aarXiv:1110.6919 [hep-ph]
Techni-Chiral-Color
Thomas W. Kephart ∗ and HoSeong La † Department of Physics and Astronomy,Vanderbilt University,,Nashville, TN 37235, USA
Chiral Color is extended by incorporating Technicolor, which induces dynamicalbreaking of the Electroweak symmetry as well as Chiral Color to QuantumChromodynamics. Gauge anomalies are cancelled by introducing two generations oftechnifermions, and the fourth generation of quarks and leptons is required. Eachtechnifermion generation is coupled to only two Standard Model generations by theYukawa interaction. Various phenomenological implications are explained.PACS: 12.60.-i, 12.90.+b, 12.15.-y ∗ [email protected] † [email protected] a r X i v : . [ h e p - ph ] M a r he model introduced here is a hybrid between Chiral Color (CC)[1, 2] and Technicolor(TC)[3, 4]. Neither of these have any evidence of their existence. However, there are goodreasons why these could be the immediate future of new physics beyond the Standard Model(SM).First, the nature of chirality has been fascinating us since the discovery of parity violationand the V-A theory. The correct identification of quarks and leptons in the SM based onanomaly cancellation proves the value of the chiral nature of the Electroweak theory. Yet,eventually the low energy world of unbroken symmetry is vector-like. One cannot help butraising the further question of why Quantum Chromodynamics (QCD) (i.e. SU(3) C ) is vector-like, while part of the Electroweak theory (i.e. SU(2) L ) is chiral. In fact, as a global flavorsymmetry, chiral symmetry is introduced in QCD to explain the origin of (light) quark masses.We can go one step further and ask if QCD itself is a result of spontaneous breaking of localchiral symmetry. Indeed this question was asked before and the model constructed is knownas the Chiral Color[1].Second, the existence of the Higgs to provide the electroweak symmetry breaking (EWSB)in terms of an elementary scalar is still elusive and it is possible that we may face the situationof no discovery. If there is no Higgs, the most obvious alternative is clearly dynamical symmetrybreaking[5] and there are active investigations going on in the context of TC[6, 7, 8].Furthermore, we will break both CC and EW symmetry dynamically at the same time. Thenwe can resolve many outstanding issues. To name a few, formulation of a realistic CC withsufficiently heavy axigluons, heavy top-quark mass and fermion mass hierarchy, small mixingbetween top-quark and others in the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix,suppression of of the flavor changing neutral current (FCNC), etc. The cost of doing thisis the addition of two technifermion generations and a fourth generation of quarks and leptons.The model we consider is based on the (relevant) gauge group G = SU(2) TC × SU(3) L × SU(3) R × SU(2) L × U(1) Y , (1)where quarks and leptons are SU(2) TC singlets and given by Q :(1 , , , , / L , (1 , , , , / R , (1 , , , , − / R ,L :(1 , , , , − / L , (1 , , , , − R (2) The gauge group is actually G × S U (2) (cid:48) . However, no matter fields carry SU(2) (cid:48) charges so that SU(2) (cid:48) isdecoupled from the matter sector. The role of SU(2) (cid:48) is to be clarified later. TC interaction, then it getsinteresting. (See [9] for another way of incorporating TC with CC.) So, we are led to include T Q ∗ : (2 , , , , − / L , (2 , , , , − / R , (2 , , , , / R ,T L : (2 , , , , / L , (2 , , , , R , (2 , , , , R , (3)where the asterisk is added to indicate 3. Notice that we still define the electric charge as Q EM = I L + Y / TC will remain unbroken but confined. Since the T Q ’s are doubletunder SU(2) TC , to cancel SU(3) anomalies we need to match one generation of technifermionswith two generations of Q, L ’s. Therefore, in total we have two generations of technifermionsand four generations of
Q, L , hence the fourth generation is needed. This distinguishes thetop-quark (and t (cid:48) ) generation from the two light generations as in the Topcolor model[10, 11].The T Q ’s and T L ’s will condense due to strong SU(2) TC interactions, and this should breakthe EW symmetry as well as CC. We will count Nambu-Goldstone bosons (NGB) slightlydifferently, but it is equivalent to a total of 255 NGB’s due to the SU (16) L × SU (16) R chiralsymmetry breaking down to SU (16) V . Because of four different colored T Q ’s, there are sixteencondensates formed by T Q ’s under SU(2) TC . However, to be more precise, all of these have(3 ,
3) degrees of freedom according to CC. Upon twisting CC to QCD times the axial SU(3),these degrees of freedom separate into SU(3) singlets and octets, 144 in total. In addition tothese, there are another fifteen condensates formed by T L ’s under SU(2) TC and sixteen coloredcondensates of (cid:104) T Q T L (cid:105) and their conjugates, in total 111. Out of sixteen SU(3) octets, one willbe eaten to break SU(3) L × SU(3) R to the SU(3) C of QCD to make eight axigluons massive.The remaining fifteen octets will form condensates due to SU(3) C which could have propertiessimilar to glueballs in QCD after confinement. Three combinations of (cid:104) T L T L (cid:105) and the singlet (cid:104) T Q T Q (cid:105) will be eaten to break the electroweak symmetry. The remainders are psuedo-Nambu-Goldstone bosons (PNGB) and techniaxions[8]. Note that the global symmetry is not SU(32) due to CC charges. The extra 240 condensates which couldhave appeared under enhanced global symmetry SU(32) breaking to Sp(32) are not Lorentz scalars, althoughthey are SU(2) singlets. So they should not be counted as NGB’s in our case.
2o match the EWSB scale the technipion (i.e. singlet) decay constant is given by F π TC =(246 GeV) / √ , which leads to the correct weak boson masses[7, 8]. Notice that,for axigluons, the octet decay constant F g TC does not have to be the same as F π TC , dependingon the detail of flavor symmetry breaking. They may be related, but we do not have anyexperimental data to use to fix parameters needed to specify the relationship at this moment.Since we can construct an effective lagrangian in which (color) singlet and octet technimesonshave independent kinetic energy terms with their own decay constants, e.g. ( F π TC / | D µ U | +( F g TC / | D µ U | , where U is given in terms of isospin triplet π T as U ≡ exp( iσ · π T /F π TC )and U is in terms of color octet Π as U ≡ exp( iλ · Π /F g TC ), we can safely assume they areindependent at this moment. So, the axigluon mass is given by m g A ∼ g s (cid:112) C F g TC (4)where g s is the QCD coupling constant and C = 3 is the second Casimir invariant for theadjoint representation of SU(3). If we choose F g TC ∼ Λ TC / ∼ . , , , , . (5)which only carries TC charge. If this scalar field interacts with fermions according to Yukawacouplings of λ Q Φ A Q a T QAa + h . c ., (6)where the label ‘ A ’ is an SU(2) TC index and ‘ a ’ is a CC index, then condensations of T Q ’sand Φ’s will generate masses. Motivated by the anomaly cancellation structure, we assumethat the first (second) generation of T Q couples only to the first and second (third and fourth)generations of Q . This assumption can be justified either by imposing restrictions on Yukawacoupling constants or by imposing a discrete symmetry. In the case of the discrete symmetryΓ, we assume that Yukawa couplings respect Γ, while the TC gauge interaction is allowed toviolate Γ because T Q and T Q are identified as weak eigenstates, so that there is no reasonwhy TC should have the same eigenstates. The simplest example is Γ = Z such that differenttechnifermions have different Z -parities and others are assigned accordingly, then no Yukawa3oupling mixing will be allowed. The scalar Φ is also assumed to be non-self-interacting forsimplicity and it will be confined.The mass matrix for, say, up-quarks, is a 4 × × M Q = ( M IJ ) , (7)where I, J identify the technifermion generations and each block is given by 2 × M IJ = ( (cid:88) (cid:96) =1 m ( (cid:96) ) ij ) , (8)where i, j identify the quark generations. Notice that for I ( J ) = 1 i ( j ) = 1 ,
2, while for I ( J ) = 2, i ( j ) = 3 , T Q Φ Q Qλ Q λ Q Figure 1: Feynman diagram for diagonal block quark mass at the lowestorder. T QI T QJ Q Q ( g Q )TC Φ Φ g TC c IJ g TC λ Q λ Q ( λ f Q ) Figure 2: An example of Feynman diagram for higher order contributions.Those inside brackets are for off-diagonal-block quark mass. c IJ indicatetechnifermion mixingings.We will consider only the two leading order contribution to the diagonal block M II = ( m (1) ij + m (2) ij ) . (9)The condensations of T Q ’s and Φ’s in the dimension eight operator of an effective actionrepresented by Fig.1 lead to the first term m (1) ij ∼ λ i λ j (cid:104) Φ (cid:105)(cid:104) T QI (cid:105) Λ ∼ π λ i λ j m T QI . (10)4 contribution for the diagonal block at the two-loop level generated by Fig.2 gives m (2) ij ∼ π c II g λ i λ j m T QI Λ TC , (11)where c II = 1.Similarly, the leading order of the off-diagonal-block components is generated at the two-loop level (Fig.2 with those fields inside the bracket) from a dimension sixteen operator suchthat M IJ = ( m (2) ij ) ∼ (cid:18) π c IJ g λ i λ j m T Q m T Q Λ TC (cid:19) , (12)where c IJ is the technifermion mixing, hence suppressed by order m T Q / Λ TC compared to thediagonal ones. For c IJ (cid:54) = 0 when I (cid:54) = J , we need T Q and T Q mixing for their couplings toTC gauge fields. This is because T Q and T Q are weak eigenstates, hence there is no reasonfor them to interact with SU(2) TC in the same way. This mixing indicates that SU(2) TC isnot the entire gauge symmetry, but in fact part of a larger gauge symmetry as follows: LetSU(2) TC be a (properly) twisted part as in SU(2) × SU(2) (cid:39) SU(2) TC × SU(2) (cid:48) with couplingconstants given by g = (1 + c ) g TC and g = (1 − c ) g TC , where SU(2) (cid:48) is confining andno matter fields carry SU(2) (cid:48) charges so that SU(2) (cid:48) decouples, then we only have SU(2) TC coupled to matter in the Λ TC region. So, the gauge invariance under SU(2) × SU(2) can bedemonstrated by untwisting as follows. Let T Q = ( T Q + T Q ) / √ (cid:101) T Q = ( T Q − T Q ) / √ g T Q A (1) µ T Q + g (cid:101) T Q A (2) µ (cid:101) T Q , then this leads to gauge invariance. Rewrite in terms of T Q and T Q and, since they only carry SU(2) TC charges, identifying A TC µ = A (1) µ = A (2) µ , weget the technifermion mixing terms. Notice that T Q ( (cid:101) T Q ) couples to SU(2) (SU(2) ) only. Thismixing is crucial to generate the desired mass matrix and CKM-like mixing, and in fact is theorigin of the mixing in the SM in this context. In the gauge sector the gauge fields of SU(2) (cid:48) behave like vector “matter” fields with respect to SU(2) TC upon twisting SU(2) × SU(2) [15].So, the evidence of SU(2) (cid:48) will show up in the TC gauge sector only, e.g., to modify Wardidentities of SU(2) TC in some cases.The mass matrix given in terms of Eqs.(10)-(12) has two zero eigenvalues. To avoid thesezero eigenvalues we need to add a flavor diagonal term and the tadpole contribution can takecare of this. So, we are led to introduce two sterile scalar fields in the same spirit of groupinggenerations as prescribed (one is sufficient too if we allow much larger hierarchy of Yukawacoupling constants) Φ I : (1 , , , ,
0) (13)5 Q Φ I T QI λ I λ T I
Figure 3: Tadpole diagram for flavor diagonal masses.such that they only interact with (techni)fermions via Yukawa couplings as λ T I Φ I T QI T QI , λ i Φ I Q i Q i . (14)Notice that Φ interacts only with Q , , etc., as before due to restrictions on Yukawa couplingconstants. Then the tadpole contribution from Fig.3 generates the diagonal contribution m ii ∼ π λ i λ T I m T QI (cid:32) Λ TC m I (cid:33) , (15)where their masses are free parameters in this context. However, the mass can be easilygenerated dynamically by assuming another strong interaction, e.g. SU(2) (cid:48) mentioned before,whose charge only these scalars carry and the radial components become Φ I .The down-quarks mass matrix can be generated similarly and the difference compared withthe up-quarks mass matrix will be due to the difference in Yukawa couplings and technifermionmasses.Let V u and U d be the matrices which diagonalize the up-quarks and down-quarks massmatrices, respectively. Since the initial mass matrices are of block-form, V u and U d will alsonaturally be of block-form. However different blocks can have different phase ambiguities sothat mixing matrix can have different phases for different blocks. Hence, we can construct themixing matrix as K = V † u U d , (16)where V u = (cid:18) e iδ V e iδ V e − iδ V e iδ V (cid:19) , U d = (cid:32) e i (cid:101) δ U e i (cid:101) δ U e − i (cid:101) δ U e i (cid:101) δ U (cid:33) . (17)In principle, a 4 × δ = (cid:101) δ and δ = (cid:101) δ , then there remain three independent phases, namely, δ − (cid:101) δ , δ − δ , and6 + δ . The first 3 × × c m T Q / Λ TC thanthe diagonal blocks, and so is the mixing matrix. This can easily explains why in the CKMmatrix, the top mixings to the first two generations are smaller by order of 10 − than the mixingof the first with the second generations. This in turn explains the suppression of FCNC.To leading order the diagonalized masses are (assuming all Yukawa couplings are of orderunity) of the form m u : m c : m t : m t (cid:48) ∼ m T Q (cid:32) Λ TC m Φ (cid:33) : m T Q : m T Q (cid:32) Λ TC m Φ (cid:33) : m T Q . (18)There are seven unknowns including m t (cid:48) and only four relations, so there are three free param-eters (in addition to Yukawa coupling constants). Using known masses of quarks and assumingΛ TC ∼ m Φ ∼
23 TeV for m u ∼ . m T Q ∼ m t (cid:48) (cid:38)
300 GeV, then m T Q ∼ m c and m Φ ∼ . λ t /λ u ∼ m t /m u as in SM, etc., can generate the samemass hierarchy. But we find this quite uninspiring. On the contrary, in our model it is easy togenerate the heavy top quark mass and explain the known mass hierarchies in terms of Yukawacoupling constants of similar order of magnitude.For lepton masses, we can use the same strongly interacting Φ such that the necessaryYukawa couplings are given by λ L Φ A LT ∗ AL + h . c , (19)where the label ‘ A ’ is an SU(2) TC index. The flavor diagonal contribution is again due tothe sterile scalars given in Eq.(13), then, the lepton masses can be generated accordingly.The difference compared to the quark masses is now that the lepton masses are generated bycondensation of technileptons T L ’s so that it can easily accommodate the quark-lepton masshierarchy. For given Λ TC and m Φ I estimated from the quark mass hierarchy, we can generate the7epton mass hierarchy based on the limit on the fourth generation leptons, without unreasonablysmall or large Yukawa coupling constants. Lepton mixing matrix can be similarly constructedfrom this lepton mass matrix.In fact, the structure we have introduced in this letter is fairly generic. As long as onetechnifermion generation (with or without technileptons) couples to two SM generations asprescribed, all phenomenological outcomes are similar. We can do this even without CC,although less motivated coupling one technifermion generations to two SM model generations.We suspect there might be a larger framework from which this structure can be inherited andjustified.The electroweak precision constraints ruled out QCD-like TC models with degenerate tech-nifermion doublets, but we have good reasons why our model should be safe. The details will bepresented elsewhere[16]. First, it is known that massive scalars and extra nondegenerate heavyfermions can contribute to meet the precision data[13, 17, 18, 19, 20]. We have three massivescalars and fourth SM generation, in addition to technifermions that can be non-degenerate.Second, the precision data test is based on TC × SM, not TC × CC × EW. So, it is possible thatCC may modify the outcome. Third, strictly speaking, our model is not QCD-like because oftechnifermion mixing.Since T Q or T L is the lightest technifermion, it is a good place to look for a signal todistinguish this model. Both interact with known world particles above the TC scale, whileemitting a strongly interacting scalar, which could lead to a monojet. One may think theirmasses may be too low, but it is acceptable because neither of them will show up as a quarkor a lepton below Λ TC due to SU(2) TC . What we observe at low energy will be their (doublyfor T Q ) confined objects, whose mass can be quite high beyond the present measured scale.Furthermore, the mass can always be raised by increasing Λ TC or adjusting Yukawa couplingconstants.Both strongly interacting and sterile neutral scalar fields could be candidates for the darkmatter. Their masses can be at least of order TeV, which is well beyond the current limit ofabout 500 - 600 GeV set by LHC[21]. These scalars interact with the known world particles onlyin terms of Yukawa interactions. The lowest level flavor changing effective Yukawa couplingsgenerated by technifermion condensation appear only at two-loop level, and their contributionto FCNC amplitudes below Λ TC is suppressed at least, even for tree-level Yukawa couplings oforder unity, by O (( m T Q / π Λ TC ) (cid:96) +1) ), where (cid:96) = 2 is the lowest number of TC loops needed.8ence, the mass bounds on these scalars are even lower by O (( m T Q / π Λ TC ) ) than those ofYukawa coupled non-Higgs scalars’. The highest mass bound of the latter from flavor physicsis about 1 TeV[22]. Then the bound on scalars in our case is at most about one tenth of that,which is lower than the bound set by LHC. Since we have SU(2) TC , we do not expect thetechnibaryon problem[23].The mixing matrix we proposed here accommodates the CKM matrix, yet goes further sothat it is a good place to look for clues of physics beyond the SM. There are two additionalCP-violating phases involving the fourth generation, hence the model allows much more roomfor CP-violation. The SM has a difficulty explaining the baryon asymmetry due to insufficientCP-violation. Our current work certainly opens the door to resolving this issue.One shortcoming of our model is that we still cannot avoid the large number of PNGB’s andtechniaxions as in most of TC models. The only way to reducing the number is reducing that oftechnifermions. The possibility of eliminating technileptons based on [15] is under investigation.Also, it will be nice to generate flavor diagonal masses without sterile scalar fields so that wecan reduce free parameters.We have presented our basic ideas briefly in this letter, but more details and variant modelswill be presented elsewhere[16].HL thanks the theory group of the Department of Physics & Astronomy at VanderbiltUniversity for their hospitality while this work is completed. We thank Alex Kagan for bringingreferences [14], [18] and [19] to our attention. TWK is supported by US DOE grant numberDE-FG05-85ER40226. References [1] P.H. Frampton and S.L. Glashow, Phys. Lett. (1987) 157; Phys. Rev. Lett. (1987) 2168.[2] J. Pati and A. Salam, Phys. Rev. D10 (1974) 275.[3] Steven Weinberg, Phys. Rev.
D13 , 974-996 (1976); Leonard Susskind, Phys. Rev.
D20 ,2619 (1979).[4] E. Eichten and K. Lane, Phys. Lett. , 125-130 (1980); Savas Dimopoulos and LeonardSusskind, Nucl. Phys.
B155 , 237 (1979).95] Y. Nambu and G. Jona-Lasinio, Phys. Rev. , 345 (1961); J. Schwinger, Phys. Rev. , 397 (1962);128, 2425 (1962); R. Jackiw and K. Johnson, Phys. Rev. D8 , 2386 (1973);J. M. Cornwall and R. E. Norton, Phys. Rev. D8 , 3338 (1973).[6] E. Farhi and L. Susskind, Phys. Rep. (1981) 277-321;[7] K. Lane, arXiv:hep-ph:9401324; arXiv:hep-ph/0202255.[8] C. T. Hill and E. H. Simmons, Phys. Rept. , 235 (2003) [Erratum-ibid. , 553(2004)] [arXiv:hep-ph/0203079].[9] W. -C. Kuo, D. Slaven, B. -L. Young, Phys. Lett. B202 , 353 (1988).[10] C. T. Hill, Phys. Lett. B , 419 (1991).[11] C.T. Hill, “Topcolor,” hep-ph/9702320 and references therein.[12] E. H. Simmons, Nucl. Phys.
B312 , 253 (1989); S. Samuel, ibid.
B347 , 625 (1990);A. Kagan and S. Samuel, Phys. Lett. B , 605 (1990); , 37 (1991);[13] C. D. Carone, H. Georgi, Phys. Rev.
D49 , 1427-1436 (1994) [arXiv:hep-ph/9308205].[14] A. Kagan,
Proceedings of the 15th Johns Hopkins Workshop on Current Problems inParticle Theory , G. Domokos and S. Kovesi-Domokos eds. (World Scientific, Singapore,1992), p.217, and
Phys. Rev.
D51 (1995) 6196, hep-ph/9409215; B. A. Dobrescu,
Nucl.Phys.
B449 (1995) 462, hep-ph/9504399.[15] H. S. La, hep-ph/0306223.[16] H. S. La, in preparation.[17] M. E. Peskin and T. Takeuchi, Phys. Rev. D , 381 (1992).[18] B. A. Dobrescu and J. Terning, Phys. Lett. B , 129 (1998) [hep-ph/9709297].[19] B. A. Dobrescu and E. H. Simmons, Phys. Rev. D , 015014 (1999) [hep-ph/9807469].[20] J. Erler and P. Langacker, in Particle Data Group (http://pdg.lbl.gov/2011/reviews/rpp2011-rev-standard-model.pdf).[21] ATLAS and CMS Talks presented at CERN on Dec. 13th, 2011.[22] J. M. Arnold, M. Pospelov, M. Trott and M. B. Wise, JHEP , 073 (2010)[arXiv:0911.2225 [hep-ph]]; B. Grinstein, A. L. Kagan, J. Zupan and M. Trott, JHEP , 072 (2011) [arXiv:1108.4027 [hep-ph]].[23] R. S. Chivukula and T. P. Walker, Nucl. Phys. B329