Unitarity Constraints on Dimension-Six Operators
aa r X i v : . [ h e p - ph ] N ov YITP-SB-14-46
Unitarity Constraints on Dimension-Six Operators
Tyler Corbett ∗ C.N. Yang Institute for Theoretical Physics, SUNY at Stony Brook, Stony Brook, NY 11794-3840, USA
O. J. P. ´Eboli † Instituto de F´ısica, Universidade de S˜ao Paulo, S˜ao Paulo – SP, Brazil.
M. C. Gonzalez–Garcia ‡ Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA),Departament d’Estructura i Constituents de la Mat`eria,Universitat de Barcelona, 647 Diagonal, E-08028 Barcelona, Spain andC.N. Yang Institute for Theoretical Physics, SUNY at Stony Brook, Stony Brook, NY 11794-3840, USA
We obtain the partial-wave unitarity constraints on dimension-six operators stemming from theanalyses of vector boson and Higgs scattering processes as well as the inelastic scattering of standardmodel fermions into electroweak gauge bosons. We take into account all coupled channels, all possiblehelicity amplitudes, and explore a six-dimensional parameter space of anomalous couplings. Ouranalysis shows that for those operators affecting the Higgs couplings, present 90% confidence levelconstraints from global data analysis of Higgs and electroweak data are such that unitarity is notviolated if √ s ≤ . O WWW , the present bounds fromtriple-gauge boson analysis indicate that within its presently allowed 90% confidence level rangeunitarity can be violated in f ¯ f ′ → V V ′ at center-of-mass energy √ s ≥ . I. INTRODUCTION
The standard model (SM) of electroweak interactions has been extremely successful in the description of the availabledata, and up to now there is no clear experimental evidence that challenges its predictions. As long as no new statehas been observed, effective lagrangians provide a well defined systematic way to parametrize departures from thestandard model. Furthermore, the recent discovery of a particle resembling a light Higgs boson indicates that the SU (2) L ⊗ U (1) Y gauge symmetry might be realized linearly in the effective theory. Therefore, we can parametrizethe effects of new physics by adding to the SM lagrangian higher dimension operators made up of the SM fields.Within the global symmetries of the SM the lowest dimension of the new operators is six, hence we include thosedimension-six operators: L eff = L SM + X n f n Λ O n (1)where, in general, the dimension-six operators O n involve gauge-bosons, Higgs doublets, fermionic fields, and(covariant-) derivatives of these fields. Each operator has a corresponding coupling f n and Λ is the characteris-tic energy scale at which new physics (NP) becomes apparent.It is well known that nonrenormalizable higher dimensional operators give rise to rapid growth of the scatteringamplitudes with energy, leading to partial-wave unitarity violation. This fact constrains the energy range where thelow energy effective theory is valid once the coefficients f n are fixed. With this aim in mind in this work we revisit thebounds from partial-wave unitarity on L eff arising from vector boson and Higgs boson scattering, as well as inelasticprocesses f ¯ f ′ → V V ′ where f ( ′ ) is a SM fermion and V ( ′ ) is an electroweak gauge boson.Previous works in the literature studied the unitarity bounds on some of the dimension-six operators either con-sidering only one non-vanishing coupling at a time, and/or they did not take into account coupled channels, or theyworked in the framework of effective vertices [1–6]. Here, we complete these previous analyses by considering the ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] effects of coupled channels leading to the strongest constraints, including both elastic and inelastic channels, andalso by analyzing the general six-dimensional parameter space of relevant anomalous couplings. Moreover, we onlyconsider contributions up to order 1 / Λ to apply systematically the effective field theory approach.The outline of this article is as follows. We summarize the formalism employed in Sec.II, in particular Sec.II Acontains the basis of operators considered, and in Sec.II B we briefly present the standard partial-wave unitarityconstraints from elastic gauge boson scattering and inelastic f ¯ f ′ → V V ′ processes. Section III contains our resultsfrom the unitarity analysis and we compare those with the presently allowed range from collider searches. In particularwe conclude that, even in the most general case, those operators affecting the Higgs couplings do not violate unitarityfor center-of-mass energies √ s ≤ . II. ANALYSES FRAMEWORK
In this section we present the effective interactions considered in this work, as well as the unitarity relations thatwe use to constrain them
A. Effective Lagrangian
We parametrize deviations from the Standard Model (SM) in terms of dimension-six effective operators as inEq. (1). The dimension-six basis contains 59 independent operators, up to flavor and Hermitian conjugation, whichare sufficient to generate the most general S-matrix elements given the SM gauge symmetry and that baryon andlepton number symmetries are obeyed by the NP [7]. Exploiting the freedom in the choice of basis, we work in thatof Hagiwara, Ishihara, Szalapski, and Zeppenfeld (HISZ) [8, 9].In what follows we consider bosonic operators relevant to two-to-two scattering processes involving Higgs and/orgauge bosons at tree level, and will impose C - and P -evenness on the operators, which leaves us with ten dimension-sixoperators. These operators can be classified into three groups according to their field content • pure gauge operators, in this class there is just one operator O W W W = Tr[ c W νµ c W ρν c W µρ ] ; (2) • gauge-Higgs operators which include O W W = Φ † c W µν c W µν Φ , (3) O BB = Φ † b B µν b B µν Φ , (4) O BW = Φ † b B µν c W µν Φ , (5) O W = ( D µ Φ) † c W µν ( D ν Φ) , (6) O B = ( D µ Φ) † b B µν ( D ν Φ) , (7) O Φ , = ( D µ Φ) † ΦΦ † ( D µ Φ) , (8) O Φ , = ( D µ Φ) † ( D µ Φ)(Φ † Φ) ; (9) • and pure Higgs operators: O Φ , = 12 ∂ µ (Φ † Φ) ∂ µ (Φ † Φ) , (10) O Φ , = 13 (Φ † Φ) , (11) We do not consider operators with higher derivative kinetic term for the Higgs and gauge bosons. They can be traded by a combinationof the operators considered plus some fermionic operators and hence do not lead to new unitarity violating effects in the scatteringamplitudes studied here; see [13] for the explicit derivation for the case of the operator with a higher derivative kinetic term for theHiggs.
V V V V V V V HV V HV V V HHV V HHH HHHH H ¯ ff O WWW
X X O WW X X X O BB X X O BW X X X X X O W X X X X X O B X X X X O Φ , X X X X X X X O Φ , X X X X X O Φ , X X O Φ , X X X X XTABLE I: Couplings relevant for our analysis that are modified by the dimension–six operators in Eqs. (2)–(11). Here, V stands for any electroweak gauge boson, H for the Higgs and f for SM fermions. where Φ stands for the Higgs doublet and we have adopted the notation b B µν ≡ i ( g ′ / B µν , c W µν ≡ i ( g/ σ a W aµν , g with g ′ being the SU (2) L and U (1) Y gauge couplings respectively, and σ a the Pauli matrices.The dimension-six operators given in Eqs. (2)–(11) modify the triple and quartic gauge boson couplings, the Higgscouplings to fermions and gauge bosons, and the Higgs self-couplings.; see Table I. Further details are presented inthe appendix A. A thorough discussion of the effects of the operators relevant to Higgs physics and anomalous gaugecouplings in the basis here employed can be found in [10–12].We notice first that operators O BW and O φ, contribute at tree level to the oblique electroweak precision parameter T (or ∆ ρ ) [14–17] . Taking into account that the present available data impose strong bounds on these parameters [18],the couplings f BW and f Φ , are severely constrained, consequently we neglect O BW and O φ, in our analyses. Thisleaves us with a basis of 8 operators. Furthermore for large center–of–mass energies ( √ s ), which we will take to mean √ s ≫ M W,Z,H for our analysis, the behavior of O φ, and O φ, is the same up to a sign for the scattering processesconsidered and as such for our discussion we can quantify their behavior by a single operator coefficient: f Φ2 , Λ ≡ f Φ , − f Φ , Λ . (12)This is expected since O Φ , + O Φ , can be traded via equations of motion by a combination of Yukawa-like operatorswhich do not contribute to the 2 → O Φ , modifies the Higgs self-couplings as well as the relation between the Higgs mass, its vev and thepotential term λ (see Appendix A). However these effects do not induce unitarity violation in the 2 → f W , f B , f W W , f BB , f W W W , and f Φ2 , . B. Partial-wave unitarity
In the two-to-two scattering of electroweak gauge bosons ( V ) V λ V λ → V λ V λ (13)the corresponding helicity amplitude can be expanded in partial waves in the in the center–of–mass system as [19] M ( V λ V λ → V λ V λ ) = 16 π X J (cid:18) J + 12 (cid:19) r δ V λ V λ r δ V λ V λ d Jλµ ( θ ) e iMϕ T J ( V λ V λ → V λ V λ ) , (14)where λ = λ − λ , µ = λ − λ , M = λ − λ − λ + λ , and θ ( ϕ ) is the polar (azimuth) scattering angle. d is theusual Wigner rotation matrix. In the case one of the vector bosons is replaced by the Higgs we can still employ thisexpression by setting the correspondent λ to zero.Partial-wave unitarity for the elastic channels requires that | T J ( V λ V λ → V λ V λ ) | ≤ , (15)where we have assumed s ≫ ( M V + M V ) . More stringent bounds can be obtained by diagonalizing T J in theparticle and helicity space and then applying the condition in Eq. (15) to each of the eigenvalues.We have also studied unitarity constraints from fermion annihilation processes [6] f σ ¯ f σ → V λ V λ . (16)In this case the partial-wave expansion is given by M ( f σ ¯ f σ → V λ V λ ) = 16 π X J (cid:18) J + 12 (cid:19) δ σ , − σ d Jσ − σ ,λ − λ ( θ ) T J ( f σ ¯ f σ → V λ V λ ) , (17)where, for simplicity, we have set ϕ = 0. These processes proceed via s-channel exchange of a J = 1 vector boson andtherefore in the limit of massless fermions those must appear in opposite helicity states, a condition which is explicitlyenforced in the expression above by the inclusion of the term δ σ , − σ .Following the procedure presented in Ref. [6] the unitarity bound on the inelastic production of gauge boson pairsin Eq. (16) is found by relating the corresponding amplitude to that of the elastic process f σ ¯ f σ → f σ ¯ f σ . (18)In this case the unitarity relation is given by2Im[ T J ( f σ ¯ f σ → f σ ¯ f σ )] = (cid:12)(cid:12) T J ( f σ ¯ f σ → f σ ¯ f σ ) (cid:12)(cid:12) (19)+ X V λ ,V λ (cid:12)(cid:12) T J ( f σ ¯ f σ → V λ V λ ) (cid:12)(cid:12) + X N (cid:12)(cid:12) T J ( f σ ¯ f σ → N ) (cid:12)(cid:12) , where as before we take the limit s ≫ ( M V + M V ) . N represents any state into which f σ ¯ f σ can annihilate whichalso does not consists of two gauge bosons. Eq. (19) is a quadratic equation for Im[ T J ( f σ ¯ f σ → f σ ¯ f σ )] whichonly admits a solution if X V λ ,V λ (cid:12)(cid:12) T J ( f σ ¯ f σ → V λ V λ ) (cid:12)(cid:12) ≤ . (20)The strongest bound can be found by considering some optimized linear combination | X i = X f ,σ x f ,σ | f σ ¯ f σ i (21)with the normalization condition P fσ | x fσ | = 1, for which the amplitude T J ( X → V λ V λ ) is largest. III. RESULTS
Let us start by considering all two-to-two Higgs and electroweak gauge-boson scattering processes. We have calcu-lated the scattering amplitudes for all possible combinations of particles and helicities generated by the SM extendedwith the dimension-six operators presented in Sec.II A. In doing so we have consistently kept the anomalous termsinduced by the dimension-six terms in linear order. It is interesting to notice that to this order there is no amplitudethat diverges as s . This is a result of gauge invariance enforcing that the corresponding triple and quartic verticessatisfy the requirements for the cancellation of the s terms to take place [20].All together we find 26 processes (in particle space) which yield some helicity amplitude that grows as s for someof the dimension-six operators while the rest are constant or vanishing at large energies. We give the correspondingexpressions of the parts of the amplitudes which grow as s in Tables II–VI. Table II displays the terms in theamplitudes that grow as s at high energies due to the contributions of the operators O Φ , and O Φ , . It is interestingto notice that these operators lead to unitarity violation only for the scattering of longitudinal gauge bosons. Thisis expected as these operators do not generate higher derivative terms beyond those already present in the SM inthe triple and quartic couplings. The amplitudes that violate unitarity at high energies due to the presence of O W ( O B ) are presented in Table III (IV), the results for O W W and O BB are contained in Table V, and those for O W W W are shown in Table VI. As we can see from these tables, for these five operators the growth as s of the amplitudesoccurs not only for the scattering of longitudinal gauge bosons but also for transversely polarized ones. Notice alsothat all amplitudes which grow with s generated by O Φ , , O Φ , , O W , O B , O W W , and O BB have only J = 0 or J = 1partial-wave projections. O W W W leads to violation of unitarity also in helicity amplitudes with projections over J ≥ J , we compute the constraints using onlythe amplitudes in J = 0 and J = 1 partial waves.With the results in Tables II–VI we proceed to build the T and T amplitude matrices in particle and parameterspace. These matrices are formed with the s-divergent amplitudes corresponding to all combinations of gauge bosonand Higgs pairs with a given total charge Q = 2 , , J which are:( Q, J ) States Total(2 , W + ± W + ± W +0 W +0 , W + ± W + ± W + ± W +0 W +0 W + ± , W + ± Z ± W +0 Z W + ± γ ± W +0 H , W +0 Z W + ± Z W +0 Z ± W + ± Z ± W +0 γ ± W + ± γ ± W +0 H W + ± H , W + ± W −± W +0 W − Z ± Z ± Z Z Z ± γ ± γ ± γ ± Z H HH , W +0 W − W + ± W − W +0 W −± W + ± W −± Z ± Z Z Z ± Z γ ± Z H Z ± H γ ± H
18 (22)where upper indices indicate charge and lower indices helicity, and taking into account the relation T J ( V λ V λ → V λ V λ ) = ( − λ − λ − λ + λ T J ( V − λ V − λ → V − λ V − λ ) . (23)We present in the right-hand side of Eq. (22) the dimensionality of the corresponding T J . For example for Q = 2, T in the basis (cid:0) W ++ W ++ , W +0 W +0 , W + − W + − (cid:1) is the 3 × π s e f W W W − e f B − e f W − f Φ , e f W W W . (24)In order to obtain the most stringent bounds on the coefficients f n / Λ we diagonalize the six T J matrices andimpose the constraint Eq. (15) on each of their eigenvalues. We find that there are 50 possible nonzero eigenvalues ofthe total 59. Considering only one operator different from zero at a time, we find that the strongest constraint arisefrom the following eigenvalues: (cid:12)(cid:12)(cid:12)(cid:12) π f Φ2 , Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ ⇒ (cid:12)(cid:12)(cid:12)(cid:12) f Φ2 , Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) . g π f W Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ ⇒ (cid:12)(cid:12)(cid:12)(cid:12) f W Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g s W ( p + 3s W )128c π f B Λ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ⇒ (cid:12)(cid:12)(cid:12)(cid:12) f B Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r g π f W W Λ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ⇒ (cid:12)(cid:12)(cid:12)(cid:12) f W W Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (25) (cid:12)(cid:12)(cid:12)(cid:12) . g π f BB Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ ⇒ (cid:12)(cid:12)(cid:12)(cid:12) f BB Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) (1 + q − s ) 3 g π f W W W Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ ⇒ (cid:12)(cid:12)(cid:12)(cid:12) f W W W Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ . Next we consider the effects of O W , O B , and O W W W on fermion scattering into gauge bosons pairs. They are dueto the induced modification of the triple gauge boson couplings. We considered the total charge Q = 0 processes l ¯ l → W + W − , ν ¯ ν → W + W − and q ¯ q → W + W − , Notice that we have introduce in Eq. (14) the symmetry factors r δ V λ V λ and r δ V λ V λ in the definition of the corresponding T J amplitude while in some other conventions they are included in the definition of the two equal gauge boson states. where l ( ν , q ) stand for SM charged leptons (neutrinos, quarks), as well as the Q = 1 reactions lν → W + Z , q u ¯ q d → W + Z , q u ¯ q d → W + γ , and lν → W + γ . Taking into account that the operators O W , O B , and O W W W do not give rise to anomalous triple neutral gaugeboson vertices we did not consider the γγ and ZZ final states.Table VII contains the unitarity violating terms for the inelastic processes above. As we can see, the operator O W W W does not contribute to the helicity amplitudes for which O W and O B do due to their different tensor structures. Inorder to impose unitarity constraints on these inelastic processes we will follow the procedure described in the previoussection [6]; see Eq. (20). We find that strongest constraints can be imposed by using two fermion states in the Q = 0( V a V b = W + W − ) combination | x i = 1 √ | N f (cid:0) − e −− e ++ + ν e − ¯ ν e + + N c u − ¯ u + − N c d − ¯ d + (cid:1) i , (26) | x i = 1 √ | N f (cid:0) − e − + e + − + N c u + ¯ u − − N c d + ¯ d − (cid:1) i , (27)where N f = 3 is the number of generations and N C = 3 the number of colours. They yield the bounds124 "(cid:12)(cid:12)(cid:12)(cid:12) g π f W W W Λ s (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) . g π f W Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ ⇒ (cid:12)(cid:12)(cid:12)(cid:12) f W W W Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) f W Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (28)121 (cid:12)(cid:12)(cid:12)(cid:12) √ s w c w g π f B Λ s (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) . g π f B Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ ⇒ (cid:12)(cid:12)(cid:12)(cid:12) f B Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ f i s/ Λ , we must consider the case where more than one of theparameters is non-vanishing. Therefore, we should search for the largest allowed value of a given parameter whilevarying over the others. Technically we obtain these generalized bounds by searching in a six-dimensional grid thewidest range of the parameters which satisfy both the elastic and inelastic partial-wave unitarity constraints. We get: (cid:12)(cid:12)(cid:12)(cid:12) f Φ2 , Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) f W Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) f B Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) f W W Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (29) (cid:12)(cid:12)(cid:12)(cid:12) f BB Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) f W W W Λ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ . It is important to stress that these results do not mean that the largest ranges for each parameter can all simultaneouslybe realized but rather they are the most conservative constraints on a given parameter allowing for all possiblecancellations with the others in the scattering amplitudes.Comparing the results in Eq. (29) with those in Eqs. (25) and (28) we find that working in the most generalsix-dimensinal space the bounds become weaker, but not substantially. Thus, even when allowing for all possiblecancellations between the contribution of the relevant dimension-six operators, partial-wave unitarity still imposesconstraints on their range of validity.We can now compare the unitarity constraints in Eq. (29) with the bounds on the corresponding coefficients fromthe global analysis of the available data from Tevatron and LHC Higgs results as well as from triple anomalous gaugecoupling bounds as updated from Ref. [11]. Mapping the allowed ranges at 90%CL of the six dimensional space fromthat analysis onto the unitarity constraints in Eq. (29) we find the lowest energy for which presently allowed values ofthe coefficients of operators affecting Higgs physics would lead to unitarity violation. For the operator O W W W whichonly affects gauge boson self-couplings we can naively estimate the bound by using the presently allowed range onthe effective parameter λ γ [14] from the PDG [21], λ γ = − . ± . O W W W by λ γ = λ Z = M W g f WWW Λ . Altogether we find: − ≤ f Φ , Λ (TeV − ) ≤ . ⇒ √ s ≤ . , − . ≤ f W Λ (TeV − ) ≤ . ⇒ √ s ≤ . , − ≤ f B Λ (TeV − ) ≤ . ⇒ √ s ≤ . , − . ≤ f W W Λ (TeV − ) ≤ . ⇒ √ s ≤ . , (30) − . ≤ f BB Λ (TeV − ) ≤ . ⇒ √ s ≤
11 TeV , − ≤ f W W W Λ (TeV − ) ≤ . ⇒ √ s ≤ . . In summary, in this work we have consistently derived the partial-wave unitarity bounds on the general spaceof dimension-six operators affecting Higgs and/or electroweak gauge boson interactions from two-to-two scatteringprocesses including vector boson and Higgs boson scattering channels, as well as inelastic processes f ¯ f ′ → V V ′ where f ( ′ ) is a SM fermion and V ( ′ ) is an electroweak gauge boson. We have found that the relevant set reduces to sixoperators and gauge invariance enforces that the corresponding amplitudes only diverge as s in the large s limit. Themost general bounds obtained in this framework are given in Eq. (29). They can be translated on the maximumcenter-of-mass energy for which the presently allowed range of the coefficients of the corresponding operators from theanalysis of Higgs and gauge-boson data will satisfy partial-wave unitarity. We find that for those operators affectingthe Higgs couplings, present 90% constrains from global data analysis of Higgs and electroweak data are such thatunitarity is not violated if √ s ≤ . O W W W , naive translation of thepresent bounds from triple-gauge boson analysis indicate that within its presently allowed 90% range unitarity canbe violated in f ¯ f ′ → V V ′ at center-of-mass energy √ s ≥ . Acknowledgments
We thank J. Gonzalez-Fraile for a careful reading of the manuscript and comments.O.J.P.E. is supported in part by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) and byFunda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo (FAPESP); M.C.G-G and T.C are supported by USA-NSFgrants PHY-0653342 and PHY-13-16617 and by FP7 ITN INVISIBLES (Marie Curie Actions PITN-GA-2011-289442).M.C.G-G also acknowledges support by grants 2014-SGR-104 and by FPA2013-46570 and consolider-ingenio 2010program CSD-2008-0037. [1] C. Bilchak, M. Kuroda and D. Schildknecht, Nucl. Phys. B , 7 (1988).[2] G. J. Gounaris, J. Layssac and F. M. Renard, Phys. Lett. B , 146 (1994) [hep-ph/9311370].[3] G. J. Gounaris, J. Layssac, J. E. Paschalis and F. M. Renard, Z. Phys. C , 619 (1995) [hep-ph/9409260].[4] G. J. Gounaris, F. M. Renard and G. Tsirigoti, Phys. Lett. B (1995) 212 [hep-ph/9502376].[5] C. Degrande, EPJ Web Conf. , 14009 (2013) [arXiv:1302.1112 [hep-ph]].[6] U. Baur and D. Zeppenfeld, Phys. Lett. B , 383 (1988).[7] W. Buchmuller and D. Wyler, Nucl. Phys. B , 621 (1986); B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek,JHEP , 085 (2010) [arXiv:1008.4884 [hep-ph]].[8] K. Hagiwara, S. Ishihara, R. Szalapski and D. Zeppenfeld, Phys. Rev. D , 2182 (1993).[9] K. Hagiwara, T. Hatsukano, S. Ishihara and R. Szalapski, Nucl. Phys. B , 66 (1997) [hep-ph/9612268].[10] T. Corbett, O. J. P. Eboli, J. Gonzalez-Fraile and M. C. Gonzalez-Garcia, Phys. Rev. D , 075013 (2012) [arXiv:1207.1344[hep-ph]].[11] T. Corbett, O. J. P. Eboli, J. Gonzalez-Fraile and M. C. Gonzalez-Garcia, Phys. Rev. D , 015022 (2013) [arXiv:1211.4580[hep-ph]].[12] T. Corbett, O. J. P. Eboli, J. Gonzalez-Fraile and M. C. Gonzalez-Garcia, Phys. Rev. Lett. , no. 1, 011801 (2013)[arXiv:1304.1151 [hep-ph]]. [13] I. Brivio, O. J. P. Eboli, M. B. Gavela, M. C. Gonzalez-Garcia, L. Merlo and S. Rigolin, arXiv:1405.5412 [hep-ph].[14] K. Hagiwara, R. D. Peccei, D. Zeppenfeld and K. Hikasa, Nucl. Phys. B , 253 (1987).[15] K. Hagiwara, S. Ishihara, R. Szalapski, and D. Zeppenfeld, Phys. Rev. D48 , 2182 (1993).[16] S. Alam, S. Dawson and R. Szalapski, Phys. Rev. D , 1577 (1998) [hep-ph/9706542].[17] A. De Rujula, M. Gavela, P. Hernandez, and E. Masso, Nucl. Phys. B384 , 3 (1992).[18] M. Baak et al. [Gfitter Group Collaboration], Eur. Phys. J. C , 3046 (2014) [arXiv:1407.3792 [hep-ph]].[19] M. Jacob and G. C. Wick, Annals Phys. , 404 (1959) [Annals Phys. , 774 (2000)].[20] C. Csaki, C. Grojean, H. Murayama, L. Pilo and J. Terning, Phys. Rev. D , 055006 (2004) [hep-ph/0305237].[21] K. A. Olive et al. [Particle Data Group Collaboration], Chin. Phys. C , 090001 (2014). Appendix A: Anomalous interactions
Here we present the anomalous interactions that are generated by the dimension–six operators in Eqs. (2)–(11).For simplicity of discussion we make use of the unitary gauge in which the Higgs doublet becomes:Φ = 1 √ v + h ( x ) ! (A1)where v is the Higgs vacuum expectation value (vev).We first note that O φ, , O φ, , and O φ, lead to corrections of the kinetic term for the Higgs field, therefore, wemake a field redefinition to obtain a canonical form for the kinetic Higgs term: H = h r v ( f Φ , + 2 f Φ , + f Φ , ) , (A2)resulting, together with O φ, , in corrections to the Higgs mass given by (to linear order) M H = 2 λv (cid:18) − v ( f Φ , + 2 f Φ , + f Φ , + f Φ , λ ) (cid:19) , (A3)where λ is the quartic scalar coupling. Additionally O BW affects Zγ mixing giving corrected mass eigenstates of theform: Z µ = (cid:20) − g g ′ g + g ′ ) v Λ f BW (cid:21) − / Z SMµ (A4) A µ = (cid:20) g g ′ g + g ′ ) v Λ f BW (cid:21) − / A SMµ − (cid:20) gg ′ ( g − g ′ )4( g + g ′ ) v Λ f BW (cid:21) Z SMµ (A5)where: Z SMµ = 1 p g + g ′ ( gW µ − g ′ B µ ) and A SMµ = 1 p g + g ′ ( g ′ W µ + gB µ ) . (A6)Furthermore, the operator O Φ , simultaneously affects the W and Z boson masses, while O Φ , and O BW only affectthe Z mass. Again to linear order we have: M Z = g + g ′ v (cid:18) v (cid:18) f Φ , + f Φ , − g g ′ g + g ′ f BW (cid:19)(cid:19) , (A7) M W = g v (cid:18) v f Φ , (cid:19) . (A8)Notice that in all expressions above v represents the vev of the Higgs field at the minimum of the potential includingthe effect of O Φ , .We will use in our analysis as inputs the measured values of G F , M Z , and α , the so called Z -scheme [17], and forconvenience we absorb the tree-level renormalization factors mentioned in equations (A7) and (A8) into the measuredvalue of M W . Through the relation G F √ = g M W and equations (A7) and (A8), we obtain the relations: v = ( √ G F ) − / (cid:18) − v f Φ , (cid:19) , (A9) M Z = ( √ G F ) − g (cid:18) v f Φ , − g g ′ g + g ′ ) v Λ f BW (cid:19) , (A10)where we have introduced the tree level weak mixing angle, c W ≡ g/ p g + g ′ .The dimension-six effective operators give rise to triple Higgs-gauge interactions, taking the following forms: L HV V eff = g Hγγ HA µν A µν + g (1) HZγ A µν Z µ ∂ ν H + g (2) HZγ HA µν Z µν + g (1) HZZ Z µν Z µ ∂ ν H + g (2) HZZ HZ µν Z µν + g (3) HZZ HZ µ Z µ (A11)+ g (1) HW W ( W + µν W − µ ∂ ν H + h . c . ) + g (2) HW W HW + µν W − µν + g (3) HW W HW + µ W − µ , where we have defined V µν = ∂ µ V ν − ∂ ν V µ , for V = A, Z, W and g Hγγ = − (cid:16) g v s (cid:17) f BB + f WW − f BW g (1) HZγ = (cid:16) g v (cid:17) s W ( f W − f B )2c W g (2) HZγ = (cid:16) g v (cid:17) s W [2s f BB − f WW +(c − s ) f BW ]2c W g (1) HZZ = (cid:16) g v (cid:17) c f W +s f B g (2) HZZ = − (cid:16) g v (cid:17) s f BB + c f WW +c s f BW g (3) HZZ = (cid:16) g v (cid:17) h v (cid:16) f Φ , + 3 f Φ , − f Φ , − g g ′ ( g + g ′ ) f BW (cid:17)i = M Z ( √ G F ) / h v ( f Φ , + 2 f Φ , − f Φ , ) i g (1) HW W = (cid:16) g v (cid:17) f W g (2) HW W = − (cid:16) g v (cid:17) f W W g (3) HW W = (cid:16) g v (cid:17) h v (3 f Φ4 − f Φ , − f Φ , ) i = 2 M W ( √ G F ) / h v (2 f Φ , − f Φ , − f Φ , ) i (A12)Quartic vertices involving Higgs and gauge bosons read: L HHV V eff = g (1) HHW W H W + µν W − µν + g (2) HHW W H ( ∂ ν H )( W − µ W + µν + h . c . )+ g (3) HHW W H W + µ W − µ + g (1) HHZZ H Z µν Z µν + g (2) HHZZ HZ ν ( ∂ µ H ) Z µν + g (3) HHZZ H Z µ Z µ (A13)+ g (1) HHZA H ( ∂ µ H ) Z ν A µν + g (2) HHZA H A µν Z µν + g (1) HHAA H A µν A µν , g (1) HHW W = − g f W W g (2) HHW W = g f W g (3) HHW W = g h v (5 f Φ , − f Φ , − f Φ , ) i = M W √ G F h v (5 f Φ , − f Φ , − f Φ , ) i g (1) HHZZ = − g Λ ( c f W W + s f BB + c s f BW ) g (2) HHZZ = − g Λ (c f W + s f B ) g (3) HHZZ = g h v (5 f Φ , + 5 f Φ , − f Φ , − g g ′ ( g + g ′ ) f BW ) i = M Z √ G F h v (4 f Φ , + 5 f Φ , − f Φ , ) i g (1) HHZA = − g s W W Λ ( f W − f B ) g (2) HHZA = − g s W W Λ (c f W W − s f BB − (c − s ) f BW ) g (1) HHAA = − g s ( f W W + f BB − f BW ) (A14)and L HV V V eff = g (1) HZW W H ( W − µ W + ν − h . c . ) Z µν + g (2) HZW W HZ µ ( W + ν W − µν − h . c . ) + g (3) HZW W ( ∂ µ H ) Z ν ( W − µ W + ν − h . c . )+ g (1) HAW W H ( W − µ W + ν − h . c . ) A µν + g (2) HAW W HA ν ( W + νµ W − µ − h . c . ) (A15)+ g (3) HAW W ( ∂ µ H ) A ν ( W − µ W + ν − h . c . ) , with g (1) HZW W = ig v W Λ (c f W − s f B + 4c f W W + 2s f BW ) g (2) HZW W = − ig v W Λ ( f W + 4c f W W ) g (3) HZW W = ig v W Λ s f W g (1) HAW W = ig v s W ( f W + f B + 4 f W W − f BW ) g (2) HAW W = − ig s W v Λ f W W g (3) HAW W = − ig v s W f W (A16)Triple gauge boson couplings are: L W W V eff = g (1) W W Z ( W + ν W − µ − h . c . ) Z µν + g (2) W W Z ( W + µν W − µ Z ν − h . c . ) + g (3) W W Z ( W + µν W − νρ − h . c . ) Z ρµ + g (1) W W A ( W + ν W − µ − h . c . ) A µν + g (2) W W A ( W + µν W − νρ − h . c . ) A ρµ , (A17)where g (1) W W Z = ig v c W ( f W + s c f B + c W f BW − e c W f Φ , ) ≡ ig c W ∆ κ Z g (2) W W Z = − ig v W Λ ( f W + c W f BW − s W e c W f Φ , ) ≡ − ig c W ∆ g Z g (3) W W Z = − ig c W Λ f W W W ≡ − ig c W M W λ Z g (1) W W A = ig v s W ( f W + f B − f BW ) ≡ ig s W ∆ κ γ g (2) W W A = − ig s W f W W W ≡ − ig s W M W λ γ (A18)where we have defined c W = cos(2 θ w ) and s W = sin(2 θ w ).Quartic gauge boson vertices read: L W W V V eff = g (1) W W W W W − µ W + ν ( W − µ W + ν − h . c . ) + g (2) W W W W W + µν W − νρ ( W + µ W − ρ − W + ρ W − µ )+ g (1) W W ZZ Z µ Z µ W + ν W − ν + g (2) W W ZZ Z µ Z ν ( W + ν W − µ + h . c . ) + g (3) W W ZZ (cid:0) W + µν Z µρ ( Z ν W − ρ − Z ρ W − ν ) + h . c . (cid:1) + g (3) W W AA (cid:0) W + µν A µρ ( A ν W − ρ − A ρ W − ν ) + h . c . (cid:1) + g (1) W W ZA W − µ W + µ Z µ A µ + g (2) W W ZA ( W − ν W + µ + h . c . ) A ν Z µ (A19)+ g (3) W W ZA (cid:0) W + µν Z µρ ( A ν W − ρ − A ρ W − ν ) + W + µν A µρ ( Z ν W − ρ − Z ρ W − ν ) + h . c . (cid:1) g (1) W W W W = e + g v ( f W + 2 s c W f BW − s W c W e f Φ , ) g (2) W W W W = − g f W W W g (1) W W ZZ = − e s − g v Λ (c f W + s W c W f BW − s W c e c W f Φ , ) g (2) W W ZZ = e c + g v Λ (c f W + s W c W f BW − s W c e c W f Φ , ) g (3) W W ZZ = − g v c f W W W g (3) W W AA = − g v s f W W W g (1) W W ZA = − e − g v s W W Λ ( f W + 2 s c W f BW − s W c W e f Φ , ) g (2) W W ZA = e + g v s W W Λ ( f W + 2 s c W f BW − s W c W e f Φ , ) g (3) W W ZA = − g s W c W f W W W (A20)Finally Higgs self interactions take the form: L HHH eff = g (1) HHH H + g (2) HHH H ( ∂ µ H )( ∂ µ H ) , (A21)(A22) L HHHH eff = g (1) HHHH H + g (2) HHHH H ( ∂ µ H )( ∂ µ H ) , (A23)where g (1) HHH = − λv + v Λ ( λ f Φ , + f Φ , + λ f Φ , + λ f Φ , )= − M H ( √ G F ) / h − v ( f Φ , + 2 f Φ , λ f Φ , ) i g (2) HHH = v Λ ( f Φ , + f Φ , + f Φ , ) g (1) HHHH = − λ + v ( λf Φ , + f Φ , + 2 λf Φ , + λf Φ , )= − M H ( √ G F ) h v ( f Φ , + λ f Φ , + f Φ , ) i g (2) HHHH = ( f Φ , + 2 f Φ , + f Φ , ) (A24) Appendix B: Helicity Amplitudes
We present here the list of unitarity violating amplitudes for all the 2 → ( × f Φ , , Λ × s ) W + W + → W + W + − W + Z → W + Z − XW + H → W + H − XW + W − → W + W − YW + W − → ZZ W + W − → HH − ZZ → HH − ZH → ZH − X TABLE II: Unitarity violating (growing as s ) terms of the scattering amplitudes M ( V λ V λ → V λ V λ ) for longitudinalgauge bosons generated by the operators O Φ , and O Φ , where X = 1 − cos θ and Y = 1 + cos θ . The overall factor extractedfrom all amplitudes is given at the top of the table. ( × e f W Λ × s )0000 00 + + 0 + 0 − − − +0 − W + W + → W + W + − X − Y − Y X W + Z → W + Z − X − W c − s X − W Y − W Y X − W W + γ → W + γ − − X − − − − W + Z → W + γ − W (3c − s )16c W s W X − W Y − − W + Z → W + H − − − W Y − W W + γ → W + H − − − W Y − − − W W + H → W + H − X − − − − X − W + W − → W + W − Y − X X − W + W − → ZZ s − c W X − W Y − W Y W X − W + W − → γγ − − − − − − − W + W − → Zγ − − W s W − W X − W Y − − W + W − → ZH − − − W Y − − W X W + W − → γH − − W Y − W X − W + W − → HH − − − − − − ZZ → ZZ − X − Y − Y X − ZZ → Zγ − − W s W W s W X − − W s W Y − − ZZ → HH − − − − − − Zγ → ZZ − − W s W X − W s W Y − − − W c W Zγ → HH − − − − − − W c W ZH → ZH − X − − − − X − ZH → γH − − − − − W s W X − TABLE III: Unitarity violating (growing as s ) terms of the scattering amplitudes M ( V λ V λ → V λ V λ ) for gauge bosonswith the helicities λ λ λ λ listed on top of each column, generated by the operator O W . X = 1 − cos θ and Y = 1 + cos θ .The overall factor extracted from all amplitudes is given on the top of the table. An entry marked as 0 means that there is no s growth for the amplitude, while we denote as − an amplitude that does not exist. ( × e f B Λ ) × s − − − +0 − W + W + → W + W + − W + Z → W + Z − W s − c X − W Y − W Y − W W + γ → W + γ − − X − − − − W + Z → W + γ − W c − W c W X − W Y − − W + Z → W + H − Y − − − W Y − W W + γ → W + H − − − W Y − − − W W + W − → W + W − Y W + W − → ZZ c − s W X − W Y − W Y W X W + W − → γγ − − − − − − − W + W − → Zγ − − W s W − W X − W Y − − W + W − → ZH − Y − − − W Y − − W X W + W − → γH − − W Y − W X − ZZ → ZZ − X − Y − Y X − ZZ → Zγ − W s W − W s W X − W s W Y − − ZZ → HH − − − − − − Zγ → ZZ − − − W s W X W s W Y − − W c W Zγ → HH − − − − − − − W c W ZH → ZH − X − − − − X − ZH → γH − − − − − − W s W X − TABLE IV: Same as Table III for the operator O B . ( × e f WW Λ × s ) ( × e f BB Λ ) × s
00 + + 0 + 0 − − − +0 − − − − +0 − W + W + → W + W + − X Y Y − X W + Z → W + Z − c X − X − s X W + γ → W + γ − − X − − − − − − X − − − − W + Z → W + γ − c W W X − − − s W W X − − − W + H → W + H − − − − − X − − − − − − W + W − → W + W − − X − X W + W − → ZZ c s W + W − → γγ − − − − − − − − − − W + W − → Zγ c W W − − − − s W W − − − W + W − → HH − − − − − − − − − − − ZZ → ZZ c − c X c Y c Y − c X c s − s X s Y s Y − s X s ZZ → γγ − − − − − − − − − − ZZ → Zγ c W W − c W W X − c W W Y − − − s W W s W W X − − s W W Y − − ZZ → HH − − − − − − c − − − − − − s Zγ → ZZ − − c W W X c W W Y − − c W W − s W W X − s W W Y − − − s W W Zγ → Zγ − − X − − − − − − X − − − − Zγ → HH − − − − − − c W W − − − − − s W W γγ → HH − − − − − − − − − − − − ZH → ZH − − − − − c X − − − − − − s X − γH → γH − − − − − X − − − − − − X − ZH → γH − − − − − c W W X − − − − − s W W X − TABLE V: Same as Table III for the operators O WW and O BB .( × e f WWW Λ × s )00 + + 0 + 0 − − − +0 − − + − −− + − +++ + − + + + −− W + W + → W + W + − Y )32s X )32s X )32s − Y )32s − W + Z → W + Z Y − X )c W X +2)c W X +2)c W Y − X )c W − X XW + γ → W + γ − − − − − − X XW + Z → W + γ − Y − X )32s − X +2)32s − − − W X W XW + Z → W + H − − X +2)c W − Y )32s − Y − X )c W − − W + γ → W + H − − X +2)32s − − − Y − X )32s − − W + W − → W + W − Y − X )32s Y )32s Y )32s Y − X )32s Y − YW + W − → ZZ Y )c W − X +2)c W − X +2)c W Y )c W − W + W − → γγ − − − − − − W + W − → Zγ Y )32s − − X )32s − − W − W W + W − → ZH − − − X )c W − − Y )c W Y − X )32s − − W + W − → γH − − − X +2)32s − − Y )32s − − − TABLE VI: Same as Table III for the operator O WWW . Process σ , σ , λ , λ Amplitude e + e − → W − W + − + 00 − ig s sin θ f W +s f B c Λ + − − ig s sin θ f B c Λ − + −− − ig s sin θ f WWW Λ − + ++ − ig s sin θ f WWW Λ ν ¯ ν → W − W + : − + 00 ig s sin θ f W − s f B c + −
00 0 − + −− ig s sin θ f WWW Λ − + ++ ig s sin θ f WWW Λ u ¯ u → W − W + − + 00 ig N c s sin θ f W +s f B + − ig N c s sin θ c f B − + −− ig N c s sin θ f WWW Λ − + ++ ig N c s sin θ f WWW Λ d ¯ d → W − W + − + 00 − ig N c s sin θ f W − s f B + − − ig N c s sin θ
12 s f B c Λ − + −− − ig N c s sin θ f WWW Λ − + ++ − ig N c s sin θ f WWW Λ e + ¯ ν → W + Z − + 00 ig s sin θ √ f W Λ + −
00 0 − + −− i c W g s sin θ √ f WWW Λ − + ++ i c W g s sin θ √ f WWW Λ e + ¯ ν → W + A : − + 00 0+ −
00 0 − + −− i s W g s sin θ √ f WWW Λ − + ++ i s W g s sin θ √ f WWW Λ TABLE VII: Unitarity violating (growing as s ) terms of the scattering amplitudes M ( f σ ¯ f λ → V λ V λ ) for fermions andgauge bosons with the helicities σ σ λ λ4