Hyperquarks and Bosonic Preon Bound States

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This is reminiscent of technicolor models and appears ces and λa are generators of the corresponding gauge to be promising. However, closer inspection shows groups. that hyperquarks are singlets under weak isospin. Although there are two degenerate types of preons Therefore, they and their scalar bound states do (T and V) there is no global SU(2) isospin symme- not have the same SU(2)WL U(1)Y group structure try on the preon level because the charged and neu- as the standard model fermions× and Higgs fields. tral preon belong to different representations in color Thus, hyperquark bound state scalars cannot give space. To be consistent with the parity assignment mass to any of the standard model gauge bosons and for the standard model fermions, the intrinsic parity fermions. In this paper, we concentrate on the spec- Π of the T and V preons must be different [4]. The trum of vector bosons. We hope to come back to the different parities of T and V preons provide a possi- issue of scalar particles and mass generation in the ble explanation for parity violation at low energies as preon model elsewhere. will be discussed in sect. IV. The paper is organized as follows. Section II gives a short review of fermionic bound states in the preon preon H C Q P ΥΠ 1 1 1 model. In sect. III, taking as straightforward a posi- T 3 3+ 3 + 3 + 3 −1 tion as possible, we discuss an extension of the stan- ¯ 1 1 V 3 3 0 + 3 − 3 +1 dard electroweak theory, making explicit the preon 1 1 V¯ 3¯ 3 0 − + −1 content of the massive electroweak gauge bosons in- 3 3 T¯ 3¯ 3¯ − 1 − 1 − 1 +1 cluding those responsible for hyperquark-quark tran- 3 3 3 sitions. A further generalization of the extended TABLE I: The color C, hypercolor H, electric charge Q, weak and the hypercolor interactions is discussed in preon number P, Υ number, and intrinsic parity Π of sect. IV. The resulting theory, which we call partial preons and antipreons (see also [4]). unification, represents a necessary step towards grand unification, as shown in sect. V. Sect. VI provides numerical results for the running couplings and unifica- Preons and their bound states are characterized by tion constraints from where mass ranges of the heavy new quantum numbers [5]. These are the preon number and Υ number, which are linear combinations bosons, the lightest neutrino, and the hyperquarks P are obtained. The predicted hyperhadron masses are of the numbers of T-preons n(T) and V-preons n(V) within reach of the Large Hadron Collider at CERN. in a given state Sect. VII contains a summary and outlook. 1 = (n(T )+ n(V )) (2.2) P 3 1 II. FERMIONIC BOUND STATES Υ = (n(T ) n(V )) . 3 − In the Harari-Shupe model [1–3] all quarks and lep- The factor 1 in Eq.(2.2) is a convention. The Υ 1 3 tons are built from just two spin- 2 fermions (preons). number is also related to the baryon (B) and lepton According to Harari and Seiberg [2] the two types of (L) numbers of the standard model as Υ = B L. preons belong to the following representations of the The antipreon numbers n(T¯) and n(V¯ ) are defined− as underlying exact gauge group SU(3)H SU(3)C n(T¯)= n(T ) and n(V¯ )= n(V ). The preon quan- 1 ¯ × × − − U(1)Q: T: (3,3) 3 and V: (3,3)0, where the first (sec- tum numbers of individual preons are summarized in ond) argument is the dimension of the representation Table I. in hypercolor (color) space and the subscript denotes There is a connection between the and Υ num- the electric charge Q. The fundamental Lagrangian bers, and the electric charge Q of theP preons of the preon model [3] reads 1 Q = ( + Υ) , (2.3) 1 2 P = T¯ ∂/ + g A/ + g A/ + e A/ T L H H C C 3 Q which can be readily verified from Table I. This gen- ¯ + V ∂/ + gH A/H + gC A/C V eralized Gell-Mann-Nishijima relation does not only 1 1 1 hold for the preons but for all bound states, such as F F F F F F , (2.1) −4 H H − 4 C C − 4 Q Q leptons, quarks, hyperquarks and their bound states, as well as the effective weak gauge bosons. µ a with A/ = γµAa λ representing the three fundamental As shown in Ref. [4] the ’t Hooft anomaly condi- gauge fields of the theory with their respective cou- tion, which demands that the anomalies on the preon pling strengths gH , gC, and e. The last three terms level match those on the bound state level, can be in Eq.(2.1) represent the kinetic energies of the gauge satisfied if in addition to leptons and quarks a third fields, where the field strength tensors F are as usual fermionic bound state type, called hyperquarks, is in- µ given in terms of the A . The γµ are the Dirac matri- troduced. Hyperquarks have the same electric charge 3

state preon content bound state P Υ B L Q Π T3 1 (VVV ) (νe,νµ,ντ ) +1 −1 0 +1 0 +1+ 2 leptons ¯ ¯ ¯ − − − 1 T T T e , µ ,τ −1 −1 0 +1 −1 +1 − 2 1 1 2 1 (TTV ) (u, c, t) +1+ 3 + 3 0 + 3 +1 + 2 quarks T¯V¯ V¯ (d,s,b) −1 + 1 + 1 0 − 1 +1 − 1 3 3 3 2 TT V¯ u,˜ c,˜ t˜ + 1 +1 +1 0 + 2 −1 0 3 3 hyperquarks TVV¯ d,˜ s,˜ ˜b + 1 −1 −1 0 − 1 +1 0 3 3 TABLE II: Allowed three-preon bound states representing leptons, quarks and hyperquarks and their quantum numbers (see also [4]). Formally, the hyperquarks are obtained from the corresponding quarks by interchanging: V ↔ V¯ (hyperquark transformation).

as the corresponding quarks. However, instead of be- III. BOSONIC BOUND STATES AND ing color triplets and hypercolor singlets as ordinary EXTENDED WEAK INTERACTIONS quarks, they are color singlets and hypercolor triplets. Moreover, because of the different parities ofu ˜ and Hyperquarks are hypercolored objects and thus d˜ one cannot define a weak SU(2) isospin symmetry cannot exist as free particles but must be confined for hyperquarks. Therefore, they do not participate into hypercolorless bound states such as hyperme- in the usual left-right asymmetric weak interaction sons and hyperbaryons. This is in complete analogy but must couple left-right symmetrically calling for a to quarks being confined into colorless mesons and left-right symmetric extension of weak interactions. baryons. Although hyperhadrons might have been The same conclusion is also obtained from the created at sufficiently high energies available in the anomaly freedom constraint of this extended elec- Early Universe they are no longer observed today and troweak theory. In the standard model the anomaly hence cannot be stable. This leads to the question contributions of quarks and leptons cancel. There which gauge bosons are responsible for their decay. are no additional fermionic bound states that could We begin our discussion with charged hypermeson cancel an anomaly contribution coming from hyper- decay, which is generated by a new gauge boson W˜ quarks. Therefore, hyperquarks must not contribute followed by a short exposition of the charged and to electroweak anomalies. The only way this can be neutral weak transitions between fermionic bound achieved is that their weak interactions be left-right states. In section IIIC we discuss the issue of hy- symmetric. Formally, hyperquarks are obtained from perquark decay and show that it is mediated by a quarks by replacing their neutral preons with their six-preon bosonic bound state, called χ, which can be antiparticles. We refer to this process as hyperquark thought of as two-neutrino bound state. The scenario transformation. of lepton number violating neutrino-antineutrino os- In summary, we can construct two groups of preon cillations characteristic of Majorana neutrinos is de- bound states (leptons and quarks) with the same in- scribed in section IIID. There, also the consequences trinsic parities and integer preon number for which a for the formulation of an extended left-right symmet- new quantum number “weak isospin” corresponding ric weak interaction theory are expounded. to an effective SU(2) chiral isospin symmetry can be defined in terms of the preon number as P

A. Weak meson and hypermeson decays into 1 T = . (2.4) leptons 3 2 P

As stated in the introduction and shown in Fig. 1, The members of the third group (hyperquarks) nec- in the preon model the weak decays of hadrons essarily have opposite intrinsic parities and fractional and hyperhadrons are mediated by composite gauge preon number and thus do not possess weak isospin. bosons. For example, the weak decay of the positively The three diﬀerent types of fermionic bound states charged π-meson into a muon and a neutrino can be and their quantum numbers are shown in Table II. schematically written as 4

+ ~ + u(TTV) µ( TTT) u(TTV) µ( TTT)

+ T T T ~ T T T W + VVV W VVV

~ d(TVV) ν(VVV) d(TVV) ν(VVV)

FIG. 1: Weak pion decay into leptons (left) and lepton number violating weak hyperpion decay (right) mediated by bosonic preon bound states.

As is obvious from the notation, the compositeness of quarks and leptons implies that the weak gauge u (T T V ) TTT bosons are composed of six preons [3], and further- π+ W + d¯(T V V ) ! −→ V V V ! more that initial and ﬁnal states correspond merely to diﬀerent arrangements of these preons. µ+ (TTT )+ ν (V V V ) . (3.1) −→

Similarly, the corresponding hyper-π-meson decay B. Charged and neutral weak transitions reads between fermionic preon bound states

In the preon model, the weak transitions among the u˜ TT V¯ TTT π˜+ W˜ + members of quark and lepton weak isospin doublets ˜¯ ¯ ¯ −→ V¯ V¯ V¯ d T V V ! ! caused by the charged weak currents of the standard + µ (TTT )+¯ ν V¯ V¯ V¯ . (3.2) model are written as −→ ¯ ¯ ¯ + TTT A comparison of the preon content of the bound u (T T V ) d T V V + W −→ V V V ! states in Eq.(3.2) and Eq.(3.1) shows that the neutral ¯ ¯ ¯ ¯ preons V are replaced by their antiparticles V when − ¯ ¯ ¯ − T T T going from quarks to hyperquarks [4] or from mesons e T T T ν (V V V )+W . (3.3) −→ V¯ V¯ V¯ ! to hypermesons (hyperquark transformation). In addition, in both cases the intermediate gauge bosons In these transitions all three preons of the final are formed by a mere rearrangement of the initial fermionic bound states are created from the vacuum, state preons. Hence, we propose that the existence of while the initial state preons and the antiparticle left-right symmetrically coupling hyperquarks entails counterparts of the three vacuum pairs merge to form the existence of a new class of composite weak gauge the weak gauge bosons. bosons, called W˜ , which couple left-right symmetri- Analogously, the corresponding weak transitions cally to fermions. among hyperquarks imply the existence of new gauge Although there is a certain analogy between weak bosons, W˜ , which may then also generate transitions meson and hypermeson decays into leptons there is within lepton doublets an important difference between them. According to the quantum number assignments in table II, the lat- TTT u˜ TT V¯ d˜ T¯ V V + W˜ + ter process simultaneously violates lepton and baryon −→ V¯ V¯ V¯ ! number (∆B = ∆L = 2), which indicates that it − − − T¯T¯T¯ occurs only at a higher energy scale where left-right e T¯T¯T¯ ν¯ V¯ V¯ V¯ + W˜ . (3.4) symmetry is restored. In sect. VI this scale is calcu- −→ V V V ! 9 lated as MP ∼= 10 GeV. Note that in both processes the preon and Υ = B L numbers remain con- In contrast to the usual electron-neutrino transition served. P − in Eq.(3.3), the W˜ induced process in Eq.(3.4) leads 5 to an antineutrino in the final state and thus vio- state content P Υ Q Π T3 lates lepton number conservation (∆L = 2). On W − (3T¯ , 3V¯ ) −2 0 −1 −1 −1 − the other hand, Υ = B L and are conserved W + (3T, 3V ) +20+1 −1 +1 because Υ = 2 and =− 0 for theP W˜ − boson ac- W0/B0 (3(T T¯), 3(V V¯ ))0 0 0 −1 0 cording to Table− III. Similarly,P in the hyperquark sec- − tor the charged weak transition violates baryon num- W˜ (3T¯ , 3V ) 0 −2 −1+1 0 ber (∆B = 2) but again Υ = B L and are W˜ + (3T, 3V¯ ) 0 +2+1+1 0 conserved. One− also notices that these− chargedP weak transitions leave the type of fermionic bound state in- TABLE III: Preon content and quantum numbers of the standard model weak gauge bosons W and B0 and the variant, i.e., a quark remains a quark, a hyperquark ˜ remains a hyperquark, and a lepton remains a lepton. newly introduced gauge bosons W . Because neutral currents do not change the inter- nal quantum numbers of the fermionic bound states involved in the transition, the W boson must be a N 0 q q linear combination of the two neutral six-preon states. For the neutral standard model gauge bosons we define

1 T¯T¯T¯ V¯ V¯ V¯ W0 = √2 " TTT ! − V V V !# U U 1 T¯T¯T¯ V¯ V¯ V¯ B0 = + . (3.5) √2 " TTT ! V V V !# Note that applying the hyperquark transformation to Eq.(3.5) leaves these states invariant so that we ˜ need not introduce additional neutral bosons W0 and l B˜0. As a linear combination of the two states in ¯ Eq.(3.5) the Z-boson is a pure (3V, 3V ) state, while FIG. 2: Fermion triangle. The three fermionic bound ¯ the orthogonal combination (3T, 3T) state can in states, hyperquarks (˜q), quarks (q), and leptons (l) and some sense be interpreted as an effective photon sim- their antiparticles are placed at the corners of a triangle. ilar to the vector meson dominance model. In this The dipreonic bound states (N, U, U˜) and their antipar- way the standard model weak forces are seen to be ticles describing transitions between these fermions are mediated by effective gauge bosons composed of six placed along the edges. preons. As far as we can see, 6 T and 6 T¯ states do not occur in any weak process and are not considered here. fermionic preon bound states. In principle, hyper- The same applies to six-preon bound states consist- quarks can decay into quarks and into leptons. How- ing of one charged preon and five neutral ones or one ever, below the grand unification (GUT) scale the neutral preon and five charged ones. In Table III we former process dominates for the following reason. list the composite gauge bosons introduced so far as The decay of quarks into leptons, as required for pro- well as their preon content and quantum numbers. ton decay is suppressed due to the heavy mass of In summary, for the weak gauge bosons W , Z, the GUT gauge bosons of the order of 1016 GeV, and W˜ , the following rules apply: (i) leptons and which corresponds to lifetimes of the order of 1035 quarks transform via W exchange; (ii) leptons and y. Because the preonic substructure of hyperquarks hyperquarks transform via W˜ exchange; (iii) leptons, and quarks are very similar, transitions from hyperquarks, and hyperquarks transform via Z exchange. quarks to leptons are also suppressed at lower en- But neither the standard model gauge bosons nor the ergies. The decay of quarks and hyperquarks into newly introduced W˜ can transform hyperquarks into leptons mediated by dipreonic bound states U and quarks. U˜ occuring at the GUT scale will be considered in section V. Clearly, the fact that hyperquarks are not observed today requires that their lifetime be much C. Weak hyperquark decays into quarks shorter than the lifetime of the Universe (1010 y), and that the gauge bosons N responsible for hyperquark Similar to the case of hypermesons discussed in decay into quarks be lighter than the GUT bosons. sect. III A, the nonobservation of hyperbaryons im- The transitions between the three types of fermionic plies that they are not stable. Consequently, there bound states mediated by dipreonic bound states N, must be transitions from hyperquarks to known U, and U˜ are shown in Fig. 2. 6

In our previous paper [4], we have seen that the hyperquark bound states, ∆˜ + u˜u˜d˜ and ∆˜ 0 u˜d˜d˜ , transition from hyperquarks to quarks is formally ob- decay via a two-step process, i.e., they first change tained by interchanging the neutral preon by its an- − into a ∆˜ ++(˜uu˜u˜) and ∆˜ (d˜d˜d˜) via W˜± emission as tiparticle (V V¯ ). Hyperquark decays into quarks in Eq.(3.4) and then decay into ordinary baryons are generated↔ via a new class of electrically neutral via χ emission as in Eq.(3.8). Because of the van- bosons, called N-bosons or neutralons N (V V ) and ishing isospin of hyperhadrons the spin-symmetric N¯ V¯ V¯ ∆˜ baryons are the lightest fermionic hyperhadron states. u˜ TT V¯ u (T T V )+ N¯ V¯ V¯ −→ d˜ T¯ V V d T¯V¯ V¯ +2N (V V ) . (3.6) −→ D. Left-right symmetric weak interactions and As color and hypercolor triplets, neutralons are neutrino-antineutrino oscillations confined particles. In order to carry away the energy and momentum made available in theq ˜ q transi- In this section we discuss in more detail how the tion, an unconfined neutral particle must→ be emitted. existence of hyperquarks leads to a left-right symmet- Such a hypercolor and color singlet can be formed by ric extension of standard model weak interactions. an NN¯ pair or three neutralons. The emission of two As explained before, hyperquarks do not have weak NN¯ pairs, as occuring in hyperproton decay accord- isospin and therefore they couple left-right symmet- ing to Eq.(3.6) seems at first sight possible. But an rically to the new gauge bosons W˜ and N. At the NN¯ state has neither weak isospin nor electric charge partial unification scale M , where these new left- and thus it cannot couple to any of the known low P right symmetric gauge bosons appear, hyperquarks energy bosons such as Z or γ into which it could an- decay into quarks as in Eq.(3.6). Therefore, at this nihilate. Therefore, direct hyperproton decay via the scale quarks must also couple left-right symmetri- emission of two NN¯ pairs does not work. However, cally to the standard model weak gauge bosons W . annihilation into the vacuum is possible for three NN¯ This entails an extension of the standard model weak pairs. In fact, 3N or 3N¯(and thus 3 NN¯) form color- isospin group SU(2) to the gauge group SU(2) less and hypercolorless bound states which can decay WL WL SU(2) . × into neutrinos and antineutrinos. In the following the WR 3N and 3N¯ states are called χ andχ ¯. At the same energy, the effective gauge boson masses M , M , M are of order M , and the More generally, we postulate that in all weak pro- WR WL W˜ P left-right symmetric gauge bosons can transform into cesses occuring below the grand unification scale of each other. In particular, the W -bosons can change 1016 GeV, such as hyperquark-quark transitions, as into the W˜ -bosons and vice versa, well as in processes involving W , W˜ and Z exchange, the generation or annihilation of preon-antipreon TTT TTT pairs is only possible for integer multiples of three W˜ + W + preon-antipreon pairs. This is symbolically written V¯ V¯ V¯ ←→ V V V ! !L/R as − T¯T¯T¯ − T¯T¯T¯ 16 W˜ W . (3.9) n(T¯ T ) + n(V¯ V ) =3k for E 10 GeV , (3.7) V V V ←→ V¯ V¯ V¯ ≤ ! !L/R where n(T¯ T ), n(V¯ V ), and k are natural numbers. Thus, in any weak interaction below the grand unifi- In addition, there are transitions between the χ, Z, cation scale, only six-preon (see previous subsection) andχ ¯ bosons or three-dipreon bosons (see next subsection) are in- χ (νν) Z (ν ν¯) χ¯ (¯ν ν¯) . (3.10) volved. We refer to this as “preon triality rule”. ←→ ←→ The simplest hyperbaryon decay process seems to proceed via a ∆˜ type hyperbaryon in which the six- The transformations in Eq.(3.9) and Eq.(3.10) are a preon bound states χ andχ ¯ are produced reflection of left-right symmetry restoration at the partial unification scale. These left-right symmetric ∆˜ ++ (˜uu˜u˜) ∆++ (uuu)+¯χ interactions also enable transitions between the left- −→ and right-handed neutrino sectors ∆˜ − d˜d˜d˜ ∆− (ddd)+2χ, (3.8) −→ ν ν¯ or νL νR (3.11) and where the latter subsequently decay into two neu- ←→ ←→ trinos (χ 2ν) or antineutrinos (¯χ 2¯ν). Thus, characteristic for two-component Majorana neutri- the transition→ from hyperquarks to quarks→ is only pos- nos [6, 7], for which the difference between neutrino sible within bound states of three hyperquarks of the and antineutrino is only one of helicity, i.e. νL ν same charge state (˜u, c,˜ t˜ or d,˜ s,˜ ˜b). Other baryonic and ν ν¯. Such neutrino-antineutrino transitions≡ R ≡ 7

corresponds to the so-called see-saw mechanism [8, 9]

2 χ me Z = mν , (3.12) e ν MP ν ν

where mνe and me are the neutrino and electron FIG. 3: Neutrino-antineutrino oscillation via χ and Z 9 mass, and MP ∼= 10 GeV is the scale of left-right emission and absorption. symmetry restoration. The extended left-rigt symmetric electroweak interactions, which go hand in hand with ν ν¯ and can be thought of as proceeding via χ Z χ¯ tran- − sitions as depicted in Fig. 3. Obviously,↔ at↔ this en- χ Z χ¯ oscillations as depicted in Fig. 3, lead to new Υ− = −B L number violating processes, such as for ergy scale the inner parity and weak isospin of preon − bound states is no longer a good quantum number. example, (i) neutrinoless hyperbaryon decay where In the preon model neutrino-antineutrino oscilla- Υ- and B- numbers are violated and lepton number tions may be allowed for the following reasons. As L is conserved and (ii) neutrinoless double beta de- discussed in sect. II, the neutral V and V¯ preons have cay where Υ- and L- numbers are violated and B is conserved different SU(3)C SU(3)H and parity assignments (see Table I). Because× none of the fundamental gauge interactions of the theory, namely SU(3) , SU(3) , ∆˜ ++ (˜uu˜u˜) ∆++ (uuu)+ Z; ∆Υ = ∆B = 2 C H −→ − and U(1) violates parity conservation, there can be − − Q ∆˜ d˜d˜d˜ ∆ (ddd)+2Z; ∆Υ = ∆B =4 no direct transitions between the neutral V preon and −→ its antiparticle V¯ . For the same reason, transitions 2n(udd) 2p+(uud)+2e−; ∆Υ= ∆L= 2. ¯ ¯ ¯ −→ − − between the neutralons N(V V ) and N(V V ) are for- (3.13) bidden. However, for bound states of three V and three V¯ preons, which are electrically neutral color and hypercolor singlets, there is from the viewpoint of The discovery of any one of these B or L violating de- the fundamental gauge symmetries no distinction be- cays would lend some support to the ideas developed tween ν(V V V )andν ¯(V¯ V¯ V¯ ) as pointed out by Harari here. and Seiberg [5]. As one can readily see from Table I, the additive and Υ quantum numbers are simultaneously violatedP IV. PARTIAL UNIFICATION in Eq.(3.11) but in such a way that the total electric charge associated with the fundamental U(1)Q gauge interaction remains conserved in accordance with the Having motivated an extension of the standard generalized Gell-Mann Nishijima relation Eq.(2.3), model weak interaction that accounts for hyperquark i.e., ∆Q = 0 which entails ∆ = ∆Υ. Further- decay, left-right symmetric weak gauge bosons, and more, it is observed that this violationP − can only oc- direct neutrino-antineutrino oscillations, we study in cur for integer values of these quantum number and this section further aspects of this extension. In par- processes involving three V or three V¯ , whereas frac- ticular, we propose that the generalized weak inter- tional values of and Υ are strictly conserved be- actions are part of a larger unification scheme for cause of their connectionP with fundamental gauge in- weak and strong interactions between preon bound teractions (see Table I). Analogous to the and Υ states that includes (i) 7 left-right symmetric stan- P 0 number violation associated with the direct neutrino- dard model weak gauge bosons WL, WR, B , trans- antineutrino oscillation, the intrinsic parity violation forming according to a simply extended rank 3 gauge in Eq.(3.11) only occurs at the level of bound states group SU(2)WL SU(2)WR U(1)Y , (ii) 20 new left- × × of at least three preons. right symmetric gauge bosons W˜ , N, N¯, (iii) 8 hy- Within each chiral sector the and Υ quantum pergluons, altogether 35 gauge bosons (see Table V) numbers are conserved as requiredP by the anomaly characteristic of an effective SU(6) gauge group. In equations [4]. For this reason, any violation of these addition, we emphasize that in the present model it quantum numbers is accompanied by a simultaneous is possible to connect the left-right symmetry of weak change of the chiral sector, which is only possible for interactions at high energies and its breaking at low massive particles. The corresponding transition rate energies to the existence of hyperquarks and the new is determined by the ratio of the neutrino mass and weak gauge bosons. Before this, we discuss the conse- the partial unification scale, where the right-handed quences of the new effective gauge interactions and of electroweak gauge bosons occur [7, 8]. The heavier hyperquarks for the momentum transfer dependence these bosons, the smaller the helicity changing tran- of the electroweak coupling strengths αW , αY and αQ sition rate and therefore the neutrino rest mass. This and calculate the corresponding Weinberg angle. 8

A. Extended electroweak couplings between them [10, 11]

In the standard model the left-handed leptons α T 2 Q := sin2 Θ = i 3 i and quarks form weak isospin doublets transform- W 2 αW i Qi ing according to the gauge group SU(2) whereas P WL α Y 2 the right-handed leptons and quarks are isospin sin- Q := cos2 Θ = Pi W i α′ W Q2 glets transforming only according to the gauge group P i i U(1)Y . The connection between electric charge Qi, 1 1 1 = + ′ , P (4.2) weak isospin T3i, and weak hypercharge YW i of par- αQ αW α ticle i is given by the Gell-Mann Nishijima relation where the third equation follows from dividing the Q = T + Y , (4.1) i 3i W i sum of the ﬁrst two by αQ. This implies the relation where the particle index i stands for a member of the 2 2 2 lepton or quark doublets. The corresponding quan- Qi = T3i + YW i, (4.3) i i i tum numbers are given in Table IV. X X X

2 2 2 which can also be obtained directly from the square Fermion T3 YW B L T3 YW Q 1 1 1 1 of the Gell-Mann Nishijima relation Eq.(4.1) because νL + 2 − 2 0 +1 4 4 0 − 1 1 1 1 the sum (over all fermions) of the mixed product eL − 2 − 2 0 +1 4 4 1 − terms vanishes. e 0 −1 0 +1 0 1 1 ′ R In must be noted that an equality of αW and α 1 1 1 1 1 4 2 2 uL + 2 + 6 + 3 0 4 36 9 cannot be obtained because T3i = i YW i. There- 2 1 4 4 6 uR 0 + 3 + 3 0 0 9 9 fore, the following coupling αY is introduced 1 1 1 1 1 1 P P dL − 2 + 6 + 3 0 4 36 9 1 1 1 1 1 1 Y 2 dR 0 − 3 + 3 0 0 9 9 i W i ′ = 2 (4.4) 2 4 4 α αY i T3i u˜L/R 0 + 3 +1 0 0 9 9 P ˜ 1 1 1 dL/R 0 − 3 −1 0 0 9 9 P which, when inserted in Eq.(4.2) gives TABLE IV: Weak isospin, weak hypercharge, baryon, and lepton quantum number assignments of the left- and 1 1 Y 2 1 = + i W i . (4.5) right-handed leptons, quarks and left-right symmetric hy- α α T 2 α perquarks. The squares of weak isospin, hypercharge, and Q W P i 3i Y electric charge are also given. P In the standard electroweak theory (without hyperquarks) we have at the weak scale M = 91.2 GeV It has been shown in section IIIB that hyperquarks Z from Eq.(4.5) do not participate in the usual charged current interactions mediated by W exchange. Therefore, with 1 1 5 1 respect to the standard electroweak gauge group, the = + . (4.6) left- and right-handed hyperquarks are isospin sin- αQ αW 3 αY glets (T3i = 0) analogous to the right-handed leptons and quarks. With these properties of the composite Above this scale, at a certain energy which will be fermions at hand, we can now proceed and discuss determined in section VI as mhq ∼= 26 TeV, the ad- how the coupling constants and their ratios are af- ditional contributions of the hyperquarks affect the fected by the presence of the hyperquarks. sums in Eq.(4.2) and Eq.(4.5) as discussed in the fol- 2 First, we note the standard definitions of the three lowing. First, the sum T3i over (left-handed) quarks ′ electroweak couplings αQ, αW and α and the relation and leptons remains unmodified

1 2 1 2 1 2 1 2 T 2 = N + + N + , (4.7) 3iL G 2 −2 C 2 −2 i " ν e u d!# X because hyperquarks do not have weak isospin. Here, the factor NC stands for the number of quark colors, and NG is the number of generations. Second, the sum of the weak hypercharge squares is modiﬁed by the 9 presence of hyperquarks as

2 2 2 1 1 2 Y = NG + + ( 1) W i −2 −2 − eR i " νL eL X 1 2 1 2 2 2 1 2 + N + + + C 6 6 3 −3 uL dL uR dR ! 2 2 1 2 2 2 1 2 + NH + + + , (4.8) 3 −3 ˜ 3 −3 ˜ u˜L dL u˜R dR !# where NH is the number of hypercolors. Third, the sum over the squared charges is

2 2 2 2 2 2 2 1 2 1 Qi =2 NG ( 1)e + NC + + NH + , (4.9) − 3 −3 3 −3 ˜ i " u d! u˜ d!# X

2 where the factor of 2 is due to equal contributions which is twice the experimental value of sin ΘW at from left-handed fermion doublets and right handed MZ = 91.2 GeV. We note that for a simply extended fermion singlets, and numerically NC = NH = NG = weak gauge group SU(2)WL SU(2)WR U(1) without 2 × × 3 according to Ref. [4]. hyperquarks one gets sin ΘW =3/4. Because hyperquarks carry electric charge and The breaking of left-right symmetry in standard weak hypercharge but no weak isospin, their inclusion electroweak theory at the MZ scale leads to a smaller in Eq.(4.2) leads to the following Weinberg angle value for the Weinberg angle which is only half of the value at M given by Eq.(4.13). This change of the α T 2 3 P sin2 Θ = Q = i 3 i = . (4.10) Weinberg angle is accompanied by a mass shift from W α Q2 13 W P i i the left-right symmetrically coupling bosons at MP down to their low energy counterparts at the Fermi Thus, we have sin2 Θ =P 3/13 = 0.231, which is W ∼ scale M . close to the experimental Weinberg angle 0.23119(14) Z at M = 91.2 GeV [12]. We take this as an indica- At the energy MP , where left-right symmetry is Z restored, the existence of an additional SU(2) tion for the physical relevance of hyperquarks. In the WR original preon model of Harari and Seiberg [2] the gauge group leads to an extended electroweak group 2 SU(2)W SU(2)W U(1)Y [5] which itself is embed- weak mixing angle was predicted as sin ΘW =1/4. L × R × Due to left-right symmetry restoration at the par- ded in an even larger partial unification group SU(6)P 9 with a single coupling constant tial unification scale MP ∼= 10 GeV discussed in sect. III D we have left-right symmetric couplings of α := 2 α . (4.14) the standard weak gauge bosons and the new gauge P W/Y bosons W˜ and χ to the fermions. Consequently, Here, α is to be evaluated at the scale of left-right above this scale, the sum over the squared isospin P symmetry restoration and the factor 2 is due to the quantum numbers for leptons and quarks must run ensuing strength doubling according to Eq.(4.11) and over both left- and right- handed fermions depicted by the short dotted line in Fig. 5. Further 2 2 2 aspects of the unification of weak with hypergluon T3i = T3iL + T3iR, (4.11) i i i interactions will be discussed next. X X X and hence is twice as large as in Eq.(4.10). Further- more, at the energy scale of M = 109 GeV, where P ∼ B. Partial unification group SU(6) left-right symmetry is restored, the electroweak cou- P pling constants will necessarily have equal strength (see Fig. 5) We have seen that on the level of preon bound states new effective gauge interactions emerge, e.g. αW/Y := αW = αY . (4.12) those mediated by the W˜ and the N bosons, and that they occur at an energy where a left-right symmet- Using these conditions in Eq.(4.10) one obtains then for the Weinberg angle at M ric extension of standard model weak interactions is P required. However, we have not yet incorporated the 2 αQ 6 new gauge bosons in an appropriately enlarged effec- sin ΘW = = , (4.13) αW/Y 13 tive gauge group. This can be achieved by combining 10 the extended weak and strong hypercolor gauge in- We mention that at energies where SU(6)P comes teractions. Such a combination is also suggested by into play, anomaly freedom of SU(6)P is guaranteed the fact that the coupling constants of SU(3)H and because by definition all its generators are left-right SU(3)C always run in parallel (see Fig. 5) and can- symmetric. Furthermore, the charge sum of the above not reach equal strength because both groups have multiplet is zero. the same structure. Consequently, one of these color groups has to be embedded in a larger group if one wishes to unify the strong and weak interactions be- C. Preon intrinsic parity and weak interaction tween preon bound states. phenomenology We propose a partial unification by embedding the extended electroweak and hypercolor interactions within a broken gauge group SU(6) In this subsection we wish to comment on the dif- P ⊃ ferent intrinsic parity assignment for T and V preons SU(3)H SU(2)WL SU(2)WR U(1)Y . Note that SU(6) is× the simplest× rank 5× group having the same (see Table I) and its consequences for the effective rank as its subgroups. At the partial unfication scale preon bound state interactions that have been discussed so far. MP , the following equality of coupling constants is required to hold As shown in Table II, leptons and quarks have the same intrinsic parity whereas the two hyperquarksu ˜ αP := αH =2 αW/Y , (4.15) and d˜ have opposite intrinsic parities. Therefore, one where the last equality follows from Eq.(4.14). The can assign a weak isospin to leptons and quarks but SU(6) group also contains the hypercolored neu- not to hyperquarks. This means that only leptons ± and quarks can be divided into left- and right-handed tralons and the W˜ bosons. The 6 6 matrix repre- sectors carrying different weak isospin T and T . senting the effective SU(6) gauge bosons× is schemat- L R ically shown below At low energies below the partial unification scale MP , where chiral isospin symmetry is broken, this GH N classification of leptons and quarks in separate chiral 0 + ˜ + sectors having different weak isospin TL = 1/2 and WR WR W GP =  −  . (4.16) T = 0 goes hand in hand with left-right asymmetric N¯ 0 + R  WR WL WL   − −  weak interactions mediated by the standard model  W˜ W B0    L   gauge bosons WL and Z. In contrast, hyperquarks   This scheme includes 3 3 matrices for the hyperglu- do not possess a weak isospin and consequently do × not participate in the standard model weak isospin ons GH and the neutralons N and N¯ transforming ac- interactions but only in left-right symmetric W˜ , N, cording to SU(3)H . For the latter only the hypercolor degree of freedom is considered when counting their and Z mediated interactions as discussed before. multiplicity. The partial unifification group is then a For energies above MP , hyperquark-quark transi- unitary group of dimension 6, comprising 8 hyperglu- tions occur, and the left-right symmetric hyperquark ons, 9 left-right symmetric weak gauge bosons, and interactions entail left-right symmetric weak interac- 18 left-right symmetric dipreonic neutralons, in total tions also for quarks and leptons coupling to the weak 35 generators as discussed above. The corresponding isospin TL and TR with equal strength because at this ˜ gauge bosons with their respective energy scales are scale the effective masses of the WL, WR, W are of listed in Table V. Note that the diagonal generators the same order as MP . At this point the inner parity 0 0 of preon bound states ceases to be a good quantum GH , WR and WL have a B0 admixture [11]. The relevant SU(6) representations for the number. fermionic preon bound states arise from the direct Thus, preon intrinsic parities reveal themselves product of three fundamental six-dimensional repre- in different effective interactions of standard model sentations, where the “6” is due to the three hypercol- fermions and of hyperquarks. At low energies of ors and the two types of preons. We then have for the 102 GeV, we have left-right asymmetric interactions fermions 6 6 6 = 20 56 70 70, where only the between particles having definite intrinsic parities. lowest dimensional⊗ ⊗ (antisymmetric)⊕ ⊕ ⊕ 20 is needed to At higher energies of 109 GeV we have left-right represent the first generation of leptons, quarks, and symmetric interactions but intrinsic parity violat- hyperquarks and their antiparticles. The 20 dimen- ing transitions involving hyperquarks and neutrino- sional representation decomposes into 4 hypercolor antineutrino pairs. neutral leptons, 4 hypercolor neutral quarks, and 12 These aspects of the extended weak interaction hyperquarks where the multiplicity of hypercolor (H) have their origin in the different intrinsic parity as- is included as indicated below signments of the fundamental preon building blocks. ¯ Moreover, the existence of left-right symmetry at high ν ν¯ u u¯ u˜ u˜ and left-right asymmetry at low energies is seen to be − ¯ . (4.17) e e+ d d¯ d˜ d˜ ! ! !H !P closely connected with the production and decay of 11 hyperquarks and hence to some extent explained in T U V the preon model developed here. U

V. GRAND UNIFICATION

U As motivated in the previous chapter we are now D D NN left with three gauge groups, namely SU(6)P , SU(3)C and U(1)Q. At the grand unification scale MG, the corresponding interactions have the same strength U and are described by a common coupling constant (see Fig. 5). U αG := αP = αH = αC = αQ. (5.1) T V At this scale the direct product, U SU(6)P SU(3)C U(1)Q is embedded in the larger gauge× group SU(9)× SU(6) SU(3) U(1) . FIG. 4: Preon square. The two fundamental preons G ⊃ P × C × Q (T,V ) and their antiparticles are placed at the corners of Thus, the unbroken fundamental gauge symmetry a square. The dipreon bound states and their antipar- SU(3) SU(3) U(1) of the preon model is also H × C× Q ticles describing transitions between preons are placed part of this larger group, which finds its expression along the edges (N, U) and diagonals (U˜). The dipreon in the equality of coupling constants according to states D(TT ) connecting T and T¯ can be represented as a Eqs.(5.1). linear combination of U˜ and U or U and N. Thus they are The grand unification group SU(9)G contains in ad- not included as independent gauge bosons in Table VII. dition to the gauge bosons already included in SU(6)P the following color anti-triplet and hypercolor singlet gauge bosons generating transitions between quarks The preon content of the dipreonic U and of the and leptons: neutralons N already introduced in sect. IV is given in Table VII. The transitions between the preons that are generated by these dipreonic bosons are graphi- T T V T¯V¯ V¯ cally depicted in Fig. 4. X = ; Y = ; T T V ! T¯V¯ V¯ ! T T V A. Nucleon decay processes U = = T V¯ , (5.2) T¯V¯ V¯ ! Nucleon decay into leptons is mediated by six where the six-preon bound state U reduces to a preon bound states and in an analogous way by dipreon bound state with respect to its quantum dipreon bound states. For example, in proton decay numbers. These bosons are responsible for proton de- p(uud) e+ + γ the preons in the two u-quarks cay, which has already been predicted by the SU(5) combine−→ into an intermediate X-boson which then grand unification [13] as well as within the preon decays by rearrangement into an d¯ quark and an e+ model [14]. Furthermore, there are new hypercolor anti-triplet T T V u (T T V ) + u (T T V ) X and color singlet gauge bosons generating transitions −→ T T V ! between hyperquarks and leptons d¯(T V V )+ e+ (TTT ) . (5.4) −→ TT V¯ T¯ V V X˜ = ; Y˜ = ; The remaining d-quark in the proton annihilates with TT V¯ ! T¯ V V ! the final state d¯-quark into a photon. An analogous proton decay process of a (u d) quark pair in the pro- TT V¯ U˜ = = (T V ) . (5.3) ton involves the dipreon U of Eq.(5.2) T¯ V V ! u (T T V ) + d T¯V¯ V¯ U T V¯ −→ Formally, the latter bosons are obtained from those ¯ ¯ ¯ + u¯T T V + e (TTT ), (5.5) in Eq.(5.2) by the hyperquark transformation. The −→ quantum numbers of the six-preon GUT bosons X where the final state is generated by creating two ad- and Y generating baryon decay into leptons and of ditional T T¯ pairs from the vacuum. Note that the the new GUT bosons X˜ and Y˜ mediating hyper- preon triality rule of Eq.(3.7) is broken at the grand baryon decay into leptons are listed in Table VI. unification scale. 12

gauge group generators gauge boson energy scale −3 U(1)Q 1 AQ ΛQ = 10 GeV ∼ SU(3)C 8 GC ΛC = 0.2 GeV − + 0 0 SU(2)WL × U(1)Y 4 WL ; WL ; WL; B MZ = 91.18 GeV ∼ SU(3)H 8 GH ΛH = 1700 GeV

SU(6)P 35 GH − + 0 − + 0 0 WL ; WL ; WL; WR ; WR ; WR; B W˜ −; W˜ + 9 N; N¯ MP = 8.3 × 10 GeV H SU(9)G 80 GP

GC X; Y ; X¯; Y¯ ; U; U¯ C X˜; Y˜ ; X˜¯; Y˜¯ ; U˜ ; U˜¯ H 16 AQ MG = 1.2 × 10 GeV

TABLE V: Gauge groups, gauge bosons, and corresponding energy scales.

state content P Υ Q Π or via the dipreon U 2 4 X (4T, 2V ) +2 + 3 + 3 + ¯ ¯ ¯ 2 4 X (4T , 2V ) −2 − 3 − 3 + u (T T V ) + d T¯V¯ V¯ U T V¯ Y (2T¯ , 4V¯ ) −2 + 2 − 2 + −→ 3 3 d¯(T V V )+¯ν V¯ V¯ V¯ (5.7) ¯ 2 2 Y (2T, 4V ) +2 − 3 + 3 + −→ ˜ ¯ 2 4 X (4T, 2V ) + 3 +2 + 3 + ˜¯ ¯ 2 4 requiring the creation of two V V¯ pairs from the vac- X (4T , 2V ) − 3 −2 − 3 + Y˜ (2T¯ , 4V ) + 2 −2 − 2 + uum. Thus, these nucleon decay processes can be de- 3 3 scribed by six-preon bound states X and Y , as well Y˜¯ (2T, 4V¯ ) − 2 +2 + 2 + 3 3 as by the dipreon bound state U(T V¯ ). TABLE VI: Quantum numbers of the six-preon GUT In analogy, hypernucleon decay processes involve gauge bosons X and Y responsible for baryon decay into the corresponding X˜, Y˜ , and U˜(T V ) bosons as dis- leptons, and of the new GUT bosons X˜ and Y˜ mediating cussed before and are obtained from Eq.(5.4)-Eq.(5.7) hyperbaryon decay into leptons. by hyperquark transformation. Common to these processes is that they cause a simultaneous violation of baryon- and lepton-numbers (∆B = ∆L = 1), dipreon P Υ Q Π whereas and Υ numbers are conserved. There− 2 2 P N (VV ) + 3 − 3 0 + are also B L violating proton decays [14] such as ¯ ¯ ¯ 2 2 +− N V V − 3 + 3 0 + p ν + π which can be thought of as being gener- → + U T V¯ 0 + 2 + 1 + ated from the B L conserving p ν¯ + π decay 3 3 via a ν ν¯ oscillation− as discussed in→ sect. III D. U¯ TV¯ 0 − 2 − 1 + 3 3 − ˜ 2 1 For completeness we mention that between the U (TV ) + 3 0 + 3 - U˜¯ T¯V¯ − 2 0 − 1 - color triplet six-preon bound states X, Y and the 3 3 dipreonic U gauge bosons there are the following TABLE VII: Dipreon bound states and their quantum weak transition processes numbers. X U + W +; Y U + W − −→ −→ − X˜ U˜ + W˜ +; Y˜ U˜ + W˜ . (5.8) Similarly, neutron decay n ν¯ + γ proceeds ei- −→ −→ ther by rearrangement −→ The reactions in the second line are generated by T¯V¯ V¯ applying the hyperquark transformation to the reac- d T¯V¯ V¯ + d T¯V¯ V¯ Y tions in the ﬁrst line, emphasizing once again that the −→ T¯V¯ V¯ ! hyperquark transformation is applicable to all preon u¯ T¯T¯V¯ +¯ν V¯ V¯ V¯ , (5.6) bound states containing neutral V preons. −→ 13

B. Grand unification group SU(9)G VI. RUNNING COUPLINGS AND ENERGY SCALES After having discussed the additional bosons needed to enable transitions from quarks and hy- On the basis of the model described so far, we dis- perquarks to leptons, we can now count the number cuss in this section the momentum dependence of of gauge bosons involved. We find that the unitary the couplings α(q2) described in sect. IV, where q2 group has to be as large as SU(9) to accomodate all is the momentum transfer exchanged in the interac- gauge bosons that have been introduced. The corre- tion. Our aim is to determine the scale, where the sponding 9 9 matrix of gauge bosons is schematically lightest hyperquark bound states appear, that is the × shown below mass scale ΛH of hyperchromodynamics, where αH is of order (1). This is analogous to QCD where the O lightest quark bound states (pions) appear at ΛC , i.e. ˜ ˜ ˜ GC XYU X Y U the scale where αC is of order (1). 2 O X¯ Y¯ U¯ GH N In analogy to QCD, αH (q ) will decrease with in-   2 0 + ˜ + creasing q and the slope bH of this decrease depends GG = WR WR W .  ˜¯ ˜¯ ˜¯ ¯ − 0 +  on the number of hyperquark flavors along the way  X Y U N  WR WL WL    − −  to higher energies in the same manner as the slope  W˜ W B0  2   L   bC of αC (q ) depends on the number of quark flavors     appearing with increasing q2. It should be clear that This scheme includes the 6 6 matrix of the SU(6)P we cannot determine the exact position of each single partial unification group G× of Eq.(4.16), the 3 3 P × hyperquark flavor but only an average value of these matrix of the QCD gluons GC transforming according hyperquark masses denoted as mhq. to SU(3) , four blocks of 3 3 matrices for the colored 2 C × To fix the high q end of the running couplings we GUT bosons X, Y , and U, their antiparticles, as well use the constraints defining partial and grand unifi- as their hyperquark transformed states. In addition, cation of Eq.(4.15) and Eq.(5.1) respectively as de- we have to include the elementary photon associated tailed in the next section, where we also study how with the group U(1)Q. The generator AQ appears the β-functions of the different effective gauge groups as an admixture in the diagonal entries and must be change when q2 crosses the hyperquark mass scale. included in the counting leading to 80 generators in total. Representations for the fermionic preon bound A. β-functions of the SU(N) and U(1) gauge states arise from the direct product of three funda- groups mental representations of dimension 9, where the 9 is due to the three color and three hypercolor degrees In non-abelian gauge theories the slope of the run- of freedom associated with each preon. Note that ning couplings is determined to first order by the β- the preon types T and V need not be included as a function b separate degree of freedom, i.e. we do not have an i SU(18) because specifying the color and hypercolor 11 2 b = N n . (6.1) representations uniquely specifies the preon type (see i 3 i − 3 f Table I). We then have 9 9 9 = 84 240 240 165, where the index i stands for the different gauge ⊗ ⊗ ⊕ ⊕ ⊕ where only the lowest dimensional (antisymmetric) groups, Ni is number of degrees of freedom of the 84 is needed to represent the three generations of SU(Ni) group, and nf is the number of fermion fla- leptons, quarks, and hyperquarks, as well as their vors. For the running coupling one has in one-loop antiparticles: approximation [15]

2 ¯ 1 1 bi q ν ν¯ u u¯ u˜ u˜ = + ln . (6.2) − ¯ . (5.9) 2 2 2 e e+ d d¯ d˜ d˜ αi(q ) αi(Λi ) 4π Λi ! !C !H !G For the present purposes higher order approximations The 84 dimensional fermion representation decom- of αi can be neglected. Eq.(6.2) connects an a priori poses as follows: 12 leptons, 36 quarks, and 36 hyper- unknown high energy scale q2 = M 2 to a known low- quarks, where the multiplicities of color (C), hyper- 2 energy scale Λ such as for example, ΛC for QCD. color (H) and generation number (G) for the bound Constraints for the high energy scale are obtained states are included. from the equality of certain coupling constants at this mass scale. 14

α H α C 20 α α W P 40 α Y α G 60 α−1 80

α 100 Q α g 120

140 2 4 6 8 10 12 14 16 18 20 1 10 10 10 10 10 10 10 10 10 10 q [GeV]

FIG. 5: Running coupling constants. The dashed vertical line indicates the expected hyperquark mass scale. Because hyperquarks are hypercolor triplets and charged weak isosinglets, they only affect the SU(3)H coupling αH and the 9 U(1) gauge couplings αQ and αY . The dotted vertical line at 10 GeV indicates the partial unification scale where the weak isospin and hypercharge couplings converge and are unified with the hypercolor interaction to a common coupling 16 αP . The three gauge couplings αP , αC , and αQ meet at the grand unification scale of 10 GeV. Finally, from the ∼ 18 grand unification scale to the point where αG meets the gravitational coupling αg (= 10 GeV), preons behave as quasifree particles.

1. The β-functions of SU(3)C and SU(3)H 2. The β-function of U(1)Q

For the non-abelian groups SU(3)C and SU(3)H The running coupling constant of QED reads and where N = N = 3 we have C H 1 1 b q2 = + Q ln . (6.6) α (q2) α (Λ ) 4π Λ2 11 2 Q Q Q Q ! bC = NC (nd + nu) 3 − 3 Here, bQ is defined as 11 2 2 bH = NH nd˜ + nu˜ . (6.3) 2 3 − 3 bQ = Q , (6.7) −3 i i X Because there are three generations for each fermionic 2 where Qi is the square of the fermion charge (see bound state type [4] Table IV), and the sum extends over all fermions. Thus, the β-function of QED is n = n = n ˜ = n =3, (6.4) d u d u˜ 2 1 4 b = 2 n +3 n + n Q −3 e 9 d 9 u we obtain for both gauge groups the same b value 1 4 +3 nd˜ + nu˜ , (6.8) bC = bH =7. (6.5) 9 9 ! 15 where ne = 3 is the number of leptonic flavors. The 4. The β-function of the partial unification group SU(6) identical contributions of the left- and right handed fermion charges are taken into account by an overall Here, we deal with the nonabelian gauge group factor 2, and the color and hypercolor multiplicities SU(6) as discussed in sect. IV. At the MP scale the are indicated by factors 3 multiplying the quark and Majorana description of the neutrinos implies that hyperquark contributions. the right-handed neutrinos are identical with antineu- At the hyperquark scale m the β-function b hq Q trinos, i.e., νR ν¯. Therefore, only the left-handed changes neutrinos are included≡ in the summation over the left- 32 and right-handed fermions bQ = (without hyperquarks), − 3 11 1 52 bP = NP nν +2 ne + nd + nu + nd˜ + nu˜ , bQ = (withhyperquarks) (6.9) 3 − 3 − 3 (6.14) which gives b = 11 for N = 6. due to the additional contribution of the hyperquarks. P P We point out that two-state Majorana neutrinos provide the only consistent description in the present framework. A four-state Dirac neutrino would re- 3. Electroweak β-functions for U(1)Y and SU(2)W duce the β-function to bP = 10. As will be seen in section VI B, the latter would lead to inacceptable For the U(1)Y group we replace in Eq.(6.6) αQ by results for the energy scales MG, MP , and mhq. In αY and bQ by bY which is defined as particular, it would imply a hyperquark mass scale 2 below the masses of W and Z bosons, and a much b = Y 2, (6.10) too short proton lifetime, both of which are in con- Y −5 i i tradiction to experimental facts. X where the sum extends over the fermionic preon bound states. In addition, we have to take into ac- 5. The β-function of the grand unification group SU(9) count that left- and right-handed fermions give different contributions. As noted before hyperquarks are weak isospin singlets (T3 = 0) and therefore have At this scale the different bosons can be accom- a left-right symmetric coupling, whereas quarks and modated into a larger gauge group SU(9). By an leptons couple left-right asymmetrically. analogous counting of the fermions as in Eq.(6.14) We then obtain for the β-function according to Ta- we obtain ble IV 11 1 bG = NG nν +2 ne + nd + nu + nd˜ + nu˜ 2 1 5 5 17 3 − 3 bY = nν + ne +3 nd + nu (6.15) −5 4 4 36 36 which gives bG = 22 for NG = 9. 2 8 +3 n + n , (6.11) 9 d˜ 9 u˜ ! B. Calculation of energy scales with nν = 3 and where the factors in front of the ni come from adding left and right handed contributions To obtain numerical values for MG, MP and mhq one starts from the following constraints. First, from in Eq.(4.8). Note that bY changes at the hyperquark Eq. (4.12) we have at MP mass scale mhq due to the contribution of the hyperquarks as 1 1 2 = 2 b = 4 (without hyperquarks), αW (MP ) αY (MP ) Y − bY = 8 (with hyperquarks). (6.12) 2 − 1 bW MP 1 2 + ln 2 = 2 αW (mt ) 4π mt αY (mt ) For the non-abelian weak isospin group SU(2)W 2 2 the corresponding β-function reads bY t mhq bY MP + ln 2 + ln 2 , (6.16) 11 1 4π mt ! 4π mhq ! b = N (n + n + n + n ) (6.13) W 3 − 3 ν e d u where the evolution starts at the top quark mass mt. where N = 2. Here, only left-handed leptons and According to Eq.(6.2) the first two terms on the right- quarks contribute and we obtain the standard model hand side can be written as 1/αY (mhq), and the third 10 value bW = 3 . term on the right-hand side describes the evolution 16 from mhq, at which point we have to include the hy- This is different from the model of Harari-Seiberg 9 perquarks to the partial unification scale MP . which gives a ΛH of order 10 GeV. The numerical Second, from Eq.(4.14) and Eq.(5.1) evaluated at values of the β-functions, coupling constants, and en- MG follows ergy scales calculated in this section are compiled in 1 1 Table VIII. = With the constraint m Λ where m de- α (M 2 ) α (M 2 ) hq1 H hq1 C G P G notes the mass of the lightest≥ hyperquark, we ob- 2 tain for the weak decay lifetime of hyperquark bound 1 bC MG + ln states, e.g., a hyperpion α (m2) 4π m2 C t t 2 4 1 1 bW M 1 MP = + ln P τπ˜ = 100 s. (6.22) 2 α (m2) 4π m2 ∼ α2 m5 ≤ W t t P hq1 b M 2 + P ln G , (6.17) For the neutrino masses according we obtain accord- 4π M 2 ing to Eq.(3.12) P −8 where the factor 2 in the denominator of the first term mνe = 3.3 10 eV, on the right-hand side comes from the strength dou- × −3 mνµ = 1.3 10 eV, bling of αW at the partial unification scale according × − m = 3.8 10 1 eV (6.23) to Eq.(4.15). ντ × Third, from Eq.(5.1) also follows −3 compared to mνµ = 8.8 10 eV and mντ = 5.0 − × × 1 1 10 2 eV obtained from upper limits of experimental 2 = 2 neutrino squared mass differences [16]. αC (MG) αQ(MG) Furthermore, the large value of the SU(9)G boson 2 16 1 bC M 1 masses M =1.2 10 GeV results in the following + ln G = G × α (m2) 4π m2 α (m2) value for the proton lifetime C t t Q t 2 2 1 M 4 bQ mhq bQ MG G 35 + ln + ln (6.18). τp = = 10 y, (6.24) 2 2 ∼ α2 m5 ∼ 4π mt ! 4π mhq ! G p The energy scales where these conditions hold can where mp = 938.3 MeV is the proton mass. This is now be determined by solving the above three equa- in good agreement with the experimental lower limit τ > 5.5 1033 y [17]. tions (6.16-6.18) with three unknowns. This gives p × m = 2.6 104 GeV hq × M = 8.3 109 GeV We close this section with a remark on the evolu- P × 16 tion of αG for still higher momentum transfers. The MG = 1.2 10 GeV. (6.19) × grand unification scale MG calculated here is near the In order to calculate the low energy scale ΛH we Planck scale of quantum gravity MPl given by use Eq.(6.2) and start the evolution at the partial ~c q2 unification scale MP where according to Eq.(4.15) the 2 MPl = , αg = 2 , (6.25) following equality of coupling constants holds αP = γG MPl αH =2αW . By backextrapolation we obtain where γG is the Newton gravitational constant. The 2 coupling αg is the only nonrenormalized coupling, 1 1 bH MP 2 = 2 ln 2 having a linear dependence on the momentum trans- αH (ΛH ) αH (MP ) − 4π mhq ! fer q2. 2 Explicit preon degrees of freedom will become im- bH1 mhq ln 2 , (6.20) portant at the fundamental preon scale defined by the − 4π ΛH ! 2 2 constraint αG(Mpr)= αg(Mpr). We can write 31 where bH1 = 3 is calculated according to Eq.(6.3) 2 2 1 bG Mpr MPl assuming n ˜ = 1 and n = 0 for the contribution d u˜ 2 + ln 2 = 2 . (6.26) of the lightest hyperquark below the average scale αG(MG) 4π MG ! Mpr mhq. The slope bH1 corresponds to the dotted line From this constraint we obtain using MPl = 1.22 segment of αH in the left upper corner of Fig. 5. We 19 × 2 10 GeV for the Planck scale then demand αH (ΛH ) = (1) which, using the nu- ∼ O 18 merical values in Eq.(6.19), leads to a prediction of Mpr =1.56 10 GeV (6.27) the infrared cut off mass of hypercolor interactions × for the scale Mpr where an asymptotically free preon ΛH ∼= 1700GeV. (6.21) dynamics is expected. 17

MZ = 91.18 ± 0.02 GeV

SU(3)C SU(2)W U(1)Y U(1)Q 23 11 103 80 bCz = 3 bWz = 3 bYz = − 30 bQz = − 9 −1 −1 −1 −1 αCz = 8.24(12) αWz = 29.57(05) αYz = 59.00(04) αQz = 127.90(02)

mt = 176.9 ± 4.0 GeV 10 32 bC = 7 bW = 3 bYt = −4 bQt = − 3 −1 −1 −1 −1 αCt = 9.05(15) αWt = 29.96(06) αYt = 58.64(05) αQt = 126.96(05)

3 3 mhq = (26.3 ± 2.3) 10 GeV (ΛH = (1.66 ± 0.34) 10 GeV)

SU(3)H SU(3)C SU(2)W U(1)Y U(1)Q 10 52 bH = 7 bC = 7 bW = 3 bY = −8 bQ = − 3 −1 −1 −1 −1 −1 αHh = 5.58(20) αCh = 14.63(18) αW h = 32.62(08) αY h = 55.45(07) αQh = 118.47(09)

9 MP = (8.31 ± 0.88) 10 GeV

SU(6)P SU(3)C U(1)Q 52 bP = 11 bC = 7 bQ = − 3 −1 −1 −1 αP = 19.66(08) αCp = 28.73(18) αQp = 83.57(09)

16 MG = (1.17 ± 0.12) 10 GeV

SU(9)G

bG = 22 −1 αG = 44.48(27)

18 Mpr = (1.56 ± 0.01) 10 GeV −1 αg = 61.20(37)

TABLE VIII: Numerical values of β functions and coupling constants at the corresponding energy scales. The indicated errors are due to the experimental input values.

C. Hyperquark and hyperhadron mass scales MeV, and ΛH = 1700 GeV, we arrive at a hyperpion mass mπ˜ = 1100 GeV. Similarly, using the proton We are now in the position to make statements mass mp = 938 MeV as input we find hyperbaryon 4 about the energy scale where the hyperquarks and masses to be of order mB˜ ∼= 10 GeV. These results their bound states presumably occur. As mentioned are compiled in Table IX. before, it is reasonable to assume that the new par- We conclude this section with some remarks per- ticles do not all appear at the same threshold level taining to the hyperquark and hyperhadron masses but that their mass values are distributed over a cer- and lifetimes. Because hyperquarks d˜ andu ˜ have tain range. Thus, the present result m = 26 TeV hq different intrinsic parities they do not form strong provides an average energy level but gives no informa- isospin doublets. Thus, unlike the quarks of the tion concerning the size of the mass range over which first generation their masses need not be close to the particles are distributed. However, because of the each other. For the lightest hyperquark (d˜) we find missing isospin degree of freedom the spectrum of hy- a constituent mass of order m ˜ = Λ = 1700 perhadrons is expected to be much wider than that d ∼ H ∼ GeV. The mass of the hyperquark (˜u) could be sub- of ordinary hadrons. stantially higher, and is conjectured to be of order For quark bound states we have the result mπ m ˜ +Λ = 3400 GeV. Consequently, unlike the Λ . Analogously, for the bound states of hyperquarks∼ d H ∼ C standard model pions, hyperpions do not form an we have m Λ . This leads to the following esti- π˜ H isospin triplet and we expect that charged hyperpions mate for the∼ hyperhadron mass are heavier than neutral ones. Furthermore, because

ΛH of the missing isospin degree of freedom, the lowest- mh˜ ∼= mh. (6.28) lying hyperhadrons have a symmetric spin conﬁgu- ΛC 3 ration corresponding to spin 2 for hyperbaryons and With the numerical values mπ = 140 MeV, ΛC = 100 spin 1 for hypermesons. 18

boson spin Π mass [GeV] lifetime [s] decay-mode Z Z − − H0 0 + =∼ 350 − 500 10 26 Z, W +W ,ff¯ − π˜0 1 - =∼ 1100 10 27 H0,Z, γ π˜± 1 + ≈ 3000 ≤ 100 ff¯ ˜ − 3 4 − ∆ 2 + ≈ 10 ≤ 1 ∆ ~ ˜ ++ 3 4 ++ π ∆ 2 - ≈ 10 ≤ 1 ∆

TABLE IX: The H0 boson and low lying hyperhadrons

tively understood as a residual force that is associated with the preon number of the bound state. On the other hand, a model which contains only the un- Z Z broken gauge interactions SU(3)H SU(3)C U(1)Q between preons does not readily lend× itself to× a quan- FIG. 6: ZZ bound state H0 generated viaπ ˜ exchange. titative description of the left-right asymmetric weak interactions between preon bound states at low energies. Furthermore, it does not provide a dynamical While neutral hyperquark bound states can decay explanation for certain new phenomena predicted by electromagnetically, the charged ones can only decay the present theory, for example, the decay of hyper- via W˜ emission as discussed in sect. III A. Therefore, quarks into quarks. Therefore, we have attempted the lifetimes of the charged hyperquark bound states to make some progress by considering approximate are according to Eq.(6.22) with τhq 100 s substan- effective gauge theories on the level of preon bound tially longer than the lifetimes of the≤ neutral ones. states. As to the experimental signature, a neutral hyperpion In particular, we have investigated the bosonic decays electromagnetically into two photons with life- sector of the preon model in some detail. We ~ −27 time τπ˜ = /Γπ˜ 10 s if one assumes a decay have shown that the introduction of hyperquarks in ≤ ΛH width Γπ˜ = 800 GeV derived from Γ˜ = Γh in h ∼ ΛC Harari’s theory requires new classes of effective gauge analogy to Eq.(6.28). bosons, called W˜ and N in order to describe weak Hypermeson masses of order TeV correspond to a transitions among hyperquarks and between hyper- short range Yukawa coupling interaction with range quarks and quarks. The presence of these additional ~ 2 −19 /mπ˜c ∼= 10 m. At energies of several tens of gauge bosons leads to an extension of the standard TeV, hyperquarks couple only to the Z boson and SU(2)WL U(1)Y electroweak theory to a larger gauge × 9 the photon. As a result, there is a coupling of the group emerging at an energy MP ∼= 10 GeV. At neutral hyperpion to the Z boson. This short-range this scale, 9 left-right symmetric weak bosons, 8 hy- interaction could lead to the formation of a scalar ZZ pergluons, and 18 neutralons provide the 35 gener- 0 bound state with spin 0, called H , as shown in Fig. 6, ators of an effective SU(6)P gauge group, refered which could be a viable candidate for solving the uni- to as partial unification group. This scheme pre- 0 2 tarity problem in W-W-scattering [18]. For the H dicts a Weinberg angle sin ΘW = 6/13 at MP ∼= boson mass and decay width an estimate according to 109 GeV, which is twice its experimental value at 2 √ −1/2 M = 91.2 GeV. Furthermore, it shows that the Ref. [18] using λ = Gf mH0 / 2, where Gf ∼= 293 Z GeV is the Fermi constant and 1 λ 2 gives breaking of left-right SU(2)WL SU(2)WR symmetry ≤ ≤ × mH0 = 350 500 GeV and ΓH0 = 40 GeV corre- into the standard model symmetry SU(2)WL U(1)Y ∼ ∼ − × − 0 ~ 0 26 is closely connected with the production and decay sponding to the lifetime τH = /ΓH ∼= 10 s. of hyperquarks and thus to some extent explained in the present model. VII. SUMMARY AND OUTLOOK 16 At the grand unification scale MG ∼= 10 GeV the spectrum of bosonic preon bound states is much The standard model leaves many questions unan- larger. In addition to the usual GUT bosons, which swered; for example, why leptons and quarks share provide transitions between quarks and leptons as in the same weak interaction, mediated by heavy vector proton decay processes, several new gauge bosons ap- bosons coupling differently to left- and right-handed pear, the counting of which leads to the grand unifi- quarks and leptons. This fact, among others, points cation group SU(9)G. Next to the 35 gauge bosons to a deeper connection between leptons and quarks. of SU(6)P , there are 8 gluons of SU(3)C , 18 colored In the preon model, weak interactions are qualita- bosons (X, Y , and U and their antiparticles), 18 hy- 19 percolored bosons (X˜, Y˜ , and U˜ and their antipar- Finally, from the unification constraints mentioned ticles), and the photon AQ of U(1)Q, altogether 80 above, the Majorana description of neutrinos is pref- generators of SU(9)G. ered. For Dirac neutrinos the present theory leads to The hyperquark mass scale has been found from a proton lifetime which is too short, and furthermore the dimension of these unified gauge groups, the num- to a hyperquark mass scale below the masses of W ber of fermionic bound states, and the requirement and Z bosons, both results contradicting experimen- that the coupling constants of the various effective tal facts. gauge interactions are equal at the two unification scales. These constraints lead to a system of three In summary, based on the introduction of hyper- equations with three unknows, allowing the determi- quarks as a new class of fermionic bound states we 9 have depicted a scenario for the unification of forces nation of both unification scales MP = 10 GeV and 16 ∼ that partly explains the left-right asymmetry of weak MG = 10 GeV, and the average hyperquark mass ∼ 4 interactions at low energies and fills the large gap scale mhq ∼= 10 GeV. To obtain a mass constraint for the lightest hyperquark bound state, the hyperpion, between the Fermi scale and partial unification scale with a wide spectrum of hyperquark bound states. we have extrapolated from the point mhq to the point ΛH where αH ∼= (1) and obtained ΛH ∼= 1700 GeV, as the typical scaleO for hyperhadron masses, which is Acknowledgments: We thank Haim Harari and within reach of the LHC at CERN. Don Lichtenberg for helpful comments.

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