Discovering the Origin of Yukawa Couplings at the LHC with a Singlet Higgs and Vector-like Quarks
Simon J.D. King, Stephen F. King, Stefano Moretti, Samuel J. Rowley
DDiscovering the Origin of Yukawa Couplings at the LHCwith a Singlet Higgs and Vector-like Quarks
Simon J. D. King (cid:63) , Stephen F. King (cid:63) , Stefano Moretti (cid:63) , Samuel J. Rowley (cid:63) , (cid:63) Department of Physics and Astronomy, University of Southampton,SO17 1BJ Southampton, United Kingdom
Abstract
Although the 125 GeV Higgs boson discovered at the LHC is often heralded as the originof mass, it may not in fact be the origin of Yukawa couplings. In alternative models,Yukawa couplings may instead arise from a seesaw type mechanism involving the mixingof Standard Model (SM) chiral fermions with new vector-like fermions, controlled by thevacuum expectation value (VEV) of a new complex Higgs singlet field (cid:104) Φ (cid:105) . For example,the largest third family ( t, b ) quark Yukawa couplings may be forbidden by a U (1) (cid:48) gaugeor global symmetry, broken by (cid:104) Φ (cid:105) , and generated effectively via mixing with a vector-likefourth family quark doublet ( T, B ) . Such theories predict a new physical Higgs singlet φ ,which we refer to as the Yukon, resulting from (cid:104) Φ (cid:105) , in the same way that the Higgs boson h results from (cid:104) H (cid:105) . In a simplified model we discuss the prospects for discovering theYukon φ in gluon-gluon fusion production, with ( t, b ) and ( T, B ) quarks in the loops, anddecaying in the channels φ → γγ, Zγ and φ → tT → tth , ttZ . The potential for discoveryof the Yukon φ is studied at present or future hadron colliders such as the LHC (Run 3),HL-LHC, HE-LHC and/or FCC. For example, we find that a 300-350 GeV Yukon φ could beaccessed at LHC Run 3 in the di-photon channel in the global model, providing a smokinggun signature of the origin of Yukawa couplings. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] a r X i v : . [ h e p - ph ] F e b Introduction
Given the failure so far of the Large Hadron Collider (LHC), superseding the TeVatron machineby a tenfold increase in both energy and luminosity, to detect any new particles Beyond theStandard Model (BSM), it is tempting to conclude that a so-called desert landscape awaits usin the search for new physics. In any case, to make progress will require substantial increase inmachine energy and/or luminosity, such as considered for a High Energy LHC (HE-LHC) [1], aHigh Luminosity LHC (HL-LHC) [1, 2] and/or a Future Circular Collider (FCC) [3].However, the conclusion that the LHC will not discover new particles is certainly prematuresince there are certain kinds of BSM physics which are perfectly consistent with the SM andonly constrained by the limits placed by direct collider searches. Indeed, such associated newparticles could be lying in wait to be discovered already at Run 3 of the LHC. Such new physics,if it is to be not in conflict with current measurements, must have a certain property known asdecoupling, such that the masses of the new heavy particles involved can be smoothly taken tobe very large without spoiling the consistency of the SM precision measurements.A well-known example of decoupling theories is supersymmetry (SUSY) [4], where thesquarks and sleptons can be made arbitrarily heavy without affecting any such precision mea-surements. Another example, not as widely studied as SUSY, which concerns us here is thatof theories with vector-like fermions, whose left- and right-handed components transform in thesame representation of the SM gauge group. Unlike a sequential chiral fourth family of quarksand leptons, which relies on the Higgs Yukawa couplings for its masses, and therefore can-not be made sufficiently heavy to have avoided collider constraints without violating some SMmeasurements (such as the SM Higgs production cross-sections and decay rates), a vector-likefourth family, having both left- and right-handed components transforming identically, may begiven large Dirac masses by hand. Furthermore, such masses may be increased at will withoutimpacting on any other SM measurement.The search for vector-like quarks (VLQs) is particularly interesting at the LHC [5, 6] as theycan readily be produced, being coloured fermions which are pair produced via their couplingsto gluons, which are universal and identical to the standard QCD ones. There are various SMassignments that the VLQs may take, the simplest case being that they have the SM assignmentsof the usual quarks, i.e. colour triplets being Electro-Weak (EW) doublets and singlets with theusual hypercharge assignments, resulting in the usual electric charges of the observed quarks.For example, the phenomenology of a VLQ EW doublet ( T, B ) with the electric charges of theusual quark EW doublet ( t, b ) has recently been studied in great detail in [7]. Indeed there aremany motivations for VLQs of all kinds arising from theories of little Higgs [8, 9], compositeHiggs [10–13], SUSY [14, 15] and quark mixing [16, 17].Here we shall focus on a quite different motivation for VLQs with distinctive experimentalimplications, namely that they may play a crucial role in the origin of all SM Yukawa couplings,in such a way that all fermion masses would be zero in the absence of mixing with vector-likefermions. The first example [18] of such a model was one in which the usual Higgs Yukawacouplings with the three families of chiral fermions were forbidden by a discrete Z symmetry,which is subsequently broken by the vacuum expectation value (VEV) of a new complex Higgssinglet field (cid:104) Φ (cid:105) , allowing the Higgs Yukawa couplings to arise effectively from mixing with avector-like fourth family. The origin of the large Yukawa couplings of the ( t, b ) quarks, withsmall mixing, was then explained by assuming that the mass of the VLQ EW doublet ( T, B ) is much lighter than the EW singlet VLQs. The origin of the Yukawa couplings of the secondfamily ( c, s ) quarks is due to their mixing with other VLQs which are EW singlets, whose massesare assumed to be much heavier than ( T, B ) . From an LHC perspective, only the lightest VLQEW doublet ( T, B ) is relevant.In a later example [19], the Z symmetry above was replaced by a gauged U (1) (cid:48) symmetry,under which the SM fermions are neutral but the Higgs doublets are charged, thereby forbid-1ing Yukawa couplings but allowing mixing with the charged vector-like fourth family. If the U (1) (cid:48) symmetry is gauged, its breaking will yield a massive Z (cid:48) gauge boson with non-universalcouplings to quarks and leptons is generated by the mixing with the fourth family [19]. Insuch a model [19], the connection between non-universal Z (cid:48) couplings and the origin of Yukawacouplings may have interesting experimental implications for the R K and R K ∗ anomalies [20],since the lightest VLQ doublet ( T, B ) will mix strongly with the ( t, b ) quarks and generateboth Yukawa and Z (cid:48) couplings for the latter. In such models, the vector-like fourth family mayemerge as a Kaluza-Klein excitation of quarks and leptons in 5d [21]. More recently a global U (1) (cid:48) version of such a model has been considered, focussing on the g − muon and electronanomalies in a two Higgs doublet model (2HDM) with fourth and fifth vector-like families [22].In the above models, the presence of the new singlet Higgs Φ associated with the breakingof the extra symmetry ( Z or U (1) (cid:48) ), with coupling to both the SM quarks ( t, b ) and VLQdoublet ( T, B ) , may have interesting implications for collider studies which have not so far beenconsidered in the literature. In particular, such theories predict a new physical Higgs singlet φ ,resulting from (cid:104) Φ (cid:105) , in the same way that the Higgs boson h results from (cid:104) H (cid:105) . The discoveryof φ with the predicted couplings to VLQs would be tantamount to the discovery of the originof Yukawa couplings, just as the discovery of the Higgs boson h was said to be equivalent tothe discovery of the origin of mass. For that reason we shall refer to Higgs singlet boson φ asthe Yukawa boson or Yukon for short. This motivates the present study in which we providethe first phenomenological analysis of the production and decay modes of the Yukon φ with thenecessary couplings to VLQs ( T, B ) in order to generate Yukawa couplings.In the present paper, then, we consider the experimental signatures associated with the originof Yukawa couplings along the lines of the above models [18, 19, 21, 22]. We propose and studya simplified model in which we shall ignore all fermions apart from the third family ( t, b ) quarks,since they mix most strongly with the lightest VLQ EW doublets ( T, B ) . In our simplifiedmodel we suppose that the direct ( t, b ) Yukawa couplings to Higgs doublets, in the limit of zeromixing with ( T, B ) , to be forbidden by either a gauge or a global U (1) (cid:48) symmetry broken by aHiggs singlet Φ . The third family quark Yukawa couplings are generated effectively after U (1) (cid:48) breaking via mixing with a vector-like fourth family quark doublet ( T, B ) . We shall focus on theresulting physical singlet Higgs Yukon φ associated with the U (1) (cid:48) breaking, with characteristiccouplings to ( t, b ) and ( T, B ) quarks, whose discovery at hadron colliders would provide evidencefor such models. We shall assume that there is negligible mixing of the Φ with the two Higgsdoublets, so that the physical φ scalar boson predominantly arises as the real component of thecomplex singlet field Φ after it develops its VEV. This is a natural assumption in the case that (cid:104) Φ (cid:105) greatly exceeds the Higgs doublet VEVs, and can be enforced by assuming certain couplingterms in the Higgs potential which couple Φ to the Higgs doublets to be small. We discuss theprospects for discovering the Yukon φ at Run 3 of the LHC, HE-LHC and HL-LHC as well as aFCC, focussing in particular on gluon-gluon Fusion (ggF) production, including both ( t, b ) and ( T, B ) quarks in the loops, of the Yukon φ which promptly decays via the channels φ → γγ, Zγ and φ → tT → tth , ttZ . The discovery of the Yukon φ through any of the described productionand decay modes at any of the mentioned colliders would shed new light on the origin of Yukawacouplings.Another interesting signature of the gauge model would be the Z (cid:48) , which could be discoveredin several similar channels to the Yukon φ , such as Z (cid:48) → tT → tth , ttZ . However the productionwill be suppressed as it can only be produced directly at the LHC through b ¯ b → Z (cid:48) , or throughggF. In this paper, we will not pursue Z (cid:48) signatures but instead focus on the Yukon φ , whichwould be a smoking gun signature of the origin of Yukawa couplings. It would be interesting todiscuss the Z (cid:48) and 2HDM signatures in a future publication.It is worth briefly comparing our study to other studies of singlet Higgs production and Hereafter, with our textual notation tt , tT , etc., we always intend the appropriate charge conjugated channels,i.e., t ¯ t , t ¯ T + ¯ tT , etc. U (1) (cid:48) symmetry is not a flavour symmetrysince chiral quarks and leptons of all three families carry zero charges under it [19]. Also, mostrealistic flavon models involve a large number of flavon fields, whereas in the type of modelconsidered here [18, 19] there is a unique complex scalar singlet Φ responsible for all quark andlepton Yukawa couplings. Moreover, in many flavon models, the top quark Yukawa couplingappears as a direct lowest order tree-level Higgs coupling as in the SM, independently of flavons,while in our model (cid:104) Φ (cid:105) is indispensable for generating all Yukawa couplings, including that ofthe top quark. It is for such reasons that we suggest to call the associated physical Higgs singletboson the “Yukon”, in order to distinguish it from “flavons” whose phenomenology is typicallyvery different.The layout of the remainder of the paper is as follows. In section 2 we discuss the simplifiedmodel that we consider consisting of third family quarks, fourth family VLQ doublets and agauge or global U (1) (cid:48) symmetry broken by a complex Higgs singlet field Φ . We show howYukawa couplings can emerge via a seesaw-type mechanism involving mixing with the VLQs.We also give an initial discussion of various couplings in the flavour basis, as well as the parameterspace of the model. In section 3 we present the couplings in the mass basis. In section 4 wesurvey the constraints on the model. In section 5 we calculate the branching ratios of T, B and φ .In section 6 we consider the hadron collider signatures of φ , including its production and decaycross-sections. Section 7 concludes the main body of the paper. In Appendix A we give the fullHiggs potential for the global U (1) (cid:48) model, as well as the Higgs mass matrices. In Appendix Bwe give the SM Higgs production cross-section in our model. In Appendix C we give the rathercomplicated formulae for various decay widths. Although the origin of Yukawa couplings remains mysterious in the SM, certain new physicsscenarios responsible for these couplings can be explored at present and/or future hadronicmachines. For example, the largest third family ( t, b ) quark Yukawa couplings to Higgs doubletsmay be forbidden by a U (1) (cid:48) gauge or global symmetry and generated effectively after U (1) (cid:48) breaking via mixing with a vector-like fourth family quark doublet Q = ( T, B ) , the latter beinga doublet of the SM EW group SU (2) W × U (1) Y .A complete theory of this kind, capable of generating Yukawa couplings for all quarks andleptons, would require at least one full vector-like fourth family of fermions, Q, U, D, L, E , andan additional U (1) (cid:48) gauge symmetry under which the three chiral families of quarks and leptonscarry zero charges [19]. However, in examples of this kind [19], it is usually assumed that M Q (cid:28) M D (cid:28) M U in order to provide a natural explanation of small quark mixing angles. Thusfrom a phenomenological point of view the lightest states Q with mass M Q will be discoveredfirst. It is then possible to drop the heavier fourth family SU (2) W singlet quarks U, D alongwith the vector-like leptons
L, E , leaving us only with the single VLQ doublet Q = ( T, B ) , wherevector-like means that both left (L) and right (R) chiralities have identical quantum numbersunder the SM gauge group SU (3) C × SU (2) W × U (1) Y . We also drop the first and second familiesof chiral SM fermions, assuming they have interactions with neither the vector-like fermions northe additional gauge/global symmetry.The resulting simplified model given in Table 1 includes Q L and Q R , the new vector-likedoublets, together with a gauged/global U (1) (cid:48) which is broken by the complex Higgs singletscalar Φ . It also includes two Higgs doublets H u , H d which are charged under U (1) (cid:48) , preventing3ukawa couplings to the chiral quarks q (cid:48) L , t (cid:48) R , b (cid:48) R , which have no conventional Yukawa couplings.The SM Yukawa couplings to Higgs doublets will arise only after mixing with the VLQs, as showndiagrammatically in a mass insertion approximation in Figure 1, which is reminiscent of theseesaw mechanism. However, due to the large top quark mass, the mass insertion approximationis not sufficient, and we need to use the full large angle mixing formalism introduced in [19], aswe now discuss.Figure 1: The origin of the third family ( t, b ) quark Yukawa couplings in this model.Field q (cid:48) L t (cid:48) R b (cid:48) R Q (cid:48) L Q (cid:48) R H u H d Φ SU (3) C SU (2) W U (1) Y
16 23 −
13 16 16 12 − U (1) (cid:48) − − Table 1: Simplified vector-like fermion and U (1) (cid:48) gauge or global model.The chiral and VLQs in Table 1 may be written out explicitly as doublets and singlets underthe SM weak gauge group SU (2) W as, q (cid:48) L = (cid:32) t (cid:48) L b (cid:48) L (cid:33) , t (cid:48) R , b (cid:48) R , Q (cid:48) L = (cid:32) T L B L (cid:33) , Q (cid:48) R = (cid:32) T R B R (cid:33) . (1)We use primes here to indicate the basis in which the fourth family Q (cid:48) L = ( T L , B L ) carries unitgauged/global U (1) (cid:48) charges as in Table 1 and the third family q (cid:48) L = ( t (cid:48) L , b (cid:48) L ) carry zero U (1) (cid:48) charges. In this basis the largest mass terms (ignoring contributions from SM Higgs masses) arethe following: L Q (cid:48) mass = x Q Φ Q (cid:48) R q (cid:48) L + M Q Q (cid:48) R Q (cid:48) L → Q (cid:48) R ( x Q (cid:104) Φ (cid:105) q (cid:48) L + M Q Q (cid:48) L ) ≡ ˜ M Q Q (cid:48) R Q L . (2)Here, the coupling to the Φ boson that breaks U (1) (cid:48) is promoted to a mass term once Φ acquiresa VEV, while the vector-like mass term M Q is a parameter of the theory. We can combine thetwo terms in the bracket in Equation (2) into one term with a mass ˜ M Q where we have defineda “heavy” field Q L (without primes) and an orthogonal light field q L as Q L = s Q q (cid:48) L + c Q Q (cid:48) L , q L = c Q q (cid:48) L − s Q Q (cid:48) L , (3)4here c Q = cos θ Q , s Q = sin θ Q , tan θ Q = x Q (cid:104) Φ (cid:105) /M Q , ( ˜ M Q ) = ( x Q ) (cid:104) Φ (cid:105) + ( M Q ) . (4)The right-handed fields do not get rotated so we drop primes and write Q R = Q (cid:48) R , t R = t (cid:48) R , b R = b (cid:48) R . In matrix notation, the results are neatly summarised as follows: (cid:32) q L Q L (cid:33) = (cid:32) c Q − s Q s Q c Q (cid:33) (cid:32) q (cid:48) L Q (cid:48) L (cid:33) , Q R = Q (cid:48) R , t R = t (cid:48) R , b R = b (cid:48) R , (5)where the “light” fields q L and the “heavy” fields Q L (without primes) are q L = (cid:32) t L b L (cid:33) , Q L = (cid:32) T L B L (cid:33) , (6)and the superscript ‘ ’s denotes the states after the θ Q rotations have been performed. Hence,these are weak eigenstates but not eigenstates of U (1) (cid:48) , due to mixing.Although there are no allowed Yukawa couplings of the chiral quarks q (cid:48) L , t (cid:48) R , b (cid:48) R to Higgsdoublets, due to the gauged/global U (1) (cid:48) , there are allowed Yukawa couplings which couple t (cid:48) R , b (cid:48) R to the VLQs Q (cid:48) L , L = − y u t (cid:48) R H u Q (cid:48) L − y d b (cid:48) R H d Q (cid:48) L + H . c . . (7)In terms of the “light” and “heavy” basis defined by Equation (5), by inverting the matrix, weobtain Q (cid:48) L = c Q Q L − s Q q L , which, once substituted into Equation (7), will yield effective HiggsYukawa couplings involving “light” and “heavy” fields, L = − y u t (cid:48) R H u ( c Q Q L − s Q q L ) − y d b (cid:48) R H d ( c Q Q L − s Q q L ) + H . c . . (8)The full matrix of mass terms, including the effective Higgs Yukawa couplings, involving the“light” and “heavy” fields in Equation (6) when inserted into Equation (8), is: L = − (cid:16) t L T L (cid:17) − s Q y u v u √ c Q y u v u √ ˜ M Q (cid:32) t R T R (cid:33) − (cid:16) b L B L (cid:17) − s Q y d v d √ c Q y d v d √ ˜ M Q (cid:32) b R B R (cid:33) + H.c. . (9)The matrices above show that the “light” and “heavy” states indicated by superscript ‘ ’s arenot yet true mass eigenstates, since there is still some small mixing between them.The mass matrices above may be diagonalised in the small angle approximation, L = − (cid:16) t L T L (cid:17) (cid:32) m t M T (cid:33) (cid:32) t R T R (cid:33) − (cid:16) b L B L (cid:17) (cid:32) m b M B (cid:33) (cid:32) b R B R (cid:33) + H.c. , (10)where mass eigenstates, eigenvalues and further small mixing angles are given by: (cid:32) t L,R T L,R (cid:33) = (cid:32) cos θ uL,R − sin θ uL,R e iϕ u sin θ uL,R e − iϕ u cos θ uL,R (cid:33) (cid:32) t L,R T L,R (cid:33) , (cid:32) b L,R B L,R (cid:33) = (cid:32) cos θ dL,R − sin θ dL,R e iϕ d sin θ dL,R e − iϕ d cos θ dL,R (cid:33) (cid:32) b L,R B L,R (cid:33) , (11) m t ≈ s Q y u v u √ , M T ≈ ( ˜ M Q ) + (cid:18) c Q y u v u √ (cid:19) , θ uL ≈ m t M T θ uR , θ uR ≈ c Q y u v u √ M Q , (12)5 b ≈ s Q y d v d √ , M B ≈ ( ˜ M Q ) + (cid:18) c Q y d v d √ (cid:19) , θ dL ≈ m b M B θ dR , θ dR ≈ c Q y d v d √ M Q . (13)These relationships lead to the left-handed mixing angles being suppressed relative to the (small)right-handed mixing angles by ratios of third generation (SM fermion) masses to fourth family(vector-like) masses. The phases ϕ u and ϕ d are irreducible transformation phases that will beimportant later. The resulting couplings of the third and fourth family quarks to
W, Z and Higgs bosons areexactly as given in [7]. However, there will be additional couplings to Φ and (in the gaugeoption) Z (cid:48) , as follows. In the original basis the Z (cid:48) couples only to Q (cid:48) L and Q (cid:48) R (not q (cid:48) L whichhave zero U (1) (cid:48) charge), like − g (cid:48) Z (cid:48) µ [ Q (cid:48) L γ µ Q (cid:48) L + Q (cid:48) R γ µ Q (cid:48) R ] . In the decoupling basis in Equation(5) we have Q (cid:48) L = c Q Q L − s Q q L and Q (cid:48) R = Q R , so the Z (cid:48) couplings may be expanded as thefollowing Lagrangians: L Z (cid:48) t T = − g (cid:48) Z (cid:48) µ [ t L γ µ X (cid:48) Ltt t L + t R γ µ X (cid:48) Rtt t R + { t L γ µ X (cid:48) LtT T L + t R γ µ X (cid:48) RtT T R + H . c . } + T L γ µ X (cid:48) LT T T L + T R γ µ X (cid:48) RT T T R ] , (14) L Z (cid:48) b B = − g (cid:48) Z (cid:48) µ [ b L γ µ X (cid:48) Lbb b L + b R γ µ X (cid:48) Rbb b R + { b L γ µ X (cid:48) LbB B L + b R γ µ X (cid:48) RbB B R + H . c . } + B L γ µ X (cid:48) LBB B L + B R γ µ X (cid:48) RBB B R ] , (15)In the chosen basis, t L and b L carry equal U (1) (cid:48) charges and t R and b R have zero U (1) (cid:48) chargessince they do not mix with Q R . Couplings denoted X (cid:48) L/Rxy are expanded in terms of mixingangles as follows: X (cid:48) Ltt = ( s Q ) , X (cid:48) Rtt = 0 , X (cid:48) LtT = − s Q c Q , X (cid:48) RtT = 0 , X (cid:48) LT T = ( c Q ) , X (cid:48) RT T = 1 , (16) X (cid:48) Lbb = ( s Q ) , X (cid:48) Rbb = 0 , X (cid:48) LbB = − s Q c Q , X (cid:48) RbB = 0 , X (cid:48) LBB = ( c Q ) , X (cid:48) RBB = 1 . (17)Transformation to the mass eigenstate basis requires additional small angle rotations as discussedin Equation (11), which are straightforward to include numerically. We emphasise that in thecase of the global U (1) (cid:48) model there is of course no Z (cid:48) so all of the above couplings are absent.The 2HDM plus Higgs singlet potential and mass matrices are given in Appendix A. Asdiscussed there, we shall assume that there is negligible mixing of the Φ with the two Higgsdoublets, so that the physical φ Yukon predominantly arises as the real component of the complexsinglet field Φ after it develops its VEV. This is a natural assumption in the case that (cid:104) Φ (cid:105) greatlyexceeds the Higgs doublet VEVs, and can be enforced by assuming certain coupling terms inthe Higgs potential which couple Φ to the Higgs doublets to be small. The Φ couplings tofermions are then obtained from rotating couplings in the U (1) (cid:48) flavour basis x Q Φ q (cid:48) L Q (cid:48) R , afterwe substitute Q (cid:48) R = Q R (unrotated) and q (cid:48) L = c Q q L + s Q Q L (from Equation (5)), yielding: L Φ = − x Q Φ[( c Q t L + s Q T L ) T R + ( c Q b L + s Q B L ) B R ] + H . c . (18)Further rotations are required to recover couplings of the complex scalar Φ , to propagatingfermions. The plethora of couplings above lead to decays such as: T → tZ (cid:48) , B → bZ (cid:48) , T → tφ , B → bφ with Z (cid:48) → tt, bb and φ → tt, bb , where φ is the physical scalar boson which we assumeto dominantly arise from the complex singlet field Φ .6 .2 Parameters In the case of the gauged model, both Higgs doublets H u and H d couple to the U (1) (cid:48) sector,each with a charge of − , leading to interesting phenomenology involving Higgs coupling to Z (cid:48) .The couplings are very similar to the Higgs coupling to the usual Z . The SM Higgs also couplesto the φ boson.We now derive the parameters of interest for later calculation of relevant observables includingmass eigenvalues of the SM top and vector-like top (VLT) in terms of model parameters, makinguse of Equations (12) and (13): m t ≈ s Q y u v u √ , M T ≈ ( ˜ M Q ) + (cid:32) c Q s Q m t (cid:33) , θ uL ≈ m t M T θ uR , θ uR ≈ c Q y u v u √ M Q , (19) m b ≈ s Q y d v d √ , M B ≈ ( ˜ M Q ) + (cid:32) c Q s Q m b (cid:33) , θ dL ≈ m b M B θ dR , θ dR ≈ c Q y d v d √ M Q , (20) tan θ Q = x Q (cid:104) Φ (cid:105) /M Q , ( ˜ M Q ) = ( x Q ) (cid:104) Φ (cid:105) + ( M Q ) , M Z (cid:48) = g (cid:48) v φ . (21)We rewrite all of these quantities without loss of generality in terms of the parameter set for thegauge U (1) (cid:48) model of: { M T , M Z (cid:48) , M φ , g (cid:48) , θ Q , tan β } and SM quantities. This is one possiblechoice of many equivalent descriptions. Note that for ( x Q , M Q ) we have solved the system ofsimultaneous equations and used the trigonometric identity sin( x ) = (cid:112) tan ( x ) / ( x ) sothat; y u ≈ √ m t s Q v sin β , ( ˜ M Q ) ≈ M T − (cid:32) c Q s Q m t (cid:33) , (22) θ uL ≈ m t M T θ uR , θ uR ≈ c Q y u v sin β √ M T = c Q mts Q mT , (23) y d ≈ √ m b s Q v cos β , ( M B ) ≈ M T − (cid:32) c Q s Q (cid:33) ( m t − m b ) , θ dL ≈ m b M B θ dR , (24) θ dR ≈ c Q y d v cos β √ M B = c Q m b s Q M T , v φ = M Z (cid:48) g (cid:48) , x Q = sin( θ Q ) M T (cid:104) Φ (cid:105) , M Q = M T cos θ Q . (25)In the case of the global model similar results apply but with the smaller parameter set: { M T , v φ , M φ , θ Q , tan β } and SM quantities. Note that in the case g (cid:48) = 1 (as we assume inthis paper) v φ = M Z (cid:48) . Thus many of the results that we present later for a particular value Z (cid:48) of mass in the gauge model are equally valid for the case of the global model with a v φ of thesame value. Also note that (cid:104) Φ (cid:105) = v φ / √ . We now present the full couplings in the mass basis. The content involving the Z (cid:48) and φ is newwhereas the remaining couplings to W, Z and h (the aforementioned SM-like Higgs state) aretaken from [7]. Here, h is the lightest Higgs state with SM-like couplings and the other Higgsstates are heavier and decoupled. 7 .1 Interactions Between Light and Heavy States The following interactions will determine the decays of the new heavy VL quark states, as wellas for the Yukon φ . The interactions with the SM gauge states W and Z are the same as in theusual one Higgs doublet model, and L W = − g √ (cid:2) ¯ T L γ µ V LT b b L + ¯ T R γ µ V RT b b R + ¯ t L γ µ V LtB B L + ¯ t R γ µ V RtB B R (cid:3) W + µ + H.c. , L Z = − g c W (cid:2) ¯ t L γ µ X LtT T L + ¯ t R γ µ X RtT T R − ¯ b L γ µ X LbB B L − ¯ b R γ µ X RbB B R (cid:3) Z µ + H.c. , L Z (cid:48) = − g (cid:48) (cid:104) ¯ t L γ µ X (cid:48) LtT T L + ¯ t R γ µ X (cid:48) RtT T R − ¯ b L γ µ X (cid:48) LbB B L − ¯ b R γ µ X (cid:48) RbB B R (cid:105) Z (cid:48) µ + H.c. . (26)Where the couplings as a function of the quark mixing angles and phases are V LT b = sin θ uL cos θ dL e − iϕ u − cos θ uL sin θ dL e − iϕ d , V RT b = − cos θ uR sin θ dR e − iϕ d ,V LtB = cos θ uL sin θ dL e iϕ d − sin θ uL cos θ dL e iϕ u , V RtB = − sin θ uR cos θ dR e iϕ u ,X LtT = 0 , X
RtT = − sin θ uR cos θ uR e iϕ u ,X LbB = 0 , X
RbB = − sin θ dR cos θ dR e iϕ d ,X (cid:48) LtT = ( s Q cos θ uL + e iϕ u c Q sin θ uL )( − c Q cos θ uL + e iϕ u s Q sin θ uL ) , X (cid:48) RtT = − sin θ uR cos θ uR e iϕ u ,X (cid:48) LbB = ( s Q cos θ dL + e iϕ d c Q sin θ dL )( − c Q cos θ dL + e iϕ d s Q sin θ dL ) , X (cid:48) RbB = − sin θ dR cos θ dR e iϕ d . (27)Here and throughout, we work in the so-called alignment limit for the 2HDM, β − α = π/ , suchthat interactions with the lightest scalar h are the same as with one Higgs doublet, as shownbelow together with the Yukon φ couplings, L h = − gM T M W (cid:0) ¯ t R Y LtT T L + ¯ t L Y RtT T R (cid:1) h − gM B M W (cid:0) ¯ b R Y LbB B L + ¯ t L Y RbB B R (cid:1) h + H.c. , L φ = − g (cid:48) M T M Z (cid:48) (cid:16) ¯ t R Y (cid:48) LtT T L + ¯ t L Y (cid:48) RtT T R (cid:17) φ − g (cid:48) M T M Z (cid:48) (cid:16) ¯ b R Y (cid:48) LbB B L + ¯ b L Y (cid:48) RbB B R (cid:17) φ + H.c. , (28)where the couplings are Y LtT = sin θ uR cos θ uR e iϕ u , Y RtT = m t M T sin θ uR cos θ uR e iϕ u ,Y LbB = sin θ dR cos θ dR e iϕ d , Y RbB = m b M B sin θ dR cos θ dR e iϕ d ,Y (cid:48) LtT = − e iϕ u s Q sin θ uR ( s Q cos θ uL + e iϕ u c Q sin θ uL ) , Y (cid:48) RtT = s Q cos θ uR ( c Q cos θ uL − e iϕ u s Q sin θ uL ) ,Y (cid:48) LbB = − e iϕ d s Q sin θ dR ( s Q cos θ dL + e iϕ d c Q sin θ dL ) , Y (cid:48) RbB = s Q cos θ dR ( c Q cos θ dL − e iϕ d s Q sin θ dL ) . (29) The usual light SM-like quark states will have modified gauge and scalar interactions to the SMcontent, in addition to new interactions with the U (1) (cid:48) gauge boson. Constraints on the SMinteractions are explored in [26, 27] and we will not discuss them here. The new interactions areas follows, L W = − g √ (cid:2) ¯ t L γ µ V Ltb b L + t R γ µ V Rtb b R (cid:3) W + µ + H.c. , L Z = − g c W (cid:2) ¯ t L γ µ X Ltt t L + ¯ t R γ µ X Rtt t R − ¯ tγ µ (2 Q t s W ) t − ¯ b L γ µ X Lbb b L − ¯ b R γ µ X Rbb t R − ¯ bγ µ (2 Q b s W ) b (cid:3) Z µ , L Z (cid:48) = − g (cid:48) (cid:104) ¯ t L γ µ X (cid:48) Ltt t L + ¯ t R γ µ X (cid:48) Rtt t R − ¯ b L γ µ X (cid:48) Lbb b L − ¯ b R γ µ X (cid:48) Rbb b R (cid:105) Z (cid:48) µ , (30)8ith the couplings as V Ltb = cos θ uL cos θ dL + sin θ uL sin θ dL e i ( ϕ u − ϕ d ) , V Rtb = sin θ uR sin θ dR e i ( ϕ u − ϕ d ) ,X Ltt = 1 , X
Rtt = sin θ uR ,X Lbb = 1 , X
Rbb = sin θ dR .X (cid:48) Ltt = 2 c Q s Q cos( ϕ u )cos θ uL sin θ uL + ( s Q ) (cos θ uL ) + ( c Q ) (sin θ uL ) , X (cid:48) Rtt = sin θ uR ,X (cid:48) Lbb = 2 c Q s Q cos( ϕ d )cos θ dL sin θ dL + ( s Q ) (cos θ dL ) + ( c Q ) (sin θ dL ) , X (cid:48) Rbb = sin θ dR . (31)Given the small size of the mixing angles, these are quite close to the usual SM interactions.The scalar interactions between light quarks and Higgs fields are as follows; L h = − gm t M W Y tt ¯ t t h − gm b M W Y bb ¯ b b h , L φ = − g (cid:48) M T M Z (cid:48) Y (cid:48) tt ¯ t t φ − g (cid:48) M T M Z (cid:48) Y (cid:48) bb ¯ b b φ . (32)with the couplings as Y tt = cos θ uR , Y bb = cos θ dR ,Y (cid:48) tt = s Q sin θ uR ( s Q sin θ uL − e iϕ u c Q cos θ uL ) , Y (cid:48) bb = s Q sin θ dR ( s Q sin θ dL − e iϕ d c Q cos θ dL ) . (33)We will explore the effect of deviations in Y tt in section 4.1. For the heavy VLQ interactions we find the following, L W = − g √ (cid:2) ¯ T L γ µ V LT B B L + ¯ T R γ µ V RT B B R + ¯ t L γ µ V LT B B L + ¯ T R γ µ V RT B B R (cid:3) W + µ + H.c. , L Z = − g c W (cid:2) ¯ T L γ µ X LT T T L + ¯ T R γ µ X RT T T R − ¯ B L γ µ X LBB B L − ¯ B R γ µ X RBB B R (cid:3) Z µ + H.c. , L Z (cid:48) = − g (cid:48) (cid:104) ¯ T L γ µ X (cid:48) LT T T L + ¯ T R γ µ X (cid:48) RT T T R − ¯ B L γ µ X (cid:48) LBB B L − ¯ B R γ µ X (cid:48) RBB B R (cid:105) Z (cid:48) µ + H.c. , L h = − gm t M W Y T T ¯ T T h − gm b M W Y BB ¯ B B h , L φ = − g (cid:48) M T M Z (cid:48) Y (cid:48) T T ¯ T T φ − g (cid:48) M T M Z (cid:48) Y (cid:48) BB ¯ B B φ . (34)The heavy-heavy interaction Lagrangians are similar the the above cases, but with largely re-placing t → T and b → B (noting the pre-factor for h interactions remains the top mass m t ). V LT B = cos θ uL cos θ dL + sin θ uL sin θ dL e − i ( ϕ u − ϕ d ) , V RT B = cos θ uR cos θ dR ,X LT T = 1 , X
RT T = cos θ uR ,X LBB = 1 , X
RBB = cos θ dR ,X (cid:48) LT T = 12 ( − c Q s Q cos( ϕ u )cos θ uL sin θ uL + cos(2 θ Q ) cos(2 θ uL ) + 1) , X (cid:48) RT T = cos θ uR ,X (cid:48) LBB = 12 ( − c Q cos( ϕ d ) s Q cos θ dL sin θ dL + cos(2 θ Q ) cos(2 θ dL ) + 1) , X (cid:48) RBB = cos θ dR ,Y T T = sin θ u R , Y BB = sin θ dR ,Y (cid:48) T T = ( s Q ) cos θ uR (cos θ uL + e iϕ u sin θ uL cot( θ Q )) ,Y (cid:48) BB = ( s Q ) cos θ dR (cos θ dL + e iϕ d sin θ dL cot( θ Q )) . (35)9 Constraints
Before considering the phenomenology of the new fields in this model, we realise that we do nothave complete freedom in our entire parameter set { M T , M Z (cid:48) , M φ , g (cid:48) , θ Q , tan β } . Thus, wetest to what extent the production of the SM Higgs is altered by the presence of mixing betweenthe SM-like and VLT. We then focus on constraints arising from Flavour Changing NeutralCurrents (FCNCs) and EW Precision Observables (EWPOs) affecting the Z’. Finally, there aretheoretical constraints from perturbativity to account for too. The presence of additional quark content in this model can alter the production cross-sectionof the SM Higgs boson at the LHC. In Figure 2, we draw the Feynman diagrams of dominantinteractions. In principle the LHC is sensitive to deviations of Y tt = cos θ uR by a few percent, butany deviation here is largely compensated by the contribution from Y T T = sin θ uR . We presenta full derivation of the production cross-section in Appendix B and summarise the result here.We include the dominant contributions from third generation top and VLT, { t, T } , yielding: σ ( gg → h ) | NLO = 27 . pb , (36)for {√ s = 14 TeV , M T = 1 . TeV , θ Q = π } . We calculated the cross-section at Next-to-LeadingOrder (NLO) in QCD via a k -factor of 1.7 [28]. This NLO cross-section is within the SM NLOerrors [29, 30], so that it presents no constraints or discovery potential . The same conclusionscan be obtained for the other choices of VLQ parameters that we will use in the remainder ofthe numerical analysis. gg h t tt gg h T TT
Figure 2: SM Higgs production dominant diagrams at the LHC. Z (cid:48) Constraints
Strict constraints apply to the new Z (cid:48) vector boson from FCNCs plus EWPOs and we take veryconservative limits here. Since the new vector boson has family non-universal couplings, this results in quark FCNCsfrom the coupling:
L ⊃ Z (cid:48) µ ( g bs ¯ s L γ µ b L ) , (37)Where; g bs ≈ g (cid:48) ( s Q ) V ts . (38)This effect has been studied previously [19] and one finds a bound; M Z (cid:48) (cid:38) g (cid:48) ( s Q ) (6 TeV ) . (39) To match the observed LHC cross-sections (see e.g. [29]) for σ ( gg → h ) , one requires larger loop ordersthan NLO, such as N3LO. In this work we fix NLO in the signal for consistency to match the backgrounds for γγ, tth, ttZ , which are only known to NLO.
10o maximise Yukawa couplings, which are proportional to the inverse of v φ , we thus wish tomaximise the ratio R ( g (cid:48) ) ≡ g (cid:48) /M Z (cid:48) . Here, we find a linear scaling and so no preferred value ofthe gauge coupling. From LEP, there is a model-dependent limit [31], from EWPO constraints observed in LEP-II,which do not require direct couplings between SM leptons and the new gauge boson. Such aconstraint comes in the form of θ (cid:48) , a hypothetical mixing angle between the SM Z and U (1) (cid:48) Z (cid:48) , such that θ (cid:48) (cid:46) g (cid:48) M Z M Z (cid:48) − M Z −→ M Z (cid:48) (cid:38) M Z (cid:18) g (cid:48) + θ (cid:48) θ (cid:48) (cid:19) . (40)Unlike in the FCNC case, the ratio R ( g (cid:48) ) no longer linearly scales and is instead optimised forlarger values of the gauge coupling. Taking both FCNC and EWPO constraints into account,we may find the optimal gauge coupling to maximise the allowed value of R g (cid:48) , which is g (cid:48) ≈ . and thus M Z (cid:48) ≈ GeV. However, here, and throughout, for simplicity, we will take the valueof g (cid:48) = 1 . This value has the additional benefit of aligning the VEV of Φ with the Z (cid:48) mass, M Z (cid:48) = v φ . In this case we find the two constraints to beFCNCs: M Z (cid:48) (cid:38) GeV , (41)EWPOs: M Z (cid:48) (cid:38) GeV , (42)and so take the stronger FCNC requirement, by fixing throughout g (cid:48) = 1 and M Z (cid:48) = 3000 GeV. φ The same FCNC limits apply to the Yukon φ , however, they place much weaker bounds than forthe Z (cid:48) due to the small coupling between φ and the SM b quark. Defining mass basis couplingsbetween SM quarks and φ , we obtain − L φqq = Y (cid:48) bb ¯ b L φb R + Y (cid:48) tt ¯ t L φt R + H . c ., (43)so that one finds the following limit from B meson oscillations [32]: M φ > Y (cid:48) bb (6 TeV ) , (44)where, using the small angle approximation, the leading order term of this coupling is Y (cid:48) ff (cid:39) ( c Q ) m f M T . (45)These extremely small couplings lead to very weak constraints. For example, with M T = 1000 GeV, c Q = 1 / √ , one finds M φ (cid:38) GeV. θ Q Requiring that the Yukawa coupling of the top quark is perturbative, we find, with tan β = 30 ,the following constraint: y u = √ m t s Q v sin β (cid:38) −→ θ Q (cid:38) . ∼ π/ . (46)Where possible, in this paper, we will present results for both θ Q = π/ (the most natural case)and the perturbative limit scenario, θ Q = π/ .11 .4.2 Perturbativity Constraints on λ In the CP-even mass matrix listed in Appendix A.2, there is a limit on the quartic potentialterm for λ to be perturbative. Assuming small mixing in the potential, we find M φ ≈ λ v φ , (47)so that by placing a conservative limit on λ (cid:46) one finds v φ (cid:38) M φ . (48) T , B and the Yukon φ T and B Decays
After decoupling the additional 2HDM content and with a too heavy (kinematically forbidden) Z (cid:48) , there are four possible decay modes for the heavy VLQs T and B , T → h t, T → Zt, T → W b, T → φt, (49) B → h b, B → Zb, B → W t, T → φb, (50)where the expressions for the various decay modes are given in Appendix C.We present the results of these Branching Ratios (BRs) of the VLT in Figure 3. We fix the Z (cid:48) mass to be M Z (cid:48) = 3000 GeV and the gauge coupling to be g (cid:48) = 1 , as discussed previously,while choosing the phase regime ϕ u = π, ϕ d = 0 , which will be motivated later. We furtherchoose an example φ mass to be smaller than the T, B quark ones (as an example, we have taken M φ = 340 GeV) as an attempt to maximise its production via VLQ decay. We can see that,for the VLT, the T → h t and T → Zt modes dominate and for the Vector-Like Bottom (VLB)the B → W t mode dominates. Given the large BRs to SM content, we may straightforwardlyextract the experimental limit on the VLT mass as M T (cid:38) TeV, from which we derive a VLBmass of M B = 956 GeV, which is stronger than the independently calculated VLB experimentallimit [7]. Henceforth, we will use these two masses as our standard parameter set. Since theBR to φ is very small in both cases, it is not possible to attempt any discovery with this massspectrum. Instead, we will focus in the remainder of the paper on the scenario where M φ > M T .Numerically, in this case we find, with M T = 1000 GeV, with negligible difference between θ Q = π/ and π/ , that BR ( T → h t ) = 0 . , (51)BR ( T → Zt ) = 0 . , (52)BR ( B → W t ) = 0 . . (53)(54) φ Decays
There are several possible decay modes of the Yukon φ . After decoupling the Z (cid:48) and 2HDMcontent and requiring a not overly heavy φ , M φ < M B , we are left with the following two-bodydecay modes: φ → bb, φ → tt, φ → tT, φ → bB, (55) φ → γγ, φ → gg, φ → Zγ, (56)where the full partial width expressions are derived in Appendix C. In Figure 4, we plot the BRsas a function of the scalar mass, M φ , for both θ Q = π/ in solid colours and θ Q = π/ in dashed12 Q = π / θ Q = π /
500 1000 1500 2000 2500 3000110 - - - M T / GeV B R ( T → XY ) T branching ratios, M ϕ =
340 GeV, M Z ' = = φ u = π , φ d = h tZtWb ϕ t θ Q = π / θ Q = π /
500 1000 1500 2000 2500 3000110 - - - - - M T / GeV B R ( B → XY ) B branching ratios, M ϕ =
340 GeV, M Z ' = = φ u = π , φ d = Wt ϕ bZb h b Figure 3: BRs of T (upper) and B (lower) as a function of the VLT mass for θ Q = π/ drawnin solid lines and θ Q = π/ drawn in dashed lines.ones. Some explanation is required for the peculiar behaviour of φ → Zγ . Unlike for γ , the Z can couple also non-diagonally SM quarks to VLQs, so the Zγ partial width has six amplitudecontributions from loops of tt, T T, tT, bb, BB, bB . As these six amplitudes smoothly changewith M φ , the sum of these (cid:80) i A i smoothly changes above and below zero several times with low φ masses. Since the partial width is the absolute square of the sum of the amplitudes, as the sumof amplitudes transitions from negative to positive, the partial width produces a cancellation tozero and then increases again, causing the oscillating behaviour. For the θ Q = π/ case, thesum of the amplitudes does not cross the axis, and no such a behaviour is seen.13
00 400 600 800 1000 1200 1400110 - - - - - - - M ϕ / GeV B R ( ϕ → XY ) ϕ branching ratios, M T = M B = M Z ' = = φ u = π , φ d = γγ ggZ γ Figure 4: BRs of the Yukon φ as a function of M φ . Here, solid lines represent θ Q = π/ whereasdashed ones denote θ Q = π/ . φ φ Production at the LHC
Using the same formulae as for the SM case, we calculate the production cross-section for thenew scalar φ from ggF, as can be seen in the Feynman diagrams in Figure 5. This productioncross-section is plotted with the change in φ mass in Figure 6. We calculate this at NLO byemploying a k -factor at NLO of . , similarly to the SM Higgs production case. As previously,we fix the gauge coupling to be unity, g (cid:48) = 1 , and take the EWPO and FCNC limits into accountby using M Z (cid:48) = 3000 GeV, in order to maximise the Yukawa couplings. In the solid colourswe have taken θ Q = π/ while in the dashed ones we have adopted θ Q = π/ . In addition,we plot two phase regimes, ϕ u = ϕ d = 0 in blue and ϕ u = π, ϕ d = 0 in red. Modifying thedown quark phase bears little impact due to the bottom quark mass suppresion (and so is notshown in the figure), but one can see that the ϕ u = π phase is clearly favoured. The t -loopcontribution is subleading so that the kink around M φ = 2 m t is due to interference effectsbetween the t -, T - and B -loop channels (in fact, we find that even for lower φ masses the VLTand VLB amplitudes still dominate), since the φ only couples to the SM-like top through massmixing. The production cross-sections correspond to those of the gauged (global) U (1) (cid:48) versionof the model on the left(right)-hand side of the frame. gg φt tt gg φT TT gg φB BB Figure 5: Dominant Yukon φ production modes at the LHC.14 Q = π / φ u = φ d = θ Q = π / φ u = π , φ d = θ Q = π / φ u = φ d = θ Q = π / φ u = π , φ d =
050 200 400 600 800 1000 1200 1400110 - - - - - - M ϕ / GeV σ N L O ( pp → ϕ ) / pb G auged U ( ) ' , M Z ' = T e V σ N L O ( pp → ϕ ) / pb G l oba l U ( ) ' , v ϕ = G e V NLO ϕ production cross - section, s =
14 TeV, g' = Figure 6: Production cross-section of the Yukon φ from ggF at the LHC with √ s = 14 TeV, via t, T, b, B loops, with parameters: θ Q = π/ in solid and θ Q = π/ in dashed. In blue we plotfor ( ϕ u = ϕ d = 0) (cid:39) ( ϕ u = 0 , ϕ d = π ) and in red for ( ϕ u = ϕ d = π ) (cid:39) ( ϕ u = π, ϕ d = 0) . Theleft scale is for the gauge model, M Z (cid:48) = 3 TeV, and the right scale is for the global model, with v φ = 625 GeV. Note that the shape of the curves for the gauge and global models is identical.
200 400 600 800 1000 1200 140010 - - - - - - - - M ϕ / GeV σ ( pp → ϕ → XY ) / pb bbtttTbB γγ ggZ γ LHC pp →ϕ→
XY cross - section, M T = M B = M Z ' = = φ u = π , φ d = s =
14 TeV
Figure 7: Rates for Higgs singlet Yukon φ production and decay at the LHC with √ s = 14 TeV.Solid lines correspond to θ Q = π/ and dashed lines to θ Q = π/ . Above 350 GeV the tt modedominates, suppressing the γγ signal, making the Yukon φ harder to discover.15 .2 Discovery Channels for the Yukon φ Now, we may collect together the results from previous sections to look at the full production anddecay chain of the Yukon φ . We utilise the set of parameters which maximises Yukawa couplingswhile complying with the EWPO/FCNC limits that have been discussed previously, along withthe optimal transformation phases required to give a maximal production cross-section for φ , asillustrated in section 6.1. The cross-section for φ production from ggF at NLO multiplied by theBR for each channel (computed at LO), with changing φ mass, is shown in Figure 7, where wehave included all possible kinematically-allowed decays of the new scalar. The cross-sections aremuch too small to compete with QCD backgrounds, so there is no chance to see a signal from bb, tt, bB, gg at any M φ . We will however study the possibility to discover the other channels,which suffer significantly less from QCD contamination, like φ → tT , γγ and Zγ , though wefound that the latter is never competitive (hence we neglect it thereafter).We will consider two Yukon φ mass regimes, both of which are designed to offer maximumsensitivity to the two interesting channels i.e., γγ and tT . By investigating Figure 7, two obviouschoices emerge. We will study γγ for M φ = 340 GeV and tT (with T → h t, tZ ) for M φ ≈ GeV. In all channels, we will consider both model configurations, the gauged as well as the globalone. Then, as collider energies, we will use √ s = 14 TeV (which is appropriate for both Run3 of the LHC and the HL-LHC) as well as √ s = 33 TeV (which is appropriate for both theHE-LHC and the first stage of the FCC). In all cases, we will use as integrated luminosity ( L int )the value of fb − , so as to ascertain the relative strength of each collider soilely in termsof energy reach. (In fact, we can anticipate that the model with a global U (1) (cid:48) symmetry offerssome sensitivity in the γγ case already with 300 fb − at the lower energy considered.)To recap, for all three channels, we will use the following common input parameters: M T = 1000 GeV , M B = 955 GeV , g (cid:48) = 1 , θ Q = π/ , ϕ u = π, ϕ d = 0 , (57)Gauged model: M Z (cid:48) = v φ = 3000 GeV , (58)Global model: v φ = 625 GeV , (59)and for specific signatures, γγ : M φ = 340 GeV , (60) T → tth , ttZ : M φ = 1250 GeV . (61)As intimated, these values correspond to the optimal sensitivity yield for all discovery channelsconsidered. Specifically, note that the global model VEV of v φ = 625 GeV is taken as thesmallest possible one (to optimise Yukawa couplings) whilst still accounting for the perturbativelimit set in Equation (48) of v φ (cid:38) M φ / GeV.
In all channels, we will perform the usual “bump hunt” in the invariant mass plots by determiningthe Gaussian significance at the resonance of the φ state, for a given mass M φ . For both signaland background, we will calculate the cross-sections at LO and approximate the NLO result byemploying the relevant k -factors. We calculate the signal cross-section as previously describedwhile using MadGraph [33] to calculate the backgrounds . For each channel we will identify thedetector resolution and find the signal and background cross-section within the relevant invariantmass window. To account for the number of observed events by the detector, we will employ anacceptance × selection efficiency factor to both signal and background. We find the Gaussian Throughout this work we have used version
MG5_aMC_v2.6.0 , with default parameters unless otherwise spec-ified. α = S √ S + B , (62)where S is the number of signal events and B is the value for the background ones after theyhave undergone a kinematical selection with an acceptance × selection efficiency rate (cid:15) ( S ) and (cid:15) ( B ) , respectively.For later convenience we summarise the cross-sections and cuts which we shall assume inthe remainder of the paper. The cross-sections for Yukon φ production and decay in variouschannels are given in Table 2. The cross-sections and cuts on SM background processes in variouschannels are shown in Table 3. These numbers will be used to calculate the final significancesobtained in the later results in Table 4–6. Channel Energy M φ (GeV) σ NLO ( pp → φ ) (pb) Branching Ratio Cuts Final Cross-Section (pb)Gauge Global Gauge Global γγ √ s = 14 TeV 340 0.0437 1.01 BR( φ → γγ ) = 0 . (cid:15) = 0 . ∗ . × − √ s = 33 TeV 0.228 5.25 0.00034 0.0078 tth √ s = 14 TeV 1250 0.000478 0.0110 BR( φ → tT → tth ) = 0 . (cid:15) ( S ) 0 . (cid:15) ( S ) 0 . (cid:15) ( S ) √ s = 33 TeV 0.00594 0.137 . (cid:15) ( S ) 0 . (cid:15) ( S ) ttZ √ s = 14 TeV 1250 0.000478 0.0110 BR( φ → tT → ttZ ) = 0 . (cid:15) ( S ) 0 . (cid:15) ( S ) 0 . (cid:15) ( S ) √ s = 14 TeV 0.00594 0.137 . (cid:15) ( S ) 0 . (cid:15) ( S ) Table 2: Table of cross-sections for Yukon φ production and decay in various channels, withcuts on the signal processes. Note all signals are calculated at LO, then multiplied by a k -factor of k NLO = 1 . to get the written NLO results: S NLO = S LO × k NLO . The parameterset used in all cases is as follows: { θ Q = π/ , ϕ u = 0 , ϕ d = π, g (cid:48) = 1 } . Gauged model fixes M Z (cid:48) = 3 TeV, whereas global model fixes v φ = 625 GeV. Results for v φ = 625 are a factor (3000 / (cid:39) larger than the M Z (cid:48) = 3 TeV ones in all cases, since σ NLO ( pp → φ ) ∝ v − φ andBRs are independent of v φ in all channels. ∗ Signal cut effect on φ → γγ determined from fraction of SM h → γγ events observed with andwithout η, p T cuts. Channel Energy Cuts (GeV, except η ) σ LO ( pp → X ) (pb) k -Factor Final Cross-Section (pb) γγ √ s = 14 TeV (cid:26) | η | < . , p T > GeV (cid:27) √ s = 33 TeV < M γγ < tth √ s = 14 TeV (cid:26) < M h t < (cid:27) † . × (cid:15) ( B ) √ s = 33 TeV < M tth < † . × (cid:15) ( B ) ttZ √ s = 14 TeV (cid:26) < M tZ < (cid:27) † . × (cid:15) ( B ) √ s = 33 TeV < M ttZ < † . × (cid:15) ( B ) Table 3: Table of cross-sections and cuts on SM background processes in various channels, whichwill compete against the φ boson signal in these channels shown in the previous table, wherethe suggested cuts are designed to enhance the signal. † The listed σ LO results are calculated using a cut on M h t (or M tZ ). To account for the al-ternative cut, on M ¯ th (or M ¯ tZ ), one should multiply the σ LO result by a factor of 2, which isincluded in the final cross-section. γγ signal We will now examine the possibility to detect the Yukon φ through ggF, as seen in the Feynmandiagrams in Figure 5, and decaying to γγ , as shown in the Feynman diagram in Figure 8. As17 γγf ff Figure 8: Feynman diagram for φ → γγ , where f = { b, t, B, T } .Model Experiment ( L int = 3000 fb − ) SignificanceGauged U (1) (cid:48) HL-LHC, √ s = 14 TeV 0.66 σM Z (cid:48) = 3000 GeV HE-LHC/FCC, √ s = 33 TeV 2.4 σ Global U (1) (cid:48) HL-LHC, √ s = 14 TeV 15 σv φ = 625 GeV HE-LHC/FCC, √ s = 33 TeV 52 σ Table 4: Significances for the gg → φ → γγ signal with M φ = 340 GeV and the parameter setupgiven in Equation (6.2) after the following cuts on both photons: | η | < . and p T > GeV.intimated, at low Yukon masses, namely for M φ < m t , this is the cleanest and simplest channelto consider. The relevant cuts to adopt are on the pseudorapidity and transverse momentumof the photons, both taken with | η | < . and p T > . To simulate the effect of these cutson our signal, we calculate the fraction of events captured by these through a Monte Carlo(MC) simulation and obtain an acceptance × selection efficiency rate of (cid:15) ( S ) = 0 . . Forthe two collider configurations considered (HL-LHC and HE-LHC/FCC), we assume a detectorresolution of . [34].Taking 340 GeV for the Yukon mass, this leads to an invariant mass window on the photonsof GeV < M γγ < GeV, over which we sample both signal and background. The latter,like the former, is also generated at LO, yet supplemented by a M γγ dependent NLO k -factorobtained from [35] (which, at M γγ = 340 GeV, gives . ) with, again computed through MCanalysis, of (cid:15) ( B ) (cid:39) .Based on the above kinematical selection, we find the significances given in Table 4. Fromhere, we can see that the gauged model would be difficult to find at the HL-LHC, whereas it cancertainly be seen at the HE-LHC and FCC. The global model, however, can easily be seen inall such collider environments. Indeed, the latter also has clear potential to be accessed by theend of Run 3 of the LHC, assuming L int = 300 fb − , as the significance rescales to a . σ signal.All this is modulo the effects of photon identification and of mistagging jets and/or electronsas photons, both of which are however expected to be marginal. Finally, notice that we refrainhere from placing exclusion limits on our model, as this is complicated by the dimensionalityof the parameter space, which is mapped in { M φ , θ Q , g (cid:48) , ϕ u , ϕ d , v φ , M T , M B } . Indeed, weleave this task for future studies. tth Signal
In addition to the previous mode, there is also the possibility to detect gg → φ → tT → tth ,as shown in Figure 9, when the new scalar state is heavy. To gain sensitivity in this channel(specifically, to the M φ resonance), we can exploit the decay of the VLT, as the invariantmass of its decay products will be equal to its mass, M h t = M T , and in turn we will alsohave M tth = M φ , up to some detector resolution. Unlike the γγ case, tth is reconstructedthrough several different channels which depend on the various possible decay paths of the top The efficiency is around 0.96 from [36], but within the accuracy of this paper we approximate this to unity. t h tT Figure 9: Feynman diagram for Yukon decay φ → tth .Model Experiment ( L int = 3000 fb − ) Significance (cid:15) ( S ) = (cid:15) ( B ) (cid:15) ( S ) = 2 (cid:15) ( B ) (cid:15) ( B ) · − (cid:15) ( B ) · − U (1) (cid:48) HL-LHC, √ s = 14 TeV 0.056 σ σ σ σM Z (cid:48) = 3000 GeV HE-LHC/FCC, √ s = 33 TeV 0.23 σ σ σ σ Global U (1) (cid:48) HL-LHC, √ s = 14 TeV 1.2 σ σ σ σv φ = 625 GeV HE-LHC/FCC, √ s = 33 TeV 4.6 σ σ σ σ Table 5: Significances for the gg → φ → tT → tth signal with M φ = 1250 GeV and theparameter setup given in Equation (6.2) for four values of acceptance × selection efficiency: onthe left the total signal efficiency is equal to the background one, (cid:15) ( S ) = (cid:15) ( B ) , while on the rightit is twice the background one, (cid:15) ( S ) = 2 (cid:15) ( B ) . For each of those cases, we take two backgroundtotal efficiencies: a conservative (cid:15) ( B ) = 5 × − and an optimistic (cid:15) ( B ) = 10 − .and Higgs states. This leads to a far worse M tth invariant mass resolution compared to the M γγ case, of 30% [37], which we will also use as the resolution for the invariant mass of theVLT decay products, M h t . Currently, at the LHC with √ s = 13 TeV, a total cross-section of σ ( tth ) = 790 +230 − fb from ATLAS [38] and σ ( tth ) = 639 +157 − from CMS has been measured[39]. However, for the luminosity recorded to date at Run 2, the event rates are too small to plotany meaningful invariant mass distribution in this channel, so we have to compute these by MCanalysis, which we do again with MadGraph . Here, we employ a constant k -factor to calculatethe NLO corrections from QCD for the background of this channel as k tth = 1 . , derived fromcomparing the LO MadGraph result to the SM prediction of σ ( tth ) = 507 +35 − fb given in [38].We then compute significances using a conservative total efficiency (fraction of events whichare recorded after all cuts have been applied) of (cid:15) ( B ) = 0 . from [38] and also display thesignificances for a more optimistic scenario where this is doubled to (cid:15) ( B ) = 0 . . This scenariorepresents a potential total efficiency when incorporating all possible decay paths, including h → ¯ bb , which was not incorporated in the above study. We also assume two scenarios for thesignal total efficiency: a conservative one, where it is equal to the background case (cid:15) ( S ) = (cid:15) ( B ) ,and another optimistic one, where the total efficiency is twice that of the background, (cid:15) ( S ) =2 (cid:15) ( B ) . We present the yields of these scenarios in Table 5, wherein it should be recalled that weare assuming resolution for both M h t and M tth . With this resolution, and having fixed M φ just above the tT threshold at 1250 GeV, the invariant mass cuts that we apply are GeV What is reported from CMS is the signal normalised to the SM prediction of 1.26 +0 . − . , from which we canextract the above cross-section. M h t < GeV and
GeV < M tth < GeV. We finally find that the significance istoo small for the signal to be seen at the HL-LHC, or even at the HE-LHC/FCC, for the gaugedmodel, though the global one will show a clear signal already at the HL-LHC with plenty ofdiscovery potential at the HE-LHC/FCC. ttZ
Signal φ t ZtT
Figure 10: Feynman diagram for Yukon decay φ → ttZ .Model Experiment ( L int = 3000 fb − ) Significance (cid:15) ( S ) = (cid:15) ( B ) (cid:15) ( S ) = 2 (cid:15) ( B ) (cid:15) ( B ) · − (cid:15) ( B ) · − U (1) (cid:48) HL-LHC, √ s = 14 TeV 0.033 σ σ σ σM Z (cid:48) = 3000 GeV HE-LHC/FCC, √ s = 33 TeV 0.14 σ σ σ σ Global U (1) (cid:48) HL-LHC, √ s = 14 TeV 0.71 σ σ σ σv φ = 625 GeV HE-LHC/FCC, √ s = 33 TeV 2.9 σ σ σ σ Table 6: Significances for the gg → φ → tT → ttZ signal with M φ = 1250 GeV and theparameter setup given in Equation (6.2) for four values of acceptance × selection efficiency: onthe left the total signal efficiency is equal to the background one, (cid:15) ( S ) = (cid:15) ( B ) , while on the rightit is twice the background one, (cid:15) ( S ) = 2 (cid:15) ( B ) . For each of those cases, we take two backgroundtotal efficiencies: a conservative (cid:15) ( B ) = 5 × − and an optimistic (cid:15) ( B ) = 10 − .We may proceed with ttZ in a similar fashion to tth , utilising now the decay path T → Zt as shown in Figure 10. With respect to the tth case, though. we may assume here a betterresolution of , for both M tZ and M ttZ . Hence, we can adopt the invariant mass cuts GeV < M tZ < GeV and
GeV < M ttZ < GeV, as we again have M φ = 1250 GeV.We employ as NLO k -factor for the background the value k ttZ = 1 . , which is constant in ourMC generation. We list the results for the same total efficiencies as in the tth case, (cid:15) ( S ) = (cid:15) ( B ) and (cid:15) ( S ) = 2 (cid:15) ( B ) , with (cid:15) ( B ) = 5 × − and (cid:15) ( B ) = 10 − , in Table 6. The significances aresimilar to the tth signature, so that the gauged model will remain difficult to trace anywhere inthe large φ mass scenario. Similarly to tth , the ttZ channel offers a small signal for the globalmodel at the HL-LHC and significant discovery potential at the HE-LHC/FCC. Owing to the fact that a significant portion of Z decays is into electrons and muons, while the SM Higgsstate essentially only decays into b ¯ b and W + W − → -fermions, i.e., predominantly into hadronic final states,which are more difficult to reconstruct in comparison. Conclusions
Amongst the many puzzles pertaining to the structure of the SM is the origin of the Yukawacouplings, responsible for the strong hierarchy of the quark and lepton masses. A possibleexplanation for this feature could be their different origin, for example, rather than emergingas direct couplings to the SM Higgs doublet, effective Yukawa couplings could be generatedvia their mixing with new vector-like fermionic states hitherto undiscovered. An intriguingpossibility in this respect could be the one realised through an additional, gauged or global, U (1) (cid:48) symmetry added to the SM gauge group, which explicitly forbids all direct Yukawa couplings atthe Lagrangian level. Such couplings are effectively generated after U (1) (cid:48) breaking, via seesawtype diagrams, involving both Higgs doublets and a new Higgs singlet, where the strength ofthe Yukawa coupling is suppressed by the mass of the intermediate vector-like fermion.The large third family quark ( t, b ) Yukawa couplings are effectively generated via mixingwith a vector-like fourth family quark EW doublet ( T, B ) , which are assumed to be relativelylight, with masses perhaps at the TeV scale. The smallness of the second family quark ( c, s ) Yukawa couplings is due to their coupling to heavier vector-like fourth family quark EW singlets,whose masses must lie well beyond current collider energies. Similar considerations apply to thelightest first family quarks ( u, d ) which couple to even heavier VLQs. The hierarchy of leptonicYukawa couplings can similarly result from vector-like leptons, although these are more difficultto produce at the LHC.Hence in this paper we have focussed on a simplified model involving only ( t, b ) and thelightest VLQ EW doublets ( T, B ) , whose masses may lie around the TeV scale where theywould be accessible to the LHC. Furthermore, following the U (1) (cid:48) breaking, a new Higgs singletstate, φ , which we refer to as the Yukon, is generated, with characteristic couplings to both ( t, b ) and ( T, B ) quarks. If the 125 GeV Higgs boson h is “the origin of mass”, then the Yukon φ is“the origin of Yukawa couplings”, and its discovery would be a smoking gun of this mechanism.In the considered simplified model, the masses of both the new fermion doublet ( T, B ) andHiggs singlet Yukon φ are essentially free parameters, so that one can attempt to access theirsignals at present and/or future hadron colliders, such as the LHC (Run 3), HL-LHC, HE-LHCand/or FCC. Presently, the lower bound on their masses, M T ∼ M B , is constrained to be atthe TeV scale or just below it, by direct searches at Run 2 of the LHC, as such vector-like(coloured) states can be copiously produced via standard QCD interactions. On the other hand,the constraints on the new Higgs singlet φ mass M φ are very weak, as the new scalar state φ does not couple directly to any SM particle but the t and b quarks, whose couplings remainsmall in comparison to those involving T and B states. Therefore, a natural way of establishingsome sensitivity to this model is to exploit signatures where all such new states, T, B and φ ,coexist. Specifically, one can look at the time-honoured production mode of a Higgs scalar athadronic machines, i.e., gluon-gluon fusion (ggF), wherein the φ state is produced primarily viathe T and B loops. Furthermore, depending on the relative values of M φ and M T ∼ M B , onecan search for φ → γγ signals if φ is lighter than T and B (so that loops of the latter also triggerthe di-photon decay of the former) or φ → tT → tth , ttZ otherwise (so that VLQs can insteadbe produced as real particles) .For the purpose of extracting such hallmark signatures of our model, following an initial in-vestigation of its parameter space compliant with both theoretical and experimental constraints,we have defined within it two benchmark points which maximise the sensitivity of the afore-mentioned hadronic machines to the processes gg → φ → γγ and gg → φ → tT → tth , ttZ in the presence of the corresponding irreducible backgrounds, for the choices M φ = 340 and1250 GeV, respectively, while setting M T ∼ M B ∼ TeV (i.e., just beyond the current limits).Upon performing a MC analysis of both, in the presence of NLO effects, we have been able to We expect the φ → bB modes to be more difficult to extract owing to a significant SM backgrounds from b quarks. σ level), with the HL-LHC, HE-LHC and FCC providing incontrovertible evidence of it, albeitlimitedly to the case of the global U (1) (cid:48) model. As for the signal yielding h , Z production inassociation with top-quark pairs, this will be very difficult to establish at the HL-LHC, thougha combination of the individual significances corresponding the two cases ( h and Z ) could pro-duce herein an excess very close to the σ level, with the HE-LHC/FCC again offering plentyof scope for discovery. However, it continues to be the case that this is true only for the global U (1) (cid:48) configuration, with the gauged U (1) (cid:48) one remaining very elusive.In conclusion, the discovery of the Higgs singlet Yukon φ , with the predicted couplings to ( t, b ) and ( T, B ) in the simplified global U (1) (cid:48) model, would provide evidence for a new theoryof Yukawa couplings. Our results indicate that the production cross-section of the Yukon witha mass around 300-350 GeV could be sufficiently enhanced by the vector-like quark doublet ( T, B ) in the loop to enable its di-photon decays to be observed at the LHC Run 3. However,the observation of a Yukon with a mass above the TeV scale, decaying in the tth and/or ttZ channels, would require a future collider such as the HE-LHC or FCC. Acknowledgements
S. F. K. and S. M. acknowledges the STFC Consolidated Grant ST/L000296/1. S. F. K. alsoacknowledges the European Union’s Horizon 2020 Research and Innovation programme underMarie Skłodowska-Curie grant agreement HIDDeN European ITN project (H2020-MSCA-ITN-2019//860881-HIDDeN). S. M. is supported in part through the NExT Institute. The authorsthank Billy Ford and Claire Shepherd-Themistocleous for useful discussions.
A Two Higgs Doublets with the Higgs singlet
The Higgs doublet and singlet VEVs are defined by H u = (cid:32) H + u v u + √ (cid:0) Re H u + i Im H u (cid:1)(cid:33) , H d = (cid:16) v d + √ (cid:0) Re H d + i Im H d (cid:1) , H − d (cid:17) (63) Φ = √ ( v φ + φ + i Im Φ) . (64) A.1 The potential for the 2HDM with the Higgs singlet
As recently discussed in [22], the scalar potential of the model under consideration takes theform: V = µ H u H † u + µ H d H † d + µ ΦΦ ∗ − µ sb (Φ + (Φ ∗ ) ) + λ ( H u H † u ) + λ ( H d H † d ) + λ ( H u H † u )( H d H † d ) + λ ( H u H † d )( H d H † u ) + λ ( ε ij H iu H jd Φ + H.c. )+ λ (ΦΦ ∗ ) + λ (ΦΦ ∗ )( H u H † u ) + λ (ΦΦ ∗ )( H d H † d ) (65)where the λ i are dimensionless and µ j ( j = 1 , , ) are dimensionful parameters. Note that the µ sb is a mass squared parameter which we include only for the global U (1) (cid:48) model, where itsoftly breaks the U (1) (cid:48) symmetry of the Higgs potential, giving non-zero mass to the would-beGoldstone boson resulting from the CP-odd mass matrix. In the gauged U (1) (cid:48) model, the µ sb term is absent and the Goldstone boson is eaten by the Z (cid:48) .22 .2 Mass matrix for CP-even, CP-odd neutral and charged scalars The squared mass matrix for the CP-even scalars in the basis (cid:0) Re H u , Re H d , φ (cid:1) takes the form: M CP − even = λ v u − λ v u v φ v u λ v φ + 2 λ v u v d √ v φ ( λ v d + λ v u ) λ v φ + 2 λ v u v d λ v d − λ v u v φ v d √ v φ ( λ v u + λ v d ) √ v φ ( λ v u + λ v d ) √ v φ ( λ v u + λ v d ) 2 λ v φ (66)From the mass matrix given above, we find that the CP-even scalar spectrum is composed ofthe GeV SM-like Higgs h and two non SM CP-even Higgses H , .The CP-odd mass matrix in the basis (cid:0) Im H u , Im H d , Im Φ (cid:1) is given by: M CP − odd = − λ v d v φ v u − λ v φ −√ λ v d v φ − λ v φ − λ v u v φ v d −√ λ v u v φ −√ λ v d v φ −√ λ v u v φ − λ v u v d − µ sb (67)In this paper we assume that there is negligible mixing of the Φ with the two Higgs doublets,so that the physical Yukon φ scalar boson predominantly arises as the real component of thecomplex singlet field Φ after it develops its VEV. This is a natural assumption in the case that v φ (cid:29) v u , v d . Alternatively this can be enforced by assuming the coupling terms in the Higgspotential which couple Φ to the Higgs doublets controlled by λ to be small.The charged Higgs mass matrix is given by: M charged = λ v d − λ v d v φ v u λ v u v d − λ v φ λ v u v d − λ v φ λ v u − λ v u v φ v d . (68) B SM Higgs Production Cross-Section
We fully simplify the partial width from The Higgs Hunter’s Guide [40], taking into account allSM Higgs VEVs and SU (2) gauge coupling factors into a compact expression: Γ( h → gg ) = α s π M h (cid:12)(cid:12)(cid:12) (cid:88) i y i m i (cid:2) − τ i ) f ( τ i ) (cid:3)(cid:12)(cid:12)(cid:12) , τ i = 4 m i M h (69) y i represents the coupling at the vertex between the two fermions of flavour i and the SM Higgsboson. The production cross-section from a hard scattering process involving two gluons is thenrelated to the decay width thus; dσdy ( AB → h + X ) = π Γ( h → gg )8 M h g A ( x A , M h ) g B ( x B , M h ) (70) g A and g B are the parton distribution functions (PDFs) of the two gluons, and the fractionalmomenta x A and x B are related to the centre-of-mass energy as per Equation (71). x A = M h e y √ s , x B = M h e − y √ s (71) y is the rapidity of the SM Higgs. This gives an expression for the differential cross-section withrespect to rapidity: dσdy ( AB → h + X ) = α s πM h (cid:12)(cid:12)(cid:12) (cid:88) i y i m i (cid:2) − τ i ) f ( τ i ) (cid:3)(cid:12)(cid:12)(cid:12) g A ( x A , M h ) g B ( x B , M h ) . (72)23n the SM, we do not to make a distinction between the Yukawa coupling y i , and the mass ofthe top mass m t , since m t = y t v/ √ . However, in the model we scrutinise here, this is no longertrue due to mixing between the chiral and VLQs. In general, one defines the Yukawa coupling y i as the interaction strength between two fermions and the scalar boson. For a generic fermion f coupling to a complex scalar φ , the Yukawa coupling is given in Equation (73). L = y f ¯ f f φ (73)In the VLQ model, one finds a modified top Yukawa coupling to be [7]: y t = √ m t v cos θ uR = y t cos θ uR , (74)which will reduce the cross-section of Higgs production due to the light top state. However,there is also a contribution in the loop due to the heavier VLT, which can be shown to have aYukawa coupling: y T = √ m t v sin θ uR . (75)In the limit that the VLT has the same mass as SM top, then one would have exact cancel-lation, and no change to the cross-section. Considering larger VLT masses, there is still somecancellation in this direction. We show the two Feynman diagrams that contribute dominantlyto the Higgs production cross-section in our model, in Figure 2.In the calculation of the SM Higgs production cross-section in the model under test weinclude contributions from t and T , along with the SM bottom b and the VLB, B . C Decay Widths
C.1 T and B decays In this work, for the VLQ decays, we use the results of [7]. We copy their results into theappendix below, and include our additional Q → φq modes.Defining r x = m x /m Q , where Q is the heavy quark and x one of its decay products, and thefunction λ ( x, y, z ) ≡ ( x + y + z − x y − x z − y z ) , (76)the partial widths for T decays are Γ( T → W + b ) = g π M T M W λ ( M T , m b , M W ) / (cid:8) ( | V LT b | + | V RT b | ) (cid:2) r W − r b − r W + r b + r W r b (cid:3) − r W r b Re V LT b V R ∗ T b (cid:9) , Γ( T → Zt ) = g πc W M T M Z λ ( M T , m t , M Z ) / (cid:8) ( | X LtT | + | X RtT | ) (cid:2) r Z − r t − r Z + r t + r Z r t (cid:3) − r Z r t Re X LtT X R ∗ tT (cid:9) , Γ( T → h t ) = g π M T M W λ ( M T , m t , M h ) / (cid:8) ( | Y LtT | + | Y RtT | ) (cid:2) r t − r h (cid:3) + 4 r t Re Y LtT Y R ∗ tT (cid:9) , Γ( T → φt ) = g π M T M Z (cid:48) λ ( M T , m t , M φ ) / (cid:110) ( | Y (cid:48) tT L | + | Y (cid:48) tT R | ) (cid:2) r t − r φ (cid:3) + 4 r t Re Y (cid:48) tT L Y (cid:48) tT R ∗ (cid:111) , (77) Note given a complex scalar which undergoes Spontaneous Symmetry Breaking (SSB), we would have φ =( ϕ + v (cid:48) ) / √ , but in this project we solely write in terms of the complex scalar field and not the real scalar field ϕ . B quark they are analogous, Γ( B → W − t ) = g π M B M W λ ( M B , m t , M W ) / (cid:8) ( | V LtB | + | V RtB | ) (cid:2) r W − r t − r W + r t + r W r t (cid:3) − r W r t Re V LtB V R ∗ tB (cid:9) , Γ( B → Zb ) = g πc W M B M Z λ ( M B , m b , M Z ) / (cid:8) ( | X LbB | + | X RbB | ) (cid:2) r Z − r b − r Z + r b + r Z r b (cid:3) − r Z r b Re X LbB X R ∗ bB (cid:9) , Γ( B → h b ) = g π M B M W λ ( M B , m b , M h ) / (cid:8) ( | Y LbB | + | Y RbB | ) (cid:2) r b − r h (cid:3) + 4 r b Re Y LbB Y R ∗ bB (cid:9) , Γ( B → φb ) = g π M B M Z (cid:48) λ ( M B , m b , M φ ) / (cid:110) ( | Y (cid:48) bBL | + | Y (cid:48) bBR | ) (cid:2) r b − r φ (cid:3) + 4 r b Re Y (cid:48) bBL Y (cid:48) bBR ∗ (cid:111) . (78) C.2 Yukon decays φ → f f Armed with these couplings, we can now consider the partial widths for φ decay modes, followingthe process in The Higgs Hunter’s Guide [40] for decay of the SM Higgs boson but adapting for φ . As such, we split possible couplings into their axial and vector pieces in Equation (79). V φf f = i ( A + Bγ ) (79)The decay of a scalar (of any charge) into two fermions with colour factor N c goes like Equation(80), where we have labelled the scalar by φ in anticipation of our application to the U (1) (cid:48) -breaking boson. Γ( φ → f ¯ f ) = N c λ (cid:0) m f , m f , M φ (cid:1) / M φ π (cid:20) ( M φ − m f − m f ) A + B − m f m f ( A − B ) (cid:21) (80)Where λ is the following kinematic function: λ = (cid:0) ( m f + m f − M φ ) − m f m f (cid:1) . (81)This reproduces the usual results for two fermions of the same mass [40] as seen in Equation(82). Γ( φ → ¯ f f ) = N c g m f πm W β M φ = N c y f π β M φ (82) m W = gv/ , m f = y f v/ √ . y f is the usual Yukawa coupling between a scalar and two fermions.The function labelled β is purely kinematic and can be expressed as the following: β = 1 − m f /M φ . (83) C.3 Yukon decays φ → V V Γ( φ → Z (cid:48) Z (cid:48) ) = g (cid:48) (cid:0) M φ − M φ M Z (cid:48) + 12 M Z (cid:48) (cid:1) πM Z (cid:48) M φ (cid:115) − M Z (cid:48) M φ , (84)Defining τ i = 4 (cid:18) M i M φ (cid:19) (85)25nd f ( τ ) ≡ arcsin ( √ τ ) , for τ ≥ − [log √ − τ −√ − τ − iπ ] , for τ < (86)then the γγ and gg decay modes are [40] Γ (cid:0) φ → γγ (cid:1) = 9 α EM π M φ (cid:12)(cid:12)(cid:12) (cid:88) i − e i y (cid:48) i m i (cid:2) (cid:0) − τ i (cid:1) f ( τ i ) (cid:3)(cid:12)(cid:12)(cid:12) , (87) Γ( φ → gg ) = α s π M φ (cid:12)(cid:12)(cid:12) (cid:88) i y (cid:48) i m i (cid:2) (cid:0) − τ i (cid:1) f ( τ i ) (cid:3)(cid:12)(cid:12)(cid:12) (88)Where e i is the QED charge of the fermion propagating in the loop which couples the scalar tophotons and y (cid:48) i is defined from L φ ⊃ − y (cid:48) i √ ψ i ψ i φ. (89) C.4 Yukon decay φ → Z γ We present here the partial width for the Yukon φ decay to γZ for a VLQ model from [41]. Thecouplings are defined from the following Lagrangian: L = − eA µ Q f ¯ f γ µ f + eZ µ ¯ f γ µ ( g fL P L + g fR ) f − m t v φ φ ¯ t ( κ t + iγ ˜ κ t ) t + φ ¯ T ( y T + iγ ˜ y T ) T + φ ¯ t ( y tTL P L + y tTR P R ) T (90) Γ( φ → γZ ) = G F α m φ √ π (1 − m Z m φ ) | A t + A T + A tT | . (91)Where A t = 2 N Ct Q t ( g tL + g tR ) κ t A f ( τ t , λ t ) = 2 N Ct Q t κ t c L + I T ( s L + s R ) − s W s W c W A f ( τ t , λ t ) , A T = − N CT Q T y T v φ M T ( g TL + g TR ) A f ( τ T , λ T ) = − N CT Q T y T v φ M T s L + I T ( c L + c R ) − s W s W c W A f ( τ T , λ T ) , A tT = − N CT Q T v φ m φ − m Z { m t Re( g tTL ( y tTL ) ∗ + g tTR ( y tTR ) ∗ )[( m φ − m Z − m t ) C (0 , m Z , m φ , m t , m t , M T ) − M T C (0 , m Z , m φ , M T , M T , m t ) − m Z B ( m φ , m t , M T ) − B ( m Z , m t , M T ) m φ − m Z − M T Re( g tTL ( y tTR ) ∗ + g tTR ( y tTL ) ∗ )[( m φ − m Z − M T ) C (0 , m Z , m φ , M T , M T , m t ) − m t C (0 , m Z , m φ , m t , m t , M T ) − m Z B ( m φ , m t , M T ) − B ( m Z , m t , M T ) m φ − m Z − } . (92)Here τ i and λ i are defined as τ f = 4 m f m φ , τ W = 4 m W m φ , λ f = 4 m f m Z , λ W = 4 m W m Z . (93)26nd the A f , A W are defined as A f ( τ f , λ f ) ≡ I ( τ f , λ f ) − I ( τ f , λ f ) ,I ( τ, λ ) = τ λ τ − λ ) + τ λ τ − λ ) [ f ( τ ) − f ( λ )] + τ λ ( τ − λ ) [ g ( τ ) − g ( λ )] ,I ( τ, λ ) = − τ λ τ − λ ) [ f ( τ ) − f ( λ )] . (94)Here f ( τ ) is defined as F f ( τ f ) ≡ − τ f [1 + (1 − τ f ) f ( τ f )] , F W ( τ W ) ≡ τ W + 3 τ W (2 − τ W ) f ( τ W ) ,f ( τ ) ≡ arcsin ( √ τ ) , for τ ≥ − [log √ − τ −√ − τ − iπ ] , for τ < . (95)and g ( τ ) is defined as g ( τ ) ≡ √ τ − √ τ ) , for τ ≥ √ − τ [log √ − τ −√ − τ − iπ ] , for τ < . (96) C (0 , m Z , m φ , m t , m t , M T ) = C (0 , m Z , m φ , M T , m t , m t )= − (cid:90) (cid:90) (cid:90) dxdydz δ ( x + y + z − yp + z ( p + p )] + ( x + y ) m t + zM T − yp − z ( p + p ) = − (cid:90) (cid:90) (cid:90) dxdydz δ ( x + y + z − yz ( m φ − m Z ) + z m φ + ( x + y ) m t + zM T − zm φ . (97) φ Zγt tt φ ZγT TT φ Zγt TT φ ZγT tt Figure 11: φ → Zγ contributions. 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