Exploring extended scalar sectors with di-Higgs signals: a Higgs EFT perspective

Journal of High Energy Physics, May 2018

Abstract We consider extended scalar sectors of the Standard Model as ultraviolet complete motivations for studying the effective Higgs self-interaction operators of the Standard Model effective field theory. We investigate all motivated heavy scalar models which generate the dimension-six effective operator, |H|6, at tree level and proceed to identify the full set of tree-level dimension-six operators by integrating out the heavy scalars. Of seven models which generate |H|6 at tree level only two, quadruplets of hypercharge Y = 3Y H and Y = Y H , generate only this operator. Next we perform global fits to constrain relevant Wilson coefficients from the LHC single Higgs measurements as well as the electroweak oblique parameters S and T. We find that the T parameter puts very strong constraints on the Wilson coefficient of the |H|6 operator in the triplet and quadruplet models, while the singlet and doublet models could still have Higgs self-couplings which deviate significantly from the standard model prediction. To determine the extent to which the |H|6 operator could be constrained, we study the di-Higgs signatures at the future 100 TeV collider and explore future sensitivity of this operator. Projected onto the Higgs potential parameters of the extended scalar sectors, with 30 ab−1 luminosity data we will be able to explore the Higgs potential parameters in all seven models.

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Exploring extended scalar sectors with di-Higgs signals: a Higgs EFT perspective

HJE Exploring extended scalar sectors with di-Higgs signals: a Higgs EFT perspective Tyler Corbett 0 1 3 7 8 9 10 Aniket Joglekar 0 1 2 3 4 5 7 8 9 10 Hao-Lin Li 0 1 3 5 7 8 9 10 Jiang-Hao Yu 0 1 3 5 6 7 8 9 10 Victoria 0 1 3 7 8 9 10 Australia 0 1 3 7 8 9 10 0 School of Physics, The University of Melbourne 1 Beijing 100190 , China 2 Enrico Fermi Institute and Kavli Institute for Cosmological Physics, University of Chicago 3 Chinese Academy of Sciences 4 Department of Physics and Astronomy, University of California-Riverside , USA 5 Amherst Center for Fundamental Interactions 6 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics 7 900 University Ave. , Riverside, CA 92521 , U.S.A 8 Chicago , IL 60637 , U.S.A 9 Amherst , MA 01003 , U.S.A 10 Department of Physics, University of Massachusetts Amherst , USA We consider extended scalar sectors of the Standard Model as ultraviolet complete motivations for studying the e ective Higgs self-interaction operators of the Standard Model e ective eld theory. We investigate all motivated heavy scalar models which generate the dimension-six e ective operator, jHj6, at tree level and proceed to identify the full set of tree-level dimension-six operators by integrating out the heavy scalars. Of seven models which generate jHj6 at tree level only two, quadruplets of hypercharge Y = 3YH and Y = YH , generate only this operator. Next we perform global ts to constrain relevant Wilson coe cients from the LHC single Higgs measurements as well as the electroweak oblique parameters S and T . We nd that the T parameter puts very strong constraints on the Wilson coe cient of the jHj6 operator in the triplet and quadruplet models, while the singlet and doublet models could still have Higgs self-couplings which deviate signi cantly from the standard model prediction. To determine the extent to which the jHj6 operator could be constrained, we study the di-Higgs signatures at the future 100 TeV collider and explore future sensitivity of this operator. Projected onto the Higgs potential parameters of the extended scalar sectors, with 30 ab 1 luminosity data we will be able to explore the Higgs potential parameters in all seven models. Beyond Standard Model; E ective Field Theories; Higgs Physics 1 Introduction 2 The e ective Lagrangian Real scalar singlet Complex scalar singlet Two Higgs doublet model Real scalar triplet Complex scalar triplet Quadruplet with Y = 3YH Quadruplet with Y = YH Summary of EFTs 3 Higgs coupling measurements at the LHC Electroweak precision measurements Higgs diphoton rate Higgs global ts 3.4 Implications for the UV physics 4 Di-Higgs production at the 100 TeV collider General formalism on di-Higgs production Di-Higgs cross section Monte Carlo simulation and validation Determination of Wilson coe cients Exploring parameter region in UV models 5 Conclusions A Unitarity considerations 1 Introduction 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3.1 3.2 3.3 4.1 4.2 4.3 4.4 4.5 The discovery of the Higgs boson at the Large Hadron Collider (LHC) marked the discovery of the last missing piece of the Standard Model (SM). Precision measurements of the Higgs couplings are a major goal for current and future high energy experiments. Current experimental results provide strong evidence that the nature of the Higgs boson is consistent with the predictions of the SM. The measurement of this behavior is entirely dependent on single Higgs phenomena through precision measurements of the Higgs couplings to the vector bosons and the SM fermions. On the other hand, the Higgs self-interactions, responsible for Electroweak Symmetry Breaking (EWSB), still remain undetermined experimentally. { 1 { electroweak phase transition (EWPT). In order to investigate the generic features of the trihiggs coupling at the LHC and a future collider we adopt an E ective Field Theory (EFT) approach [1{4]. In doing so we assume some possible new physics beyond the SM which modi es the Higgs couplings and is heavy with, for example, new physics scales such as NP TeV. The e ects of the new physics are parametrized by higher dimension e ective operators, and the dimension-six QH = (HyH)3 operator is the leading operator which modi es the momentum independent Higgs self-couplings at low energy. The QH operator remains the only operator related to the Higgs sector unconstrained by current experiment. In order to motivate this study of the e ective operator QH we consider ultraviolet (UV) complete models which may generate this operator at tree level and therefore with a larger Wilson coe cient. This requirement combined with Lorentz invariance then limits our consideration to extended scalar sectors.1 Additionally the new scalar must not be charged under SU(3)c as closure of color indices requires QH be generated at one loop. Such scalar extensions of the SM constitute relatively simple scenarios beyond the SM which are also well-motivated by studies of the electroweak phase transition and baryogenesis [5, 6], having dark matter candidates [7{9], or mechanisms for neutrino mass generation [10{13]. The complete list of the scalar extensions which generate a tree-level QH are real [5, 7, 8, 14] and complex singlets [6], the two Higgs doublet model (2HDM) [15{17], real [9, 18] and complex [10{13] triplets, and complex quadruplets. Assuming the new scalars in these models are heavy, we utilize an EFT approach to study their e ects on electroweak precision tests, modi cations of the single Higgs couplings, and the di-Higgs production process in a model-independent and predictive way. Many new physics models with SM-compatible single Higgs phenomena could exhibit di-Higgs phenomenology distinct from that of the SM [19, 20]. The modi cations of the Higgs trilinear couplings can only be directly observed in Higgs boson pair production, therefore the di-Higgs process at the LHC and future colliders is the only direct way to measure the Wilson coe cient of the e ective QH operator. Alternatively the trilinear Higgs coupling can be studied indirectly [21, 21, 22, 22{25]. However our paper will focus on the direct constraints, we discuss the indirect constraints brie y at the end of section 2. The di-Higgs production mechanism at hadronic colliders is dominated by the gluon fusion process which includes the triangle and the box contributions from the top quark. Due to destructive interference between these two contributions, the di-Higgs production cross section in the SM is typically small and thus challenging to observe in the near future. However, in the scalar extended models, the di-Higgs cross section may be increased considerably making measurement a possibility at the proposed 100 TeV collider [26, 27]. In this paper we investigate the di-Higgs production cross sections in the EFT framework, and study the discovery potential of the Wilson coe cients in the EFT at the proposed 100 TeV collider. 1Requiring closure of spinor and Lorentz indices implies fermions may only generate the QH operator Lagrangians after integrating out the new heavy degrees of freedom. Then in section 3 we study the implications of single-Higgs measurements on the corresponding EFTs as well as the implications of these constraints on the UV complete models. In section 4 we study these EFTs' impacts on di-Higgs production at the proposed 100 TeV collider. Finally our conclusions are found in section 5. 2 The e ective Lagrangian We consider all ultraviolet (UV) complete models which include one additional heavy scalar which generate, after integrating out the new scalar, dimension-six operators a ecting the trihiggs vertex at tree level. In order to generate tree level dimension-six operators one needs a term H2S or H3S, where S is the new heavy scalar, this is a result of all other models having an additional Z2 symmetry due to the requirements of the gauge symmetry and renormalizability.2 The relevant theories are then, real and complex scalar singlets, the two-Higgs doublet model, real and complex scalar triplets of SU(2)L with hypercharge Y = 0 and Y = 1 respectively, and nally complex scalar quadruplets of SU(2)L with either Y = 3=2 or Y = 1=2. For each model we write down the Lagrangians for each UV-model along with the corresponding e ective eld theory (EFT) to dimension-six at tree level, we will only write the new terms in addition to the standard model terms for convenience. In writing the EFTs we will follow the procedure of Henning et al. [28, 29]. To clarify our notation and conventions, we write here the general Lagrangian for all UV complete models, neglecting SM fermionic and gauge boson terms, considered: L = (D H)y(D H) 2(HyH) (HyH)2 + L : Where L contains all terms containing new elds (in the case of the models we consider this is one new scalar multiplet of SU(2) which may or may not have hypercharge). becoming negative signals spontaneous symmetry breaking leading to the massive gauge bosons of the SM. After deriving the EFTs we employ the Warsaw basis [2] for the dimension-six operators, translations between the various bases are included throughout much of the recent literature including a package for relating the bases [30]. The operators which are relevant to our analyses are: QH = (HyH)3 ; QH = (HyH) (HyH) ; QHD = (D H)yHHy(D H) ; QeH = (HyH)(LeRH) ; QuH = (HyH)(QuRH~ ) ; QdH = (HyH)(QdRH) : The fermionic operators should be summed over each generation with an appropriate Wilson coe cient. In general the fermionic operators can have o diagonal components, however for the models considered this is only possible for the two-Higgs doublet model and 2An exception to this is the HS3 or HS2 vertex, however the HS3 vertex will not generate operators at tree level below dimension-eight and the HS2 vertex does not exist for any representations given the Higgs is a doublet of SU(2)L. (2.1) 2 (2.2) { 3 { HJEP05(218)6 we will employ particular choices of the fermionic matrices in the model to suppress o diagonal components, as is motivated by studies of avor changing neutral currents, and therefore assume these operators to be diagonal. After integrating out the S eld we nd the EFT: We note that there are corrections to the renormalizable jHj4 vertex, which we will nd is a common feature of integrating out scalars in our models, as well as the dimensionsix operators QH and QH which a ect the trihiggs couplings. Additionally the term ggH3S=3=M 6 appears to be of the next order in the EFT expansion, we will retain these terms in the text, however in our summary tables 2 and 3 we neglect such corrections. 2.2 Complex scalar singlet For the complex scalar singlet we consider the case of Y = 0. While the complex scalar singlet is technically the same as introducing two real singlets, and therefore doesn't t our criteria for considered models, we consider it here as it has been studied extensively in the literature. Some examples from the literature which study the complex singlet case and its implications for in ation, the electroweak phase transition, enhancement of the di-Higgs signal, and vacuum stability include [6, 33{36]. The Lagrangian is then: (2.3) (2.4) M 2j j 2 0 4 2 j j 4 gHS(HyH) + h:c: H (HyH) 2 + h:c: 4 1 ( y)3 + h:c: : (M 0)2 2 3 g 3 + h:c: 2 + h:c: g0 3 0 2 H (HyH)j j 2 ( y)2 + h:c: 4 4 + h:c: (2.5) The M 0 term corrects the dimension-six operator coe cients with terms proportional to M 0=M which must be small for the validity of the EFT so we neglect them.3 Integrating 3M 0 is the parameter which dictates the size of the mass splitting between the components of the complex scalar eld. If M 0 were to become large it is possible that the lighter resonances would enter the low energy spectrum and invalidate our EFT approach. Therefore it is a requirement of our EFT approach that this parameter be small. For the same reason we will neglect the e ects of Y3 in the 2HDM below. { 4 { Again we induce corrections to the jHj4 vertex as well as the e ective operators QH and QH . 2.3 Two Higgs doublet model Of the many extended scalar sectors studied in the literature the two Higgs doublet model is the most well studied, reviews on the status of the model from the UV perspective have a long history (some extensive reviews include [15{17]), the two Higgs doublet model has also recently been studied in the EFT framework in great detail [ 32, 37, 38 ] including comparisons between the phenomenological aspects of both the UV complete and EFT frameworks at tree and one-loop levels [39, 40]. We begin in the \Higgs basis", where the doublets have already been rotated to a basis where the physical CP even state is the observed 125 GeV Higgs. This rotation is performed by rotation of H1 and H2 by the angle . We follow the notation of [ 37 ]. Note the Yukawa couplings are entered generically and later will be recast in terms of each of the four \types" usually considered to evade avor changing neutral currents when we write the EFT. These various types considered are outlined in table 1. L = (D H2)y(D H2) M 2jH2j2 Y3(H1yH2 + h:c:) Z22 jH2j4 2 Z5 (H1yH2)(H1yH2) Z7jH2j2(H1yH2) Z 2 Z7 jH2j2(H2yH1) Z3jH1j2jH2j2 Z4(H1yH2)(H2yH1) 5 (H2yH1)(H2yH1) Z6jH1j2(H1yH2) Z6 jH1j2(H2yH1) H2;iQj YuuR ij + H2;iQiYddR + H2;iLiYleR + h:c: : (2.7) The e ective Lagrangian for each \type" of 2HDM is then given below. Note we have neglected terms suppressed by Y3=M 2 as explained above in the complex scalar discussion. We adopt the notation cos = c and sin = s , where the mixing angle is the angle which diagonalizes the mass matrices of the charged scalars and pseudoscalars, to allow us to rewrite the Higgs-fermion couplings in terms of the mixing angle and the parameter Z6. { 5 { We see that the 2HDM only induces one purely bosonic operator, QH , at leading order in Y3=M 2, and induces various combinations of rescalings of the Yukawa couplings, i.e. the operators QeH , QuH , and QdH . The only di erence between the various realizations of the 2HDM considered are di erences in the weight of the fermionic operators, i.e. by tan or cot . To make manifest the mass dependence of the Higgs couplings to fermions above we have expanded the fermionic dimension-six operators (in the unitary gauge for convenience) to recast the couplings of H1 to fermions in terms of their masses, Z6, and the mixing angle . In particular the rst line of each expression indicates the shift of the Higgs-fermion couplings relative to the SM prediction, p 2m v LH = h R L : (2.12) Another unique feature of the 2HDM e ective Lagrangians is that they also contain 4Fermi operators. These are not relevant to our analysis and, as they are weighted by the square of the Yukawa, are unlikely to have large Wilson coe cients except possibly in the case of the top quark which has Yt { 6 { The real scalar triplet model [18, 41, 42] has been studied in the literature with ambitions of making the electroweak phase transition rst order, e.g. in [43], with the possibility of the neutral component being a dark matter candidate [9], as well as from an EFT point of view in [28, 44]. The relevant Lagrangian is given by, (2.13) HJEP05(218)6 Integrating out the heavy triplet then gives the e ective Lagrangian: L = g 2 8M 2 (HyH)2 g 2 2M 4 QHD g 2 8M 4 QH g 2 + 2M 4 (HyH)(D H)y(D H) g jHj2(D H)y(D H) for the other dimension-six operators at the cost of an error of the next order in the EFT (i.e. O(1= 4)). While it is frequently simpler to maintain the basis obtained after integrating out the heavy states [45], for the sake of this work which will consider many UV completions and their e ective eld theories we choose to project onto a common basis. Discussions of the validity of this method including proofs of the invariance of the S-matrix can be found in [46{49]. We perform the change of basis by using the Higgs equation of motion, scaled up to dimension-six through multiplication by additional Higgs elds, (HyH)(D H)y(D H) = Rv2(HyH)2 + QH + 2 RQH 1 2 + 1 2 (YlQlH + YdQdH + YuQuH + h:c:) + O(1= 4) ; (2.15) 4M 4 (YlQlH +YdQdH +YuQuH +h:c:) : where we have called the renormalized (HyH)2 coupling, (HyH)2 coupling of eq. (2.1), yielding the new form of eq. (2.14): R = + g2=8=M 2 with the v 2 2M 2 g2v2 16M 4 (HyH)2 g 2 2M 4 QHD + 8M 4 QH g 2 g 2 (2.16) Consistent with our other examples we have again generated the QH and QH operators, however interestingly we have also generated the QHD operator which will have important phenomenological implications which we discuss in section 3. { 7 { Charging the Scalar Triplet under hypercharge, Y = 1, has important uses in the Type II seesaw [10{13]. The relevant UV complete Lagrangian is then, L = jD Integrating out the heavy complex triplet yields the e ective Lagrangian, HJEP05(218)6 2 M 2 L = j H3 j (HyH)3 : 2 6j H3 j jHyD Hj2jHj2 : Note that for a quadruplet we expect a contribution to the T -parameter. This operator does not occur at dimension-six, but does at dimension-eight. Deriving only the dimension-eight operator contributing to the T -parameter yields: Here we have con rmed the sign of [51]. We will see in the case of Y = YH we obtain a di erent sign from this work. L = 2jMgj22 (HyH)2 + Mjgj24 (HyH)(D H)y(D H) + Mjgj24 QHD jgj2 2M 4 H 2 + 0 4 which after applying the equation of motion from eq. (2.15) (notice here R = +jg2j=2=M 2) gives the nal form for the e ective Lagrangian: L = jgj2 H 4 + jgj2v2 2M 4 2 0 8 (HyH)2 + 2jMgj24 QH + Mjgj24 QHD jgj2 (2.19) This e ective Lagrangian and the e ective operators it contains are consistent with our expectations from the other models, particularly the real scalar triplet. 2.6 Quadruplet with Y = 3YH given by: For the two quadruplet models we follow the notation of [50], the UV Lagrangian is then jklHl ( Integrating out the quadruplet leads to the simple EFT, { 8 { (2.17) QH ; (2.18) (2.20) (2.21) (2.22) The UV complete Lagrangian is given by, jklHl ( 2 2j H3 j jHyD Hj2jHj2 : which we supplement with the dimension-eight T -parameter operator. This expression agrees with [51] up to a sign. As the sign of the dimension-eight T parameter operators in each quadruplet model come purely from the covariant derivative term of the Lagrangians (other contributions cancel) they should be the same in both eqs. (2.22) and (2.24). 2.8 Summary of EFTs Finally after deriving the corresponding EFTs for each model we may construct a table with the Wilson coe cients for each operator for each model considered. We summarize the renormalization of the (HyH)2 term in table 2 and the Wilson coe cients of the dimensionsix operators in table 3. While it appears that of all the theories the 2HDM is the only which does not generate a correction to the renormalizable (HyH)2, this is a re ection of neglecting terms suppressed by Y3=M 2, these corrections are generated rst at O(Y3=M 2). Unsurprisingly neither the 2HDM nor the two singlet models generate QHD, also referred to as the T -parameter operator as they are known not to shift the relation between the W and Z-masses. It is, however, expected from studies of the dynamics of the triplet models below EWSB that the triplet models considered in this work correct the T -parameter. This is consistent with our ndings in Equations (2.16) and (2.19). In the case of the quadruplet we found they were unique in that at dimension-six they generate only one operator, QH , and that the T -parameter operator was generated at dimension-eight. Additionally, as there are no allowed tree level couplings to Fermions in any of the theories except the 2HDM none of the other theories generate the fermionic operators, however after trading the operator (HyH)(D H)y(D H) in the triplet models via the EOM we do generate the fermionic operators for the two triplet models. The case of the quadruplets is particularly interesting as studies which indirectly probe the Higgs self coupling, such as [21], only allow the SM coupling to vary. Our work indicates that, within the assumptions of our EFT,4 such a study corresponds to a very speci c 4For example relaxing the assumptions of a single new multiplet one could envision a scenario with multiple quadruplets in which the T -parameter bounds may be evaded allowing for a sizable H6 operator coe cient and no other operators at dimension-six. In the case where only the H6 operator is generated the indirect constraints may be more stringent than those of di-Higgs production [24]. { 9 { (2.23) (2.24) nal renormalized (HyH)2 coupling (i.e. after shifting the operators by the EOM) including from eq. (2.1). In this table, as mentioned in the text in the Real Scalar singlet discussion, we neglect terms which are of O(g4=M 6). UV complete scenario, in the case where one expects the NP to come from dimension-six operators this corresponds to the quadruplets. In the case of the quadruplets the shift in due to the e ective operators is restricted to be extremely small since the same UV parameter that generates the operator QH contributes to the strongly constrained T -parameter. This demonstrates that indirect probes of the Higgs self coupling which don't vary other Higgs couplings are incomplete or correspond to speci c UV completions which do not satisfy the criterion of the UV complete models considered. Other studies which vary these additional couplings of the Higgs such as [22, 23] indicate the bounds on the Higgs self coupling are weakened or even lost without the inclusion of the direct di-Higgs probe. It is useful to project these e ective Lagrangians into Lorentz forms relevant to the di-Higgs analysis performed. We do so here, from the perspective of arbitrary Wilson coefcients, when the nal analyses are performed we use the expressions for the Wilson coefcients expressed in table 3. We assume that only the heaviest generation for each fermion has a non-negligible contribution to the EFT. Starting from the e ective Lagrangian, L = (D H)y(D H) + j j2(HyH) RF (HyH)2 + cH QH + cH QH + cHDQHD + ceH QeH + cuH QuH + cdH QdH ; (2.26) we can proceed to expand the operators to nd the relevant Lorentz forms. Here we have used RF to represent the nal renormalized coe cient of the (HyH)2 operator, the expression for RF may be found in table 2 in terms of of eq. (2.1) and the parameters of each UV-model. This involves nite eld renormalizations as the operators QH and QHD both alter the Higgs kinetic term below EWSB. Details of this procedure may be found in, for example, [52{54]. Below EWSB expanding out the Lorentz forms we nd (employing the unitary gauge): L = gH(3Z)Z hZ Z +gHW W hW +W + gHeheLeR +gHuhuLuR +gHdhdLdR +h:c: + gHHuh2uLuR +h:c: + : (2.27) HJEP05(218)6 normalizations corresponding to each coupling constant should be read directly from the relevant Lagrangians in text. In this table, as mentioned in the text in the Real Scalar singlet discussion, we neglect terms which are of O(g4=M 6). While the operator QHD is not generated in the quadruplet models we have entered the contributions to the T parameter in terms of an e ective coe cient for this operator into the table. Here \ " indicates the various operators and Lorentz forms which have no impact on our analysis. The coe cients of the terms in the Lagrangian of eq. (2.27) are given by: HJEP05(218)6 cHD 4cH + cH ; (2.28) and c H as placeholders for the relevant fermion type (i.e. e, u, or d), and in this analysis we only consider couplings to the third generation of each. We have only included gHHu and its corresponding operator as only the top quark h 2 operator will have an e ect on our analyses as it is proportional to the top-quark Yukawa coupling which is the only large Yukawa in the SM. It is possible to remove the v 2 4 v 2 4 v 2 4 v 2 4 gH(2H)H operator by a eld rede nition of h, however as pointed out in [55] removing this operator by a eld rede nition of h (not the full doublet H) requires a nonlinear eld rede nition which may prove to make one loop calculations di cult and if done incorrectly gauge dependent. Therefore we retain the g(2) HHH coupling in favor of easier comparison with other works, such as those which study globally the constraints on the h3 coupling via one loop dependent processes [21, 22, 24, 25, 56, 57]. 3 Higgs coupling measurements at the LHC In this section we consider important constraints on our EFTs in section 2. We begin by considering the constraints from electroweak precision data along with a discussion of the loop order at which the S- and T -operators are generated either explicitly via integrating out at the mass scale of the extended scalar sectors or via operator mixing in the EFT while running down to the Higgs mass scale. Next we introduce the e ective h coupling in order to add an additional constraint to our global t to single Higgs processes. We delegate to appendix A unitarity considerations from the EFT perspective, where many amplitudes grow with the square of the center-of-mass energy S, as they do not add additional constraints to our models. Finally with our precision constraints on the EFTs we project these constraints into the UV complete models parameter spaces, this is especially useful in helping to limit the size of the cH coupling which is partially dependent on the same couplings as the h e ective coupling. 3.1 Electroweak precision measurements Electroweak precision data (EWPD) provide very strong constraints on the Wilson coe cients of e ective operators. We note that the operator QHD contributes at tree level to the T -parameter, while the operator, QHW B = HyB W H ; contributes to the S-parameter at tree level. However, the only operators contributing to EWPD that are generated at tree- or one-loop level in our theories are QH and QHD the operator QHW B is only generated at two-loop or higher order. From Jenkins et al. [54, 58, 59] we have the elements of the anomalous dimension matrix for each of these operators: c_H = c_H = c_HD = cHW B + 4g1g2yhcHB + 4g1g2yhcHW : + ( 6g22 + 24g12yh2)cHD + ; (3.1) (3.2) Where we have introduced the U(1)Y , SU(2)L, and SU(3)C couplings g1, g2, and g3 respectively, ng is the number of active generations at the relevant energy scale, the operators corresponding to the wilson coe cients cW , cHB and cHW are given by, QW = ijkW i; W j; W k; ; QHB = (HyH)B B ; QHW = (HyH)W i W i; ; and \ " represents other operators not generated at tree-level in our EFTs. The nal line of eq. (3.2) is included to indicate that cHW B is not generated at 1-loop by operator mixing and therefore must be generated at two- or higher loop order. However, the T -parameter is generated at tree-level by the triplet models, and one-loop by any theory which induces cH (namely all but the 2HDM). In the quartet models, since the only dimension-six operator is the H6 operator, there is no contribution to S and T from the H6 operator. However, the T -parameter can be generated at tree-level by dimension-eight operators. Including both the one-loop and running e ects we have for the S and T parameters (see e.g. [60] and [52]): S = sin 2 W v2cHW B T = 2 1 v2cHD + 3 e 2 4c2 16 2 2v2cH log 1 e 2 6 16 2 4v2cH log M 2 m2H M 2 m2H + + ; : Again we have used to represent operators generated at higher loop order in our theories. For the quadruplet models, the dimension-eight operators generate the following T -parameter T ' v 4 4 cT 8; cT 8 = 2j H3 j M 4 2 & cT 8 = 6j H3 j2 ; M 4 for the Y = YH and Y = 3YH quadruplet models respectively. Note that the coe cients cT 8 depend on the same quadruplet parameters as the operator H6, and therefore the Wilson coe cient of H6 is also strongly constrained through this correlation. From GFitter [61] we have the central values of the S and T parameters with correlation matrix as follows, S ! T = 0:06 0:10 0:09 0:07 ! ; = When considering all of the operators discussed above one may perform a sophisticated t to the EWPD of the many operator coe cients (see e.g. [62]), however for our study we where we have de ned the Wilson coe cient cT 8 to be the coe cient of the T -parameter operator at dimension-eight. This coe cient cT 8 is then given by, (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) 2HDM and cHD as discussed above. Therefore performing a simpli ed chisquare t relevant to our EFTs, we obtain constraints on the Wilson coe cients (cHD; cH ): v2cHD v2cH ! = 0:003654 0:002677 8:935 9:086 ! ; = 1:00 0:97 We note that cHD is tightly constrained while cH is not as its contribution to S and T is generated at one-loop. 3.2 Higgs diphoton rate In section 2, only the leading tree-level e ective operators are written when integrating out the heavy scalars. The leading e ective operators which contribute to the Higgs diphoton signature are not included in our framework as they originate from the one-loop contributions. However because of the precision of the H ! measurements we will include them in this section. Note that after integrating out the heavy scalars at one loop one may expect contributions to the H dimension-six operators, ! coupling from the following gauge-invariant LH = cHB(HyH)B B + cHW (HyH)W i; W i + cHW B(Hy iH)W i B : (3.10) However, since we are only interested in the diphoton rate, and not in corrections to the h ! ZZ and h ! W W rates we may simplify the calculation of the Wilson coe cients by only considering one e ective operator in the broken phase: LH ! 4 c h 2v F F : The general Higgs diphoton Wilson coe cient c for new scalars and fermions at one loop may be found in, e.g. [63]. For the UV complete models considered in section 2 we nd the wilson coe cients in table 4. As mentioned in the previous section the Wilson coe cients of the Quadruplet model are all proportional to the parameters contributing to the T -parameter. As such we will not consider the Quadruplet models for the rest of this section. Finally the diphoton rate relevant to our models is, (3.11) (3.12) (h ! ) = 128p2 3 j 2GF m3h c SM + c j2 ; Where we have de ned c SM = X f=t;b; Nc;f Qf2 A1=2( f ) + A1( W ); (3.13) as the SM part of the h ! width taking into account shifts in the couplings of the Higgs to the t-quark and W -bosons due to the e ective Lagrangian of eq. (2.27). Here the loop functions A1=2( ) and A1( ) are de ned in ref. [63]. The Run-I Higgs measurements [64{67] provide constraints on some Wilson coe cients in the e ective Lagrangian. For convenience we reproduce our e ective Lagrangian below HJEP05(218)6 L = gH(3Z)Z hZ Z + gHW W hW +W + g(1) h 2v + gHeheLeR + gHuhuLuR + gHdhdLdR + h:c: + c F F : (3.14) The corresponding Wilson coe cient dependence can be found in eq. (2.28) while the Wilson coe cients for each model can be found in tables 2, 3, and 4. We note that the modi ed Yukawa coupling of the top-quark also causes a shift the Higgs-digluon e ective coupling which we have taken into account in our analyses. These Wilson coe cients contribute to the Higgs signal strengths tracted from the Higgs coupling data, where A is the product of the acceptance and the e ciency. Since the Higgs discovery global ts to the e ective operators relevant to Higgs physics have become an important area of research [52, 68, 69] and recently they have gone beyond simple inclusion of signal strengths to inclusion of kinematic variables and o -shell measurements [70, 71]. They have also been considered in scenarios where EWSB is not linearly realized [72{74]. However for the sake of our analyses we require a much smaller set of e ective operators, therefore we perform a simpli ed global t to the Higgs signal strengths i using the program Lilith [75]. In Lilith, all the Run I LHC Higgs measurements [64{67] are taken into account, and a likelihood statistical procedure is performed to obtain the constraints on the signal strengths. It is based on the assumption that the Higgs measurements are approximately Guassian and thus the likelihood function L( ) could be simply reconstructed. Under this assumption adapted by Lilith, the 2 log L( ) follows a 2 law for each observable, ; where ^i is the theoretical prediction of the measured Higgs signal strengths i with Gaussian uncertainty i. The full likelihood L( ) = Qi L( i) is de ned as 2 log L( ) = 2 ( ) = ( ^)T C 1 ( ^); where C 1 is the inverse of the n n covariance matrix, with Cij = cov[^i; ^j ]. 61 41 81 0 21 L 01 ol2 8 ¡( 6 ¢ ) g .04 .02 .0 other operators are xed to be the local best values. Then the constraints on the signal strengths are recast as bounds on the Wilson coe cients. We perform a global t on these Wilson coe cients (cHD; cH ; c ; ciH ) with i = t; b; , and then project our results into the sub-space in each scalar model. First we perform the six-parameter t, and obtain 0 B 0 v2 ctH 1 c C A B v 2 cbH CC BB v B v22 ccHHD CCCC = BBBB 0:02993 BB B 0:1399 0:3373 0:02224 0:111 0:60 1:00 0:38 0:19 0:40 0:38 1:00 0:29 0:21 0:19 0:29 1:00 where is the correlation matrix for this global t. These Wilson coe cients are typically small due to suppression by Mv22 . However from subsection 3.1 we know we must also consider the EWSB constraints. Assuming equal weight and combining with the constraints coming from the S and T parameters, we nd that CHD is very tightly constrained: 0 v2 ctH 1 B v BB v B 2 cbH CC 2 c H CC = B B B B B v2 cHD CC B C A B B B 0:04967 0:121 0:003816 0:4551 0:5917 0:4722 0:0004666 0:02302 0:1513 0:0003861 C 0:2184 1:891 1 C C C C ; C C A = B 0:43 0:44 0:58 1:00 0:35 We also obtain that v2 (cHD 4cH ) = 0:09256 0:8731, which by eq. (2.28) we see is a very important constraint on both the momentum dependent and momentum independent tri-higgs couplings. In gure 1 we show the v2(cHD 4cH ) cscalar plane where we have marginalized over the parameters not shown. We see from gure 1 that the independent constraint on c provides an important constraint in the space of Wilson coe cients which will translate to a constraint on the Model N/A 4 4 25:29 19:97 N/A 5 H + 0=2 = 22 141:3 C Quadruplet (Y = 1=2 & Y = 3=2) H3 =M 2 or 3 H3 =M 2= 0:00 0:053 TeV 2 We also limit the range of the dimensionless Higgs couplings to be less than 4 . various four scalar couplings of the UV models and therefore through their correlation with the Wilson coe cient cH on the a ects of the QH operator. We project these constraints in the EFT framework onto the UV complete model parameters in the next subsection. Implications for the UV physics to be O(0:1 In the global tting procedure, all the Wilson coe cients are assumed to be independent. We know from section 2 that in the speci c scalar extended models some Wilson coe cients are correlated and some Wilson coe cients may be absent altogether. These correlations and absences may be seen in table 3. Therefore, it proves useful to recast the global t results to obtain constraints on the UV model parameters in each model. We perform the global t using the Lilith program in each scalar extended model. In gure 2, we show the 1, 2, and 3 contours on the model parameters in the real and complex singlet, Type-I doublet, and complex/real triplet models. At the same time, we also show the central values and errors for the model parameters in table 5. These plots exhibit similar features. First, the Higgs-Higgs-scalar coupling g=M 2 or Z6=M 2 is constrained 1) by the Higgs gauge boson couplings in the singlet and doublet models, while in the triplet models the T -parameter puts tighter constraints on the parameter g=M 2. Secondly, for the doublet and triplets, the Higgs to diphoton rate puts additional constraints on the couplings which contribute to the c . Converting to the couplings in the UV model, we are not further able to constrain the Higgs-Higgs-scalar-scalar couplings of the triplet models H and 0, because the constraints shown in gure 2 and table 5 are very loose. Even the perturbativity constraint, shown as the blue dashed lines in gure 2, is tighter than the constraint from the global t. So to place constraints on the Wilson coe cients of QH for the 2HDM and triplet models, we have to rely on di-Higgs collider constraints. Finally, we note that although the global t cannot constrain the renormalizable Higgs self coupling , it is able to constrain the dependence of the h(@h)2 e ective coupling indirectly. We have neglected to project our global t into the parameter space of the quadruplet as it is so strongly constrained by the T -parameter and the triplet serves as an example of the a ects. While these indirect constraints on the UV models from the global t are interesting and useful for our di-Higgs analysis in the following section, stronger constraints may of course be found in UV complete considerations of these models. The ability to loosely p .04 .02 .0 .02 .04 21 gL 0 = ‚ ‚ 2 + ' H 5 20 0 .03 .03 .10 4 2 61 41 81 0 4 2 0 61 41 02 81 21 gL 01 2lo 8 ¡( 6 ¢ ) 21 gL 01 2lo 8 ¡( 6 ¢ 04 02 02 04 ‚ H' 0 .04 .02 .0 (complex) singlet model. In the others, we show the 1, 2, and 3 contours on the model parameters in the Type-I 2HDM (top right), the real triplet (bottom left) and complex triplet model (bottom right). The colored contours show the log likelihood values in the global t. The blue dashed lines denotes the perturbativity bounds of the dimensionless scalar couplings: 4 . constrain numerous models at once from simple Higgs global ts is nonetheless intriguing and (especially in the advent of a signi cant deviation from the SM expectation) a useful way to direct UV complete searches of greater depth in the future. 4 Di-Higgs production at the 100 TeV collider The measurement of the triple Higgs coupling using non-resonant di-Higgs production at both the LHC and future 100 TeV collider has been studied in great detail in the literature which was recently reviewed in [27]. Among all the channels for the Higgs decay nal state, the bb channel [76{83] is the most promising due to the combination of large h ! bb branching ratio and more accurate reconstruction of photon momentum compared with other channels which helps reduce the backgrounds. Three di erent topologies of Feynman diagrams of the pp ! hh process via the gluon fusion production are shown in gure 3. Due to the destructive interference between the triangle and box diagram for the di-Higgs production in the gluon fusion channel, it is believed that at 14 TeV LHC with 3 ab 1 luminosity, the triple Higgs coupling gH(1H)H = SM v where SM p Mhh = L = g(1) + gtH htLtR + gbH hbLbR + gHHthhtLtR + gHHbhhbLbR + h:c: ; 4cH + cH ; (cHD p 2 ; with the SM vacuum expectation value v 2(p2G1F )1=2 and the SM dimensionless coupling 2GF m2H . From the above Lagrangian, we note that in the Warsaw basis, in addition to the SM trihiggs couplings, we also have derivative triple-Higgs couplings, which may contribute di erently to the distribution compared with solely non-derivative couplings. According to gure 3, the parton amplitude of the di-Higgs production g(p1)g(p2) ! h(p3)h(p4) via the gluon fusion process is would be constrained to only [ 0:8; 7:7] at 95% CL [84]. In all models considered in this article, the Wilson coe cients of the jHj6 operator cannot be chosen arbitrarily large. Based on the considerations of the validity of EFT and perturbative constraints, we estimate the value of the modi ed trilinear Higgs coupling to be within the range ( 0:1 SM; 2 SM), and take the cuto scale to be 2 TeV. The higher the cuto scale, we expect the narrower range of the trilinear Higgs coupling. On the other hand, at a 100 TeV collider with 30 ab 1 luminosity, the SM value of the triple Higgs coupling can be measured with around 10% uncertainty [27], and even around 4% based on the latest study [85]. Therefore, we expect that 100 TeV collider provides a good opportunity to explore the Wilson coe cients cH in various models we have considered.5 General formalism on di-Higgs production In our EFT framework, the e ective Lagrangian relevant to the di-Higgs production is large enough to modify the di-Higgs production cross section to give some evidence in 14 TeV LHC, yet other models we considered de nitely need the help of 100 TeV collider to probe, due to small or zero ctH . ss^ ab 4 v2 (" a (p1) b (p2) gHtv gH(1H)H mt v s^ + gH2tv2 mt2 G B 3m2H m2H ) ; gH(2H)H v s^ s^ + 2m2H + m2H 2v2 mt ! gHHt F 4 + gH2tv2 mt2 F # A (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (a) gluon fusion box (b) gluon fusion triangle (c) gluon fusion tthh where the Lorentz structures are A B = g = g + p p 1 2 ; p 2 T ; 4 are the form factors for triangle and box diagrams which can be found in ref. [86]. Correspondingly, the di erential cross-section for di-Higgs production is d (pp ! hh) ds^dt^ = 1 S Lgg S s^ p ; s^ jMhhj2 ; 32 s^ where S is the center-of-mass energy squared of the proton-proton system, s^ = (p1 + p2)2, t^ = (p1 p3)2 and the parton luminosity function is de ned as Lgg(y; F ) = Z 1 dx y x fg=p(x; F )fg=p y x ; F ; with fg=p the gluon distribution function, and F the factorization scale. As we have previously noted, the triangle diagram and box diagram interfere destructively and the smallest cross section is obtained when g(1) HHH =v 2:5 SM assuming no derivative interaction and no corrections to the quark-Higgs couplings. Due to this fact, the variation in the gluon fusion to di-Higgs cross section about the SM value of gH(1H)H = When gH(1H)H decreases, the total cross section decreases, till g (1) SM v is not symmetric. HHH reaches 2:5 SM . Any further decrease in g imum value at gH(1H)H =v (1) lower than HHH results in increasing of the cross section with respect to its min 2:5 SM eventually surpassing the SM value for gH(1H)H values 5 SM . On the other hand as gH(1H)H increases from zero, the total cross section increases. In our case, the situation is more complicated, we now have both an additional vertex and corrections to the quark Higgs couplings. (4.7) (4.8) (4.9) (4.10) HJEP05(218)6 2 dFeingoutreed 4b.yTthhee rdaatisoheodf tbhleuecrcoossntsoeucrtsionins otfheth(egpH(1pH)!H;hgHh(2H)pHrocess to the SM di-Higgs cross section ) plane, the plots from left to right correspond to three di erent value of ctH = 0; 0:4; 0:4. We adopt the NNLL matched NNLO SM di-Higgs cross section: 1:75 pb [27]. σ(pp>hh)/σSM(pp>hh) 100TeV v2ctH=0 3 gure 4 we show the cross section contours of the pp hh process in the ! v) plane with three di erent values of ctH . To evaluate the range of tri HHH v, we rst use the eq. (2.28) and table 3 to express the and g(2) HHH ( 0:015; 0:015). two couplings in terms of the parameters in the UV model, then varies the dimensionless parameters in the UV models within the range 4 , couplings with mass dimension to be in the range 1 TeV, and the cuto scale are set to be 2 TeV. These values are chosen such that our EFT matching procedure is valid (dimension-eight operators will not be enhanced by the factor g2=M 2) and the contribution of the kinematic region larger than cuto scale to the total rate is negligible due to the suppression of the parton luminosity. After these consideration, we choose relatively loose ranges for the two couplings: g(1) HHH ( 0:36; 0:07) For ctH = 0, the anomalous Higgs fermion coupling gHHt in eq. (4.5) vanishes and the corrections to the quark Higgs couplings are proportional to cHD 4cH . In such a case, only the rst triangle and box diagrams of gure 3 contribute to the cross section with approximate SM quark Higgs couplings. Hence, one can nd that, along the positive vertical direction, given a xed value of gH(2H)H , the cross section increases. Along the gH(2H)H direction, one can nd that a positively increasing value of gH(2H)H will lead to an increase in the total cross-section. This can be understood from eq. (4.6), where we observe that, with a positive g(2) HHH , the second term inside the bracket in front of the F 4 which is induced by the derivative interaction will add destructively with the rst term which is induced by the ordinary triple Higgs interaction, such that the e ect of destructive interference between the box and triangle diagrams is alleviated. In the case of ctH = 0:4, the cross section increases signi cantly when compared with the cross section for ctH = 0, this can also be understood from eq. (4.6) and eq. (4.4): the positive ctH will decrease the magnitude of gtH and also gives a new positive term generated by tthh vertex, which will alleviate the destructive interference. In the case of ctH = 0:4, the cross section will reach some minimum value between gH(1H)H =v = 0:15 due to the destructive interference. Below the miminum points, for a xed g(1) the triangle diagram becomes dominant, increasing g(2) HHH will decrease the cross section, because at this point the amplitude from HHH will decrease the magnitude of the term inside the bracket in front of the F4, thereby decreasing the cross section. Monte Carlo simulation and validation In order to perform our simulations we begin by using FeynRules [87] to generate an UFO model le adding the e ects of the dimension-six operators in eq. (4.1). We then modify the model le to include the full triangle and box form factors as computed in [88]. Then we implement MadGraph 5.2.4.3 [89] to generate events. We use Pythia 6 [90] for the parton shower and the FCC card in Delphes 3.4 [91] for simulating the detector. The following analysis is only concerned with statistical uncertainties as the systematical uncertainties are unknown at the moment. When taken into account they will lower the signi cance levels given in this section. We refer to the cuts applied while generating the events in MadGraph/Delphes as preselection cuts in the table 6. They are as follows:6 j j;b; j < 2:5; Rjj;j > 0:4; pTj;b > 20 GeV; pT > 10 GeV: (4.11) Important irreducible backgrounds consist of Z(bb)h( ), tth( ), bbh( ), bb production. Apart from these, there are bbj , jj , cc and bbjj channel that can potentially have a contribution to the background. Jet fake rates to photons are taken to be 0:012%, while jet and charm mistagging rates to bottom quarks are taken to be 1% and 10% respectively [ 92 ]. The backgrounds can be greatly reduced by vetoing extra jets, i.e., by demanding exact two b-tagged jets in each event. This is particularly helpful in reducing the tth background. Applying a Higgs mass window cut of 112:5 < mbb < 137:5 GeV, to the invariant mass of b-jets results in a large reduction in the Zh background due to exclusion of the Z-peak region. The Higgs mass window cut for the di-photon invariant mass is sharper than that for the invariant mass of b-jets and helps to reduce the background in all the channels. Furthermore, from the normalized distributions for b-jet-pair pT and di-photon pT in gure 5 indicate that the signal is favored for pT values larger than 150 GeV and 140 GeV respectively. Therefore, we further apply these cuts in order to enhance the statistical signi cance. The resulting e ciencies and cross sections at each stage due to these cuts in our analysis for leading backgrounds and three benchmark (BM) points for the signal are tabulated in table 6. We rst investigate the sensitivity of the trilinear Higgs coupling in the absence of the derivative Higgs coupling g(2) HHH . In this case, we recover the scenario widely discussed in the literature: how to probe the deviation of the HHH from its SM 6For bb and bbj events, we also implement the 50 < mbb < 250GeV and 90 < mj ; < 160GeV to increase the e ciency of the sample, and we found that the events outside these cuts contribute negligibly HHH = gH(1H)H =v to the nal results. Channel Pre-selection pTbb > 150 GeV pT + #bjet=2;# =2 120 < m < 130 GeV E ciency (fb) E ciency (fb) E ciency (fb) E ciency (fb) Bckgs bb bbj cc jj bbh( ) tth( ) Zh( ) bbjj Total SM BM1 BM2 BM3 Sig. BMs 50500 8424a bIncluding fake rate of c ! b: 10%. cIncluding fake rate of j ! b: 1%. dIncluding fake rate of j ! : 0.012%. : 0.012%. Cut- ow table for the analysis we perform. Basic cuts refer to generator level cuts described in eq. (4.11). In the cross sections we have multiplied by the following NLO k-factors [27]: kzh = 0:87, ktth = 1:3, kbbjj = 1:08, kjj = 1:43. Signal benchmarks in the (gH(1H)H =v; gH(2H)H v) plane are as follows: BM1=(0:0225; 0), BM2=( 0:032; 0:0152), and BM3=( 0:141; 0:0152). PT(bb) PT(γγ) 50 100 150 200 250 300 50 100 150 200 250 300 grounds as described in the legend. Black solid histogram corresponds to the SM distribution for di-Higgs production. Remaining solid histograms correspond to the three signal benchmarks (BMs) considered. Dashed histograms correspond to various SM backgrounds as indicated in the legend. 0..08 U . A 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0 jbγbγh(aγnγd) ccγγ ZH ttH bbj bbγγ bbjγ BM1 BM2 BM3 SM 350 PT(bb) [GeV] 400 .U0.09 . A0.08 ttH bbj bbγγ bbjγ BM1 BM2 BM3 SM 350 PT(γγ) [GeV] 400 λHHH= λSM g(1)HHH vSMλSM Δσs/ σs=10% Δσs/ σs=4% 12% 11% 10% 9% /SS8% δ 7% 6% 5% 4% 3030 B /25 S 20 15 100TeV30ab- 1 λHHH= λSM g(1)HHH vSMλSM 0 0 0.5 HJEP05(218)6 HHH = HHH =v assuming that the derivative Higgs coupling g(2) HHH is zero. The orange and green bands correspond to the 1 uncertainty in the S=p B with assumptions of the theoretical uncertainty for the di-Higgs production cross-section to be 4% and 10% respectively. Right panel: the percentage uncertainties on the measured number of signal events varies with the value of trilinear Higgs coupling. Orange and green lines correspond to theoretical uncertainties of 4% and 10% respectively. p value SM at the future collider. Compared with the work in [80], we obtain comparable signi cance of about 8:25 for the SM di-Higgs production for luminosity of 3 ab 1. This corresponds to the signi cance of left panel of gure 6, where we plot the S= p 26 for 30 ab 1 as can be seen from the black line in the B for 30 ab 1 and zero derivative interaction. We also estimate the uncertainty in the value of S= B by taking into account the statistical uncertainty for the signal and background as well as the theoretical uncertainty on the di-Higgs production cross-section. It turns out that for a 30 ab 1 luminosity, the statistical uncertainty in the number of signal events due to Poisson uctuations is around 3%, which is less than the 10% theoretical uncertainty coming from the in nite top mass approximation, the scale, and the PDF uncertainties [27]. The 1 uncertainty due to this is denoted by the green band in the left panel of gure 6. However, the latest estimation on the theoretical uncertainties places them as low as 4% [85]. Therefore, we also include this case denoted by orange band in the plot shown in the left panel of gure 6. The right panel of gure 6 represents the percentage uncertainties for the measured number of signal events as a function of the ratio of the triple Higgs coupling to its SM predicted value. Orange and green lines here correspond to the theoretical uncertainty of 4% and 10% respectively. As expected from the above quoted numbers, the theoretical uncertainty dominates except where the ratio of triple Higgs couplings is close to 2.5, where the cross section for di-Higgs production is the lowest leading to enhanced uncertainty due to Poisson uctuations. Here we comment on the validity of EFT in our collider analysis. The EFT breaks down when the parton collision center of mass energy approaches the scale of the cuto scale M = 2 TeV. Therefore, we should in principle add a cut on the kinematic variables 1 2 -0.9 mhh > 1TeV mhh > 1:5 TeV mhh > 2TeV like invariant mass of di-Higgs to only keep the events produced in low energy regime to make our EFT analysis valid. The di-Higgs spectrum is peaked at an invariant mass mhh near the two higgs threshold indicating our EFT approach should be valid (i.e. the processes considered have energy well below our cuto of 2 TeV). Additionally we have investigated the number of events below 1 TeV, 1.5 TeV, and 2 TeV for three benchmark points: the SM, HHH = SM = 2, HHH = SM = 0:9, and nding the results in table 7. As there are only a small number of outlying events with higher energies these numbers support the assertion that the EFT approach is valid in our Monte Carlo simulation. One should note that even if the heavy particles were to be discovered at higher energies that in order to extract the trilinear couplings of the SM Higgs one would still employ an EFT. Such a procedure is analogous to the use of an e ective four fermion theory for avor physics where the heavy W s have been integrated out of the theory in favor of unrenormalizable operators. Determination of Wilson coe cients Equation (2.28) and table 3 demonstrate it is necessary to investigate the discovery potential at the 100 TeV collider when both the deviation of the HHH coupling from the SM value, and non-zero g(2) HHH exist. Turning on the derivative Higgs coupling g(2) HHH will change the signi cance of the di-Higgs signatures. In gure 7 we present the reach of the 100 TeV collider with integrated luminosity of 30 ab 1 in the space of g(1) the left panel as well as in the space of Wilson coe cients cH and cHD panel, each with ctH = 0. The left and right panels of the gure 7 are not independent. Their values are connected by eq. (2.28), where the contours in the right panel are essentially rotated around the SM values as governed by the eq. (2.28). This represents only a class of models, in which ctH is not important, for example, singlet, triplet and quadruplet models. We plot the statistical signi cance contours for 2HDMs in ctH cH space as HHH 4cH gH(2H)H in in the right shown in separate plots of gure 8. ctH = 0 corresponds to tan the experimental bounds on tan in 2HDMs. ! 1, which is outside Figure 7 shows the allowed parameter regions in singlet, triplet and quadruplet models, which overlap within the signi cance contours. In these models, according to table 3, the Wilson coe cients cH and cHD 4 cH are not independent. More speci cally, they are related by linear relations such as cH ' HS( )(cHD 4 cH ). This linear relation then implies that the boundaries of these regions are governed by the input perturbative limit j HS( )j 4 and are straight lines as can be seen in gure 7. The values of the dimensionless Higgs scalar couplings, such as HS , H , determine the slopes of the HJEP05(218)6 0.2 complex singlet Real singlet Quadruplet Y=1/2 Quadruplet Y=3/2 S/√B = S/√B(SM) (0; 0) with S=pB at 100 TeV with integrated luminosity of 30 ab 1 . Left panel: the signi cance contours are plotted in the g(1) HHH =v vs. g(2) HHH plane, the shaded region is constrained by dimensionless couplings in the Lagrangian within the range 4 for couplings with mass dimension within the range 1 TeV and cuto scale M = 2 TeV. The light and dark shaded brown and blue regions are allowed by all the global t constraints. The Red line and magenta line corresponds to quadruplet model with Y = 1=2 and 1=3 respectively. Orange and green regions correspond to the 1 uncertainty on the signi cance with assumptions of the theoretical uncertainty for the di-Higgs production crosssection to be 4% and 10% respectively. Right panel: the signi cance contours are plotted in the v2cH vs. v2(cHD 4cH ) plane. The darker brown and light brown dotted lines on the right panel correspond to the Wilson coe cient constraints from the Higgs coupling measurements and the T -parameter in the real and complex triplet models. Shaded regions on the right have the same meaning as in the left panel. Both plots are with ctH = 0 and the SM limit in both is located at parameter region in each model. For example, in the real singlet case, along the boundary of the parameter region, the Higgs scalar coupling HS should be around 4 . In the region far from the boundary, the dimensionless Higgs scalar couplings appearing in cH should be small. We choose ctH to be equal to zero in these two plots. This condition is automatically satis ed by singlet and quadruplet models, and also approximately satis ed by triplet models. This is because ctH in triplet models is suppressed by the coupling g2 which is constrained to be very small by EWPD due to its relation to the T -parameter. In addition to the allowed region in each model, we also illustrate the region that will generate the expected signi cance within the 2 uncertainties around SM value. In principle one should derive the prospective con dence level contour for the parameter space that are consistent with SM prediction in the future experiments. However, this requires the scanning of a ne grid to obtain the selection e ciency of each point, which is beyond the scope of our study. Therefore we simply estimate that this 2 region roughly gives the region that is hard to di erentiate from the SM in the future experiments. 0.2 0.15 H c 2 v S/ B 100TeV Type-II 2HDM 30ab-1 signal at 100 TeV with integrated luminosity of 30 ab 1 . The left and right plots represents Type-I and Type-II 2HDM respectively. The light blue regions correspond to the parameter regions in tan which has been ruled out by experimental data from avor physics. The orange and green regions are within the SM 2 uncertainty with assumption of the percentage uncertainty of di-Higgs production cross section equal to 4% and 10% respectively. One can observe that, the future di-Higgs experiment is not sensitive to the cH and cHD which have already been strongly constraint by the EWPD. On the other hand, it can constrain the value of cH . Depending on the theoretical uncertainties that can be achieved, it may also be possible to exclude some parameter space of the singlet models, which represents the region outside the 2 region. bb of the 2HDM. The case of the 2HDM is much more promising for distinguishing between the SM and the NP model as ctH is non-zero. We demonstrate the signi cance for 100 TeV collider at 30 ab 1 integrated luminosity in v2ctH vs v2cH plane, shown in gure 8. Both v2ctH vs v2cH depend on tan . Here we choose the range of tan such that it satis es the constraints from in avor physics according to ref. [93]. This rules out some parameter regions as shown gure 8 by blue regions. We note that the signi cance in the 2HDM is generally larger than that of the singlet, triplet and quadruplet models due to typical enhancement from the Yukawa couplings, and it is very likely to observe a signi cant deviation from the SM signal. We also nd that, unlike the singlet and triplet, signal signi cances in the 2HDM are much more enhanced compared to the ones in the SM. The plots also show that the contours of signi cance of two types of 2HDMs are di erent despite the coupling to up-type quarks being the same in both Type I and Type II, the reason being that we are using the nal state and the branching ratio of h ! bb are di erent between the two versions From gure 7 and gure 8, if we limit ourselves in these models with all the heavy particles integrated out, the di-Higgs process puts additional constraints on the scalar model 5 SH 0 λ -5 -10 H λ -5 -10 complex (right panel) singlet models. The contours correspond to the signi cance given integrated luminosity of 30 ab 1 . The orange and green regions are with in the SM 2 uncertainty with assumption of the percentage uncertainty of di-Higgs production cross-section equal to 4% and 10% respectively. parameters. Our analysis in gure 7 and gure 8 shows that the Complex singlet and 2HDM (triplet and quadruplet) scalar models are the most (least) sensitive, among those resulting from the models under consideration, to the collider search. As a consequence, the di-Higgs process probes the allowed region of cH , and thus the Higgs scalar couplings in the UV models. Exploring parameter region in UV models models are within the 1 hard to di erentiate from the SM. We project the sensitivity of the Wilson coe cients into the parameter space corresponding to the models under consideration. In the real singlet model, the parameter space of the with S=p e ective coe cients allowed is indicated by the light blue region in gure 7, can be probed resulting from integrating out the complex singlet can be probed to S=p B values higher B more than 25, while in the complex singlet model, the Wilson coe cients than even 40. In gure 9, we show the possible reach of the model parameters ( HS ; gHS ) in the real singlet model, and ( HS + 0HS =2; gHS ) in the complex singlet model, given 30 ab 1 luminosity data set. One can see that, most of the region in the singlet and triplet uncertainty band for S=p B reach for the SM, so that they are The 2HDM, owing to its preservation of custodial symmetry, resides on the line cHD = 4cH = 0 (up to the assumptions made in this paper, that is a tree-level dimensionsix analysis). Therefore, the Higgs coupling measurements and the electroweak precision tests do not place strong constraints on the model parameters. On the other hand, the 10. /)M5. 1 1 V V e e T T 30. β β 40. 50. S/ B 100TeV 30ab-1 Type-I 2HDM S/ B 100TeV 30ab-1 Type-II 2HDM 25. Z6/M [TeV -1] 25. Z6/M [TeV -1] -2. -1.5 -1. -0.5 0 -2. -1.5 -1. -0.5 0 Type-II (right panel) 2HDM. The contours correspond to the signi cance given integrated luminosity of 30 ab 1 . The orange and green regions are with in the SM 2 uncertainty with assumption of the percentage uncertainty of di-Higgs production cross-section equal to 4% and 10% respectively. di-Higgs signature starts to provide a strong constraint on cH . In gure 10 we show the signi cance contour on the model parameter Z6 vs tan plane for Type-I model and TypeII model with the 30 ab 1 luminosity. Note that when Z6 = 0, the SM limit is recovered (see table 3). We also nd that in the Type-II model, for negative Z6 and large tan (left top corner in the right plot in gure 10), the signi cance approaches to the SM value. This is because the decreasing of the Higgs to b quark coupling reduces the Higgs to b decay branching ratio, which ameliorate the increasing of the di-Higgs production rate. In the real and complex triplet models, both cHD and cH in the EFTs obtained by integrating out real and complex triplet models are very tightly constrained as shown by the vertical dashed lines, shown in gure 7 (right panel). These vertical darker and lighter green lines represent the 3 bounds allowed by the Higgs data global t on the Wilson coe cient linear combination of cHD 4cH for the real and complex triplet model respectively. The reason that these stringent bounds only exist for the triplets and not the singlets is that the coe cient cHD is connected with custodial symmetry breaking and is tightly constrained by the electroweak precision parameter T . As table 3 denotes, the cHD and cH are tightly related for the triplet models and therefore the stringent bounds on cHD translate into stringent bounds on the cHD 4cH as well. In the case of the singlet models, there are no couplings of the singlets to the gauge bosons resulting in cHD being identically zero as indicated in table 3, liberating them from these constraints su ered by the triplet models. As a result of these, cH is also strongly constrained from the small allowed values of cHD 4cH , as shown in gure 7 (right panel). However, the dimensionless Higgs potential parameters, such as H and , are still very 4.5 4. 35. M S/ B Quadruplet Y=1/2 30 ab-1 S/ B Quadruplet Y=3/2 30 ab-1 3. 2.5 26. 2-.5. -4. -3. -2. -1. 0 λΦ3H dashed black contours correspond to the S=p B values for an integrated luminosity of 30 ab 1. The blue region is excluded by constraints from the electroweak precision tests. The orange and green regions are within the SM 2 uncertainty with an assumption of the percentage theoretical uncertainty of di-Higgs production cross-section equal to 4% and 10% respectively. loosely constrained due to cH g2 M4 H . Therefore, it is very hard for us to extract the Higgs scalar couplings from the cH operator, because the deviation of the Higgs coupling from the SM value is very small in the triplet case. For the quadruplet model, at dimension-six, only the Wilson coe cient of QH operator is non zero. However, we include the cHD generated by dimension 8 operator because it is strongly constraint by EWPD. In the left plot in gure 11, the allowed region for two types of quadruplet models are denoted by two lines with di erent slopes. The reason can be seen from table 3, the cH and cHD are correlated, all proportional to the coupling j 3H j2. So given a by a single parameter j xed cut o scale M , both g(1) HHH and g(2) HHH can be parameterized 3H j2. In the right plot in gure 11, we nd that the allowed parameter space from the global t to EWPD for quadruplet models is tightly constrained, and almost becomes a point near the SM value. In gure 9, we show the signi cance of the model parameter 3H vs new physics scale M varies with the 30 ab 1 in contours, while the blue region is excluded by the constraint on cHD from EWPD. One could observe that, the T-parameter constraint on 3H is very sensitive to the cuto scale, the reason is that the cHD is generated by dimension-eight operator so that it is proportional to the fourth power of (v=M ). Our collider analysis demonstrates that the potential of the 100 TeV collider in probing the Wilson coe cients resulting from the ve scenarios considered here is very promising with the 2HDM. The singlet, triplet and quadruplet models on the other hand are restricted due to electroweak precision measurements and their e ective coe cients will have less sensitivity. These restrictions also manifest in the constraints on the deviation of the triple Higgs couplings in such models owing to the direct relation between cH and the triple and quadruplet Higgs coupling as shown in eq. (2.28). An interesting consequence of our analysis is that, due to the di erence in the allowed region for each model under the theoretical bound and the global t constraints, it is possible to di erentiate the 2HDM model from singlet, triplet and quadruplet models with the observation of a large deviation of the signal rate from the SM expectation. If a future experiment detects a signi cantly larger signal rate compared with the expected SM model value, then it should favor the presence of an extended scalar sector consisting of the 2HDM the assumptions of this work. If the future experiment does not detect a signi cant deviation from the SM expectation, then one may have hard time to di erentiate SM from all the models considered here as well as models where the wilson coe cients are induced at loop level. Both a reduction in the theoretical uncertainty estimation and higher luminosities will be needed to make a more precise measurement of the di-Higgs signal rate. 5 We began by motivating a study of the dimension-six Higgs self-interaction operator QH = (HyH)3 in the Standard Model e ective eld theory because of its importance to studying the nature of electroweak symmetry breaking and the nature of the electroweak phase transition. We noted that the largest Wilson coe cients can be obtained by considering extended scalar sectors which are the only models which admit a tree level QH operator. After identifying all possible SU(2)L representations which allow for a tree level QH along with the corresponding hypercharge Y we wrote the ultraviolet complete Lagrangians for each model. Finally, assuming that the new scalars are heavy we integrated them out of the theory obtaining the dimension-six e ective Lagrangian at tree-level. Of the seven models which generate QH at tree level all but two generate more than one e ective operators. Those which generate only QH at dimension-six are plagued by strong constraints coming from the dimension-eight T -parameter operator. This helps put into context that any study performed by shifting a single parameter of the model is not model independent and, in the case of shifts in the self coupling of the Higgs coming from UV physics generating dimension-six operators as leading e ects, not well justi ed. After identifying the full set of tree-level dimension-six operators for the extended scalar sectors we proceeded to consider the constraints on the e ective theories given single Higgs measurements from run I of the LHC as well as the electroweak oblique parameters S and T . In order to fully take advantage of the single Higgs measurements we also derived the Wilson coe cient for the e ective Higgs coupling to photons although it enters at loop level after integrating out the new heavy charged scalars. For the constraints from the single Higgs measurements we implemented the tool Lilith to perform a global t to the Wilson coe cients other than the QH . Having constrained the Wilson coe cients we then projected those constraints back into the parameters of the ultraviolet models deriving relations between various parameters of the models. It is the multi-Higgs measurement instead of the single Higgs measurement which determines the Wilson coe cient of the Higgs self-interaction operator QH . As such, we We have converted the discovery reach of the QH operator into the Higgs potential HJEP05(218)6 have investigated the dependence of the coe cient cH on the di-Higgs production cross section, and studied simulations of the di-Higgs process for the proposed future 100 TeV collider. We then obtained the sensitivity contours of the Wilson coe cients in the general e ective theory framework as the luminosity varies at the future 100 TeV collider. Finally, we reduced the g(1) HHH gH(2H)H plane, for various cases of the UV complete extended scalar sector models, to its subspace for the cut-o scale of 2 TeV and in the perturbative regime for dimensionless Higgs couplings and demonstrated that most of these regions can be probed to the statistical signi cance of more than 5 using the di-Higgs signatures in a future 100 TeV collider. parameters in seven UV models. Among the models considered, the Higgs self coupling in the singlet and doublet models could have large deviation from the standard model prediction, while the triplet and quadruplet models can only have very small deviation, due to the strong correlation between the T parameter and the Wilson coe cient of the jHj6 operator. We showed that for the projected data collected for an integrated luminosity of 30 ab 1 at the proposed 100 TeV collider, the trilinear Higgs coupling in the all scalar models with a single heavy scalar integrated out could be fully explored. If a signi cant deviation in the trilinear Higgs coupling is observed, it will rule out the possibilities of the triplet and quadruplet models. On the other hand, if there is only small or no deviation, it will strongly constrain the Higgs potential parameters in the singlet and doublet models. Therefore, combined with electroweak precision data, the di-Higgs search can e ectively di erentiate singlet and 2HDM models from triplet and quadruplet models, within the framework of e ective eld theory of new scalar models. Overall, the di-Higgs process provides a unique opportunity to probe the cH operators which cannot be obtained by single Higgs phenomena and the electroweak precision tests. These future experimental measurements on the Higgs self-interaction operator QH will provide important information on the shape of the scalar potential under the assumption that the new scalars are very heavy. This provides a method complementary to direct phenomenological searches to nd evidence of additional scalars in these extended scalar models. Acknowledgments J.H.Y. is supported by the CAS Pioneer Hundred Talents Program. T.C. is supported by the Australian Research Council. A.J., H.L.L. and J.H.Y. are supported under U.S. Department of Energy contract DE-SC0011095. A.J. is also partially supported by the United States Department of Energy through Grant No. DE FG02 13ER41958. A Unitarity considerations Following the discussion of [53, 94], we nd unitarity requires that the partial waves for 2 ! 2 elastic scattering of bosons be bounded by, T J (V1 1 V2 2 ! V1 1 V2 2 )j 2 ; (A.1) where we may freely substitute Vi i ! h for the Higgs boson. Considering only amplitudes which grow with the square of the center of mass energy S in the above cited works the authors found that the operator QH is not bounded by unitarity considerations for 2 ! 2 and QHD only result in unitarity violation for the purely longitudinal case. Note that as the 2HDM does not generate either QH or QHD at leading order in the Y3 expansion it does not generate operators which violate unitarity with growing S. It was found that for one operator non-zero at a time the bounds were given by, (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) jcHDSj jcH S jcHDSj 67 ; 50 : 67 ; 67 : p p Scrit Scrit r 67 cH r 67 2 gHS 2M 4 A simultaneous search of the bounds allowing for cancellations between the two e ective couplings yields the bounds: It should be noted these constraints indicate the largest allowed values of the two operator coe cients allowing for cancellations between them and not that all values within these bounds will be simultaneously allowed. 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Tyler Corbett, Aniket Joglekar, Hao-Lin Li, Jiang-Hao Yu. Exploring extended scalar sectors with di-Higgs signals: a Higgs EFT perspective, Journal of High Energy Physics, 2018, 61, DOI: 10.1007/JHEP05(2018)061