Lattice-Friendly Gauge Completion of a Composite Higgs withTop Partners

Helene Gertov1,2,∗ Ann E. Nelson2,† Ashley Perko3,‡ and Devin G. E. Walker3§1CP3-Origins, University of Southern Denmark, Campusvej 55, 5230 Odense, Denmark

2Department of Physics, Box 1560, University of Washington, Seattle, WA 98195-1560 USA and3Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755 USA

We give an explicit example of a composite Higgs model with a pseudo-Nambu-Goldstone Higgsin which the top Yukawa coupling is generated via the partial compositeness mechanism. Thismechanism requires composite top partners which are relatively light compared to the typical massscale of the strongly coupled theory. While most studies of the phenomenology of such models havefocused on a bottom-up approach with a minimal effective theory, a top-down approach suggeststhat the theory should contain a limit in which an unbroken global chiral symmetry protects themass of the top partners, and the spectrum of the partners satisfies ‘t Hooft matching conditions.We find that the relatively light fermions and pseudo-Goldstone bosons fall into complete multipletsof a large approximate global symmetry, and that the spectrum of particles lighter than a few TeVis non-minimal. Our example illustrates the likely features of a such a composite Higgs theory andalso serves as an example of a non-chiral theory with a possible solution to the ‘t Hooft matchingconditions. We find in this example that for some low-energy parameters in the effective theorythe top partners can decay into high-multiplicity final states, which could be difficult for the LargeHadron Collider (LHC) to constrain. This may potentially allow for the top partners to be lighterthan those in more minimal models.

I. INTRODUCTION

One of the most important questions facingparticle physicists is understanding the mecha-nism of electroweak symmetry breaking. Theminimal Standard Model does a great job of de-scribing this in a manner which is in good agree-ment with all collider data, but theoretically thesize of the Higgs mass is puzzling. Precisionelectroweak corrections and the absence of anynew particle discovery at the LHC suggests thatthe Standard Model may be a good descriptionof particle physics up to a cutoff of at least afew TeV. Dimensional analysis would then givea Higgs mass of order of this cutoff unless thereis either some fine-tuning or some symmetry rea-son for it to be lighter. One possible reason for alight Higgs is that it is an approximate Nambu-Goldstone boson, whose mass is protected byan approximate non-linearly realized symmetry.However, such a symmetry must be broken to al-low non-derivative couplings. The largest suchcoupling is to the top quark, and the the topYukawa coupling leads to large corrections to theHiggs mass squared. In theories with light toppartners, such corrections can be partially can-celed and the Higgs can be naturally light com-pared to the cutoff.

∗ gertov@cp3-origins.net† aenelson@uw.edu‡ perko@dartmouth.edu§ devin.g.walker@dartmouth.edu

A new gauge theory which confines the Higgsand the top partners has to have certain dy-namical features which are different from thoseof QCD. One non-QCD-like feature is that thetheory should have composite fermions which arelight compared with the compositeness scale toserve as top partners. One possibility is a theorywhich has a limit in which it confines but does notbreak all of the chiral symmetry, and which con-tains massless fermions which match the ‘t Hooftanomaly conditions that may serve as top part-ners. A second feature is that the theory shouldbe near a limit in which the chiral global symme-try which is protecting the mass of the top part-ners does spontaneously break, in order to pro-duce a pseudo-Nambu-Goldstone boson multipletcontaining the composite Higgs which coupled tothe top partners. In order for a small perturba-tion to result in chiral symmetry breaking, theunperturbed theory must be near a second-orderphase transition, and the spectrum of the unper-turbed theory would then be expected to containa relatively light scalar which is not a Nambu-Goldstone boson.

The first example of an ultraviolet (UV) com-pletion of a theory with light composite top sat-isfying ’t Hooft anomaly matching conditions foran approximate symmetry was studied in [1],which supplied a composite UV completion tothe SU(5)/SO(5) “Littlest Higgs” version of acomposite Higgs model [2, 3]. Other potentialexamples have been given in [4]. The assump-tion that such dynamics are possible is motivatedby a recent lattice study. In [5], evidence of an

arX

iv:1

901.

1045

6v2

[he

p-ph

] 1

3 Fe

b 20

19

2

anomalously light scalar was found for a system of12 fermion flavors coupled to SU(3) gauge fields.This scalar could be anomalously light by beingan order parameter for a second-order phase tran-sition. It would be interesting to see whetherlattice studies could provide evidence for anoma-lously light fermions in some model as well. Lat-tice studies of dynamics are limited to vector-likegauge theories with real positive fermion determi-nant, and therefore theories that have no scalarswith Yukawa couplings. In our model the confin-ing group is a vector-like gauge theory.

II. DESCRIPTION OF MODEL

Our composite pseudo-Nambu-GoldstoneHiggs model is based on the SU(4)/Sp(4)symmetry-breaking pattern [6] of the “Interme-diate” composite Higgs model [7]. We assumetop partners which are in a multiplet of anapproximate chiral global symmetry. We alsoassume that the chiral symmetry-breaking scaleis low compared to the confinement scale, andthe global chiral anomalies are matched bycomposite fermions. Provided our assumptionsabout the dynamics are correct, our model canprovide a realistic composite Higgs with a topYukawa coupling and top partners.

We need a model with a large enoughglobal symmetry to embed the Standard Modelgauge group, in order to allow for colored andelectroweak-charged composite fermions, as wellas the non-linearly realized symmetry protectingthe Higgs mass. The minimal model we foundwhich also allows for a solution to the ’t Hooftmatching conditions has 10 fermion flavors cou-pled to a symplectic gauge group, as well as anadjoint fermion. The fermion content is that ofa supersymmetric theory. From previous workon supersymmetric theories, we know that theSU(10)×U(1) global symmetry is confining withanomaly matching between the UV and the in-frared (IR) confined phase when the gauge groupis Sp(6) [8]. Giving mass to the supersymmet-ric scalars does not change the anomaly match-ing, although it might change the dynamics in thelimit where the scalars are heavy compared withthe confinement scale. The Sp(6) gauge groupis automatically free of gauge anomalies as longas there are an even number of pseudo-real rep-resentations. The fundamental fermions, whichwe will call preons, will be confined at a scaleΛ into composite scalars (mesons) and fermions(baryons), which furnish the composite Higgs aswell as a top partner that will mix with the fun-damental top through the mechanism of partial

compositeness. One possibility is to assume thesupersymmetry of this theory is broken and thescalar superpartners have mass below the com-positeness scale. This would allow us to perform asystematic analytic understanding of the dynam-ics, but simulating the the theory on the latticewould suffer from a sign problem and would notbe feasible. In this work we assume the super-symmetric scalars are at or well above the com-positeness scale or are even absent.

According to the Vafa-Witten theorem [9], invector-like gauge theories with no scalars coupledto the fermions, there can be no massless compos-ites with massive constituents, ie. there can beno violation of the persistent mass condition [10].Our composite top partners violate this conditionas their mass is protected by the SU(10) globalchiral symmetry, but they contain a fermion ad-joint which can be given a mass without violat-ing the symmetry. We therefore expect that inthe highly non-supersymmetric limit in which thescalar masses which are heavy compared to thecompositeness scale, the chiral global symmetriesof the theory will spontaneously break to a sub-group which allows fermion masses. We specu-late that the chiral symmetry-breaking scale andthe fermion masses in this case could be belowthe compositeness scale, which would imply thatthe chiral symmetry-breaking must be describ-able by a linear sigma model containing compos-ite scalars which are anomalously light comparedto the compositeness scale as well.

In order to produce a pseudo-Goldstone Higgsand masses for the top partners, the SU(10) sym-metry must spontaneously break. We assume thisbreaking scale is lower than the compositenessscale, which requires that even in the absenceof symmetry breaking, the theory contains a rel-atively light scalar multiplet with a non-trivialSU(10) transformation. We assume that a rel-atively small mass for the scalar arises becausethe theory with this matter content is near asecond-order phase transition. Because this is anon-supersymmetric, vector-like theory, this as-sumption can be checked on the lattice. TheSU(10) × U(1) global symmetry will also be ex-plicitly broken by preon mass terms and StandardModel gauge interactions.

It is convenient to analyze the particle contentunder an SU(6) × SU(4) × U(1) subgroup. TheSU(4) will contain the electroweak and custo-dial symmetries, and the SU(6) will contain thecolor group in the diagonal subgroup of SU(3)×SU(3) ⊂ SU(6). When an effective theory forthe light mesons and baryons of this theory is an-alyzed, we will find that for some parameters wecan trigger spontaneous breaking of the SU(4) to

3

Sp(4) at a lower scale f . This coset structure wasexplored in [7], and is known to be the minimalcoset that can form a pseudo-Goldstone compos-ite Higgs which admits a fermionic UV comple-tion. UV completions of this coset have previ-ously been studied in [11–17]. A SU(6)× SU(4)global symmetry structure was previously studiedin the context of a Sp(6) gauge group in [15], butthis work did not give a dynamical explanationfor the mass and couplings of the top partner,which is provided here by the SU(10) breaking.Furthermore, that work assumed a different con-densate structure that breaks the global SU(6) toSO(6), rather than the Sp(6) that we will find, aswell as having the composite fermions in a differ-ent representation of the global SU(6)× SU(4).

III. UV THEORY

In the UV, we consider a model with a newSp(6) gauge group that confines at the scale Λ,which will be of order 10 TeV. The matter contentabove Λ consists of fermions F in the fundamen-tal representation of Sp(6), which have a globalflavor symmetry in the fundamental of SU(10), aswell as A, a fermionic adjoint of Sp(6) that is asinglet under the flavor symmetry and is neededto form the composite fermions. There is alsoan anomaly-free global U(1) symmetry, and ananomaly-free discrete symmetry. As we can showthat that any operator allowed by the anomaly-free continuous global symmetries is allowed bythe anomaly-free discrete symmetry, we will notdiscuss the discrete symmetry any further. Thismatter content is summarized in Tab. I.

Sp(6) SU(10) U(1)

F − 45

A 1 1

TABLE I. Matter content in UV (above confinementscale Λ)

Anomaly matching between the UV and theIR effective theory below the confinement scale,where only the flavor symmetry remains, will de-termine the low-energy degrees of freedom. Welist some of the possible composite particles inthis picture, following [18]:

Tk = TrAk, k = 2, 3

Mk = FAkF, k = 0, 1, 2 . (1)

The Tk are singlets under SU(10), and the Mk

are the antisymmetric part of the product of

two fundamentals of SU(10), which makes them45-plets. The composite matter quantum num-bers are summarized in Tab. II. The compos-

SU(10) U(1)

Tk 1 k

Mk − 85

+ k

TABLE II. Matter content in IR (below confinementscale Λ)

ite fermion M1 = FAF , which we will call Ψ,matches all of the SU(10) × U(1) global anoma-lies. If we assume that this theory has a phasewhich confines without chiral symmetry breaking,then Ψ must be massless. We will also assumethat the composite scalar M0 = FF is anoma-lously light, and this meson we will call Φ. Weassume the rest of the composite matter contenthas masses at the composite scale and thus weignore them in the low-energy analysis.

In order to write down the low-energy effectivetheory we will need to consider the sources of ex-plicit SU(10)×U(1) breaking. Part of the SU(10)is weakly gauged under the Standard Modelgauge group. Global baryon number symmetry isalso embedded in the SU(10). The fermions andtheir Sp(6)× SU(3)× SU(2)w × U(1)Y × U(1)Bcharges are listed in Tab. III, where the subscriptsdenote the component of F that transforms inthat representation under the Standard Modelsymmetries.

Sp(6) SU(3)c SU(2)w U(1)Y U(1)B

F3 1 16

13

F3 1 − 16

− 13

F2 1 0 0

F+ 1 1 + 12

0

F− 1 1 − 12

0

A 1 1 0 0

TABLE III. Preon Charges

The following SU(10)×U(1) global symmetry-breaking mass terms are consistent with thegauge symmetries and U(1)B :

4

L ⊃ mAAA+m3F3F3+m2F2F2+m1F+F−+h.c.(2)

We will assume the mA and m3 terms are smallbut non-negligible and the m2 and m1 terms arevery small. The SU(4) subgroup of the SU(10)is thus mostly only explicitly broken by the weakgauge couplings. The mA and m3 terms will leadto important spurions in the low-energy effectivetheory.

IV. EFFECTIVE THEORY BELOWCOMPOSITENESS SCALE

We will now analyze the effective dynamics ofthis model in the low-energy limit. The m3 massterm in the UV for the Q preons results in a spu-

rion which explicitly breaks the SU(6) part ofthe SU(10) symmetry to Sp(6) in the compos-iteness Lagrangian, leaving the SU(4) invariant.The mA term does not break the SU(10) but doesbreak the U(1), and will be treated as a spurionwith U(1) charge -2.

Using the spurions to restore the SU(10)×U(1)symmetry, we write the non-kinetic terms in thecompositeness Lagrangian that are invariant un-der this symmetry. In Eq. (IV) we give the termsallowed through dimension 4 for the scalars andup to dimension 7 for terms containing a fermionbilinear, where M is the Sp(6)-preserving spurionwhose SU(6) part is given by

M6×6 = m3

(0 I3−I3 0

). (3)

L ⊃ (M210)TrΦ†Φ− Λ2TrM†Φ + λ1Tr(Φ†Φ)2 + λ2Tr(Φ†ΦΦ†Φ) +

mA

Λ2

[gΨTrΦ†ΨΦ†Ψ

+g0TrΦ†ΨTrΦ†Ψ + g1TrM†ΨTrΦ†Ψ + g2TrM†ΨΦ†Ψ + g3TrM†ΨM†Ψ + g4TrM†ΨTrM†Ψ]

+λ3Tr(M†ΦM†Φ) + λ4Tr(M†Φ)Tr(M†Φ) + λ5Tr(M†ΦΦ†Φ) + λ6Tr(M†Φ)Tr(Φ†Φ)

+λ7Tr(MM†Φ†Φ) + λ8Tr(MΦ†)Tr(M†Φ) +g7m

∗A

Λ3εabcdefghijΦabΦcdΦefΨghΨij + h.c. (4)

Note that the term M210 will depend on mAm

†A

and on TrM†M . This term is allowed by all thesymmetries. If it is as large as the compositenessscale, then the power-counting of the effectivetheory will not be useful as either all the scalarswould be too heavy to allow an effective field the-oretic treatment or the chiral symmetry-breakingscale will be at the confinement scale and the toppartners will be too heavy to belong in the effec-tive theory. If, however, M10 is light comparedwith the compositeness scale, then an effectivefield theoretic treatment keeping all the scalarsand fermions can be quantitatively useful. Fur-thermore, this limit is technically natural as longas M2

10 is no lighter than a loop correction to thismass squared. Since the effective theory does notcontain any couplings larger than O(1), M2

10 canbe naturally be smaller than the compositenessscale by a factor of O(1/16π2). In the followinganalysis, we will assume that M2

10 is positive andthat the chiral symmetry-breaking which leads toa pseudo-Goldstone boson Higgs is triggered bythe explicit chiral symmetry-breaking, as wouldbe the case if the theory were approximately su-persymmetric. A analysis with a negative butsmall M2

10 would lead to a similar spectrum.

Under SU(6)× SU(4), the meson field decom-poses as

Φ =

(φX φT−φTT φY

). (5)

The spurion M produces a tadpole that will leadto VEV for some components of φX ,

φ0 =m3Λ2

2M210

, (6)

which must be smaller than Λ in order for theeffective compositeness Lagrangian to be useful.

Expanding around this VEV, the φX particleshave mass

m2φX

= 8(M2

10 + λ6m23 + 2m3φ0(λ4 + 3λ5)

−4φ20(3λ1 + λ2)

). (7)

The φT also get a mass from this VEV,

m2φT

= 4M210 + 2m2

3λ6 + 4φ0m3(λ4 + 6λ5)

−8φ20(6λ1 + λ2) , (8)

as do the φY ,

m2φY

= 4M210 + 24φ0m3λ5 − 48φ2

0λ1 . (9)

5

The φY mass-squared can be taken to be negativeso that φY obtains a VEV, spontaneously break-ing SU(4) to its Sp(4) subgroup. This will resultin five pseudo-Nambu-Goldstone bosons, four ofwhich will play the role of our composite Higgsdoublet.

V. SPONTANEOUS SYMMETRY-BREAKING EFFECTIVE THEORY

A. SU(4)/Sp(4) Higgs

We can now analyze the spontaneous symme-try breaking in the Higgs sector. First, we expandφY around the Sp(4)-preserving VEV

Σ0 =

(iσ2 00 iσ2

), (10)

which also preserves the electroweak gauge group.As in [7], the Nambu-Goldstone bosons of

SU(4)/Sp(4) symmetry breaking can be writtenas

〈φY 〉 = Σ = feiΠ/fΣ0eiΠ/f Π =

(A HH† −A

),

(11)where

H =

(h0 + ih3 ih2 + h1

ih2 − h1 h0 − ih3

)A =

(η 00 η

),

(12)and f is the decay constant of the sigma model,which here is f = φ0.

In unitary gauge, where h1, h2, h3 are eaten bythe W and Z bosons, Σ is given as:

Σ =

0 cosα 0 i sinα− cosα 0 −i sinα 0

0 i sinα 0 cosα−i sinα 0 − cosα 0

, (13)

where α =√h2 + η2/

√2φ0 and h2 =

∑i h

2i .

We implement the symmetry breaking by set-ting φY → φ0Σ in Eq. (4), the composite La-grangian. We will see that the Higgs VEV

θ =〈h〉√2φ0

, (14)

will give mass to the top partners in the followingsection.

B. Top Partner Embedding

The symmetry breaking from the scalar sec-tor will propagate to the fermion sector via the

scalar-fermion couplings in Eq. (4), resulting inmasses and Yukawa couplings for the compositefermions.

We can decompose the composite fermion Ψ interms of SU(6)× SU(4) as

Ψ =

(X S−ST Y

). (15)

The top partners are in S, which as a (6,4) ofSU(6)×SU(4), decomposes into four componentsin terms of the global SU(6) and the global SU(4)which contains the custodial SU(2)L × SU(2)R.It is given by

S =

(Q′ P ′

Q′ P ′

). (16)

Eq. (17) gives the decomposition in terms ofSU(3)L × SU(3)R × SU(2)L × SU(2)R and intoSU(3)D×SU(2)L×U(1)Y , where U(1)Y is a lin-ear combination of the T3 generator of SU(2)Rand an SU(10) generator.

Q′ = (3, 1, 2, 1)→ (3, 2, 1/6)

P ′ = (3, 1, 1, 2)→ (3, 1, 2/3), (3, 1,−1/3)

Q′ = (1, 3, 2, 1)→ (3, 2,−1/6)

P ′ = (1, 3, 1, 2)→ (3, 1, 1/3), (3, 1,−2/3) .(17)

We will gauge SU(3)D for QCD and SU(2)L forthe weak group. Q′ is the top partner which hasthe same gauge quantum numbers as the left-handed top quark and P ′ contains a particle withthe same gauge quantum numbers as the left-handed anti-top, which we will call T ′.

C. Light Composites

We assume that below the compositeness scale,the low-energy effective theory contains the Stan-dard Model gauge bosons, three generations ofquarks and leptons, composite fermions whichform a 45-plet of the SU(10), and compositescalars which also form a 45-plet of the SU(10).While the fermions are necessarily light due tothe approximate SU(10) × U(1) global symme-try, there is no symmetry protecting the mass ofthe scalars. A description of such a symmetry-breaking transition may be made by introducingscalars, and if the critical value for the preon massterms is low and the transition is second orderthen the scalars must be light relative to the com-positeness scale. The preon charges in Table IIIresults in the charges for the composite fermionsand bosons given in Tables IV and V.

The fermion U(1) charges are chosen so thatwe have an appropriately charged Q and T . Note

6

that because baryon number and lepton numberare conserved, we have a conserved discrete “R-parity”, (−1)3B+L+2S ≡ RP , which we list aswell.

Field SU(3)c SU(2)L U(1)Y U(1)B RP

Li 1 2 -1/2 0 1

QF,i 3 2 1/6 1/3 1

UF,i 3 1 -2/3 -1/3 1

Di 3 1 1/3 -1/3 1

Ei 1 1 1 0 1

Y 1 2 1/2 0 -1

Y 1 2 -1/2 0 -1

ψ1 1 1 0 0 -1

ψ2 1 1 0 0 -1

Q′ 3 2 1/6 1/3 1

T ′ 3 1 2/3 1/3 1

B′ 3 1 −1/3 1/3 1

Q′ 3 2 −1/6 -1/3 1

T ′ 3 1 −2/3 -1/3 1

B′ 3 1 1/3 -1/3 1

D′ 3 1 1/3 2/3 -1

D′ 3 1 −1/3 −2/3 -1

ψ3 1 1 0 0 -1

X 8 1 0 0 -1

TABLE IV. Spin-1/2 field content of the low-energyeffective theory. All fields are left-handed 2 compo-nent Weyl spinors.

Field SU(3)c SU(2)L U(1)Y U(1)B Rp

h 1 2 1/2 0 1

η 1 1 0 0 1

q′ 3 2 1/6 1/3 -1

t′ 3 1 2/3 1/3 -1

b′ 3 1 −1/3 1/3 -1

q′ 3 2 −1/6 -1/3 -1

t′ 3 1 −2/3 -1/3 -1

b′ 3 1 1/3 -1/3 -1

d′ 3 1 1/3 2/3 1

d′ 3 1 −1/3 −2/3 1

ρ 1 1 0 0 1

x 8 1 0 0 1

TABLE V. Spin-0 field content of low-energy effectivetheory.

D. Yukawas and Partial Compositeness

We have obtained masses for the compositetop partners via symmetry breaking, but it re-mains to propagate this to the fundamental top

via partial compositeness [19]. This entails lin-early coupling the composite quark and top part-ners to the fundamental ones. We embed thefundamental degrees of freedom in an incompleteSU(6)× SU(4) multiplet:

TF =

(QF (TF 0)0 0

), (18)

and couple it linearly to the quark doublet part-ner Q′ and the anti-top partner T ′:

L ⊃ g5φ0TFT′ + g6φ0QF Q

′ . (19)

This gives the top partners a mass obtained fromthe charge-2/3 mass matrix m2/3 given in Ta-ble VI, where we have defined gt ≡ 2mA(g2m3 +2gψφ0)/Λ2.

Q′ T ′ TF

Q′ gtφ0 cos θ igtφ0 sin θ 0

T ′ igtφ0 sin θ gtφ0 cos θ g5φ0

QF g6φ0 0 0

TABLE VI. Charge-2/3 mass matrix

Expanding to first order in θ, we have

L ⊃φ0Q′(gtQ

′ + g6QF ) + φ0T′(gtT

′ + g5TF )

+ igtφ0T′Q′θ + igtφ0Q

′T ′θ . (20)

We see that Q′ pairs with the following linearcombination of fields:

gtQ′ + g6QF√g2t + g2

6

(21)

to gain a mass.Similarly, T ′ pairs with the combination:

gtT′ + g5TF√g2t + g2

5

(22)

to become massive.The linear combinations of fields that are light

are given by:

TL ≡g6Q

′ − gtQF√g2t + g2

6

TR ≡g5T

′ − gtTF√g2t + g2

5

. (23)

These light quarks have mass term:

〈h〉√2ε5ε6gtTLTR , (24)

7

where we have defined the mixing angles

ε5≡g5√g2t + g2

5

ε6≡g6√g2t + g2

6

. (25)

In contrast, the other quarks have massesφ0

√g2t + g2

4 and φ0

√g2t + g2

6 , making them muchheavier due to the fact that 〈h〉 � φ0.

E. Higgs Effective Potential

As in [7], the contribution of the top to theHiggs effective potential (which dominates overthe gauge boson contribution) is

−∑i

3|m2

i |2

16π2log |m2

i |, (26)

where |m2i | are the eigenvalues of the charge-2/3

mass matrix m2/3m†2/3 obtained from Eq. (19).

The gauging of the EW symmetry does notbreak the U(1) symmetry under which the η sin-glet transforms, so it does not get a potential fromgauge loops [19]. A massless η would be ruled outby Kaon decays, so we must give it a small massby adding another spurion

φ30mηTr(Σ†0Σ) . (27)

This term also contributes to the Higgs mass, soits contribution must be smaller than the elec-troweak breaking scale so as not to reintroducefine-tuning.

VI. PHENOMENOLOGY

A. Precision-Electroweak and FlavorConstraints

Any composite Higgs model must satisfy pre-cision electroweak constraints. The most danger-ous couplings are those of the bottom quark. Atthis point, the fundamental bottom quark is stillmassless. In order to give the bottom quark amass we will need to couple the fundamental bot-tom to the composite quarks, for example with acoupling gBBFB. In analogy to the generationof the top mass in Eq. (20), this coupling willgenerate a mass for the bottom which is propor-tional to gB . Since gB must be small in order toproduce the observed small value of the bottommass, the corrections to the Zbb coupling will besuppressed, as noted in [20, 21].

The S is proportional to 〈h〉2/φ20 [22], so it

does not receive too-large corrections as long thecompositeness scale is larger than about 1 TeV.The custodial symmetry protects the T param-eter from large corrections [23], but since theYukawa couplings g5, g6, gB break the custodialsymmetry we must consider how large these ef-fects are. The T parameter is modified by one-loop diagrams of the top partners and StandardModel fermions, which is parametrically of order

∆T ∼ m2t

m2W

ε2B〈h〉2

φ20

logg2t 〈h〉2

m2t

(28)

in the limit gB � g5, g6 [21]. We conclude thatthe modifications to the T parameter are smallbecause εB � 1.

We next turn to flavor constraints. Because thetop partners are weakly coupled to the Higgs viathe partial compositeness mechanism, the scaleof flavor violation can be decoupled from thecompositeness scale. As shown in [24], we canavoid large flavor-changing neutral currents byassuming that the fundamental Standard Modelfermions are coupled to the Sp(6) preons via four-fermion operators at some scale much larger thanthe compositeness scale Λ.

B. Dark Matter Candidate

As we discussed in section V C, this theory hasa conserved discrete symmetry related to lepton-and baryon-number symmetry analogous to an“R-parity” for a supersymmetric theory, whichwe have termed Rp. This symmetry was previ-ously discussed in the context of composite Higgsmodels in [1] and was termed “dark matter par-ity”. Neutral particles that are odd under thisdiscrete symmetry are good dark matter candi-dates. From Table IV we see that the neutral,Rp-odd particles are the fermion singlets ψa. Thelightest of these Rp-odd singlets could be the darkmatter.

C. Particle Spectrum

Since at the compositeness scale the completeSU(10) fermion multiplet is massless and onlygains mass from soft breaking terms, while thescalar has mass M10 at the compositeness scaleand gets additional soft corrections, we will as-sume all the composite fermions are lighter thanthe scalars except for the pseudo-Goldstones.Thus, to determine the novel phenomena of thismodel at the LHC we only need to consider the

8

decays of the top partners to other fermions andto h and η.

Particle SU(3)c SU(2)L U(1)Y U(1)B Rp

Y 1 2 1/2 0 -1

Y 1 2 -1/2 0 -1

ψ1 1 1 0 0 -1

ψ2 1 1 0 0 -1(TL

BL

)3 2 1/6 1/3 1

TR 3 1 2/3 1/3 1

BR 3 1 −1/3 1/3 1(T ′LB′L

)3 2 1/6 1/3 1

T ′R 3 1 2/3 1/3 1

B′R 3 1 −1/3 1/3 1(T ′′LB′′L

)3 2 −1/6 -1/3 1

T ′′R 3 1 −2/3 -1/3 1

B′′R 3 1 1/3 -1/3 1

D′ 3 1 1/3 2/3 -1

D′ 3 1 −1/3 −2/3 -1

ψ3 1 1 0 0 -1

X 8 1 0 0 -1

TABLE VII. Fermionic Particle Content (Mass Basis)

As summarized in Table VII, our theory con-tains the light top quark T and two heavy quarksT ′ and T ′′, as well as the bottom B and itspartners B′ and B′′, just like the SU(4)/Sp(4)intermediate Higgs model [7], but it also con-tains the uncolored weak doublets Y and Y , thecolor triplets with exotic baryon number D′ andD′, a color octet X, and three neutral singletsψ1, ψ2, ψ3. In addition, we have the SU(4)/Sp(4)Higgs doublet h and the additional scalar singletη.

The phenomenology of this model will dependon the particle spectrum. Masses for the compos-ite fermions are generated by the spurion M andthe VEV of φ:

φ0 =m3Λ2

2M210

< Λ . (29)

From this expression we see that m3/φ0 <Λ2/2M2

10, and so m3 cannot be parametricallylarger than φ0 unless M10 is larger than Λ, whichwould break our perturbative expansion. Thusfrom Eq. (4), we expect that the new fermionswill all have mass of order mAφ

20/Λ

2. In contrast,the scalars have typical mass-squared of order φ2

0,so since we must have mA � Λ and φ0 < Λ inorder for the effective compositeness Lagrangian

to be consistent, this means that the non-pseudo-Goldstone scalars are generically heavier than thefermions. We can therefore neglect them in ouranalysis of the low-energy phenomenology.

The partial compositeness described in sectionVI A mixes T ′, T ′, T ′′, T ′′, Q, Q with the funda-mental quarks TF , QF , resulting in the physicalquarks TL and TR as well as two heavier speciesof quarks. This mechanism results in a top massof order

mT ∼ ε5ε6mAφ0

Λ2〈h〉 , (30)

which must be matched to the observed value.The fermions which are not top partners havemasses of order

mexotic ∼ g∗mA

Λ2φ2

0 , (31)

where g∗ is a linear combination of the couplingparameters gψ, g0, g1, g2, g3, g4.

In addition to the top and bottom partners, theparticles most relevant for phenomenology are theHiggs and the singlet η, where mη < mh. Theratio of the exotic fermion masses to the Higgssector masses is

mexotic

〈h〉∼ g∗

mA

Λ

φ0

Λ

φ0

〈h〉, (32)

where φ0/〈h〉 > 1, φ0/Λ < 1, and mA/Λ � 1.Note that it is possible for some of the exoticfermions to be lighter than h and η, but since ex-otic decays of the Higgs are strongly constrainedby the LHC, we will not consider this case.

The masses of the exotic fermions are given interms of the couplings in the Lagrangian as:

mY = mY =2gψmAφ

20

Λ2

mψ1= mY

mψ2 = mY +8g0mAφ

20

Λ2

mX =2mA

Λ2

(g3m

23 + g2φ0m3 + gψφ

20

)mD = mD = 2mX

mψ3= mX +

12mA

Λ2

(g4m

23 + g1φ0m3 + g0φ

20

),

(33)

and the heavy top partners have masses

mT ′ = φ0

√g2t + g2

5

mT ′′ = φ0

√g2t + g2

6 , (34)

where gt = 2mA(g2m3 + 2gψφ0)/Λ2.

9

1. Spectrum for m3 � φ0

For simplicity we will first study the regimewhere m3/φ0 � 1 so that at leading order wecan neglect the terms proportional to m3 in themasses. We see that in this limit, mX ≈ mY andmψ3

≈ mY + 12g0φ20mA/Λ

2. Thus Y, Y , ψ1, andX have mass mY while D′D′ are heavier, witha mass of approximately 2mY . The remainingfermions ψ2 and ψ3 are either lighter or heavierthan mY , depending on the sign of g0. If g0 < 0,then mψ3

< mψ2< mY , and mψ3

is the darkmatter candidate. The six parameters gi havecollapsed in this limit to two independent param-eters, mY and mψ3 because the terms involvingg1, g2, g3, g4 are negligible.

Finally, we need to consider how the heavytop partners fit into the mass hierarchy. Sinceφ0/m3 � 1, we find that

gt ≈4gψmAφ0

Λ2. (35)

Thus, from Eq. (34) we see that in the limit of nomixing with the fundamental quarks (g5 = g6 =0), T ′ and T ′′ both have mass 2mY , which is equalto that of the exotic triplets D′, D′. When g4 andg5 are nonzero, Q′ and Q′′ become heavier thanD′, D′ because

√g2t + g2

i > gt. This makes themthe heaviest of the exotic fermions.

The masses of heavy top partners are boundedfrom above by the inverse of the fine-tuning pa-rameter 〈h〉/φ0. In the absence of fine-tuningthey should be lighter than about 800 GeV, andthey are bounded from below by LHC searches.Since the X particle is lighter than the top part-ners in this case, there are strong bounds fromlight gluino searches on this regime of the model,which we will discuss in section VI D.

2. Spectrum for m3 ∼ φ0

The other regime of our model that is allowedby perturbativity is where m3 and φ0 are compa-rable. To understand what this relation impliesabout the masses in the theory, we can rewriteM10 and m3 in terms of dimensionless parame-ters, M10 = εΛ, m3 = βΛ. Then the constraintsimposed by requiring our theory to be perturba-tive are φ0 < Λ, M10 < Λ, and m3 < Λ. In termsof the dimensionless parameters, this is ε < 1,β < 1, β

2ε2 < 1. Since m3/φ0 = 2ε2, we can takem3 ∼ φ0 as long as ε is a fraction of order oneand β � 2ε2. This corresponds to M10 close to(but less than) the compositeness scale, and m3

much lighter.

In the limit m3 ∼ φ0, the masses mY , mψ2,

mX , andmψ3 are independent parameters. In ad-dition to these mass parameters, we have the topsector masses, mT , mT ′ , and mT ′′ , which are alsoindependent parameters because they depend ong2, gψ, g5, and g6 in the Lagrangian. Thus wecan trade seven of the eight original dimension-less parameters gi in the Lagrangian for the sevenmasses. Only the pair (g1, g4) remains degeneratein Eq. (33).

D. LHC Phenomenology

The strongest phenomenological constraints onour scenario comes from the LHC. The phe-nomenology of the Higgs plus singlet arising fromthe SU(4)/Sp(4) coset structure was previouslystudied in [25, 26], but these analyses did notinclude top partners. Vector-like quark partnerscoupling to the top that are singlets or doubletsunder the custodial SU(2)×SU(2) have been con-strained to have masses greater than around 800GeV after the first run of the LHC [27], but addi-tional decay modes of the top partners will opennew channels for searches at the LHC. In addi-tion, the possibility of producing η will affect thedecays of these exotic vector-like quarks.

The standard channels to look for new vector-like quarks are through the decays T → bW ,T → tZ, and T → th. Because our model hasa light pseudoscalar η, we have the additionalprocesses T → tη and B → bη. These decayswere studied in [28] with the conclusion that thebranching ratios to these exotic decays can belarge and of the order of the Standard Model de-cays. Since limits on the top partner mass de-pend on the branching ratios to Standard Modelparticles, large branching ratios to η can signif-icantly affect these limits. In addition to η, theexotic fermions D′, D′, X, and the ψa will affectsearches for the top partners.

To see this, consider Table VIII, which liststhe leading interactions involving the decay of thefermions in our model up to dimension 6 (the fulllist of interactions including processes other thandecays is given in Table IX and Table X in Ap-pendix A). In addition to these, the dimension-4 Higgs sector interactions hhηη, hhhh, ηηηη arerelevant for LHC phenomenology.

1. LHC constraints for m3 � φ0

The decay processes depend on the masshierarchy between the exotic fermionsY,X, ψa, D

′, D′. We will first explore the

10

dim 4 hY ψa

hY X

ηψaψb

dim 5 Y ψahη

Y Xhη

ψaψbηη

dim 6 ψaDTiLB

jL

ψaDTiRB

jR

ψaXTiLT

jL

ψaXBiLB

jL

ψaXTiRT

jR

ψaXBiRB

jR

ψaψbXX

ψaψbDD

ψaψbY Y

ψaψbψcψd

ψaψbTiLT

jL

ψaψbBiLB

jL

ψaψbTiRT

jR

ψaψbBiRB

jR

TABLE VIII. Leading interactions producing de-cay of the exotic fermions. Here, the superscript irefers to unprimed, primed, or double primed for thethree generations of top and bottom quarks, and thesubscript a ranges over 1, 2, 3 for the three neutralfermions.

case m3 � φ0. From the discussion in sec-tion VI C, we know that if g0 < 0 then the massspectrum is mT ′′ ∼ mT ′ > mD > mY ∼ mX ∼mψ1

> mψ2> mψ3

. Any colored particle will beproduced from gluons at the LHC, so we expectthe color octet X and the D′, D′ to be producedas well as the top partners.

Since decays of X to Y are not kinematicallyallowed, the dominant decay process of X is tottψa at dimension 6. This means that X acts likea gluino in the case of a heavy stop, and so itdecays to t, t and a neutralino LSP, which in ourcase is the dark matter candidate ψ3 [29].

The main signatures of this model is largemissing energy plus multi-jets, which is similarto searches for natural SUSY [30, 31]. Thesesearches constrain the gluinos to be heavier thanaround 1.2 TeV. This is larger than the up-per bound on the top partner mass, so thisregime is essentially ruled out. One considerationthat tends to weaken the constraint from gluinosearches is that if the parameter g0 is small,then there is a small mass difference betweenthe “gluino” and the “neutralino”, which leads toless energetic jets and missing transverse energy,which can weaken the limits on the “gluino” mass[32]. However, we do not expect this weakening

to accommodate the required mass of less than800 GeV for the X. Furthermore, the additionaldegrees of freedom of the Weyl fermion X withrespect to the scalar gluino serve to strengthenthe limits. Thus, the m3 � φ0 regime of ourmodel is not viable in a natural mass regime.

2. LHC constraints for m3 ∼ φ0

We turn to the regime where m3 is of the sameorder as φ0. We have several experimental con-straints to bound the seven free masses that de-termine the theory in the case m3 ∼ φ0: mT isfixed by its observed value, and mT ′ and mT ′′

are constrained by naturalness to be less thanabout 800 GeV. Since in this case mX is a freeparameter, it can be larger than the mass of theheavy top partners, which would alleviate thestrong bounds from light gluinos. In addition,since the singlets and the Y could be lighter thanthe X, there can be additional decay modes tothese particles plus the Higgs starting at dimen-sion 4, which we can see from Table VIII. SincemD is fixed by this model to have mass 2mX ,D′, D′ will be heavier than the top partners andcan decay to them via dimension-6 operators.

The singlet masses mψ2and mψ3

are notstrongly constrained by the LHC, and differenthierarchies of these masses can open up addi-tional decay modes for the top partners in Ta-ble VIII. Note that the other singlet mψ1

is fixedto have the same mass as Y . If either of ψ2 orψ3 is the lightest of the Rp-odd fermions in thismodel, it can be a dark matter candidate.

This regime of our model has a particle and in-teraction content that combines aspects of SUSYwith that of composite Higgs models. TheSUSY-like features, include gluino-, higgsino-,and neutralino-like particles,and a conserved darkmatter parity. Composite-like features includetwo vector-like heavy top partners and an addi-tional pseudoscalar in the Higgs sector.

VII. DISCUSSION

The discovery of the Higgs boson at the LHCin the absence of any other new particles hasleft us few clues to understand its unnaturallysmall mass. One possibility is that the Higgs bo-son is light because it is a composite particle ofsome underlying strongly-coupled gauge theory.In the strongly-coupled regime perturbative cal-culations cannot be made but lattice calculationscan shed light on its spectrum. The low-energyspectrum of such a theory is expected to have

11

complex phenomenology due to the large numberof possible composite particles of the fundamen-tal degrees of freedom, in contrast to the simplestcomposite Higgs models which do not address theUV completion of the theory. Thus we expect re-alistic models of a composite Higgs to have richerphenomenology than minimal models.

We find that at low energies, this model in-deed has several new fermions: a color octet, anelectroweak doublet, three scalars, two vector-likequark partners, and a pair of color triplets withexotic baryon number. This theory exhibits phe-nomenology that combines aspects of SUSY theo-ries with aspects of composite Higgs theories, andcan be studied on the lattice to see whether ourexpectations about the low-energy spectrum areborne out. A recent lattice study of partial com-positeness in a SU(4) gauge theory suggests thatthe spectrum of our model can indeed be probed[33]. They found that the SU(4) model motivatedby [12, 13] is not able to produce a realistic topmass and attribute this to the fact that the theorydid not have the required near-conformal dynam-ics to generate small composite fermion masses.In our model, the small fermion mass is due to thetheory being confining without breaking the chi-ral symmetries and having matter content whichis near a second-order phase transition so thatthe chiral symmetry can be spontaneously bro-ken. These assumptions can be checked on thelattice since our fundamental matter content isvector-like. Thus, our model is a good candi-date to produce realistic top couplings in a latticestudy like [33].

Since this model is the minimal one that canserve as a UV completion of the composite Higgswith top partners and which has a fundamen-tal matter content that can be simulated on thelattice, models with larger symmetry groups willlikely provide novel phenomenology accessible tothe LHC. A promising avenue for future work is toexplore such non-minimal “lattice-friendly” mod-els.

ACKNOWLEDGMENTS

A.N. is funded in part by the DOE under grantDE-SC0011637, and by the Kenneth K. YoungChair. H.G. acknowledges partial support fromthe Danish National Research Foundation grantDNRF:90, the Fulbright commission and the OleRømer foundation.

Appendix A: Interactions of CompositeFermions up to Dimension 6

Here we list all the allowed operators includingthe composite fermions up to dimension 6. Thesubset of these that are relevant for decays aregiven in Table VIII. Table IX lists all dimension-4 and dimension-5 operators involving compositefermions, and Table X lists the dimension-6 op-erators.

Dim 4 Dim 5

hT iRT

jL DDηη

hBiLB

jR Y Y ηη

hY ψa ψaψbηη

hY X XXηη

ηY Y Y Y hh

ηψaψb T iLT

jLηη

ηXX T iRT

jRηη

ηDD BiRB

jRηη

ηBiLB

jL Bi

LBjLηη

ηBiRB

jR Bi

RTjRhh

ηT iLT

jL T i

RTjLhη

ηT iRT

jR Bi

LBjRhη

Y ψahη

Y Xhη

TABLE IX. Dimension-4 and dimension-5 operatorsof fermions for LHC phenomenology. Here, the super-script i refers to unprimed, primed, or double primedfor the three generations of top and bottom quarks,and the subscript a ranges over 1, 2, 3 for the threeneutral fermions.

DDXX ψaψbTiLT

jL Y Y T i

LTjL T i

LTjRT

kRT

mL

Y Y XX ψaψbBiLB

jL Y Y Bi

LBjL T i

LTjLT

kLT

mL

ψaψbXX ψaψbTiRT

jR Y Y T i

RTjR T i

RTjRT

kRT

mR

Y Y DD ψaψbBiRB

jR Y Y Bi

RBjR Bi

LBjLB

kRB

mR

ψaψbDD ψaDTiLB

jL Y Y T i

RBjR Bi

LBjLB

kLB

mL

XXXX ψaDTiRB

jR Y Y Bi

RTjR Bi

RBjRB

kRB

mR

DDDD DDT iLT

jL Y Y T i

RBjR T i

LTjLB

kRB

mR

Y Y ψaψb DDBiRB

jR XXT i

LTjL T i

RTjRB

kLB

mL

ψaψbψcψd DDT iRT

jR XXBi

LBjL T i

RTjRB

kRB

mR

Y Y Y Y DDBiLB

jL XXT i

RTjR T i

LTjLB

kLB

mL

ψaXBiLB

jL ψaXB

iRB

jR XXBi

RBjR T i

LTjRB

kLB

mR

ψaXTiRT

jR ψaXT

iLT

jL T i

LTjRB

kLB

mR

TABLE X. Dimension-6 operators of fermions forLHC phenomenology. Here, the superscript i refersto unprimed, primed, or double primed for the threegenerations of top and bottom quarks, and the sub-script a ranges over 1, 2, 3 for the three neutralfermions.

12

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