Constraints on the trilinear Higgs self coupling from precision observables
RRM3-TH/17-1
Constraints on the trilinear Higgs self couplingfrom precision observables
G. Degrassi a , M. Fedele b , P.P. Giardino c (a) Dipartimento di Matematica e Fisica, Universit`a di Roma Tre andINFN, sezione di Roma Tre, I-00146 Rome, Italy(b) Dipartimento di Fisica, Universit`a di Roma “La Sapienza” andINFN, sezione di Roma, I-00185 Rome, Italy(c) Physics Department, Brookhaven National Laboratory,Upton, New York 11973, US Abstract
We present the constraints on the trilinear Higgs self couplingthat arise from loop effects in the W boson mass and the effectivesine predictions. We compute the contributions to these preci-sion observables of two-loop diagrams featuring an anomaloustrilinear Higgs self coupling. We explicitly show that the sameanomalous contributions are found if the analysis of m W andsin θ lepeff is performed in a theory in which the scalar potentialin the Standard Model Lagrangian is modified by an (in)finitetower of (Φ † Φ) n terms with Φ the Higgs doublet. We find thatthe bounds on the trilinear Higgs self coupling from precisionobservables are competitive with those coming from Higgs pairproduction. a r X i v : . [ h e p - ph ] M a y Introduction
The discovery of a new scalar resonance with a mass around 125 GeV atthe Large Hadron Collider (LHC) [1, 2] opened a new era in high-energyparticle physics. The study of the properties of this particle provides strongevidence that it is the Higgs boson of the Standard Model (SM), i.e. , a scalarCP-even state whose coupling to the other known particles has a SM-likestructure and a strength proportional to their masses [3–5]. At present,the combined analysis based on 7 and 8 TeV LHC data sets [5] shows thatthe couplings with the vector bosons are found to be compatible with thoseexpected from the SM within a ∼
10% uncertainty, while in the case of theheaviest SM fermions (the top, the bottom quarks and the τ lepton) thecompatibility is achieved with an uncertainty of ∼ − √ s = 13 −
14 TeV center-of-mass-energy, the fit of the Higgsboson couplings to the vector bosons is expected to reach a ∼
5% precisionwith 300 fb − luminosity, while the corresponding ones for the fermions,with the exception of the µ lepton, can reach ∼ −
15% precision. Similarestimates for the end of the High Luminosity option indicate a reduction ofthese numbers by a factor ∼ V ( φ ) = m H φ + λ vφ + λ φ (1)where the Higgs mass ( m H ) and the trilinear ( λ ) and quartic ( λ ) inter-actions are linked by the relations λ SM4 = λ SM3 = λ = m H / (2 v ), where v = ( √ G µ ) − / is the vacuum expectation value, and λ is the coefficientof the (Φ † Φ) interaction, Φ being the Higgs doublet field.The experimental verification of these relations, that fully characterizethe SM as a renormalizable Quantum Field Theory, relies on the measure-ments of processes featuring at least two Higgs bosons in the final state.However, since the cross sections for this kind of processes are quite small,constraining the Higgs self interaction couplings within few times their pre-dicted SM value is already extremely challenging. In particular, informa-tion on λ can be obtained from Higgs pair production with the presentbounds on this reaction from 8 TeV data that allow to constrain λ within O ( ± (15 − λ SM3 ) [8–11]. At √ s = 13 TeV, the Higgs pair production crosssection, in the SM, is around 35 fb in the gluon-fusion channel [12–21] and2ven smaller in other production mechanisms [22, 23] that suggests, assum-ing an integrated luminosity of 3000 fb − , that it will be possible to excludeat the LHC only values in the range λ < − . λ SM3 and λ > . λ SM3 via the b ¯ bγγ signatures [24] or λ < − λ SM3 and λ > λ SM3 includingalso b ¯ bτ ¯ τ signatures [25]. Concerning the quartic Higgs self-coupling λ ,its measurement via triple Higgs production seems beyond the reach of theLHC [26,27] due to the smallness of the corresponding cross section (around0 . λ coupling arestudied. This approach builds on the assumption that New Physics (NP)couples to the SM via the Higgs potential in such a way that the lowest-order Higgs couplings to the other fields of the SM (and in particular tothe top quark and vector bosons) are still given by the SM prescriptionsor, equivalently, modifications to these couplings are so small that do notswamp the loop effects one is considering. This strategy was first applied to ZH production at an e + e − collider in Ref. [28] and later to Higgs productionand decay modes at the LHC [29–31].The aim of this work is twofold. On the one side we apply the samestrategy to the precise measurements of the W boson mass, m W , and theeffective sine, sin θ lepeff . In order to constrain λ we look for effects inducedby an anomalous Higgs trilinear coupling at the loop level in the predictionsof m W and sin θ lepeff . Following the approach of Ref. [29] we parametrize theeffect of NP at the weak scale via a single parameter κ λ , i.e. the rescalingof the SM trilinear coupling λ SM3 , so that the φ interaction in the potentialis given by V φ = λ v φ ≡ κ λ λ SM3 v φ , λ SM3 ≡ G µ √ m H , (2)and compute, in the unitary gauge, the effects induced by κ λ in the two-loop W and Z boson self-energies, which are the relevant quantities entering inthe two-loop determination of m W and sin θ lepeff . On the other side we specifybetter the anomalous coupling approach employed above by showing that,at the order we are working, i.e. at the two-loop level, it is equivalent to theuse of a SM Lagrangian with a scalar potential given by an (in)finite towerof (Φ † Φ) n terms. Furthermore, we show that the use of the unitary gauge inthe anomalous coupling approach does not introduce any gauge-dependentproblematics. 3he paper is organised as follows. In Section 2 we discuss the contribu-tions induced by an anomalous Higgs trilinear coupling in m W and sin θ lepeff .Section 3 is devoted to show that the addition to the SM Lagrangian of(Φ † Φ) n terms gives rise to the same contributions. In the following sectionwe discuss the constraints on λ that can be obtained from the current data.In the last section we summarise and draw our conclusions. λ -dependent contributions in m W and sin θ lepeff We consider a Beyond-the-Standard-Model (BSM) scenario, described atlow energy by the SM Lagrangian with a modified scalar potential. Wefurther assume that only Higgs self couplings will be affected by this mod-ified potential while the strength of the couplings of the Higgs to fermionsand vector bosons will not change with respect to its SM value, or, equiv-alently, that any modification of these couplings is going to induce effectsmuch smaller than the ones coming from the “deformation” of the Higgs selfcouplings.In the
M S formulation of the radiative corrections [32–34] the theoreticalpredictions of m W and sin θ lepeff are expressed in terms of the pole mass ofthe particles, the M S
Weinberg angle ˆ θ W ( µ ) and the M S electromagneticcoupling ˆ α ( µ ), defined at the ’t-Hooft mass scale µ , usually chosen to beequal to m Z . In particular, given the radiative parameters ∆ˆ r W , ∆ ˆ α , Y MS defined through (sin ˆ θ W ( m Z ) ≡ ˆ s ) [35] G µ √ π ˆ α ( m Z )2 m W ˆ s (1 + ∆ˆ r W ) , ˆ α ( m Z ) = α − ∆ ˆ α ( m Z ) , ˆ ρ ≡ m W m Z ˆ c = 11 − Y MS , (3)with ˆ c = 1 − ˆ s , m W is obtained from m Z , α, G µ via m W = ˆ ρ m Z (cid:34) − A m Z ˆ ρ (1 + ∆ˆ r W ) (cid:35) / , (4)where ˆ A = ( π ˆ α ( m Z ) / ( √ G µ )) / , while the effective sine is related to ˆ s viasin θ lepeff = ˆ k (cid:96) ( m Z )ˆ s , ˆ k (cid:96) ( m Z ) = 1 + δ ˆ k (cid:96) ( m Z ) , (5)4here ˆ k (cid:96) ( q ) is an electroweak form factor (see Ref. [36]) andˆ s = 12 − (cid:34) − A m Z ˆ ρ (1 + ∆ˆ r W ) (cid:35) / . (6)In our BSM scenario the modifications of the scalar potential affect theradiative parameters ∆ˆ r W and Y MS at the two-loop level while ∆ ˆ α and δ ˆ k (cid:96) ( m Z ) are going to be affected only at three loops. Recalling that thepresent knowledge of m W and sin θ lepeff in the SM includes the completetwo-loop corrections, we are going to discuss only the modifications inducedin ∆ˆ r W and Y MS . The two-loop contribution to ∆ˆ r W and Y MS can beexpressed as [35]∆ˆ r (2) W = Re A (2) WW ( m W ) m W − A (2) WW (0) m W + . . . (7) Y (2) MS = Re (cid:34) A (2) WW ( m W ) m W − A (2) ZZ ( m Z ) m Z (cid:35) + . . . (8)where A WW ( A ZZ ) is the term proportional to the metric tensor in the W ( Z )self energy with the superscript indicating the loop order, and the dots rep-resent additional two-loop contributions that are not sensitive to a modifi-cation of the scalar potential.From the knowledge of the additional contributions induced in ∆ˆ r (2) W and Y (2) MS one can easily obtain the modification of the radiative parameters∆ r and κ e ( m Z ) of the On-Shell (OS) scheme [37]. Considering only newcontributions from the modified scalar potential one can write∆ r (2) = ∆ˆ r (2) W − c s Y (2) MS , (9)where c ≡ m W /m Z , s = 1 − c with ∆ r being the radiative parameterentering the m W − m Z interdependence. The effective sine is related to s in the OS scheme via sin θ lepeff = κ e ( m Z ) s and for the new contributions in κ e ( m Z ) one can write κ (2) e ( m Z ) = 1 − c s Y (2) MS . (10) In our MS formulation the top contribution is not decoupled. Then ˆ k is very closeto 1 and sin θ lepeff can be safely identified with ˆ s [36]. φ W W W W W W φ φ W Wφ + φ φ φ φ φ φ = φ a ) b ) c ) d ) φ φ φ φ φ φ φ e ) e ) e ) W W W
Figure 1: Two-loop λ -and- λ -dependent diagrams in the W self-energy,in the unitary gauge. The dark blob represent the insertion of the modifieddiagrams in the one-loop Higgs self energy, shown in the second row. Theblack point represents either an anomalous λ or λ .The new contribution in the self energies in eqs. (7,8) can be parametrizedjust by a modification of the trilinear coupling as described in eq. (2). Inorder to correctly identify the effects related to the φ interaction we followRef. [29] and work in the unitary gauge. Here we discuss the W self energybut an identical analysis can be done also for the Z self energy.The two-loop diagrams in the W self energy that are sensitive to a mod-ification of the Higgs self couplings are depicted in fig. 1. The dark blob indiagrams 1 a ), 1 d ) represents the one-loop Higgs self energy or the one-loopHiggs mass counterterm that in our scenario gets modified with respect tothe SM result in the unitary gauge by the diagrams in fig. 1 e ). The am-plitudes of the diagrams in fig. 1 were generated using the Mathematicapackage FeynArts [38] and reduced to scalar Master Integrals using privatecodes and the packages FeynCalc [39, 40] and Tarcer [41]. After the reduc-tion to scalar integrals we were left with the evaluation of two-loop vacuumintegrals and two-loop self-energy diagrams at external momenta differentfrom zero. The former integrals were evaluated analytically using the resultsof Ref. [42]. The latter ones were instead reduced to the set of loop-integralbasis functions introduced in Ref. [43]. For their numerical evaluation weused the C program TSIL [44]. Our results are expressed in terms of the OSHiggs mass that specifies the Higgs mass counterterm.Few observations are in order: i) the insertion of the “cactus” diagram e ) in diagrams a ) and d ) in fig. 1 gives rise to a contribution proportional tothe quartic Higgs self couplings on which we did not make any assumption.6owever, this contribution is exactly cancelled by the corresponding Higgsmass counterterms diagram so that the final result does not depend on λ . This finding is general and does not depend on the particular schemeused to define the Higgs mass. Using a different Higgs mass definition,like, e.g., an M S
Higgs mass, ˆ m H , the expression for the W self-energyacquires an explicit λ dependence. However, this dependence is going tobe cancelled by the λ dependence of ˆ m H , when the latter is extracted froma physical quantity like the OS mass. ii) We expect the modified potentialto contain Higgs self interactions with a number of φ fields larger than 4(quintic, sextic, etc. interactions). However, none of these interactions isgoing to contribute to the W self energy at the two-loop level . Thus thenew contributions induced by our BSM scalar potential at the two-loop levelare only functions of κ λ . iii) The contribution to the physical observablesgiven by the diagram 1 d ) vanishes in the differences of self energies (seeeqs. (7,8)).As in the case of single Higgs processes the λ -dependent contributionscan be divided into a part quadratically dependent on λ and another lin-early proportional to λ . The former is due to the diagram 1 a ) with theinsertion of diagram 1 e ) and of its corresponding Higgs mass counterterm.The latter is given by diagrams 1 b ), 1 c ). (Φ † Φ) n theory In this section we show that the results presented in section 2, where nospecific assumption on the BSM scalar potential was made, can be obtainedusing a SM Lagrangian with a scalar potential of the form V NP = N (cid:88) n =1 c n (Φ † Φ) n , Φ = (cid:18) φ +1 √ ( v + φ + iφ ) (cid:19) , (11)where N can be a finite integer or infinite, and in the latter case we assumethe series to be convergent. This is the only constraint we impose on the c n coefficients, in particular we do not assume an effective-field-theory (EFT)scaling on them, i.e. c n +2 ∼ c n / Λ with Λ the scale of NP. The SM poten-tial is recovered setting N = 2 in eq. (11) with c = − m and c = λ where − m is the Higgs mass term in the SM Lagrangian in the unbroken phase. A quintic self interaction gives rise to a two-loop tadpole. However, tadpole contri-butions cancel in eqs. (7,8). φ u = φ + φ − + φ the n -th term in the series can be writtenas(Φ † Φ) n = n (cid:88) k =0 k (cid:88) j =0 j (cid:88) h =0 (cid:18) nk (cid:19)(cid:18) kj (cid:19)(cid:18) jh (cid:19) φ n − k u (cid:18) v (cid:19) k − j (cid:18) φ (cid:19) j − h ( vφ ) h , (12)with (cid:18) nk (cid:19)(cid:18) kj (cid:19)(cid:18) jh (cid:19) = n !( n − k )!( k − j )!( j − h )! h ! , (13)and its contribution to any Higgs self interaction can be labelled by thetriplet { k, j, h } . For example, the minimum of the potential can be obtainedfrom the triplet { n, , } : d V NP d φ (cid:12)(cid:12)(cid:12)(cid:12) φ =0 = v N (cid:88) n =1 c n n (cid:18) v (cid:19) n − = 0 , (14)while the Higgs mass is given by the two triplets { n, , } and { n, , } .However, due to the condition in eq. (14), the first one is giving a vanishingcontribution so that m H = v N (cid:88) n =1 c n n ( n − (cid:18) v (cid:19) n − . (15)The potential V NP up to quartic interactions can be written as V NP φ = m H v (cid:20) φ + φ − ( φ + φ − + φ ) + 14 φ (cid:21) + (cid:18) m H v + dλ (cid:19) φ + (cid:18) m H v + 3 dλ (cid:19) φ (cid:20) φ + φ − + 12 φ (cid:21) + (cid:18) m H v + v dλ (cid:19) φ + m H v φ (cid:0) φ + 2 φ + φ − (cid:1) + 12 m H φ . (16)with dλ = 13 N (cid:88) n =3 c n n ( n − n − (cid:18) v (cid:19) n − , (17) dλ = 23 N (cid:88) n =3 c n n ( n − n − (cid:18) v (cid:19) n − . (18)It is worth noting that in eq. (16) only few couplings are modified withrespect to their SM values. In particular, concerning the unphysical scalars,8 φ W W W WW W W WW W W W W Wφ , φ + φ , φ + W W φ φ φ φ φ φ φ φ φ + a ) b ) c ) d ) Wφ + φ + φ + φ + φ + φ + φ + φ + φ + φ φ γ, Z φ , φ φ + h ) g ) f ) e ) Figure 2: Two-loop diagrams in the W self-energy, involving unphysicalscalars where modified couplings (black points) from V NP φ appear. The darkblob represents the insertion of the relevant one-loop self energy (see fig. 3).only the coupling of φ u φ is modified, with a deformation that is relatedto the deformation of λ .In order to show that the result for the two-loop W self energy computedusing V NP is egual to the one obtained assuming an anomalous λ workingin the unitary gauge, we have to analyze the two-loop diagrams that aremodified with respect to their SM result working in a generic R ξ gauge.Besides the ones in fig. 1, now computed in an R ξ gauge, the diagramscontaining unphysical scalars, shown in fig. 2, should be taken into account.In the latter figure the dark blob represents the insertion of the relevantone-loop self energy. In fig. 3 we show for the various self energies andthe tadpole only the diagrams that are modified with respect to their SMresult due to the new scalar potential V NP . It easy to show that the onlynon-vanishing contributions in figs. 1 a ), 1 d ), 2 a ) come from the insertion ofdiagram e ) in fig. 1 plus its corresponding counterterm diagram while allthe other insertions being of the cactus type (see 1 e ) and 3 a )) are cancelledagainst the corresponding Higgs mass counterterm diagrams. Furthermorethe sum of diagrams 1 a ) and 2 a ) is gauge invariant. Similarly one can provethat the sum of diagrams 1 b ), 1 c ), 2 b ), and 2 c ), is gauge invariant.To complete our proof about the equivalence of the two computationswe have to show that the additional contributions with respect to the SMresults in the diagrams 2 d )–2 h ) and in the corresponding counterterm di-agrams must vanish. Diagram 2 d ) is automatically zero, while in the re-maining diagrams a self energy of an unphysical scalar is always present.9 ) φ a ) a ) = φ φ φ φ φ + φ φ + ; b ) = φ b ) φ φ φ + d ) = φ + φ + φ φ + d ) c ) = φ φ φ φ φ c ) ; Figure 3: One-loop self energy and tadpole diagrams that contain modifiedcouplings with respect to the SM.According to V NP φ the only modified contributions in the one-loop self en-ergies of the unphysical scalars are given by diagrams 3 c ) and 3 d ). Tothe contribution of diagrams 2 e )–2 h ) with the insertion of 3 c ) or 3 d ) onehas to add the counterterm diagrams. The counterterm associated to therenormalization of the mass of an unphysical scalar contains a term relatedto the mass of the corresponding vector boson plus a term that is related tothe renormalization of the vacuum. The former is not affected by our mod-ified scalar potential. The latter, when v is identified with the minimum ofthe radiatively corrected potential, is given by the tadpole contribution [45].Then the only modified contribution in the mass renormalization of the un-physical scalars is given by diagram 3 b ). Thus, the additional contributionswith respect to the SM result in the diagrams 2 e )-2 h ) are exactly cancelledby the additional contributions in the unphysical scalar mass countertermdiagrams. The key point in this cancellation is the fact that the modifica-tion in the vertex with three physical Higgses and the one in the verticescontaining two physical and two unphysical Higgses are related by a factor3 /v as shown in eq. (16).We have shown that in a theory with a scalar potential given by eq. (11)the two-loop W self energy is modified with respect to its SM value byadditional contributions that are gauge-invariant. Then, one can directlycompute them in the unitary gauge, that corresponds to the computationwith an anomalous λ once the identification κ λ = 1 + 2 v /m H dλ is made.10 C m W . × − − . × − sin θ lepeff − . × − . × − Table 1: Values of the coefficients C and C . The analytic expressions for the contributions induced in ∆ˆ r (2) W and Y (2) MS by an anomalous λ are reported in the Appendix. These contributions aregoing to modify the SM predictions for m W and sin θ lepeff via eqs. (3–6).Denoting as O either m W or sin θ lepeff one can write O = O SM (cid:2) κ λ − C + ( κ λ − C (cid:3) , (19)with the values of the coefficients C and C reported in Table 1.Let us comment on the validity of eq. (19). At the two-loop level we areworking, the contributions induced by an anomalous Higgs trilinear couplingin the precision observables are finite (see table 1 or the Appendix), i.e.they are not sensitive to the NP scale Λ associated with the modificationof the potential. This situation is analogous to what happens in singleHiggs processes where new contributions induced by an anomalous λ at theNLO are also finite [29]. As in single Higgs processes if NNLO effects areconsidered, one expects that at three or more loops the modified potential isgoing to induce contributions not only proportional to λ but also to quartic,quintic etc. Higgs self interactions and moreover these contributions will besensitive to the NP scale.The constraints on κ λ we are going to derive below assume the validity ofa perturbative approach. Then, we expect any higher-order contribution tobe subdominant with respect to the effects we are computing. This impliesthat these higher-order contributions should not contain any large amplify-ing factor related to the scale Λ, or equivalently that Λ cannot be too farfrom the Electroweak scale. Furthermore, since at the three-loop level oneexpects the anomalous contribution from the trilinear coupling to grow as κ λ , a restricted range of κ λ should also be imposed. Following Ref. [29] weconsider | κ λ | (cid:46)
20 as a range of validity of our perturbative approach.In order to set limits on κ λ from the analysis of precision observables,we perform a simplified fit. We define the best value of κ λ as the one that11 gF (cid:43) VBFM w (cid:43) Sin eff ggF (cid:43)
VBF (cid:43) M w (cid:43) Sin eff (cid:45) (cid:45)
10 10 20 Κ Λ (cid:68)Χ ggF (cid:43) VBFM w (cid:43) Sin eff ggF (cid:43)
VBF (cid:43) M w (cid:43) Sin eff (cid:45) (cid:45)
10 10 20 Κ Λ p (cid:45) value Figure 4: Left: χ for the different sets of observables described in thetext, the two horizontal lines represent ∆ χ = 1 and ∆ χ = 3 .
84. Right:corresponding p -value, the horizontal line is p = 0 . χ ( κ λ ) function defined as χ ( κ λ ) ≡ (cid:88) ( O exp − O the ) ( δ ) , (20)where O exp refers to the experimental measurement of the observable O , O the is its theoretical value obtained from eq. (19) and δ is the total uncertainty,that we take as the sum in quadrature of the experimental and theory errors.In order to ascertain the goodness of our fit, we also compute the p -value asa function of κ λ : p -value( κ λ ) = 1 − F χ n ) ( χ ( κ λ )) , (21)where F χ n ) ( χ ( κ λ )) is the cumulative distribution function for a χ distri-bution with n degrees of freedom, computed at χ ( κ λ ).In the fit we consider not only the two precision observables but also thesignal strength parameter for single Higgs production in gluon fusion ( gg F)and vector boson fusion (VBF). The latter observables were indicated as the P set in Ref. [29] where it was shown that they were returning the moststringent bound on κ λ . We then considered three set of data: • The P set in Ref. [29]. The experimental results are presented inTab. 8 of Ref. [5]. See Ref. [29] for more details. • The W mass and effective sine. For the W mass we use the latest resultby the ATLAS collaboration m W = 80 . ± .
019 GeV [46]. Thisnumber, although it has a slightly larger uncertainty with respect to12he world average m W = 80 . ± .
015 GeV [47], it is closer to the SMprediction m W = 80 . ± . ± .
003 where the errors refer to theparametric and theoretical uncertainties [35]. Concerning the effectivesine, we use the average of the CDF [48] and D0 [49] combinationssin θ lepeff = 0 . ± . θ lepeff = 0 . ± . ± . • The combination of these two sets of data.The χ ( κ λ ) and p -value functions for the three sets are reported in fig. 4.In particular for the combination we find κ best λ = 0 . , κ σλ = [ − . , . , κ σλ = [ − . , . , (22)where the κ best λ is the best value and κ σλ , κ σλ are respectively the 1 σ and 2 σ intervals. We identified the 1 σ and 2 σ intervals assuming a χ distribution.The comparison between the numbers in eq. (22) and the corresponding onesfor the gg F+VBF case [29], namely κ best λ = − . , κ σλ = [ − . , . , κ σλ = [ − . , .
0] ( P set) , (23)shows that the inclusion of the precision observables reduces the allowedrange for κ λ . Similarly, looking at the solid black line in the p -value partof fig. 4, we can exclude at more than 2 σ models with κ λ in the regions κ λ < − . κ λ > . m W and sin θ lepeff with singleHiggs processes could be very powerful in constraining the allowed regionfor κ λ , in particular the region of positive κ λ . In this work we have discussed how the predictions of the W boson massand the effective sine are affected by loops featuring an anomalous trilinearHiggs coupling. Following Ref. [29] we have chosen to present our results inthe contest of the κ -framework, parametrising the effect of NP at the weakscale via a single parameter, κ λ , i.e. the rescaling of the SM Higgs trilinearcoupling. Indeed, given a generic scalar potential constructed using only theHiggs doublet field, at the two-loop level these precision observables are onlysensitive to the modification of the trilinear Higgs coupling. As in Ref. [29]13e worked in the unitary gauge to easily identify the effects we were lookingfor. We proved that the latter choice is just a technical trick and does notintroduce any gauge-dependent issues. In fact, we have explicitly shown thatour approach is equivalent, to the order we were working, to an analysis of m W and sin θ lepeff in a generic R ξ gauge performed in a theory described bya SM Lagrangian in which the scalar potential is modified by the additionof an (in)finite tower of (Φ † Φ) n terms.Concerning this scalar potential, one important point to remark is thefact that we did not make any assumption on the size of the coefficients ofthe various terms in the potential, so that in principle we do not have apriori any restriction on κ λ , apart from the requirement of perturbativity.This is at variance with an EFT approach based on the addition to the SMLagrangian of a dimension six (Φ † Φ) term [30, 31] where the requirementof v being a global minimum constraints κ λ < c n coefficients in the potential should be assumed. In particular, either oneassumes that c n exhibit a scaling with the order n so that the couplings ofthe interactions with a large number of φ do not grow (as an example dλ does not become larger than dλ , see eqs. (17,18)) or that the various c n are related to each other enforcing cancellations among the various terms inthe potential.As we said a theory with a modified trilinear coupling is expected to bevalid up to a scale Λ that cannot be too far from the Electroweak scale.An estimate of Λ can be obtained by looking when perturbative unitarityis lost in processes like e.g. the annihilation of longitudinal vector bosonsinto n Higgs bosons, V L V L → n φ [53]. A preliminary study on this subjectindicates that Λ ∼ − m W and sin θ lepeff to an anomaloustrilinear coupling via a one-parameter fit. We have also shown that whenthe analysis of the precision observables is combined with the one fromsingle Higgs inclusive measurements at the LHC 8 TeV, a restricted rangeof allowed κ λ is found. The range found is actually competitive with thepresent bounds obtained from the direct searches of Higgs pair production. Acknowledgements
Two of us (G.D., P.P.G.) want to thank their collaborators F. Maltoni andD. Pagani together with Xiaoran Zhao for useful discussions and commu-14ications. G.D. wants to thank R. Rattazzi for an enlightening “endless”discussion. The work of P.P.G. is supported by the United States Depart-ment of Energy under Grant Contracts de-sc0012704.15 nomalous contributions in ∆ˆ r (2) W and Y (2) M S
Here we give the analytic expressions for the additional contributions in-duced in ∆ˆ r (2) W and Y (2) MS by an anomalous λ . In the formulae below ζ W = m H m W , ln( x ) = log (cid:18) xµ (cid:19) , (A.1)with µ the ’t-Hooft mass scale. We find for the κ λ contributions∆ˆ r (2 ,κ λ ) W = (cid:18) ˆ α πs (cid:19) (cid:40)(cid:34) ζ W (cid:0) − ζ W + 49 ζ W + 18 (cid:1) + ζ W ζ W − ζ W + 616 ( ζ W −
1) ln (cid:0) m W (cid:1) + (cid:18) − ζ W
16 ( ζ W − ζ W + − ζ W + 9 ζ W − ζ W + 6032 ( ζ W − ζ W ln (cid:0) m W (cid:1)(cid:19) ln (cid:0) m H (cid:1) + 2 ζ W − ζ W + 46 ζ W − ζ W − ζ W ln (cid:0) m W (cid:1) +3 2 ζ W − ζ W − ζ W + 18 ζ W − ζ W − ζ W ln (cid:0) m H (cid:1) + (cid:32) (cid:0) ζ W − ζ W − (cid:1) ζ W + 18 ( ζ W − ζ W ln (cid:0) m W (cid:1) −
18 ( ζ W − ζ W ln (cid:0) m H (cid:1) (cid:33) B (cid:0) m W , m H , m W (cid:1) + 18 ( ζ W − ζ W B (cid:0) m W , m H , m W (cid:1) + ζ W − m W ζ W S (cid:0) m W , m W , m H , m H (cid:1) −
12 ( ζ W − ζ W T (cid:0) m W , m H , m W , m H (cid:1) + 14 ( ζ W − ζ W U (cid:0) m W , m H , m W , m H , m W (cid:1) − ζ W (cid:0) ζ W + ζ W − (cid:1) U (cid:0) m W , m W , m H , m H , m H (cid:1) + 116 m H (cid:0) ζ W − ζ W + 24 (cid:1) M (cid:0) m W , m H , m H , m W , m W , m H (cid:1) +3 − ζ W + ζ W + 4 ζ W + 2464 ( ζ W − ζ W φ (cid:18) (cid:19) + 3 4 ζ W − ζ W + 1032 ( ζ W − ζ W φ (cid:18) ζ W (cid:19) − ζ W (cid:0) ζ W − ζ W + 18 ζ W + 40 ζ W − (cid:1)
64 ( ζ W − φ (cid:18) ζ W (cid:19) (cid:35) κ λ (cid:34) (cid:0) − ζ W + 2403 ζ W − ζ W + 1652 ζ W + 120 (cid:1)
256 ( ζ W −
4) ( ζ W − ζ W +3 ( ζ W − ζ W + 39 ζ W − ζ W + 1232 ( ζ W −
4) ( ζ W − ζ W ln (cid:0) m W (cid:1) +9 (cid:18) ζ W − ζ W + 80 ζ W − ζ W + 4832 ( ζ W −
4) ( ζ W − ζ W − ζ W
32 ( ζ W − ln (cid:0) m W (cid:1)(cid:19) ln (cid:0) m H (cid:1) − ζ W − ζ W + 117 ζ W − ζ W + 12064 ( ζ W −
4) ( ζ W − ζ W ln (cid:0) m H (cid:1) + (cid:32) ζ W (cid:0) ζ W − ζ W + 12 (cid:1) − ζ W (cid:0) ζ W − ζ W + 12 (cid:1) ln (cid:0) m H (cid:1) − ζ W −
32 ( ζ W − ζ W B (cid:0) m H , m H , m H (cid:1) (cid:33) B (cid:0) m W , m H , m W (cid:1) + (cid:32) ζ W − ζ W + 33 ζ W − ζ W + 3264 ( ζ W −
4) ( ζ W − ζ W +9 ζ W − ζ W + 14 ζ W − ζ W + 832 ( ζ W −
4) ( ζ W − ζ W ln (cid:0) m W (cid:1) − ζ W − ζ W + 19 ζ W − ζ W + 2032 ( ζ W −
4) ( ζ W − ζ W ln (cid:0) m H (cid:1) (cid:33) B (cid:0) m H , m H , m H (cid:1) +9 ζ W − ζ W + 832 m W ( ζ W − ζ W S (cid:0) m W , m W , m H , m H (cid:1) − ζ W − ζ W + 16 ζ W −
124 ( ζ W − ζ W T (cid:0) m W , m H , m W , m H (cid:1) + 7 ζ W − ζ W + 52 ζ W + 2432 ( ζ W − ζ W U (cid:0) m W , m W , m H , m H , m H (cid:1) +3 7 ζ W − ζ W + 99 ζ W − ζ W + 8464 ( ζ W −
4) ( ζ W − ζ W φ (cid:18) (cid:19) +27 4 ζ W −
164 ( ζ W − ζ W φ (cid:18) ζ W (cid:19) (cid:35) κ λ (cid:41) , (A.2)17 (2 ,κ λ ) MS = (cid:18) ˆ α πs (cid:19) (cid:40) (cid:20) f (cid:18) m H m W (cid:19) − c f (cid:18) m H m Z (cid:19)(cid:21) κ λ + (cid:20) f (cid:18) m H m W (cid:19) − c f (cid:18) m H m Z (cid:19)(cid:21) κ λ (cid:41) , (A.3)where we have defined the functions f , f as f ( ζ ≡ m H /m ) = 132 (cid:34) − (cid:0) ζ − ζ − (cid:1) ζ + 4(2 ζ − ζ ln (cid:0) m (cid:1) + ( ζ − ζ ln (cid:0) m (cid:1) − (cid:18) ζ + ( ζ − ζ ln (cid:0) m (cid:1) (cid:19) ln (cid:0) m H (cid:1) + 3 ζ ln (cid:0) m H (cid:1) +4 (cid:18) − − ζ + ζ + ( ζ − (cid:0) m (cid:1) − ( ζ − ζ ln (cid:0) m H (cid:1) (cid:19) ζB (cid:0) m , m H , m (cid:1) +4( ζ − ζB (cid:0) m , m H , m (cid:1) +4( ζ − ζ m H S (cid:0) m , m , m H , m H (cid:1) − ζ − ζ T (cid:0) m , m H , m , m H (cid:1) +8( ζ − ζU (cid:0) m , m H , m , m H , m (cid:1) − (cid:0) ζ + ζ − (cid:1) ζU (cid:0) m , m , m H , m H , m H (cid:1) +2 (cid:0) ζ − ζ + 24 (cid:1) m H M (cid:0) m , m H , m H , m , m , m H (cid:1) − ζ φ (cid:18) (cid:19) − ( ζ − ζ − ζφ (cid:18) ζ (cid:19) (cid:35) , (A.4) f ( ζ ≡ m H /m ) = 1128 (cid:34) − (cid:0) ζ − ζ + 1660 ζ + 60 (cid:1) ζζ − (cid:0) ζ − ζ + 12 (cid:1) ζζ − (cid:0) m (cid:1) +36 5 ζ − ζ + 32 ζ − ζ ln (cid:0) m H (cid:1) − ζ − ζ + 36 ζ − ζ ln (cid:0) m H (cid:1) +36 (cid:32) ζ − ζ + 12 ζ − ζ − ζ + ζ − ζ + 8 ζ − ζ ln (cid:0) m (cid:1) ( ζ − ζ − ζ ln (cid:0) m H (cid:1) − ( ζ − ζ − ζB (cid:0) m , m H , m (cid:1) (cid:33) B (cid:0) m H , m H , m H (cid:1) +4 (cid:0) ζ − ζ + 12 (cid:1) (cid:18) − (cid:0) m H (cid:1) (cid:19) ζB (cid:0) m , m H , m (cid:1) +36 ζ − ζ + 8 m H ( ζ − ζ S (cid:0) m , m , m H , m H (cid:1) − ζ − ζ + 16 ζ − ζ − ζT (cid:0) m , m H , m , m H (cid:1) +4 7 ζ − ζ + 52 ζ + 24 ζ − ζU (cid:0) m , m , m H , m H , m H (cid:1) +6 7 ζ − ζ + 36 ζ − ζ φ (cid:18) (cid:19) (cid:35) . (A.5)In eqs.(A.2–A.5) φ ( x ) = 4 (cid:114) x − x Im(Li ( e i √ x ) )) , (A.6)and, following Refs. [43, 44], we define the d -dimensional functions B ( s, x, y ) = lim (cid:15) → (cid:20) B ( s, x, y ) − (cid:15) (cid:21) = − (cid:90) dt ln[ tx + (1 − t ) y − t (1 − t ) s ] , (A.7) S ( s, x, y, z ) = lim (cid:15) → (cid:20) S ( s, x, y, z ) + x + y + z (cid:15) + s − x − y − z (cid:15) − A ( x ) + A ( y ) + A ( z ) (cid:15) (cid:21) , (A.8) T ( s, x, y, z ) = − ∂∂x S ( s, x, y, z ) , (A.9) U ( s, x, y, z, u ) = lim (cid:15) → (cid:20) U ( s, x, y, z, u ) + 12 (cid:15) − (cid:15) − B ( s, x, y ) (cid:15) (cid:21) , (A.10) M ( s, x, y, z, u, v ) = lim (cid:15) → [ M ( s, x, y, z, u, v )] , (A.11)with d = 4 − (cid:15) and A ( x ) = − i (2 πµ ) (cid:15) π (cid:90) d d k (cid:16) k − x (cid:17) , (A.12) B ( s, x, y ) = − i (2 πµ ) (cid:15) π (cid:90) d d k (cid:16) k − x (cid:17)(cid:16) k − y (cid:17) , (A.13)19 ( s, x, y, z ) = − (cid:18) (2 πµ ) (cid:15) π (cid:19) (cid:90) (cid:90) d d k d d k (cid:16) k − x (cid:17)(cid:16) k − y (cid:17)(cid:16) k − z (cid:17) , (A.14) U ( s, x, y, z, u ) = − (cid:18) (2 πµ ) (cid:15) π (cid:19) (cid:90) (cid:90) d d k d d k (cid:16) k − u (cid:17)(cid:16) k − x (cid:17)(cid:16) k − z (cid:17)(cid:16) k − y (cid:17) , (A.15) M ( s, x, y, z, u, v ) = − (cid:18) (2 πµ ) (cid:15) π (cid:19) (cid:90) (cid:90) d d k d d k (cid:16) k − x (cid:17)(cid:16) k − y (cid:17)(cid:16) k − z (cid:17)(cid:16) k − u (cid:17)(cid:16) k − v (cid:17) , (A.16)where we introduced the notation k = k − p , k = k − p , k = k − k , (A.17)with p = s . References [1]
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