Higgs mass from compositeness at a multi-TeV scale
NNovember 22, 2013; revised June 3, 2014 FERMILAB-Pub-13-350-T
Higgs mass from compositeness at a multi-TeV scale
Hsin-Chia Cheng (cid:5) , Bogdan A. Dobrescu (cid:63) , Jiayin Gu (cid:5)(cid:5)
Department of Physics, University of California, Davis, CA 95616, USA (cid:63)
Theoretical Physics Department, Fermilab, Batavia, IL 60510, USA
Abstract
Within composite Higgs models based on the top seesaw mechanism, we showthat the Higgs field can arise as the pseudo Nambu-Goldstone boson of the broken U (3) L chiral symmetry associated with a vector-like quark and the t - b doublet.As a result, the lightest CP-even neutral state of the composite scalar sector islighter than the top quark, and can be identified as the newly discovered Higgsboson. Constraints on weak-isospin violation push the chiral symmetry breakingscale above a few TeV, implying that other composite scalars are probably too heavyto be probed at the LHC, but may be within reach at a future hadron collider withcenter-of-mass energy of about 100 TeV. Contents U (3) L × U (2) R symmetric dynamics 4 U (3) L breaking from electroweak interactions 114 Higgs mass and the heavy state spectrum 135 Phenomenology 206 Conclusions 22A Renormalization group running of couplings 23 a r X i v : . [ h e p - ph ] A ug Introduction
The discovery of the Higgs boson at the Large Hadron Collider (LHC) [1] representsan important step towards understanding the origin of electroweak symmetry breaking.Its mass around 126 GeV has major implications for different models or mechanisms ofelectroweak symmetry breaking.A pressing question is whether the Higgs boson is an elementary particle or a compositeone. Current data cannot distinguish between an elementary Higgs boson (up to veryhigh energies) and a composite Higgs boson arising from a theory with a decoupling limitidentical to the Standard Model (SM). Nonetheless, the requirement that the Higgs bosonmass M h is near 126 GeV poses model building challenges on SM extensions.A composite Higgs boson has generically a large quartic coupling and hence is ex-pected to be heavy, unless its mass is protected by some symmetry. One possibility isthat the Higgs field arises as a pseudo-Nambu-Goldstone boson (pNGB) of some brokenglobal symmetry [2]. Many types of models which realize the Higgs multiplet as a pNGBhave been proposed, including Little Higgs [3] and models that invoke the AdS/CFT cor-respondence [4]. In these effective theories, M h is quite model-dependent. The quarticcoupling arises at tree level in some models, and from loops in others. The relatively largetop quark mass, m t , can be accommodated through partial compositeness [5], i.e. , mix-ing of the elementary top fields with composite vector-like fermions. In a class of pNGBcomposite Higgs models where the dominant explicit breaking of the global symmetrycomes from these mixings [4], there are strong correlations between M h and the masses ofcomposite vector-like fermions. Typically, M h ≈
126 GeV requires at least one vector-liketop partner well below 1 TeV [6] which is in some tension with the 8 TeV LHC data, andwill be thoroughly tested by LHC searches in the near future. An additional predictionof these models that will be soon tested is that higher dimensional operators modify theHiggs couplings [7].A more direct way of obtaining a heavy top quark is the top condensation mechanism,where the Higgs field is a top-antitop bound state [8, 9]. The top quark, however, wouldhave in this case a mass around 600 GeV if the compositeness scale is not much above theelectroweak scale. Furthermore, the leading number of colors ( N c ) approximation gives M h ≈ m t . Both results are in conflict with the data. The top and Higgs masses can bereduced, at the expense of fine-tuning, if the compositeness scale is raised; but even withcompositeness at the Planck scale, m t and M h would still be above 200 GeV [9].2n attractive solution to the top mass problem in top-condensation models is throughthe top-seesaw mechanism [10–13]. The heavy top given by top condensation mixes witha massive vector-like top partner, resulting in a light mass eigenstate of m t ≈
173 GeV,identified as the observed top quark. Although the composite Higgs boson is likely tobe heavier than the weak scale, for some range of parameters it is possible to obtain arelatively light Higgs boson [11, 12, 14]. Generically, the composite Higgs sector includestwo doublets and two singlets [12], and a light Higgs boson could arise due to the mixingamong the four CP-even scalars.In this paper we show that a composite Higgs boson can naturally have M h ≈ v is fixed at 246 GeV) in a top-seesaw model where the strongdynamics preserves a global U (3) L symmetry acting on the top-bottom doublet ( t L , b L )and the left-handed component of the vector-like quark, χ L . The composite Higgs doubletis the collection of pNGB’s of the spontaneous U (3) L → U (2) L breaking. The explicitsymmetry breaking required for the seesaw mechanism also controls M h . As a conse-quence, the mass of the composite Higgs boson is correlated with m t , and has an upperbound around m t . The electroweak interactions further reduce M h so that generically itis substantially below m t .The low energy effective theory is given by the SM plus the heavy states arising fromtwo Higgs doublets and two complex singlet scalars, together with the vector-like toppartner. The strongest constraint on this model is due to weak-isospin violation, whichrequires the vector-like quark and consequently most of the scalars other than the SMHiggs boson to have masses of order 10 TeV. This implies some tuning to obtain the weakscale at v ≈
246 GeV. Nevertheless, given that no new physics has been found at the LHCso far, some tuning seems inevitable in any theory which attempts to explain electroweaksymmetry breaking.The strong dynamics responsible for compositeness and chiral symmetry breaking maybe an asymptotically-free gauge interaction that is spontaneously broken near the scale Λwhere it becomes strong. Various theories of these type have been proposed [15,16]. It mayeven be possible that a strongly coupled gauge interaction breaks its own gauge symmetry[17,18]. The consequence would be a non-confining strong interaction whose bound statesare present together with their constituents at energies below the compositeness scale Λ.Its effects at the scale Λ may be parametrized by certain 4-fermion interactions. Sucha theory is viable because in the decoupling limit, where the scale of chiral symmetrybreaking f is much larger than v , the deviations from the SM are O ( v /f ). In practice,3he hierarchy of scales v < f < Λ should not be too large because both f /
Λ and v/f require fine-tuning.In Section 2 we present the effective theory of the composite Higgs model below thecompositeness scale. We discuss the U (3) L × U (2) R symmetric terms and also their explicitbreaking terms. In Section 2.1 we find the vacuum which breaks the electroweak symmetryand produces the top-seesaw mechanism. In Section 2.2 we derive an approximate analyticformula for M h , and find that its maximum value is near m t . In Section 3 we discuss theeffects of electroweak gauge loops and show that they always reduce M h . In Section 4 weperform a detailed numerical study of M h as a function of the parameters in this model.We show that the 126 GeV Higgs boson can be easily accommodated as a composite fieldin our model. After fixing M h ≈
126 GeV, we calculate the spectrum of the heavy states,and then in Section 5 we comment on phenomenological implications. The conclusionsare drawn in Section 6. U (3) L × U (2) R symmetric dy-namics We consider an effective theory at a scale Λ (cid:29) SU (2) W -singlet vector-like quark, χ , of electric charge +2 /
3, and some4-fermion interactions suppressed by Λ (which presumably arise due to an asymptotically-free gauge interaction that is spontaneously broken near the scale Λ where it becomesstrong). This theory does not include a Higgs doublet nor any elementary scalars. Weassume that some of the 4-fermion interactions involving third generation quarks and χ are attractive and sufficiently strong to form quark-antiquark bound states [12]. Thesestrong interactions are not confining, because at distances longer than 1 / Λ the effectsof 4-fermion interactions (other than the presence of bound states) are exponentiallysuppressed.The low-energy theory includes the composite fields that are deeply bound such thattheir masses are less than the compositeness scale Λ. As in the Nambu-Jona-Lasinio(NJL) model [19], if the coefficient of a 4-fermion interaction is larger than a certaincritical value, then the squared mass of the scalar bound state becomes negative, andthe scalar acquires a vacuum expectation value (VEV). Furthermore, as the coefficientcrosses the critical value, there is a second order phase transition so that the VEV can inprinciple be much smaller than Λ [16]. 4oncretely, we assume the constituents of the deeply bound states to be only χ L , χ R ,the right-handed top quark t R , and the left-handed top-bottom doublet ψ L = ( t L , b L ). Inthe limit where the electroweak interactions are ignored, the kinetic terms of these quarkshave an U (3) L × U (2) R chiral symmetry, which we assume to be approximately preservedby the 4-fermion interactions. The U (3) L × U (2) R symmetric interactions give rise to thefollowing Yukawa couplings of the composite scalars (collectively labelled by Φ) to theirconstituents: L Yukawa = − ξ (cid:0) ¯ ψ L , ¯ χ L (cid:1) Φ (cid:18) t R χ R (cid:19) + H . c . (2.1)Here ξ is a dimensionless coupling whose value at scale Λ, upon integrating out Φ, matchesthe coefficient of the 4-fermion interactions. The scalar field Φ is a 3 × t , Φ χ ) , (2.2)where the scalar fields Φ t and Φ χ are the bound states of the U (3) L triplet ( t L , b L , χ L )with t R and χ R , respectively:Φ t ∼ ¯ t R (cid:18) ψ L χ L (cid:19) , Φ χ ∼ ¯ χ R (cid:18) ψ L χ L (cid:19) . (2.3)At scales µ < Λ, the Yukawa couplings (2.1) give rise to the following potential for Φ: V Φ = λ (cid:2) (Φ † Φ) (cid:3) + λ (cid:0) Tr[Φ † Φ] (cid:1) + M Φ † Φ . (2.4)The quartic couplings λ and λ depend on the scale µ ; if the kinetic term for Φ iscanonically normalized, then λ becomes non-perturbative near Λ. In the large N c limit, λ is generated by a fermion loop, while λ vanishes. Scalar loops, however, generate anon-zero value for λ , so that λ (cid:29) | λ | . In the Appendix we use 1-loop renormalizationgroup (RG) equations to estimate the λ /λ ratio.The squared mass of Φ is assumed to satisfy | M | (cid:28) Λ , and if the 4-fermion in-teractions are super-critical, then M < U (3) L × U (2) R .We assume that there are additional explicit U (2) R breaking effects which distinguish t R and χ R . Given that | M Φ | (cid:28) Λ, such effects could induce a large relative splitting ofthe masses for Φ t and Φ χ . We parametrize these effects by V U (2) = δM tt Φ † t Φ t + δM χχ Φ † χ Φ χ + ( M χt Φ † χ Φ t + H . c . ) (2.5)5hese U (2) R breaking masses can be diagonalized by a U (2) rotation. As we will seelater, it is convenient to work in a different basis from the one that diagonalizes thesemass terms so we will keep Eq. (2.5) general.The U (2) R breaking effects may also split the quartic couplings in (2.4), but this doesnot make any qualitative difference (unless the changes in couplings are larger than orderone), because our discussion mostly relies on the U (3) L symmetry. For simplicity we donot include such effects.Gauge invariant masses for the SU (2) W -singlet quarks can be present at the scale Λ: L mass = − µ χt ¯ χ L t R − µ χχ ¯ χ L χ R + H . c . (2.6)We assume that µ χt , µ χχ (cid:28) Λ (which is technically natural because there is an enhancedchiral symmetry in the µ χt , µ χχ → U (3) L × U (2) R down to U (2) L × U (1) R . Below Λ, these fermion masses map to tadpole terms for the SU (2) W -singlet scalars: V tadpole = − (0 , , C χt )Φ t − (0 , , C χχ )Φ χ + H . c . (2.7)Matching at the scale Λ, we have C χt (cid:39) µ χt ξ Λ , C χχ (cid:39) µ χχ ξ Λ . (2.8)Note that when the scalars are integrated out at the cutoff scale (where M ∼ Λ ), thefermion mass terms (2.6) are recovered.The effective potential of the scalar sector below the compositeness scale is given by V scalar = λ + λ (cid:104) (Φ † t Φ t ) + (Φ † χ Φ χ ) (cid:105) + λ | Φ † t Φ χ | + λ (Φ † t Φ t )(Φ † χ Φ χ )+ M tt Φ † t Φ t + M χχ Φ † χ Φ χ + ( M χt Φ † χ Φ t + H . c . ) − (0 , , C χt ) Re Φ t − (0 , , C χχ ) Re Φ χ , (2.9)where M tt and M χχ are the sums of the mass terms in Eqs. (2.4) and (2.5). The parameters M χt , C χt , and C χχ can be chosen to be real without loss of generality. As mentioned earlier, M χt can be removed by a U (2) rotation. Then C χt and C χχ can be made real by phaseredefinitions of Φ t and Φ χ respectively. The effective field theory below the cutoff Λ isthus described by Eqs. (2.1) and (2.9). 6he SU (2) W × U (1) Y gauge symmetry is a subgroup of the U (3) L × U (2) R chiralsymmetry. Thus, the electroweak interactions explicitly break the U (3) L symmetry. The U (3) L triplets Φ t and Φ χ can be written in terms of fields belonging to electroweak rep-resentations: Φ t = (cid:18) H t φ t (cid:19) , Φ χ = (cid:18) H χ φ χ (cid:19) . (2.10) H t and H χ transform under SU (2) W × U (1) Y as the SM Higgs doublet, while φ t and φ χ are SU (2) W × U (1) Y singlets. The effects of the electroweak interactions will be discussedin Sec. 3. H χ ( t ) , φ χ ( t ) will get VEVs due to negative squared masses and tadpole terms.Expanding H χ ( t ) , φ χ ( t ) around their VEVs in terms of fields of definite electric charges,we can write H t = √ v t + h t + iA t ) H − t , H χ = √ v χ + h χ + iA χ ) H − χ ,φ t = 1 √ u t + ϕ t + iπ t ) , φ χ = 1 √ u χ + ϕ χ + iπ χ ) . (2.11)The VEVs v t , v χ , u t and u χ are real, and some of them may vanish, depending on theparameters of the effective potential. We use the notation v t + v χ = v , u t + u χ = u , and f = √ u + v (2.12)is the scale of U (3) L breaking. The measured Fermi constant requires v ≈
246 GeV.We now analyze the low energy effective theory given by Eqs. (2.9) and (2.1).
It is convenient to perform an U (2) R transformation (which rotates t R and χ R , as wellas Φ t and Φ χ ) to go to a basis where v t = 0 and v χ = v . For simplicity, we will use thesame notation in this basis as in Eqs. (2.9)-(2.11). In this basis we define u t = u sin γ and u χ = u cos γ , and the short-hand notation s γ = sin γ and c γ = cos γ .The extremization conditions for V scalar relate the parameters from the effective po-tential to the VEVs: v (cid:18) M χt + λ u s γ c γ (cid:19) = 0 ,v (cid:18) M χχ + λ (cid:0) u c γ + v (cid:1) + λ (cid:0) u + v (cid:1)(cid:19) = 0 , (2.13)7or the derivatives with respect to h t and h χ , and C χt = u √ (cid:20) M χt c γ + (cid:18) M tt + λ u + λ (cid:0) u + v (cid:1)(cid:19) s γ (cid:21) ,C χχ = u √ (cid:20) M χt s γ + (cid:18) M χχ + λ + λ (cid:0) u + v (cid:1)(cid:19) c γ (cid:21) , (2.14)for the derivatives with respect to ϕ t and ϕ χ . Eqs. (2.13) and (2.14) have a solution for v = 0, and a different solution for v >
0. When the latter is a minimum ( i.e. , the squaredmasses of all spin-0 states are positive), we find that it is also the global minimum of theeffective potential while v = 0 is a saddle point.For v >
0, Eqs. (2.13) imply M χt = − λ u s γ c γ ,M χχ = − λ (cid:0) u c γ + v (cid:1) − λ (cid:0) u + v (cid:1) . (2.15)Substituting these into Eqs. (2.14) gives C χt = u s γ √ (cid:20) M tt + λ u s γ + λ (cid:0) u + v (cid:1)(cid:21) ,C χχ = 0 . (2.16)Thus, the basis where (cid:104) H t (cid:105) = 0 and (cid:104) H χ (cid:105) (cid:54) = 0 is the one where C χχ = 0 (or equivalently,where t R and χ R are defined such that µ χχ = 0). Since the electroweak symmetry isbroken only by the VEV of H χ , the eaten Nambu-Goldstone bosons are contained in H χ only. The charged Higgs boson resides entirely within H t , and its mass squared is M H ± = M tt + λ u s γ + λ (cid:0) u + v (cid:1) . (2.17)The nonzero tadpole coefficient is then related to M H ± by C χt = u s γ M H ± / √
2. Note that v (cid:54) = 0 requires M χχ <
0, but does not restrict the sign of M tt . In Section 4, however, wewill show that the Higgs boson would be lighter than about 100 GeV unless M H ± > u ,which in turn requires M tt > v (cid:28) u . Asa result, the U (3) L breaking scale, defined in Eq. (2.12), is given by f (cid:39) u . Eqs. (2.15) and(2.16) have solutions for v , u and s γ that satisfy v/u (cid:28) M χχ , M χt , M tt and C χt parameters.8eglecting the mixing of the charm and up quarks with t and χ , the mass terms ofthe heavy charge-2/3 quarks, arising from Eq. (2.1), are given by − ξ √ (cid:0) t L , χ L (cid:1) (cid:18) vus γ uc γ (cid:19) (cid:18) t R χ R (cid:19) + H . c . (2.18)Diagonalizing this matrix gives the masses of the top quark t and the new quark, whichwe label by t (cid:48) in the mass eigenstate basis. Keeping only the leading nonvanishing termin v /f , we find the mass of the top quark, m t (cid:39) ξ √ vs γ . (2.19)Thus, ξ and s γ can be related to the top Yukawa coupling y t by s γ (cid:39) y t ξ . (2.20) ξ is expected to have a value around 3 or 4 (see Section 4) and y t ∼
1, so s γ ∼ O (0 . v /f , the mass of the new quark is given by m t (cid:48) (cid:39) ξ √ f . (2.21)while the mixing angle θ L , which rotates the t L and χ L gauge eigenstates into the masseigenstate quarks, is given by sin θ L ≡ s L (cid:39) vf . (2.22) Substituting Eqs. (2.15)-(2.17) back into the scalar potential (2.9), we find that the 4 × h t , h χ , ϕ t , ϕ χ ) is given by M H ± + λ v − λ uvc γ − λ uvs γ λ + λ ) v λ uvs γ ( λ + λ ) uvc γ − λ uvc γ λ uvs γ M H ± + (cid:20) λ (cid:18) − c γ (cid:19) + λ s γ (cid:21) u (cid:18) λ λ (cid:19) u s γ c γ − λ uvs γ ( λ + λ ) uvc γ (cid:18) λ λ (cid:19) u s γ c γ (cid:20) λ (cid:18) − s γ (cid:19) + λ c γ (cid:21) u . (2.23)9he lightest Higgs boson mass-squared ( M h ) is given by the smallest eigenvalue of themass matrix (2.23). Using Eq. (2.20), keeping the leading order in v /f , and expandingin s γ , we find M h = λ v s γ M H ± M H ± + λ u (cid:20) λ + 2 λ )( M H ± + λ u ) ( λ + λ ) M H ± (2 M H ± + λ u ) s γ + O ( s γ ) (cid:21) . (2.24)The Higgs boson mass is suppressed by s γ , because in the limit of ξ → ∞ or m t → U (3) L breaking tadpole terms C χt , C χχ vanish and the H χ and π χ fields becomeNambu-Goldstone bosons. Keeping only the leading term in s γ , M h (cid:39) λ ξ (cid:18) λ m t (cid:48) ξ M H ± (cid:19) − y t v = λ h v , (2.25)where λ h is the Higgs quartic coupling. In the fermion-loop approximation of NJL, theratio of couplings λ / (2 ξ ) is equal to 1. Scalar and gauge boson loops reduce this ratio.In the Appendix we show that it has a quasi-infrared fixed point value ∼ .
4, so we expect λ / (2 ξ ) between 0.4 and 1.Since both y t and λ h are obtained from integrating out the heavy quark, Eq. (2.25)relates the quartic Higgs coupling and the top Yukawa coupling at the scale m t (cid:48) , implying λ h < y t at that scale. For m t (cid:48) ∼
10 TeV, we find y t ∼ .
6. Evolving λ h down to the scale v we obtain an upper limit on the Higgs mass, M h ∼ <
185 GeV . (2.26)This is an interesting result, in contrast to the na¨ıve expectation in many composite Higgsmodels that the Higgs boson is heavier than the weak scale. We see that the Higgs bosonis light because it is a pNGB of the U (3) L → U (2) L symmetry breaking. In Section 4 weperform a more refined analysis, and as a result the above upper limit is further reduced.There is one additional pNGB, A , mostly given by the π χ field, with small admixturesof the other neutral CP-odd scalars in Eq. (2.11). Its squared mass is M A = 14 (cid:0) M H ± + λ f (cid:1) (cid:34) − (cid:115) − λ M H ± f s γ (cid:0) M H ± + λ f (cid:1) (cid:35) , (2.27)where we neglected terms suppressed by v /f . Expanding in s γ , and using Eq. (2.25),we find M A (cid:39) fv M h . (2.28)Even though A is much heavier than the Higgs boson, it is substantially lighter than theother composite scalars. 10 U (3) L breaking from electroweak interactions In Section 2 we have assumed that the mass and quartic terms in the potential respect the U (3) L symmetry, and the only explicit U (3) L breaking comes from tadpole terms. Otherexplicit U (3) L breaking effects, such as the SU (2) W × U (1) Y gauge interactions, can feedinto the mass and quartic terms through loops. In this section we study the effect of theseadditional U (3) L breaking effects on M h .We parametrize the U (3) L breaking mass and quartic terms as∆ V breaking = κ (cid:104) ( H † t H t ) + ( H † χ H χ ) + 2( H † t H χ )( H † χ H t ) (cid:105) + κ (cid:16) H † t H t + H † χ H χ (cid:17) + κ (cid:48) (cid:104) H † t H t φ † t φ t + H † χ H χ φ † χ φ χ + (cid:16) H † t H χ φ † χ φ t + H . c . (cid:17)(cid:105) + κ (cid:48) (cid:16) H † t H t + H † χ H χ (cid:17) (cid:16) φ † t φ t + φ † χ φ χ (cid:17) + ∆ M tt H † t H t + ∆ M χχ H † χ H χ + (cid:0) ∆ M χt H † χ H t + H . c . (cid:1) , (3.1)where we assumed again, for simplicity, that the quartic terms are U (2) R symmetric. Itis straightforward to repeat the analysis of Section 2 by including Eq. (3.1). To leadingorder in s γ , v /f , and U (3) L breaking, the correction to M h is∆ M h (cid:39) (cid:18) κ − κ (cid:48) − ∆ M χχ f (cid:19) v , (3.2)where κ ≡ κ + κ , κ (cid:48) ≡ κ (cid:48) + κ (cid:48) . The corrections from ∆ M tt and ∆ M χt are suppressedbecause the Higgs boson resides mostly in H χ . The correction of Eq. (3.2) can be mosteasily seen in the limit s γ → M χt = 0, where ϕ t and h t decouple and the massmatrix (2.23) becomes block-diagonal; we then only need to diagonalize the 2 × (cid:32) ( λ + κ ) v ( λ + κ (cid:48) ) uv ( λ + κ (cid:48) ) uv − ∆ M χχ + (cid:0) λ − κ (cid:48) (cid:1) u − ( κ − κ (cid:48) ) v (cid:33) , (3.3)where λ ≡ λ + λ . In order to keep the Higgs boson light, and to have the correctvacuum, we need (cid:12)(cid:12)(cid:12)(cid:12) κ − κ (cid:48) − ∆ M χχ f (cid:12)(cid:12)(cid:12)(cid:12) < λ ξ . (3.4)The contribution due to the mass-squared splitting between φ χ and H χ was presented inRefs. [11, 12]. 11n our model, the additional U (3) L breaking effects (besides the tadpole terms) comefrom the SU (2) W × U (1) Y gauge interactions. They contribute to both the mass andquartic terms. In the case where the gauge loops are the dominant source of U (3) L breaking, we have ∆ M tt = ∆ M χχ and ∆ M χt = 0 because the gauge interactions are U (2) R symmetric. Only H χ and H t transform under SU (2) W × U (1) Y while φ χ and φ t aresinglets. The SU (2) W × U (1) Y gauge loops split the masses of H χ ( t ) and φ χ ( t ) , analogouslyto the mass splitting between the charged and neutral pions due to the electromagneticinteraction. This contribution is quadratically divergent and needs to be cut off. In thecase of π + − π mass difference, the cutoff is effectively provided by the ρ meson mass [20]from the Weinberg sum rules [21]. Based on this analogy, we denote the cutoff by M ρ ,which is set by the mass of some (presumably vector) state in this theory. The 1-loopsplitting is then given by ∆ M χχ = 364 π (cid:0) g + g (cid:1) M ρ , (3.5)where g and g are the SU (2) W × U (1) Y gauge couplings. This mass splitting impliesthat M h receives a correction (at the chiral symmetry breaking scale f ) of∆ M h (mass) ≈ − (cid:18) . v M ρ f (cid:19) , (3.6)where the gauge couplings are evaluated at 10 TeV. This effect reduces the Higgs bosonmass and can be quite significant if M ρ (cid:29) f . For example, for M ρ = 5 f this reduces theeffective quartic Higgs coupling, λ h , by 0 .
16 at the chiral symmetry breaking scale f .The SU (2) W × U (1) Y gauge interactions also generate the additional quartic inter-actions involving H t and H χ . The dominant 1-loop contribution can be estimated tobe κ λ (cid:39) κ (cid:48) λ (cid:39) π (cid:0) g + g (cid:1) ln (cid:18) M ρ µ (cid:19) , (3.7)where we have assumed the same cutoff as in Eq. (3.5), and the renormalization scale µ should be taken around the heavy scalar states in the spectrum. We have also neglectedthe small g , contributions. Note that Eq. (3.7) is valid only for µ < M ρ . The correctionsto the quartic couplings give a correction to the Higgs squared mass:∆ M h (quartic) (cid:39) − π λ (cid:0) g + g (cid:1) v ln (cid:18) M ρ µ (cid:19) ≈ − . v λ ξ (cid:18) ξ . (cid:19) ln (cid:18) M ρ µ (cid:19) . (3.8)12his contribution is also negative, so that the electroweak interactions only reduce theHiggs boson mass. Thus, the Higgs mass formula obtained in the previous section [seeEqs. (2.24)-(2.26)] provides an upper bound on the Higgs mass in the absence of other U (3)breaking effects. If the cutoff M ρ is too large, the effective quartic coupling of the lightHiggs can turn negative which implies that we are expanding around the wrong vacuum.This puts an upper limit on M ρ , which depends on other parameters, as discussed in thenext Section. In the previous sections we derived the approximate expression for the Higgs mass andthe corrections from the electroweak gauge loops. In this section we perform a numericalstudy of the Higgs mass and show that the exact numerical result agrees well with theanalytic approximations. After fixing M h = 126 GeV, we calculate the masses of theother composite scalars.We start with an enumeration of the parameters of this model. In Eqs. (2.1) and (2.9),our model contains the following parameters: ξ, λ , λ , M tt , M χχ , M χt , C χt , C χχ . (4.1)One of the parameters ( M χt or C χχ ) is not independent due to the freedom of the U (2) R ro-tation. After minimizing the potential, the mass and tadpole terms can be written in termsof the VEVs, quartic couplings and the charged Higgs mass M H ± through Eqs. (2.15)-(2.17). The explicit U (3) L breaking from the electroweak gauge loops discussed in Sec-tion 3 introduces one more parameter M ρ , the cutoff of the electroweak gauge loop. As aresult, the spectrum is fully determined by the following eight parameters: ξ, λ , λ , M H ± , v, f, s γ , M ρ . (4.2)Two of these eight parameters are fixed by the weak scale and the top mass. To producethe correct m t , we use the SM 1-loop RG equations to evolve the top Yukawa coupling y t to the scale of the heavy fermion mass m t (cid:48) , and use it to solve for s γ , which in the lowestorder is given by Eq. (2.20). The running top Yukawa coupling in the MS scheme at thescale m t corresponds to m t ( µ = m t ) ≈
160 GeV [22]. For a set of input values of theother six parameters, which are taken to be ξ, λ / (2 ξ ) , λ /λ , f, M H ± /f, M ρ /f, (4.3)13ne can calculate the masses and couplings at scale m t (cid:48) . We choose the ratios of couplingsor mass scales as the independent parameters because they are more convenient and betterconstrained. To calculate M h , we match the theory to the SM at scale m t (cid:48) , compute thequartic Higgs coupling λ h , and then evolve λ h down to the weak scale.To make a generic prediction for M h , we first examine the expected ranges of inputparameters listed in Eq. (4.3). The U (3) L symmetry breaking scale must satisfy f > v .In fact, as we will see later, f is constrained to be much larger than v by the precisionmeasurement of weak-isospin violating effects. On the other hand, larger f also meansmore fine tuning for the weak scale. We will consider f up to 10 TeV to avoid excessive fine-tuning. For the effective theory to be a valid description, the states in the theory shouldhave masses below the cutoff scale. Therefore we will take M H ± < πf ∼ Λ. Similarly,the cutoff of electroweak gauge loops also satisfies M ρ < πf . The ranges of the couplingratios are discussed in the Appendix and they are expected to be 0 . ∼ < λ / (2 ξ ) ∼ < − . ∼ < λ /λ ∼ <
0. The Yukawa coupling ξ is expected to be ∼ − Z Φ (Λ) = 0 gives ξ (cid:39) π N ln(Λ /m t (cid:48) ) , (4.4)where N = N c = 3 if only fermion loops are included, and N = N c + 5 / / /m t (cid:48) = 3 (10), ξ ≈ . .
4) for N = 3, and ξ ≈ . .
5) for N = 11 /
2. In our numerical study, we use ξ = 2 π/ √ ≈ . m p / ≈
313 MeV corresponds to a Yukawa coupling ∼ .
4, which is close toour estimate.In Fig. 1 we show the Higgs boson mass as a function of the dimensionful parameters M H ± , f and M ρ by fixing the dimensionless couplings to some typical values, ξ = 3 . λ / (2 ξ ) = 0 . λ /λ = 0. We see that M h = 126 GeV can be obtained withreasonable parameters of our model. Higgs mass increases as M H ± increases, as expectedfrom the approximate analytic formula Eq. (2.25). The dependence on f is mild becauseit only enters the higher order corrections for fixed M H ± /f and affects the starting pointof RG running of the couplings. On the other hand, M h decreases as the cutoff M ρ ofthe electroweak gauge loops increases due to its negative contribution to λ h . For this setof { ξ, λ , λ } , we see that the correct Higgs boson mass is close to the upper bound for M ρ ≈ f and hence we cannot have M ρ > f . This upper bound depends on the valuesof the coupling parameters, which will be examined later.14
00 126 15790 126 14580 100 1210 2 4 6 8 10 12246810 M H (cid:177) (cid:144) f f (cid:72) T e V (cid:76) T (cid:61) (cid:61) M h (cid:72) GeV (cid:76) Λ Ξ (cid:61) Ξ (cid:61) Ρ (cid:61) Ρ (cid:61) Ρ (cid:61) Figure 1: Contours of Higgs boson mass (labelled in GeV), for ξ = 3 . λ / ξ = 0 . λ = 0. The solid (blue) lines correspond to no electroweak corrections ( M ρ = 0, seeSection 3), while the dashed (red) and dotted (brown) lines correspond to M ρ /f = 3 and 5,respectively. The dark and light shaded regions correspond to T > . . < T < . T -parameter given by Eq. (4.5).In Fig. 1 we also include contours of the T parameter [24], which measures the weak-isospin violation and constitutes the strongest constraint on this model. The Higgs fieldarises as the pNGBs of the broken U (3) L symmetry, which does not contain a custodial SU (2) symmetry. As a result, the dimension-6 operator | H † χ D µ H χ | , which representsthe weak-isospin violation, is expected based on na¨ıve dimensional analysis to have acoefficient ∼ π / Λ ∼ /f . In our model, the scalar fields are composite degrees offreedom and only exist below the compositeness scale. All low-energy operators containingderivatives of scalars, including the kinetic term and the weak-isospin violating operator | H † χ D µ H χ | operator, are generated by fermion loops [15, 25], which can be calculated inthe leading N c approximation. The leading contribution to the T parameter is thereforecaptured by the loops involving the t (cid:48) quark [10–12], and is given by [11] T = 3 s L π αv (cid:20) s L m t (cid:48) + 4(1 − s L ) m t (cid:48) m t m t (cid:48) − m t ln (cid:18) m t (cid:48) m t (cid:19) − (2 − s L ) m t (cid:21) , (4.5)where s L is the sine of the left-handed mixing angle, given in Eq. (2.22). This is equivalentto calculating the coefficient of the | H † χ D µ H χ | operator, with the Higgs field replacedby its VEV. Contributions to the T parameter from loops with heavier scalars (whichalso come from fermion loops in the UV theory, but subleading in N c ) are very smallcompared to the contribution from fermion loops calculated here. The contribution to15 .025 0.05 0.1 0.20.4 Ξ f (cid:72) T e V (cid:76) T Figure 2: Contour plot of the T parameter in the ξ − f plane, given by Eq. (4.5). The T parameter is roughly proportional to 1 /f . The constraint from the electroweak fit( T < .
15 at the 95% CL) implies f (cid:29) v .the S parameter is negligible because we have added only vector-like quarks to the SM.In the above discussion of the T parameter we have assumed that the strong dynamicsresponsible for the composite Higgs sector includes only 4-fermion operators of the NJLtype [12], which are equivalent to the product of a left-handed current and a right-handedone (LR). If these arise from a spontaneously broken gauge symmetry, then they areaccompanied by LL and RR 4-fermion operators that contribute to the coefficient ofthe | H † χ D µ H χ | operator [26]. The ensuing correction to the T parameter, coming fromdiagrams with two or more fermion bubbles, is expected to be somewhat smaller thanour result in Eq. (4.5). In what follows we will ignore these model-dependent correctionsrelated to UV physics.Eq. (4.5) only depends on ξ , f , v and m t , where the last two are fixed by theirexperimental values. With m t (cid:48) ≈ ξf / √ s L ≈ v/f [Eqs. (2.21), (2.22)], one can see thatall three terms in Eq. (4.5) are roughly proportional to 1 /f , while they have differentdependence on ξ . Eq. (4.5) can be rewritten in terms of ξ and f as T ≈ π αf (cid:20) v ξ m t ln (cid:18) ξf √ m t (cid:19) − m t (cid:21) . (4.6)By using the low energy values of the Yukawa coupling ξ , the running effect from thecompositeness scale to the fermion masses is also included. This effect reduces the T parameter somewhat compared to the na¨ıve estimated value [ ∼ v / (2 αf )] and rendersa slightly milder constraint. Contours of T in the ( f, ξ ) plane are shown in Fig. 2. From16
00 126 17090 12690 126 150 M H (cid:177) (cid:144) f Λ (cid:144) (cid:72) Ξ (cid:76) M h (cid:72) GeV (cid:76) f (cid:61) TeV
Ξ (cid:61) Ρ (cid:61) Ρ (cid:61) Ρ (cid:61) Figure 3: Contours of Higgs boson mass (labelled in GeV), for ξ = 3 . λ = 0, and f = 4 TeV. The solid (blue) lines correspond to no electroweak corrections ( M ρ = 0, seeSection 3), while the dashed (red) and dotted (brown) lines correspond to M ρ /f = 3 and5, respectively.the current electroweak fit [27], the 68% and 95% bound roughly correspond to T = 0 . T = 0 .
15 (for S = 0). For ξ = 3 .
6, these bounds translate to f (cid:38) . f (cid:38) . λ / (2 ξ ), M H ± /f and M ρ /f . The dependence on the other parametersare expected to be mild. To study the M h dependence on the more sensitive parameters,we fix ξ = 3 . λ = 0, and f = 4 TeV (which corresponds to T =0.12, close to its lowerbound), and make the contour plot of the Higgs boson mass in the λ / (2 ξ )-versus- M H ± /f plane for several different values of M ρ /f (Fig 3). As expected, a larger M h occurs forlarger M H ± /f , λ / (2 ξ ) and smaller M ρ /f . There is an upper limit M h (cid:46)
175 GeV evenfor extreme values of these parameters. This is close to our estimate from the analyticformula, Eq. (2.26).A lower bound on M h follows from the condition that the quartic coupling λ h ispositive at the matching scale m t (cid:48) . Otherwise of our vacuum is not a minimum of thetree-level potential, and the universe is more likely to end up in wrong vacuum. Imposingthe boundary condition λ h = 0 at the scale m t (cid:48) (cid:39) ξf / √
2, and using the SM 1-loopRG equations to evolve λ h down to the weak scale, we find that the physical M h grows17 M H (cid:177) (cid:144) f M h (cid:72) G e V (cid:76) Λ (cid:144)(cid:72) Ξ (cid:76) (cid:61) Ρ (cid:61) Ξ (cid:61) (cid:61) Λ (cid:61) Ξ (cid:61) (cid:61) Λ (cid:61) Ξ (cid:61) (cid:61) Λ (cid:61) Ξ (cid:61) (cid:61)
10 TeV, Λ (cid:61) Ξ (cid:61) (cid:61) Λ (cid:61)(cid:45) Λ Figure 4: Higgs boson mass as a function of M H ± /f for various values of ξ, f, λ , when λ / ξ = 0 . M ρ /f = 3. The different curves are obtained by varying one parameterat a time with respect to the solid (orange) curve.monotonically from 80 GeV for m t (cid:48) = 6 TeV (corresponding, e.g. , to f = 3 . ξ = 2 . m t (cid:48) = 25 TeV. Thus, the lower bound on M h is around 80 GeV. We concludethat, in our composite Higgs model, the Higgs boson mass is constrained to be in the80 GeV < M h <
175 GeV (4.7)range, with the upper limit significantly tighter for most of the parameter space (as shownin Figs. 1 and 3); the measured 126 GeV Higgs mass sits comfortably in the middle ofthis range.So far we have considered the Higgs boson mass dependences on the more sensitiveparameters. To check how the Higgs mass varies with the less sensitive parameters,we show in Fig. 4 the Higgs mass as a function of M H ± /f for several different sets of { ξ, f, λ } , by fixing λ / (2 ξ ) = 0 . M ρ = 3 f . We see that indeed the dependenceson these parameters are rather mild. Among them, the Yukawa coupling ξ , which entersboth the higher order correction in Eq. (2.25) (through m t (cid:48) ) and the electroweak gaugeloop correction in Eq. (3.8) [for a fixed value of λ / (2 ξ )], has a slightly larger effect. The λ dependence is almost negligible.In addition to M h , we are also interested in other predictions of this model, such asthe spectrum of heavy states. This model contains two doublet and two singlet scalars in18he effective theory. After electroweak symmetry breaking, apart from the eaten Nambu-Goldstone modes and the SM-like Higgs boson, there are 3 neutral CP-even scalars (de-noted by H , , according to the ascending order of their masses), 3 neutral CP-odd scalars(denoted by A , , ), and a complex charged Higgs boson ( H ± ). One CP-odd scalar ( A ) islighter than other heavy states because it is also the pNGB of the broken U (3) L symmetryand hence its mass is also suppressed by s γ . However, its mass is controlled by f ratherthan v , so it is still quite heavy compared to the Higgs boson. The masses of the statescoming from Φ t receive contribution from both M tt and λ f . In the limit of M tt (cid:29) λ f ,all these states are expected to be around M H ± and decouple from low energy physics.We use the mass of the discovered Higgs boson to fix one more parameter. We chooseit to be M ρ and plot its required values to obtain the correct Higgs boson mass in the M H ± /f − λ / (2 ξ ) plane in the first panel of Fig. 5. The contour M ρ /f = 0 representsthe case where the explicit U (3) L breaking from the electroweak gauge loops is absent. M h = 126 GeV cannot be obtained in the region to the left of that contour, unless thereis additional explicit U (3) L breaking from the cutoff scale that has positive contributionsto λ h . For small values of M H ± or λ / (2 ξ ), M ρ /f also needs to be small, which meansthat additional states responsible for cutting off the electroweak gauge loop contributionare probably required to appear near or even below the scales of our heavy scalars.Assuming that M ρ takes the value which produces M h ≈
126 GeV, we can calculatethe masses of the heavy scalars in this theory. We show the masses of the lightest CP-oddscalar ( A ) and the first two heavy CP-even neutral scalars ( H , H ) in the other threepanels of Fig. 5, for the same parameters as in Fig. 3. For fixed M H ± /f , the heavy statesscale proportionally to f . Since f = 4 TeV is close to the lower bound allowed by the T parameter, the masses in these plots are close to their lower limits in this model. We seethat A is indeed much lighter compared to other heavy states, though it is still above1 TeV. For small values of M H ± /f , H mostly comes from Φ t and its mass is close to M H ± , so the contours run vertically. On the other hand for large M H ± /f , H dominantlyconsists of ϕ χ and its mass is proportional to λ , and is independent of M H ± . The secondpseudoscalar A is closely degenerate with H ± , and the masses of H , A are a little bithigher but are also close. 19 M H (cid:177) (cid:144) f Λ (cid:144) (cid:72) Ξ (cid:76) M Ρ (cid:144) f M H (cid:177) (cid:144) f Λ (cid:144) (cid:72) Ξ (cid:76) M A (cid:72) TeV (cid:76) M H (cid:177) (cid:144) f Λ (cid:144) (cid:72) Ξ (cid:76) M H (cid:72) TeV (cid:76)
15 20 30 37 M H (cid:177) (cid:144) f Λ (cid:144) (cid:72) Ξ (cid:76) M H (cid:72) TeV (cid:76)
Figure 5: Contour plots for M ρ /f , the mass of the lightest CP-odd neutral scalar ( M A )and the masses of the second and third lightest CP-even neutral scalars ( M H and M H ).We have fixed f = 4 TeV, ξ = 3 . λ = 0, which corresponds to m t (cid:48) = 10 . T = 0 .
12. The shaded regions are not consistent with M h = 126 GeV (unless there isadditional U (3) L breaking with effects opposite to the electroweak corrections). Within this composite Higgs model, the lightest particle beyond the SM is the CP-oddscalar A . Its mass is larger than M h by a factor of f /v , as shown in Eq. (2.28). Takinginto account the scale dependence of M h , the A mass M A is slightly reduced (see secondpanel of Fig. 5). The smallest value of f allowed by the electroweak data, f = 3 . M A = 1 . A may be singly produced at hadron colliders through gluonfusion. Given that A is mostly part of the φ χ singlet, its coupling to the top quarkis suppressed. The dominant contribution to gluon fusion is a t (cid:48) loop, relying on the( ξ/ √ A ¯ t (cid:48) γ t (cid:48) coupling. For f = 3 . A production cross section is ∼ . A may decayinto t ¯ t or hZ , but in either case the cross section for pp → A X is too small, even for aluminosity of 3000 fb − [30].The other composite scalars as well as the vector-like quark are too heavy to be probedat the LHC. A hadron collider at a center-of-mass energy of O (100) TeV would be neededin order to produce them. For a typical m t (cid:48) = 9 TeV, the t (cid:48) ¯ t (cid:48) production cross sectionat a 100 TeV pp collider (we refer to it as the VLHC) is 0.12 fb, based on MadGraph 5,with a 50% increase to account for higher order effects. The VLHC cross section can beas large as 1.8 fb, for m t (cid:48) = 6 TeV. Besides the usual t (cid:48) → W b, th, tZ decays [31], there isan interesting t (cid:48) → tA decay, followed by A → t ¯ t or hZ . Given that the backgroundsrelevant at an invariant mass (cid:38) t (cid:48) quark atthe VLHC even with the initial luminosity of 2 × cm − s − [32].Single A production at the VLHC through gluon fusion would have a cross sectionaround 100 fb, for M A = 1 . A production leads to promising channels.The heaviness of the particles beyond the SM puts this model close to the decouplinglimit, so the Higgs couplings are close to the SM predictions. The main correction comesfrom mixing with the singlet scalar fields, which can be seen in Eq. (2.23). We haveverified numerically that throughout the allowed parameter space, the couplings of theHiggs boson to SM particles are given, to a very good approximation, by their SM valuestimes the factor cos( v/f ) (cid:39) − v / (2 f ), which is the fraction of the doublet componentin the Higgs boson. The deviation from the SM couplings is only 0.2% for f = 4 TeVand is inversely proportional to f . The modifications to the branching fractions are evensmaller because the dominant correction is universal. Such small deviations can not betested at the LHC and are probably even beyond the reach of a future e + e − collider.Nevertheless, a precise determination of the custodial SU (2) breaking T parameter canhelp to probe or constrain this model further.Since the SM-like Higgs field is composite and made of the top quark and the newquark χ , the light SM fermion masses presumably come from some 4-fermion interactions21enerated above the cutoff scale. Constraints on flavor changing neutral currents (FCNC)limit the coefficients and patterns of these 4-fermion interactions [12]. Below the cutoff,our model has two Higgs doublets. General couplings of SM fermions to the two Higgsdoublets lead to tree-level FCNCs. However, it is reasonable to assume that the couplingsof the SM fermions to heavy Higgs states have a hierarchical structure correlated withthe corresponding Yukawa couplings of the SM Higgs boson, due to some approximateflavor symmetries [33]. In that case, the FCNC constraints require the scalars from theother Higgs doublet to be heavier than a few hundred GeV to ∼ µ → eγ . (See Ref. [34]for a comprehensive study and review.) With the heavy states around 10 TeV in ourmodel, these FCNC constraints are easily satisfied. Only A is relatively lighter, but it ismostly part of the φ χ singlet, so that it does not induce any significant FCNC effects. We have studied a composite Higgs model based on non-confining dynamics, in which thenewly discovered Higgs boson is a bound state of a vector-like quark and the left-handedtop quark. The strongly coupled 4-quark interactions that describe the non-confiningdynamics at the compositeness scale Λ produce scalar bound states which consist of two SU (2) W -doublets and two gauge singlets [12]. We have shown that if the underlying strongdynamics preserves an approximate U (3) L chiral symmetry, a SM-like Higgs doublet arisesnaturally as the pNGB of U (3) L → U (2) L breaking.Explicit U (3) L breaking terms produce the correct m t through the top-seesaw mecha-nism. They also give the mass to the SM-like Higgs boson. As a result, the Higgs and topmasses are tightly correlated, and satisfy M h (cid:46) m t . Electroweak effects further reduce M h , so that it is easily compatible with the measured Higgs mass within the natural rangeof parameter space.The strongest constraint on this model comes from weak-isospin violation due to heavyquark loops. Requiring T (cid:46) .
15 pushes the U (3) L symmetry breaking scale f above 3.5TeV, so that some fine-tuning is needed to obtain the weak scale v ≈
246 GeV (cid:28) f .It also means that most of the new states beyond the SM (except the lightest CP-oddscalar) will have masses around or above 10 TeV, beyond the reach of the LHC. Thecorrections to the SM Higgs couplings are tiny as the new sector is close to the decouplinglimit. Nevertheless, the fact that no new particles or any deviation from the SM has been22iscovered at the LHC so far suggests that the SM Higgs sector is somewhat tuned, andthe scale of new physics may be higher than previously thought. Our model is certainlyconsistent with the current experimental observations. It would require a collider beyondthe LHC with a center-of-mass energy O (100) TeV to probe the heavy states in this modeldirectly.On the other hand, the scale of U (3) L chiral symmetry breaking may be significantlylowered if the contribution to T from the heavy quark can be cancelled by some additionalcontribution. This would make the model less tuned and the new states lighter and moreaccessible. If the model can be extended to include an approximate custodial SU (2)symmetry, for example, by adding a new vector-like quark to mix with the bottom quark,then f can be lowered to (cid:46) Acknowledgements : We would like to thank Sekhar Chivukula, Markus Luty andJohn Terning for useful discussions. H.-C. Cheng and J. Gu are supported by the Depart-ment of Energy (DOE) under contract no. DE-FG02-91ER40674. Fermilab is operatedby the Fermi Research Alliance under Contract No. De-AC02-07CH11359 with the DOE.H.-C. Cheng would like to acknowledge the hospitality of National Center for TheoreticalSciences (North) in Taiwan where part of this work was done.
A Renormalization group running of couplings
In Sec. 2.2 the light Higgs boson mass is shown to be proportional to the ratio of thecouplings, (cid:112) λ / (2 ξ ), in the absence of SU (2) W × U (1) Y gauge interactions. In thefermion-loop (bubble) approximation, this ratio is predicted to be 1, which correspondsto the well-known result of m φ = 2 m f in the NJL type model [19], where m φ is the massof the composite scalar and m f is the constituent fermion mass after chiral symmetrybreaking. The fermion-loop approximation neglects the gauge loop corrections and theback reaction of the scalar self-interactions. It can be viewed as the leading N c resultif the gauge interactions are ignored. The presence of the other interactions will modifythis ratio. If the chiral symmetry breaking scale f is tuned to be much smaller than thecompositeness scale Λ, this ratio can also be well-determined due to the infrared fixedpoint structure of the RG equations [9]. Quasi-infrared fixed points have been used topredict top quark and Higgs boson masses in some theoretical models with large couplings23t high scales [18, 35]. For f not much smaller than Λ as in the case we are interested,one cannot trust the RG analysis because the couplings are strong and the logarithms areonly O (1). Nevertheless, finding the infrared fixed point of the RG equations may stillprovide us some ideas of the possible range of the relevant coupling ratio λ / (2 ξ ). In thefar infrared the couplings become perturbative, and the fixed point can be determined by1-loop RG equations presented below.To be general and to identify the fermion and scalar loops, we write down the coupledRG equations of the couplings ξ , λ , λ , and QCD strong coupling g for an U ( N L ) L × U ( N R ) R theory:16 π dg dt = − (cid:18) − N f (cid:19) g , (A.1)16 π dξdt = (cid:18) N L + N R N c (cid:19) ξ − N c − N c g ξ, (A.2)16 π dλ dt = 2( N L + N R ) λ + 4 λ λ + 4 N c ( ξ λ − ξ ) , (A.3)16 π dλ dt = 4 λ + 4( N L + N R ) λ λ + 2 N L N R λ + 4 N c ξ λ , (A.4)where we have ignored the electroweak couplings g , g , and the light fermion Yukawacouplings. N f is the number of quark flavors. These equations can be inferred fromRef. [36].These RG equations are in the mass-independent scheme. Near the composite scalewhere the composite scalars dissolve, the scalar masses are large and the scalar loopsshould decouple [37]. This justifies the fermion-loop approximation near the composite-ness scale if the gauge couplings are relatively small. If we drop the scalar loop contribu-tions ( i.e. , terms without the N c factor) and ignore the gauge couplings, we obtain16 π ddt ln (cid:18) λ ξ (cid:19) = 2 N c ξ (cid:18) − ξ λ (cid:19) . (A.5)We see that the infrared fixed point corresponds to λ = 2 ξ , agreeing with the result ofthe fermion-loop approximation. On the other hand, λ is not generated by the fermionloops. Note that the fermion-loop approximation sums fermion loops to all orders so itapplies even to large couplings. As a result, one may treat λ = 2 ξ , λ = 0 as the initialcondition when the scalar loops become relevant.It is instructive to derive the approximate IR fixed point analytically for the ratios ofcouplings. For simplicity we first neglect the QCD coupling g because it is much smaller24han the other couplings near the cutoff scale. We obtain the RG equations for r ≡ λ /λ and s ≡ λ /ξ by combining Eqs. (A.2)-(A.4):16 π d ln rdt = 2 λ (cid:20) r + N L + N R + ( N L N R − r + 2 N c s (cid:21) , (A.6)16 π d ln sdt = ξ (cid:20) N L + N R + 2 r ) s + 2 N c − N L − N R − N c s (cid:21) . (A.7)The infrared fixed point is reached when the right-hand side of the equations vanishes:2 r + N L + N R + ( N L N R − r + 2 N c s = 0 , N L + N R + 2 r ) s + 2 N c − N L − N R − N c s = 0 . (A.8)There are multiple solutions to these polynomial equations. The actual IR fixed pointhas | r | (cid:28)
1, so we can further simplify the equations by ignoring the terms proportionalto positive powers of r , then Eq. (A.8) gives s ∗ (cid:39) (cid:16) − x + √ x + x (cid:17) , (A.9)where x ≡ N c / ( N L + N R ). We have chosen the positive solution because both ξ and λ stay positive. Substituting it into Eq. (A.8), we obtain r ∗ (cid:39) − xN c (1 + x/s ∗ ) . (A.10)For N L = 3 , N R = 2 and N c = 3, we have s ∗ ≈ r ∗ ≈ − . . (A.11)To check the accuracy of the analytical approximation of the IR fixed point solution,we solve the 1-loop RG equations (A.1)–(A.4) numerically. We set the initial condition λ = 2 ξ , λ = 0 and choose several different initial values for ξ . The results of 1-loopRG running are shown in Fig. 6. We see that the ratios of couplings are quickly driven tothe approximate fixed point values given by Eq. (A.11), though we should not trust theexact evolution in the beginning due to potentially large higher loop contributions. Theinfrared value of r is a bit smaller than the approximate result in Eq. (A.11) due to thegauge loop contribution from g .If the chiral symmetry breaking scale is not far below the compositeness scale, we cannot trust the 1-loop RG results. However, if we assume a smooth evolution, the ratios of25 (cid:45) (cid:45) log (cid:72) Μ (cid:76) c oup li ng r a ti o s Λ (cid:144)(cid:72) Ξ (cid:76) Ξ (cid:72) (cid:76) (cid:76) (cid:61) ,5 20 Λ (cid:144) Λ Ξ (cid:72) (cid:76) (cid:76) (cid:61) ,5 20 (cid:76)(cid:61) GeV
Figure 6: One-loop RG evolutions of the coupling ratios λ / (2 ξ ) and λ /λ for initialvalues λ / (2 ξ ) = 1, λ /λ = 0 and ξ = 5 or 20. The horizontal axis is the logarithm ofthe energy scale.couplings are expected to lie in between their initial values and the infrared fixed pointvalues: 0 . (cid:46) λ ξ (cid:46) , − . (cid:46) λ λ (cid:46) . (A.12)We use these ranges in Sections 2 and 4. References [1] G. Aad, et al. [ATLAS Collaboration], “Observation of a new particle in the searchfor the SM Higgs boson”, Phys. Lett. B , 1 (2012) [arXiv:1207.7214].S. Chatrchyan, et al. [CMS Collaboration], “Observation of a new boson at a massof 125 GeV”, Phys. Lett. B , 30 (2012) [arXiv:1207.7235].[2] D. B. Kaplan and H. Georgi, “ SU (2) × U (1) breaking by vacuum misalignment”,Phys. Lett. B , 183 (1984); D. B. Kaplan, H. Georgi and S. Dimopoulos, “Com-posite Higgs scalars,” Phys. Lett. B , 187 (1984); T. Banks, “Constraints on SU (2) × U (1) breaking by vacuum misalignment”, Nucl. Phys. 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