Probing the fermionic Higgs portal at lepton colliders
PPrepared for submission to JHEP
Probing the fermionic Higgs portal at leptoncolliders
Michael A. Fedderke, a,b
Tongyan Lin, b and Lian-Tao Wang a,b a Department of Physics, The University of Chicago, Chicago, Illinois, 60637, USA b Enrico Fermi Institute and Kavli Institute for Cosmological Physics, The University of Chicago,Chicago, Illinois, 60637, USA
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We study the sensitivity of future electron-positron colliders to UV comple-tions of the fermionic Higgs portal operator H † H ¯ χχ . Measurements of precision electroweak S and T parameters and the e + e − → Zh cross-section at the CEPC, FCC-ee, and ILC areconsidered. The scalar completion of the fermionic Higgs portal is closely related to thescalar Higgs portal, and we summarize existing results. We devote the bulk of our analysisto a singlet-doublet fermion completion. Assuming the doublet is sufficiently heavy, weconstruct the effective field theory (EFT) at dimension-6 in order to compute contributionsto the observables. We also provide full one-loop results for S and T in the general massparameter space. In both completions, future precision measurements can probe the newstates at the (multi-)TeV scale, beyond the direct reach of the LHC. Keywords:
Higgs Physics, Effective Field Theories
ArXiv ePrint: a r X i v : . [ h e p - ph ] M a y ontents S and T
25B A clarification on matching in the presence of mixing 27
B.1 Example 1: two-point functions of the UV-theory n field and the EFT field χ Z -boson coupling to the EFT field χ Among the particularly compelling scenarios in which new physics couples to the StandardModel (SM) are the so-called vector, neutrino and Higgs “portals,” which involve the threelowest-dimension SM-singlet operators which can couple to new physics (see, e.g., ref. [1]and references therein). Of these, the Higgs portal, which encompasses operators of theform H † H O NP where O NP is a SM-singlet operator built from new fields, has been thefocus of much interest in the literature (e.g., refs. [2–15]), especially in the wake of the2012 discovery [16, 17] of a light Higgs boson with SM-like properties [18, 19]. There are anumber of reasons for its appeal aside from its low dimensionality. The Higgs portal providesa possible coupling of the dark matter (sector) to the SM, as we discuss in more detail below.Some well-studied models of this class—such as H † HS with S a scalar singlet—have theability to effect non-trivial modification of the dynamics of electroweak symmetry breaking(EWSB), rendering a first-order phase transition possible, with interesting implicationsfor electroweak baryogenesis [20–25]. Additionally, such models may characterize the low-energy physics in theories which ameliorate the SM hierarchy problem [26].While the simple picture of only considering the lowest-dimension fermionic Higgs por-tal operator in a bottom-up analysis is of course instructive, much more can be said about– 1 – h S χ ¯ χ hh ¯ χχF Figure 1 . Two possible UV completions of the fermionic Higgs portal operator, H † H ¯ χχ . Inthe “scalar completion” (left diagram), a singlet scalar S has a renormalisable coupling to the H doublet and a Yukawa coupling to the singlet χ field; in the “fermionic completion” (right diagram),a Yukawa-like coupling exists between a vector-like SU (2) doublet F and the H and χ fields. particular possible UV completions of this model. In this paper we study two such com-pletions (see figure 1) for the CP-even fermionic Higgs portal operator, H † H ¯ χχ , and theirprecision constraints. In the first model, a heavy singlet S acts as a mediator, with a Yukawacoupling to the fermion χ (which we take to be Dirac) and renormalisable couplings to theHiggs-sector of the SM. This is essentially an extension of the singlet-scalar Higgs portalscenario. In the second model, we introduce a new heavy vector-like SU (2) fermion doublet F and couple both the singlet and doublet fermions to the SM Higgs: − κH ¯ F χ + h.c.. Suchsinglet-doublet models have also been studied in the past [27–29].Since the new states in either completion are coupled to the Higgs-sector of the SM (andin the latter scenario, also have non-trivial charges under the SM gauge group), precisionelectroweak and Higgs physics will be important probes of these potential UV completions;for example, measurements of S , T and the Higgsstrahlung cross-section σ Zh are particularlysensitive. Although the present electroweak precision constraints have been known for along time to be stringent, the current round of conceptual studies for future precision high-energy e + e − machines—e.g., the ILC, CEPC, and FCC-ee —have resulted in significantlysharper projections for how the constraints could improve. This brings into even clearerfocus the discovery and exclusion potential possible through a study of these probes. Itis the major goal of this paper to more fully understand how the projected sensitivities ofthese machines will allow stringent constraints to be placed on the UV completions of theHiggs portal that we consider.Although we do not impose a dark-matter interpretation on the singlet χ in our models,it is worth noting this possibility has received much attention recently. For the dimension-five fermionic Higgs portal, H † H ( a ¯ χχ + ib ¯ χγ χ ) , which comprises both CP-even and CP-odd coupling terms, a thermal-relic dark-matter interpretation is tightly constrained in theCP-even scenario by spin-independent direct detection constraints, whereas for the CP-oddcoupling the direct detection constraints are alleviated and much parameter space remains The collider formerly known as TLEP. – 2 –pen (see, e.g., refs. [8, 15, 30–32] and references therein). Turning to the UV completion ofthese operators, additional parameter space for thermal-relic dark matter is available whenthe new mediator mass (either the doublet fermion or singlet scalar) is similar to or belowthe mass scale of χ . For example, the relic density may be set by coannihilation of thecharged and neutral fermions [29, 33], or by dark matter annihilation into lighter scalars(see refs. [7, 34–37] for studies of the singlet scalar UV completion). These scenarios canhave small couplings and thus evade electroweak precision and other constraints. As ourfocus is not on the dark-matter interpretation of χ , but rather on precision probes of thefermionic Higgs portal coupling and its UV completions generally, we restrict ourselves tothe regime where the new mediator is heavy and the EFT is valid.The remainder of this paper is structured as follows: we begin in section 2 summariz-ing the experimental sensitivities on measurements of S , T and σ Zh which are currentlyprojected to be attainable at future colliders. In section 3 we introduce and briefly discussthe singlet-scalar UV completion, summarizing existing results in the literature. In section4 we introduce the fermionic UV completion and discuss our general expectations. We thendetail the construction of the EFT we used to analyse this model in section 4.1, and give theEFT results for the precision electroweak and Higgsstrahlung constraints in sections 4.1.1and 4.1.2, respectively. In section 4.2 we discuss the one-loop computation of the precisionelectroweak limits in the general mass parameter space for this model. We conclude insection 5. Appendix A contains technical details on the one-loop computation of S and T ,while Appendix B contains a clarification on a technical point of the EFT matching. Several proposed e + e − collider experiments which can make advances in precision elec-troweak and Higgs physics measurements are currently under consideration: the Inter-national Linear Collider (ILC) [38], the Future Circular Collider (FCC-ee) [39] and theCircular Electron Positron Collider (CEPC) [40]. We summarize the sensitivities for vari-ous present and future measurements of the electroweak S and T parameters [41, 42], andthe Higgsstrahlung cross-section σ Zh , along with the references on which they are based, intable 1. For each collider, we include several scenarios for the run parameters in order tocompare the improvements in sensitivity that are possible with collider upgrades.For the S and T electroweak precision obserable (EWPO) limits, we present a parametri-sation of the covariance matrix used to construct the 68% coverage likelihood contours pre-sented graphically in the references. We made the simplifying assumption of exact Gaussianuncertainties centered at ( S, T ) = (0 , , and have parametrised the covariance matrix as Σ = (cid:32) σ s ρ st σ s σ t ρ st σ s σ t σ t (cid:33) , (2.1)from which the likelihood function is given by − L / L ] = ∆ T Σ − ∆ , where ∆ T =( S, T ) . The parameters ( σ s , σ t , ρ st ) were obtained by least-squares fitting of the constraintequation − L / L ] = 2 . to a large number of ( S, T ) co-ordinates read from the relevantgraphically-presented 68% coverage ellipses.– 3 – able 1 . Experimental sensitivities for current and future measurements of ( S, T ) and σ Zh assumed in this paper. The σ Zh sensitivities are quoted as 68% confidence percentage uncertainties,while the ( S, T ) limits are shown as the parameters ( σ s , σ t , ρ st ) which define the covariance matrixin eq. (2.1), and are extracted from the 68% coverage likelihood contours ( − L / L ] = 2 . ) inthe indicated references. † The individual ILC Higgsstrahlung constraints must be combined wherenecessary by summing the relevant − L / L ] values to obtain the constraints for the threescenarios we consider in this paper: 250/fb @ 250 GeV, 250/fb @ 250 GeV + 500/fb @ 500 GeV,and 1.15/ab @ 250 GeV + 1.6/ab @ 500 GeV. Precision Electroweak ( S, T ) Scenario ( σ s , σ t , ρ st ) Reference
Current ( 8.62 × − , 7.37 × − , 0.906 ) [43], figure 1CEPC “Baseline” ( 2.39 × − , 1.93 × − , 0.844 ) [43], figure 4CEPC “Improved Γ Z , sin θ ” ( 1.14 × − , 8.79 × − , 0.518 ) [43], figure 4CEPC “Improved Γ Z , sin θ, m t ” ( 1.12 × − , 7.26 × − , 0.779 ) [43], figure 4ILC ( 1.71 × − , 2.14 × − , 0.891 ) [43], figure 1FCC-ee-Z (aka TLEP–Z) ( 9.32 × − , 8.70 × − , 0.440 ) [43], figure 1FCC-ee-t (aka TLEP–t) ( 9.24 × − , 6.18 × − , 0.794 ) [43], figure 1 Higgsstrahlung σ Zh Scenario (∆ σ Zh ) /σ Zh (%) Reference CEPC: 5/ab @ 240 GeV 0.5% [40]ILC: 250/fb @ 250 GeV 2.6% [44, 45]ILC: 500/fb @ 500 GeV 3.0% [44, 45]ILC: 1.15/ab @ 250 GeV 1.2% [44, 45]ILC: 1.6/ab @ 500 GeV 1.7% [44, 45]FCC-ee: 10/ab @ 240 GeV (4IP) 0.4% [39] † In this parametrisation, the T -axis intercepts of the 68% coverage likelihood contour for two parametersare at S = 0 , T = ± σ t (cid:2) − ρ st (cid:3) / · (cid:2) (∆ χ ) (cid:3) / where (∆ χ ) = 2 . , and similarly for the S -axisintercepts. In particular, this means that for S = 0 , the coverage in the T parameter is about a factor of 2better when going from the FCC-ee-Z to the FCC-ee-t scenario, despite the modest 30% decrease in σ t . The precision electroweak sensitivity is based primarily on Z -pole measurements atthe various colliders. For the CEPC, the “baseline” is a Z mass threshold scan. The“improved” scenarios consider the gains if the collider is upgraded to allow for improvedmeasurements of Γ Z (with the resonant depolarization method for energy calibration), sin θ , or possibly m t (from an ILC top threshold scan). The FCC-ee-Z sensitivities are for Z -pole measurements with polarised beams (denoted “TLEP-Z” in ref. [43]). The FCC-ee-tscenario includes Z -pole measurements with polarised beams along with threshold scansfor W W and t ¯ t (denoted “TLEP-t” in ref. [43]) and has improved sensitivity owing to moreprecise measurements of the W and top masses.As has recently been emphasized in refs. [26, 46] (and investigated further in, e.g.,refs. [47–51]), the “Higgsstrahlung” process e + e − → Zh can be measured at the percent or– 4 –ub-percent level at future colliders and is a sensitive probe of new physics. Among thevarious ways a new-physics model modifies the Higgsstrahlung cross-section, a modificationto the wavefunction renormalization of the Higgs (i.e., a modification of the momentum-dependent part of the Higgs two-point function) may be induced, leading to a “Higgsoblique” correction.The limits on ∆ σ Zh /σ Zh are quoted in table 1 as the 68% percentage confidence boundsgiven in the various references. We consider a CEPC projection with 5/ab of data at √ s =
240 GeV [40] and an FCC-ee projection with the assumption that there are 4 interactionpoints with 10/ab of combined integrated luminosity collected at √ s = 240 GeV [39].Various configurations of proposed runs for the ILC have been analyzed in the literature[44, 45, 58]; we consider only the configurations where (a) 250/fb of data are collected at √ s = 250 GeV, (b) 500/fb of data at √ s = 500 GeV are added to the 250/fb of datacollected at √ s = 250 GeV, and (c) the integrated luminosity is increased to a total of1.15/ab at √ s = 250 GeV and 1.6/ab at √ s = 500 GeV.
The simplest possible UV completion of the CP-even fermionic Higgs portal is to takethe Standard Model (SM) augmented by a vector-like Dirac fermion SM-singlet χ and aSM-singlet scalar S . These couple via the following Lagrangian L = L sm + i ¯ χ /∂χ − m χ ¯ χχ + 12 ( ∂ µ S ) − m S S − b m S S − λ S S + a m S S | H | + (cid:15) S S | H | − κ S S ¯ χχ, (3.1)where a, b, (cid:15) S , κ S , m S and m χ are real parameters. To be explicit, our sign and normalisationconventions for the SM Higgs-sector are L sm ⊃ | D µ H | + µ | H | − λ | H | , (3.2)and our unitary-gauge normalization conventions are such that H = (0 , ( v + h ) / √ , with v ≈ GeV.This model is just the renormalisable scalar Higgs portal model (see, e.g., ref. [50]for detailed discussion closely related to the present work), augmented with the singlet- χ Yukawa coupling. If we consider the limit where bm S , (cid:15) S (cid:28) am S , κ S , and further assumethat m S (cid:29) v, m χ , the mediator S can be integrated out at tree-level to give rise to thefollowing effective field theory: L EFT ⊃ L sm + i ¯ χ /∂χ − m χ ¯ χχ + 12 a | H | − aκ S m S H † H ¯ χχ + a m S
12 ( ∂ µ | H | ) + 12 κ S m S ( ¯ χχ ) + · · · , (3.3) See also refs. [52–57] for previous work on constraining anomalous Higgs couplings with the Hig-gsstrahlung cross-section. – 5 –here · · · represents terms proportional to various powers of (cid:15) S , b and neglected higher-dimensional operators. Allowing terms proportional to (cid:15) S , bm S leads, at dimension-6, toonly one additional operator, | H | , which taken together with the ability to change the signof the | H | operator in the SM-Higgs Lagrangian and yet maintain a stable minimum tothe Higgs potential, can have interesting implications for the order of the electroweak phasetransition [50, 59].For the purposes of the discussion here, the | H | operator can simply be absorbedinto an unobservable shift of the SM Higgs quartic-coupling: λ → λ + a . The fermionicHiggs portal operator H † H ¯ χχ leads to a variety of effects (see, e.g., ref. [31] for a detailedanalysis): upon EWSB, it contributes to the mass of the χ field, and allows both h ¯ χχ and h ¯ χχ couplings. The former coupling allows for χ - χ scattering via Higgs exchange, andthere is a further scattering contribution from the ( ¯ χχ ) operator.It is possible in this scenario to have the χ field play the role of the stable dark matter(DM), saturating the relic density. If the mediator S is assumed to be somewhat heavierthan the χ , an EFT analysis is applicable [30, 31]: the relic abundance requirement fixes m S / ( aκ S ) ∼ . TeV, while satisfying the LUX direct detection bounds [60] requires thatthe χ be fairly massive, m physical χ (cid:38) TeV. These requirements are actually in tension withthe assumptions of an EFT analysis as they demand a very large coupling aκ S (cid:38) π toavoid m χ > m S and the resulting issues with perturbative unitarity [61–63] for the EFTdescription of the non-relativistic DM freeze-out. This essentially means that, with thefinely-tuned exception of m χ ∼ m h / [30, 31], the heavy-mediator scenario is ruled outhere. On the other hand, the scenario with a light mediator, m S < m χ , is still viable as thedirect detection constraints can be avoided while maintaining the correct relic abundance[30]. Of course, if we were to abandon the DM interpretation for χ , and allow it to decaysufficiently quickly via some suitable modification to this model, the couplings a and/or κ S can be freely dialed down without overclosing the universe, thereby alleviating the directdetection constraints and allowing light m χ .In the EFT description for this model, the operator ( ∂ µ | H | ) leads to a “Higgsoblique” correction to the Higgstrahlung process; i.e., the operator induces a Higgs wave-function renormalization δZ h = a v /m S , which modifies the cross-section by ∆ σ Zh /σ Zh = δZ h [26, 49]. With a sensitivity of ∆ σ Zh /σ Zh ∼ . % (corresponding closely to the CEPCand FCC-ee sensitivities in table 1), values of m S /a (cid:46) . TeV could be ruled out at 95%confidence, with 5- σ discovery reach up to m S /a ∼ . TeV [49]. (The limits on m S /a justquoted are slightly weaker than those in figure 1 of ref. [50], wherein a combined fit to allprojected ILC Higgs coupling measurements is considered.)In this model, the electroweak precision observables S and T are generated by one-loop running of the operators between µ = m S and µ = m W , leading to a mixing of theHiggs oblique operator with the operators responsible for giving the S and T parameters.This one-loop running has been computed in ref. [50], which considers the UV completionabove in the limit that κ S → (and with no assumption on the sizes of (cid:15) S , b ). Theleading contributions to the S and T parameters are found to be log-enhanced, but linearlyproportional to the high-scale ( µ = m S ) value of the Wilson coefficient of the Higgs obliqueoperator, and suppressed by a loop factor. – 6 –o summarize the results of ref. [50], the electroweak precision constraints on the modelare marginally weaker than those from the Higgsstrahlung measurement, if the most opti-mistic Higgsstrahlung precision (with FCC-ee or CEPC) is compared to the most optimisticelectroweak precision constraints (with FCC-ee-t). Current constraints on S and T exclude m S /a (cid:46) . TeV at 95 % confidence.In the discussion above, we have neglected the impact of the neutral χ field that isstill present in the EFT given by eq. (3.3). This objection notwithstanding, provided that m χ (cid:29) E ∗ , where E ∗ is the energy scale for some process of interest, one can also integratethe χ out at one-loop, in which case the Wilson coefficient of the ( ∂ µ | H | ) operator willshift by a subdominant amount; here, the shift will be smaller by a factor of ∼ κ S / π than the leading contribution. Therefore turning on κ S as a non-zero weak coupling, andintegrating the χ out at the scale µ = m χ , will give a subdominant one-loop shift to σ Zh , andhigher-loop contributions to S and T . The ratio σ Zh : S : T should not change significantly,thus roughly preserving the relative strengths of the limits when χ is added to the theory.The foregoing limits on m S /a are therefore expected to be fairly accurate. They arise asan effect caused by the mediator, and are fairly insensitive to the presence of the portalcoupling per se .In summary, the precision electroweak and Higgsstrahlung constraints on this scenarioare very similar to those of the model in which the χ is simply absent, and this case hasalready received extensive attention in the literature. We devote the bulk of this paper to the singlet-doublet UV completion of the fermionicHiggs portal operator. The model consists of the Standard Model augmented by the samevector-like Dirac fermion SM-singlet χ as before, as well as a vector-like Dirac fermion SU (2) -doublet F = ( C, N ) transforming under the SM gauge group as ( , , +1 / . Thecoupling of these particles to the SM is taken to be L = L sm + i ¯ χ /∂χ − m χ ¯ χχ + i ¯ F /DF − M F ¯ F F − κ ¯ F Hχ − κ ¯ χH † F, (4.1)where D µ ≡ ∂ µ − igW µa t a − ig (cid:48) B µ Y , and t a = σ a . Without loss of generality we mayabsorb the phase of κ into the definition of χ and/or F , and so throughout we take κ tobe a non-negative real parameter. Without much modification (it would suffice, e.g., toreplace m χ ¯ χχ → m χ ¯ χ l χ r + h.c., with arg m χ (cid:54) = 0 ) it would also be possible to generate theCP-odd Higgs portal operator H † H ¯ χiγ χ ; we do not, however, consider this case further,and take all parameters to be real.In the parameter region where M F (cid:29) m χ , the heavy doublet F can be integrated out atthe scale µ = M F , and the leading correction to the SM Lagrangian is the fermionic Higgsportal operator. We will consider constraints on this UV completion both in the regimewhere M F (cid:29) m χ , constructing an EFT to analyze the low-energy effects, as well as in themore general mass parameter space. For general masses, this model does not necessarilyprovide the UV completion to the fermionic Higgs portal operator as it is defined here– 7 –nd our EFT is not valid, so we instead perform direct one-loop computations of relevantobservables.Before discussing the computations and results in detail, it is worth laying out ourgeneral expectations. The Yukawa-like coupling κ between the F, χ and H fields in eq. (4.1)is a hard breaking of the accidental global SU (2) V custodial symmetry [64] of the SM,and as such we expect fairly large corrections to the precision electroweak T parameter.Additionally, the mass-splitting (i.e., weak iso-spin breaking) in the F doublet which arisesfrom the mixing of the neutral fermions N and χ after electroweak symmetry breaking(EWSB), leads us to expect that there will additionally be contributions to the electroweak S parameter.Note that, although we do not consider such a case in this paper, custodial symmetrycould be restored—or broken in a controlled fashion—by augmenting the field content withan additional positively-charged vector-like fermion ψ ∼ ( , , +1) with the same Yukawa-like coupling to the H and F fields; the mass-splitting | m ψ − m χ | then controls the degree towhich the symmetry is broken (being restored in the degenerate-mass limit). There wouldhowever be additional experimental handles in this case as the new direct coupling of theHiggs doublet to electrically-charged fermions would lead to one-loop corrections to the h → γγ rate.We also expect that, in the model defined by eq. (4.1), the Higgs oblique correction willbe generated (by closed N - χ loops on the Higgs propagator, in the full-theory picture), sothat even if we were to tune away the large corrections to T by, e.g., the method indicatedin the previous paragraph, significant constraints would still remain on this model. To beclear, however, the Higgs oblique correction is by no means the only contribution to theshift in the Higgsstrahlung cross-section which is induced in this model; indeed, non-zero S and T parameters themselves will also generate a shift to σ Zh (see ref. [49]), and we takeall such shifts into account. It is instructive to begin our analysis by integrating out the new field content, assumingit is heavy, and considering the low-energy effects of the dimension-6 operators that aregenerated. In many popular bases of gauge-invariant dimension-6 operators which extendthe SM (see, e.g., refs. [65–68]), one can immediately read off the S and T parameters asthe Wilson coefficients of certain operators. In the so-called “HISZ” basis [67], S is simplyproportional to the Wilson coefficient of the operator H † ˆ B µν ˆ W µν H , and T is likewise pro-portional to the Wilson coefficient of the operator | H † D µ H | ; the observable corrections tothe momentum-dependent part of the Higgs two-point function would arise from the opera-tors | H † D µ H | and ( ∂ µ | H | ) in this basis, but the full σ Zh corrections require significantlymore work to obtain [49].In the following, we first integrate out the F doublet at the “high-scale” µ = M F ,performing the one-loop matching onto an effective theory with the Standard Model fieldcontent plus the singlet χ . We then repeat the procedure, integrating out the χ field at the“low-scale” µ = m χ and matching at one-loop onto a new EFT with the SM field contentonly. We assume the mass hierarchy M F (cid:29) m χ (cid:29) v ∼ m W , where M F (cid:29) m χ is imposed to– 8 – able 2 . Operators appearing in the effective theory below the scale M F (see text for sign andnormalisation conventions). For operators with a χ bilinear, we have performed tree-level matchingat the high-scale M F up to and including dimension-10 operators in order to be able to computecorrections to fairly high order in m χ /M F when later matching at µ = m χ , but omit these lengthyresults here as we used only selected terms from them as necessary (see text). Operators containingonly SM content appear in the one-loop matching both at the high scale µ = M F and at the low scale µ = m χ . The naming convention for the operators with only n Higgs doublets H and m (covariant)derivatives is O ( n,m ) ; other operators are named on an ad-hoc basis. We define ˆ W µν ≡ igW aµν t a and ˆ B µν ≡ ig (cid:48) B µν Y . Note that the operator O (2 , is non-standard in the literature and can beeliminated using eq. (4.2). Tree-level — operators containing a χ bilinearName Operator Name Operator O H † H ¯ χχ O A ¯ χγ µ χ i ( H † D µ H − h.c. ) O B H † H i ( ¯ χ /∂χ − h.c. ) O A H † H ( ¯ χ (cid:3) χ + h.c. ) O B | D µ H | ¯ χχ O C i (cid:0) ( D µ H ) † D ν H − h.c. (cid:1) ¯ χσ µν χ O D (cid:0) H † D µ H − h.c. (cid:1) ( ¯ χγ µ /∂χ − h.c. ) etc. One-loop — operators with only SM content O (2 , | D µ H | O (2 , | H | O (4 , | H | O (4 , ,A ( ∂ µ | H | ) O (4 , ,B | H † D µ H | O (4 , ,C | H | | D µ H | O (6 , | H | O (2 , H † D µ D ν D ν D µ H + h.c. O ( W W ) H † ˆ W µν ˆ W µν H O ( BB ) H † ˆ B µν ˆ B µν H O ( BW ) H † ˆ B µν ˆ W µν H O ( DW ) Tr (cid:104) [ D µ , ˆ W νρ ][ D µ , ˆ W νρ ] (cid:105) O ( DB ) − ( g (cid:48) ) ( ∂ µ B νρ )( ∂ µ B νρ ) O ( W ) ( D µ H ) † ˆ W µν ( D ν H ) O ( B ) ( D µ H ) † ˆ B µν ( D ν H ) O ( W W W ) Tr (cid:104) ˆ W µν ˆ W νσ ˆ W µσ (cid:105) guarantee that the leading correction upon integrating out the F is the CP-even fermionicHiggs portal operator; this is not a necessary assumption of the model itself.In performing the matching, we find it convenient to match onto the basis of dimension-6 operators with SM fields shown in table 2. This basis of operators is not one of the stan-dard ones and in particular certain operators (e.g., O (2 , ) are redundant in the sense thatthey are related to other operators by SM classical equations of motion (EOM). The redun-dant operators can be eliminated by field re-definitions whose effect is exactly equivalent tomaking replacements using the classical EOM (see, e.g., refs. [65, 69]); this is correct even atthe quantum level if we consistently neglect higher-dimension operator corrections. We notefor now that the operators in table 2 form a convenient and sufficient set to match onto; wewill return below to the treatment of the various redundancies in order to transform intothe operator bases most convenient for calculating the S and T parameters, and σ Zh .Upon integrating out F , we find the operators and Wilson coefficients shown in tables2 and 3. The only operators of dimension 4 or less appearing in table 2 – O (2 , , O (4 , ,– 9 – able 3 . Wilson coefficients of the operators appearing in table 2, evaluated at µ = M F , the“high scale.” Our sign convention is that the Lagrangian term containing any operator of dimensiongreater than 4 which is shown in table 2 appears multiplied by its corresponding Wilson coefficientand the appropriate inverse power of the EFT cutoff-scale M F ; however , for operators of dimension 4or less, the sign appearing in front of the operator follows the SM conventions and any multiplyingpower of M F is absorbed into the Wilson coefficient (see discussion in text). These expressionsassume a hierarchy of scales M F (cid:29) m χ , and are correct to tree-level for the c i and to one-loop forthe B ( j ) . We present results to fairly high order in m χ /M F , although not all of these terms arenumerically necessary to obtain our results. Tree-level — operators containing a χ bilinearCoefficient Value Coefficient Value c + κ c A + κ c B + κ c A − κ c B + κ c C − κ c D − κ etc. One-loop — operators with only SM contentCoefficient Value B (2 , − ( κ / π ) M F (cid:2) m χ /M F + m χ /M F + m χ /M F + m χ /M F + m χ /M F + m χ /M F + · · · (cid:3) B (2 , +( κ / π ) (cid:2) − m χ /M F − m χ /M F − m χ /M F − m χ /M F − m χ /M F − m χ /M F + · · · (cid:3) B (4 , − ( κ / π ) (cid:2) m χ /M F + 9 m χ /M F + 16 m χ /M F + 25 m χ /M F + · · · (cid:3) B (4 , ,A − ( κ / π ) (cid:2) m χ /M F − m χ /M F (cid:3) B (4 , ,B − ( κ / π ) (cid:2) − m χ /M F − m χ /M F + · · · (cid:3) B (4 , ,C − ( κ / π ) (cid:2) − m χ /M F − m χ /M F + · · · (cid:3) B (6 , − ( κ / π ) (cid:2) m χ /M F + 48 m χ /M F · · · (cid:3) B (2 , +( κ / π ) (cid:2) − m χ /M F − m χ /M F − m χ /M F − m χ /M F + · · · (cid:3) B ( W W ) − ( κ / π ) (cid:2) − m χ /M F + 3 m χ /M F + 10 m χ /M F + 68 m χ /M F + · · · (cid:3) B ( BB ) − ( κ / π ) (cid:2) − m χ /M F + 3 m χ /M F + 10 m χ /M F + 68 m χ /M F + · · · (cid:3) B ( BW ) − ( κ / π ) (cid:2) − m χ /M F + 3 m χ /M F + 10 m χ /M F + 68 m χ /M F + · · · (cid:3) B ( DW ) +(1 / π ) B ( DB ) +(1 / π ) B ( W ) +( κ / π ) (cid:2) − m χ /M F − m χ /M F + · · · (cid:3) B ( B ) +( κ / π ) (cid:2) − m χ /M F − m χ /M F + · · · (cid:3) B ( W W W ) Operator generated, but coefficient not computed.Irrelevant to the observables of interest. – 10 –nd O (2 , – already appear in the SM. Our convention for these operators is such that theWilson coefficients B (2 , , B (4 , , and B (2 , simply add positively to the coefficients alreadypresent for these operators in the SM: i.e., L ⊃ (1 + B (2 , ) O (2 , + ( µ + B (2 , ) O (2 , − ( λ + B (4 , ) O (4 , . On the other hand, all operators with dimension greater than 4 are taken toappear in the Lagrangian as L ⊃ c i M − d i F O i where c i is the Wilson coefficient and d i isthe dimension of the operator O i .We worked to tree-level for the Wilson coefficients for any operator containing a χ -field bilinear, and to one-loop in obtaining the Wilson coefficients for operators containingonly SM field content. The rationale is that whenever the χ later appears in any diagramcomputed in a consistent one-loop matching to obtain a Wilson coefficient of a dimension-6operator containing only SM content, it will appear as a closed χ loop, so only the tree-level χ -SM couplings are required.The tree-level matching was performed by inverting the classical equation of motionfor the heavy doublet to solve for F in terms of H and χ , and expanding in powers of D µ /M F (some lower-dimensional results were checked diagrammatically). The one-loopmatching was performed diagrammatically utilizing dimensional regularization and the MSrenormalisation scheme. Calculations were done in the broken electroweak phase, and onthe EFT-side of the one-loop matching computations we consistently included diagramswith one or more tree-level χ -SM coupling(s) and a single closed χ loop. As a check, someone-loop computations were performed both in the non-mass-diagonal basis of new neutralfermions, N and χ , and in the mass-diagonalised basis. In the latter case, expansion ofthe result in powers of the Higgs vev v allows the comparison of these two methods andthe extraction of the Wilson coefficients. Finally, we have checked that, to the order wework, the EFT and full theory one-loop computations reproduce the same IR-divergent logarithms ( ∼ log (cid:2) m χ /M F (cid:3) at µ = M F ) as is of course required.The EFT power-counting we employed is such that we kept only leading terms in v/M F ,but retained terms at higher order in m χ /M F , although this was not always numericallynecessary to obtain the results we present below. In order to compute these higher-orderterms in the mass-ratio m χ /M F in our one-loop matching, we have systematically carriedout the tree-level matching up to and including dimension-10 operators so as to capturethe operators with higher numbers of derivatives acting on the χ field since, in dimensionalregularization and MS, such higher-derivative operators contribute positive powers of m χ to the S -matrix element when appearing in conjunction with a closed χ -loop. This latterpoint is perhaps opaque, and is illuminated by a simple concrete example: the term in B ( W W ) suppressed by m χ /M F relative to the leading term can be found from, e.g., the v ( p /M F )( m χ /M F ) part of the ZZ two-point function at one-loop (along with other V V Mixing-angle effects are necessarily small in the range of EFT validity, being suppressed by at least κv/M F for M F (cid:29) m χ , κv (see eq. (A.7)). It is well-known (see, e.g., §3.3 (pp. 37–38) of ref. [70], andreferences therein) that physical effects which in the full theory would be ascribed to these small mixingangles are instead captured in the EFT description through the inclusion of higher-order operators, whoseWilson coefficients have their genesis (in part) in the UV-theory mixing angles. Readers unfamiliar withthis point may wish to consult Appendix B for further comments. In the sense that they diverge if m χ → . – 11 –wo-point functions so as to break the degeneracy between B ( W W ) , B ( BW ) , and B ( BB ) ).Simple power-counting then indicates that, owing to the M − F suppression, this requires inturn knowledge of the subset of the operators at dimension-10 in the tree-level matchingwhich contain two Z boson fields, two Higgs fields, one ¯ χ field, one χ field, two derivativesacting on the Z fields, and–to be dimensionally correct–one derivative acting on either ofthe χ or ¯ χ fields. When the χ loop is closed and the Higgs field replaced by its vev, suchoperators contribute to the ZZ two-point function in the EFT with a term proportional to ( v p /M F ) (cid:82) d d k k / ( k − m χ ) ∼ ( v p /M F ) m χ , as desired.Having obtained the mixed tree-level/one-loop matching conditions, one should byall rights run the theory down from the high-scale µ = M F to the “low-scale” µ = m χ ,integrate the χ out at this scale, and then run the theory to the scale required to considerany process of interest (i.e., the matching conditions supply only the initial conditionsfor the renormalisation group equation (RGE) running). However, we will choose to notrun the operators using the RGE. The errors one makes in ignoring the running between µ = M F and µ = m χ are proportional to log (cid:2) m χ /M F (cid:3) ; provided we do not take toolarge a hierarchy, this factor is not necessarily very large. Furthermore, for the electroweakprecision observables S and T , we expect the effect of the running is either higher-loop orpower-suppressed in m χ /M F and therefore a subdominant modification to the coefficientsgenerated at the high-scale. This is in contrast to, e.g., the case considered in ref. [50],where the operators responsible for S and T are not generated at the high-scale, and arisepurely from running: in that case, it is important to find the contributions from runningto the corresponding Wilson coefficients because S and T constraints are so strong. Ourneglect of the RGE in this model is also numerically justified, at least for the S and T parameters, as we will show in comparing EFT and full-theory one-loop results.We thus move on to the matching at the low-scale µ = m χ where we integrate the χ out, working consistently correct to one-loop in the matching. The operators O i for i ∈{ , A, B, A, etc.} are thereby removed from the Lagrangian and the Wilson coefficients B i for the dimension-6 operators with only SM field content receive new contributions inthe EFT valid below µ = m χ . We call these shifted Wilson coefficients in the low-scaleEFT C i ; they are shown in table 4 assuming no running or operator mixing between thescales µ = M F and µ = m χ .While the basis of operators shown in table 2 was convenient for the matching com-putation, it still contains the various redundancies we mentioned earlier. It also does nothappen to be the most convenient for the computation of the S and T parameters, and σ Zh . The S and T parameters are each simply related to the Wilson coefficient of a singleoperator when these redundancies are removed. In addition, ref. [49] contains a computa-tion of σ Zh in terms of a different basis of operators, and it useful to transform to this basisto enable us to perform a cross-check on our own independent computation of σ Zh .As O (2 , is non-standard in the literature, we eliminate it via the identity O (2 , = 2 | D H | − O ( BB ) − O ( BW ) − O ( W W ) − O ( B ) − O ( W ) , (4.2)plus irrelevant total-derivative terms. We are also free to make use of the SM EOM for H to trade out | D H | for corrections to the coefficients of the operators | H | , | H | , | H | – 12 –nd the SM Yukawa Lagrangian L Yukawasm , in addition to two new types of dimension-6operators— H † H × L Yukawasm and a number of four-fermion contact operators proportionalto two powers of the Yukawa couplings—as well as dimension-8 corrections, the latter ofwhich we consistently neglect. The corrections to the | H | , | H | and L Yukawasm terms areabsorbed into unobservable shifts of the SM parameters µ , λ and Γ i (the SM Yukawacoupling matrices), but the remaining additional terms here lead in principle to observableeffects.Our last step is to rescale the H field to obtain a canonically normalized covariantderivative term (i.e., we absorb the coefficient of | D µ H | , (1 + C (2 , ) , into the H field)and re-define the SM couplings [ µ , λ , and Γ i ] to absorb both the shifts proportional to C (2 , , as well as the shifts to the SM parameters arising from C (4 , and C (2 , . We notethat in the coefficients of all dimension-6 operators, the rescaling of the H field can simplybe ignored as the leading coefficients are already linear in the C i : any correction resultingfrom the H rescaling is therefore proportional to C (2 , × C i and thus formally counted as atwo-loop effect, since a single C i insertion is formally counted as a one-loop correction. Inother words, the coefficients C (2 , , C (4 , and C (2 , can simply be ignored in an analysiscorrect to one-loop (this assumes a degree of tuning in the bare value of µ such that westill obtain the correct measured Higgs mass). Having eliminated O (2 , using eq. (4.2), the S and T parameters can be read off from thenew coefficients of the operators O ( BW ) and O (4 , ,B , respectively. These operators modifythe gauge boson vacuum polarization amplitudes Π µνV V ( q ) = ig µν Π V V ( q ) + ... . Correctto linear order in the Wilson coefficients C i , we find that the new physics contributions to S and T are: S ≡ c W s W α e (cid:20) Π ZZ ( M Z ) − Π ZZ (0) M Z − c W − s W c W s W Π Zγ ( M Z ) M Z − Π γγ ( M Z ) M Z (cid:21) (4.3) ≈ c W s W α e (cid:20) Π (cid:48) ZZ (0) − c W − s W c W s W Π (cid:48) Zγ (0) − Π (cid:48) γγ (0) (cid:21) (4.4) = − c W s W α e Π (cid:48) W B (0) (4.5) ≈ − π v M F (cid:2) C ( BW ) − C (2 , (cid:3) , (4.6) Again, we do not run the operator coefficients between m χ and the Z -pole. The operators O ( DB ) , ( DW ) contribute only terms ∼ p to the gauge-boson two-point functions andhence do not contribute to the S , and T parameters (at least if S is defined per eq. (4.4)); instead, theseoperators give rise to the ˆ Y and ˆ W operators of ref. [71]. However, we find the corresponding limits arenot competitive with those from S and T and do not pursue this further. – 13 – able 4 . Wilson coefficients of the operators appearing in table 2, evaluated at µ = m χ , the “lowscale,” after the χ has been integrated out. These expressions are correct under the assumption ofno operator running or mixing between the scales µ = M F and µ = m χ . Our sign conventions areas detailed in table 3. These expressions assume a hierarchy of scales M F (cid:29) m χ . Coefficient Value C (2 , − ( κ / π ) (cid:2) M F + M F m χ + m χ (cid:3) ( this appears to be exact ) C (2 , +( κ / π ) (cid:2) − m χ /M F − m χ /M F − m χ /M F − m χ /M F − m χ /M F − m χ /M F + · · · (cid:3) C (4 , − ( κ / π ) (cid:2) m χ /M F + 8 m χ /M F + 12 m χ /M F + 16 m χ /M F + · · · (cid:3) C (4 , ,A − ( κ / π ) (cid:2) m χ /M F − m χ /M F + · · · (cid:3) C (4 , ,B − ( κ / π ) (cid:2) − m χ /M F − m χ /M F + · · · (cid:3) C (4 , ,C − ( κ / π ) (cid:2) − m χ /M F − m χ /M F + · · · (cid:3) C (6 , − ( κ / π ) (cid:2) m χ /M F + 51 m χ /M F + · · · (cid:3) C ( W W ) − ( κ / π ) (cid:2) − m χ /M F + 3 m χ /M F + 10 m χ /M F + 41 m χ /M F + · · · (cid:3) C ( BB ) − ( κ / π ) (cid:2) − m χ /M F + 3 m χ /M F + 10 m χ /M F + 41 m χ /M F + · · · (cid:3) C ( BW ) − ( κ / π ) (cid:2) − m χ /M F + 3 m χ /M F + 10 m χ /M F + 41 m χ /M F + · · · (cid:3) C (2 , +( κ / π ) (cid:2) − m χ /M F − m χ /M F − m χ /M F − m χ /M F + · · · (cid:3) C ( DB ) +(1 / π ) C ( DW ) +(1 / π ) C ( W ) +( κ / π ) (cid:2) − m χ /M F − m χ /M F + · · · (cid:3) C ( B ) +( κ / π ) (cid:2) − m χ /M F − m χ /M F + · · · (cid:3) C ( W W W ) Operator generated, but coefficient not computed.Irrelevant to the observables of interest. and T = ρ − α e (4.7) ≡ α e M W (cid:20) Π W + W − (0) − Π W W (0) (cid:21) (4.8) = 1 α e M W (cid:20) Π W + W − (0) − c W Π ZZ (0) (cid:21) (4.9) ≈ − α e v M F C (4 , ,B , (4.10)where in arriving at Eqs. (4.4) and (4.9), we have made use of Π Zγ (0) = Π γγ (0) = 0 . The U parameter is identically zero at dimension 6; the first non-zero contribution occurs atdimension 8. – 14 –ubstituting the explicit forms of the Wilson coefficients, we have S ≈ κ π v M F (cid:34) − m χ M F − m χ M F − m χ M F − m χ M F + · · · (cid:35) (4.11) T ≈ α κ α e κ π v M F (cid:34) − m χ M F − m χ M F + · · · (cid:35) where α κ ≡ κ π (4.12) U = 0 + (dim-8) (4.13)The parametric enhancement (or suppression, depending on the relative sizes of α κ and α e ) of T compared to S can be understood as follows. For M F (cid:29) m χ , Π W + W − (0) − c W Π ZZ (0) ∼ g ( κ v )( κ v /M F ) , while Π (cid:48) ZZ (0) ∼ ( g + ( g (cid:48) ) )( κ v /M F ) . This is deter-mined by the mass dimension of the quantity concerned and the fact that, in our model,a diagrammatic new-physics contribution proportional to the Higgs vev v must always ac-tually be proportional to κv owing to the structure of the H - F - χ coupling. Therefore, T ∼ ( g v /M W )( κ /α e )( κ v /M F ) ∼ ( α κ /α e )( κ v /M F ) while S ∼ ( c W s W /α e )Π (cid:48) ZZ (0) ∼ κ v /M F since α e /c W s W ∼ g + ( g (cid:48) ) . The parametric dependence on κ can also be readoff from the form of the dimension-6 operators giving rise to S and T . Upon EWSB, the S parameter arises from O ( BW ) = H † ˆ B µν ˆ W µν H . As a result, diagrammatic contributions to S come with two powers of the Higgs vev v and hence two powers of κv , since the two deriva-tives in the field strengths are removed in the definition of S as a momentum derivative ofthe two-point function. The T parameter arises, upon EWSB, from O (4 , ,B = | H † D µ H | ;the diagrammatic contributions to T thus come with four powers of the Higgs vev v andhence four powers of κv (two of the Higgs vev factors are later cancelled by the M − W in thedefinition of T ). Therefore, T ∼ κ and S ∼ κ .Note that an indirect effect of the dimension-6 operators may be to imply shifts to theLagrangian coupling constants (e.g., the gauge couplings) away from their SM referencevalues, as we will discuss when we turn to the Higgsstrahlung process in the followingsection. However, since the leading results for S and T are already proportional to C i andhence formally one-loop, any coupling shift can be dropped as a higher-order correction( S = T = 0 in the SM at tree-level). Moreover, the coupling constant shifts will imply ashift in the numerical values of the one-loop computations of the SM contributions to the S and T parameters, but these are again formally at least two-loop effects. Since we workto one-loop accuracy, we neglect these contributions and the value of α e appearing in T istaken to be its SM value (we take it at the Z pole).In figure 2, we show the 95 % confidence exclusion regions in the mass parameter spacefrom measurements of the electroweak precision variables ( S, T ) , for fixed representativevalues of κ . These results are computed in the EFT, and are shown for both current limits(LEP+SLD) and for the proposed CEPC, ILC and FCC-ee colliders with the sensitivitiesgiven in table 1.In the region of parameter space under consideration in figure 2, the limits are drivenalmost exclusively by the size of the T parameter, owing to its parametric enhancement ∼ α κ /α e over the S parameter. In the regime where the EFT is valid, m χ (cid:46) M F , the– 15 – m χ [GeV] M F [ G e V ] . . . . . . . . . CurrentCEPC “Baseline”CEPC “Improved Γ Z , sin θ ”CEPC “Improved Γ Z , sin θ, m t ” m χ [GeV] M F [ G e V ] . . . . . . . . . CurrentILCFCC-ee–ZFCC-ee–t Figure 2 . Two-parameter 95% confidence exclusion regions ( − L / L ] (cid:38) . ) from measurementof the precision electroweak variables ( S, T ) , using results computed in the EFT. We present these asboundaries in the allowed mass parameter space for fixed representative values of κ , as annotated on eachline; the unshaded region to the lower-left of these lines is excluded for the given value of κ . To compare EFTresults with the full-theory calculation, see figure 4. The light shaded region, M F / (cid:46) m χ (cid:46) M F / , denotesthe region where the EFT begins to break down: the error in the EFT result for T compared to the v /M F piece of the full result (i.e., the “dimension-6 part of” the full result) is ∼ at m χ ∼ M F / , reaches ∼ at m χ ∼ M F / , and becomes > before m χ ∼ M F / . In the dark-shaded region, m χ (cid:38) M F / ,the results have consequently been masked as they are invalid. The various line styles correspond to thecurrent constraints and various projected constraints on ( S, T ) for the proposed CEPC collider (left plot)and the proposed ILC and FCC-ee colliders (right plot); see table 1. exclusion reach is also largely insensitive to the value of m χ , as one might expect. Alreadywith current (LEP+SLD) constraints on S, T , masses M F below about 675 GeV can beruled out for κ ∼ ; due to the κ dependence of T , this lower limit increases to around2.9 TeV for a coupling κ ∼ , but there is essentially no limit in the regime where the EFTis valid if κ ∼ . .With even the “baseline” proposal for the CEPC, these lower limits increase by a factorof approximately 1.75 owing to the factor ∼ . times stronger limits on S and T in thisscenario as compared to the current bounds [43]. For the CEPC scenario with improvedmeasurements of Γ Z and sin θ only, the lower limits on M F increase by a factor of ∼ compared to the present limits, rising to ∼ . TeV (6 TeV) for κ ∼ . (2.0), in the limitwhere m χ (cid:28) M F . This results from the tightening in the limits on T (and S ) by factorof ∼ compared to the present constraints [43]. For the best-case scenario for the CEPC,with improved measurements of Γ Z , sin θ , and m t , the lower limits on M F increase by afactor of ∼ . compared to the present limits (i.e., about 30% over the previous scenario),rising to ∼ . TeV (7.7 TeV) for κ ∼ . (2.0), in the limit where m χ (cid:28) M F . This resultsfrom the tightening in the limit on T by factor of ∼ , and the limit in S by a factor of ∼ compared to the present limits (i.e., the m t measurement roughly halves again the– 16 –easurement uncertainty on T compared to the previous scenario, and also improves the S constraint) [43].The projected sensitivities of the ILC and FCC-ee show improvements in the lowerlimits on M F which are broadly similar to the various CEPC scenarios. The ILC projectionand CECP “baseline” projection are very similar, as are the FCC-ee-Z limits and the CEPClimits with improved measurements of Γ Z and sin θ only. Finally, the FCC-ee-t lower-limitson M F (for κ ∼ . and κ ∼ . they are . TeV and . TeV, respectively) are slightlystronger than the best-case CEPC scenario, but are nevertheless still broadly similar.
The computation of shifts to the Higgsstrahlung cross-section σ Zh is more involved thanthat required to obtain the S and T parameters. There are a large number of operatorswhich directly contribute to a shift in the cross-section, both by way of contributions towavefunction renormalization of the h, Z fields in the hZ µ Z µ coupling, and the introductionof additional diagrams. Furthermore, σ Zh is nonzero at tree-level in the SM and so theone-loop new-physics shifts to the relationships between SM input parameters and SMLagrangian parameter values must be accounted for here to maintain a result consistentlycorrect to one-loop order: the standard set of SM input parameters for high-precision workare ( G F , m Z , α e ) and the dimension-6 operators impact the processes (e.g., muon decay at q = 0 ) used to relate the numerical values of these parameters to the values of Lagrangianparameters, leading to shifts in the Lagrangian coupling constants away from their SMreference values [72].The requisite computation of σ Zh has actually recently appeared in the literature inref. [49]. We have nevertheless independently repeated the computation of σ Zh in the EFTas a cross-check; we find complete agreement with their results. In order for us to makereference to the results of ref. [49], we transform to the same basis of operators used there.In addition to the elimination of O (2 , described above, four further sets of manipulationson our basis of operators are required: (a) we re-write the operator O (4 , ,B = | H † D µ H | = −
14 ( H † ↔ D µ H ) + 14 ( ∂ µ | H | ) ; (4.14)(b) we rewrite the operators O ( DB ) and O ( DW ) , up to total derivative terms, as O ( DB ) = − ( g (cid:48) ) ∂ µ B νρ )( ∂ µ B νρ ) = − ( g (cid:48) ) ( ∂ µ B µν ) , and (4.15) O ( DW ) = Tr (cid:104) [ D µ , ˆ W νρ ][ D µ , ˆ W νρ ] (cid:105) = 2 Tr (cid:104) [ D µ , ˆ W µρ ][ D ν , ˆ W νρ ] (cid:105) + 4 Tr (cid:104) ˆ W νρ ˆ W µν ˆ W ρµ (cid:105) , (4.16)and utilise the SM EOM for ˆ B µν and ˆ W µν to rewrite the first term on the RHS of each ofthese relationships; and (c) we make two manipulations to remove O (4 , ,C : H † H | D µ H | = −
12 ( ∂ µ | H | ) − H † H (cid:104) H † ( D H ) + ( D H ) † H (cid:105) , (4.17)– 17 –ollowed by the substitution of the SM EOM to rewrite D H . Finally, (d) we rewrite theoperators O ( B ) and O ( W ) , up to total-derivative terms, as O ( B ) = 12 O ( BB ) + 12 O ( BW ) − g (cid:48) (cid:20) H † i ↔ D ν H (cid:21) ( ∂ µ B µν ) , and (4.18) O ( W ) = 12 O ( W W ) + 12 O ( BW ) − g (cid:20) H † iσ a ↔ D ν H (cid:21) ( D µ W a, µν ) , (4.19)and then utilise the SM EOM for ˆ B µν and ˆ W µν to re-write the last term on the RHS ineach line, and the SM EOM for H to re-write some factors of D H which appear upondoing so. This is followed by a final re-definition of the SM Higgs-Lagrangian parameters,to absorb some unobservable shifts to the SM parameters which occur during this process.When the dust settles, we are left with L = L sm + (cid:88) j C j M F O (6) j , (4.20)where the dimension-6 operators O (6) j and Wilson coefficients C j , correct to linear order inthe original C ( ··· ) , are listed in table 5, and L sm takes the same form as the SM Lagrangian,except that all the parameters are now understood to be defined so as to absorb any ofthe unobservable shifts which occurred as a result of the manipulations just described. Asfar as the e + e − → Zh process is concerned, all of the operators in table 5 listed above thehorizontal line contribute, either through a shift in the SM input parameters or throughthe addition of a new term in the amplitude itself. The results of ref. [49] can then be used to read off the value of σ Zh ; in table 6, we supplyan explicit dictionary to transform our Wilson coefficients into the notational conventionsof ref. [49]. In all our results, we utilize the ( G F , M Z , α e ) input-parameter set.Apart from the independent re-computation of σ Zh beginning from the basis of opera-tors and Wilson coefficients in table 5 to cross-check against the results of ref. [49], we alsoverified the correctness of the fairly complicated manipulations which were used to removethe operator O ( DW ) (see eq. (4.16) and the text following) in arriving at eq. (4.20). To doso, we augmented the SM Lagrangian with O ( DW ) alone and considered its effects on σ Zh directly, rather than eliminating this operator by EOM.Furthermore, since the ILC measurement projections assume a polarized electron beamwith ( P e − , P e + ) = ( − . , +0 . [44], in the course of our independent computation, wealso computed the polarised-beam cross-section. In the notation of ref. [49], the polarized These unobservable shifts can consistently be ignored; they are not the same as the physically relevantshifts which need to be accounted for in relating the input parameters to the values of the Lagrangianparameters. Note that C ( WW ) = C ( BB ) = C ( BW ) and C ( W ) = C ( B ) in our model, so the Wilson coefficients of theoperators H † ˆ W µν ˆ W µν H and H † ˆ B µν ˆ B µν H remain equal, and equal to one-half of the Wilson coefficientof the operator H † ˆ B µν ˆ W µν H . This ultimately follows from the fact that there are no new Higgs-charged-charged vertices in the full theory picture, and guarantees that there is no leading-order (one-loop) correctionto the h → γγ decay rate. This would not be true if we had introduced a charged partner to the χ tomaintain custodial symmetry, as we discussed earlier. – 18 – able 5 . Operators O (6) j and corresponding Wilson coefficients C j appearing in the effectiveLagrangian which we utilise to compute σ Zh [see eq. (4.20)]. Only the operators listed above thehorizontal line contribute a shift at leading order to e + e − → Zh , either through a shift in therelationships between the SM input parameters and the Lagrangian parameters, or through theaddition of a new term in the amplitude itself. λ, µ and the Yukawa matrices Γ i are understood totake their unobservably shifted values. We define ˆ W µν ≡ igW aµν t a , ˆ B µν ≡ ig (cid:48) B µν Y , ↔ D µ ≡ D µ − ← D µ and σ a ↔ D µ ≡ σ a D µ − ← D µ σ a . Superscript p, q are generation indices; i, j are SU (2) -fundamentalindices; and a is an SU (2) -adjoint index—all are summed over if repeated. Operator O (6) j Wilson coefficient C j ( ∂ µ | H | ) − C (4 , ,C + C (4 , ,B + C (4 , ,A − g C ( W ) + g C ( DW ) + g C (2 , ( H † ↔ D µ H ) − C (4 , ,B − ( g (cid:48) ) C ( B ) + ( g (cid:48) ) C ( DB ) + ( g (cid:48) ) C (2 , H † ˆ W µν ˆ W µν H C ( W W ) + C ( W ) − C (2 , H † ˆ B µν ˆ B µν H C ( BB ) + C ( B ) − C (2 , H † ˆ B µν ˆ W µν H C ( BW ) + (cid:0) C ( W ) + C ( B ) (cid:1) − C (2 , ( H † i ↔ D µ H ) (cid:16)(cid:80) f L ,f R Y f ¯ f γ µ f (cid:17) − ( g (cid:48) ) C ( DB ) + ( g (cid:48) ) (cid:0) C ( B ) − C (2 , (cid:1) ( H † σ a i ↔ D µ H ) (cid:0) ¯ L pL γ µ σ a L pL (cid:1) − g C ( DW ) + g (cid:0) C ( W ) − C (2 , (cid:1)(cid:0) ¯ L pL γ µ σ a L pL (cid:1) − g C ( DW ) | H | + C (6 , + λ (cid:16) C (2 , + 2 C (4 , ,C − g C ( DW ) + g C ( W ) − g C (2 , (cid:17) Tr (cid:104) ˆ W µν ˆ W νσ ˆ W µσ (cid:105) C ( W W W ) + 4 C ( DW ) ( H † H ) (cid:34) (cid:0) ¯ L j Γ e e (cid:1) H j + (cid:0) ¯ Q j Γ d d (cid:1) H j + (cid:0) ¯ Q j Γ u u (cid:1) (cid:15) jk (cid:0) H † (cid:1) k (cid:35) + h.c. C (4 , ,C − g C ( DW ) + g C ( W ) + (cid:0) λ − g (cid:1) C (2 , ( H † σ a i ↔ D µ H ) (cid:0) ¯ Q pL γ µ σ a Q pL (cid:1) − g C ( DW ) + g (cid:0) C ( W ) − C (2 , (cid:1)(cid:16)(cid:80) f L ,f R Y f ¯ f γ µ f (cid:17) − ( g (cid:48) ) C ( DB ) (cid:0) ¯ Q pL γ µ σ a Q pL (cid:1) − g C ( DW ) (cid:0) ¯ L pL γ µ σ a L pL (cid:1) (cid:0) ¯ Q qL γ µ σ a Q qL (cid:1) − g C ( DW ) ( ¯ L Γ e e )(¯ e Γ e † L ) + ( ¯ Q Γ d d )( ¯ d Γ d † Q )+( ¯ Q Γ u u )(¯ u Γ u † Q )+(( ¯ L Γ e e )( ¯ d Γ d † Q ) + h.c. ) − (( ¯ L j Γ e e ) (cid:15) jk ( ¯ Q k Γ u u ) + h.c. ) − (( ¯ Q j Γ d d ) (cid:15) jk ( ¯ Q k Γ u u ) + h.c. ) C (2 , cross-section is obtained from the unpolarized cross-section by making the following simplereplacements in the quantities defined either in their eq. (2.3) or in their table 2: multiplyevery appearance of g L in their F sm and F , , as well as every appearance of g L in their– 19 – able 6 . A dictionary to convert our Wilson cofficient results from table 4 to the Wilson coefficientsdefined in ref. [49]. Wilson Coefficient in ref. [49] Value per our table 4 c W W − (cid:0) C ( W W ) + C ( W ) − C (2 , (cid:1) c BB − (cid:0) C ( BB ) + C ( B ) − C (2 , (cid:1) c W B − (cid:0) C ( BW ) + C ( W ) + C ( B ) − C (2 , (cid:1) c H − C (4 , ,C + C (4 , ,B + C (4 , ,A + g C ( DW ) − g (cid:0) C ( W ) − C (2 , (cid:1) c T − C (4 , ,B + ( g (cid:48) ) C ( DB ) − ( g (cid:48) ) (cid:0) C ( B ) − C (2 , (cid:1) c (3) lL − g C ( DW ) + g (cid:0) C ( W ) − C (2 , (cid:1) c (3) lLL − g C ( DW ) c lL ( g (cid:48) ) C ( DB ) − ( g (cid:48) ) (cid:0) C ( B ) − C (2 , (cid:1) c eR ( g (cid:48) ) C ( DB ) − ( g (cid:48) ) (cid:0) C ( B ) − C (2 , (cid:1) F , , by (1 − P e − )(1 + P e + ) . Similarly, multiply every appearance of g R in their F sm and F , , as well as every appearance of g R in their F , , by (1 + P e − )(1 − P e + ) . No changes areneeded to the factors appearing explicitly in their eq. (3.9).In figure 3 we show the regions in the mass parameter space ( m χ , M F ) which, forthe given fixed value of κ , would yield a value of σ Zh in conflict with the projected 95%confidence limits as given in table 1. We can see that the best sensitivity comes from theCEPC and FCC-ee. At the CEPC, the lower bounds on M F range from 590 GeV to 7 TeVfor κ ∼ . − . in the limit where m χ (cid:28) M F . These lower limits rise to approximately660 GeV to 10 TeV for m χ ∼ M F / , which is roughly where we begin questioning thevalidity of the EFT. (The limits on M F in the region where the EFT results are validare more sensitive to the value of m χ than the electroweak precision limits.) The FCC-eeconstraints are again a little stronger than CEPC constraints. Lower limits on M F in thelimit where m χ (cid:28) M F range from approximately 630 GeV to 7.8 TeV for κ ∼ . − . .The constraints obtained at the ILC are somewhat weaker in all three of the consideredscenarios than either the CEPC or FCC-ee constraints. Note that care was taken here incombining limits where runs are at different energies as (∆ σ Zh ) /σ sm Zh is energy-dependent(see comment in caption on table 1). The most optimistic ILC scenario with several ab − ofdata yields lower limits on M F for m χ (cid:28) M F that are approximately 490 GeV and 5.9 TeV,respectively, for κ ∼ . and 4.0.In none of these cases are the limits from the precision Higgsstrahlung measurementcompetitive with the electroweak precision programs at these future colliders in imposingconstraints on this specific model; nevertheless, these results do demonstrate that the σ Zh measurement would provide a strong complimentary constraint on closely allied modelswhere the T parameter is dialed away, as we discussed previously. κ = 4 . is a fairly large coupling; we display these results only with the explicit point of indicating thata very large coupling is required to probe this region of parameter space. – 20 – m χ [GeV] M F [ G e V ] . . . . CEPC m χ [GeV] M F [ G e V ] . . . . . . . . . . . . ILC 250/fb @ 250GeVILC + 500/fb @ 500GeVILC 1.15/ab @ 250GeV + 1.6/ab @ 500GeVFCC-ee 10/ab @ 240GeV (4IP) Figure 3 . One-parameter 95% confidence exclusion regions ( − L / L ] (cid:38) . ) from precisionmeasurements of σ Zh , using results computed in the EFT. We present these as boundaries in theallowed mass parameter space for fixed representative values of κ , as annotated on each line; the unshaded region to the lower-left of these lines is excluded for the given value of κ . Absent the fullloop computation, it is not possible to quote an error on this EFT-based result, but based on thecomparison of the EFT and full-theory computations for the EWPO results, it is probable that theEFT results here are questionable in the light shaded region, M F / (cid:46) m χ (cid:46) M F / , and are almostcertainty invalid in the dark-shaded region, m χ (cid:38) M F / , where the results have consequently beenmasked. See table 1 for the assumed sensitivities of the experiments. Although a large number of operators contribute to σ Zh , a partial and heuristic un-derstanding of the generic difference in the strength of these limits can be obtained byexamining a subset of the operators relevant for the generation of T and ∆ σ Zh . Sup-pose L = L sm + a ( H † D µ H − h.c. ) + b ( ∂ µ | H | ) . It can then be easily shown that T = ( av ) / ( α e Λ ) ∼ a ( v / Λ ) , and a little more work shows that ∆ σ Zh /σ Zh ≈− ( v / Λ )( b + 0 . a ) ; note that the two operators contribute almost equally to the Hig-gstrahlung cross-section. The anticipated one-parameter 95% confidence measurement un-certainties on T (restricted to S = U = 0 ) are . × − , . × − , and . × − forthe CEPC baseline, “Improved Γ Z , sin θ ”, and “Improved Γ Z , sin θ, m t ” scenarios, respec-tively. The resulting 95% confidence lower bounds on Λ / (cid:112) | a | are approximately 20 TeV,23 TeV and 29 TeV, respectively. On the other hand, the 95% confidence measurementuncertainty on the Higgsstrahlung cross-section at CEPC is projected to be ∼ % (corre-sponding to the 68% confidence projection ∆ σ Zh /σ Zh = 0 . in table 1), which yields thelimit Λ / (cid:112) | b + 0 . a | (cid:38) . TeV. It is clear that the latter bounds are significantly weakerthan the EWPO constraints for roughly equally-sized Wilson coefficients, which is the sce- The ‘0.83’ here is the numerical value of a fairly complicated function of the gauge couplings, assuming g = 0 . , g (cid:48) = 0 . . These are obtained from the values in table 1 as σ
95% CL T, = 1 . σ T (cid:112) − ρ ST . – 21 – m χ [GeV] M F [ G e V ] . . . . . . . . . . . . CurrentCEPC “Baseline”CEPC “Improved Γ Z , sin θ ”CEPC “Improved Γ Z , sin θ, m t ” m χ [GeV] M F [ G e V ] . . . . . . . . . . . . CurrentILCFCC-ee–ZFCC-ee–t
Figure 4 . As for figure 2, two-parameter 95% confidence exclusion regions ( − L / L ] (cid:38) . )from measurement of the S and T parameters, except these results are computed using the full one-loop results described in section 4.2 and hence are valid for the full range of masses shown. Notethat m χ and M F are the input mass-parameters, and are not the physical masses of the neutralfermions; see Appendix A. nario one would expect when these operators are generated at the same loop order (providedof course that there is no symmetry—e.g., custodial—which prevents this). Parametrically,it is clear that the relative strength of the limits can be traced directly to the enhancementof T by a factor of /α e . By design, the EFT we have constructed is valid only in the region of parameter spacewhere M F (cid:29) m χ , and it is in this region of parameter space where this model supplies aUV completion of the fermionic Higgs portal operator H † H ¯ χχ . One could of course alsoconstruct other EFTs valid in the region m χ , M F (cid:29) v ∼ m W , but for similar particlemasses, m χ ∼ M F , or for the opposite mass hierarchy, m χ (cid:29) M F . Given the amountof work required to obtain the results already presented, it appears to be more efficientto simply perform the full one-loop computations of S, T ; this computation is detailed inAppendix A and was performed in the mass-eigenstate basis of neutral fermions, accountingfor mixing angle effects. The full loop computation of σ Zh is beyond the scope of this paper,as we have already established from the EFT results that the Higgsstrahlung bounds areexpected to be somewhat weaker than those from EWPO.The exclusion regions due to precision electroweak constraints arising from the fullloop computation of S, T are shown in figure 4. In the limit where m χ (cid:28) M F it is clearthat the results agree well with the EFT computation (c.f., figure 2), the small differencesbeing ascribable to our neglect of dimension-8-and-higher operators, as well as the runningbetween scales, in the EFT. Note that if the full result for S, T is expanded out to O ( v /M F ) – 22 –o find the contribution to the result ascribable to dimension-6 operators, we find all thesame analytic (in the masses) pieces of the result as in the EFT computation, but naturallyalso find non-analytic factors ∼ log[ m χ /M F ] which are absent in our EFT result as weneglected the running of the Wilson coefficients between scales µ = M F and µ = m χ ;however, for the contributions to S and T , these are suppressed by at least the third powerof m χ /M F and are thus negligibly small whenever the EFT which we constructed is valid.As we have already discussed the results in the region m χ (cid:28) M F in connection withthe EFT results, we focus here on the parameter space not covered by the EFT. Of coursein this region, the theory does not provide a UV completion of the fermionic Higgs portaloperator H † H ¯ χχ , although in the limit m χ (cid:29) M F , it would provide a completion of thefermionic-doublet Higgs portal operator H † H ¯ F F .The most obvious point is that a significantly larger region of the m χ parameter spaceat small M F is ruled out than vice versa . For example, the fully improved CEPC resultsindicate that if M F = 100 GeV, then m χ < TeV can be constrained for κ ∼ . ,compared to M F < . TeV being constrained if m χ = 100 GeV. This pattern is genericfor all of the experiments; it traces its origin to the fact that for m χ (cid:28) M F , S and T are both positive and, roughly speaking, | S | ∼ . | T | near the exclusion limit, whereasfor M F (cid:28) m χ , T is positive and S is negative, with | S | ∼ | T | near the exclusion limit(indeed, | S | > | T | is possible here). The larger deviation from ( S, T ) = (0 , in this regionleads to the stronger limits. There is also a stronger dependence of the boundary of theexclusion region on M F at large m χ than vice versa . Both of these behaviours are due tothe fact that, parametrically, S ∼ ∆ M/M F where ∆ M is the mass-splitting in the doublet: ∆ M ≡ M neutral F − M charged F = v ( κ / v/ ( M F − m χ )] . This parametric dependence arisesbecause S is zero in the weak-isospin-symmetric limit of the theory, and because it is always F fermions which couple directly to W and Z . This implies that the leading contributionsare S ∼ κ v /M F for m χ (cid:28) M F , but S ∼ − κ v / ( M F m χ ) for M F (cid:28) m χ ; clearly then, atlight M F and heavy m χ , | S | will be larger than if the mass hierarchy were reversed.The slightly different shapes of the ILC and FCC-ee exclusion regions when m χ (cid:38) M F ,as seen in the right plot of figure 4, arise from the different alignments of the 68% coveragelikelihood contours in the ( S, T ) plane [43] for the ILC compared to those for FCC-ee, whichin turn is due to the different projected sensitivites to m W , sin θ l eff and Γ Z at each collider(see also ref. [73]). With reference to figure 1 in ref. [43], it is clear that if S, T > and | S | (cid:28) | T | , as is the case for m χ (cid:28) M F , the limits from FCC-ee-Z (TLEP-Z) are moreconstraining than those for the ILC; whereas if T > , and S ∼ − T , as is the case for for M F (cid:28) m χ , the ILC and FCC-ee-Z limits will be about equally constraining, with the ILCactually marginally better, as is visible in the right panel of our figure 4 for M F (cid:46) GeV.
In this paper we examined two possible models which UV-complete the CP-even fermionicHiggs portal operator H † H ¯ χχ , and investigated how future precision electroweak and pre-cision Higgsstrahlung measurements at the proposed ILC, FCC-ee and CEPC high-energy e + e − colliders could constrain these models. In the first model, the “scalar completion,”– 23 – scalar singlet acts as a mediator between vector-like Dirac fermion χ and the SM Higgsfield H , with a Yukawa coupling κ S S ¯ χχ to the χ and a coupling a m S S | H | to the Higgsdoublet. In the second model, the “fermionic completion,” a vector-like SU (2) doublet ofDirac fermions couples to the χ and H field with a Yukawa-like interaction κ ¯ F Hχ + h.c..For the scalar completion, we estimated that the effect of the new χ particle willbe subdominant (i.e., loop-suppressed) for the computation of the electroweak precisionvariables S and T and the Higgs couplings to the SM particles, compared to the dominanteffect of the singlet mediator itself. The limits which can be placed on the SM augmentedwith such a singlet scalar mediator have already been extensively studied in the literature(e.g., ref. [50]): a precision Higgsstrahlung cross-section measurement (∆ σ Zh ) /σ Zh ∼ . can place a 95% confidence upper limit on m S /a around 2.5 TeV, while the precisionelectroweak limits are only marginally weaker (despite S and T only being loop-induced byrunning).Our main focus was on the fermionic completion. We constructed an EFT valid in thelimit v (cid:46) m χ (cid:28) M F and examined the 95% exclusion reach on the mass parameter space,for a variety of projected sensitivities for the precision electroweak and Higgsstrahlungmeasurements. Provided the coupling constant κ is O (1) , we found that the precisionelectroweak limits are very powerful and primarily driven by T ∼ ( α κ /α e ) S (cid:29) S , owingto the violation of custodial symmetry. Already with current limits, the 95% confidenceexclusion reach on M F for m χ (cid:28) M F , is up to M F ∼ GeV (2.9 TeV) for κ = 1 . . .For the most optimistic projections we consider for the various possible configurations ofthe ILC, CEPC and FCC-ee colliders, these lower limits rise to 1.2 TeV (5.3 TeV), 1.8 TeV(7.7 TeV), and 2.0 TeV (8.5 TeV), respectively (see figure 2).For this model, the precision electroweak limits are generically more powerful than theprecision Higgsstrahlung cross-section measurement due to the parametric enhancement of T over (∆ σ Zh ) /σ Zh by ∼ /α e . Nevertheless, precise measurement of the latter also yieldsgood complementary exclusion reach, which is useful because closely-allied UV completionsin which the T parameter can be made small will therefore still be fairly strongly constrainedby this measurement (other probes, such as h → γγ , may also become important). The mostoptimistic 95% confidence exclusion lower limits from Higgsstrahlung on M F , in the limit m χ (cid:28) M F , for the various possible configurations of the ILC, CEPC and FCC-ee colliders,are 490 GeV (1.5 TeV), 590 GeV (1.9 TeV), and 620 GeV (2.1 TeV), for κ = 1 . . ,respectively (see figure 3).We also considered the full one-loop computation of the electroweak precision observ-ables in the more general mass parameter space where this model does not necessarilyprovide a UV completion for the CP-even fermionic Higgs portal as we have defined it. Theone-loop computation was found to agree well with the EFT computation for m χ (cid:28) M F .The model is significantly more constrained for the ‘opposite’ mass hierarchy m χ (cid:29) M F owing to the fact that S and T are comparably large in this region; for example, we find95% confidence lower limits on m χ for M F = 100 GeV are typically a factor of 3–5 timeshigher than the corresponding lower limits on M F for m χ = 100 GeV (see figure 4).Additionally for the fermionic completion, the appearance in the EFT analysis of oper-ators modifying the coupling of the Higgs to SM fermions (see table 5) raises the prospect– 24 –f a modification to the rate for h → b ¯ b which will be measured to fairly high accuracy atlepton colliders; we defer investigation of this point to future work.Overall, we see that the sensitivities possible at future e + e − machines through mea-surements of precision electroweak observables and the Higgsstrahlung cross-section willallow significant improvements in the exclusion reach for the CP-even fermionic Higgs por-tal over current limits, pushing into the (multi-)TeV range of particles masses, well beyondthe direct reach of the LHC. Acknowledgments
We would like to thank Nathaniel Craig for useful correspondence. This work was supportedin part by the Kavli Institute for Cosmological Physics at the University of Chicago throughgrant NSF PHY-1125897 and an endowment from the Kavli Foundation and its founder FredKavli. L.-T.W. is supported by the DOE Early Career Award under grant de-sc0003930.
A One-loop computation of S and T The one-loop computation of S and T was performed in dimensional regularization in theMS scheme. The IJ self-energy (where I and J are SM vector bosons) arises from diagramssuch as those in figure 5. For particles A and B with masses M A and M B , respectively,running in the loop, we find a contribution Π IJ ; AB ( p ) = g IAB · g JAB π A (cid:0) M A (cid:1) + A (cid:0) M B (cid:1) − ( M A + M B ) + 13 p − p B (cid:0) p , M A , M B (cid:1) + 12 B (cid:0) p , M A , M B (cid:1) (cid:0) ( M A − M B ) − M A M B (cid:1) + 12 B (cid:0) p , M A , M B (cid:1) − B (cid:0) , M A , M B (cid:1) p (cid:0) M A − M B (cid:1) + counter-terms (A.1)where A and B are the usual Passarino-Veltman scalar integrals [74]. Expanding Π( p ) in powers of the gauge-boson momentum as Π( p ) = Π(0) + p Π (cid:48) (0) + · · · , we find, at the Figure 5 . A one-loop contribution to the IJ self-energy function arising from particles A and B running in the loop. ABkk + ppI α J β α β – 25 –S scale µ , Π IJ ; AB (0) = g IAB · g JAB π × M A − M A M B + M B + 13 (cid:0) M A − M B (cid:1) − M A M B log (cid:0) M B /M A (cid:1) − M A (cid:0) M A M B − M A + M B (cid:1) log (cid:0) µ /M A (cid:1) +2 M B (cid:0) M A M B − M B + M A (cid:1) log (cid:0) µ /M B (cid:1) (A.2) −→ as M A → M B , (A.3)and Π (cid:48) IJ ; AB (0) = g IAB · g JAB π × (cid:18) − (cid:19) (cid:2) M A − M B (cid:3) − × (cid:0) M A − M B (cid:1) M A (cid:0) M A − M B (cid:1) log (cid:0) µ /M A (cid:1) +6 M B (cid:0) M B − M A (cid:1) log (cid:0) µ /M B (cid:1) +9 M A M B − M A M B + 9 M A M B + 2 M A + 2 M B +6 M A M B (cid:0) − M A M B + M A + M B (cid:1) log (cid:0) M A /M B (cid:1) (A.4) −→ g IAB · g JAB π × (cid:18) − (cid:19) log (cid:0) µ /M (cid:1) as M A → M B ≡ M. (A.5)In the broken phase of the theory, the mass-eigenstate basis of new fermions is { n , n , C } ,where (cid:32) n n (cid:33) = (cid:32) cos θ − sin θ sin θ cos θ (cid:33) (cid:32) χN (cid:33) . (A.6)Here the mixing angle is given by tan 2 θ = √ κvM F − m χ , (A.7)and these particles have eigenmasses M = 12 (cid:34) m χ + M F − (cid:20) ( M F − m χ ) + 2 κ v (cid:21) / (cid:35) , (A.8) M = 12 (cid:34) m χ + M F + (cid:20) ( M F − m χ ) + 2 κ v (cid:21) / (cid:35) , and M C = M F . (A.9)The W + W − , ZZ , Zγ and γγ self-energy functions and their derivatives can be immediatelyfound by summing the contributions from eqs. (A.2)–(A.5) over all the allowed new-physics– 26 – able 7 . The new-physics particles ( A, B ) which run in the self-energy loop shown in figure 5. Self-energy IJ Particles ( A, B ) in loop W + W − ( C, , ( C, ZZ (1,1), (2,2), (1,2), (2,1), ( C, C ) Zγ ( C, C ) γγ ( C, C ) particles which run in the loop (see table 7), using the following results for the couplings: g ZCC = g − ( g (cid:48) ) (cid:112) g + ( g (cid:48) ) , g γCC = gg (cid:48) (cid:112) g + ( g (cid:48) ) , (A.10) g Z = − (cid:112) g + ( g (cid:48) ) θ, g W + C = g W − C = − g √ θ, (A.11) g Z = − (cid:112) g + ( g (cid:48) ) θ, g W + C = g W − C = g √ θ, (A.12)and g Z = g Z = (cid:112) g + ( g (cid:48) ) θ cos θ. (A.13) S and T can then be computed from their definitions, eqs. (4.4) and (4.9), and theresults shown here; we have checked explicitly that all terms depending on µ cancel in theresults for S and T , as they should. Note that T arises as a small difference between twonumerically large terms, and so care is required to obtain a numerically accurate result ifthese terms are not manually cancelled. B A clarification on matching in the presence of mixing
This Appendix aims to clarify a point about the matching at the scale µ = M F whichmay not be immediately familiar to the reader: when (cid:104) H (cid:105) (cid:54) = 0 , the mass-eigenstate χ fieldwe write in the intermediate-energy EFT should not be thought to be identified withthe (mass-eigenstate admixture) χ field in the UV theory; rather the χ field in the EFTshould instead be identified with the light-mass-eigenstate n in the UV theory. This factnotwithstanding, it is common practice, although something of an abuse of notation, to stillcall the field in the EFT ‘ χ ’.Consider first a general argument. We have presented both the EFT and UV theoryin terms of SU (2) × U (1) Y –symmetric operators. However, because (a) the limit (cid:104) H (cid:105) → is smooth in the sense that the F doublet in the UV theory remains sufficiently massive ( M F (cid:29) κv, m χ ) to be integrated out, and that the EFT power-counting is in κ (cid:104) H (cid:105) /M F , sono EFT operators or Wilson coefficients diverge as (cid:104) H (cid:105) → ; (b) the only source of EWSBin either the UV theory or the EFT is the existence of a non-zero (cid:104) H (cid:105) ; and (c) the EFTis explicitly constructed to match the amplitudes of the UV theory at µ = M F , it follows In this Appendix, ‘EFT’ always refers to the intermediate-energy EFT, after the F field is integratedout at µ = M F , but before the χ field is integrated out at µ = m χ . This is the EFT described by theoperators and coefficients in Tables 2 and 3. – 27 –hat when (cid:104) H (cid:105) (cid:54) = 0 , the EFT must by construction capture all the (cid:104) H (cid:105) -dependent effectswhich are present in the UV theory, at each order in the EFT power counting.Put another way, up to the truncation order of the EFT, one obtains the same physicalpredictions from the EFT by (a) matching in the broken phase ( (cid:104) H (cid:105) (cid:54) = 0 ) of the UV-theoryby integrating out the heavy-mass-eigenstates n and C , and writing the EFT in termsof the light-eigenstate n and the physical Higgs field h ; or (b) matching in the unbrokenphase ( (cid:104) H (cid:105) = 0 ), integrating out the F doublet whole and writing the EFT in terms of(the EFT field) χ and the SU (2) -doublet field H , and then turning on a non-zero (cid:104) H (cid:105) inthe EFT after the matching. Once (cid:104) H (cid:105) is turned on, the EFT automatically includes thecorrections to the χ field which allow its identification with the same physical degree offreedom represented by the light-mass-eigenstate n of the UV theory.We emphasize that it is by construction that the EFT must reproduce the UV-theoryamplitudes and so must capture the mixing of the fermions χ and N in the UV-theory.In the SU (2) -symmetric EFT, this mixing is present in the form of the higher-dimensionoperators of the EFT. We now show examples of this by explicit computation. B.1 Example 1: two-point functions of the UV-theory n field and the EFTfield χ The clearest demonstration that the EFT field χ represents the same physical degree offreedom as the UV-theory field n is to show that these two fields have the same two-point function; or, equivalently, that when they both have canonical kinetic terms in theirrespective Lagrangians, they have the same physical mass (up to corrections at the truncatedorder in the EFT power-counting).The UV-theory n field has a canonical kinetic term and mass given in eq. (A.8).Expanding this result out for M F (cid:29) κv, m χ gives M ≈ m χ − κ v M F − m χ κ v M F + κ v − m χ κ v M F + O ( M − F ) . (B.1)The EFT χ field does not have canonical kinetic terms. The relevant higher-dimensionaloperators in the EFT are (see Tables 2 and 3) L EFT ⊃ i ¯ χ /∂χ − m χ ¯ χχ + c M F O + c B M F O B + c A M F O A + O ( M − F ) (B.2) = i ¯ χ /∂χ − m χ ¯ χχ + κ M F ( H † H ) ¯ χχ + 12 κ M F ( H † H ) i ( ¯ χ /∂χ − h.c. ) − κ M F ( H † H )( ¯ χ (cid:3) χ + h.c. ) + O ( M − F ) . (B.3)Breaking electroweak symmetry by setting (cid:104) H (cid:105) = (0 , v/ √ T , the relevant terms become L EFT ⊃ i ¯ χ /∂χ (cid:20) κ v M F (cid:21) − ¯ χχ (cid:20) m χ − κ v M F (cid:21) − κ v M F (cid:104) ¯ χ ← (cid:3) χ + ¯ χ → (cid:3) χ (cid:105) + O ( M − F ) , (B.4)– 28 –his can be re-cast to canonical form by performing a field-redefinition, which does notchange the physical content of the theory: χ → (cid:20) − κ v M F − m χ κ v M F (cid:21) χ − κ v M F i /∂χM F + O ( M − F ) . (B.5)In terms of the redefined field, we have L EFT ⊃ i ¯ χ /∂χ − ¯ χχ (cid:34) m χ − κ v M F − m χ κ v M F + κ v − m χ κ v M F (cid:35) + O ( M − F ) . (B.6)Comparing eqs. (B.1) and (B.6), we have established the desired result. B.2 Example 2: Z -boson coupling to the EFT field χ We now look at an example for the EFT χ interaction terms. Consider that the UV-theory χ field is an exact SM-singlet, while N ⊂ F has SM gauge-couplings. When (cid:104) H (cid:105) (cid:54) = 0 ,the mass-basis rotation/mixing, eq. (A.6), generates SM gauge-couplings for both the n , mass-eigenstate fermions; see eqs. (A.10) to (A.13).If the EFT field χ were to be identified exactly with the χ field in the UV theory, itwould necessarily follow that the former should have no SM gauge-couplings; if instead theEFT field χ were identified with the light-mass-eigenstate n of the UV theory, we wouldexpect the former to couple to, e.g., the Z -boson. The second alternative is the correct one,as the following short argument shows.Consider that in the UV theory the coupling constant of the n vector current to the Z -boson is g Z = − (cid:112) g + ( g (cid:48) ) sin θ where tan 2 θ = √ κv/ ( M F − m χ ) ; see eqs. (A.7)and (A.11). In the limit where M F (cid:29) κv, m χ , we have g Z ≈ − (cid:112) g + ( g (cid:48) ) κ v M F + · · · . (B.7)Meanwhile, in the EFT the following operator is present: L EFT ⊃ c A M F O A ⊃ − (cid:112) g + ( g (cid:48) ) κ v M F ¯ χγ µ χ Z µ + · · · . (B.8)(Note: the field-redefinition in Example 1 above does not modify this leading-order cou-pling.) One can readily see that the leading-order coupling of the Z -boson with the vector-current of the EFT χ field has the same sign and magnitude, and appears at the sameorder in the EFT power-counting, as the leading-order coupling of the Z -boson with thevector-current of the UV-theory n field. Again, we draw the reader’s attention to the factthat it is a higher-dimension operator in the EFT that has captured an effect of UV-theoryfermion mixing when (cid:104) H (cid:105) (cid:54) = 0 . References [1] B. Batell, M. Pospelov, and A. Ritz,
Exploring Portals to a Hidden Sector Through FixedTargets , Phys.Rev.
D80 (2009) 095024, [ arXiv:0906.5614 ]. – 29 –
2] J. McDonald,
Gauge singlet scalars as cold dark matter , Phys.Rev.
D50 (1994) 3637–3649,[ hep-ph/0702143 ].[3] C. Burgess, M. Pospelov, and T. ter Veldhuis,
The Minimal model of nonbaryonic darkmatter: A Singlet scalar , Nucl.Phys.
B619 (2001) 709–728, [ hep-ph/0011335 ].[4] B. Patt and F. Wilczek,
Higgs-field portal into hidden sectors , hep-ph/0605188 .[5] Y. G. Kim, K. Y. Lee, and S. Shin, Singlet fermionic dark matter , JHEP (2008) 100,[ arXiv:0803.2932 ].[6] C. Englert, T. Plehn, D. Zerwas, and P. M. Zerwas, Exploring the Higgs portal , Phys.Lett.
B703 (2011) 298–305, [ arXiv:1106.3097 ].[7] S. Baek, P. Ko, and W.-I. Park,
Search for the Higgs portal to a singlet fermionic darkmatter at the LHC , JHEP (2012) 047, [ arXiv:1112.1847 ].[8] A. Djouadi, O. Lebedev, Y. Mambrini, and J. Quevillon, Implications of LHC searches forHiggs–portal dark matter , Phys.Lett.
B709 (2012) 65–69, [ arXiv:1112.3299 ].[9] B. Batell, S. Gori, and L.-T. Wang,
Exploring the Higgs Portal with 10/fb at the LHC , JHEP (2012) 172, [ arXiv:1112.5180 ].[10] A. Greljo, J. Julio, J. F. Kamenik, C. Smith, and J. Zupan,
Constraining Higgs mediateddark matter interactions , JHEP (2013) 190, [ arXiv:1309.3561 ].[11] Z. Chacko, Y. Cui, and S. Hong,
Exploring a Dark Sector Through the Higgs Portal at aLepton Collider , Phys.Lett.
B732 (2014) 75–80, [ arXiv:1311.3306 ].[12] J. M. Cline, K. Kainulainen, P. Scott, and C. Weniger,
Update on scalar singlet dark matter , Phys.Rev.
D88 (2013) 055025, [ arXiv:1306.4710 ].[13] D. G. E. Walker,
Unitarity Constraints on Higgs Portals , arXiv:1310.1083 .[14] T. Robens and T. Stefaniak, Status of the Higgs Singlet Extension of the Standard Modelafter LHC Run 1 , Eur.Phys.J.
C75 (2015) 104, [ arXiv:1501.02234 ].[15] A. Freitas, S. Westhoff, and J. Zupan,
Integrating in the Higgs Portal to Fermion DarkMatter , arXiv:1506.04149 .[16] ATLAS Collaboration , G. Aad et al.,
Observation of a new particle in the search for theStandard Model Higgs boson with the ATLAS detector at the LHC , Phys.Lett.
B716 (2012)1–29, [ arXiv:1207.7214 ].[17]
CMS Collaboration , S. Chatrchyan et al.,
Observation of a new boson at a mass of 125GeV with the CMS experiment at the LHC , Phys.Lett.
B716 (2012) 30–61,[ arXiv:1207.7235 ].[18]
ATLAS Collaboration , Measurements of the Higgs boson production and decay rates andcoupling strengths using pp collision data at sqrt(s) = 7 and 8 TeV in the ATLASexperiment , Tech. Rep. ATLAS-CONF-2015-007, CERN, Geneva, Mar, 2015.[19]
CMS Collaboration , V. Khachatryan et al.,
Precise determination of the mass of theHiggs boson and tests of compatibility of its couplings with the standard model predictionsusing proton collisions at 7 and 8 TeV , Eur.Phys.J.
C75 (2015) 212, [ arXiv:1412.8662 ].[20] J. R. Espinosa and M. Quiros,
Novel Effects in Electroweak Breaking from a Hidden Sector , Phys.Rev.
D76 (2007) 076004, [ hep-ph/0701145 ]. – 30 –
21] M. Gonderinger, Y. Li, H. Patel, and M. J. Ramsey-Musolf,
Vacuum Stability, Perturbativity,and Scalar Singlet Dark Matter , JHEP (2010) 053, [ arXiv:0910.3167 ].[22] D. J. Chung, A. J. Long, and L.-T. Wang,
125 GeV Higgs boson and electroweak phasetransition model classes , Phys.Rev.
D87 (2013) 023509, [ arXiv:1209.1819 ].[23] M. Fairbairn and R. Hogan,
Singlet Fermionic Dark Matter and the Electroweak PhaseTransition , JHEP (2013) 022, [ arXiv:1305.3452 ].[24] T. Li and Y.-F. Zhou,
Strongly first order phase transition in the singlet fermionic darkmatter model after LUX , JHEP (2014) 006, [ arXiv:1402.3087 ].[25] S. Profumo, M. J. Ramsey-Musolf, C. L. Wainwright, and P. Winslow,
Singlet-catalyzedelectroweak phase transitions and precision Higgs boson studies , Phys.Rev.
D91 (2015)035018, [ arXiv:1407.5342 ].[26] N. Craig, C. Englert, and M. McCullough,
New Probe of Naturalness , Phys.Rev.Lett. (2013), no. 12 121803, [ arXiv:1305.5251 ].[27] R. Enberg, P. Fox, L. Hall, A. Papaioannou, and M. Papucci,
LHC and dark matter signalsof improved naturalness , JHEP (2007) 014, [ arXiv:0706.0918 ].[28] G. Cynolter and E. Lendvai,
Electroweak Precision Constraints on Vector-like Fermions , Eur.Phys.J.
C58 (2008) 463–469, [ arXiv:0804.4080 ].[29] T. Cohen, J. Kearney, A. Pierce, and D. Tucker-Smith,
Singlet-Doublet Dark Matter , Phys.Rev.
D85 (2012) 075003, [ arXiv:1109.2604 ].[30] L. Lopez-Honorez, T. Schwetz, and J. Zupan,
Higgs portal, fermionic dark matter, and aStandard Model like Higgs at 125 GeV , Phys.Lett.
B716 (2012) 179–185, [ arXiv:1203.2064 ].[31] M. A. Fedderke, J.-Y. Chen, E. W. Kolb, and L.-T. Wang,
The Fermionic Dark Matter HiggsPortal: an effective field theory approach , JHEP (2014) 122, [ arXiv:1404.2283 ].[32] A. De Simone, G. F. Giudice, and A. Strumia,
Benchmarks for Dark Matter Searches at theLHC , JHEP (2014) 081, [ arXiv:1402.6287 ].[33] C. E. Yaguna,
Singlet-Doublet Dirac Dark Matter , Phys. Rev.
D92 (2015), no. 11 115002,[ arXiv:1510.06151 ].[34] S. Baek, P. Ko, W.-I. Park, and E. Senaha,
Vacuum structure and stability of a singletfermion dark matter model with a singlet scalar messenger , JHEP (2012) 116,[ arXiv:1209.4163 ].[35] S. Baek, P. Ko, and W.-I. Park, Invisible Higgs Decay Width vs. Dark Matter DirectDetection Cross Section in Higgs Portal Dark Matter Models , Phys. Rev.
D90 (2014), no. 5055014, [ arXiv:1405.3530 ].[36] V. V. Khoze, G. Ro, and M. Spannowsky,
Spectroscopy of scalar mediators to dark matter atthe LHC and at 100 TeV , Phys. Rev.
D92 (2015), no. 7 075006, [ arXiv:1505.03019 ].[37] S. Baek, P. Ko, M. Park, W.-I. Park, and C. Yu,
Beyond the Dark matter effective fieldtheory and a simplified model approach at colliders , arXiv:1506.06556 .[38] H. Baer, T. Barklow, K. Fujii, Y. Gao, A. Hoang, et al., The International Linear ColliderTechnical Design Report - Volume 2: Physics , arXiv:1306.6352 .[39] TLEP Design Study Working Group , M. Bicer et al.,
First Look at the Physics Case ofTLEP , JHEP (2014) 164, [ arXiv:1308.6176 ]. – 31 – The CEPC-SPPC Study Group , M. Ahmad et al.,
CEPC-SPPC preliminary conceptualdesign report , Tech. Rep. IHEP-CEPC-DR-2015-01, IHEP-EP-2015-01, IHEP-TH-2015-01,March, 2015.[41] M. E. Peskin and T. Takeuchi,
A New constraint on a strongly interacting Higgs sector , Phys.Rev.Lett. (1990) 964–967.[42] M. E. Peskin and T. Takeuchi, Estimation of oblique electroweak corrections , Phys.Rev.
D46 (1992) 381–409.[43] J. Fan, M. Reece, and L.-T. Wang,
Possible Futures of Electroweak Precision: ILC, FCC-ee,and CEPC , arXiv:1411.1054 .[44] D. Asner, T. Barklow, C. Calancha, K. Fujii, N. Graf, et al., ILC Higgs White Paper , arXiv:1310.0763 .[45] S. Dawson, A. Gritsan, H. Logan, J. Qian, C. Tully, et al., Working Group Report: HiggsBoson , arXiv:1310.8361 .[46] C. Englert and M. McCullough, Modified Higgs Sectors and NLO Associated Production , JHEP (2013) 168, [ arXiv:1303.1526 ].[47] J. Cao, C. Han, J. Ren, L. Wu, J. M. Yang, et al.,
SUSY effects in Higgs productions at highenergy e + e − colliders , arXiv:1410.1018 .[48] A. Katz and M. Perelstein, Higgs Couplings and Electroweak Phase Transition , JHEP (2014) 108, [ arXiv:1401.1827 ].[49] N. Craig, M. Farina, M. McCullough, and M. Perelstein,
Precision Higgsstrahlung as a Probeof New Physics , JHEP (2015) 146.[50] B. Henning, X. Lu, and H. Murayama,
What do precision Higgs measurements buy us? , arXiv:1404.1058 .[51] M. Beneke, D. Boito, and Y.-M. Wang, Anomalous Higgs couplings in angular asymmetriesof H → Z(cid:96) + (cid:96) − and e + e − → HZ , JHEP (2014) 028, [ arXiv:1406.1361 ].[52] K. Hagiwara, S. Ishihara, J. Kamoshita, and B. A. Kniehl,
Prospects of measuring generalHiggs couplings at e+ e- linear colliders , Eur.Phys.J.
C14 (2000) 457–468,[ hep-ph/0002043 ].[53] S. Dawson and S. Heinemeyer,
The Higgs boson production cross-section as a precisionobservable? , Phys.Rev.
D66 (2002) 055002, [ hep-ph/0203067 ].[54] V. Barger, T. Han, P. Langacker, B. McElrath, and P. Zerwas,
Effects of genuinedimension-six Higgs operators , Phys.Rev.
D67 (2003) 115001, [ hep-ph/0301097 ].[55] S. S. Biswal, R. M. Godbole, R. K. Singh, and D. Choudhury,
Signatures of anomalous VVHinteractions at a linear collider , Phys.Rev.
D73 (2006) 035001, [ hep-ph/0509070 ].[56] J. Kile and M. J. Ramsey-Musolf,
Fermionic effective operators and Higgs production at alinear collider , Phys.Rev.
D76 (2007) 054009, [ arXiv:0705.0554 ].[57] S. Dutta, K. Hagiwara, and Y. Matsumoto,
Measuring the Higgs-Vector boson Couplings atLinear e + e − Collider , Phys.Rev.
D78 (2008) 115016, [ arXiv:0808.0477 ].[58] M. E. Peskin,
Estimation of LHC and ILC Capabilities for Precision Higgs Boson CouplingMeasurements , arXiv:1312.4974 . – 32 –
59] C. Grojean, G. Servant, and J. D. Wells,
First-order electroweak phase transition in thestandard model with a low cutoff , Phys.Rev.
D71 (2005) 036001, [ hep-ph/0407019 ].[60]
LUX Collaboration , D. Akerib et al.,
First results from the LUX dark matter experimentat the Sanford Underground Research Facility , Phys.Rev.Lett. (2014) 091303,[ arXiv:1310.8214 ].[61] G. Busoni, A. De Simone, E. Morgante, and A. Riotto,
On the Validity of the Effective FieldTheory for Dark Matter Searches at the LHC , Phys.Lett.
B728 (2014) 412–421,[ arXiv:1307.2253 ].[62] G. Busoni, A. De Simone, T. Jacques, E. Morgante, and A. Riotto,
On the Validity of theEffective Field Theory for Dark Matter Searches at the LHC Part III: Analysis for the t -channel , JCAP (2014) 022, [ arXiv:1405.3101 ].[63] G. Busoni, A. De Simone, J. Gramling, E. Morgante, and A. Riotto,
On the Validity of theEffective Field Theory for Dark Matter Searches at the LHC, Part II: Complete Analysis forthe s -channel , JCAP (2014) 060, [ arXiv:1402.1275 ].[64] P. Sikivie, L. Susskind, M. B. Voloshin, and V. I. Zakharov,
Isospin Breaking in TechnicolorModels , Nucl.Phys.
B173 (1980) 189.[65] B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek,
Dimension-Six Terms in theStandard Model Lagrangian , JHEP (2010) 085, [ arXiv:1008.4884 ].[66] W. Buchmuller and D. Wyler,
Effective Lagrangian Analysis of New Interactions and FlavorConservation , Nucl.Phys.
B268 (1986) 621–653.[67] K. Hagiwara, S. Ishihara, R. Szalapski, and D. Zeppenfeld,
Low-energy effects of newinteractions in the electroweak boson sector , Phys.Rev.
D48 (1993) 2182–2203.[68] S. Willenbrock and C. Zhang,
Effective Field Theory Beyond the Standard Model , Ann.Rev.Nucl.Part.Sci. (2014) 83–100, [ arXiv:1401.0470 ].[69] C. Arzt, Reduced effective Lagrangians , Phys.Lett.
B342 (1995) 189–195, [ hep-ph/9304230 ].[70] W. Skiba,
Effective Field Theory and Precision Electroweak Measurements , in
Physics of thelarge and the small, TASI 09, proceedings of the Theoretical Advanced Study Institute inElementary Particle Physics, Boulder, Colorado, USA, 1-26 June 2009 , pp. 5–70, 2011. arXiv:1006.2142 .[71] R. Barbieri, A. Pomarol, R. Rattazzi, and A. Strumia,
Electroweak symmetry breaking afterLEP-1 and LEP-2 , Nucl.Phys.
B703 (2004) 127–146, [ hep-ph/0405040 ].[72] C. Burgess, S. Godfrey, H. Konig, D. London, and I. Maksymyk,
Model independent globalconstraints on new physics , Phys.Rev.
D49 (1994) 6115–6147, [ hep-ph/9312291 ].[73] S. Mishima, “Sensitivity to new physics from TLEP precision measurements.” talk at 6thTLEP workshop, CERN, Oct 16, 2013.[74] G. Passarino and M. Veltman,
One Loop Corrections for e + e − Annihilation Into µ + µ − inthe Weinberg Model , Nucl.Phys.
B160 (1979) 151.(1979) 151.