JHEP10(2019)255 Springer August 21, 2019 October 15, 2019 October 28, 2019 : : : Received Accepted Published ). Conversely, analyticity πv ), quite independently of the (4 πv O (4 O corresponds to the more familiar one Published for SISSA by https://doi.org/10.1007/JHEP10(2019)255 (1) deviations of the cubic coupling, compatible b O non-analytic [email protected] , and analytic . 3 and Riccardo Rattazzi 1902.05936 a The Authors. c Effective Field Theories, Higgs Physics

We classify effective field theory (EFT) deformations of the Standard Model , adam.falkowski@cern.ch . Our distinction in H We illustrate the physical distinction between these two EFT families by discussing Laboratoire de Physique Th´eorique(UMR8627), CNRS, Univ. Paris-Sud,91405 Universit´eParis-Saclay, Orsay, France Theoretical Particle Physics Laboratory (LPTP),Lausanne, Institute Switzerland of Physics, EPFL, E-mail: b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: the direct LHC reach. Larger deviationsabout are higher-dimensional possible, but operators subject in tocubic less the coupling robust Higgs is assumptions produced potential. by acalculation On non-analytic that the deformation the of other the theory hand, SM, reachesmagnitude when we strong of show the by coupling the an at cubic explicit enhancement. occurs when new physics can be pushed parametrically aboveHiggs the boson electroweak self-interactions. scale. Inthe the Higgs potential, analytic there case, exists at spacewith for the single price Higgs of and some electroweak un-naturalness precision in measurements, and with new particles out of cal insight. From the UVmass perspective, from non-analyticity electroweak occurs symmetry whenhigh breaking, scales. the new and This states is thus reflected acquire for cannot in processes the be IR involving decoupled by many thewith to Higgs anomalous a arbitrarily growth bosons breakdown of and the of interaction longitudinally the strength polarized EFT description massive below vectors, a scale Abstract: (SM) according to the analyticity propertyblet of the Lagrangian as a functionbetween of linearly the Higgs and dou- non-linearly realized electroweak symmetry, but offers deeper physi- Which EFT Adam Falkowski JHEP10(2019)255 symmetry that Q U(1) × C electroweak (EW) gauge invariance, as in 8 Y 4 – 1 – U(1) × W to other SM fields. To carry out this construction there is no h 1 12 In the construction of effective theories, symmetries play a central role. For instance, lagrangian invariant under the colorcouples and the electromagnetic Higgs SU(3) boson need whatsoever for manifestthe SU(2) broken theory one can alwaysthe pick particle the unitary content gauge. explicit, makes But unitary the gauge, structure while of making interactions less transparent. Indeed in the veryto case tame of flavor EFTs changingsubtle, for neutral as currents. the they Higgs The mostlythe role control sector, EFT. of the flavor Our strength gauge symmetries main of symmetriesfollows. point, are the is which The interaction perhaps obviously concerns most and more crucial precisely general the these EFT range deformation aspects, of of validity can of the be SM summarized Higgs as sector is given by a general Along these lines,extensions many authors of the have pursued SM.view of a Those the relevant variety well known for of conceptualscalar the effective problems particle. associated Higgs field This with sector theory the paperclassification existence are makes (EFT) of of a particularly an these simple elementary motivated EFTs. observation, in which provides a sharp structural distances. Whether that isof the case physics or at not, lower it energiesSM. is will We expect quite not certain a be that muchan limited richer the infinite to structure effective set the of deforming description non-renormalizable few the interactions.at leading renormalizable The renormalizable the couplings lack SM of of LHC direct through the evidence has of indeed new physics boosted the relevance of indirect searches for such deformations. this perspective, renormalizablescale QFT is is either exponentially but large,asymptotically a at free least useful theories. at The idealization weak Standardinteractions, coupling, where Model can or (SM), the when even indeed limited infinite, UV be to inraising its extrapolated cut-off the the renormalizable to conceptual case possibility energy of that scales the next of layer the in particle order physics be of at the such ultrashort Planck scale, 1 Introduction According to the modern Wilsonianbe viewpoint, viewed any as Quantum an Field effective Theory description (QFT) valid should below some physical energy cut-off scale. From 3 Non-analytic Higgs potential (HEFT) 4 Conclusions Contents 1 Introduction 2 Analytic Higgs potential (SMEFT) JHEP10(2019)255 , h (1.1) (10%) accu- O , together with v ∗ g ∼ 246 GeV denotes the ∗ it is either analytic or ≈ m : v H doublet. We however believe immediately imply the upper W π : 4 . Y ! . h ∗ U(1) g ). Even when deviations in the single 0 + 3 TeV, where = 0. In other words, non-analytic EFTs × v 1.1 ∼ W H

) i σ i included. This is the more familiar case, where πv is kept manifest so as to form, together with – 2 – iπ (4 e H in eq. ( Z , we can always form such a linear multiplet. Two 2 O 1 h √ H and ≡ W H term in the SM lagrangian is completely determined by two 3 h being or not being part of a SU(2) eaten by i h π = 0. More precisely: either the lagrangian is analytic, possibly after a of the = 0, signaling the breakdown of the EFT, must be associated to some 3 H λ H transforming linearly under SU(2) for the mass defining the UV cut-off. Our distinction between analytic and non- H πv ], probing the Higgs self-interactions remains challenging. The ongoing experimental 1 The interest in measuring Higgs self-interactions is fueled by the hope that it may The classification we advertise is generally applicable to EFT extensions of the SM. In sophisticated scenario. The currentThe efforts coefficient are mostlyprecisely focused measured on observables: the the Higgs cubic bosonother mass self-coupling. SM and predictions the Fermi in constant. theracy While [ Higgs many sector have been successfully tested with us to make theresults discussion relevant for very the concrete ongoing and explorations focused. of the cubic Ascontain Higgs a a self-coupling. clue bonus, about wepotential the will is more derive arguably fundamental useful the theorythat most underlying the ad-hoc the true SM. element dynamics of Indeed, driving the the the SM, Higgs Higgs and it field is to reasonable acquire to a suspect VEV is described by a more between linear (so-called SMEFT)equivalently between and non-linear (so-calledour classification HEFT) is effective more theory, adequate and or enlightening from athis physical paper point we of shall view. illustrate it in the specific case of the Higgs potential. That will allow simply correspond to the presencethe of Higgs VEV. new The massive familiar statesthe relation whose naive between mass dimensional coupling is analysis and fullybound mass (NDA) controlled 4 expectation by analytic lagrangians coincides with the distinction, in use in the Higgs EFT community, or double Higgs productionprocesses happen involving many to Higgs bosons be andIn small, longitudinally hindsight polarized the massive this low vector bosons.singularity cut-off result at will has become a manifestheavy simple in degree interpretation of from freedom becoming a massless top-down at perspective. The small deviations from the renormalizable SMtechnically, are compatible the with ultimate a large cut-off cut-offThe scale. grows situation More like is an sharply inversedetail, different power in the the of cut-off non-analytic the basically case.vacuum size reduces expectation There, of to value as the (VEV) we deviation. shall of illustrate in We stress that, whatever thepossibilities origin are of then givennon-analytic for at the lagrangian asfield a redefinition, function or there of between is these no two field possibilities redefinition isthe that not lagrangian renders aesthetic is polynomial it but in analytic. purely all dynamical. fields, The In distinction the analytic case of Goldstone bosons a doublet our sharp structural classification of EFTs is most succinctly formulated when the triplet JHEP10(2019)255 , ], 3 3 , 2 ], where 4 can be of relative . 13 , ξ | is equivalent 3 12 ∆ . We start with | H † 1. In this setting 1 can be as large H . † 0 H − H . is moderately strong. 3 SM ξ ] and in EW precision , λ 3 8 ∗ λ – m 5 ≡ 3 2 (1.2) . 3 ∆ – 3 – powers of the BSM scale. We review the power in the UV theory at . 4 is a relevant coupling that becomes strong when ∗ 0 − g 3 . In parallel, the cubic can be constrained through D ∗ λ 3 m λ , and instead new degrees of freedom must appear at v is an analytic or non-analytic function of and outside the LHC reach, and 2) the magnitude and h h . This case is equivalent to the so-called SMEFT [ ∗ m m . It is possible to add to the SM lagrangian terms of the form ]. However, all of these methods currently leave room for a large , which in the unitary gauge yield Higgs boson self-interactions 2 m n H 11  † relative to the SM prediction. ] in single Higgs production at the LHC [ ∗ , using the language of a non-linearly realized EW symmetry, see e.g. H 3 4 ], or through tree-level effects in single Higgs production in association m λ H ] for a review). This framework naively offers more freedom to arrange 10 14 can largely exceed the relative corrections to single Higgs couplings. This moderately strong and generic coefficients of higher-dimensional operators we relax the assumption that the scalar potential is a polynomial or , 9 3 ∗ 3 . In particular, we can arrange such non-analytic terms to contribute to ∆ g terms suppressed by n with integer D . Technically, this happens due to the wrong (inconsistent with perturbative ≤ 2 k πv n/ 4 0, with the cubic coefficient in the potential held fixed. More precisely we find that
is much bigger than (1) when the coupling strength ∗ In section This paper discusses the range of the Higgs cubic coupling that can be generated by a H . → † O with m ∗ h (10) deviation of k H section II.2.4 of [ for a large cubic Higgs couplingWe without violating will theoretical argue and phenomenological however bounds. parametrically that in separate the presencem of the non-analytic terms it is impossible to h with or without affecting otherthe Higgs (self-)interaction SM terms. local An symmetry EFTto and lagrangian the that degrees has so-called of HEFT freedomHiggs doublet but framework field is (which non-analytic is in usually formulated without introducing the in the Higgs potential. Largerstringent assumptions or in negative order corrections to are ensureobtained possible, vacuum for but stability. a are Overall, subject reasonable we find to hypothesis more about dimension-8 operators inanalytic the function Higgs of potential. m the cubic enhancement in the range is possible for corrections to the HiggsLagrangian. cubic We are demonstrate generated that atas the the cubic level enhancement ∆ ofRemarkably, dimension-6 ∆ operators in the can be understood by noting that counting that controls the coefficients ofand various phenomenological terms in constraints the on potential, theseHiggs stability coefficients mass conditions, from and the couplings. LHC We measurements1) are of interested in the a phenomenologically viableBSM scenario corrections where to single Higgs couplings satisfies the LHC bounds dynamics beyond the SM (BSM).energy The scales below analysis depends onthe whether former the case Higgs in potential section various at terms in the lagrangian aredimension organized according to their canonical dimensions, with which is sensitive atits tree one-loop level effects to [ measurements [ with two W/Z bosons [ O effort in this direction consist in measuring the Higgs boson pair production rate [ JHEP10(2019)255 = πv and 4 4 (2.1) (2.2) (1.3) λ 2 1. We . π , ∗ h 2 λ m |  3 q ) even when ∆ bosons. That | πv . Z = 3 (4 2 or O h /v . v 2 h m n 3 W h m 4 2 − ≡ n h n ≡ λ m h ! SM λ , n 3 λ range roughly between 3 =5 ∞ h X n = λ , + ) 3 } 4 λ takes the form h ] suggests the lower of the values, while the 2 4 n h ) {z 15 λ H h . .| . h 1 ) hit strong coupling at 4! † } L H + 1.3 ( 3 h – 4 – m 4 {z h ...V λ 3 1 of the cubic coupling. In the SM the observed L . This however brings us back to the SMEFT case λ + V | h ( − H 3! † H m † M 2, which is well within the perturbative regime. Indeed vanish. Our goal is to set a theoretical bound on the H SM , stands for longitudinally polarized + / H 3 4 Choosing for definiteness a reference strong coupling value 1 2 2 h L 8 operators on the stability of the Higgs potential. 2 h 1 /λ 01. The SM quartic is thus about two orders of magnitude V n> ) can be controlled only when the Higgs potential is well- m ' . 3 2 h ≤ λ ). 0 2 λ h − m 1.3 D λ ] yields the upper value. ∼ 1.2 ≡ Consider the SMEFT arising as a low-energy approximation of a 16 h ) = 3 3, and ) = ¯ 2 λ H h / † ( ≥ , while h imply h 1, contrary to the assumptions of this analysis. The wrong behavior λ V H n 2 λ v ( ]. Our analysis offers a different perspective, emphasizing the dependence on the 3 ∼ R 17 V = ξ and 2 h 2, and h 2 m v we have 3 m ≥ 2 π m ≡ 4 in accordance with NDA. In this section we tackle this question in the framework of the SMEFT with higher- 2 More precisely the RG evolution estimateSee used also in ref. [ [ ≡ SM 1 2 π , h 4 ¯ dimensional operators. scattering phase method of [ microscopic properties of the UVin theory our and discussion fine-tunings the required impact by phenomenology. of Moreover we include λ below its perturbativemagnitude upper below bound, its while perturbativerange the upper can cubic be bound. covered is by accordingly A plausible extensions fair about of question one the is order SM. what of portion of this λ relative deformation ∆ values of standard estimates of the perturbative10 upper bound of In the SM this arises by expanding around the vacuum the potential The SM cubic and quartic couplings take values generality, the potential for the Higgs boson field approximated by a polynomial in and to the bound in eq. ( 2 Analytic Higgs potentialLet (SMEFT) us first define our notation and introduce the relevant physical quantities. In complete deformation; in fact, the amplitudesnon-analytic in terms eq. generate ( a relativelyconclude small that correction the presence to of theand non-analytic cubic typically terms term, in theof Higgs potential the leads amplitudes to in eq. ( where conclusion depends very weakly (logarithmically) on the magnitude of the non-analytic unitarity) behavior of the tree-level amplitudes of the form JHEP10(2019)255 . 2 h is f φ m ) 1 is 2 ∼ (2.4) (2.3) a H 2, the † v (1).  / 2 ξ O H in which for v 2 ∗ . We will is the top 2 g π ∗ = ∗ t ξ + /g , /g i ≡ y , focussing in ∗ h = 2 2 ∗ 2 ∗ H g v λ h m † m implies precisely can range from 0 2 ¯ λ − H 2 k h √ , so that according simply corresponds α . We cannot offer a p + ( ∗ f 2 H n g n † v φ ≡ , where ∼ k 2 ∗ ∗ H n g ∗ f → 2 2 g /g 2 a ∗ 3 ∗ ]. g ≡ m 19 = 2 = ξ for any 2 ∗ appears more plausible than the V . In view of these properties this 2 m ξ h 2 π in ref. [ λ a 16 . But such a generic theory is at odds / = f 2 t ,X . ) 2 h y ∼ 2 ∗ coefficients corresponds to the absence of and maximal coupling size g m X v ( ∼ 2 n ∗ is strong, a tuning of order a a P n m ∗ 2 . Some couplings could consistently, naturally a = – 5 – g ∗ X 2 h m 2 ∗ 4 ∗ . Indeed a redefinition λ g ∗ (1) and m 4 01, corresponds to the case ) implies g . O 0 ≡ 2.3 = ∼ n is more structural. In our experience ) we can write n 2 X a n a accidentally light Higgs n 2.2 a , which up to a constant coefficient produces the same scaling. 2 ]. Assuming the existence of a minimum at at the scale 2 depending on the model. Now, the factor − ≥ , several independent dimension-6 SMEFT operators, such as e.g. ∞ ∗ n 20 X n 2 g , would produce deformations of single Higgs couplings of relative α – k 3 2 ∗ 4 ∗ 1 in the discussion below. In concrete models the relation 1, corresponding to operators with many legs. It would be nice to n in eq. ( /f . g 18 a 2 0 m 4 h v  and HQ λ , and estimate the size of various Wilson coefficients using the usual . → (1) in a generic theory. Indeed one should be more careful especially k R ∗ n ≡ , in which case one can organize the SMEFT operators in a meaningful ¯ t ξ ) = n h 2 ξ a | H /m m we would expect ( ) is generated by either tree or one-loop graphs. One finds the rough scaling ∼ O H (1). The upper bound on the | = 1 would produce a maximally strong ∗ V , with  n 2 2.3 α m a or ∗ a − ≤ O 2 ). In view of the agreement of the LHC Higgs data with the SM predictions we m n ) ξ n 2 n ( | and k a 2. In particular, a simple UV model with potential O Before proceeding we would like to make a little digression concerning the naive ex- Another independent tuning may be needed to ensure that the Higgs boson mass ∗ / H | g ∼ µ n ∂ a to the ambiguity inthe the redefinition definition of The power-law dependence on to 5 pectation when considering have the analogue of NDAgeneral including self-consistent an analysis estimate along forwhere these the eq. scaling lines, ( with but we can discuss a few simple models, This shows that, whenneeded. the The strongest UV tuning, coupling a generic scenario was referred to as an matches the observed value.to Eq. the ( definition of Indeed, defining ( size will thus assume typically achieved by fine tuning.tuning This of single multiple coefficients tuning required of to match Higgs data in a theory with or unnaturally, be tunedpseudo-Nambu-Goldstone to Higgs be small. we have Yukawa coupling. For instance On the in other hand the forby simplest an ordinary instance scalar in of a compositewith generic theory phenomenology characterized and some tuning is always necessary. Consider first the relation with couplings stronger than particular on the Higgsassume potential. It isexpansion also in convenient to 1 define power counting rules [ potential has the general form microscopic theory with fundamental scale JHEP10(2019)255 2 3 0 ). π for 8 ≥ ∞ = 0. √ (2.6) (2.7) (2.5) ) + i , , ≈ ... X ∗ 1 3 ( H , g + λ † 2 ∗ − 2 P g 2 2. In both a 2 H 2 / h (1), one can X , which implies √ a √ 2 2 4 O 3 3 a = 5 = = ξ/a . + 3 . α 4 a 2 SM SM X , , − , a 3 3 4 vanishes. Unless this n 3 a ξ = 0), we readily obtain a = 0. These expressions 2 n V + a a X 1, 2 , λ 2 a ≡  n>  2 a n ξ c 4 2 2 (i.e. , ξ ) = , a a 2 / 2 a are both of order X  ). This happens because the cubic is increased. v ( 0, with the cubic coefficient in the ∗ 4 1 2 . = 0, where g ξ λ P +16 ( 0 with → ξ i ≡ that, keeping all other terms fixed, the ξ/a O X 3 2 3 = 0. This result simply follows from the h , λ 4 H a a a   † m α  , produce a series with and to 4 ξ 2 H ) with 01 , which is defined in the domain [ . 3 3 2 φ h a ... up to 0 λ 2 ∗ a a X ( g 1+12 3 + – 6 –  λ X/ξ P  2 3 + . More specifically, by considering the couplings 2 ]. a ˜ ξ 1+2 3 X = propagator. On the other hand, other UV variants SM encode effects of dimension-8 and higher SMEFT X 4 19 , φ  20 4 c ∗ ˜ φ 3 moderately strong) one can obtain sizable deviations X λ ∝ ≈ m + (10) for a sufficiently strong coupling in the UV theory. SM , ∗ n> ∗ = 3 3 , a series with g V O ˜ c g X
λ φ ∆ corresponding to the EW preserving vacuum 3 2 + c ξ = ξ 4 2 a  − φ ξ 2 ∗ = 0. In the following we quantify the stability constraints. 2 3 = , while m a ) we have a is consistent with the breakdown of the low-energy expansion for h 3 +16 ˜ 2 c ) = 1 + , one sees that, within the range √ ξ X + X n ∗ ( 1 (thus for 3 ˜ (1) so as to enhance and becomes strong when ) around its minimum at a X g + a ˜ 1 the relative corrections to P ( O 2 H = 2 2 ). In order to make the discussion more transparent it is convenient ˜  Hφ of the linear term is directly related to the correction to the cubic coupling ( P a  a † cannot be made arbitrarily large without generating a second minimum = 3 3 ∞ 2 V ξ √ +12 c 3 H 2 a 2 + ), with ∗  a h ) = a ∗ , which is physically expected. ξ, m relevant √ ∞ 2 ∗ g , are reached more or less simultaneously as X ): ∆ ∗ 2 ∗ 2 − ( 2 0 we have a realistic local minimum at + g [ g m ξ/ π 3 3 P 8 √ 3 2 2.5 ξ, > a ∼ = ∈ ≈ − = = 2 2 [ | 4 3 4 V Our potential has the form Expanding a X λ Note that for Nevertheless this was overlooked in ref. [ λ λ ∈ H 3 4 | 2 ∗ in eq. ( that in principle theand two numerical approach factors the disturb strong thisand coupling NDA, differently. and However, as phenomenological a constraints result the respective strongly coupled values, to work with theWriting rescaled variable The coefficient X For minimum is also global,metastability it thus basically will coincides be withfor destabilized the by condition for vacuum absolute tunneling. stability: The condition for The above conclusion, however,stability does of the not EW take vacuum.coefficient into of Indeed account it the isdeeper obvious requirement than of the absolute one at coupling is potential held fixed. Numerically, theSM correction one to is the cubic given Higgs by coupling relative to the and naively it can be larger than where we factored outshow the that SM for resultfrom obtained the in SM the even limit written for in relatively terms small of choose the low-energy Higgs self-couplings. In particular for the cubic and quartic we find geometrical series generated bylike the cases the scaling of g produces, upon integrating out JHEP10(2019)255 4 (2.8) (2.9)

. It is Instability | 3 c 2. Thus, | / 2 2 2. Left: the v / 2 1 the function v = ≤ π ≤2π * i * relative to the SM = |  H 3 i 3 3 † 0 Δ . c H | (1), the experimental H † h O (1) H h Forbidden forg Forbidden forg . , n  O , a -2 4 (10) − (1) 1 under different hypotheses about n . O O ξ (1) may lead to an instability. , i.e. within the physical domain , 2 3 2 , which encodes effects of dimension-8 Instability . O = 0 . 2 ˜  X /c ξ 2  1 4 /a 1 -4 . ξ 1 c  4 . 0 ξ

are suppressed with respect to

1 0

1.0 0.8 0.6 0.4 0.2 0.0

) than the one at 2 4 4 . ξa ξ = / c a a + ξ

| 4 the bound coincides with the condition ). It follows that for 2 0 3 ≡ − H ˜ =   X < (   2.7 c n> 3 4 = 1 and 3 2 c V c ˜ 01   01 under different hypotheses about the parameter | 4 X 2 . ∆ a . 01 ∆ a . – 7 – 0 a 2 1 2 01 0 . ' a  < = 0 0  2 ˜ 4 =  X a 2

= 1 + =

− Instability = 10 n 2 ˜ P ξ ξ ξ | | | 01 imply 2 2 3 2 4 2 n . a a a a a a = 1 and 0 , stability is an issue. The behavior of the potential at small | | | | 4 3 a & 1 the parameters = = = ∆ 2 | | | | 3 a 0 3 4 4  Δ of the BSM theory underlying the SM. c c | | ξ are anyway expected to be suppressed. The resulting constraints are ∗ n> g c | | 4 Forbidden forξ≥ 0.1 Forbidden forξ≥ 0.08 Forbidden forξ≥ 0.05 1 and . Outside the region 0 . n> -2 0 c 1 | .

Instability ξ , which characterizes the size of the corrections to the single Higgs boson couplings to . Parameter space for the cubic Higgs self-coupling deformation ∆ -4 ), leading to a deeper minimum of 2 ∞ /f 2 + To make the bound more precise, it is quantitatively adequate to focus on the case v

,

will cross zero near the origin at is dominated by the first two terms in eq. ( 0.0 0.6 0.4 0.2 1.2 1.0 0.8

2 4 4 1

/ ξ = a a

c = 2 ˜ − ˜ given that the shown in figure now clear why, for large X P [ the correction to the Higgs cubic coupling larger than We conclude that for operators in the Higgs potential.constraints Now, under the assumption as the potential containspurple areas a are deeper excluded minimum for ξ that the EWmatter. vacuum Right: at the bluethe areas coupling are strength excluded for Figure 1 value. The allowed regionSMEFT depends operators on the in value the Higgs potential. The gray area is excluded by stability considerations, JHEP10(2019)255 . n ξ 2, ∼ h ∗ ≤ (3.1) (2.10) m 3 ∆ = 2. 3 ≤ ) we obtain 2.7 . Furthermore, as π . For more moderate ∗ 4 can only be achieved = in eq. ( , which follows from the m 4 > ∗ H Higgs couplings to matter c g | 3 . ∆ 4 | h . 4 for single 2 2 h . v  2 m 8 1. In particular, for 0 1 1 one can reach up to ∆ . . ξ − 0 . +  3 3 = 0 h 1) and with a fundamental scale ∆ ξ . 2 ) 01 ˜ 0 . 3 a X < 0 . . r 7 2 can still be covered even for relatively weak . 4 – 8 – 1 and ξ are possible, subject to assumptions about the 1 . a ≤ (1 + ∆ − 3 √ . Using the definition of 3 2 h 4 v = 0 4 2 c ∆ m 2 . 4, the bound is weaker corresponding to cases where a | 16 can be obtained consistently with present data and for ≤ 3 < 3 ) = ) is satisfied. In particular < 2 can be obtained under the very conservative assumption ∆ 3 h | 2 3 ( = 1, ≤ ∆ 1 one has 3 V 2.10 3 a : ∆ < few. In particular we showed that, at the price of a tuning of ∆ < (1) deviations in its self-couplings but compatible with all single ˜ P | O above the present LHC reach. Essential in the derivation was the ≤ . 4 ∗ a | | , the bound can be tightened if the 3 m 1 ∆ | 5 are absent. Note that such a pattern cannot be obtained from any (1) deviation ∆ O ≥ such that eq. ( n 4 (1) which seems less plausible. The maximal value is reached for a maximally a , an 5 TeV out of reach of present LHC direct searches. 2 O v . & ξ 0. Larger or smaller values of ∆ 4 √ Consider, for concreteness, a simple scenario of an EFT where the Higgs boson self- One way to read our results is that there exists space for a strongly coupled accidentally and a > is even allowed to vanish. All in all, we conclude that a correction to the Higgs cubic becomes negative in the unphysical region v/ 2 h 4 ∗ 4 ˜ interactions are described by the potential where the only deviationterms from with the SM resides in the cubic coupling. In particular all section we discuss theremoved. cubic self-coupling We shall in see a that,the setting going beyond separation where the between the SMEFT, the analyticity thereto assumption BSM is much an is and more obstruction EW severe to scales. constraints achieving coming This from other direct scenario and indirect is searches therefore for subject new physics. the Higgs boson: m a new physics scale analytic dependence of the lagrangianassumption on that the the Higgs heavy doublet states field are massive regardless of EW symmetry breaking. In this 3 Non-analytic Higgs potentialIn (HEFT) the previous sectionparametric we separation have between demonstrated BSMmental that, constraints and in lead EW to the scales, an SMEFT upper theoretical framework bound with arguments on the and the magnitude experi- of the cubic self-coupling of couplings. For instance for light Higgs with sizable Higgs and EWg precision measurements ( (and perhaps more realistic) couplings,under the bound the is condition correspondingly stronger;illustrated for in example figure are better constrained byNotice experiment, however that leading the to region a 0 stronger bound on the parameter coupling in the range 0 a coefficient for strongly coupled BSM theory completing the SMEFT at the scale On the other hand,P for 0 c for absolute positivity of JHEP10(2019)255 . , n v ) H . It † and 2 (3.3) (3.4) (3.2) ∼ / L H h/v 1 ( ) f n /Z c H L † , the terms =2 W 2 H ) belongs to ∞ n ( G P 3.1 ⊃ ) represents one 4 . In particular, it ) is not important v ) in V , invariant language: 3.1 H ], which in a certain 3 = † 3.2 Y  ], these correspond to 3.2 21 v H V . 22 n − U(1)  = 0, this potential reduces H × , i h † v 3 ), where integrating Φ at tree π W . −  H c . v 2  √ − bosons at high energies. This way, =2 +h ∞  2 X n Z 2 h H 2 v G † 2 m G + 3 ). Another is a model with the second 2 h (Φ and 2 µ 4 ) H m † v 3 + W + ∆ 3 4 H + | 2  Φ h | – 9 – Then, outside the unitary gauge, the lagrangian 2 ( = ∆ log( κ v 5 2 p . 2 v ) = −  ) in a manifestly SU(2) h ! 2 + H 2 h ). In the unitary gauge, 3 v H † 3.1 2 UV † G h 2 ) in the low-energy effective theory. However, there do m iG V H 1.1 G 2 h 3 H ( v tadpole in the effective potential, thus 2 3.2 + ∆ m 4 +  3 ⊃ is allowed by the symmetries, and eq. ( h h 1 ⊃ 3 2 2 h n v + V ∆ iG c m 8 V v . Yet another example is the model of ref. [ 3 ⊃ /

2 ) = V ) 2 1 √ H H transform non-linearly under the full EW symmetry, while the Higgs ( † = V invariant potential that is an analytic function of Z , and we do not display the Goldstone-Higgs interactions originating from H i ( Y H G i ⊃ and G V U(1) ) contains interactions between the Higgs and the Goldstones: ≡ W is the Higgs field in eq. ( × is a perfect singlet. As a consequence, the general potential 2 ). We should mention that we are not aware of a concrete UV-complete model 3.2 part of the EW symmetry is linearly realized. In the HEFT, the Goldstone bosons H G h W 3.1 Q For this discussion it is more convenient to work with the linear parametrization of the It is illuminating to rewrite eq. ( This is because we assumed no modifications to other Higgs couplings. Then, in the linear parametriza- 5 tion, the Goldstone bosons do not have derivative couplings, which simplifies the analysis. where the analytic SM part of theinteractions potential. of longitudinal By components the of equivalencethe the theorem non-analytic [ terms effectively introducean hard arbitrary contact interactions number between ofwith Higgs two bosons. Goldstone In boson particular, fields expanding are eq. ( in eq. ( parametric limits leads towill an be clear from the followingfor discussion our that argument, the as precise long form of as eq. the ( potential is described byHiggs a doublet: non-analytic function of exist familiar examples where integrating outeffective heavy interactions. degrees of One freedoms is yieldsout non-analytic the at SM one plus loop, aHiggs doublet chiral generates Φ 4th and generation the potential which,level yields when integrated where eq. ( that would lead to exactly eq. ( eaten by boson with arbitrary coefficients particular direction within the HEFT parameter space. cannot be obtained inoperators the contributes SMEFT, unless to thewhich the entire is potential. infinite phenomenologically tower verythe of That implausible. higher-dimensional parameter situation space however On ofU(1) corresponds the the to other so-called hand, HEFT, eq. which is ( an effective theory where only the SU(2) JHEP10(2019)255 ). 3. n h ≥ (3.5) (3.6) n matrix → . we have coincide: S πv =0 4 T I iT ] . ∗ GG ([ m -matrix, we know = 1 + S M ]. In what follows we S π 1 , √ amplitudes with 28 4 2 h – is perfectly well-behaved n . It is thus clear that an m ! 26 n v n hh v → ) = 3 2 n h → √ : the scattering phase shift is 3 π 3 → GG ∆ with ∆ a Hermitian operator. The =0 =0 +1 ∆ l I 2 i ] n π e . 1) ], it is instead the presence of operators n . = ], the loss of perturbativity is driven by GG − h ) contains higher-order interactions of the i ( ([ S 25 23 , Ψ | → 3.4 ≈ M 0 24 ) T – 10 – } Higgs bosons is given by † 0 ) must have a low cut-off scale, GG T . The regime of weak coupling can thus be defined n n | {z , eq. ( π (10). In order to quantify the validity regime of an 3.1 3 Ψ h . .| . h h ), the leading high-energy contribution to the inelastic O ), we have to investigate 2 , so that in the Born approximation ∆ and 4 in the lagrangian, carrying coefficients with negative → . 3.4 3.4 | > ... 3 =0 2 that is not suppressed at large energies leads to onset of I + ] ∆ | 2 . Considering elastic 2-to-2 scattering one can easily check that non-zero ∆ n > GG i iT ([ Ψ for the eigenvalues of ∆. Now, the computation of the − | any M . From eq. ( . This is conceptually fine as long as one does not interpret the issue T π ), the tree-level 2-two-2 amplitude i i G . 3.4 i 3 ) = i G √ | δ i iT P implies. The point is simply and purely the breakdown of perturbation theory amplitude with i ≡ . A rough but reasonable way to require perturbativity is thus to ask for ln(1 + n 0 i =0 T I − ] → Consider a family of scattering amplitudes of the isospin-0 two-Goldstone state Before proceeding we would like to briefly review the logic of the standard estimates of The need for a UV-completion below a certain scale manifests itself as a breakdown = 0 GG [ and the corresponding s-waveA amplitude 2 is for any incoming state this prescription produces the usualshall NDA simply bounds apply on this couplings [ to the processed | amplitude for scattering this state into manifestly satisfied order by∆ order = in perturbation∆ theory. Writing only issue concerns thephase ability operator, to whose computemaximized eigenvalues ∆ when are an in defined eigenvalue perturbation equals moduloby theory. the 2 request ∆ isin a perturbation scattering theory can be phrased as a computation of ∆. In so doing unitarity is perturbative unitarity of unitarity too strictly.perturbative Of course there isassociated never to an the issue onset withof of unitarity, course a as that strong the unitarity coupling adjective is regime. guaranteed, that Focussing on is the Indeed, from eq. ( and perturbative as longEFT as with the interactions in eq. ( the validity of the EFT. These are normally done by invoking the notion of breakdown of mass dimension. Thederivatives critical or operator powers of dimension fields. 4of In can longitudinal the be more vectors overcome familiar inderivative by case, interactions either like the which for powers make Higgsless instance of amplitudes 2-to-2 SMfor grow scattering with massive [ energy. fermions In inwith the the an case Higgsless arbitrarily at large SM hand, number like [ of legs that causes the breakdown of perturbation theory. Higgs and Goldstone bosonEFT suppressed with only the scalar by potential the in EW eq. scale ( of perturbation theory around thatinteraction scale. terms of This always dimension happens because of the presence of We can see that, for JHEP10(2019)255 ] . i ]. ), ∼ 30 Ψ , πv | 31 . 2 3.9 (3.7) (3.8) (3.9) πv π 2 , while , where i ∼ pairs are Ψ 2 πv | | 4 0 ) few, in which n T GG † . ∼ 0 h T , and its terms ∗ | → m max H Ψ † n h . Indeed taking =0 =0 H s l I ] ) is easily seen to read , √ 2 π GG 3.5 ([ . ∼ 1 we have that cannot be removed by ] of the n-body phase space &  |M ], which claims the onset of strong (1) 29 | ) 2 H 3 ∗ ) 1. In this bound we have only † 30 will depend little on the precise 2 ∗ ∆ | H ∗ (Λ πv Λ ∼ O n m (4 . However, one can show that the unitarity V |   ! 3 n amplitudes at the scale of order 4 2  1 . For , which were discussed in section / n ∆ , can the validity regime of the EFT be 1  h n | | h 2 3 / m =2 exp ∞ 1 the bound in eq. ( πv m | X n → ∆ 3 4 | i  πv  ∆ h 4 = | is the volume [ 2  ) h ∗ =0 =0 ∗ m ∗ 3 2 ∗ ` I m ] – 11 – m − πv  Λ  even for m n =Λ 2 2 s (4 ) of the center-of-mass energy / GG √ π 1 [

∗ πv | 4 log 2 4 | with 2 + ) ∼ − 2)!(4 ∗ n n 2 log  2 − 2 ) h x n 2 ∗ state 4 . πv Λ /m v 4 h → 2 (4 1)!( ∗ m π − πv GG ∼ m n =0 4 Inserting the explicit form of the amplitude and performing the =0 27 l I 2( ] 512 6 max 2 3 0. , from which we obtain the unitarity bound on the BSM scale: = n scattering of special multi-particle Higgs and Goldstone states [ ∆ In fact, that scale is only logarithmically sensitive to the magnitude GG ∗ -wave n s ([ Λ → 7 n Π final states. In reality, final states involving any number of h d ≤ |M n → ). A similar bound can be derived whenever the potential (or any other m ∗ R n h n m Π 3.1 d 1 the maximum scale of the UV completion is parametrically of order 4 ) = , the above condition reduces to x n Z ( ! n | ∼ 1 n , and thus remains of order 4 3 V | 3 ∆ It is clear from our argument that the bound on | =2 ∞ We stress that the effect we discuss is unrelated to the one in [ This approximation clearly breaks down for large enough ∆ X n | 7 6 , leading to an onset of strong coupling in 2 bounds are dominated by case the effect of the Higgs mass oncoupling the within phase the space integral SMa at in still high multi-Higgs perturbative energies amplitudes contact canand interaction near be arising way the safely from above neglected. production the threshold threshold. interplay and between is Our (large) free non-perturbative effect from amplitude arises the and from subtleties (small) existing phase in space. [ scattering amplitudes. Only whenorganized the as EFT an expansion lagrangian in isparametrically 1 extended analytic above in the EWof scale. BSM Such models an with EFT the is scale a separation low-energy approximation part of the lagrangian)field contains redefinitions terms or non-analytic equationsterms in of between Higgs motion. and Goldstone Inv bosons such are a suppressed only case, bySuch higher-dimensional powers a of interaction set-up the is EW equivalent scale gauge to the invariant operators SMEFT with with the large expansion canonical parameter dimensions may dominate contributions to by exploiting These improvements do notand change are the not essential parametric for dependence our of argument. the limitform in of eq. eq. ( ( 3 TeV, as expected. of considered the equally important. Our computationthe thus true represents upper a bound on lower the bound cut-off of is lower. Further optimization of the bound is possible By definition For in the limit the sum over to coincide with the where strong coupling at some finite value Λ JHEP10(2019)255 h 1 . 0 m ∼ ξ 4 appear rather (1), can never be & O | . That is completely 3 = f 3 ∆ | . The bottom line is that, and (10) even if corrections to 1 ∗ O 5 TeV for strong coupling. A m & 2 is possible under very broad ∼ , values 3 ξ , indeed ∆ 3 ≤ 3 like for all other Higgs couplings. Our ∆ ξ ≤ ∼ above the weak scale crucially relies on 3 . Corrections to the cubic couplings arise at ∗ 2, vacuum stability depends on the pattern of ∗ , as visible in figure m . ξ – 12 – ≤ 3 should be at least a few percent, which implies /m 3 to the cubic Higgs coupling can be enhanced when ξ , which is the central result of this paper. ∆ 3 than our accidentally light Higgs. For this reason we 1 ≤ 3 1 escaping, via multiple tunings, all phenomenological . In view of that, it is impossible to derive sharp bounds, ∗ ∼ m 1 TeV for weak coupling to (10%). However, this simple estimate ignores the issue of ξ while keeping or for smaller ∼ O 3 ∗ g is strongly coupled, such that ∆ ∗ 1 is measured by experiment, we immediately learn important facts m of BSM particles is hierarchically larger than the EW scale. Under 1 TeV or ∗ . m |  3 ∗ ranging from ∆ m | ∗ m ), that is from dimension-6 operators in the SMEFT lagrangian. Power counting hold for moderate being suppressed with respect to their natural values, 2 3 ∗ − v Obviously, our bounds are not set in stone. There is always the possibility of a theory It is important to stress that the upper values of ∆ Enhancement of the cubic in the range 0 m ( constraints from Higgs andgiven EW the outcome precision of measurements directmeasurements and BSM that searches from returned at direct results the consistent searches. LHC,be with as a the Still, well less SM as likely predictions, a option webelieve wide to believe that range enhance that the of ∆ to bounds precision presented in this paper are robust. of generic composite Higgsscenario models for one maximizing has ∆ ∆ and consistent, but necessarily accidental or fine tuned. with either will translate into a stronger bound on ∆ obtained in the moresupersymmetric natural models. models of In EW thoseis symmetry models, a breaking, even symmetry when like controlling the composite the Higgs Higgs size is or of strongly all coupled, terms there in the Higgs potential. Indeed in the case in the case about the microscopic theorycoupled. underlying the Furthermore, SM. the Firstthat of parameter BSM all, deviations it in hasThe single to flip be Higgs side rather boson of strongly couplings that may last also statement be is within that the improved LHC limits reach. on the single Higgs couplings SMEFT operators with dimensionsof higher the than BSM six, theory at whichhowever, the in scale given turn the depends present onimplausible, experimental the even constraints details allowing on for aon ∆ maximally strongly coupled BSM theory. Stronger limits assumptions. In particular, correctionsand in with this range aresignificant robustly portion compatible of with thisthe region other is hand, therefore outside the outside range the 0 present reach of LHC data. On suggests that the relative correctionthe ∆ BSM theory at other Higgs couplingsvacuum are stability. Taking thatparameter carefully regions into displayed account, in we figure found the allowed and excluded In this paper we derivedthe bounds on mass the scale Higgs bosonthis self-interactions, assumption which the are valid low-energy when SMEFT, EFT describing organized Higgs as interactions anO at expansion the in EW 1 scale is the 4 Conclusions JHEP10(2019)255 bosons Z and while robustly symmetry acts 3 W Y is an EW singlet. h U(1) × W from the EW scale, leading to ∗ m -point amplitudes in this model, similarly = 0. The resulting SMEFT class contains n W,Z would be classified as SMEFT according to the Therefore, the HEFT is an appropriate parametrically larger than the EW scale m 2 | 8 ∗ . m DH . We studied corrections to the cubic Higgs πv | ] = 0. In our classification analyticity versus non 2 H / † – 13 – 3 symmetry, provided one allows in the lagrangian ) H H invariant language, are described by a non-analytic H Y † Y = 0 and thus unavoidably associated to a low cut-off scale. For H (  H U(1) around U(1) × × H W = [1 + W L makes manifest, via the equivalence theorem, the existence of the ] may allow one to circumvent these phenomenological constraints. H amplitudes for scattering of longitudinally polarized n 32 , → . At first sight, this scenario may offer more freedom to arrange for a large 21 , modulo field redefinitions, is what distinguishes SMEFT from HEFT. In ] and as HEFT according to ours. Indeed the H amplitudes mentioned in the previous paragraph, which prohibit extending H 33 † n H ] proposed another criterion to distinguish SMEFT and HEFT. That criterion states that 3 Higgs bosons become strong and violate perturbative unitarity around the scale → 33 3 TeV. Therefore, in this scenario it is impossible to have a sizable ∆ ≥ n ≈ Our analysis exemplifies the physical difference between Higgs EFTs with analytic and We also investigated a more general EFT where the Higgs potential at the EW scale 1. Namely, 2 without violating stability or experimental constraints. We have shown however that Ref. [ 8 3 ∼ πv SMEFT corresponds to theelectroweak symmetry special is subclass restored, ofthe or, SMEFT HEFT equivalently, class where for defined by which ourtive there criterion. lagrangians exists However that it a are seems to non-analytic pointinstance, us at it in it seems is field strictly to larger, space uscriterion as that where of it e.g. also ref. includes [ effec- to the case studied in this paper, imply a low cut-off scale, which makes our criterion appear more physical. Conversely, BSM models with theare mass described scale at low energies by the SMEFT. classification is not just athe matter non-analyticity of in aesthetics andstrong directly 2 concerns the dynamics.the Indeed validity of thatlow-energy HEFT description above for the non-decoupling BSM scale models 4 with the mass scale close to a TeV. with a linearly realized SU(2) terms that are non-analytic in analyticity in this paper we discussedto only distinguish the SMEFT Higgs vs potential, HEFT but at the the same level classification of can Higgs be interactions used with other fields. Our Both of these EFTs haveis the usually same introduced particle as spectrum a (thatin of more a the general non-linear SM), theory way however whereThis the on the results HEFT the in SU(2) more Goldstone freedom bosons,expansion. in while writing In the the Higgs this Higgs potential paperthe at boson we the SMEFT provided leading and a order in clear the the and HEFT. EFT intuitive We dynamical argued distinction that between the HEFT can be equivalently formulated non-analytic potential. In thethe standard SMEFT nomenclature, and the these HEFT, EFTsries respectively. go was Previously, described under the in distinction the a between less names the intuitive of language two of theo- linearly or non-linearly realized symmetries. ξ into 4 satisfying all the constraints fromdirect single searches. Higgs processes, Again, EW itmodel precision is building measurements not [ and completely excluded that multiple tunings and/or clever cannot be written asself-coupling a that, in power a series SU(2) function in of ∆ in such a setting there is an obstruction to decoupling JHEP10(2019)255 ]. D ]. , SPIRE (2017) ]. and IN SPIRE h ][ IN ]. Phys. Rev. → D 95 ][ SPIRE , collisions at gg IN ][ pp SPIRE (2017) 887 IN ][ Phys. Rev. A global view on the Higgs C 77 Electroweak oblique , ]. (2019) 121803 arXiv:1606.02266 , CERN, Geneva, Switzerland ]. 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