing |λi| also an energetially favoured [7℄ massive solution emergesas in the original Nambu Jona-Lasinio model. Now we explore the parameterspae λi to nd aeptable phyial masses.Let the ondensates be approximated by free eld propagators〈ΨiαΨiβ〉=δαβ4Ii = −δαβ8π2Mi(Λ2 −M2i ln(1 +Λ2M2i)), i = 1, 2,+, (31)where M+ = m+. Here Λ is a four-dimensional physial uto, it sets the saleof the new physis responsible for the non-renormalizable operators. From thepoint ov view of symmetry breaking Λ an be hosen arbitrary large (below theGUT or Plank sale), but higher Λ implies stronger ne tuning of λ3, see (32),to keep the new fermion masses in the eletroweak range. To avoid ne tuningand allow reasonable fermion masses Λ is expeted to be a few TeV, typiallyaround 3 TeV [5℄.For the eletroweak symmetry breaking the most important equation is (27),it triggers mixing between the dierent representations of the weak gauge group.Applying (31) it reads0 = (M1 −M2) c · s 1λ3+Λ2π2−M31 ln(1 + Λ2M21)−M32 ln(1 + Λ2M22)M1 −M2 . (32)(32) always has a symmetri solution (M1 −M2) c · s = 0, implying sin 2φ = 0for M1 6= M2 ,essentialy no mixing, M1 = M2 is disussed after (22). If |λ3| isgreater than a ritial value |λc3| = π2Λ2 there also exists a symmetry breakingsolution (M1 6= M2), whih always has lower energy if the massive solutionexists [7℄. Equation (32) has a solution with moderate masses (M1,2 < 0.7Λ )if λ3 is negative. In the small mass limit the parantheses in (32) simplies to5050010001500M 2500 1000 1500M1(GeV)(GeV)−20−15−12−10Figure 3. Constant λ3 ontours in the M1-M2 plane forλ3 = {−10,−12,−15,−20} · 1/Λ2, Λ = 3 TeV.1λ3+ Λ2π2 −(M21 +M1M2 +M22) (ln(Λ2)− ln(M̃2))where M̃ ≃ max(M1,M2).If |λ3| is slightly larger than its ritial value, then we generally get small massesompared to Λ, M21 + M1M2 + M22 ≪ Λ2. The ritial oupling agrees withthe original Nambu-Jona Lasinio value, only a fator of two oming from thedenition in the Lagrangian (1), λ3 also dened dierently in [5℄. If |λ3| <|λc3| then the parantheses does not vanish in (32), the ondensate a3 is notformed and (M1 −M2) c · s = 0. The physial solution is c · s = 0, thereis no meaningful mixing, ΨS , ΨD are the physial mass eigenstates, and theeletroweak symmetry is not broken.Despite the ompliated struture of the non-linear equations (27-30) we geta relatively simple gap equation for λ1, similar to (32), from (16) 2λ1 (a1 − a+) =m1 −m+. In the physial elds we haveM+ − c2M1 − s2M2 = 2λ1(I+ − c2I1 − s2I2). (33)It inludes four unknowns, therefore it annot be analyzed diretly. We get auseful restrition solving (27) and (28) for λ1 and substituting it to (33), relatingM1,M2,M+ and c2independently of the λi's. Requiring that 0 ≤ c2 ≤ 1 we getM1 ≤ M+ ≤ M2. (34)As a result of the logaritmi terms in Ii, M+ is nonlinear in c2, while m1 =c2M1 + s2M2. We remark that though (27) and (33) are very similar, formoderate masses λ3 is always negative, while λ1 is positive (also λ2 > 0). In thec2 = 1 (0) limit M+ = M1 (M2) and there are anelations in (27-30). Turning6bak to the symmetri solution of (32) the relation (34) gives M+ = M1 = M2and the rest of the gap equations set the ommon mass equal to zero unless thespeial relation 6(λ3 − λ2) = 8(λ3 − λ1) holds to provide anellations.To nd the ritial value for λ1 and λ2 we onsidered the limit M+ → M2 =M and M1 → 0 thenλ1 =17π2Λ2 −M2 ln(1 + Λ2M2) , λ2 =43π2Λ2 −M2 ln(1 + Λ2M2) . (35)We get the same NJL type expression if we take the limit M+ → M1 = M andM2 → 0. (35) provides massive solutions if λ1 ≥ 17 π2Λ2 and λ2 ≥ 43 π2Λ2 . Numerialsans show that these are the minimal, ritial values for the ouplings and anbe approximated in speial limits. Numerial solutions are shown in Table 1.for uto Λ = 3 TeV. The role of M1 and M2 an be exhanged together withc2 ↔ s2, therefore we hose M1 < M2 without the loss of generality. As theuto is not too high, 3 TeV, there is no serious ne tuning in the λi's to ndrelatively small masses.To understand the signs and roughly the fators in λc1,2 onsider the limitM1 ≃ M2 ≃ M+ ≃ M . If M ≪ Λ then λ3 ≃ λc3 = − π2Λ2 , though in the exatlimit (27) beomes singular. We get from (27-30) the relation 14λ1 = 6λ2 +8λ3and a single gap equation ( I = IM in (31) )M = − (14λ1 + 8λ3) I. (36)Small mass solution requires λ̃ = 14λ1 + 8λ3 to be lose to it's ritial value2π2/Λ2 and provides rough estimates λ1 ∼ 57 π2Λ2 and also λ2 ∼ 3 π2Λ2 to generatesmall masses. Numerial solutions also provide general (M+ not lose to M1 orM2) small masses for ouplings lose to these values, see Table 1.In the strongest small mass limit one neglets the logaritmi terms in theondensates (31), and equations (27-30) redue to a linear homogeneous systemof equations. Two onditions emerge to nd a nonzero solution for the physialmasses(1 + 2λ1Λ2π2)(M+ − c2M1 − s2M2)= 0 (37)(1− 14λ1Λ2π2)(1− 6λ2Λ2π2)= 128(λ3Λ2π2)2(38)Remember that m+ = M+ and m1 = c2M1 + s2M2, m2 = s2M1 + c2M2. Inase λ1Λ2π2 = − 12 (38) beomes(1− 6λ2 Λ2π2)= 16(λ3Λ2π2)2and there is a massrelation m1 +m+ =(2λ3Λ2π2)m2. If λ1Λ2π2 6= − 12 , we get two relations for themasses, m+ = m1 = c2M1 + s2M2 andm1m2= 1−6λ2Λ2/π216λ3Λ2/π2= 8λ3Λ2/π21−14λ1Λ2/π2.We have alulated and presented briey in [5℄ the onstraints on the pa-rameters of the model from perturbative unitarity [8, 9℄. Consider the ampli-tudes of two partile(Ψ(+)D ,Ψ(0)D or ΨS)elasti sattering proesses and impose|ℜa0| ≤ 1/2 for the J = 0 partial wave amplitudes. The ontat graph gives thedominant ontribution, negleting the fermion masses for the Ψ(+)D Ψ(−)D satter-ing gives an upper bound on λ1 oupling, |λ1| s ≤ 8π, where s is the maximalenter of mass energy(M2+ ≪ s ≤ Λ2).7λ1(π2Λ2)0.546 0.740 0.496 0.380 0.502 0.468 0.419λ2(π2Λ2)2.540 3.11 (!) 2.403 2.120 2.457 2.455 2.451λ3(π2Λ2)-1.031 -1.041 -1.042 -1.070 -1.083 -1.178 -1.330M1 (GeV) 100 148 100 100 150 200 200M2 (GeV) 150 150 200 300 300 500 800M+ (GeV) 149 149 190 290 290 490 790Table 1: Solutions of the gap equations for the uto Λ =3 TeV, λi are given inunits ofπ2Λ2 . In the seond olumn λ2 violates perturbative unitarity.We annot always use the small mass limit, as the solution of the gapequations provide higher λi's for signiantly higher masses. Therefore wehave alulated dierent heliity amplitudes [10℄ for non-vanishing masses. ForΨa(1)Ψ̄a(2) → Ψa(3)Ψ̄a(4), (a = 0, s,+),M = λi [(v̄2u1) (ū3v4)− (ū3u1) (v̄2v4)],where λi=1,2,3 is the only relevant four-fermion oupling. We onsider ΨS =sΨ1 + cΨ2 sattering as a linear ombination in the oupled Ψ1, Ψ2 hannelsto employ only λ2 (and simiarly Ψ(0)D to onstrain λ1). The ontributions ofthe γ, Z exhange graphs are negligible(O(g2)≪ 8π)beause of the extrapropagator. There are three dierent heliity hannels, we give the represen-tative heliity amplitudes, these are maximal for the bak to bak sattering(θ{13}scattering = π)M ((+−) → (+−)) = λi(s− 4M2i), (39)M ((++) → (−−)) = λis, (40)M ((+−) → (−+)) = λi4M2i . (41)For other sattering angles |M | is smaller than in (40), for example the maximumfor θ = 0 is λi4M2i . The mass dependent unitarity bound agrees with the rstestimateλis ≤ 8π, (42)where (i=1,2,3) and s ≤ Λ2 is the enter of mass energy. The unitarity on-straints are most stringent for λ2, even the equal small mass limit (36) wouldset λ2 ≃ 3π2/Λ2 whih is above the maximum allowed by unitarity 8π/Λ2 ≃2.55 · π2Λ2 . As an example we show a non-physial nearly equal mass solution inthe seond olumn of Table 1., whih is not allowed by perturbative unitarity.(42) implies an absolute upper bound on the smaller neutral mass, M1 ≤ 230GeV for Λ = 3TeV and generally pushes up the harged mass lose to M2. Theλ2 ontours are drawn in the M2−M1plane for M+ = M2− 10GeV in Figure 4.In onlusion, we explored and solved the gap equations in an extensionof the reently proposed fermion ondensate model of eletroweak interations[5℄. The original fermions mix via the non-diagonal four-fermion term in thenon-renormalizable Lagrangian. The gap equations an be formulated in thephysial degrees of freedoms and we have given examples of physial masses forvarious oupling onstants. We determined the ritial ouplings to dene theregion of massive solutions, λ3 < −π2/Λ2 is required to break the eletroweaksymmetry. With a few TeV uto the ouplings do not have to be ne tuned to8172125020406080100120140160180200220(GeV)M1200 400 600 800 1000 1200 1400 1600 (GeV)M2Figure 4. Constant λ2 ontours in the M2-M1 plane forλ2 = {25, 21, 17}1/Λ2, the allowed region is below the urves, Λ = 3 TeV.imply masses of few hundred GeV. In the spetrum three new fermions appear,between the lightest and the heaviest neutral ones there is a harged partile.Perturbative unitarity sets (via the oupling λ2) the mass of the harged fermionrelatively lose to the heavier neutral one and the lightest fermion mass must bebelow an upper bound, M1 ≤ 230 GeV for Λ = 3 TeV. The lightest new fermionis stable and a good dark matter andidate as the new fermions interat onlyin pairs and weakly with standard partiles. The model an be tested soon atLHC, numbers of pairs of new partiles are expeted with masses of few hundredGeV, ross setions for eletron-positron olliders are presented in [5℄.AknowledgementThe authors thank George Pósik for valuable disussion and ollaboration on[5℄.Referenes[1℄ Dynamial Symmetry Breaking, World Sienti, 1992, Ed. K. Yamawaki;C.T. Hill and E. H. Simmons, Phys. Rept. 381, 235 (2003); Erratum-ibid.390, 553 (2004).[2℄ W.A. Bardeen, C.T. Hill and M. Lindner, Phys.Rev.D41 1647 (1990); C.T.Hill, Phys.Lett. B266 , 419 (1991); M. Lindner and D. Ross, Nul.Phys.B370 , 30 (1992).9[3℄ G. Cynolter, E. Lendvai and G. Pósik, Eur. Phys. J. C38: 247 (2004);Mod.Phys.Lett A9, 1701 (1994).[4℄ C. Csáki, C. Grojean, H. Murayama, L.Pilo and J. Terning, Phys. Rev.D 69, 055006 (2004); C. Csáki, C. Grojean, L.Pilo and J. Terning, Phys.Rev. Lett. 92, 101802 (2004); K. Agashe, A. Delgado, M.J. May and R.Sundrum, JHEP 0308, 050 (2003).[5℄ G. Cynolter, E. Lendvai and G. Pósik, Eur. Phys. J. C46: 545 (2006).[6℄ Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); Y. Nambuand G. Jona-Lasinio, Phys. Rev. 124, 246 (1961).[7℄ S. P. Klevansky, Rev. Mod. Phys. 64, No. 3 (1992).[8℄ B.Lee, C.Quigg and H.Thaker, Phys. Rev. D16, 1519 (1977); D.Dius andV.Mathur, Phys. Rev. D7, 3111 (1973).[9℄ G. Cynolter, A. Bodor and G. Pósik, Heavy Ion Phys. 7, 245 (1998).[10℄ T. Appelquist, Mihael S. Chanowitz, Phys.Rev.Lett. 59, 2405 (1987),Erratum-ibid. 60,1589 (1988).10

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