A Near-Conformal Composite Higgs Model
AA Near-Conformal Composite Higgs Model
Thomas Appelquist, James Ingoldby, and Maurizio Piai Department of Physics, Sloane Laboratory, Yale University, New Haven, Connecticut 06520, USA Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151, Trieste, Italy Department of Physics, College of Science, Swansea University, Singleton Park, Swansea, Wales, UK
We analyze a composite Higgs model based on the confining SU (3) gauge theory with N f = 8Dirac fermions in the fundamental representation. This gauge theory has been studied on the latticeand shown to be well described by a dilaton effective field theory (EFT). Here we modify the EFTby assigning standard-model quantum numbers such that four of the composite pseudo-Nambu-Goldstone boson (pNGB) fields form the standard-model Higgs doublet, by coupling it to the topquark, and by adding to the potential a term that triggers electroweak symmetry breaking. Themodel contains a pNGB Higgs boson, a set of heavier pNGBs, and an approximate dilaton in thesame mass range. We study the phenomenology of the model, and discuss the amount of tuningrequired to insure consistency with current direct and indirect bounds on new physics, highlightingthe role of the dilaton field. I. INTRODUCTION
Lattice studies of the SU (3) gauge theory with N f = 8Dirac fermions in the fundamental representation showevidence of a light scalar singlet [1–5]. (Similar resultshold with N f = 2 fermions in the symmetric represen-tation [6–11].) The suggestion that this state might bea dilaton has fueled a revival of interest in the dilatoneffective field theory (EFT). Its history dates back to dy-namical symmetry breaking [12–14], well before this re-cent lattice-driven activity [15–26]. Existing lattice data,analyzed via the dilaton EFT [20, 21] yielded the firstmeasurement of a key, large anomalous dimension relatedto the fermion bilinear condensate [27]. The results areconsistent with earlier expectations [28] and with recenthigh-loop perturbative studies [29, 30].This theory, with a global SU (8) × SU (8) symmetry,broken to the diagonal SU (8), is a natural candidateto build a composite Higgs model (CHM) [31–33], (seealso [34–40] and references therein). Lattice studies of SU (2) [41–47], SU (4) [48–52], and Sp (4) [53–55] gaugetheories have explored the possible origin of CHMs. The SU (3) gauge theory has distinctive features: the presenceof a light scalar singlet which modifies the EFT descrip-tion of the CHM (see also Ref. [56]), and the presenceof large anomalous dimensions. Furthermore, ordinarybaryons can give rise to top compositeness [34].In this paper, we show that the presence of the dilatonfield in the EFT allows us to construct an appealing CHMbased upon the SU (8) × SU (8) /SU (8) coset. We demon-strate that observables such as the ratio of the mass of theHiggs boson, m h (cid:39)
126 GeV, to the electroweak vacuumexpectation value (VEV), v (cid:39)
246 GeV, and to the massof the additional heavy scalars, are substantially alteredwith respect to generic CHM expectations. We highlighthow current lattice studies might already be exploringphenomenologically relevant regions of parameter space.These statements depend on the value of a (currently)unknown scaling dimension w , which in principle can bemeasured on the lattice. Fermion SU(2) L U(1) Y SU (3) c SU (3) L α R , (cid:32) / − / (cid:33) T S SU (3) c × SU (2) L × U (1) Y is the SM gauge group,while SU (3) is the strongly coupled gauge symmetry. Wedenote with α = 1 , SU (2) L index. The fermions de-noted by R , form a fundamental representation of the global SU (2) R custodial symmetry. A model with similar assign-ments has been considered in Ref. [34]. II. THE MODEL
We assign to the eight Dirac fermions the quantumnumbers indicated in Table I. The global SU (8) × SU (8)symmetry group is broken both explicitly (by a diagonalmass term) and spontaneously (by the strong dynam-ics) to its diagonal SU (8) subgroup. The gauge group ofthe standard model (SM) is a subgroup of the unbroken SU (8). The EFT description contains 63 pseudo-Nambu-Goldstone Bosons (pNGBs) denoted as π a , and one ad-ditional SU (8) singlet, the dilaton, which we denote as χ . We ignore the U (1) A meson, which is a singlet andhas a large mass, due to the anomaly.The CHM construction starts from the observationthat 8 of the pNGBs have the correct quantum numbersto form two copies of the Higgs doublet of the standardmodel. We further modify the dilaton EFT of Refs. [20–22] by adding two terms: a coupling of one of these twodoublets to the top quark, and a related potential termfor the pNGBs. In this paper, we ignore all SM fermionsother than the top; the generalization to include otherSM fermions within the EFT framework is straightfor-ward. a r X i v : . [ h e p - ph ] J a n The EFT Lagrangian density that results from thisconstruction is the following: L = 12 ( ∂ µ χ ) + L π + L M − V ( χ )+ L Y − V t + L , (1)where the dilaton field χ acts as a conformal compen-sator, coupling to EFT operators in such a way as torestore scale invariance in Eq. (1). It acquires a VEV (cid:104) χ (cid:105) ≡ F d , breaking scale invariance spontaneously.The kinetic term for the pNGBs is L π = F π (cid:18) χF d (cid:19) Tr (cid:104) D µ Σ ( D µ Σ) † (cid:105) , (2)where F π enters the EFT as the scale of spontaneousbreaking of SU (8) × SU (8). The matrix–valued fieldΣ represents the 63 pNGBs spanning the SU (8) × SU (8) /SU (8) coset. The covariant derivatives describecouplings to the SM gauge bosons, following the embed-ding identified in Table I. Their kinetic terms and selfinteractions are the standard ones, which we include in L . Σ satisfies the nonlinear constraint ΣΣ † = .The Dirac mass given to the fermions of the new strongsector leads directly to the following term in the EFT L M = M π F π (cid:18) χF d (cid:19) y Tr (cid:2) Σ + Σ † (cid:3) , (3)and breaks the global symmetry. The quantity M π setsthe scale for the masses of the 59 pNGBs besides thosethat become the Higgs doublet. The parameter y hasbeen interpreted as the scaling dimension of the fermionbilinear condensate in Ref. [27]. Its value is y = 2 . ± .
05 [22].The scalar potential V ( χ ) describes the self-interactions of the dilaton field. It encodes both the spon-taneous and explicit breaking of scale symmetry origi-nating from the underlying gauge theory. We provideda general form for this potential in Ref. [22], where itplayed a key role. Here we will not find it necessary tofurther invoke the explicit form of V ( χ ).At the level of the EFT, we describe the mass of thetop quark using the Yukawa interaction L Y = y t F π (cid:18) χF d (cid:19) z (cid:0) ¯ Q αL t R (cid:1) Tr [ P α Σ] + h.c. . (4)The underlying gauge theory determines the scaling di-mension z . H α ≡ T r [ P α Σ] transforms as the Higgs Dou-blet, with quantum numbers (2 , − /
2) under SU (2) L × U (1) Y . Here α is the index of SU(2) L . We take theprojectors P α to be the following 8 × P α = (cid:18) ˜ P α (cid:19) , (5) with˜ P = 12 − , ˜ P = 12 . (6)The operator in Eq. (4) breaks the SU (8) × SU (8)global symmetry. In the underlying theory, interac-tions responsible for generating Eq. (4) also generate an SU (8) × SU (8) breaking contribution to the potential ofthe form V t = − C t (cid:18) χF d (cid:19) w (cid:88) α =1 | Tr [ P α Σ] | . (7)The (unknown) scaling dimension w derives from the un-derlying gauge theory. This potential has non-Abelianglobal symmetry SU (2) L × SU (2) R × SU (4), the SU (4)remaining due to the vanishing entries in P α . We notethat the Yukawa interaction in Eq. (4) breaks SU (2) R explicitly. There are subleading contributions to this po-tential that break the SU (2) R symmetry but they aresmaller than V t , and we will not consider them. Simi-larly, the gauging of the SM subgroup breaks the globalsymmetries, and leads to additional contributions to thepotential, but they are smaller and we neglect them inthis analysis.A loop of top quarks can also generate the interac-tion in Eq. (7), naturally making the constant C t posi-tive. In addition, partial top compositeness can generateEq. (4)—and hence Eq. (7)—in which case the field Q αL couples linearly to the baryon operator B αL = L α ( T R ) † + R ( T L ) † α and t R to B R = 2 R ( T R ) + L ( T L ) in theunderlying theory. (Parentheses indicate contraction ofspinor indices.) Alternatively, Eq. (4) can be generatedby coupling the elementary fermion bilinear (cid:0) ¯ Q αL t R (cid:1) tothe mesonic operator O αM = ( L α R ) − (cid:15) αβ (cid:0) R L β (cid:1) . III. THE VACUUM
We first analyze the vacuum of the EFT. Both thepNGB that we identify with the Higgs field and the dila-ton have nontrivial vacuum values, which must be cal-culated simultaneously. Then we determine the mass ofthe composite Higgs boson in this vacuum, emphasizingthe significant role played by the dilaton field.The three terms without derivatives or the top quarkfield in the Lagrangian of Eq. (1) define a potentialfor both the pNGBs and the dilaton. It is helpful toparametrize the Σ field asΣ = exp iθ × − i × i × × × × × , (8)where only the degree of freedom corresponding to thepNGB component of the Higgs boson (represented by θ )is shown, for simplicity. The potential then reads W ( χ, θ ) = V ( χ ) − C t (cid:18) χF d (cid:19) w sin θ − M π F π (cid:18) χF d (cid:19) y (1 + cos θ ) , (9)Minimizing this potential determines the vacuum value F d of χ , and the vacuum value of θ (the misalignmentangle). We henceforth use θ to denote this vacuum valuerather than the dynamical pNGB field. The electroweakscale v (cid:39)
246 GeV is related to the misalignment anglethrough v = √ F π sin θ . The top acquires the mass m t = y t v/ √ θ = M π F π C t , (10)and0 = ∂V∂χ (cid:12)(cid:12)(cid:12)(cid:12) F d − yM π F π F d − M π F π F d (cid:18) w sin θ cos θ − y (1 − cos θ ) (cid:19) . (11)These equations determine θ in terms of C t and providea relation between F d and other EFT parameters withthe dilaton potential V .To comport with the SM at currently accessible ener-gies, we must find a small misalignment angle θ (cid:28)
1, thatis, a large separation between v and F π . This is achievedby tuning C t in Eq. (10).For θ (cid:28)
1, Eq. (11) determining F d simplifies in anessential way. The second line is suppressed, and may beneglected in first approximation. The resulting equationis precisely the one used in Refs. [20, 21] to relate F d tothe other parameters in the EFT employed there. ThatEFT, with no potential term proportional to C t , was usedto fit lattice data for the SU (3) gauge theory with N f =8. The functional form of the scalar potential V ( χ ) wasconstrained in that fit.The mass matrix for the χ and θ degrees of freedom isapproximately given (for small misalignment angle) by M = (cid:32) M d θ √ M π F π ( y − w ) F d θ √ M π F π ( y − w ) F d θ M π (cid:33) , (12)The (1,1) entry is the second derivative of W ( χ, θ ) withrespect to χ at χ = F d in the limit θ →
0. It is express-ible in terms of the scalar potential V ( χ ) and other EFTparameters by M d = ∂ V∂χ (cid:12)(cid:12)(cid:12)(cid:12) F d − y ( y − F π F d M π , (13)which was employed in Refs. [20, 21] to fit lattice data.In the present context, M d is the approximate mass ofthe heavy scalar eigenstate, composed principally of χ . By diagonalizing the mass matrix in Eq. (12), we findthat the mass m h of the lightest eigenstate (correspond-ing to the Higgs boson) is given by m h v = M π F π (cid:18) − M π F π ( y − w ) M d F d (cid:19) , (14)up to O ( θ ) corrections. The second term in the paren-theses, arising from the presence of the dilaton field, is adistinctive feature of this model. Its presence allows us toaccommodate the measured ratio m h /v ≈ .
26, drawingdirectly on lattice data for which M π / F π is typically anorder of magnitude larger. IV. PHENOMENOLOGY
In this section we examine the spectrum of particlesof our CHM and the conditions under which constraintsfrom collider experiments are satisfied. We also discussthe amount of fine tuning needed for the model to satisfythe experimental constraints.The spin-0 part of the spectrum of our EFT consistsof 63 NGBs and pNGBs associated with the spontaneousbreaking of the SU (8) × SU (8) symmetry of the underly-ing gauge theory, along with a scalar state of approximatemass M d . The 3 massless NGBs are eaten by the W ± and Z . One state is the relatively light pNGB Higgs bo-son of mass m h (Eq. (14)), while 59 are heavier pNGBstates with their mass scale set by M π . One additionalheavier state has mass M d . The quantities F π and F d are decay constants associated with these states. To setthe relative size of M π , M d , F π , we draw directly fromthe LSD lattice measurements [5].Neglecting the SM gauge interactions, the EFT hasapproximate SU (2) L × SU (2) R × SU (4) global symmetry.The pNGB multiplet decomposes into representations ofthis symmetry as follows63 = (3 , ,
1) + (1 , ,
1) + (1 , ,
1) + (2 , , , ,
15) + (2 , ,
4) + (1 , ,
4) + (1 , , . (15)The misaligned vacuum breaks SU (2) L × SU (2) R spon-taneously to its diagonal subgroup SU (2) D . As a result,the composite spectrum is organized in a set of (approx-imate) multiplets of SU (2) D × SU (4).To determine the spectrum, we first specify the quanti-ties { M π , M d , F π , F d , y, m h , v } . We use data from latticestudies of the N f = 8 gauge theory. We extract the ratios M π /F π and M d /F π from Tables. III and IV of Ref. [5] forfive different constituent fermion masses m fi . We thentake y = 2 .
06 and F π /F d = 0 .
086 from Ref. [22], and set m h and v to their experimentally determined values. Fi-nally, we must set the overall scale for the new compositesector. As a benchmark, we take M π = 4 TeV, to ensurethat the 59 heavier pNGBs lie outside the reach of directsearches. The strongest bounds coming from searches forcolor octet scalars require M π (cid:38) . SU (2) D SU (4) Mass (TeV) m f m f m f m f m f SU (2) D × SU (4) quantum numbers, shown in theleft-hand column. The m fi refer to the 5 different values forthe constituent fermion mass appearing in the lattice studyof Ref. [5] (arranged in ascending order) that is used as aninput into these estimates. independent of which lattice point ( m fi value) we usein the analysis. The small variation in the mass of theheaviest singlet state (the dilaton) is mostly due to fluc-tuations in the lattice measurement of M d /M π .In calculating the spectrum, the quantities w (forwhich there is no existing determination from latticedata) and C t are chosen to reproduce the aforementionedconstraints. We find that C t ≈ (2 TeV) , with the precisedetermination depending on the lattice point considered.Similarly, F π ≈ w . For illustration purposes, we take latticedata for M π /M d at the second fermion mass point m f from Ref. [5] (along with values for y and F π /F d fromRef. [22]) accounting for their uncertainties and shade inyellow the allowed ranges for m h /v and w . We see thatif we require m h /v (cid:39) .
26 for consistency with experi-ment, w could lie anywhere in the range 4 . < w < . m f . If lattice simulations are able to mea-sure w with some precision (and measure M π F π /M d F d with a similar precision), then meeting the requirementthat m h /v (cid:39) .
26 could require an m f value outside therange of Table II.Obtaining a spectrum with a realistic hierarchy m h (cid:28) M π does require tuning C t . For the parameter choicesrequired to produce the spectrum in Table II, the mis-alignment angle satisfies θ (cid:39) .
19, implying a tuning for C t of order 2% through Eq. (10). The dilaton indirectlyreduces the requisite tuning through Eq. (14): the effectof the term in parentheses (which depends on the dila- m h /v w FIG. 1. The ratio m h /v , as a function of w , for y =2 . ± .
05 and F π /F d = 0 . ± . m f ) in therange [5]. The horizontal black dashed line represents the ex-perimental value m h /v (cid:39) .
26. The yellow shaded region isthe uncertainty, which is dominated by the substantial uncer-tainty in the measurement of the mass M d of the scalar, andin the quantity F π /F d . ton) is to further suppress m h /M π , helping the coloredpNGBs evade direct detection bounds.Couplings between the Higgs and pairs of W or Z gaugebosons, as well as couplings between the Higgs and pairsof top quarks, deviate from their SM predictions in thisCHM. These couplings include contributions from boththe pNGB and dilaton components of the Higgs parti-cle, and the expressions for them take the same form asthose derived in the literature (see e.g. Ref. [38] and refs.therein). In the limit θ →
0, the Higgs would couple tothe gauge bosons and top with the same strength as theSM Higgs.Using the benchmark M π = 4 TeV and the values forparameters selected by the lattice data of Ref. [5], we findthat the ratio between the Higgs couplings to W and Zbosons and their SM values is approximately 0 .
98. Thecoupling to top pairs has additional weak dependence onthe unknown scaling dimension z , which arises becausethe dilaton component of the Higgs boson has a couplingto the top that is z dependent from Eq. (4). For the value z (cid:39) −
3, the coupling strength to the top becomes thesame as in the SM. Over a plausible range of values for z , the top coupling deviates from its SM value only by afew percent.The amplitudes h → gg and h → γγ also deviateslightly from their SM values. New electrically chargedand colored pNGBs contribute to these amplitudes atloop level, but are sufficiently heavy for our choice ofbenchmark that their contributions are negligible.Considering all of these deviations, the signal signifi-cance for Higgs boson production in all observable chan-nels would deviate from the SM prediction by no morethan a few percent. Given the current accuracy of theHiggs measurements, which is no better than 8% [58],these effects will lie within experimental bounds, and amore precise analysis can be deferred.The masses of composite states which are not includedin the EFT have also been calculated in the N f = 8gauge theory on the lattice, in Ref. [5]. In particular,this data allows us to estimate the masses of the vector( ρ ) and axial ( a ) mesons. Using lattice measurementsfor ratios M ρ /M π and M a /M π , we estimate that the ρ would have a mass in the 6–8 TeV range and the a amass in the 9–11 TeV range, for our choice of benchmark M π = 4 TeV. We therefore do not expect them to be de-tectable at the LHC. Given the small deviations in Higgscouplings, as well as the large ρ and a masses, precisionelectroweak observables such as the S parameter will liewithin current experimental bounds. V. SUMMARY
We have argued that the SU (3) gauge theory with N f = 8 fundamental fermions provides an attractiveultraviolet completion for a realistic composite Higgsmodel. This model has the distinctive feature that thenear–conformal behavior of its underlying dynamics hasbeen revealed by lattice studies.We have drawn on such lattice results to compute sev-eral observable quantities within the model. These in-clude the misalignment angle in the vacuum of the theory,the mass of the Higgs boson, and the spectrum of heavyscalars. We have also examined the Higgs boson cou-plings and its production rates at the LHC. The modelpasses all the direct and indirect tests currently available,at the price of a moderate amount of fine-tuning for oneof the coefficients in the EFT potential. We have described the model in terms of the dilatoneffective field theory (EFT) from Refs. [20–22], requiringonly a simple addition to realize the Higgs doublet ascomposite pseudo Nambu-Goldstone bosons. Because ofthe approximate scale invariance of the dilaton EFT, itis possible to accommodate a realistic value of the massof the composite Higgs boson even for the values of theratio M π /F π currently available from lattice studies. Asa consequence, the mass of the Higgs boson is suppressedby an order of magnitude with respect to that of the otherpNGBs and dilaton in the EFT.We look to future lattice studies for a determinationof the scaling dimensions z and w , which play importantphenomenological roles. It will also be interesting to per-form a more detailed study of the precision electroweakobservables and explore the rest of the parameter spaceof this theory. ACKNOWLEDGMENTS
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