arXiv:1010.3736v1 [hep-ph] 18 Oct 2010 estso ehiemosta break con- that producing technifermions scale, of TeV the densates at vecto- coupled be- the strongly interaction comes these, gauge technicolor In free asymptotically theories. rial, (TC) technicolor by taken saa opnnso)toHgscia uefilswith superfields chiral Higgs the T two of of) VEVs components nonzero (scalar to breaking symmetry attributes (MSSM) electroweak Model Standard supersymmetric mal uhta steter vle rmsm ihen- high some from evolves the energies, theory lower to the scale as ergy that such fteLgaga,o h form the symmetry of gauge class Lagrangian, direct-product a the a consider of shall with We theories study. gauge a of such paper this out In carry shall approaches. these we symmetries of gauge both into of insights breaking gives the to approaches namical symme- electroweak of origin breaking. the try to of underway question currently the the are answer at (LHC) Experiments Collider Hadron dimensions. 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In induced consider. of we type that the breaking of symmetry examples gauge illustrative two review where − lyarl ntednmclbekn of breaking dynamical the in role a play , b UV UV nodrfrtednmclsmer nE.(1.2) Eq. in symmetry dynamical the for order In n a eeaieteaayi ute oda with deal to further analysis the generalize can One hsppri raie sflos nSc.I we II Sect. In follows. as organized is paper This ( 1 µ / 4 = ) µ G ymtytu eursta h emosta are that fermions the that requires thus symmetry scia nalo h ae htw osdr The consider. we that cases the of all in chiral is k F − erae;(ii) decreases; togyculdgueinteractions gauge coupled strongly 1 / g 2 b ( G ≃ µ lrvoe fteform the of ultraviolet e )Tecniinta the that condition The .) ) 5 e hr ti rkn oee,our However, broken. is it where GeV 250 2 b G / k on G (4 b 2. = G osdrdb tef(ihthe (with itself by considered , UV π G b nrae sterfrneenergy reference the as increases ) G aezr nrni masses. intrinsic zero have b = n Λ and b G osdrdb tef savecto- a is itself, by considered , G bekn emo condensates fermion -breaking odnae that condensates n ⊗ k ,btwl locomment also will but 1, = b [ b i Y =1 eest hi oe nthe in roles their to refers k ssmlrt h atthat fact the to similar is symmetry l G G b b i G ] neato evolves interaction G , ⊗ G b ohvectorial both , G G ag interac- gauge b , neato is interaction G G b ewill We . i α 1 , b G ( µ ≤ inter- G in- ) (1.3) b i ≤ is 2 we carry out a comparative study of the breaking of an The technicolor gauge interaction is associated with the SU(3) gauge symmetry to SU(2) by a Higgs field in the group Gb = GTC . Typically, GTC = SU(NTC ) with some fundamental representation and by a dynamical mecha- value of NTC such as 2, so these models can be described nism. We also discuss how color SU(3)c would be broken in the notation of Eqs. (1.1) and (1.2) by in a modified Standard Model with a strongly coupled SU(2)L interaction. In Sect. IV we carry out a compar- G = GSM , Gb = SU(NTC ) , H = SU(3)c U(1)em . ative study of the breaking of an SU(3) gauge symmetry ⊗ (2.2) by a Higgs field in the adjoint representation and by a The (vectorial, asymptotically free) technicolor gauge in- dynamical mechanism. This is generalized to SU(N) in teraction produces condensates of technifermions F¯ F = Sect. V. Some further discussion and our conclusions are F¯ F + h.c. that transform as weak T = 1/2, hY =i 1 h L Ri | | given in Sects. VI and VII. and hence break GEW to U(1)em, as indicated in Eq. (2.2). Technicolor models are embedded in extended technicolor (ETC) in order to communicate the elec- II. SOME EXAMPLES OF INDUCED troweak symmetry breaking to the quarks and leptons. DYNAMICAL SYMMETRY BREAKING These TC/ETC theories are subject to a number of con- straints from induced flavor-changing neutral processes, A. QCD Breaking Electroweak Symmetry precision electroweak data, and limits on pseudo-Nambu- Goldstone bosons (PNGBs). As background for our work, we first briefly review Technicolor models can be classified into two generic two examples of induced dynamical symmetry breaking types: (i) one-family models, in which the technifermions of weakly coupled gauge symmetries by strongly coupled comprise one SM family, and (ii) one-doublet models, in gauge interactions. In addition to the physics that is re- which, among the technifermions, there is only a single sponsible for the main electroweak symmetry breaking at electroweak doublet. One-family (but not one-doublet) the scale 250 GeV, there is another source of EWSB, technicolor models feature a color-octet technivector me- albeit at∼ a much smaller mass scale. This is quantum son resonance, as well as a color-nonsinglet PNGB’s. Many searches for technihadrons have been carried out chromodynamics (QCD). The color SU(3)c gauge inter- action produces bilinear quark condensates at a scale [3]. Recent LHC results from the ATLAS and CMS ex- periments have set lower limits of order 1.5 TeV on a ΛQCD 250 MeV, in the most attractive channel (MAC) ∼ color-octet technivector meson [4, 5]. 3 3¯ 1, of the form qq¯ = q¯LqR + h.c. Because these× quark→ condensatesh transformi h asi weak T = 1/2, Y = 1 quantities, they break G to electromagnetic | | EW U(1)em. Indeed, one could imagine a hypothetical world III. BREAKING AN SU(3) GAUGE in which the electroweak symmetry were not broken at SYMMETRY TO SU(2) the normal scale, but instead remained valid all the way down to the QCD scale. In this world (assuming that In this section we shall compare Higgs and dynamical the SU(3)c, SU(2)L, and U(1)Y running gauge couplings mechanisms for breaking an SU(3) gauge symmetry to had approximately their usual values), QCD would be SU(2). We assume that the fermion content of the the- the main source of EWSB [1, 2]. Such a theory would be ory is such that the fermions that are only nonsinglets of the form of Eqs. (1.1) and (1.2), with under SU(3) form a vectorlike sector. We shall begin by considering an abstract asymptotically free SU(3) theory G = GEW , Gb = SU(3)c , H = U(1)em . (2.1) at a sufficiently high scale that it is weakly coupled.
In this hypothetical world the W and Z would pick up 2 2 2 2 2 ′2 2 masses given by mW = g fπ/4 and mZ = (g +g )fπ/4, A. Higgs Mechanism to break SU(3) to SU(2) ′ where g and g are the SU(2)L and U(1)Y running gauge couplings at the scale ΛQCD, and fπ is the pion decay constant. The requisite breaking can be accomplished by includ- ing a Higgs field φ that transforms as a fundamental (triplet) representation of the SU(3) group, with a po- tential B. Electroweak Symmetry Breaking by Technicolor µ2 λ V = φ†φ + (φ†φ)2 , (3.1) Technicolor models embody the idea of dynamical elec- 2 4 troweak symmetry breaking [1] (recent reviews include [3]). In these models, the gauge symmetry that is bro- where µ2 < 0 (and λ > 0 in order for V to be bounded ken is (the electroweak part of) the SM gauge group from below). This potential is minimized for a nonzero G = G = SU(3) SU(2) U(1) . At the scale value of the φ vacuum expectation value. Without loss SM c ⊗ L ⊗ Y where GSM is broken, all of the three gauge interactions of generality, one can use the SU(3) gauge invariance to corresponding to its factor groups are weakly coupled. define directions in SU(3) space so that this has the form 3
φ = (0, 0, 1)T v, and one can perform a global phase re- where t = ln µ. Because of this, if one fixes λ at the scale definitionh i on φ to make v real. This breaks SU(3) to the v, say, then one must worry about a possible Landau pole SU(2) subgroup generated by the the SU(3) generators in λ that could occur at a scale µ >> v. An equivalent Ta with a = 1, 2, 3 (in the usual Gell-Mann ordering of way to phrase this is that if one fixes λ at a high scale these generators). Of the six real components of the φ in the ultraviolet, then λ decreases as µ decreases and is field, five are Nambu-Goldstone bosons and are absorbed subject to an upper bound at a much lower scale such as by the five gauge bosons in the coset space SU(3)/SU(2) v [7]. to form the longitudinal polarization states of the resul- tant massive vector bosons. The resultant vector boson masses are g3(v)v, where g3 g3(µ) is the running B. Induced Dynamical Breaking of SU(3) to SU(2) SU(3) gauge∝ coupling at the scale≡µ = v. The sixth com- ponent of the φ field forms a physical Higgs boson with a In this subsection we discuss how one can produce the mass √λ v. This is a singlet under the residual SU(2) breaking of the SU(3) symmetry to SU(2) in a dynami- ∼ gauge interaction. cal manner. For Gb we choose the smallest non-Abelian As noted above, we assume that this breaking occurs Lie group, SU(2)b, so that GUV = SU(3) SU(2)b, in ⊗ at a scale v that is large compared with the scale where the notation of Eq. (1.1). To the set of fermions f 2 { } the running SU(3) gauge coupling α3(µ) = g3(µ) /(4π) transforming vectorially under SU(3) we add the follow- would have grown to O(1) and the theory would thus have ing chiral fermions (where a and α denote SU(3) and become strongly coupled. This assumption is necessary SU(2)b gauge indices, respectively and the numbers in for this model to fall under the class of theories that are parenthese denote the dimensionalities of the representa- aα α considered in this paper. If one were to relax this as- tions of GUV ): (i) ζL : (3, 2); (ii) ηL : (1, 2); and a sumption, the analysis would become more complicated, (iii) χp,R : (3, 1) with p = 1, 2. This set of fermions because one would not be able to perform a perturbative is similar to the set that one of us used in Ref. [6]. analysis of the Higgs sector. (For a recent discussion of Since the SU(2)b gauge interaction is asymptotically free, this strongly coupled case and further references, see [6]). as the reference energy scale µ decreases from large val- Below the scale v, the resultant SU(2) theory would have ues, the running coupling αb(µ) increases. The SU(2)b- a fermion sector consisting of the SU(2)-nonsinglet com- nonsinglet fermions comprise four chiral Weyl fermions ponents of the original SU(3) fermion sector, together or equivalently, two Dirac fermions. This is well below with the SU(2) gluons, with a gauge coupling inherited the estimated critical number Nf,cr 8 beyond which ∼ from the original SU(3) theory. This SU(2) theory would the SU(2)b theory would evolve into the infrared in a then evolve further into the infrared. With a sufficiently chirally symmetric manner [8]. Therefore, we can con- small fermion content f , the SU(2) coupling would clude that as µ decreases to the scale µ = Λb such that { } eventually increase to O(1) at a lower scale Λ , where αb(µ) O(1), the SU(2)b interaction produces bilinear 2 ∼ the SU(2) interaction would confine and produce bilin- fermion condensates. The most attractive channel for the ear fermion condensates. There would thus be a spec- fermion condensation is 2 2 1. One such condensate aα T ×bβ → trum of SU(2)-singlet meson and (bosonic) baryons, to- is of the form ǫαβ ζ C ζ , where ǫαβ is the antisym- h L L i gether with glueballs (which would mix with the mesons metric tensor density for SU(2)b. This is automatically to produce mass eigenstates) at this lower scale Λ2. antisymmetrized in the SU(3) indices a,b and hence is There are several properties of this Higgs mechanism proportional to that will be contrasted with the induced dynamical ǫ ǫ ζaα T C ζbβ , (3.2) breaking mechanism to be discussed next. First, a priori, h abc αβ L L i one has the freedom to choose the coefficient µ2 in the where ǫabc is the antisymmetric tensor density for SU(3). Higgs potential (3.1) to be positive or negative. Since one The condensate (3.2) transforms as conjugate fundamen- wants to construct the Higgs mechanism to break SU(3), tal (3)¯ representation of SU(3) and therefore dynamically 2 one chooses µ < 0, but this sign choice could be con- breaks SU(3) to an SU(2) subgroup. A second conden- sidered to be ad hoc, since one does not give any deeper sate formed by the SU(2)b interaction is explanation for this choice. Second, the Higgs mechanism predicts physical pointlike Higgs particle(s), whereas in a aα T β ǫαβ ζL CηL . (3.3) dynamical mechanism, although the Gb interaction leads h i to various Gb-singlet bound states, including some with This transforms as a fundamental representation of SU(3) angular momentum J = 0, the properties of these states and hence also breaks it to an SU(2) subgroup. One can are not, in general, the same as those of a Higgs par- use vacuum alignment arguments [9] to infer that these ticle. Third, as is well known, this potential is unsta- SU(2) subgroups are the same. Then, without loss of ble to large loop corrections and is thus sensitive to the generality, one may choose the index c = 3 in the con- nature of the ultraviolet completion of the theory (i.e., densate (3.2) and a = 3 in the condensate (3.3). The has a hierarchy problem). A fourth and related point is residual SU(2) subgroup preserved by these condensates that the Higgs sector is not asymptotically free, i.e., the is thus the one generated by Ta, a =1, 2, 3 in SU(3). The a α α beta function for the quartic coupling, dλ/dt, is positive, fermions ζL and ηL with a = 1, 2, 3, α = 1, 2 involved 4 in these condensates gain dynamical masses of order Λb naturally explained in the Standard Model Higgs mecha- and are integrated out of the low-energy effective field nism and also in technicolor theories. One is led, then, to theory that is operative as scales µ< Λ2. The two copies ask how general this property is in quantum field theory. a of χp,L decompose as two doublets under the resultant That is, can one construct a model that exhibits dynami- SU(2) for a =1, 2 (while the a = 3 components form two cal breaking of a vectorial gauge symmetry? Clearly, this singlets). The fermion content of this low-energy SU(2) requires more than one gauge interaction to be present, theory thus consists of these two doublets, together with since if one has just a single vectorial gauge interaction the SU(2)-nonsinglet components of the set f . With and it becomes strongly coupled and produces conden- { } an asymptotically free SU(2), the coupling α (µ), which sates, then the most attractive channel is Ri R¯i 1 2 × → is inherited from the weakly coupled SU(3) theory will for the one or more fermion representations Ri in the increase as µ decreases below Λb, and if the fermion con- theory, so it does not self-break [10, 11]. tent is sufficiently small so that α2(µ) grows to O(1) at Let us thus consider a theory with the same gauge a lower scale Λ2, the SU(2) gauge interaction will con- group, GSM , but make two changes: (i) first, we re- fine and form bilinear fermion condensates at this scale. move the usual breaking of SU(2)L at the 250 GeV Given that α2 is small at the scale Λb and increases only scale, and (ii) we arrange the values of the gauge cou- logarithmically, it follows that Λ2 << Λb. plings so that at a scale Λ2 considerably larger than If one were to turn off the SU(3) gauge interaction, ΛQCD, where SU(3)c (and U(1)Y ) are weakly coupled, the SU(2)b theory would have a classical U(4) or equiv- the SU(2)L interaction becomes strongly coupled, with 2 alently SU(4) U(1) global chiral symmetry. The U(1) α2(Λ2) = g(Λ2) /(4π) of order unity. The SU(2)L sec- ⊗ is broken by SU(2)b instantons, so that the actual non- tor contains Ngen(Nc + 1) = 12 chiral fermion doublets anomalous global chiral symmetry would be the SU(4) (where Ngen denotes the number of SM fermion gener- a α α (generated by global transformations of the ζL and ηL ations), so that the SU(2)L gauge interaction is asymp- among each other for a fixed α.) The bilinear condensates totically free, with leading coefficient would break this to Sp(4), with the resultant appearance of five Nambu-Goldstone bosons. Turning on the SU(3) 1 (b1)SU(2)L = [22 (Nc + 1)Ngen] . (3.4) gauge interaction explicitly breaks this global symmetry, 3 − although the breaking is weak, since α3 is small. Given that there is no breaking of GEW , the fermions We contrast this dynamical breaking with the corre- are massless, so they all contribute to the SU(2)L beta sponding Higgs mechanism presented above. First, the function. To illustrate this dynamical breaking in the dynamical mechanism is more predictive, in the sense simplest context, we assume Ngen = 1, so that there that once one has specified the gauge interaction Gb and are four chiral SU(2)L doublets, or equivalently, Nf =2 the fermion content, the resulting fermion condensation Dirac doublets. This is well within the phase in which and symmetry breaking follow automatically; one does SU(2)L confines and spontaneously breaks global chiral not have to make an ad hoc choice of a parameter, as symmetry. The model thus contains one family of SM 2 a one does with the choice µ < 0 in the Higgs potential ai u a a i νe fermions: Q = a , u , d , L = , and e , (3.1). Second, the theory does not suffer from a hier- L d L R R L e L R where a and i are SU(3) and SU(2) gauge indices, re- archy problem, i.e., is not sensitively dependent on an c L spectively. ultraviolet completion, in contrast to the Higgs mecha- The most attractive channel for the strongly coupled nism. Third, by construction, both the SU(3) and the SU(2) interaction is 2 2 1, and it produces several SU(2) sectors are asymptotically free, again in contrast L b condensates in this channel.× → The first of these is of the with the Higgs mechanism, in which the quartic coupling form ǫ Qa,k T CQb,ℓ , where ǫ is the antisymmetric is not asymptotically free. h kℓ L L i kℓ tensor density for SU(2)L. This is automatically anti- symmetric in SU(3)c indices and hence is proportional to C. Induced Breaking of SU(3)c in a Modified Standard Model ǫ ǫ Qa,k T CQb,ℓ =2 ǫ ua T Cdb , (3.5) h abc kℓ L L i h abc L Li
Here we discuss another way to break an SU(3) gauge This transforms as a (3 3)as = 3¯ under SU(3)c (where symmetry dynamically. In this case we will take the the subscript as stands× for antisymmetric) and hence SU(3) to be the color SU(3)c group of the Standard breaks SU(3)c to a subgroup SU(2)c. It also breaks Model. The point here is that with a modification of the U(1)Y . As is clear from the fact that electric charge properties of the Standard Model, color SU(3)c would, in satisfies Q = T3 +(Y/2) and the fact that the condensate fact, be dynamically broken by the SU(2)L gauge interac- (3.5) is invariant under SU(2)L, it also violates electric tion. Our analysis also addresses a fundamental question charge invariance. Without loss of generality, we choose in particle physics. One of the profound properties of the breaking direction of SU(3)c as the third direction, a a nature is the fact that it is the chiral part of GSM that is so that the uL and dL quarks with color indices a =1, 2 broken, leaving as a residual exact subgroup a symmetry occur in the condensate (3.5) and hence gain dynamical that is vectorial, namely H = SU(3) U(1) . This is masses of order Λ . c ⊗ em 2 5
a a The strong SU(2)L interaction would also produce the transformations that mix up the uR and dR fields (for condensate fixed a). The U(1) is broken by SU(2)c instantons, so that the actual non-anomalous global chiral symmetry a,k T ℓ a T a T ǫkℓQ CL = u CeL d Cνe,L . (3.6) is SU(2). In general, an SU(2) gauge theory with N h L Li h L − L i d massless chiral Weyl fermions transforming according to This also breaks SU(3) to an SU(2) subgroup and vio- c c the fundamental representation (with Nd = 2k even to lates hypercharge and electric charge. As in our discus- avoid a global Witten anomaly) has an SU(2k) global sion above, a vacuum alignment argument can be used to chiral symmetry corresponding to transformations that infer that the condensate (3.6) breaks SU(3)c to the same mix up the 2k chiral doublets. Formation of condensates SU(2)c as the condensate (3.5), so that the color index involving these doublets breaks this global symmetry to in Eq. (3.6) has the value a = 3. This SU(2)c is the one Sp(2k). Since the orders of these groups are 4k2 1 generated by the color generators (Ta) with a = 1, 2, 3. and k(2k + 1), respectively, this entails the breaking− of Thus, this model is of the form in Eqs. (1.1) and (1.2) 2k2 k 1 generators of SU(2k), and the resultant ap- with pearance− − of this number of massless Nambu-Goldstone bosons. In this SU(2) theory, there are N = 2 chiral G = SU(3) U(1) , G = SU(2) , H = SU(2) . c d c ⊗ Y b L c fermions, i.e., k = 1, so the SU(2) global chiral symmetry is equivalent to Sp(2), and there is no chiral symmetry (3.7) breaking due to the formation of the condensate (3.10). In addition to breaking these gauge symmetries, the con- densate (3.5) breaks baryon number by ∆B =2/3, while the condensate (3.6) breaks B by ∆B = 1/3 and lep- a a It is also worthwhile to comment on the situation con- ton number L by ∆L = 1. The quarks uL, dL with cerning global chiral symmetry at the higher scale, above a = 1, 2, 3 and the leptons eL, and νe,L involved in Λ2. In the present model with its one generation of SM these condensates gain dynamical masses of order Λ2. fermions, if one turns off the SU(3)c and U(1)Y cou- (The actual mass eigenstates involve linear combinations plings, then, an an energy above Λ2, the SU(2)L theory of these fields.) Similarly, the five gluons in the coset has a non-anomalous global SU(Nd) symmetry, where SU(3)c/SU(2)c corresponding to the broken generators Nd = Nc + 1 = 4. The condensates (3.5) and (3.6) of SU(3)c gain dynamical masses of order break this to Sp(4), leading to the appearance of five Nambu-Goldstone bosons. Since the NGBs couple in a m g (Λ )Λ (3.8) g ∼ 3 2 2 derivative manner, their scattering amplitudes are sup- pressed by powers of center-of-mass energy √s/Λ2 and and the abelian U(1)Y gauge boson B gains a mass hence they are progressively more weakly coupled as the ′ mB g (Λ2)Λ2 . (3.9) energy scale decreases further below Λ2 [13]. Turning ∼ on the SU(3)c and U(1)Y couplings explicitly breaks the Since by our assumptions, SU(3)c and U(1)Y are weakly global SU(4) symmetry, but also the would-be NGBs are coupled at this scale, the masses of these five gluons and absorbed to form the longitudinal components of the five of the one B boson are smaller than the dynamically vector bosons in the coset SU(3)c/SU(2)c. This process is produced fermion masses. reminiscent of the mechanism by which technicolor gives Of the quarks and leptons in this Ngen = 1 model, masses to the W and Z bosons. all of the components of the Nc + 1 = 4 SU(2)L dou- blets are involved in the condensates (3.5) and (3.6) and gain dynamical masses of order Λ2. These fermions are thus integrated out of the low-energy effective theory be- Our analysis here answers the question that we posed at the beginning of this subsection concerning the break- low Λ2. The SU(2)c gauge symmetry of this low-energy effective field theory remains exact. The content of non- ing of a vectorial, as contrast to a chiral, gauge sym- a metry. Our answer is that it is certainly possible for a singlet fermions in this low-energy theory consists of uR and da with a = 1, 2, which form two Weyl fermions, vectorial gauge symmetry to be broken, if this break- R ing is induced by another strongly coupled interaction. or equivalently, one Dirac fermion. The SU(2)c gauge The reason that SU(2) does not break SU(3) in nature coupling α2c is inherited from the SU(3)c theory and is L c is a consequence of the fact that SU(2) is broken well small at Λ2, but eventually grows to O(1) at a much lower L above the scale where its coupling would have become Λ2c << Λ2. At this lower scale Λ2c, the SU(2)c theory confines and produces a bilinear fermion condensate, large enough to produce the condensates (3.5) and (3.6). The resultant W and Z are massive and weakly coupled a T b ǫab uR CdR , (3.10) and their interactions are too weak to induce such con- h i densates. Indeed, even if the SU(2)L symmetry were not where here ǫab is the antisymmetric tensor density of broken at this higher scale, it would be broken by the SU(2)c. This SU(2)c theory has a classical U(2), or equiv- quark condensates at the QCD scale, as discussed above, alently, SU(2) U(1) global chiral symmetry defined by before it could become strong enough to break SU(3) . ⊗ c 6
IV. INDUCED DYNAMICAL BREAKING OF A basis. The corresponding components φa are Nambu- GAUGE SYMMETRY BY ADJOINT FIELDS: AN Goldstone bosons and are absorbed to become the longi- ILLUSTRATIVE MODEL WITH G = SU(3) tudinal components of the massive vector bosons. Four physical Higgs fields remain, with masses √λ v. Of ∼ A. Higgs Mechanism these, φa, a = 1, 2, 3 transform as the adjoint represen- tation of the residual SU(2) gauge interaction and, as- We next consider induced breaking of a gauge sym- suming that it confines, they are thus confined in SU(2)- metry by fields that transform as the adjoint represen- singlet bound states. Since we have assumed that the tation of the gauge group. In this section we discuss SU(2) theory is weakly coupled at the scale µ = v, SU(3) because of some special properties that it has, and since its coupling increases only logarithmically, the and in the next section we discuss SU(N) for general SU(2) confinement scale Λ2 is much smaller than v. In N 4. We begin by constructing a Higgs mechanism passing, we note that although a Higgs VEV of the form ≥ φ = diag(a, b, (a + b)), with a = b is, a priori, for this breaking. We assume that the theory contains − | | 6 | | a Higgs field φ transforming according to the adjoint possible, and would break SU(3) to U(1) rather than SU(2) U(1), it does not minimize the Higgs potential. (i.e., octet) representation of SU(3), with an appropri- ⊗ ate Higgs potential. We will write the components of φ as φi , 1 i, j 3; these are subject to the trace con- j ≤ ≤ dition Tr(φ) = 3 φi = 0. In general, when using the B. Dynamical Breaking Mechanism with Adjoint i=1 i Fields adjoint representationP of SU(N), in addition to the no- i tation φj with 1 i, j N, it will also be convenient ≤ ≤ 2 To study the dynamical breaking of the SU(3) sym- to use an equivalent notation φa, with 1 a N 1, that indicates the 1–1 correspondence with≤ the≤ N 2− 1 metry by an SU(Nb) gauge interaction, we must choose i − a value of Nb and a requisite sector comprised of one or generators Ta of SU(N). Thus the φj form the entries of 2 more fermion fields that transform as nonsinglets under √ N −1 a matrix given by 2 a=1 φaTa. both G = SU(3) and Gb = SU(Nb). For our model we We will require thatP the Higgs part of the Lagrangian choose a chiral fermion that transforms as an adjoint of be invariant under the replacement φ φ. It fol- SU(3) and a fundamental representation of SU(Nb): lows that the Higgs potential contains→ only − quadratic i α and quartic terms in φ. For a general SU(N) theory (ψj,L) : (8,Nb) , (4.5) with a Higgs field in the adjoint representation, there are two independent quartic terms, proportional to [Tr(φ2)]2 where the numbers in parentheses are the dimensions of 4 and Tr(φ ). For the special values N = 2 or N = 3, the representation under GUV = SU(3) SU(Nb), the 2 2 4 ⊗ [Tr(φ )] = 2Tr(φ ), so there is only one independent indices i, j are SU(3) indices, and α = 1, ..., Nb is an i α quartic term. For the present case of SU(3), the Higgs SU(Nb) index. Because (ψj,L) transforms according to potential may thus be written as a self-adjoint representation of SU(3), it does not con- tribute any gauge anomaly to the SU(3) theory. We take µ2 λ V = Tr(φ2)+ [Tr(φ2)]2 . (4.1) Nb = 2, the minimal value, so Gb = SU(2)b. As above, 2 4 we will also use the equivalent notation φa,L, 1 a 8. i α ≤ ≤ Here we take µ2 < 0 to get the symmetry breaking. This The (ψj,L) form the components of a matrix given by 8 α potential is minimized for a Higgs field vacuum expecta- √2 a=1 ψa,LTa. tion value (VEV) of the form TheP SU(2)b gauge interaction is asymptotically free, with the leading beta function coefficient b1 = 14/3 (see φ = T8 v (4.2) Appendix II for notation). The fermion (ψi )α amounts h i j,L to 8 Weyl doublets, or equivalently, 4 Dirac doublets, of where v can be made real by a global rephasing of φ SU(2). Since this number is well below the estimated crit- and T8 is the second member of the Cartan subalgebra ical value Nf,cr 8 separating the (zero-temperature) of SU(3), phase with confinement≃ and spontaneous chiral symme- try breaking from the chirally symmetric phase [8], we 10 0 1 can conclude that the SU(2) interaction confines and pro- T = 01 0 (4.3) 8 duces bilinear condensates. These occur in the most at- 2√3 0 0 2 − tractive SU(2)b channel (MAC), which is 2 2 1, with a condensate of the form ǫ (ψi )α T C(ψ×k )→β , where The VEV (4.2) breaks SU(3) according to the pattern h αβ j,L ℓ,L i here ǫαβ is the antisymmetric tensor density for SU(2)b. SU(3) SU(2) U(1) . (4.4) The ǫαβ contraction antisymmetrizes the bilinear fermion → ⊗ product, so that in the full Clebsch-Gordan decomposi- Since SU(3) has order eight, while SU(2) U(1) has order tion, four, there are four broken generators of⊗ SU(3), namely the T with a = 4, 5, 6, 7 in the standard Gell-Mann 8 8=1 +8 +8 + 10 + 10 + 27 (4.6) a × s s a a a s 7
(where the subscripts s and a denote symmetric and an- V. INDUCED BREAKING OF AN SU(N) tisymmetric combinations), the above condensate must SYMMETRY BY ADJOINT FIELDS be one of the antisymmetric products, namely 8a or 10a. We next use a vacuum alignment argument [9], according A. General to which the symmetry breaking should preserve as large a subgroup symmetry as possible. Now, relative to the In this section we carry out a comparative study of a maximal subgroup SU(2) U(1), an octet of SU(3) has ⊗ Higgs mechanism versus dynamical breaking of an SU(N) the decomposition gauge symmetry with N 4 by fields transforming ac- cording to the adjoint representation≥ of this group. Two 8SU(3) =30 +21 +2−1 +10 , (4.7) general types of breaking patterns of the SU(N) sym- where the numbers on the right-hand side are the dimen- metry will be relevant. Both of these involve breaking sionalities of the SU(2) representations and the subscripts to a maximal subgroup of SU(N), with the same rank are the hypercharges with the Gell-Mann normalization. (dimension of the Cartan subalgebra of mutually com- In contrast, the decuplet has the decomposition muting generators) as SU(N), namely N 1. However, these subgroups have different orders (numbers− of gen- 10SU(3) =41 +30 +2−1 +1−2 . (4.8) erators). The first of these symmetry-breaking patterns Of these, only the octet contains a piece that is a singlet is under SU(2) U(1). Using a vacuum alignment argu- ⊗ ment, we therefore can infer that the condensate trans- SU(N) SU(N 1) U(1) . (5.1) → − ⊗ forms as the 10 piece of the octet of SU(3) and hence has the form The residual symmetry group has order ǫ (ψi )α T C(ψj )β (T )i Λ3 . (4.9) h αβ j,L ℓ,L i∝ 8 ℓ b o SU(N 1) U(1) = (N 1)2 (5.2) In the equivalent notation using ψα with 1 a 8, h − ⊗ i − a,L ≤ ≤ the condensate has the form ǫ f (ψ )α T C(ψ )β , so the symmetry reduction in Eq. (5.1) involves the h αβ ab8 a,L b,L i where the fabc are the structure constants of the SU(3) breaking of Lie algebra. The nonzero structure constants fab8 with ∆o = 2(N 1) (5.3) a
As we will discuss further below in the context of dy- N 2 1 = − =2k(k + 1) , (5.9) namical symmetry breaking, a vacuum alignment argu- 2 ment prefers a symmetry-breaking pattern that yields the largest residual symmetry. The size of the subgroup that so that (5.8) involves the breaking of constitutes the residual symmetry can be characterized by its rank and order. All of the patterns above sat- N 2 1 ∆o = − =2k(k + 1) (5.10) isfy the condition that the rank of the residual symme- 2 try group should be maximal, i.e., the same as that of SU(N), namely N 1. Concerning the differences in the generators of SU(N). The symmetry breaking patterns − (5.5) and (5.8) can be expressed in a unified manner as orders of the various possible subgroups resulting from the symmetry breaking of SU(N), we calculate, for even SU(N) SU(N ℓ) SU(ℓ) U(1) , (5.11) N =2k, the difference → − ⊗ ⊗ where ℓ = [N/2]ip and [ν]ip denotes the integral part of the real number ν.
(N 2)2 o SU(N 1) U(1) o SU(N/2) SU(N/2) U(1) = − = 2(k 1)2 . (5.12) h − ⊗ i − h ⊗ ⊗ i 2 − This difference is positive semidefinite, and positive-definite for k 2, i.e., N 4. For odd N =2k + 1, ≥ ≥ (N 1)(N 3) o SU(N 1) U(1) o SU((N + 1)/2) SU((N 1)/2) U(1) = − − =2k(k 1) . (5.13) h − ⊗ i − h ⊗ − ⊗ i 2 −
This difference is also positive semidefinite, and positive- where we take µ2 < 0 to produce the symmetry breaking. definite for k 2, i.e., N 5. Hence, as these calcu- Since Φ is a hermitian matrix, it can be diagonalized by lations show, a≥ vacuum alignment≥ argument prefers the a unitary transformation. It follows that one can write breaking pattern (5.1) for both even and odd N 4. The ≥ i i special case N = 3 has been analyzed above, and leads Φj = δj φj (no sum on j) , (5.15) to breaking of the SU(3) group to SU(2) U(1), which is also of the form (5.1) with N = 3. × for 1 i, j N. Substituting Eq. (5.15) into Eq. (5.14) gives ≤ ≤ There are other symmetry-breaking patterns that could, a priori, occur. SU(N) could, in principle, break µ2 N λ N λ N to a non-maximal subgroup, i.e., a subgroup with rank V = φ2 + 1 ( φ2)2 + 2 φ4 . (5.16) 2 j 4 j 4 j smaller than the rank of SU(N), namely N 1. For Xi=j Xj=1 Xj=1 example, in principle SU(3) could, a priori −break to U(1), SU(4) could break to SU(2) U(1), and so forth. Since Tr(Φ) = 0, the φj satisfy the condition However, in the context of the Higgs⊗ mechanism, these symmetry-breaking patterns do not occur as minima of N the Higgs potential, and in the dynamical symmetry φj =0 . (5.17) breaking context, they are disfavored by vacuum align- Xj=1 ment arguments. Hence, φ only involves N 1 independent fields, and V only depends on N 1 of− the components φ , which we − j take to be φj with j =1, ..., N 1. Now B. SU(N) Breaking with an Adjoint Higgs Field − [Tr(Φ2)]2 Tr(Φ4) , (5.18) ≥ First, we discuss the mechanism for breaking an SU(N) as can be seen from the explicit expression for the differ- gauge symmetry with a Higgs field Φ in the adjoint rep- ence, resentation [14]. The components of the Higgs field are i 2 2 4 denoted Φj . We impose a Φ Φ symmetry. Then the [Tr(Φ )] Tr(Φ )= Higgs potential has the general→− form − − − N 1 N 1 2 2 2 2 2 µ λ λ 2 φ φ + φ φj V = Tr(Φ2)+ 1 [Tr(Φ2)]2 + 2 Tr(Φ4) , (5.14) i j i 2 4 4 1≤i This VEV (5.23) breaks SU(2k + 1) according to (5.8). 0 . (5.19) The value of the potential at the minimum is ≥ 2 2 As noted above, if N is equal to 2 or 3, then [Tr(Φ )] = 4 4 µ 2Tr(Φ ), so that there is only one independent quartic Vmin = − (5.25) 2 term, and its coefficient, (1/4)[λ1 +(λ2/2)], must be pos- N +3 4 λ1 + N(N+1)(N−1) λ2 itive. For N 4, the two quartic terms are independent, and the condition≥ that V be bounded from below requires that λ > 0. 1 It is possible for λ to have a restricted range of nega- For λ > 0, it is again convenient to consider the cases 2 2 tive values [15], of even N = 2k and odd N = 2k + 1 separately. For λ2 > 0 and even N =2k, V is minimized if the VEV of Φ has the form given by N(N 1) − λ < λ < 0 . (5.26) − N 2 3N +3 1 2 v 1 for1 i k For λ in this range,− V is minimized if Φ has the VEV φi = ≤ ≤ . (5.20) 2 h i √2N × 1 for k +1 i 2k n − ≤ ≤ v 1 for1 i N 1 The normalization of v in Eq. (5.20) and in the equa- φi = ≤ ≤ − tions below is determined by the definition Tr(Φ2) = h i 2N(N 1) × (N 1) for i = N − − − (1/2)v2, analogous to the usual normalization condition p (5.27) Tr(TaTb)=(1/2)δab for the generators of SU(N). At this where minimum, one finds 2µ2 2 2 2 2µ v = − . (5.28) v = − . (5.21) N 2−3N+3 λ2 λ1 + N(N−1) λ2 λ1 + N h i The VEV (5.20) breaks SU(2k) according to (5.5). The The VEV (5.27) breaks SU(N) according to Eq. (5.1). value of the potential at the minimum is The value of V at this minimum is µ4 Vmin = − . (5.22) µ4 λ2 Vmin = − . (5.29) 4 λ1 + N h i N 2−3N+3 4 λ1 + N(N−1) λ2 For λ2 > 0 and odd N = 2k + 1 with k 2, V is minimized if the VEV of Φ has the form ≥ Note that all three of the minimal values (5.22), (5.25), k 1/2 and (5.29) have the form φ = v h ii 2(k + 1)(2k + 1) × µ2v2 V = (5.30) min 8 1 for1 i k +1 k+1 ≤ ≤ . (5.23) for k +2 i 2k +1 2 × − k ≤ ≤ for the respective three values of v . The lower limit on the allowed negative range of λ in Eq. (5.26) is evident (The special case k = 1, i.e., N = 3, was dealt with 2 from Eq. (5.29), since in this equation Vmin as above.) The minimization condition determines v ac- λ approaches this lower limit from above. The→ fact −∞ that cording to 2 λ2 = 0 is the boundary between the two types of minima 2µ2 is evident from the difference between the values of the v2 = − . (5.24) minima for even N, N 2+3 λ1 + N(N+1)(N−1) λ2 2 4 (N 2) λ2 µ V 2 V 2 == − − (5.31) min, λ >0 − min, λ <0 4 λ + λ2 N(N 1)λ + (N 2 3N + 3)λ 1 N − 1 − 2 10 and for odd N, 3 4 N (N 1)(N 3) λ2 µ V 2 V 2 = − − − . (5.32) min, λ >0 − min, λ <0 4 N(N 2 1)λ + (N 2 + 3)λ N(N 1)λ + (N 2 3N + 3)λ − 1 2 − 1 − 2 Both of these differences are proportional to λ2, explicitly showing the switch in global minimum as λ2 reverses sign. The reason for the residual U(1) invariance in these where here and below, α is the Gb gauge index. Thus, 2 i α symmetry-breaking patterns obtained with a Higgs field we assign each of the N 1 components of (ψj )L to Φ transforming according to the adjoint representation transform according to the− fundamental representation of SU(N) is that since Φ can be diagonalized, as noted of SU(Nb). The numbers in parentheses in Eq. (5.34) are above, its VEV can be expressed as a linear combination the dimensions of the representations with respect to the of coefficients multipled by the N 1 diagonal Cartan factor groups in Eq. (5.33). The Nb copies of fermions generators of SU(N). Indeed, without− loss of generality, in the adjoint representation of SU(N) contribute zero one can define axes in SU(N) space so that it points en- gauge anomaly to SU(N). As stated earlier, these and tirely along one such Cartan generator, which can be de- the other fermions that we include are taken to have zero noted as T . Then exp(iθT ) commutes with Φ , yield- Lagrangian masses since mass terms would violate the C C h i ing the U(1) invariance. full GUV symmetry, which is chiral. From the formulas for ∆o, the number of broken gen- The choice of the rest of the Gb-nonsinglet fermions erators for the various symmetry-breaking patterns, one in the model depends on the value of Nb. We first con- can immediately infer the number of gauge bosons of sider the possibility that Nb = 2. Now, N is even 2 ⇐⇒ SU(N) that become massive. Thus, for λ2 > 0 and even N 1 is odd. The SU(2)b theory must have an even N =2k, the symmetry breaking (5.5) involves the break- number− of chiral doublet fermions in order to avoid a 2 2 2 2 i α ing of N /2=2k generators, so that of the N 1 (real) global anomaly, so if N is odd, the N 1 (ψj )L form an 2 − − components of Φ, N /2 are absorbed to become the lon- acceptable SU(2)b fermion sector by themselves, while if gitudinal components of the gauge bosons corresponding N is even, then we obtain an acceptable fermion sector to these broken generators, which pick up masses gv. by adding an odd number of additional SU(2)b doublets. The remaining N 2/2 1= 2k2 1 real components∝ We shall choose this odd number to be the minimal value, − − of Φ are physical Higgs bosons. For λ2 > 0 and odd namely, one, with the fermion N = 2k + 1, the symmetry breaking (5.8) involves the 2 α breaking of (N 1)/2=2k(k + 1) generators, so that ωL included for even N. (5.35) of the N 2 1 (real)− components of Φ, (N 2 1)/2 are absorbed to− become the longitudinal components− of the For these two cases, the SU(2)b beta function has as its gauge bosons corresponding to these broken generators. leading coefficient The remaining (N 2 1)/2=2k(k + 1) real components − 1 (23 N 2) for N odd of Φ are physical Higgs bosons. For λ2 < 0, the sym- 3 − metry breaking (5.1) involves the breaking of 2(N 1) b1 = (5.36) − 1 (22 N 2) for N even generators, and an equal number of Nambu-Goldstone 3 − bosons, which are absorbed to become the longitudinal components of the gauge bosons in the coset (5.4). The The requirement that the SU(2)b theory be asymptoti- remaining (N 1)2 real components of Φ are physical cally free is thus that N < √23 for odd N and N < √22 Higgs bosons. − for even N. These amount to the possibilities N = 3 for odd N and N = 2, 4 for even N. We have already dealt with the case N = 3 above, so here we focus on C. Dynamical Mechanism for SU(N) Breaking by the case N = 4. As discussed in the introduction, these an Adjoint Field are necessary but not sufficient conditions; we also must require that the fermion content of the SU(2)b theory is For the analysis of dynamical symmetry breaking of sufficiently small that as the reference energy scale µ de- SU(N) by an adjoint field, we analyze a model of the creases, the coupling αb(µ) will increase sufficiently so form of Eq. (1.1), in which that the SU(2)b gauge interaction will produce bilinear fermion condensates instead of evolving in a chirally sym- G = SU(N) , Gb = SU(Nb) . (5.33) metric manner into the infrared. For SU(2), the critical number of Dirac fermions, Nf,cr, below which this con- For the fermions that transform under both SU(N) and densation will occur is estimated to be Nf,cr 8 [8]. ≃ Gb we use Because SU(2) has only (pseudo)real representations, we can rewrite the theory with a given number of chiral Weyl (ψi )α : (N 2 1,N ) , (5.34) doublets as a theory with half this number of Dirac dou- j L − b 11 2 blets. For N = 4, we would have N = 16 chiral dou- value of Nb, the fermion content of the SU(Nb) sector blets, or eight Dirac doublets, which is marginal. Assum- must be small enough so that as the theory evolves down ing that the SU(2)b sector does, indeed, produce bilinear in energy scale, it produces condensates instead of evolv- fermion condensates, these would occur in the most at- ing into the infrared in a chirally symmetric (conformal) tractive channel, which is 2 2 1 in SU(2) . These manner. For a vectorial asymptotically free SU(N) gauge × → b would have either the form theory with Nf copies of Dirac fermions (with zero La- grangian masses) in the fundamental representation, if α T β ǫαβ[ψ Cψ ]as (5.37) N is smaller than a critical value, N , then as the h a,L b,L i f f,cr reference scale decreases from large values, the coupling or the form will eventually grow large enough to form condensates α T β which generically break the global chiral symmetry. In ǫαβψ Cω , (5.38) h a,L Li contrast, if Nf >Nf,cr then the theory will evolve from the ultraviolet to the infared without any spontaneous where in Eq. (5.37) the symbol [...] means an anti- as chiral symmetry breaking, yielding conformal behavior. symmetric SU(4) combination of the two adjoint fermion A combined analysis of the beta function and solutions fields. In both cases, the condensate thus transforms as of the Dyson-Schwinger equation for the fermion propa- an adjoint of SU(4). A vacuum alignment argument im- gator in the approximation of one-gluon exchange yields plies that the condensates form in such a way as to pre- the result [8] serve the largest subgroup in SU(4). The order of the 2 subgroup SU(3) U(1) is 9, which is greater than the 2Nb(50N 33) N = b − . (5.42) order of the subgroup⊗ SU(2) SU(2) U(1), which is f,cr 2 5(5Nb 3) 7. Hence, from a vacuum alignment⊗ argument,⊗ one may − Although the Dyson-Schwinger analysis does not directly infer that the condensate is proportional to the SU(4) incorporate either effects of confinement or instantons, it generator T = (2√6)−1 diag(1, 1, 1, 3), leading to the 15 has been shown that these two effects affect N in op- N = 4 special case of the symmetry-pattern− pattern f,cr posite ways, so that neglecting both of them can still yield (5.1). a reasonably accurate result [16]. Recent lattice simula- We next consider possible values N 3 for the gauge b tions of SU(3) gauge theory with variable numbers N group symmetry SU(N ) responsible for≥ the dynamical f b of light fermions in the fundamental representation are breaking of SU(N). In this case, for the rest of the G - b (taking account of theoretical uncertainties in both Eq. nonsinglet fermions we choose (5.42) and the lattice work) broadly consistent with Eq. α 2 ¯ (5.42) [17]. Although this does not test the prediction for ωp,L : (N 1)(1, Nb) , (5.39) − Nb = 3, it makes it plausible that this prediction could also6 be reasonably accurate. For these values of N , Eq. where the notation N¯ means the conjugate fundamental b b (5.42) rapidly approaches the asymptotic large-N form representation and here the copy number takes on the b N 4N . We thus require that N is sufficiently values 1 p N 2 1. This ensures that the SU(N ) f,cr b b b large that≃ the SU(N ) theory with its N = N 2 1 Dirac theory has≤ zero≤ gauge− anomaly. With the fermions (5.34) b f fermions will exhibit spontaneous chiral symmetry− break- and (5.39) (and with the SU(N) interaction taken as neg- ing and confinement instead of evolving down in energy ligibly weak), the SU(N ) theory is vectorlike. This is in b in a chirally symmetric non-Abelian Coulomb (confor- accord with one of the conditions that we imposed above, mal) phase. Using the prediction of Eq. (5.42), we thus which guarantees that the G symmetry does not self- b obtain the lower bound N 4N >N 2 1, i.e., break when it becomes strongly coupled. Expressed in f,cr ≃ b − manifestly vectorial form, it has N 2 1 Dirac fermions (N 2 1) − N > − . (5.43) transforming according to the fundamental representa- b 4 tion of SU(N ). b With the fermion content as specified via Eqs. (5.34) The beta function for the SU(N ) coupling has leading b and (5.39), and in the approximation that one turns off coefficient the SU(N) gauge interaction, the SU(Nb) sector has a 2 1 2 classical global symmetry of the form U(N 1)ψ (b1)SU(N ) = [11Nb 2(N 1)] . (5.40) 2 2 −2 ⊗ b 3 − − U(N 1)ω, or equivalently, SU(N 1)ψ SU(N 1)ω U(1) − U(1) , where the subscripts− indicate⊗ which− fields⊗ ψ ⊗ ω The requirement that the SU(Nb) theory be asymptoti- are involved in the respective symmetry transformations. cally free is thus Both the U(1)ψ and U(1)ω are broken by SU(Nb) instan- tons, but the linear combination corresponding to the 2 2(N 1) difference of the currents for the ψ and ω fields is con- Nb > − . (5.41) 11 served in the presence of instantons. We will denote this symmetry as U(1)′. The actual (non-anomalous) global As noted above, this is a necessary, but not sufficient, symmetry of the Gb theory at the high scale is thus condition for the SU(Nb) theory to produce the requi- site condensates. We must also require that, for a given SU(N 2 1) SU(N 2 1) U(1)′ . (5.44) − ψ ⊗ − ω ⊗ 12 We comment on this further below. the form that preserves the largest residual symmetry, Now we turn on the SU(N) gauge interaction. This SU(N 1) U(1). These MAC and vacuum alignment explicitly breaks the above global chiral symmetry. How- properties− would⊗ be manifest if one were to explicitly cal- ever, just as the breaking of chiral SU(2)L SU(2)R sym- culate the effective potential for the composite operator metry in QCD by electroweak interactions⊗ is weak, so represented by the condensate, along the lines of Ref. also here this breaking is weak, since αG is small at the [18]. In this context, one may recall that the Higgs po- scale Λb. We can fix the initial value of αb(µ) at a high tential was partially motivated by the original Ginzburg- value of µ so that as µ decreases to the scale Λb, this cou- Landau free energy functional in phenomenological mod- pling grows sufficiently large to produce bilinear fermion els of superconductivity, and retrospectively, from the condensates. These condensates will occur in the most perspective of the Bardeen-Cooper-Schrieffer theory and attractive channel, which, for the above fermion content, the Cooper pair condensate, one may view the Ginzburg- is Nb N¯b 1. In general, these condensates would be Landau free energy functional as an approximate way × → α T of the form ψa,L Cωp,α,L . A vacuum alignment argu- to represent the physics of this Cooper pair condensate. ment impliesh that these condensatesi will form in a man- This is, of course, not a precise isomorphism, but only a ner such as to preserve the largest residual gauge sym- partial correspondence. As recalled above, there are im- metry. We regard this implication as very plausible, but portant differences between a Higgs and dynamical mech- add the obvious caveat that one must remember the the- anism for breaking a gauge symmetry. To the extent that oretical uncertainties that are present in such a strongly one may regard a Higgs potential as embodying some of coupled theory. Combining this implication from the vac- the same physics as an effective potential for a composite uum alignment argument with our discussion above, we operator represented by bilinear fermion condensate(s), infer that the symmetry-breaking pattern is that SU(N) one may observe that the pattern of symmetry break- breaks to the maximal subgroup SU(N 1) U(1) as in ing inferred from the dynamical approach makes definite Eq. (5.1), so that the condensate would− have⊗ the form predictions for the coefficients in the corresponding Higgs potential. First, because the Lagrangian in the dynam- α T 2 ψa,L Cωp,α,L with a = N 1 . (5.45) ical model is invariant under the separate global trans- h i − α α α α formations ψa,L ψa,L and ωp,L ωp,L, it follows That is, it would transform like the last of the generators that an analogous→ effective − potential→ for − the condensate in the Cartan algebra of SU(N), (5.45) should not contain odd powers of this condensate. i Our dynamical model for the symmetry breaking of an (T 2 ) = a=N −1 j SU(N) gauge theory using fermions transforming as an adjoint representation of SU(N) then predicts that in 1 i 1 for1 i N 1 δj ≤ ≤ − a corresponding Higgs approach, in order to obtain the 2N(N 1) (N 1) for i = N − − − same pattern of symmetry breaking, the coefficients of p (5.46) the Higgs potential should have the following properties: (i) µ2 < 0, for symmetry breaking; (ii) λ < 0, yielding Here we emphasize an important contrast between this 2 the specific symmetry-breaking pattern (5.1); and the dynamical symmetry breaking mechanism and the Higgs stability properties that (iii) λ > 0 and (iv) λ satisfies mechanism. In our introductory discussion above, we 1 2 the lower bound in Eq. (5.26). have already noted a number of the differences between the Higgs mechanism and a dynamical mechanism for breaking a gauge symmetry. Among other differences, for example, the Higgs mechanism leads to the appear- ance of at least one physical pointlike Higgs field, whereas With the symmetry-breaking pattern as given by (5.1), a dynamical mechanism does not yield such a particle there are then 2(N 1) broken generators of SU(N), and (although it may yield composite J = 0 bound states). Nambu-Goldstone− modes formed from the fermion con- Furthermore, if one uses a Higgs mechanism to break densates are absorbed by the gauge bosons correspond- SU(N), then by appropriate choices of the parameters, ing to these broken generators, forming the longitudi- one can guarantee that the minimim of the potential oc- nal components of the resultant massive vector bosons. curs for a Higgs VEV of the form (5.20) or (5.23), so These masses are of order gΛb. This is reminiscent of the that the symmetry breaking is of the type (5.5) or (5.8), process whereby Nambu-Goldstone modes in the techni- rather than (5.1). However, in the dynamical approach color mechanism for electroweak symmetry breaking are to SU(N) breaking, once one specifies the gauge and absorbed to give the W ± and Z bosons their masses. fermion content, there are no free parameters, and the The SU(Nb)-nonsinglet fermions involved in the conden- theory is, in principle, completely predictive. Although sate (5.45) gain dynamical masses of order Λb and are the dynamical symmetry-breaking mechanism involves a integrated out of the low-energy effective theory that is strongly coupled gauge sector, one can use most attrac- operative at scales µ below Λb. Since, by construction, tive channel criteria and vacuum alignment arguments to the SU(Nb) theory confines, the spectrum of the SU(Nb) make a plausible inference about what form the bilinear theory includes a set of SU(Nb)-singlet mesons, baryons, fermion condensate will take, namely, as discussed above, and glueballs that form at the scale Λb. 13 VI. REMARKS ON OTHER DIRECTIONS OF and gauge coupling. Indeed, such an induced GUT sym- STUDY metry breaking scenario appears problematic, since in or- der to produce the requisite bilinear fermion condensates, We comment here on some other related directions of the Gb interaction would necessarily have to confine, and study that could be interesting to pursue. One could this would generically lead to stable Gb-singlet baryons construct models with dynamical symmetry breaking of with masses of order MGUT . With plausible estimates other gauge symmetries and compare results with those for the relevant reaction cross sections, one finds that obtained via Higgs scenarios. An example of this would these Gb-singlet baryons would contribute far too much be models with extended electroweak gauge groups such to the dark matter in the universe [26]. Interestingly, even if a dynamical approach to breaking a GUT sym- as G = SU(3)c SU(2)L SU(2)R U(1)B−L and ⊗ ⊗ ⊗ metry were not excluded by its production of excessive G = SU(4) SU(2)L SU(2)R, for which dynamical mechanisms⊗ were presented⊗ in Ref. [19]. In a more dark matter, it would predict that a GUT group such as abstract direction, one could consider groups such as SU(5) would preferentially break to SU(4) U(1) rather than the SM group, SU(3) SU(2) U(1)⊗ . In the G = SO(N). One could also study the breaking of SU(N) c ⊗ L ⊗ Y by fields transforming according to representations other conventional SU(5) GUT, the latter breaking to GSM is than the fundamental and adjoint, such as the rank-2 obtained by a Higgs mechanism with a Higgs field trans- symmetric and antisymmetric tensor representations. forming as the adjoint representation [22]. Modern GUT One could also study situations in which the G gauge theories also make use of string-inspired mechanisms for interaction is not weakly coupled at the scale Λ where the GUT gauge symmetry breaking, including higher- b dimension operators and Wilson lines [24]. the Gb interaction becomes strongly coupled, so that there is generically a combination of self-breaking of G and induced breaking of G by Gb. Indeed, in reason- ably ultra-violet-complete extended technicolor (ETC) VII. CONCLUSIONS theories, the sequential breaking of the ETC gauge sym- metry down to the residual exact technicolor symmetry In conclusion, in this paper we have constructed and typically involves both self-breaking of ETC, which is a analyzed theories with a gauge symmetry in the ultravio- strongly coupled, chiral gauge symmetry, and induced let of the form G G , in which the vectorial, asymptoti- ⊗ b breaking by an auxiliary gauge interaction called hyper- cally free Gb gauge interaction becomes strongly coupled color in [20]. A similar statement applies to ultravio- at a scale where the G interaction is weakly coupled and let completions of topcolor-assisted technicolor models produces bilinear fermion condensates that dynamically that include the necessary additional gauge interactions break the G symmetry. We have compared the results to to produce the required symmetry breakings [3, 21]. those obtained with a Higgs mechanism. There are many Although our study is primarily intended as a com- interesting contrasting properties of these two approaches parison of gauge symmetry breaking by dynamical and to breaking a gauge symmetry. The Higgs mechanism is Higgs mechanisms in a general field theoretic context, perturbative, and one has the freedom, by appropriate it is appropriate to address the question of possible dy- choices of parameters in the Higgs potential, to determine namical symmetry breaking of a grand unified symme- whether and, in general, how the symmetry breaks. In try. We recall that there has long been interest in grand contrast, the dynamical approach is arguably more pre- unified theories (GUTs) which embed the three factor dictive, in the sense that, provided that one has chosen groups of GSM in a single group, since this would unify the gauge and field content of the Gb sector appropri- quarks and leptons, predict the ratios of the three SM ately, there are no free parameters to vary; the Gb gauge gauge couplings, and quantize electric charge [22]-[24]. interaction will confine and produce fermion condensates Much work on GUTs has been done in a supersymmetric that break the G symmetry. Most attractive channel and context, since supersymmetry remedies the gauge hierar- vacuum alignment arguments provide a plausible guide to chy problem of the Standard Model and since the MSSM enable one to infer which channel(s) have fermion conden- naturally yields gauge coupling unification. There have sation, and what the form of this condensation is, thereby also been studies of the question of whether some type of predicting the resultant pattern of symmetry breaking. grand unification could feasibly be achieved in a theoret- In the dynamical models that we have constructed, we ical context involving dynamical electroweak symmetry produce this breaking by introducing fermions that are breaking [25]. It is natural to ask whether one could use nonsinglets under both G and Gb. In the course of our induced dynamical breaking of a GUT gauge symmetry analysis, we have discussed how the gauge symmetry G such as SU(5) or SO(10), which is weakly coupled at can be broken not just for the case where it is chiral (as the GUT scale, MGUT , using a (vectorial non-Abelian, in electroweak symmetry breaking), but also for the case asymptotically free) Gb gauge interaction that becomes where it is vectorial. We have compared Higgs and dy- strongly coupled at this scale. One could, of course, ar- namical mechanisms for breaking SU(3) via a Higgs field gue that such an approach differs from the original pur- or condensate transforming according to the fundamen- pose of the grand unification, which was to obtain an tal or adjoint representation. We have also carried such ultraviolet-scale theory with only a single gauge group an analogous study for SU(N) with N 4. Our present ≥ 14 study helps to elucidate the differences between the Higgs as gj(µ), where µ is the Euclidean reference momentum, 2 and dynamical mechanisms for breaking a gauge symme- and we denote αj (µ) = gj(µ) /(4π). The beta function try. We believe such theoretical studies are useful since is βGj = dgj /dt, where dt = d ln µ. 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