Improved unitarity constraints in Two-Higgs-Doublet-Models
KKA-TP-11-2018
Improved unitarity constraints in Two-Higgs-Doublet-Models
Mark D. Goodsell ∗ and Florian Staub
2, 3, † Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589,Sorbonne Université et CNRS, 4 place Jussieu, 75252 Paris Cedex 05, France Institute for Theoretical Physics (ITP), Karlsruhe Institute of Technology, Engesserstraße 7, D-76128 Karlsruhe, Germany Institute for Nuclear Physics (IKP), Karlsruhe Institute of Technology,Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-Leopoldshafen, Germany
Two-Higgs-Doublet-Models (THDMs) are among the simplest extensions of the standard modeland are intensively studied in the literature. Using on-shell parameters such as the masses of theadditional scalars as input, corresponds often to large quartic couplings in the underlying Lagrangian.Therefore, it is important to check if these couplings are for instance in agreement with perturbativeunitarity. The common approach for doing this check is to consider the two-particle scatteringmatrix of scalars in the large centre-of-mass energy limit where only point interactions contribute.We show that this is not always a valid approximation: the full calculation including all tree-levelcontributions at finite energy can lead to much more stringent constraints. We show how the allowedregions in the parameter space are affected. In particular, the light Higgs window with a secondHiggs below
GeV completely closes for large values of the Z breaking parameter | M | . Wealso compare against the loop corrected constraints, which use also the large √ s approximation, andfind that (effective) cubic couplings are often more important than radiative corrections. I. INTRODUCTION
The discovery of a scalar boson at the Large HadronCollider with a mass of around 125 GeV was a mile-stone for particle physics [1, 2]. This state has all ex-pected properties of the long searched-for Higgs boson,and all particles predicted by the standard model of par-ticle physics (SM) have finally been found. Even if noadditional, fundamental scalar has been observed so farat the LHC, it is much too early to give up the possi-bility that more Higgs-bosons exist which are involvedin electroweak symmetry breaking (EWSB). There areseveral possibilities what the origin and the properties ofsuch states could be. A very attractive and well stud-ied scenario is that a second Higgs doublet exists. AfterEWSB, the two Higgs doublets yield one particle whichhas all the properties of the discovered state, but theyalso predict the presence of one charged and two neutraladditional bosons. There exist several constraints on thiskind of models: the LHC measurements must be repro-duced, the absence of any other signal must be explained,including modifications to rare decay processes. From thetheoretical point of view, these models are usually con-fronted with two conditions: (i) the electroweak vacuummust be stable or at least sufficiently long-lived[3–12], (ii)unitarity should not be violated[13–19]. In order to probeunitarity in BSM models, the standard procedure in theliterature is to calculate the scattering matrix for → processes involving scalars. Usually, only point interac-tions are included, which do not vanish for very largescattering energies √ s . For extensions of the StandardModel, the contributions from scalar trilinear couplings ∗ [email protected] † fl[email protected] have only been considered for singlet extensions and theminimal supersymmetric standard model[20–22]. There-fore, it is time to check if the large √ s approximation inTHDMs is valid or under which circumstances it mightgive misleading results.This letter is organised as follows: we show our conven-tions for THDMs in sec. II, before we briefly summariseour approach to calculate the tree-level unitarity con-straints in sec. III. The impact on the parameter space isdiscussed in sec. IV. In sec. V, we compare against pre-viously derived one-loop results; and rederive the con-straints for different unitarity conditions. We concludein sec. VI. II. MODEL
The scalar potential of a CP conserving THDM withsoftly broken Z symmetry reads V Tree = λ | H | + λ | H | + λ | H | | H | + λ | H † H | + m | H | + m | H | + (cid:18) m H † H + 12 λ ( H † H ) + h.c. (cid:19) (1)After EWSB, the neutral components of the two Higgsstates receive vacuum expectation values (VEVs) of H i = (cid:18) H + i √ ( φ i + iσ i + v i ) (cid:19) i = 1 , (2)with (cid:112) v + v = v (cid:39) GeV and tan β = v v . Themass spectrum consists of superposition of these gaugeeigenstates, i.e. ( φ , φ ) → ( h, H ) , ( σ , σ ) → ( G, A ) and ( H +1 , H +2 ) → ( G + , H + ) . Here, G and G + are theGoldstone modes of the Z and W boson. The mixing inthese sectors is fixed by tan β , while in the CP-even sector a r X i v : . [ h e p - ph ] M a y a rotation angle α defines the transition from gauge tomass eigenstates. In practical applications, one can tradethe physical masses m h , m H , m A and m H + as well as tan β and tan α for the quartic couplings. The necessaryrelations are λ = 1 + t β t α ) v (cid:0) m H + m t β + t α ( m h + m t β ) (cid:1) (3) λ = 1 + t β t α ) t β v (cid:0) m + m t α + t β ( m h + m H t α ) (cid:1) (4) λ = 1(1 + t α ) t β v (cid:104) m h t α + 2 m H + (1 + t α ) t β + m h t α t β − m H t α (1 + t β ) + m (1 + t α )(1 + t β ) (cid:105) (5) λ = 1 t β v (cid:0) − m + m A t β − m H + t β − m t β (cid:1) (6) λ = 1 t β v (cid:0) − m − m A t β − m t β (cid:1) (7)with t β = tan β and t α = tan α . This has the advantagethat physical observables instead of Lagrangian param-eters can be chosen as input. However, one needs to becareful since a randomly chosen set of masses could eas-ily correspond to a problematic set of quartic couplings:for very large couplings perturbativity will be spoilt andalso unitarity can be violated. III. UNITARITY CONSTRAINTS
Perturbative unitarity constraints come from applyingthe unitarity of the S-matrix for → scalar field scat-tering amplitudes. We calculate a matrix a ba given by a ba ≡ π (cid:114) | p b || p a | δ δ s (cid:90) − d (cos θ ) M ba (cos θ ) , (8)which is derived proporional to the zeroth partial wave ofscattering pairs of scalars a to pairs b having matrix ele-ment M (cos θ ) , where θ is the angle between the incom-ing and outgoing three-momenta ( p a , p b respectively) inthe centre-of-mass frame. The factor δ ( δ ) is if par-ticles { , } ( { , } ) are identical, and zero otherwise. Wethen find the eigenvalues of this matrix, which we denote a i , and insist that they must satisfy | Re( a i ) | ≤ . (9)Classic unitarity constraints for the THDM have beencalculated in the limit of large scattering energies, inwhich case only the quartic couplings contribute to scat-tering and the momentum dependence of the prefactorof the integrand in (8) disappears; moreover all diagramswith propagators are suppressed by the collision energy squared and can be neglected, so the final result appearssuperficially independent of the scattering energy. Thishas been applied at tree [] and one-loop [] level. Thelimits on the quartic couplings at tree level in this ap-proximation areMax (cid:110) | λ ± λ | , (cid:12)(cid:12)(cid:12)(cid:12) λ + λ ± (cid:113) ( λ − λ ) + λ (cid:12)(cid:12)(cid:12)(cid:12) , | λ ± λ | , (cid:12)(cid:12)(cid:12) λ + λ ) ± (cid:112) λ − λ ) + (2 λ + λ ) (cid:12)(cid:12)(cid:12) , | λ + 2 λ ± λ | , (cid:12)(cid:12)(cid:12)(cid:12) λ + λ ± (cid:113) ( λ − λ ) + λ (cid:12)(cid:12)(cid:12)(cid:12) (cid:111) < π. (10)However, it has not been tested if the large s approx-imation is valid in all BSM models in which it is ap-plied. It could be that large contributions are presentat smaller s which then rule out given parameter regionsin the considered model. The theory could develop aLandau pole before s is sufficiently large to neglect themasses, or could be defined with a low cutoff. And atlarge values of the couplings, their running is usually suf-ficiently fast so that the values of the couplings at anenergy scale √ s are vastly different from those at lowerenergies. So in order to be able to test unitarity at fi-nite s , the Mathematica package
SARAH has now beenextended. The salient features are: (i) all tree-level dia-grams with internal and external scalars are included tocalculate the full scattering matrix; (ii) We neglect allgauge couplings, and treat Goldstone bosons as physicalparticles with mass equal to the gauge boson; (iii) thecalculation is done in terms of mass eigenstates, i.e. thefull VEV-dependence is kept; (iv) the numerical evalua-tion is done with the Fortran code
SPheno [23, 24]; (v)large enhancements close to poles are cut in order notto overestimate the limits. This is demonstrated at one-example in Fig. 1. More details and derivations of ourfull procedure are given in the accompanying paper [25].
IV. RESULTS
In this section we shall study the impact of the im-proved unitarity constraints on the two Higgs doubletmodel at tree level. We have chosen for our discussiontype–I, but the results hold also for other models, be-cuase our we omit fermions from our scattering processes.Hence there is only an indirect difference between theconstraints for type–I and type–II: the limits from flavourobservables are stronger for light charged Higgs massesfor type–II. Hence, the m H + must be larger in generalfor type–II [26]. On the other hand, we include the con-straints from Higgs searches via HiggsBounds [27–29],which can vary to a lesser extent between type I and IImodels.Our numerical analysis is based on the
SPheno [23, 24]interface of
SARAH [30–34]. By default,
SPheno calculates
FIG. 1. s -dependence of the maximal scattering eigenvalue.The black lines indicate the kinematic thresholds while thered region is cut about because of s -channel resonance withheavy charged and pseudo-scalar Higgs. the mass spectrum at the full one-loop level and includesall important two-loop corrections to the neutral scalarmasses [35–37]. However, we shall not make use of theseroutines in the following but work at tree-level, or equiva-lently under the assumption that an OS calculation worksin principle (with all the caveats discussed in Ref. [38]).This is because we cannot (yet) calculate quantum cor-rections to unitarity at finite s , and when the couplingsare large in almost all cases the quantum corrections tomasses/couplings become very large: this gives furthermotivation for including only constraints at finite s ! A. The light Higgs window
We start with a discussion of the effects in the casethat both CP even Higgs states have masses of 125 GeVor below. A comparison between the ‘classic’ – equa-tion (10) – and new constraints is given in Fig. 2. Obvi-ously, one finds much stronger constraints in two differ-ent cases once finite s is considered: (i) for smallish | M | the wedge M A = M H + (cid:29) m H disappears; (ii) for larger | M | the scattering amplitude in the overall ( m A , m H + grows signifcantly. The responsible channels and bestscattering energies causing these effects are quite differ-ent: • Small | M | : consider the following simplified hier-archy, m H = m h = − (cid:112) | M | (cid:28) m A = m H + ∼ √ s (11)together with tan β = − / (tan α ) = 1 . The domi-nant channels are those with heavy external statesand a light Higgs exchange. For instance, the am- plitude AA → AA can be approximated as a ( AA → AA ) = m A (cid:16) − s log (cid:16) m h − m A + m h + s (cid:17)(cid:17) πsv (cid:112) s ( s − m A ) (12)From that, we get that the ratio compared to theold constraints a max0 a s →∞ = − m A log (cid:16) m h − m A + m h + s (cid:17) (cid:112) s ( s − m A ) (13)This ratio becomes maximal slightly above thekinematic threshold s Threshold = 4 m A and an en-hancement of 2–3 is possible. Thus, the best scat-tering energy √ s is around 1–2 TeV. • Large | M | : in this case we can consider the fol-lowing, simplified hierarchy m H = m h ∼ √ s (cid:28) m A = m H + = (cid:112) | M | (14)Now, the dominant scattering processes are thosewith light external scalars only. The maximaleigenvalue of the full scattering matrix is roughlygiven by diagonalising the submatrix with CP-evenstates only hh → hh hh → HH hh → hHHH → HH HH → hHhH → hH (15)By doing that, we find that the ratio between theold and new results scales as a max0 a s →∞ = 2 M log (cid:16) m h s − m h (cid:17)(cid:112) s ( s − m h ) ∼ | M | m h (16)Thus, this ratio grows very quickly with increasing M and one finds very strong unitarity constraintsalready at scattering energies √ s of a few hundredGeV. B. Heavier Scalar a. Stronger Constraints
We turn now to the casethat all new scalars are heavier than the SM-like Higgs.We start with a short analytical estimate for parameterregions in which difference between the our calculationand previous results show up. This is, for instance, thecase for the configuration m A ∼ M H ∼ M H + (cid:29) (cid:112) | M | . (17)Assuming again the tan β = − / tan α = 1 for the mo-ment, the maximal eigenvalue for the scattering matrixin the large s limit is a max ,s →∞ = 116 πv (8 M + 4 m A + 5 m h ) (cid:39) πv m A (18)
100 200 300 400 500 600 700300400500600700 100 200 300 400 500 600 700300400500600700 100 200 300 400 500 600 700300400500600
100 200 300 400 500 600 700300400500600700
100 200 300 400 500 600 700300400500600700 100 200 300 400 500 600 700300400500600
FIG. 2. Comparison between the old and new unitarity constraints for a second light CP even scalar for three different valuesof M . The figures in the first row show the ratio of points which pass the old unitarity constraints but are ruled out by thenew ones. The second row shows the average enhancement in the maximal scattering element. The other parameters werevaried in the ranges m H ∈ [60 , GeV, m A ∈ [30 , GeV, m H + ∈ [250 , GeV, tan α ∈ [ − . , − . , tan β ∈ [1 , . We want to compare this with the scattering HA → HA scattering process which includes diagrams with the SM-like Higgs in the propagator. We find a ( HA → HA ) (cid:39) m h πv (cid:112) s ( s − m A ) × (cid:104) (cid:0) m A − s (cid:1) − (cid:0) M + 2 m A + m h (cid:1) log (cid:18) m h − m A + m h + s (cid:19) (cid:105) (19) (cid:39) m A log (cid:16) m h − m A + s + m h (cid:17) πv (cid:112) s ( s − m A ) (20)Thus, for s = 5 m A close to the kinematic threshold wefind an enhancement of roughly | √ log m h m A | compared tothe large s approximation. For m A = 700 GeV this cor-responds to nearly a factor of 2. We can confirm this bymaking use of the full numerical machinery. In Fig. 3 weshow the impact on the maximal allowed value for m H in the ( m A , m H + ) plane while scanning over all other pa-rameters as indicated in the caption. We see that thisvalue shrinks significantly and a large region of the planewhich is allowed by the old constraints is no longer ac-cessible. b. Weaker Constraints If we consider the scatteringup to a finite √ s , we can find that the scatter eigenval-ues become smaller compared to the limit √ s → ∞ forseveral reasons: (i) the dominant channels can be kine-matically forbidden; (ii) there can be a negative interfer-ence between the point interactions and the propagatordiagrams; (iii) the dominant channels can be cut out be-cause of possible resonances in order not to overestimatethe unitarity constraints. Due to these effects, one needsto ask the question to which energy scale we have actu-ally probed scattering processes of scalars at the LHC.Of course, the LHC is running with √ s = 14 TeV. How-ever, it is unrealistic to assume that the full energy isavailable in the → scattering of scalars. Moreover,there are different options to handle the t - and u -channelpoles, which can appear if internal states become on-shell, depending on how aggressive or conservative thelimits should be: if we remove these poles either com-pletely or only by a partial diagonalisation of the scat-tering matrix, large contributions to the scattering can bedropped at small s . We demonstrate via one example inFig. 4 where the maximal eigenvalue as a function of √ s isshown. If we completely ignore the t - and u poles we seea huge enhancement close to some kinematic thresholds.In contrast, if we work with a partial diagonalisation as
200 300 400 500 600 700300400500600700 200 300 400 500 600 700300400500600700
FIG. 3. The maximal value of m H + when using the large s approximation (first row) or the full calculation (secondrow). Here, we varied m H + ∈ [250 , GeV, M ∈ [ − , − ] GeV , tan α ∈ [ − . , − . , tan β ∈ [1 , . proposed in Ref. [22] we see that we find the eigenvalueof the large s approximation only for √ s > TeV. Thismight be rather surprising since all involved masses arebelow 1 TeV!
V. COMPARISON WITH LOOP CORRECTIONS
Since one of our motivations for considering finite s scattering is that the quantum corrections to masses andcouplings become large as we increase the scattering en-ergy, it is also important to examine the effect of loopcorrections to unitarity. Moreover, the boundary of uni-tarity may also coincide with a loss of perturbativity. In FIG. 4. s -dependence of the maximal scattering eigenvalue.Here, we have used two possibilities how to deal with the t -and u -channel poles. The black lines indicate the kinematicthresholds while the red region is cut about because of s -channel resonances. The vertical lines indicate limits at which t - and u -pole disappear. The two options for dealing witha t / u -channel poles are: (i) the poles are ignored and thefull scattering-matrix is taken into account (option 0); (ii)the elements affected by the poles are dropped and a partialdiagonalisation of the remaining matrix is performed (option2). general, loop corrections to unitarity have been very lit-tle studied in BSM models; however, in the context ofthe THDM, they were considered in Ref. [39] in the limitof √ s much larger than the masses in the theory. Wecan therefore make a direct comparison. In that paper,they presented general formulae for the loop correctionsto a in terms of the quartic couplings of the theory eval-uated at the scale √ s, which are effectively independentof the particle masses. Results for two scenarios, onewith an SO (3) symmetry and another with “MSSM-like”couplings, were presented.We shall make our comparison with the “MSSM-likecouplings”; in SARAH conventions this means λ = λ , λ = − λ − λ , λ = 0 . (21)With these restrictions the ‘classic’ tree-level constraintsof equation (10) simplify to | λ − λ | ≤ π, | λ + 2 λ | ≤ π, (22)which describe a rhombus inlcuding the origin. Requiringstability of the potential requires λ > , λ > − λ , (23)which, when we combine the two, leaves a portion of theparameter space where λ is at most π , and λ < π. In the previous sections, we applied the unitarity con-straint | Re( a i ) | < / , but in [39] they apply a differentconstraint, which we shall now examine. The startingpoint is the equation Im( a i ) ≤| a i | , (24)(for an elementary derivation see [25]). Naively this givessimply | a i | ≤ , which is a constraint sometimes appllied,but with a little rearranging we have Re( a i ) ≤ | Im( a i ) | (1 − | Im( a i ) | ) (25)which gives the classic limit (9). This limit makes noassumption of perturbativity, and indeed when Re( a i ) obtains its maximum value then Im( a i ) = | Re( a i ) | . Since
Im( a i ) is only generated at first at one loop order, thensaturating this bound would potentially require violatingperturbativity. On the other hand, rearranging again, wecan write the above as | a i − i | ≤ . (26)If we have complete ignorance of Im( a i ) then we justrecover the same constraint as above. However, if we havecalculated a at one loop and assume that perturbativityholds, then we can use our calculated values for the realand imaginary parts of a i and use the above constraint.Focussing on one eigenvalue, let us write a i ≡ a + b R + ib I (27)and expand eq. (26) then we find ( a ) + b R + 2 a b R − b I + b I ≤ . (28)Now Ref. [39] then appeal to perturbation theory so that b I = ( a ) + higher order terms (29)and then obtain a b R ≤ − b R + ... → | a | ≥ | b R | . (30)This can then be a very strong constraint. In Fig. 5we show the constraints from applying eq. (26) as doneby Ref. [39], with the constraints from our trilinear cou-plings and the tree-level constraints for comparison. Thetree-level quartic-only and one-loop constraints are in-dependent of tan β and all of the mass scales (exceptthat they should be interpreted as couplings evaluatedat a renormalisation scale √ s ), whereas for our scan wechoose two values of tan β (marked on the plot) and fixthe tree-level lightest Higgs mass to be GeV – this isenough to determine all of the remaining free parametersonce λ and λ are specified.We see from Fig. 5 that even though the loop-levelconstraints seem extremely severe, our tree-level trilinearconstraint still removes a significant chunk of the remain-ing parameter space.However, these one-loop constraints have the curiousfeature of excluding couplings near the origin, whicharises from the regions where one scattering eigenvaluevanishes at tree level. Indeed, from eq. (30) we see thatif a = 0 (as can happen for linear combinations of thecouplings) then unitarity is apparently violated. In the FIG. 5. Tree-level and one-loop constraints on λ and λ inthe “MSSM-like” THDM. Quartic-only tree-level constraintsare shown as black dot-dashed lines, the vacuum stability con-straint λ ≥ − λ is the red dashed line; our tree-level con-traints including trilinears are labelled with tan β = 2 and tan β = 30 . The one-loop allowed region from [39] is thewhite region enclosed by the solid purple and orange curves.The second plot is a zoom into the first one. notation of Ref. [39] the purple curve corresponds to theeigenvalue a , which derives from the scattering of (cid:15) αβ Φ α τ Φ β → (cid:15) αβ Φ α τ Φ β (31)where τ = (cid:18) − (cid:19) and gives the scattering eigen-value at tree level of a = 2 λ + 2 λ . The orange curve corresponds to a − from scatter-ing Φ † i Φ i → Φ † j Φ j , i = { , } , (32)which give the scattering eigenvalues at tree level of a = {− λ + λ , − λ − λ } . We therefore see that the one-loop constraints arisestarting from the lines λ + λ = 0 and λ + λ = 0 . The reason for this is, however, assuming that the higher-order terms in eq. (29) are not important. Indeed, inthe cases where a = 0 for λ i (cid:54) = 0 we would apparentlybadly violate perturbation theory – but this is just be-cause we have only computed up to one loop, and have atuned cancellation at tree level. Since eq. (30) compares atree-level and one-loop amplitude this seems particularlybad. Hence, if we examine the perturbation series moreclosely, specialising to the case of only quartic couplingsfor simplicity, and define λ to be a number of O ( λ i ) as aperturbation series parameter, so that b R ≡ b ,R λ + b ,R λ + ...b I ≡ ( a ) λ + b ,I λ + ... (33)we see that a → n is nonzero first for → processesat order λ . Hence defining (cid:80) n> | a → n | ≡ | X | λ wehave, order by order in perturbation theory up to λ : a b ,R − b ,I =0 (34) ( a ) + b ,R + 2 a b ,R − b ,I + | X | =0 . (35)We see that the origin of eq. (30) depends on neglecting b ,I , but if we include the information from eq. (34) thenwe would have obtained instead of eq. (30): b R + higher order terms of indeterminate sign = 0 . Furthermore, when a = 0 we simply recover b ,I ≥ and b ,I = 0 , which we could surmise from a being of O ( λ ) and the standard unitarity relation. We do notobtain any new constraint beyond | Re( a ) | ≤ . Hence in Fig. 6 we recompute the constraints at one-loop applying instead | Re( a ) | ≤ for Re( a ) = a + b ,R for the same scattering processes listed above. Weuse the expressions in the appendix from Ref. [39] toobtain the one-loop scattering amplitudes neglecting thewavefunction renormalisation contributions. These aremass-dependent and were found to be small in Ref. [39].The reason is that, in the limit that √ s is much largerthan all masses, only diagonal self-energies appear in theresults which consist of expressions of the form z / ii ∼ ( vλ ) dds ( B ( s, m , m )) ∼ ( vλ ) s → . (36)Due to the presence of the trilinear couplings in theseterms they appear at the same order as box and trianglediagrams.The one-loop constraint is then stronger than the“naive” tree-level one in some cases, and weaker in oth-ers; but we find that our tree-level constraints includingthe effect of trilinears are stronger than both in almostall cases.For comparison, one could also check the one-loop al-lowed region for the sometimes used criterion | a | ≤ .These are almost universally weaker than the tree-levelconstraints, implying that they are not sufficiently con-servative, as can be expected. FIG. 6. Tree-level and one-loop constraints on λ and λ inthe “MSSM-like” THDM. Quartic-only tree-level constraintsare shown as black dot-dashed lines, the vacuum stability con-straint λ ≥ − λ is the red dashed line; our tree-level con-traints including trilinears are labelled with tan β = 2 and tan β = 30 . The one-loop allowed region applying the con-straint | Re( a ) | ≤ is the white region enclosed by the solidorange curve. VI. CONCLUSION
We have revised the tree-level perturbative constraintsin THDMs by including the contributions from (effec-tive) trilinear couplings, and provide an extension of thepackage
SARAH which makes it possible to include theseconstraints in phenomenological studies in THDMs andmany other BSM models. We found that the obtainedlimits can be significantly stronger than the ones usuallyapplied in literature which are only correct in the limit oflarge scattering energies √ s . The importance of the im-proved constraints has been demonstrated by two chosenexamples: (i) it was shown that the values of M arehighly constrained in the light Higgs windows; (ii) onefinds a stronger upper limit for the CP-even Higgs massin scenarios with (cid:112) | M | < m A , m H + . On the otherside, we have also discussed that the restriction to max-imal scattering energies of a few TeV can revive pointswhich violated unitarity only at much higher energies.We also made comparison with previous constraints de-rived at one-loop level in the large s approximation. Ourresults indicate that the tree-level constraints includingtrilinear couplings are the most important for this classof models, and are not superseded by the one-loop large-momentum constraints; instead, it would be a very inter-esting if rather complicated task to include the effects ofthe trilinear couplings at one-loop order, which could po-tentially strengthen constraints on these models further.In other BSM models similar – or even larger – differencesbetween the full calculation and the large s approxima-tion can be seen. This is discussed for example in Ref. [40]for several triplet extensions. ACKNOWLEDGEMENTS
We thank Marco Sekulla for helpful discussions. FS issupported by ERC Recognition Award ERC-RA-0008 ofthe Helmholtz Association. MDG acknowledges supportfrom the Agence Nationale de Recherche grant ANR-15- CE31-0002 “HiggsAutomator”, and the Labex “InstitutLagrange de Paris” (ANR-11-IDEX-0004-02, ANR-10-LABX-63). We would like to thank Sophie Williamsonand Manuel Krauss for helpful discussions and collabo-ration on related topics. [1] G. Aad et al. (ATLAS), Phys. Lett.
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