Composite Leptons and Quarks from Hexad Preons

arXiv:1112.0181v1 [hep-ph] 1 Dec 2011 eto ewe rcinleeti hre n colored and A charges con- electric the level. fractional is preon model between Harari-Shupe the nection the no on of are is feature symmetry they there important isospin that framework SU(2) so this Within global preons three singlets. of hypercolor composed are leptons fqak n etn[] sal ti sue that assumed Z, is W, it the Usually (even compositeness leptons the leptons[2]. and be quarks may and prob- model quarks the standard of to the explanations the in ultimate to lems led the have mixings that and conjecture masses their of hierarchy the te n snurl h w ye fpen belong preons of types local unbroken two group the gauge The of representations fundamental the neutral. to is one other rwa ymtybekn,frinms generation elec- mass violation for CP mechanism fermion mixing, standard fermion the and breaking, the yet, found symmetry in been troweak boson not accuracy[1]. Higgs has high model very important to the physics particle However, stan- of the by model described electromagnetic be dard can and particles found weak these among been strong, forces have antiparticles The their and experimentally. , leptons six and ftepen are lcrccag of charge electric carries One preons charges. the preons. color and spin-1/2 of hypercolor elementary both carry two preons has The which 5], model[4, together. preons symmetry[3]. gauge bind to interactions quarks strong exist Leptons, of may should kind Preons new hypercolor A called preons. bosons. or entities, fermions fundamental be more of ites ∗ [email protected] h biu elcto fqaklpo aiisand families quark-lepton of replication obvious The pt o,tev lmnayprils i quarks six particles, elementary twelve now, to Up n ftesmls ro oe sHarari-Shupe is model preon simplest the of One ea rostasomunder transform Preons Hexad SU E n C r ie yeetosrn interaction electro-strong by given are QCD and and leptons q of QED six numbers exactly quantum are All There (generations); families quarks: and leptons of compositeness nsadr oe r utrsueo ros nadto,ap a addition, In mixing, b lepton preons. symmetry and electroweak dipreo of quark the of and reshuffle like questions preons just Other between are presented. forces model Waals” standard der in ”Van residual is ASnumbers: PACS SU ea ro oe hr etn,qak n oosaeco are bosons Z W and quarks leptons, where model Preon Hexad A (3) .INTRODUCTION I. (3) etc. C H ⊗ ⊗ r igesudrtehypercolor the under singlets are SU opst etn n ursfo ea Preons Hexad from Quarks and Leptons Composite SU (3) (3) f ⊗ 12.10.-g12.60.Rc C U ⊗ etc (1) eigUiest,Biig 081 P.R.China 100871, Beijing, University, Peking U eanunresolved. remain . w hnhiUiest fEgneigScience, Engineering of University Shanghai (1) etc. l ain etrso h tnadmdlcnb bandf obtained be can model standard the of features salient All . etc. Q r compos- are ) e/ ursand Quarks . 2 1 r loaddressed. also are , U etrfrHg nryPhysics, Energy High for Center olg fFnaetlStudies, Fundamental of College ,wiethe while 3, hnhi 060 P.R.China 201620, Shanghai, (3) ⊗ hnZiWang Shun-Zhi U 3 oa ag ru hc sietfidwith identified is which group gauge local (3) fietclcags ros(n nidpen)belong anti-dipreons) (and representations complex Preons are to antipreons, charges. two is identical other which the of by partner, formed ”supersymmetric” anti-dipreon the its and preon ”dipreon”s. spin-0 Each into forces spin-dependent by bounded asbtensm neutrinos Cabibbo- some the between cays de- standard conservation, and oscillations the number mixings, give in (CKM) Kobayashi-Maskawa lepton can questions like trinity such Preon model of charges. understanding same some hyper- the ”supersymmetric” any its have of and partner singlet preon overall the noted since be be dynamics com- should not color It are can quarks here anti-dipreon. an quarks the and that while preon dipreon singlet a a color of and model. posed are preon quark they one symme- original of that the states SU(3) so sta- of bound preon-flavour are that be leptons to global to The similar a assumed is are which is try preons there dynamics The hypercolor and The ble specified. QCD. not ordinary was the of group n + and oke rosa ela ur n etnbudstates bound lepton order and In quark as symmetries. well as chiral preons of guar- keep generally existence to is the masses TeV. by few fermion anteed a of than that disappearance greater indicates be The leptons Indeed, must and scale quarks compositeness compositeness. are the the quarks of of radius and scale small leptons the the of than masses smaller the much that fact the from bosons. Z In W, and apart quarks model. states leptons, composite this ordinary exotic within the many generation predicts it or addition, family of concept t fpen spooe[] nti oe,teeare there model, this + In charges electric with proposed[6]. preons spin-1/2 is three preons of ity first the of states excitation fermions. other as the generation with treated only leptons, generations and accommodate quarks two can of generation model first this the However, fermions. h omndffiut npenmdlbidn comes building model preon in difficulty common The nodrt oti i ursadsxlpos trin- a leptons, six and quarks six contain to order In 1 U , 2 e/ urscnb ie u fta fpreons; of that of out given be can quarks (1) ∗ ak n i etn iheietthree evident with leptons six and uarks ekn,tesi ffrin,teorigin the fermions, of spin the reaking, ,rsetvl.Piso rosmyb tightly be may preons of Pairs respectively. 3, Q s ti hw htalprocesses all that shown is It ns. ⊗ sil akmte addt is candidate matter dark ossible SU (3) C pst spooe.Six proposed. is mposite h ekinteraction weak The ; fthe of 3 ¯ etc oee,teei no is there However, . U SU o the rom (1) (3) Q ⊗ C oa gauge local , e/ 3, − 2 e/ 3 2 massless, the chiral symmetries respected by the strong model electromagnetism is intimately related with strong preon dynamics must remain unbroken as the compos- interaction and has no link to weak interaction. In deed, ites form. As ’t Hooft pointed out in a classic paper[7] phenomenologically the laws electromagnetism follow are that in order to preserve above chiral symmetries com- almost the same as that of strong interaction in contrast posite fermions must yield the same chiral anomalies as to that of weak interaction. Therefore there is no prob- those appearing in the underlying preon theory. It is this lem of electroweak symmetry braking here. anomaly matching condition that makes most of existing Hexad preons fall into two groups: χa (a =1, 2, 3) and models rather complex and cumbersome. φa (a = 1, 2, 3). χa (a = 1, 2, 3) belong to the funda- In some cases, the ’t Hooft anomaly constraint can be mental representations of the underlying exact electro- avoided. For example, it is shown that in some higher di- strong gauge group U(1)Q ⊗ SU(3)C with electric charge mensional supersymmetric unified gauge theories, proper e/3. φa (a = 1, 2, 3) belong to the fundamental rep- Sherk-Schwarz compactification makes chiral multiplets resentations of the underlying exact weak gauge group of composite quarks and leptons massless in four dimen- SU(3)f ⊗ U(1)w with a new weak charge 1/3. The preon sions with all unwanted states (residing in the bulk) being quantum numbers are summarized in TABLE I. The an- still massive[8]. Since we know nothing exactly about the tipreons have the same quantum numbers with opposite nature of space-time at high energy, we may in general sign. owe the masslessness of leptons and quarks to the po- tential energy of the superstrong forces which bind the preons together. TABLE I. The quantum numbers of Hexad Preons. In this paper, we follow the line of Harari-Shupe model Rep. of Rep. of and preon trinity, putting the emphasis on the inner sym- Preons Q SU(3)C (T3,Y )C SU(3)f (T3,Y )f w metries of preons. We propose that there are six anti- χ1 1/3 (1/2, 1/3) commutative fields called Hexad Preons. The underlying χ2 1/3 3 (−1/2, 1/3) preon dynamics is U(3) ⊗ U(3) local gauge theory which χ3 1/3 (0, −2/3) may be identified with U(1)Q⊗SU(3)C ⊗SU(3)f ⊗U(1)w. φ1 0 (1/2, 1/3) 1/3 We will show that all salient features of the standard φ2 0 3(−1/2, 1/3) 1/3 model can be understood from the compositeness of lep- φ3 0 (0, −2/3) 1/3 tons and quarks. Most of the defects of Harari-Shupe model and preon trinity mentioned above can be reme- died within this Hexad Preon model. In addition, we The preons are assumed to be stable. They can be also offer some new point of view to such questions like produced or destroyed only in preon-antipreon pairs. So the electroweak symmetry breaking, the spin of preons, apart from the above local charges in TABLE I, each φ the nature of weak interaction and the origin of fermion preon and anti-dipreon is assigned a flavor number P as mixing. follows: ¯ ¯ ¯ ¯ ¯ ¯ 1 (φ2φ3), (φ3φ1), (φ1φ2): P = w +1= 3 ; (4) 1 II. HEXAD PREONS φ1, φ2, φ3 : P = w = 3 . (5)

Hexad Preon model consists of six anticommutative In the following, we will show that there are just six fields quarks and six leptons with evident three families (gen- erations). All quantum numbers of leptons and quarks χa, φb, (a,b =1, 2, 3) (1) can be given out of that of preons. Neutrinos are bound states of two φ antipreons, i.e. and their complex conjugates (antipreons). They satisfy anti-dipreons. the following anticommutative relations ¯ ¯ ¯ ¯ ¯ ¯ 2 2 νe : (φ2φ3), νµ : (φ3φ1), ντ : (φ1φ2). (6) χa = φa =0, χaχb + χbχa =0, (2) φaφb + φbφa =0, χaφb + φbχa = 0 (3) Because of the anticommutative relations (3), there are exactly three families of neutrinos. Since there is no com- (a,b = 1, 2, 3) as well as their complex conjugate rela- ponent of χ preons, neutrinos are electric neutral and can tions. only take part in weak interactions. The underlying preon dynamics is U(3) ⊗ U(3) local Each charged lepton consists of three χ antipreons and gauge theory. One U(3) stands for electro-strong inter- one φ preon. action while the other is responsible for the weak sec- tor. In other word, the gauge group U(3) ⊗ U(3) may e− :(¯χ1χ¯2χ¯3)φ1, µ− :(¯χ1χ¯2χ¯3)φ2, τ − :(¯χ1χ¯2χ¯3)φ3. be identified with U(1)Q ⊗ SU(3)C ⊗ SU(3)f ⊗ U(1)w. Here Q is electric charge, SU(3)C is local gauge group Leptons are color singlets but have one unit electric of the ordinary QCD, SU(3)f is local preon-flavour sym- charge. Therefore leptons can take part in both elec- metry, w is a new weak charg. Therefore in Hexad Preon tromagnetic and weak interactions. 3

The lepton numbers are given as follows: (S) etc. may be given as in the standard model since they just reflect the fact of preon number conservation. Yf L = T3 + + P, (7) The weak isospin for all quarks can be given by e f 2 Yf B Lµ = −T3f + + P, (8) T3 = − − w. (13) 2 w 2 Lτ = −Yf + P. (9) where B is baryon numbers for quarks. It is obvious that lepton number conservation is the con- The quantum numbers of quarks are summarized in sequence of SU(3)f ⊗ U(1)w symmetry. TABLE III. The weak isospin for all leptons can be given by

L TABLE III. The preon components and quantum numbers of T3 = − − w. (10) w 6 quarks. where L is lepton numbers for leptons. quarks components Q B T3w ¯ ¯ The quantum numbers of leptons are summarized in u ((χ2χ3), (χ3χ1), (χ1χ2))(φ¯2φ¯3) 2/3 1/3 1/2 TABLE II. d (¯χ1, χ¯2, χ¯3)φ1 -1/3 1/3 -1/2 ¯ ¯ c ((χ2χ3), (χ3χ1), (χ1χ2))(φ¯3φ¯1) 2/3 1/3 1/2 TABLE II. The preon components and quantum numbers of s (¯χ1, χ¯2, χ¯3)φ2 -1/3 1/3 -1/2 leptons. ¯ ¯ t ((χ2χ3), (χ3χ1), (χ1χ2))(φ¯1φ¯2) 2/3 1/3 1/2 Leptons components Q Le Lµ Lτ T3w ¯ ¯ b (¯χ1, χ¯2, χ¯3)φ3 -1/3 1/3 -1/2 νe (φ¯2φ¯3) 0 1 0 0 1/2 − e (¯χ1χ¯2χ¯3)φ1 -1 1 0 0 -1/2 ¯ ¯ νµ (φ¯3φ¯1) 0 0 1 0 1/2 According to above patterns , it is reasonable that − a new composite state, (χ1χ2χ3)(φ¯1φ¯2φ¯3), should exist. µ (¯χ1χ¯2χ¯3)φ2 -1 0 1 0 -1/2 ¯ ¯ This state is SU(3)C and SU(3)f singlet with overall ντ (φ1φ2) 0 0 0 1 1/2 U(1) charge neutral. it is a possible dark matter candi- − τ (¯χ1χ¯2χ¯3)φ3 -1 0 0 1 -1/2 date. The spin of preons is related to the space-time at the scale of compositeness and will not be discussed here. Quarks contain both χ and φ preons. Three up quarks The leptons and quarks are composed of two or four pre- are formed by two χ preons and two φ antipreons. ons and (or) aitipreons. It is this point that makes Hexad Preon model economical and elegant. In order to under- u : (χ2χ3)(φ¯2φ¯3), (χ3χ1)(φ¯2φ¯3), (χ1χ2))(φ¯2φ¯3); stand the spin of leptons and quarks, let’s consider the ¯ ¯ ¯ ¯ ¯ ¯ c : (χ2χ3)(φ3φ1), (χ3χ1)(φ3φ1), (χ1χ2))(φ3φ1); situation in quantum field theory where Weyl and Dirac t : (χ2χ3)(φ¯1φ¯2), (χ3χ1)(φ¯1φ¯2), (χ1χ2))(φ¯1φ¯2). spinors are composed of two and four components. The spin of spinors is determined meaningfully there and we Up quarks transform as color triplet with 2/3 unit electric have never worried about the spin of their components. charge. Perhaps the status of our preons here is similar to that of Three down quarks are composed of one χ antipreon spionor components rather than fermion or boson fields. and one φ preon. This may be an indication that the nature of space-time at the scale of compositeness should be much different d :χ ¯1φ1, χ¯2φ1, χ¯3φ1; from that of ordinary four dimensional space-time con- s :χ ¯1φ2, χ¯2φ2, χ¯3φ2; (11) tinuum. b :χ ¯1φ3, χ¯2φ3, χ¯3φ3. Down quarks also carry color charge but have -1/3 unit III. THE WEAK INTERACTION AND electric charge. Therefore quarks can take part in all FERMION MIXING three kind of interactions. The baryon numbers of quarks are just the preon flavor number: In the Hexad Preon model, electro-strong interaction 1 U(1)Q ⊗ SU(3)C gives the same dynamics as that of or- B = P = . (12) dinary QED and QCD. However, the gauge interaction 3 of weak sector SU(3)f ⊗ U(1)w is obviously broken at During the processes of strong interaction, φ pre- low energy. This could be realized by the condensate of ons (dipreons) can be produced only in preon-antipreon φ preons. Pairs of φ preons may be tightly bounded into (dipreon-antidipreon) pairs. The quark numbers which ”dipreon”s. The weak interactions are just residual ”Van are conserved only in strong interactions like strangeness der Waals” forces between preons and dipreons. 4

Given above preon assignment of quarks and leptons, where U is a unitary matrix which is often called the all processes in standard model are just reshuffle of pre- Pontecorvo-Maki-Nakagawa-Sakata (PMNS) or Maki- ons. For example, muon decay Nakagawa-Sakata (MNS) mixing matrix. [9–11]. The charged current of lepton weak interaction is as follows: µ− → νµ + e− +ν ¯e (14) Ue1 Ue2 Ue3 ν1 + + + +     is as follows: J = e µ τ Uµ1 Uµ2 Uµ3 ν2 (26). l
 Uτ1 Uτ2 Uτ3   ν3  [(¯χ1χ¯2χ¯3)φ2] → [(φ¯3φ¯1)]+[(¯χ1χ¯2χ¯3)φ1]+[(φ2φ3)] . (15) In terms of preons , the above expression takes the form: In the case of neutrino lepton interaction, e. g. + ¯ ¯ ¯ ¯ ¯ ¯ Jl = Ue1(χ1χ2χ3)φ1(φ2φ3)+ Uµ1(χ1χ2χ3)φ2(φ2φ3) νµ + e− → µ− + νe, (16) + Uτ1(χ1χ2χ3)φ¯3(φ¯2φ¯3)+ Ue2(χ1χ2χ3)φ¯1(φ¯3φ¯1) we have + Uµ2(χ1χ2χ3)φ¯2(φ¯3φ¯1)+ Uτ2(χ1χ2χ3)φ¯3(φ¯3φ¯1)

+ Ue3(χ1χ2χ3)φ¯1(φ¯1φ¯2)+ Uµ3(χ1χ2χ3)φ¯2(φ¯1φ¯2) [(φ¯3φ¯1)]+[(¯χ1χ¯2χ¯3)φ1] → [(¯χ1χ¯2χ¯3)φ2]+[(φ¯2φ¯3)] . (17) + Uτ3(χ1χ2χ3)φ¯3(φ¯1φ¯2) . (27) The fact that weak interactions involving quarks have the same patten as that of leptons can be understood The elements of lepton mixing matrix, Uβj, character- ¯ now at the preon level. Nuclear beta decay ize the strength of interactions between (χ1χ2χ3)φ1 and (φ¯2φ¯3), etc. d → u + e− +¯νe (18) The origin of quark flavor mixing is in common with lepton mixing. The d-type quarks given by (11) are also is just the process mass eigenstates. They are related with quark weak in- ¯ ¯ teraction eigenstates by the unitary quark flavor mixing [¯χ1φ1] → [(χ2χ3)(φ2φ3)]+[(¯χ1χ¯2χ¯3)φ1]+[(φ2φ3)] . (19) matrix, CKM matrix V [12, 13]. The charged current of Here is another example, Pion decay: quark weak interaction is given by i j J − =u ¯ V d , i = u,c,t; j = d, s, b . (28) π−(¯ud) → µ− +¯νµ , (20) q ij [(¯χ2χ¯3)(φ2φ3)][¯χ1φ1] → [(¯χ1χ¯2χ¯3)φ2] + [(φ3φ1)] .(21) In terms of preons , the quark charged current takes 0 the form: The W ± and Z are also composite:

1 J − = {Vud(¯χ2χ¯3)(φ2φ3)¯χ1φ1 + Vus(¯χ2χ¯3)(φ2φ3)¯χ1φ2 W − = √3 (¯χ1χ¯2χ¯3){(φ2φ3)φ1 + (φ3φ1)φ2 + (φ1φ2)φ3} , 0 + Vub(¯χ2χ¯3)(φ2φ3)¯χ1φ3 + Vcd(¯χ2χ¯3)(φ3φ1)¯χ1φ1 Z ∼{φ¯1φ1 + φ¯2φ2 + φ¯3φ3 + Vcs(¯χ2χ¯3)(φ3φ1)¯χ1φ2 + Vcb(¯χ2χ¯3)(φ3φ1)¯χ1φ3 + (φ2φ3)(φ¯2φ¯3) + (φ3φ1)(φ¯3φ¯1) + (φ1φ2)(φ¯1φ¯2)} (22). + Vtd(¯χ2χ¯3)(φ1φ2)¯χ1φ1 + Vts(¯χ2χ¯3)(φ1φ2)¯χ1φ2 The fermion mixing originates from flavor symmetry + Vtb(¯χ2χ¯3)(φ1φ2)¯χ1φ3} . (29) breaking. Look at the lepton mixing first. The notion of flavor is dynamical. For example, νe is the neutrino The elements of quark mixing matrix, Vij , characterize + which is produced with e , or produces an e− in charged the strength of interactions between (¯χ2χ¯3)(φ2φ3) and current weak interaction processes: χ¯1φ1, etc. + + W → lα + να, α = e, µ, τ. (23) IV. SUMMARY AND DISCUSSIONS If the symmetry SU(3)f ⊗U(1)w was unbroken, φ¯1, for example, could only interact with (φ¯2φ¯3) so that lepton numbers are conserved. However, if the flavor symme- In this paper, we propose that there are six anti- try is broken by the condensate of two φ¯ preons, φ¯1 can commutating fields called Hexad Preons. Quarks, lep- also interact with (φ¯3φ¯1) and (φ¯1φ¯2), resulting in flavor tons and W Z bosons are composite states with even mixing. number of preons and antipreons (complex conjugates In this sense the three states given by (6) are neutrino of Hexad Preons). It is shown that there are exactly mass eigenstates: six quarks and six leptons with evident three families (generations) as well as a possible dark matter candi- ν1 = (φ¯2φ¯3), ν2 = (φ¯3φ¯1), ν3 = (φ¯1φ¯2). (24) date. The underlying preon dynamics is supposed to The weak interaction eigenstates are superpositions of be U(3) ⊗ U(3) local gauge theory which is identified above three neutrinos νj (j =1, 2, 3): with U(1)Q ⊗ SU(3)C ⊗ SU(3)f ⊗ U(1)w. All quantum numbers of leptons and quarks can be given out of that |ν i = U |ν i β = e, µ, τ, (25) of preons. Electro-strong interaction U(1)Q ⊗ SU(3)C β X βj j β gives the same dynamics as in ordinary QED and QCD. 5

Weak interactions has no link to electromagnetic inter- breaking will not only give the clues to the difference action so that there is no question of electroweak sym- between the two U(3) gauge symmetries, but also help metry breaking. However, the symmetry of weak sector, understanding the nature of gravity. SU(3)f ⊗ U(1)w, is broken by the condensate of pairs of The second problem is the mechanism for SU(3)f ⊗ preons. The weak interaction is just residual ”Van der U(1)w gauge symmetry breaking. The perfect resolution Waals” forces between preons and dipreons. Both quark of this problem should be able to recover all phenomeno- flavor mixing and lepton mixing are manifestation of this logically successful aspects of the standard electroweak symmetry breaking. theory. By the way, the mechanism for fermion mixing Our present discussion is limited to the qualitative con- will also be elucidated. sequences of the model. More quantitative analysis, espe- The last question is on the role of U(1)w symmetry. cially on the dynamics aspects of the model, is obviously The problem of fermion masses and mixing have been needed. In the following, we will give some comments on one of the outstanding puzzles in the standard model. three related problems. One promising approach is the idea of family symme- The first problem is the difference between the electro- try, in particular the idea of a U(1) family symmetry[15]. strong gauge interaction and that of the weak sector. In this respect, U(1)w symmetry may play a important The electro-strong interaction among χ preons seems role here. In addition, it has been shown that models to be more ”democratic”, while the flavor gauge inter- which satisfy the Gatto-Sartori-Tonin relations [16] must action among φ preons prefer to bind two preons into have both positive and negative abelian charges[17]. In dipreons. Since they both transform under the gauge our model, quarks have both U(1)Q electric charge and group U(3), this problem must be related to the origin U(1)w weak charge with opposite sign. Therefore the re- of the U(3) ⊗ U(3) gauge group. In fact, the author has lation between U(1)w symmetry and the fermion mass suggested that at very high energy Hexad Preons may generation, fermion mixing needs further investigation. carry hypercolor degree of freedom transforming under U(3, 3) gauge group[14]. After the emergency of metric, the gauge group U(3, 3) is broken down to its maximal ACKNOWLEDGMENTS compact subgroup U(3) ⊗ U(3). This means that the gravity may be viewed as a symmetry-breaking effect in This work is supported by Shanghai University quantum field theory. So the careful examination of the of Engineering Science under grant No.11XK11 and mechanism for the U(3, 3) hypercolor gauge symmetry No.2011X34.

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