EWPD in the SMEFT to dimension eight
PPrepared for submission to JHEP
CALT-TH/2021-007
EWPD in the SMEFT to dimension eight
Tyler Corbett, a Andreas Helset, b Adam Martin, c and Michael Trott a a Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen, Denmark b Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA91125, USA c Department of Physics, University of Notre Dame, Notre Dame, IN, 46556, USA
Abstract:
We calculate the O ( (cid:104) H † H (cid:105) / Λ ) corrections to LEP electroweak precision datausing the geometric formulation of the Standard Model Effective Field Theory (SMEFT).We report our results in simple-to-use interpolation tables that allow the interpretation ofthis data set to dimension eight for the first time. We demonstrate the impact of thesepreviously unknown terms in the case of a general analysis in the SMEFT, and also in thecases of two distinct models matched to dimension eight. Neglecting such dimension-eightcorrections to LEP observables introduces a theoretical error in SMEFT studies. We reportsome preliminary studies defining such a theory error, explicitly demonstrating the effect ofpreviously unknown dimension-eight SMEFT corrections on LEP observables. a r X i v : . [ h e p - ph ] F e b ontents L (6) L (8) R (cid:96) Interpreting current experimental results while allowing for the Standard Model (SM) tobreak down at higher energies in future experimental studies is a key task in particle physics.This can be done in a way that is agnostic about new physics at higher energies by usingan effective field theory (EFT). In this approach we are Taylor expanding in the low-energymeasurement scale(s) divided by the scale of some new physics effects. This defines the“power counting” of the EFT. When combined with the assumed symmetries, the low-energyfield content, and the representations of the fields under these symmetries, this defines anEFT. The power of EFT studies of data sets resides in the fact that such an approach issystematically improvable with quantum loop corrections, and corrections that are higherorder in the power counting without knowledge of the UV completion of the EFT.When the particle spectrum includes an SU(2) L scalar doublet ( H ), and the mass scaleof heavy new physics is parametrically separated from the electroweak scale, the Standard– 1 –odel Effective Field Theory (SMEFT) is the appropriate EFT for data with a measurementscale proximate to the electroweak scale. The Large Electron Positron (LEP) collider provided a series of precise measurementson the properties of the SM states interacting at energies proximate to the Z mass [12].Over a decade after the conclusion of the LEP experimental program, no consistent andcomplete analysis of this data in terms of the SMEFT, extended to sub-leading order in thepower counting, had been developed until the geometric formulation of the SMEFT definedthe relevant formalism not only to sub-leading order, but to all orders in the expansionin (cid:112) (cid:104) H † H (cid:105) / Λ in Refs. [13–16]. To develop this approach the SMEFT was reformulatedgeometrically as the Higgs field space geometry plays a key role in this EFT [17–20]. Equallyimportant was the development and use of Hilbert series techniques in Refs. [21–24].In this paper, we present a complete set of explicit results that allow the study of LEPElectroweak Precision Data (EWPD) constraints to dimension eight for the first time. Wealso study the effect of previously unknown and neglected dimension-eight corrections wheninterpreting LEP data. The SMEFT Lagrangian is L SMEFT = L SM + L ( d ) , L ( d ) = (cid:88) i C ( d ) i Λ d − Q ( d ) i for d > . (2.1)The higher-dimensional operators Q ( d ) i are constructed out of the SM fields. The particlespectrum includes an SU(2) L scalar doublet ( H ) with hypercharge y h = 1 /
2. The operators Q ( d ) i are labelled with a mass dimension d superscript and multiply unknown Wilson coef-ficients C ( d ) i . We define ˜ C ( d ) i ≡ C ( d ) i ¯ v d − T / Λ d − and use the Warsaw basis [33] for L (6) andRefs. [15, 34] for L (8) results. Our remaining notation is defined in Refs. [1, 27].The geometric formulation of the SMEFT (geoSMEFT [13–16]) organizes the theory interms of field-space connections. This approach builds on the results reported in Refs. [17–20, 35, 36]. Using this formulation, the SMEFT was consistently formulated at all orders inthe expansion in O ( v/ Λ) for two- and three-point functions. In particular, the theory was For a review on EFT and the SMEFT, see Ref. [1]. Subtleties in mixing of heavy and light states canpotentially lead to the HEFT [2–6]. These subtleties do not change our conclusions; see Refs. [1, 7–11] forrelated scientific discussions. Here (cid:112) (cid:104) H † H (cid:105) ≡ v T is defined to be the vacuum expection value (vev) of the Higgs field in the SMEFT.In this paper we generally do not draw a distinction between the notation ¯ v T and v , using the latter fornotational brevity at times. An exception to this rule, where the distinction between the SM classical vev v and the minimum of the potential in the SMEFT ¯ v T is important, is discussed in Section 6. The results in this work extend previous results [25–29] in a consistent Effective Field Theory extensionof the SM (nowadays called the SMEFT). We also note that the first work to stress the need to characterise atheory error due to the neglect of dimension-eight operators when interpreting EWPD is Ref. [30]. This pointhas also been stressed in several recent studies, see Refs. [31, 32]. – 2 –bservable { ˆ α, ˆ m Z , ˆ G F } inputs { ˆ m W , ˆ m Z , ˆ G F } inputs Exp. result [12]Γ e,µ [MeV] 83.978 ± ± ± τ [MeV] 83.788 ± ± ± ν [MeV] 167.166 ± ± ± u [MeV] 299.91 ± ± c [MeV] 299.84 ± ± ± d,s [MeV] 382.77 ± ± b [MeV] 375.88 ± ± ± Z [MeV] 2494.4 ± ± ± R (cid:96) ± ± ± R c ± ± ± R b ± ± ± A (cid:96)F B ± ± ± A cF B ± ± ± A bF B ± ± ± σ [pb] 41,491 ± ± ± Table 1 . Predictions for LEPI observables in the two input parameter schemes. The { ˆ m W , ˆ m Z , ˆ G F } scheme results are derived using [37–40]. In particular A cF B is derived using Ref. [37]. We havecompared the results for A (cid:96)F B using Refs. [37, 38] and the results agree within quoted errors. consistently formulated for these n -point functions to O ( v / Λ ) in Ref. [16], including inputparameter shifts. This is sufficient to examine the effect of heretofore unknown dimension-eight corrections on EWPD observables. We seek to interpret the results in Table 1 in the SMEFT consistently to O ( v / Λ ). Forthe predictions of these measurements, we need numerical values of Lagrangian parameters.These are defined in an input parameter scheme. Such a scheme is a free choice, and twochoices are in common use in the literature. These are the { ˆ m W , ˆ m Z , ˆ G F } and { ˆ α, ˆ m Z , ˆ G F } schemes. We report results in both of these schemes, and use the numerical input parameterresults in Table 2 to fix values of Lagrangian parameters to this end. For the SM resultsfor the observables in each scheme, we update the theoretical predictions. For partial widthsand ratios of partial widths we update the results beyond those quoted in Ref. [29]. Thesenumerical values were determined using the interpolation formula in Refs. [39, 40]. In addition,we use the expansion formula in Refs. [37, 38] to determine up-to-date numerical values ofthe A iF B pseudo-observables in the { ˆ m W , ˆ m Z , ˆ G F } scheme, leading to Table 1. We once again thank A. Freitas for helpful comments and advice on using the results of Refs. [37–40]. – 3 –nput parameters Value Ref.ˆ m Z [GeV] 91 . ± . m W [GeV] 80 . ± .
016 [42]ˆ m h [GeV] 125 . ± .
14 [41]ˆ m t [GeV] 172 . ± . m b [GeV] 4 . ± .
03 [43]ˆ m c [GeV] 1 . ± .
02 [43]ˆ m τ [GeV] 1 . ± . G F [GeV − ] 1.1663787 · − [43, 44]ˆ α EW α . ± . α s . ± . m ˆ αW . ± .
01 –∆ α ˆ m W . ± . Table 2 . Input parameter values used to predict EWPD theory predictions for both schemes. m ˆ αW isthe value of m W inferred in the { ˆ α, ˆ m Z , ˆ G F } scheme using the interpolation formula of Refs. [37–40],which includes SM loop corrections, while ∆ α ˆ m W is the shift in the value of alpha due to hadroniceffects for the { ˆ m W , ˆ m Z , ˆ G F } scheme. For an introductory discussion on the use of ∆ α relating lowscale measurements of ˆ α and higher scale values above the hadronic resonance region see Ref. [45].Note that we use a tree level value of m W , not m ˆ αW , in calculating the numerical coefficients for theshifts due to the SMEFT in EWPD. We present our results for SMEFT corrections normalized to SM predictions. Our resultscan be modified to take into account new SM predictions by multiplying by the ratio of theSM prediction in Table 1 divided by the new SM prediction.The observables are¯Γ i = ˆ m Z N ic π (cid:16) | g Z ,i L eff | + | g Z ,i R eff | (cid:17) (1 − m i ˆ m Z ) / , ¯Γ had = ¯Γ u + ¯Γ d + ¯Γ c + ¯Γ s + ¯Γ b , (3.1)¯ R c,b = ¯Γ c,b ¯Γ had , ¯ R (cid:96) = ¯Γ had ¯Γ (cid:96) , (3.2)¯ σ had = 12 π ˆ m Z ¯Γ e ¯Γ had ¯Γ Z , ¯ A ,fF B = 34 ¯ A (cid:96) ¯ A f , (3.3)where ¯ A i = ( g Z ,i L eff − g Z ,i R eff )( g Z ,i L eff + g Z ,i R eff )( g Z ,i L eff ) + ( g Z ,i R eff ) . (3.4)The bar notation indicates theoretical predictions in a canonically normalized SMEFT tomass dimension d . A hat indicates an experimentally measured quantity, or a numericallydefined quantity using measured input parameters. g Z ,i L eff are defined in Eq. (4.1).– 4 – EWPD results
The results of Refs. [15, 16] allow EWPD to be studied to O ( v / Λ ) in the SMEFT. Weneglect in our results corrections further suppressed by SM masses, and proportional to thesmall decay width of the Z compared to its mass, as LEP data is strongly peaked at p ∼ m Z .We also neglect self-interference effects in the decay due to dipole operators squared. Bothof these corrections in the SMEFT are calculable and neglected here largely for brevity ofpresentation. These extra effects only further support our main point, calling for a cautiousinterpretation of LEP constraints in the SMEFT. In both input parameter schemes, { ˆ m Z , ˆ G F } are used to fix the dimensions, so the ob-servables are defined in terms of these dimensionful parameters, with shifts due to correctionsto the input observables conventionally included in the shifts to the effective Z couplings g Z ,ψ eff , pr . Here p, r are flavor labels. For the observables we study, the energy scale is fixed tobe p (cid:39) ˆ m Z and the SMEFT corrections scale as O ( v n / Λ n ). Once the corrections to theeffective Z couplings in an input scheme are known to an order in this expansion, EWPD canbe analyzed to the same order.The effective couplings are defined at all orders in v/ Λ to be [15, 16] g Z ,ψ eff , pr = ¯ g Z (cid:104) (2 s θ Z Q ψ − σ ) δ pr + ¯ v T (cid:104) L ψ,pr , (cid:105) + σ ¯ v T (cid:104) L ψ,pr , (cid:105) (cid:105) = (cid:104) g Z ,ψ SM , pr (cid:105) + (cid:104) g Z ,ψ eff , pr (cid:105) O ( v / Λ ) + (cid:104) g Z ,ψ eff , pr (cid:105) O ( v / Λ ) + . . . (4.1)Here ψ L = { q L , (cid:96) L } , while ψ R = { u R , d R , e R } and σ = 1 for u L , ν L while σ = − d L , e L . L , , L , are geoSMEFT field space connections defined in Refs. [15, 16]. L (6) It is straightforward to derive (cid:104) ¯Γ SMEFT i (cid:105) = ˆΓ SM i + (cid:104) ¯Γ i (cid:105) O ( v / Λ ) + . . . (4.2)for each EWPD observable in the SMEFT. By Taylor expanding the predictions to linearorder in the corrections to the partial widths via the effective couplings, we have that (cid:104) ¯Γ Z→ ¯ ψ p ψ p (cid:105) O ( v / Λ ) ˆΓ SM Z→ ¯ ψ p ψ p = 2 Re (cid:104) (cid:104) g Z ,ψ L SM , pp (cid:105) (cid:104) g Z ,ψ L eff , pp (cid:105) O (v / Λ ) (cid:105) |(cid:104) g Z ,ψ L SM , pp (cid:105)| + |(cid:104) g Z ,ψ R SM , pp (cid:105)| + 2 Re (cid:104) (cid:104) g Z ,ψ R SM , pp (cid:105) (cid:104) g Z ,ψ R eff , pp (cid:105) O (v / Λ ) (cid:105) |(cid:104) g Z ,ψ L SM , pp (cid:105)| + |(cid:104) g Z ,ψ R SM , pp (cid:105)| = N ψ R (cid:104) g Z ,ψ R eff , pp (cid:105) O ( v / Λ ) + N ψ L (cid:104) g Z ,ψ L eff , pp (cid:105) O ( v / Λ ) . (4.3)The N ψ R/L are numerical coefficients that are reported in Table 3. For example, for Γ Z→ ¯ uu , N u R = 2 .
66 while N u L = − .
29 in the { ˆ m W , ˆ m Z , ˆ G F } input parameter scheme. For high enough Λ these neglected effects, along with loop corrections, might be on the same order orlarger than the dimension-eight corrections. In that case a more comperehensive analysis of EWPD is calledfor. See Refs. [25, 26, 29, 30, 46] for past analyses consistent with these results. – 5 –ach partial width ¯Γ i , and the sum of partial widths ¯Γ had , ¯Γ Z , are defined at linear orderin SMEFT perturbations via Table 3. Linear perturbations in the partial widths then define¯ R c,b,(cid:96) and ¯ σ had via¯ R c,b ˆ R SM c,b = 1 + (cid:104) ¯Γ c,b (cid:105) O ( v / Λ ) ˆΓ SM c,b − (cid:104) ¯Γ had (cid:105) O ( v / Λ ) ˆΓ SM had + . . . (4.4)¯ R (cid:96) ˆ R SM (cid:96) = 1 + (cid:104) ¯Γ had (cid:105) O ( v / Λ ) ˆΓ SM had − (cid:104) ¯Γ (cid:96) (cid:105) O ( v / Λ ) ˆΓ SM (cid:96) + . . . (4.5)¯ σ , SMEFT had ˆ σ , SM had = 1 + (cid:104) ¯Γ e (cid:105) O ( v / Λ ) ˆΓ SM e + (cid:104) ¯Γ had (cid:105) O ( v / Λ ) ˆΓ SM had − (cid:104) ¯Γ Z (cid:105) O ( v / Λ ) ˆΓ SM Z + . . . (4.6)The remaining observables, ¯ A ,fF B for f = { (cid:96), c, b } have the leading SMEFT perturbation (cid:104) ¯ A i (cid:105) O ( v / Λ ) ˆ A SM i = 4 (cid:104) g Z ,i L SM (cid:105) (cid:104) g Z ,i R SM (cid:105)(cid:104) g Z ,i L SM (cid:105) − (cid:104) g Z ,i R SM (cid:105) (cid:104) (cid:104) g Z ,i R SM (cid:105)(cid:104) g Z ,i L eff (cid:105) O ( v / Λ ) − (cid:104) g Z ,i L SM (cid:105)(cid:104) g Z ,i R eff (cid:105) O ( v / Λ ) (cid:105) . (4.7)The required numerical coefficients to construct these observables are given in Table 4. Forexample, for the bottom quark in the { ˆ m W , ˆ m Z , ˆ G F } scheme we find that (cid:104) ¯ A b (cid:105) O ( v / Λ ) ˆ A SM b = = 2 . (cid:104) g Z ,d R eff (cid:105) O ( v / Λ ) + 0 . (cid:104) g Z ,d L eff (cid:105) O ( v / Λ ) . (4.8)The ¯ A ,fF B follow directly via( ¯ A ,fF B ) SMEFT ( ˆ A ,fF B ) SM = 1 + (cid:104) ¯ A e (cid:105) O ( v / Λ ) ˆ A SM e + (cid:104) ¯ A f (cid:105) O ( v / Λ ) ˆ A SM f + . . . (4.9)Each of the effective couplings is expanded into SMEFT Wilson coefficients in Table 5. L (8) Defining EWPD to dimension eight in the SMEFT requires an expansion of the observablesto O ( v / Λ ), and the definition of the effective couplings to O ( v / Λ ). The latter is defined inTables 6 and 7. Expressing the results compactly, we build upon the presentation in Ref. [16].Expanding to second order the partial widths (cid:104) ¯Γ SMEFT i (cid:105) = ˆΓ SM i + (cid:104) ¯Γ i (cid:105) O ( v / Λ ) + (cid:104) ¯Γ i (cid:105) O ( v / Λ ) + . . . (4.10)where (cid:104) ¯Γ SMEFT Z→ ¯ ψ p ψ p (cid:105) O ( v / Λ ) ˆΓ SM Z→ ¯ ψ p ψ p = 2 Re (cid:104) (cid:104) g Z ,ψ L SM , pp (cid:105) (cid:104) g Z ,ψ L eff , pp (cid:105) O (v / Λ ) (cid:105) |(cid:104) g Z ,ψ L SM , pp (cid:105)| + |(cid:104) g Z ,ψ R SM , pp (cid:105)| + 2 Re (cid:104) (cid:104) g Z ,ψ R SM , pp (cid:105) (cid:104) g Z ,ψ R eff , pp (cid:105) O (v / Λ ) (cid:105) |(cid:104) g Z ,ψ L SM , pp (cid:105)| + |(cid:104) g Z ,ψ R SM , pp (cid:105)| + |(cid:104) g Z ,ψ L eff , pp (cid:105) O ( v / Λ ) | + |(cid:104) g Z ,ψ R eff , pp (cid:105) O ( v / Λ ) | |(cid:104) g Z ,ψ L SM , pp (cid:105)| + |(cid:104) g Z ,ψ R SM , pp (cid:105)| . (4.11)– 6 –here is dependence on the (cid:104) g Z ,ψ L/R eff , pp (cid:105) O ( v n / Λ n ) at each order in the expansion n . The(pseudo)-observables also have dependence on the squared dimension-six effective couplings.The required numerical coefficients are given in Table 3. As an example, for decays to upquarks in the { ˆ m W , ˆ m Z , ˆ G F } scheme we have that (cid:104) ¯Γ SMEFT Z→ ¯ u p u p (cid:105) O ( v / Λ ) ˆΓ SM Z→ ¯ u p u p = − . (cid:104) g Z ,u L eff , pp (cid:105) O ( v / Λ ) + 2 . (cid:104) g Z ,ψ R eff , pp (cid:105) O ( v / Λ ) + 12 . |(cid:104) g Z ,ψ L eff , pp (cid:105) O ( v / Λ ) | + 12 . |(cid:104) g Z ,ψ R eff , pp (cid:105) O ( v / Λ ) | (4.12)The observables ¯ R c,b , ¯ R (cid:96) , ¯ σ had are determined from the expansion of the Γ i directly via(¯Γ i / ¯Γ j ) SMEFT (ˆΓ i / ˆΓ j ) SM = 1 + (cid:104) ¯Γ i (cid:105) O ( v / Λ ) ˆΓ SM i − (cid:104) ¯Γ j (cid:105) O ( v / Λ ) ˆΓ SM j + (cid:32) (cid:104) ¯Γ j (cid:105) O ( v / Λ ) ˆΓ SM j (cid:33) (4.13)+ (cid:104) ¯Γ i (cid:105) O ( v / Λ ) ˆΓ SM i − (cid:104) ¯Γ j (cid:105) O ( v / Λ ) ˆΓ SM j − (cid:104) ¯Γ i (cid:105) O ( v / Λ ) (cid:104) ¯Γ j (cid:105) O ( v / Λ ) ˆΓ SM i ˆΓ SM j . For ¯ A ,fF B we expand directly in terms of the effective couplings via (cid:104) ¯ A SMEFT i (cid:105) O ( v / Λ ) ˆ A SM i = 2 (cid:104) g Z ,i R SM (cid:105) (cid:104) g Z ,i L eff , pp (cid:105) O ( v / Λ ) [ (cid:104) g Z ,i L SM (cid:105) + (cid:104) g Z ,i R SM (cid:105) ] (cid:32) (cid:104) g Z ,i R SM (cid:105) − (cid:104) g Z ,i L SM (cid:105) (cid:104) g Z ,i L SM (cid:105) − (cid:104) g Z ,i R SM (cid:105) (cid:33) − (cid:104) g Z ,i L SM (cid:105) (cid:104) g Z ,i R eff , pp (cid:105) O ( v / Λ ) [ (cid:104) g Z ,i L SM (cid:105) + (cid:104) g Z ,i R SM (cid:105) ] (cid:32) (cid:104) g Z ,i L SM (cid:105) − (cid:104) g Z ,i R SM (cid:105) ( (cid:104) g Z ,i L SM (cid:105) − (cid:104) g Z ,i R SM (cid:105) ) (cid:33) + 8 (cid:104) g Z ,i R eff , pp (cid:105) O ( v / Λ ) (cid:104) g Z ,i L eff , pp (cid:105) O ( v / Λ ) (cid:32) (cid:104) g Z ,i L SM (cid:105) (cid:104) g Z ,i R SM (cid:105) ( (cid:104) g Z ,i L SM (cid:105) + (cid:104) g Z ,i R SM (cid:105) ) (cid:33) (4.14)+ 4 (cid:104) g Z ,i L SM (cid:105)(cid:104) g Z ,i L eff , pp (cid:105) O ( v / Λ ) (cid:104) g Z ,i L SM (cid:105) − (cid:104) g Z ,i R SM (cid:105) (cid:104) g Z ,i R SM (cid:105) − (cid:104) g Z ,i R SM (cid:105)(cid:104) g Z ,i R eff , pp (cid:105) O ( v / Λ ) (cid:104) g Z ,i L SM (cid:105) − (cid:104) g Z ,i R SM (cid:105) (cid:104) g Z ,i L SM (cid:105) . The numerical dependence of EWPD observables on the SMEFT induced effective couplingin the Γ i is largely scheme independent and is given in Table 3. The numerical dependenceon the SMEFT induced effective couplings for the ¯ A SMEFT i are given in Table 4.The numerical dependence of the effective couplings on the Wilson coefficients are re-ported at O ( v / Λ ) , O ( v / Λ ) in Tables 5, 6, and 7. This expansion of the effective couplingsin terms of the individual Wilson coefficients carries a significant SMEFT input parameterscheme dependence, which increases at higher orders in the v n / Λ n expansion [16]. This isexpected due to the decoupling theorem and represents the effect of new physics being ab-sorbed into the lower energy measured input parameters. This is a more significant issue forthe SMEFT compared to many EFTs, due to the presence of a Higgs field.– 7 –umerical dependence of (cid:104) ¯Γ i (cid:105) / ˆΓ SM i in the { ˆ m W , ˆ m Z , ˆ G F } / { ˆ α, ˆ m Z , ˆ G F } schemes O ( v / Λ ) (cid:104) ¯Γ u (cid:105) / ˆΓ SM u (cid:104) ¯Γ ν (cid:105) / ˆΓ SM ν (cid:104) ¯Γ (cid:96) (cid:105) / ˆΓ SM (cid:96) (cid:104) ¯Γ d,b (cid:105) / ˆΓ SM d,b (cid:104) ¯Γ Z (cid:105) / ˆΓ SM Z (cid:104) ¯Γ had (cid:105) / ˆΓ SM had (cid:104) g Z ,u R eff , pp (cid:105) (cid:104) g Z ,d R eff , pp (cid:105) -1.04/-1.08 3(-0.160/-0.166) 3(-0.229/-0.236) (cid:104) g Z ,(cid:96) R eff , pp (cid:105) -4.75/-4.93 3(-0.160/-0.166) (cid:104) g Z ,u L eff , pp (cid:105) -6.29/-6.19 2(-0.756/-0.745) 2(-1.08/-1.07) (cid:104) g Z ,d L eff , pp (cid:105) (cid:104) g Z ,(cid:96) L eff , pp (cid:105) (cid:104) g Z ,ν L eff , pp (cid:105) -5.36/-5.36 3(-0.359/-0.359) O ( v / Λ ) (cid:104) ¯Γ u (cid:105) / ˆΓ SM u (cid:104) ¯Γ u (cid:105) / ˆΓ SM ν (cid:104) ¯Γ (cid:96) (cid:105) / ˆΓ SM (cid:96) (cid:104) ¯Γ d,b (cid:105) / ˆΓ SM d,b (cid:104) ¯Γ Z (cid:105) / ˆΓ SM Z (cid:104) ¯Γ had (cid:105) / ˆΓ SM had (cid:104) g Z ,u R eff , pp (cid:105) (cid:104) g Z ,d R eff , pp (cid:105) -1.04/-1.08 3(-0.160/-0.166) 3(-0.229/-0.236) (cid:104) g Z ,(cid:96) R eff , pp (cid:105) -4.75/-4.93 3(-0.160/-0.166) (cid:104) g Z ,u L eff , pp (cid:105) -6.29/-6.19 2(-0.756/-0.745) 2(-1.08/-1.07) (cid:104) g Z ,d L eff , pp (cid:105) (cid:104) g Z ,(cid:96) L eff , pp (cid:105) (cid:104) g Z ,ν L eff , pp (cid:105) -5.36/-5.36 3(-0.359/-0.359) (cid:104) g Z ,u R eff , pp (cid:105) (cid:104) g Z ,d R eff , pp (cid:105) (cid:104) g Z ,(cid:96) R eff , pp (cid:105) (cid:104) g Z ,u L eff , pp (cid:105) (cid:104) g Z ,d L eff , pp (cid:105) (cid:104) g Z ,(cid:96) L eff , pp (cid:105) (cid:104) g Z ,ν L eff , pp (cid:105) Table 3 . Dependence of the partial widths on the effective couplings, scaled to the SM prediction ofthe partial width. For the columns ¯Γ (cid:96) , ¯Γ u , ¯Γ d,b the individual partial widths are reported. The sum overflavors is explicit in the contribution to ¯Γ had , ¯Γ Z . The top section of the Table reports the dependenceon (cid:104) g Z ,ψ eff , pp (cid:105) O ( v / Λ ) . The middle section of the Table reports the dependence on (cid:104) g Z ,ψ eff , pp (cid:105) O ( v / Λ ) , whilethe bottom section is the dependence on (cid:104) g Z ,ψ eff , pp (cid:105) O ( v / Λ ) . In the quoted results, (cid:104) ¯Γ d,b (cid:105) / ˆΓ SM d,b wasdetermined using numerical values of light quarks d, s for the partial width. ˆΓ SM d,s / ˆΓ SM b differs at thepercent level in the SM. This leads to numerical differences, when combined with rounding effects,in the results quoted that should be incorporated as a simple rescaling based on Table 1. An emptyentry indicates no dependence on the relevant effective coupling. – 8 – ¯ A i (cid:105) / ˆ A SM i in the { ˆ m W , ˆ m Z , ˆ G F } / { ˆ α, ˆ m Z , ˆ G F } schemes O ( v / Λ ) (cid:104) ¯ A (cid:96) (cid:105) / ˆ A SM (cid:96) (cid:104) ¯ A c (cid:105) / ˆ A SM c (cid:104) ¯ A b (cid:105) / ˆ A SM b (cid:104) g Z ,u R eff , pp (cid:105) -6.71/-7.23 (cid:104) g Z ,d R eff , pp (cid:105) (cid:104) g Z ,(cid:96) R eff , pp (cid:105) (cid:104) g Z ,u L eff , pp (cid:105) -2.84/-3.22 (cid:104) g Z ,d L eff , pp (cid:105) (cid:104) g Z ,(cid:96) L eff , pp (cid:105) O ( v / Λ ) (cid:104) ¯ A (cid:96) (cid:105) / ˆ A SM (cid:96) (cid:104) ¯ A c (cid:105) / ˆ A SM c (cid:104) ¯ A b (cid:105) / ˆ A SM b (cid:104) g Z ,u R eff , pp (cid:105) -12.0/-10.7 (cid:104) g Z ,d R eff , pp (cid:105) -17.8/-17.8 (cid:104) g Z ,(cid:96) R eff , pp (cid:105) (cid:104) g Z ,u L eff , pp (cid:105) -13.0/-14.7 (cid:104) g Z ,d L eff , pp (cid:105) -1.77/-1.94 (cid:104) g Z ,(cid:96) L eff , pp (cid:105) -75.4/-106 (cid:104) g Z ,u R eff , pp (cid:105)(cid:104) g Z ,u L eff , pp (cid:105) -35.9/-37.8 (cid:104) g Z ,d R eff , pp (cid:105)(cid:104) g Z ,d L eff , pp (cid:105) -13.2/-13.9 (cid:104) g Z ,(cid:96) R eff , pp (cid:105)(cid:104) g Z ,(cid:96) L eff , pp (cid:105) -56.3/-57.3 (cid:104) g Z ,u R eff , pp (cid:105) -6.71/-7.23 (cid:104) g Z ,d R eff , pp (cid:105) (cid:104) g Z ,(cid:96) R eff , pp (cid:105) (cid:104) g Z ,u L eff , pp (cid:105) -2.84/-3.22 (cid:104) g Z ,d L eff , pp (cid:105) (cid:104) g Z ,(cid:96) L eff , pp (cid:105) Table 4 . Numerical coefficients defining the dependence on the SMEFT effective couplings in forwardbackward asymmetries. The expressions are normalized to the tree level SM values in each inputparameter scheme: ˆ A SM c = 0 . / .
70, ˆ A SM b = 0 . / .
94, ˆ A SM (cid:96) = 0 . / .
15. The significant schemedependence of ˆ A SM (cid:96) follows from the accidental numerical suppression of the value of the vectorialleptonic coupling, rendering it more sensitive to scheme dependence. – 9 –MEFT corrections in the { ˆ m W , ˆ m Z , ˆ G F } / { ˆ α, ˆ m Z , ˆ G F } scheme O ( v Λ ) (cid:104) g Z ,u R eff , pp (cid:105) (cid:104) g Z ,d R eff , pp (cid:105) (cid:104) g Z ,(cid:96) R eff , pp (cid:105) (cid:104) g Z ,u L eff , pp (cid:105) (cid:104) g Z ,d L eff , pp (cid:105) (cid:104) g Z ,(cid:96) L eff , pp (cid:105) (cid:104) g Z ,ν L eff , pp (cid:105) δG (6) F -0.08/0.15 0.04/-0.07 0.12/-0.22 0.18/0.41 -0.22/-0.34 -0.15/-0.49 0.26/0.26˜ C (6) HD -0.22/0.05 0.11/-0.03 0.33/-0.08 -0.13/0.15 0.02/-0.12 0.24/-0.17 0.09/0.09˜ C (6) HW B -0.21/0.39 0.10/-0.19 0.31/-0.58 -0.21/0.39 0.10/-0.19 0.31/-0.58˜ C (6) Hψ C , (6) Hψ -0.37/-0.37 0.37/0.37 0.37/0.37 -0.37/-0.37 Table 5 . The effective couplings expanded to O ( v / Λ ) in each input parameter scheme. δG (6) F is defined in Ref. [16]. Reported is the numerical coefficient multiplying each SMEFT correction. p = { , , } is a flavor index. The operator subscript labels ψ take on the values { u R , d R , (cid:96) R , q L , (cid:96) L } ,with the effective coupling ψ label dictating the value of ψ . ¯ R (cid:96) We illustrate the use of the formula we present defining EWPD to O ( v / Λ ) using the exampleof ¯ R (cid:96) . First we use the result for the expansion of this observable to second order¯ R SMEFT (cid:96) ˆ R SM (cid:96) = (ˆΓ had / ˆΓ (cid:96) ) SMEFT (ˆΓ had / ˆΓ (cid:96) ) SM (4.15)= 1 + (cid:104) Γ SMEFT had (cid:105) O ( v / Λ ) ˆΓ SM had − (cid:104) Γ SMEFT (cid:96) (cid:105) O ( v / Λ ) ˆΓ SM (cid:96) + (cid:32) (cid:104) Γ SMEFT (cid:96) (cid:105) O ( v / Λ ) ˆΓ SM (cid:96) (cid:33) + (cid:104) Γ SMEFT had (cid:105) O ( v / Λ ) ˆΓ SM had − (cid:104) Γ SMEFT (cid:96) (cid:105) O ( v / Λ ) ˆΓ SM (cid:96) − (cid:104) Γ SMEFT had (cid:105) O ( v / Λ ) (cid:104) Γ SMEFT (cid:96) (cid:105) O ( v / Λ ) ˆΓ SM had ˆΓ SM (cid:96) . Then we substitute the values from Table 5 into this expression, using the { ˆ m W , ˆ m Z , ˆ G F } input parameter scheme results, finding¯ R (cid:96) / ˆ R SM (cid:96) = 1 + (cid:104) . (cid:104) g Z ,d L eff (cid:105) − . (cid:104) g Z ,d R eff (cid:105) − . (cid:104) g Z ,(cid:96) L eff (cid:105) + 4 . (cid:104) g Z ,(cid:96) R eff (cid:105) − . (cid:104) g Z ,u L eff (cid:105) + 0 . (cid:104) g Z ,u R eff (cid:105) (cid:105) + (cid:104) . (cid:104) g Z ,d L eff (cid:105) − . (cid:104) g Z ,d R eff (cid:105) − . (cid:104) g Z ,(cid:96) L eff (cid:105) + 4 . (cid:104) g Z ,(cid:96) R eff (cid:105) − . (cid:104) g Z ,u L eff (cid:105) + 0 . (cid:104) g Z ,u R eff (cid:105) (cid:105) O ( v / Λ ) + (cid:104) . (cid:104) g Z ,d L eff (cid:105) + 6 . (cid:104) g Z ,d R eff (cid:105) + 21 (cid:104) g Z ,(cid:96) L eff (cid:105) + 8 . (cid:104) g Z ,(cid:96) R eff (cid:105) + 4 . (cid:104) g Z ,u L eff (cid:105) + 4 . (cid:104) g Z ,u R eff (cid:105) (cid:105) + (cid:104) − (cid:104) g Z ,d L eff (cid:105) (cid:104) g Z ,(cid:96) L eff (cid:105) + 19 (cid:104) g Z ,d L eff (cid:105) (cid:104) g Z ,(cid:96) R eff (cid:105) + 4 . (cid:104) g Z ,d R eff (cid:105) (cid:104) g Z ,(cid:96) L eff (cid:105) − . (cid:104) g Z ,d R eff (cid:105) (cid:104) g Z ,(cid:96) R eff (cid:105) (cid:105) + (cid:104) − (cid:104) g Z ,(cid:96) L eff (cid:105) (cid:104) g Z ,(cid:96) R eff (cid:105) + 13 (cid:104) g Z ,(cid:96) L eff (cid:105) (cid:104) g Z ,u L eff (cid:105) − . (cid:104) g Z ,(cid:96) L eff (cid:105) (cid:104) g Z ,u R eff (cid:105) − (cid:104) g Z ,(cid:96) R eff (cid:105) (cid:104) g Z ,u L eff (cid:105) (cid:105) + (cid:104) . (cid:104) g Z ,(cid:96) R eff (cid:105) (cid:104) g Z ,u R eff (cid:105) (cid:105) . (4.16)All of the g eff appearing in Eq. (4.16), excepting those on the second line, are (cid:104) g Z ,ψeff (cid:105) O ( v / Λ ) .Those on the second line are (cid:104) g Z ,ψeff (cid:105) O ( v / Λ ) , as indicated.– 10 –MEFT corrections in { ˆ m W , ˆ m Z , ˆ G F } / { ˆ α, ˆ m Z , ˆ G F } scheme O ( v Λ ) (cid:104) g Z ,u R eff , pp (cid:105) (cid:104) g Z ,d R eff , pp (cid:105) (cid:104) g Z ,(cid:96) R eff , pp (cid:105)(cid:104) g Z ,ψ eff (cid:105) C HB ˜ C HW B -0.21/0.39 0.10/-0.19 0.31/-0.58˜ C HD C HD ˜ C (6) Hψ -0.83/-0.19 -0.83/-0.19 -0.83/-0.19˜ C HD ˜ C HW B C HD (cid:104) g Z ,ψ eff (cid:105) C (6) Hψ ) C HW B ˜ C (6) Hψ -0.69/0.58 -0.69/0.58 -0.69/0.58˜ C (6) Hψ (cid:104) g Z ,ψ eff (cid:105) -6.7/-5.8 13/12 4.5/3.9˜ C HW B (cid:104) g Z ,ψ eff (cid:105) C HW ˜ C HW B -0.21/0.39 0.10/-0.19 0.31/-0.58˜ C (8) HD -0.014/0.026 0.0069/-0.013 0.021/-0.040˜ C (8) HD, -0.21/0.026 0.10/-0.013 0.31/-0.040˜ C (8) Hψ C (8) HW, -0.38/0 0.19/0 0.58/0˜ C (8) HW B -0.10/0.19 0.051/-0.097 0.15/-0.29 δG (8) F -0.078/0.15 0.039/-0.075 0.12/-0.22( ˜ C (6) HW B ) Table 6 . The effective couplings expanded to O ( v / Λ ) in each input parameter scheme. δG (8) F andthe remaining operator forms are defined in Ref. [15, 16]. (cid:104) g Z ,ψ eff , pp (cid:105) is understood to be (cid:104) g Z ,ψ eff , pp (cid:105) O ( v / Λ ) in the left most column. Reported is the numerical coefficient multiplying each SMEFT correction. p = { , , } is a flavour index. We have eliminated δG (6) F in favor of introducing (cid:104) g Z ,ψ eff , pp (cid:105) O ( v / Λ ) inthese expressions. The structure of the corrections in Eq. (4.16) in the O ( v / Λ ) expansion makes a numberof points. The numerical coefficients of the corrections in the first and second line are identical.This is due to both of these terms coming about due to linear interference with the SMamplitude, and is consistent with a naive expectation that corrections in the SMEFT follow anumerical pattern of the form n (cid:0) x + x + · · · (cid:1) , with n a numerical coefficient, and x a powercounting expansion. The first three lines follow the pattern expected from chiral symmetry,as we are neglecting light fermion masses. The full O ( v / Λ ) result includes the last fourlines. Note the fact that g ψ L eff × g ψ (cid:48) R eff interference terms are present in the full result and this isnot inconsistent with chiral symmetry at second order in the SMEFT expansion even thoughwe are neglecting light quark masses. Here ratios of observables are considered.The SMEFT is useful so long as there is at least one small power counting expansion– 11 –MEFT corrections in the { ˆ m W , ˆ m Z , ˆ G F } / { ˆ α, ˆ m Z , ˆ G F } scheme O ( v Λ ) (cid:104) g Z ,u L eff , pp (cid:105) (cid:104) g Z ,d L eff , pp (cid:105) (cid:104) g Z ,(cid:96) L eff , pp (cid:105) (cid:104) g Z ,ν L eff , pp (cid:105)(cid:104) g Z ,ψ eff (cid:105) -5.8/-0.92 4.8/1.9 7.3/0.40 -4.1/-4.1˜ C HB C HW B -0.21/0.39 0.10/-0.19 0.31/-0.58˜ C HD -0.0073/-0.073 0.0060/0.060 0.084/0.086 -0.046/-0.046˜ C HD ˜ C (6) Hψ C HD ˜ C HW B C HD (cid:104) g Z ,ψ eff (cid:105) -0.97/0.50 -0.11/0.50 -2.3/0.50 0.50/0.50( ˜ C (6) Hψ ) -0.26/0.11 0.22/-0.026 0.33/-0.14 -0.19/-0.19˜ C HW B ˜ C (6) Hψ C (6) Hψ (cid:104) g Z ,ψ eff (cid:105) C HW B (cid:104) g Z ,ψ eff (cid:105) -1.6/2.3 -0.65/1.7 -3.0/2.5˜ C HW C HW B -0.21/0.39 0.10/-0.19 0.31/-0.58˜ C (8) HD C (8) HD, -0.16/0.073 0.057/-0.060 0.26/-0.086 0.046/0.046˜ C (8) Hψ C (8) HW, -0.38/0 0.19/0 0.58/0˜ C (8) HW B -0.10/0.19 0.051/-0.097 0.15/-0.29 δG (8) F C (6) HW B ) -0.081/-0.20 0.017/-0.017 0.23/0.49˜ C (6) HD ˜ C , (6) Hψ -0.088/0.19 -0.072/-0.19 0.33/-0.19 0.19/0.19˜ C (8) Hψ, -0.19/-0.19 0.19/0.19 0.19/0.19 -0.19/-0.19( ˜ C , (6) Hψ ) -0.26/0.11 0.22/-0.026 0.33/-0.14 -0.19/-0.19˜ C (6) Hψ ˜ C , (6) Hψ C (6) HW B ˜ C , (6) Hψ -0.29/0.62 0.12/-0.49 0.56/-0.62˜ C , (6) Hψ (cid:104) g Z ,ψ eff (cid:105) -2.8/-0.042 -2.3/-0.64 -3.6/0.24 -2.0/-2.0˜ C , (8) Hψ -0.19/-0.19 0.19/0.19 0.19/0.19 -0.19/-0.19 Table 7 . The effective couplings expanded to O ( v / Λ ) in each input parameter scheme. δG (8) F andthe remaining operator forms are defined in Ref. [15, 16]. (cid:104) g Z ,ψ eff , pp (cid:105) is understood to be (cid:104) g Z ,ψ eff , pp (cid:105) O ( v / Λ )in the left most column. Reported is the numerical coefficient multiplying each SMEFT correction. p = { , , } is a flavour index. We have eliminated δG (6) F in favor of introducing (cid:104) g Z ,ψ eff , pp (cid:105) O ( v / Λ ) inthese expressions. parameter. For EWPD observables, this condition is v/ Λ (cid:28)
1. All of the O ( v / Λ ) cor-rections, beyond those in the first line of Eq. (4.16) are further suppressed. This numericalfactor is absorbed into the presentation of the results via the notation ˜ C ( d ) i ≡ C ( d ) i ¯ v d − T / Λ d − .– 12 –ypically, due to the constraints of direct searches, a working hypothesis is v/ Λ (cid:46) .
1. Thenthe second to sixth lines are suppressed, and expected to be percent level corrections to theleading perturbation. Even so, accidental numerical enhancements occur. This occurs in thisobservable, note the coefficient of 56 for (cid:104) g Z ,(cid:96) L eff (cid:105) (cid:104) g Z ,(cid:96) R eff (cid:105) . Although we have shown explicitresults for the { ˆ m W , ˆ m Z , ˆ G F } scheme, these points all hold for the { ˆ α ew , ˆ m Z , ˆ G F } scheme aswell.Fundamentally, scheme dependence is very significant in the SMEFT. Expanding theeffective couplings in terms of the individual Wilson coefficients and δG (6) F can be done usingTables 3,6,7. The resulting expressions for EWPD are not transparent in interpretation andare lengthy. The dimension-eight terms are expected to be suppressed by (cid:46) − comparedto the dimension-six terms due to the power counting expansion. On the other hand, thecalculable numerical coefficients of dimension-eight terms compared to dimension-six termsare ∼ in some cases. In ¯ R (cid:96) / ˆ R SM (cid:96) , this is reflective of the numerical accident in the enhancedcoefficient of (cid:104) g Z ,(cid:96) L eff (cid:105) (cid:104) g Z ,(cid:96) R eff (cid:105) . It does not follow that EWPD offers no constraint on theSMEFT parameter space. Such numerical accidents in one observable are also expected to beless relevant once multiple measurements are combined in the SMEFT. This observation doesencourage reasonable caution on over-interpreting LEP constraints on L (6) Wilson coefficientsin naive leading-order analyses of LEP data.
To visually illustrate the O ( v / Λ ) effects, we need to assign numbers for the unknown Wilsoncoefficients. Such a numerical output requires a scheme for numerical inputs. This is trueif constraints on a UV model are studied through its matching to the SMEFT in a globalanalysis, or if the SMEFT is studied bottom up as a model-independent EFT. In the lattercase, a rough estimate of the impact of these effects is developed in this section.As a first example, we calculate the L (8) contributions to each EWPD observable relativeto the SM EWPD values, δ O i, dim − / O i, SM . The L (8) EWPD contributions are a functionof ¯ v T / Λ and the ˜ C (8) i . After choosing a Λ, we select values for the coefficients using thesame scheme as in Ref. [16]. Specifically, we draw random coefficient values according togaussian distributions with zero mean and root mean square equal to 1 for ‘tree-level’ Wilsoncoefficients and 0.01 for ‘loop-level’ Wilson coefficients as classified by Refs. [47–49]. SelectingΛ = 1 TeV, the results for the partial width ratios R (cid:96) , R b , R c for both input schemes and 5000random coefficient selections are shown in Fig. 1; the asymmetries A (cid:96)F B and A cF B are showin Fig. 2, and the remaining EWPD observables are shown in Appendix B.The resulting distributions have widths of roughly 0 .
01 for the partial width ratios and0 . L (8) contribution scales as ¯ v T / Λ . While the numerical results shown are specific to The same narrowing/widening would occur if we fixed Λ and drew the dimension eight8 coefficients from – 13 – ���� - ���� ���� ���� ���������������������������� ( δ � � ) ���� / � � � �� � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� � � ������� Λ = � ��� - ���� - ���� ���� ���� ���������������������������� ( δ � � ) ���� / � � � �� � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� α ������� Λ = � ��� - ���� - ���� ���� ���� �������������������������������� ( δ � � ) ���� / � � � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� � � ������� Λ = � ��� - ���� - ���� ���� ���� �������������������������������� ( δ � � ) ���� / � � � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� α ������� Λ = � ��� - ���� - ���� ���� ���� �������������������� ( δ � � ) ���� / � � � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� � � ������� Λ = � ��� - ���� - ���� ���� ���� �������������������� ( δ � � ) ���� / � � � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� α ������� Λ = � ��� Figure 1 . Contributions to the R (cid:96) , R b and R c EWPD from L (8) operators relative to the SM value.Here Λ = 1 TeV. The histograms are formed by selecting random values for the coefficients 5000 timesfollowing the scheme described in the text. how we chose dimension-eight coefficients, other coefficient choices can easily be tested usingthe formulae in Sec. 4. Finally, it is important to remember that the L (8) terms are only aportion of the O ( v / Λ ) contribution. However, if we repeat the simple calculation abovewith both L (6) and L (8) operators, the O ( v / Λ ) effects are correlated with the O ( v / Λ ) a thinner/fatter distribution, given that the combination C (8) i / Λ is what appears in all observables. By thislogic, the distributions for C (8) ∼ / .
01 and Λ = 1 TeV are the same as C (8) ∼ . / × − , Λ = 0 . ∼ / . , Λ = 2 TeV. – 14 – ��� - ��� ��� ��� ����������������������� ( δ � �� � ) ���� / � ��� �� � � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� � � ������� Λ = � ��� - ��� - ��� ��� ��� ��������������������������� ( δ � �� � ) ���� / � ��� �� � � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� α ������� Λ = � ��� - ��� - ��� ��� ��� ����������������������������������� ( δ � �� � ) ���� / � ��� �� � � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� � � ������� Λ = � ��� - ��� - ��� ��� ��� ��������������������������� ( δ � �� � ) ���� / � ��� �� � � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� α ������� Λ = � ��� - ��� - ��� ��� ��� ����������������������������������� ( δ � �� � ) ���� / � ��� �� � � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� � � ������� Λ = � ��� - ��� - ��� ��� ��� ����������������������� ( δ � �� � ) ���� / � ��� �� � � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� α ������� Λ = � ��� Figure 2 . Contributions to the A (cid:96)F B and A cF B EWPD from L (8) operators relative to the SM value.Here Λ = 1 TeV. The histograms are formed by selecting random values for the coefficients 5000 timesfollowing the scheme described in the text. effects. In order to make sure we are not biased by scenarios with O ( v / Λ ) effects that areexperimentally excluded, a more careful calculation is necessary.As a second example, we will zero all L (6) coefficients except for two, then calculate the χ bounds on that 2-d coefficient space with and without O ( v / Λ ) effects. Said more plainly,we want to see how the S-T analysis [50–59] (and other, less famous 2-d slices of coefficientspace) fare at O ( v / Λ ).Using the expressions in Sec. 4, we form the full χ following the procedure laid out– 15 –n Refs. [30, 60] for O ( v / Λ ) SMEFT; we take the experimental correlation matrix fromRef. [61] and assume theoretical uncertainties are completely uncorrelated. We also calculatethe χ using EWPD observables calculated to O ( v / Λ ) only.Next, we zero all L (6) Wilson coefficients except two: our first choice is to zero allcoefficients but C HW B and C HD – S and T up to normalization factors, while for our secondchoice we zero all but C (6) H(cid:96) and C HD . C (6) H(cid:96) affects the coupling of Z to leptons and istherefore assumed to be tightly constrained by LEP, hence it is an interesting candidate tostudy including O ( v / Λ ) effects. At this point, χ O ( v / Λ ) is a function of the two nonzerocoefficients and the scale Λ, while χ O ( v / Λ ) also depends on the L (8) Wilson coefficients.Rather than rely on random coefficients, we adopt a simpler approach here – setting alltree-level L (8) Wilson coefficients to 1 and all loop-level to 0.01.For a given Λ, we determine the minima of χ O ( v / Λ ) and χ O ( v / Λ ) and plot the ∆ χ contours in Fig. 3. The green, yellow, grey regions correspond to the 68% ,
95% and 99 . χ O ( v / Λ ) . The regions correspondto χ = χ min + ∆ χ with ∆ χ = 2 .
30 (1 σ , green), 6 .
18 (2 σ ,yellow), 11 .
83 (3 σ , grey) definedvia the Cumulative Distribution function for a two-parameter fit. The same ∆ χ regions areshown in red for the fit using χ O ( v / Λ ) (inner contour is 68% CL, intermediate is 95% andthe outer is 99 .
9% CL).The difference between the contours shows the effect of going from O ( v / Λ ) to O ( v / Λ ).For Λ = 1 TeV, the shift is striking, to the extent that the different order contours don’t evenoverlap for the C HD − C (6) H(cid:96) case. The effect is smaller for Λ = 2 TeV, unsurprising giventhat the difference between the contours scales as v / Λ , but it is not negligible. Of course,the details of how the fit shifts depends strongly on our treatment of the dimension-eightoperators, so these results should be viewed as qualitative. However, we emphasize that thechoice of 1 / .
01 for tree/loop level dimension-eight coefficients was made for simplicity andnot to amplify the effect. Repeating the study using random coefficients for the dimension-eight coefficients (following the procedure used earlier in the section), we observe a wide rangeof shifts, from significantly smaller to significantly larger than what is shown in Fig. 3.
The expressions for EWPD to dimension eight in the SMEFT are lengthy. While a bottom upanalysis in the SMEFT is reported in Section 5, it is also useful to examine some cases wheremodels are matched to dimension eight, and EWPD constraints are studied. Restricting toUV models with few parameters allows the results to be visually represented. In the followingsections we explore two such UV models – the U(1) model developed to dimension eight inmatching in Ref. [16], and a model containing a scalar triplet. The details of matching thetriplet model to dimension eight can be found in Appendix A.– 16 – ��� - ��� ��� ��� ��� - ��� - ��� - ������������ � �� � ��� Λ = � ��� - ��� - ��� - ��� ��� ��� ��� - ��� - ��� - ������������ � �� � ��� Λ = � ��� - ���� - ���� - ���� - ���� - ���� ���� ���� ���� - ���� - ���� - ���� - ������������ � �� � ( � ) �� Λ = � ��� - ��� - ��� ��� ��� ��� - ���� - ���������������� � �� � ( � ) �� Λ = � ��� Figure 3 . The green/yellow/gray contours correspond to the 68% / / .
9% CL two parame-ter fit determined by ∆ χ O ( v / Λ ) , while the red rings correspond to the same CL determined using∆ χ O ( v / Λ ) . In the top panels the free parameters are C HD and C HW B , while in the bottom panelsthe free parameters are C HD and C (6) H(cid:96) . Note that the axes ranges vary from panel to panel. In theleft panels, we have taken the scale Λ = 1 TeV, while in the right panels Λ = 2 TeV. All calculationsuse the ˆ m W scheme. U(1) kinetic mixing
In this model, a heavy U(1) gauge boson K µ with Stueckelberg mass [62] m K kineticallymixes with B µ , the U(1) Y gauge boson in the SM. The SM Lagrangian is extended with theUV Lagrangian ∆ L = − K µν K µν + 12 m K K µ K µ − k B µν K µν , (6.1)where the field strength is K µν = ∂ µ K ν − ∂ ν K µ . Integrating out the heavy K µ field, thematching pattern in the SMEFT, with geoSMEFT operator form conventions, is given inTable 8 and Table 9. This weakly coupled, renormalizable model has one scale and onecoupling, but its matching pattern does not follow the pattern claimed to follow from aUV of this form in some literature. The matching pattern is consistent with the results ofRef. [16, 47–49]. – 17 – ψ DC , (6) H(cid:96) − y (cid:96) g m K b C (6) He − y e g m K b C , (6) Hq − y q g m K b C (6) Hu − y u g m K b C (6) Hd − y d g m K b H D C (6) H (cid:3) − g k m K C (6) HD − g k m K ψ : ( ¯ LL )( ¯ LL ) C (6) (cid:96)(cid:96) − g k m K C , (6) qq − g k m K C , (6) (cid:96)q g k m K ψ : ( ¯ RR )( ¯ RR ) C (6) ee − g k m K C (6) uu − g k m K C (6) dd − g k m K C (6) eu g k m K C (6) ed − g k m K C , (6) ud g k m K ψ : ( ¯ LL )( ¯ RR ) C (6) (cid:96)e − g k m K C (6) (cid:96)u g k m K C (6) (cid:96)d − g k m K C (6) qe g k m K C , (6) qu − g k m K C , (6) qd g k m K Table 8 . L (6) matching coefficients in the U(1) model [16], here b = k − λ ( k − k ) ¯ v T m K . Ournotation is such that y i is the hypercharge of field i . H ψ DC , (8) H(cid:96) y (cid:96) g m K k − g y (cid:96) m K ( k − k )(2 λ + g + g ) C , (8) He y e g m K k − g y e m K ( k − k )(2 λ + g + g ) C , (8) Hq y q g m K k − g y q m K ( k − k )(2 λ + g + g ) C , (8) Hu y u g m K k − g y u m K ( k − k )(2 λ + g + g ) C , (8) Hd y d g m K k − g y d m K ( k − k )(2 λ + g + g ) C , (8) H(cid:96) − g g m K ( k − k ) C , (8) Hq − g g m K ( k − k ) C , (8) H(cid:96) − g g m K ( k − k ) C , (8) Hq − g g m K ( k − k ) H D C (8) H,D g k m K − g g m K ( k − k ) C (8) HD g k m K − g g m K ( k − k ) X H C (8) HB − g m K ( k − k ) C (8) HW g g m K ( k − k ) Table 9 . Matching coefficients onto operators in L (8) [16]. In the U(1) model, in addition to thesematching contributions, there are four-fermion operators and four-point contributions. In the SM we have the leading-order effective couplings given in Ref. [16]. Using the– 18 –esults of Ref. [16] and this work, the O ( v / Λ ) corrections are: (cid:104) g Z ,u R eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = − [0 . / . k ¯ v T m K + [0 . / . λ ( k − k ) v ¯ v T m K , (cid:104) g Z ,d R eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [0 . / . k ¯ v T m K − [0 . / . λ ( k − k ) v ¯ v T m K , (cid:104) g Z ,(cid:96) R eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [0 . / . k ¯ v T m K − [0 . / . λ ( k − k ) v ¯ v T m K , (6.2) (cid:104) g Z ,u L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [0 . / − . k ¯ v T m K + [0 . / . λ ( k − k ) v ¯ v T m K , (cid:104) g Z ,d L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / + 0 . k ¯ v T m K + [0 . / . λ ( k − k ) v ¯ v T m K , (cid:104) g Z ,(cid:96) L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / . k ¯ v T m K − [0 . / . λ ( k − k ) v ¯ v T m K , (cid:104) g Z ,ν L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [0 . / . k ¯ v T m K − [0 . / . λ ( k − k ) v ¯ v T m K ,, (6.3)where the first value in square brackets is the value in the ˆ m W scheme and the second is theˆ α ew scheme value. The λ that appears is the Higgs quartic coupling, which arises becausethe Higgs EOM is needed to massage the operators one gets from integrating out K µ into thegeoSMEFT basis. The O ( v / Λ ) corrections are: (cid:104) g Z ,u R eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [3 . / − . − k ¯ v T m K + [ − . / . − k ¯ v T m K − [3 . / . − λ ( k − k ) v ¯ v T m K , (cid:104) g Z ,d R eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / . − k ¯ v T m K + [1 . / − . − k ¯ v T m K + [1 . / . − λ ( k − k ) v ¯ v T m K , (cid:104) g Z ,(cid:96) R eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / . − k ¯ v T m K + [5 . / − . − k ¯ v T m K + [4 . / . − λ ( k − k ) v ¯ v T m K , (6.4) (cid:104) g Z ,u L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [3 . / − . − k ¯ v T m K − [3 . / − . − k ¯ v T m K − [7 . / . − λ ( k − k ) v ¯ v T m K , (cid:104) g Z ,d L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / . − k ¯ v T m K + [1 . / − . − k ¯ v T m K − [7 . / . − λ ( k − k ) v ¯ v T m K , (cid:104) g Z ,(cid:96) L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / . − k ¯ v T m K + [5 . / − . − k ¯ v T m K + [2 . / . − λ ( k − k ) v ¯ v T m K , (cid:104) g Z ,ν L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = − [2 . / . − k ¯ v T m K + [1 . / . − k ¯ v T m K + [2 . / . − λ ( k − k ) v ¯ v T m K , (6.5)– 19 –ote that the λ dependence, which is a basis dependent artifact in this matching, cancelsexactly between the O ( v / Λ ) and O ( v / Λ ) contributions to the effective couplings (e.g.when one adds Eq. (6.2), (6.3) to Eq. (6.4), (6.5)). These terms come about due to correlatedmatching at L (6) , and L (8) in the SMEFT in both models. This occurs quite generally,due to the presence of the classical dimensionful parameter, SM Higgs vev v in the EFT.Matching contributions that are naively assumed restricted to L (8) corrections descend downin mass dimension to give matching contributions to L (6) Wilson coefficients. These matchingcontributions can be overlooked until matching results are developed to L (8) and are anexample of the intrinsic ambiguity in a L (6) SMEFT treatment related to higher-order termsin the power counting expansion.This cancelation in λ dependence in observable quantities has an important implication.Quantities such as the Z effective couplings, and subsequently the amplitudes they define arenot exact in the SMEFT, but are only defined order by order systematically in 1 / Λ. In thiscase an ambiguity is present of order ¯ v T / Λ in the matching to L (6) . This leads to an intrinsicambiguity in an amplitude that depends on C (6) i parameters of order 1 / Λ . The square of theSM perturbed with such a correction is then ambiguous and not precisely defined at order1 / Λ . Due to the classical presence of a parameter in the theory carrying mass dimension,the SM Higgs vev, all contributions to observables at each order in 1 / Λ are required to obtainbasis independent and well-defined results in an observable. Although we have discussed thispoint considering L (8) corrections leading to matching ambiguities in L (6) operators of order1 / Λ , the same effect is present for all higher order L (6+2 n ) matching corrections with n > Z was reported in Ref. [16] in each input parameter scheme. These numerical resultsdiffer in the last significant digit compared to Ref. [16]. This is due to the use of updated SMpredictions produced and reported in Table 1 and numerical approximations differing in thiswork. The total width is (cid:80) ψ ¯Γ SMEFT , ˆ α ew Z→ ¯ ψ p ψ p (cid:80) ψ ˆΓ SM , ˆ α ew Z→ ¯ ψ p ψ p = 1 + 4 . × − ¯ v T k m K − . × − ( k − . k ) ¯ v T m K , (6.6) (cid:80) ψ ¯Γ SMEFT , ˆ m W Z→ ¯ ψ p ψ p (cid:80) ψ ˆΓ SM , ˆ m W Z→ ¯ ψ p ψ p = 1 − . × − ¯ v T k m K + 7 . × − ( k − . k ) ¯ v T m K . (6.7)The remaining EWPD observables, with numerical values ordered in the [ ˆ m W , α EW ]schemes, are¯ R SMEFT c ˆ R SM c = [2 . / × − k ¯ v T m K + [0 . / . k ¯ v T m K + [ − . / . k ¯ v T m K , (6.8)– 20 – R SMEFT b ˆ R SM b = − [4 . / × − k ¯ v T m K − [1 . / . × − k ¯ v T m K + [1 . / − . × − k ¯ v T m K , (6.9)¯ R SMEFT (cid:96) ˆ R SM (cid:96) = [7 . / × − k ¯ v T m K + [4 . / − . × − k ¯ v T m K + [ − . / . × − k ¯ v T m K , (6.10)(¯ σ had ) SMEFT (ˆ σ had ) SM = − [2 . / × − k ¯ v T m K − [8 . / . × − k ¯ v T m K + [8 . / × − k ¯ v T m K , (6.11)( ¯ A ,cF B ) SMEFT ( ˆ A ,cF B ) SM = [ − . × − / . k ¯ v T m K + [0 . / − . k ¯ v T m K + [ − . / . k ¯ v T m K , (6.12)( ¯ A ,bF B ) SMEFT ( ˆ A ,bF B ) SM = [ − . × − / . k ¯ v T m K + [0 . / − . k ¯ v T m K + [ − . / . k ¯ v T m K , (6.13)( ¯ A ,(cid:96)F B ) SMEFT ( ˆ A ,(cid:96)F B ) SM = [ − . × − / . k ¯ v T m K + [0 . / . k ¯ v T m K + [ − . / . k ¯ v T m K . (6.14)There are several generic expectations on how SMEFT constraints projected onto a spe-cific UV model come about in a global analysis. Input parameter scheme dependence, andthe effects of L (8) corrections are both expected to be significant in some observables due tothe decoupling theorem and numerical accidents. Both effects are expected to be reduced asmore observables are consistently combined in a global SMEFT fit, and this expectation isborn out in the models we study in this section. In the case of the U(1) kinetic mixing model,the results for each observable are shown in Figs. 4,5,6,7,8. The plots show m K in the range[350 , n C (4) H κ m (cid:16) − v λm (cid:17) + v λ κ m C (6) H (4 λ − η ) κ m − κ +2 v λ (4 λ − η )) m + v λκ m C (8) H λ − η ) κ m + (40 η − λ − λ Φ ) κ m + κ m H n +1 ψ C (6) ψH y ψ κ m (cid:16) − v λm (cid:17) C (8) ψH y ψ κ m (cid:16) λ − η − κ m (cid:17) H D C (6) H (cid:3) κ m (cid:16) − v λm (cid:17) C (6) HD − κ m (cid:16) − v λm (cid:17) H D C (8) HD − κ m C (8) HD η − λ ) κ m + κ m Table 10 . Matching for the electroweak triplet scalar up to L (8) . These matching results are consistentwith geoSMEFT conventions on operator forms and are sufficient to examine observables constructedvia two and three point contributions. shows the largest effects at low mass scales. We make these same choices in the scalar tripletmodel. Larger suppression scales are considered in the previous section. Note the large x axis in the case of the A c,bF B constraints at L (6) in the m W scheme. This isdue to an accidental numerical suppression of the L (6) perturbations in these observables. Forsuch large coupling values, perturbation theory is breaking down, so the axis was expandedsimply to illustrate the allowed parameter space.The effect of the L (8) corrections in the U(1) model damps in the allowed parameter spacewhen all LEP observables are combined, as expected. Input parameter scheme dependenceis reduced as observables are combined, but remains even in the global fit combination.This is driven by the accidental numerical scheme dependence in the observables ( A c,bF B ) thatintroduce the largest pulls in the EWPD fit. In this model, we introduce a scalar Φ which is an electroweak triplet with zero hyperchange.The SM Lagrangian is then augmented by the Lagrangian for the new heavy scalar, L Φ = 12 ( D µ Φ a ) − m Φ a Φ a + 2 κH † τ a H Φ a − η ( H † H )Φ a Φ a − λ Φ (Φ a Φ a ) . (6.15)We now integrate out the heavy triplet scalar, with the details given in Appendix A. Theresulting matching pattern for the L (6) and L (8) Wilson coefficients in the SMEFT is givenin Table 10. For the effective couplings we find: The lower bound on m K > ¯ v T is imposed in the global minimum found to define the best fit region as aprior assumption. – 22 – g Z ,u R eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [0 . / − . κ ¯ v T m + [ − . / . λ κ v ¯ v T m , (cid:104) g Z ,d R eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / . κ ¯ v T m + [0 . / − . λ κ v ¯ v T m , (cid:104) g Z ,(cid:96) R eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / . κ ¯ v T m + [2 . / − . λ κ v ¯ v T m , (6.16) (cid:104) g Z ,u L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [0 . / − . κ ¯ v T m + [ − . / . λ κ v ¯ v T m , (cid:104) g Z ,d L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / . κ ¯ v T m + [0 . / − . λ κ v ¯ v T m , (cid:104) g Z ,(cid:96) L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / . κ ¯ v T m + [1 . / − . λ κ v ¯ v T m , (cid:104) g Z ,ν L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / − . κ ¯ v T m + [0 . / . λ κ v ¯ v T m , (6.17)with O ( v / Λ ) corrections: (cid:104) g Z ,u R eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / . κ ¯ v T m + [ − . / . η κ ¯ v T m + [1 . / − . λ κ v ¯ v T m , (cid:104) g Z ,d R eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [0 . / − . κ ¯ v T m + [0 . / − . η κ ¯ v T m + [ − . / . λ κ v ¯ v T m , (cid:104) g Z ,(cid:96) R eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [2 . / − . κ ¯ v T m + [1 . / − . η κ ¯ v T m + [ − . / . λ κ v ¯ v T m , (6.18) (cid:104) g Z ,u L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [ − . / . κ ¯ v T m + [ − . / . η κ ¯ v T m + [1 . / − . λ κ v ¯ v T m , (cid:104) g Z ,d L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [0 . / − . κ ¯ v T m + [0 . / − . η κ ¯ v T m + [ − . / . λ κ v ¯ v T m , (cid:104) g Z ,(cid:96) L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [1 . / − . κ ¯ v T m + [0 . / − . η κ ¯ v T m + [ − . / . λ κ v ¯ v T m , (cid:104) g Z ,ν L eff , pp (cid:105) [ ˆ m W / ˆ α ew ] O ( v / Λ ) = [0 . / . κ ¯ v T m + [0 . / . η κ ¯ v T m + [ − . / − . λ κ v ¯ v T m , (6.19)Again we note that the λ dependence exactly cancels between matching effects at L (6) and L (8) in observables. This occurs at the effective coupling level, as the on shell Z effectivecouplings are closely related to observable quantities.– 23 –he total width is (cid:80) ψ ¯Γ SMEFT , ˆ α ew Z→ ¯ ψ p ψ p (cid:80) ψ ˆΓ SM , ˆ α ew Z→ ¯ ψ p ψ p = 1 + 1 . κ ¯ v T m − . η κ ¯ v T m − . κ ¯ v T m , (6.20) (cid:80) ψ ¯Γ SMEFT , ˆ m W Z→ ¯ ψ p ψ p (cid:80) ψ ˆΓ SM , ˆ m W Z→ ¯ ψ p ψ p = 1 + 0 . κ ¯ v T m − . η κ ¯ v T m + 1 . κ ¯ v T m . (6.21)The remaining EWPD observables, with numerical values ordered in the [ ˆ m W , α EW ]schemes, are¯ R SMEFT c ˆ R SM c = [ − . / . κ ¯ v T m + [0 . / − . η κ ¯ v T m + [2 . / − . κ ¯ v T m , (6.22)¯ R SMEFT b ˆ R SM b = [0 . / − . κ ¯ v T m + [ − . / . η κ ¯ v T m + [ − . / − . κ ¯ v T m , (6.23)¯ R SMEFT (cid:96) ˆ R SM (cid:96) = [ − . / . κ ¯ v T m + [0 . / − . η κ ¯ v T m + [ − . / − . κ ¯ v T m , (6.24)(¯ σ had ) SMEFT (ˆ σ had ) SM = [ − . / − . κ ¯ v T m + [0 . / . η κ ¯ v T m + [7 . / . κ ¯ v T m , (6.25)( ¯ A ,cF B ) SMEFT ( ˆ A ,cF B ) SM = [ − / κ ¯ v T m + [63 / − η κ ¯ v T m + [190 / − κ ¯ v T m , (6.26)( ¯ A ,bF B ) SMEFT ( ˆ A ,bF B ) SM = [ − / κ ¯ v T m + [57 / − η κ ¯ v T m + [85 / − κ ¯ v T m , (6.27)( ¯ A ,(cid:96)F B ) SMEFT ( ˆ A ,(cid:96)F B ) SM = [ − / κ ¯ v T m + [110 / − η κ ¯ v T m + [950 / κ ¯ v T m . (6.28)The effect of the L (8) corrections in the scalar triplet model also damp in significanceas more observables are consistently combined in the SMEFT into a global fit. Once again,significant input parameter scheme dependence remains in the global fit combination. Theresults are shown for the value η = 0 .
1, and are simply illustrative. The same numericalbehavior is present for other values of η . – 24 – Discussion and Conclusions
In this paper we have developed and reported the first analysis of EWPD in the SMEFTto dimension eight. This result was enabled by the geoSMEFT formulation of the SMEFTreported in Refs. [15, 16]. The interpretation of EWPD in the SMEFT has been subject tosignificant controversy over the years. A cautious interpretation of EWPD has been advocatedin some works, when determining constraints on L (6) parameters in the SMEFT, due to theneglect of loop corrections, and dimension-eight operator effects in a leading order SMEFTanalysis of LEP data. More aggressive interpretations of LEP data in the SMEFT havealso been advanced. All of the unknown corrections leading to past differences of opinionare calculable in a well-defined formulation of the SMEFT. Recently, loop corrections forLEP observables in the SMEFT have been reported in Refs. [63–65]. In this paper we havereported the dimension-eight corrections. The key point is that such calculable correctionsintroduce more parameters into the predictions of LEP observables, compared to the numberof parameters present in a naive L (6) SMEFT analysis. We encourage the reader to draw theirown conclusion on how the interpretation of LEP EWPD is impacted by the dimension-eightcontributions that are now known, and reported in this work.
Acknowledgments
We thank Chris Hays for insightful discussions. M.T. acknowledges support from the VillumFund, project number 00010102. T.C. acknowledges funding from European Union’s Horizon2020 research and innovation programme under the Marie Sklodowska-Curie grant agreementNo. 890787. The work of A.M. is partially supported by the National Science Foundationunder Grant No. Phy-1230860. A.H. is supported by the U.S. Department of Energy (DOE)under Award Number DE-SC0011632 and by the Walter Burke Institute for TheoreticalPhysics.
A Matching the electroweak triplet scalar model
Matching to dimension six of the real scalar triplet model has been considered in Refs. [66–68]. We follow the notation of Ref. [66] and extend the matching results to dimension eight.We can write the Lagrangian in Eq. (6.15) in a different form, using vector notation, L = 12 (cid:126) Φ T (cid:0) P − m − U (cid:1) (cid:126) Φ + (cid:126) Φ · (cid:126)B − λ Φ ( (cid:126) Φ · (cid:126) Φ) , (A.1)where U = 2 η ( H † H ), (cid:126)B = 2 κH † (cid:126)τ H , and P µ = iD µ . The normalization of the SU(2) L matrices is τ a = σ a /
2, where σ a are the Pauli matrices. The equation of motion is (cid:126) Φ c = − P − m − U (cid:126)B + 1 P − m − U λ Φ ( (cid:126) Φ c · (cid:126) Φ c ) (cid:126) Φ c . (A.2)– 25 –lugging this back into the Lagrangian, we find that L = 12 (cid:126) Φ Tc (cid:0) P − m − U (cid:1) (cid:126) Φ c + (cid:126) Φ c · (cid:126)B − λ Φ ( (cid:126) Φ c · (cid:126) Φ c ) = 12 m (cid:126)B · (cid:126)B + 12 (cid:126)B T m ( P − U ) 1 m (cid:126)B + 12 (cid:126)B T m ( P − U ) 1 m ( P − U ) 1 m (cid:126)B − m λ Φ ( (cid:126)B · (cid:126)B ) + O ( κ /m ) . (A.3)We simplify the operators in the expansion. The Lagrangian involving the Higgs field is L H =( D µ H † )( D µ H ) − λ (cid:18) H † H − v (cid:19) − H † Y − Y † H + a O (4) H + a O (6) H + a O (6) H (cid:3) + a O (6) HD + b ( H † H )( (cid:3) H † H + H † (cid:3) H ) + a O (8) H + a O (8) HD + a O (8)4 DH, + a O (8)4 HD, + b ( H † H ) ( (cid:3) H † H + H † (cid:3) H ) + b ( (cid:3) H † H + H † (cid:3) H ) + b ( D µ H † D µ H )( (cid:3) H † H + H † (cid:3) H ) + b ( H † H )( (cid:3) H † (cid:3) H ) + b ( H † (cid:3) H )( (cid:3) H † H )+ b (cid:104) ( D µ H † (cid:3) H )( H † D µ H ) + ( D µ H † H )( (cid:3) H † D µ H ) (cid:105) . (A.4)where a = κ m , a = − ηκ m , a = κ m , a = − κ m ,a = (cid:18) η κ m − κ m λ Φ (cid:19) , a = 4 ηκ m , a = − κ m , a = 4 κ m ,b = − κ m , b = 2 ηκ m , b = κ m , b = − κ m ,b = 2 κ m , b = − κ m , b = 4 κ m , (A.5)and O (8)4 DH, =( D µ H † D µ H )( D ν H † D ν H ) , (A.6) O (8)4 DH, =( D µ H † D ν H )( D ν H † D µ H ) . (A.7)Here we have written the Yukawa terms compactly as − H † Y + h . c . We need to remove the higher-derivative operators to find a suitable form of the La-grangian. This we achieve by redefining the Higgs field as H → H + b ( H † H ) H + O (8) , (A.8)where O (8) = b ( (cid:3) H † H + H † (cid:3) H ) H + b ( D µ H † D µ H ) H + 12 b ( H † H ) (cid:3) H + 12 b ( H † (cid:3) H ) H + b ( D µ H † H ) D µ H + d v λ ( H † H ) H + d ( H † H ) H − b ( Y † H + H † Y ) H − b ( H † H ) Y − b ( H † Y ) H, (A.9)– 26 –nd d = (cid:18) b − b + 12 ( b + b ) (cid:19) , (A.10) d = (cid:18) b + 4 b + 4 a b − a b + 2( a − λ ) (cid:18) b + 12 b + 12 b (cid:19)(cid:19) . (A.11)The final result for the matching (contributing to three-point couplings) is given in Table 10. B Additional EWPD, coefficient scan
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Santiago, Effective description of generalextensions of the Standard Model: the complete tree-level dictionary , JHEP (2018) 109[ ]. – 30 – � - � - � � � � ������������� � � � � ( � ) ����� �� Γ � ����������� α ������ - � - � - � � � � ������������� � � � � ( � ) ����� �� Γ � ����������� α ������ - � - � - � � � � ������������� � � � � ( � ) ����� �� Γ � ����������� � � ������ - � - � - � � � � ������������� � � � � ( � ) ����� �� Γ � ����������� � � ������ - � - � � � ������������� � � � � ( � ) ����� �� � � ����������� α ������ - � - � � � ������������� � � � � ( � ) ����� �� � � ����������� α ������ - � - � � � ������������� � � � � ( � ) ����� �� � � ����������� � � ������ - � - � � � ������������� � � � � ( � ) ����� �� � � ����������� � � ������ Figure 4 . Constraints from EWPD observables in the U(1) mixing model. The results are organisedso that increasing the precision of the theoretical predication from O ( v / Λ ) to O ( v / Λ ) from left toright. Both the α and m W schemes results are shown, and individual observables carry a significantscheme dependence. Shown are the constraints on the model space from the Γ Z and R (cid:96) observables. – 31 – � - � � � ������������� � � � � ( � ) ����� �� � � ����������� α ������ - � - � � � ������������� � � � � ( � ) ����� �� � � ����������� α ������ - �� - �� - �� � �� �� �������������� � � � � ( � ) ����� �� � � ����������� � � ������ - � - � � � ������������� � � � � ( � ) ����� �� � � ����������� � � ������ - � - � � � ������������� � � � � ( � ) ����� �� � � ����������� α ������ - � - � � � ������������� � � � � ( � ) ����� �� � � ����������� α ������ - �� - �� - � � � �� �������������� � � � � ( � ) ����� �� � � ����������� � � ������ - � - � � � ������������� � � � � ( � ) ����� �� � � ����������� � � ������ Figure 5 . Shown are the constraints on the model space from the R c and R b EWPD observables. – 32 – ��� - ��� ��� ��� ��������������� � � � � ( � ) ����� �� � �� � ����������� α ������ - ��� - ��� ��� ��� ��������������� � � � � ( � ) ����� �� � �� � ����������� α ������ - �� - �� � �� �������������� � � � � ( � ) ����� �� � �� � ����������� � � ������ - � - � � � ������������� � � � � ( � ) ����� �� � �� � ����������� � � ������ - � - � � � ������������� � � � � ( � ) ����� �� � �� � ����������� α ������ - � - � � � ������������� � � � � ( � ) ����� �� � �� � ����������� α ������ - ��� - �� � �� ��������������� � � � � ( � ) ����� �� � �� � ����������� � � ������ - � - � � � ������������� � � � � ( � ) ����� �� � �� � ����������� � � ������ Figure 6 . Shown are the constraints on the model space from the A (cid:96)F B and A cF B EWPD observables. – 33 – ��� - ��� ��� ��� ��������������� � � � � ( � ) ����� �� � �� � ����������� α ������ - ��� - ��� ��� ��� ��������������� � � � � ( � ) ����� �� � �� � ����������� α ������ - ��� - �� � �� ��������������� � � � � ( � ) ����� �� � �� � ����������� � � ������ - � - � � � ������������� � � � � ( � ) ����� �� � �� � ����������� � � ������ - � - � � � ������������� � � � � ( � ) ����� �� σ ��� ����������� α ������ - � - � - � � � � ������������� � � � � ( � ) ����� �� σ ��� ����������� α ������ - � - � � � ������������� � � � � ( � ) ����� �� σ ��� ����������� � � ������ - � - � - � � � � ������������� � � � � ( � ) ����� �� σ ��� ����������� � � ������ Figure 7 . Shown are the constraints on the model space from the A bF B and σ had EWPD observables. – 34 – ��� - ��� ��� ��� ��������������� � � � � ( � ) ����� �� ����� α ������ - ��� - ��� ��� ��� ��������������� � � � � ( � ) ����� �� ����� α ������ - ��� - ��� ��� ��� ��������������� � � � � ( � ) ����� �� ����� � � ������ - ��� - ��� ��� ��� ��������������� � � � � ( � ) ����� �� ����� � � ������ Figure 8 . Combined constraints from the full set of EWPD LEP observables in the U(1) mixingmodel. – 35 – ��� - ��� - ��� ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� Γ � ����������� α ������ - ��� - ��� - ��� ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� Γ � ����������� α ������ - ��� - ��� - ��� ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� Γ � ����������� � � ������ - ��� - ��� - ��� ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� Γ � ����������� � � ������ - � - � - � � � � ������������� κ / � � ϕ ������� ����� �� � � ����������� α ������ - � - � - � � � � ������������� κ / � � ϕ ������� ����� �� � � ����������� α ������ - � - � - � � � � ������������� κ / � � ϕ ������� ����� �� � � ����������� � � ������ - � - � - � � � � ������������� κ / � � ϕ ������� ����� �� � � ����������� � � ������ Figure 9 . Constraints from EWPD observables in the scalar triplet model. Same conventions as inprevious plots. Shown are the constraints on the model space from the Γ Z and R (cid:96) observables. – 36 – � - � � � ������������� κ / � � ϕ ������� ����� �� � � ����������� α ������ - � - � � � ������������� κ / � � ϕ ������� ����� �� � � ����������� α ������ - � - � � � ������������� κ / � � ϕ ������� ����� �� � � ����������� � � ������ - � - � � � ������������� κ / � � ϕ ������� ����� �� � � ����������� � � ������ - � - � � � ������������� � � � ������� ����� �� � � ����������� α ������ - � - � � � ������������� κ / � � ϕ ������� ����� �� � � ����������� α ������ - � - � � � ������������� κ / � � ϕ ������� ����� �� � � ����������� � � ������ - � - � � � ������������� κ / � � ϕ ������� ����� �� � � ����������� � � ������ Figure 10 . Shown are the constraints on the model space from the R c and R b EWPD observables. – 37 – ��� - ��� - ��� ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� � �� � ����������� α ������ - ��� - ��� - ��� ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� � �� � ����������� α ������ - ��� - ��� - ��� ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� � �� � ����������� � � ������ - ��� - ��� - ��� ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� � �� � ����������� � � ������ - � - � � � ������������� κ / � � ϕ ������� ����� �� � �� � ����������� α ������ - � - � � � ������������� κ / � � ϕ ������� ����� �� � �� � ����������� α ������ - � - � � � ������������� κ / � � ϕ ������� ����� �� � �� � ����������� � � ������ - � - � � � ������������� κ / � � ϕ ������� ����� �� � �� � ����������� � � ������ Figure 11 . Shown are the constraints on the model space from the A (cid:96)F B and A cF B EWPD observables. – 38 – ��� - ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� � �� � ����������� α ������ - ��� - ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� � �� � ����������� α ������ - ��� - ��� ��� ��� ��������� ��� ��� κ / � � ϕ ������� ����� �� � �� � ����������� � � ������ - ��� - ��� ��� ��� ��������� ��� ��� κ / � � ϕ ������� ����� �� � �� � ����������� � � ������ - ��� - ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� σ ��� ����������� α ������ - ��� - ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� σ ��� ����������� α ������ - ��� - ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� σ ��� ����������� � � ������ - ��� - ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� σ ��� ����������� � � ������ Figure 12 . Shown are the constraints on the model space from the A bF B and σ had EWPD observables. – 39 – ��� - ��� - ��� ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� ����� α ������ - ��� - ��� - ��� ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� ����� α ������ - ��� - ��� - ��� ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� ����� � � ������ - ��� - ��� - ��� ��� ��� ��� ��������������� κ / � � ϕ ������� ����� �� ����� � � ������ Figure 13 . Combined constraints from the full set of EWPD LEP observables in the scalar tripletmodel with η = 0 . – 40 – ���� - ���� ���� ���� �������������������������������� ( δΓ � ) ���� / Γ � � �� � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� � � ������� Λ = � ��� - ���� - ���� ���� ���� �������������������������������� ( δΓ � ) ���� / Γ � � �� � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� α ������� Λ = � ��� - ���� - ���� ���� ���� �������������������� ( δσ ��� ) ���� / σ ���� �� � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� � � ������� Λ = � ��� - ���� - ���� ���� ���� �������������������� ( δσ ��� ) ���� / σ ���� �� � � �� �� ��� � � � � � � � � � � � � � � � �� �� � ��� - � ����� α ������� Λ = � ��� Figure 14 . Contributions to the Γ Z and σ had from dimension-eight operators relative to the SMvalue. Here Λ = 1 TeV. The histograms are formed from selecting random values for the coefficients5000 times following the scheme described in the text.from dimension-eight operators relative to the SMvalue. Here Λ = 1 TeV. The histograms are formed from selecting random values for the coefficients5000 times following the scheme described in the text.