Unitarity constraints in triplet extensions beyond the large s limit
KKA-TP-12-2018BONN-TH-2018-04
Unitarity constraints in triplet extensions beyond the large s limit Manuel E. Krauss ∗ and Florian Staub
2, 3, † Bethe Center for Theoretical Physics & Physikalisches Institut der Universität Bonn, Nußallee 12, 53115 Bonn, Germany 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
Triplet extensions are attractive alternatives to the standard model (SM) of particle physics.While models with only one triplet are highly constrained by electroweak precision observables, thisis not necessarily the case once several triplets are present as in the Georgi-Machacek model. Asin all other BSM models, the parameter space of triplet extensions is constrained by the conditionthat perturbative unitarity is not violated. For this purpose, limits on the eigenvalues of the scalar → scattering matrix are set. It is very common in the BSM literature that the scatteringmatrix is calculated under one crucial assumption: the scattering energy s is so large that onlypoint interactions involving quartic couplings provide non-negligible contributions. However, it isnot given that this approximation is always valid – in fact, diagrams involving propagators can playan important role. We discuss at the examples of (i) the SM model extended by a real triplet, (ii)the Y = 1 triplet extension of the SM, and (iii) the Georgi-Machacek model, how the tree-levelunitarity constraints are affected once the large s approximation is given up. For all models we findthat the impact of (effective) cubic couplings can be crucial. I. INTRODUCTION
While the LHC continues ruling out more and moreparameter space of models beyond the SM (BSM), a cen-tral question remains unanswered: whether the measuredHiggs boson is the only electroweak- to TeV-scale scalarparticle, or if there is more in the scalar sector whichtakes part in electroweak symmetry breaking (EWSB).Many attractive models have been proposed which areeither motivated from theory as they solve or ameliorateproblems in the standard model, or from experimentalreasons in the sense that they provide new interesting sig-nals which could be measured at colliders. One of thesepossibilities is the presence of one or more triplet scalars.Those could either be introduced for implementing a see-saw mechanism of type II for generating small Majorananeutrino masses [1, 2], or for providing an alternative toEWSB where triplets actively contribute by developingnon-negligible vacuum expectation values (VEVs) like inthe so-called Georgi-Machacek (GM) model [3].Before analysing its properties for the LHC, eachmodel’s parameter space has to be confronted with the-oretical constraints. Among the most stringent are theconditions that perturbative unitarity of scalar → scattering must not be violated. For the simplest tripletextension, for instance, those have been derived in Ref. [4]and for the GM model in Refs. [5, 6]. These derivations,as almost all other unitarity constraints on BSM modelswhich are found and applied in the literature, make useof the limit that the scattering energy s is much largerthan all involved masses. This has the benefit that alldiagrams containing propagators can be disregarded and ∗ [email protected] † fl[email protected] only quartic point interactions need to be taken into ac-count.Recently, it has been pointed out in Ref. [7] thatalso checks of the perturbative behaviour of a non-supersymmetric model should be taken seriously. It wasshown at the example of the GM model how large loopcorrections can be triggered by large scalar trilinear cou-plings, ultimately casting doubt about the perturbativetreatment in large regions of parameter space. Theselarge couplings might not be visible at first glance whentrading the corresponding parameters for the (tree-level)masses and therefore removing them from the list of “in-put” parameters.However, large trilinear couplings do not only affectthe loop corrections in a model, but also the → scat-tering processes at √ s not much larger than the involvedmasses. As a result, the amplitude at finite √ s mightbe significantly larger than in the typical limit √ s → ∞ ,therefore also affecting the perturbative unitarity con-straints! Because of that, we check here the full scatter-ing matrix for all possible → processes with externalscalars. We do so for a simple real triplet extension, acomplex triplet extension, and ultimately the GM model.We do not only include the possibility of finite √ s butalso the effects of electroweak symmetry breaking. Whilethis kind of calculation was done for the SM decadesago [8], the impact of trilinear couplings on the unitar-ity constraints in BSM models were to our knowledgeonly checked for the minimal supersymmetric SM [9] andsinglet extensions [10–12]. We show how the unitarityconstraints at finite energies cut deeply into the other-wise allowed parameter space of all considered examplemodels and compare them, for the GM model, with the In Ref. [13], the effect was included for obtaining bounds on theHiggs trilinear coupling. a r X i v : . [ h e p - ph ] A ug loop-improved unitarity and perturbativity checks dis-cussed in Ref. [7].This paper is organised as follows: in sec. II we showthe main ingredients for the calculation of the perturba-tive unitarity checks. In secs. III, IV and V, we presentthe three example models and show the resulting addi-tional constraints coming from the inclusion of the im-proved treatment of the scalar scattering amplitudes. Weconclude in sec. VI. II. TREE-LEVEL PERTURBATIVE UNITARITYCONSTRAINTSA. Approximations vs. full calculation
Perturbative unitarity constraints consider the → scalar field scattering amplitudes. This means that the0th partial wave amplitude a must satisfy either | a | ≤ or |R e [ a ] | ≤ . The matrix a is given by a ba = 132 π (cid:114) | (cid:126)p b || (cid:126)p a | δ δ s (cid:90) − d (cos θ ) M ba (cos θ ) , (1)where (cid:126)p a ( b ) is the center-of-mass three-momentum ofthe incoming (outgoing) particle pair a = { , } ( b = { , } ) , θ is the angle between these three-momenta and M ba (cos θ ) is the scattering matrix element. The expo-nents δ ij are if the particles i and j are identical, andzero otherwise.At the tree level, the → amplitudes are real,which is why one usually uses the more severe constraint |R e [ a ] | ≤ , which leads to |M| < π in the limit s → ∞ . This must be satisfied by all of the eigenvalues ˜ x i of the scattering matrix M . M must be derived byincluding each possible combination of two scalar fieldsin the initial and final states.For analysing whether perturbative unitarity is givenor not, it is common to work in the high energy limit,i.e. the dominant tree-level diagrams contributing to |M| involve only quartic interactions. All other dia-grams with propagators are suppressed by the collisionenergy squared and are neglected. Moreover, effects ofelectroweak symmetry breaking (EWSB) are usually ig-nored, i.e. Goldstone bosons are considered as physicalfields.However, it is hardly tested if the large s approxima-tion is valid in all BSM models in which it is applied. Itcould be that large contributions are present at small s which then rule out given parameter regions in the con-sidered model. Just consider for instance a large TeV-scale cubic scalar interaction κφ i φ j φ k . Above the reso-nance, a typical diagram would therefore scale with κ /s and hence be relevant for √ s of O (TeV) .In order to be able to apply these tests, the Mathematica package
SARAH has now been extended bythe functionality to derive more reliable unitarity limits by giving up the large s approximations. Details of thisimplementation in SARAH are given elsewhere [14]. Weonly want to summarise the main aspects: • All tree-level diagrams with internal and externalscalars are included to calculate the full scatteringmatrix • The calculation is done in terms of mass eigen-states, i.e. the full VEV dependence is kept • All necessary routines for a numerical evaluationwith
SPheno are generated • Very large enhancements close to poles or kinematicthresholds are cut in order not to overestimate thelimits • Renormalization group equation (RGE) runningcan be included to obtain an estimate of the higherorder corrections
B. Analysis Setup
We are going to study the impact of the improved uni-tarity constraints on three triplet extensions of the SM:(i) with one real triplet, (ii) with one complex triplet, aswell as (iii) with both one complex and one real triplet,the Georgi-Machacek model. Our numerical analysis willbe based on the
SPheno [15, 16] interface of
SARAH [17–21]. By default,
SPheno calculates the mass spectrum atthe full one-loop level and includes all important two-loopcorrections to the neutral scalar masses [22–24]. How-ever, we are not making use of these routines in thefollowing but work under the assumption that an on-shell (OS) calculation is working in principle (with all thecaveats discussed in Ref. [7]). Thus, only the tree-levelmasses are calculated. These are then used to calculatethe perturbative unitarity constraints for a given scatter-ing energy √ s . The constraints from Higgs searches areincluded via HiggsBounds [25–27].
III. THE REAL TRIPLET EXTENDEDSTANDARD MODELA. Model description
We start with a rather simple BSM model: the SMextended by a real scalar SU (2) -triplet T without hyper-charge T = (cid:18) T / √ T − ( T − ) ∗ − T / √ (cid:19) (2)The scalar potential of this model is given by V = m H | H | + 12 m T Tr ( T ) + 12 λ H | H | + 12 λ T Tr ( T )+ 12 λ HT | H | Tr ( T ) + κH † T H (3)After EWSB, both the Higgs as well as the neutral com-ponent of the triplet receive a vacuum expectation value: (cid:104) T (cid:105) = 1 √ v T , (cid:104) H (cid:105) = 1 √ v . (4)The scalar mass eigenstates are two CP even stateswhich are a mixture of H and T with masses m h and m H , as well as a (physical) charged Higgs boson H ± with mass m H + which is a mixture of H + and T − . Therotation in the neutral Higgs sector is fixed by an angle α . Therefore, it is possible to trade the four Lagrangianparameters λ i ( i = H, T, HT ) and κ for the three scalarmasses and one rotation angle. The relations are: κ = 2 m H + v T ˜ v , (5) λ H = m h + m H t α ( t α + 1) v , (6) λ HT = 1 vv T (cid:32) √ t α ( m h − m H ) t α + 1 + 2 m H + vv T ˜ v (cid:33) , (7) λ T = ˜ v m H − m H + (cid:0) t α + 1 (cid:1) v + m h t α ˜ v ( t α + 1) v T ˜ v , (8)where we have defined ˜ v = (cid:112) v T + v . Since v T mustbe small in order not to be in conflict with electroweakprecision data [28–31], m H and m H + must always beclose in order to avoid too large quartic couplings. Inaddition, κ needs to be small. The absolute values of theeigenvalues of the scattering matrix in the limit of large s are given by: π > Max (cid:110) | λ H | , | λ HT | , | λ T | , (cid:12)(cid:12)(cid:12) − λ H − λ T ± (cid:112) λ H − λ T λ H + 12 λ HT + 25 λ T (cid:12)(cid:12)(cid:12) (cid:111) . B. Results
In order to show the importance of additional tree-level contributions to the scattering matrix as a functionof tan α and m H , we force λ T to be small. This can bedone by setting m H + to m H + = (cid:112) m H + m h t α ˜ v (cid:112) t α v . (9)Thus, λ T vanishes and the largest eigenvalue is approxi-mately given by a s →∞ (cid:39) m h + m H (cid:16) v T t α +2 √ √ t α v − √ t α v T v +2 v T (cid:17) v T πv (10)Here, we assumed tan α (cid:28) and m H (cid:29) m h . We learnfrom this that unitarity tends to be violated for increas-ing values of m H and tan α and decreasing values of v T . hHH HH FIG. 1. Diagram contributing to the scalar scattering matrixas finite s . The large s approximation needs to be compared for in-stance with the diagram shown in Fig. 1. The s -channeldiagrams is of the form | c | m h − s (11)where the vertex c in the limit λ HT (cid:29) λ T , λ H is givenby c = 18 (cid:0) √ α − α ))( κ + λ HT v T ) + 2 λ HT v cos( α )+6 λ HT v cos(3 α ) (cid:1) , (12)which for small tan α can be further simplified to c (cid:39) −√ t α ( κ + λ HT v T ) + λ HT v (cid:39) λ HT v . (13)Thus, for √ s not much larger than m h one can expectthat this diagram scales as λ HT v m h . Therefore, althoughthe cubic Lagrangian parameter is small, the EWSB-generated terms lead to sizeable contributions by dia-grams of the type of Fig. 1. Actually, a more carefulcalculation including also the crossed t - and u -channeldiagrams results in a ( HH → HH ) (cid:39)− m H (cid:0) t α v − √ v T (cid:1) πv v T (cid:112) s ( s − m H ) ×× (cid:18) s log (cid:18) m h m h − m H + s (cid:19) − m H + s (cid:19) . (14)For somewhat larger s the dominant contribution to thefull scattering matrix comes from the process hH → hH which can be approximated to a ( hH → hH ) (cid:39) m H πsv v T ( m H − s ) (cid:16) (cid:0) s − m H (cid:1) t α v (cid:2) m H (cid:16) t α v − √ v T (cid:17) −√ sv T (cid:3) + 2 m H s log (cid:18) m H s (cid:19) (cid:16) t α v − √ v T (cid:17) (cid:17) . (15) FIG. 2. Comparison of the approximated values for the HH → HH and hH → hH scatterings at finite √ s withthe full numerical calculation and the approximated resultfor s → ∞ . We have used here m H = 400 GeV , tan α =0 . , v T = 3 GeV, and m H + was fixed by the condition λ T = 0 . Of course, one needs to keep in mind that these are justsingle entries in the scattering matrix which needs to bediagonalised. However, we will not quote here any an-alytical approximations when doing that since they arenot very helpful because of their length. We compare inFig. 2 the analytical approximations for a ( hH → hH ) , a ( HH → HH ) and a s →∞ with the full numerical cal-culation. As an example we have chosen here m H = 400 GeV , tan α = 0 . , v T = 3 GeV . (16)The range of √ s is chosen such that all possible reso-nances are avoided – and starts about an order of mag-nitude above the s -channel resonance of Fig. 1 whichleads to the largest contribution at small scattering en-ergies. We can see that for small s the full numericalresult agrees quite well with the approximation for the s -channel HH → HH scattering and is significantly largerthan the limit a < – i.e., unitarity is violated! Forlarger √ s , the full numerical result approaches this ap-proximate asymptotic value, i.e. the diagrams includ-ing trilinear interactions become suppressed. If one usesonly the approximation of large s , it would seem that a < is fulfilled, therefore underestimating the actualconstraints. In the end, this particular combination ofmodel parameters is forbidden since it violates perturba-tive unitarity – which is only seen by including the effectsof EWSB-generated trilinear scalar interactions at finitescattering energies.We now check how the difference between the full tree-level calculation and the large s approximation is affectedby the different parameters. For this purpose, we show inFig. 3 the maximal allowed value of tan α for given valuesof m H and the triplet VEV v T . We find as expectedthat the maximal value of tan α quickly drops for larger m H and smaller v T for both calculations, the one with FIG. 3. First row: Comparison between the old (dashed lines)and new (full lines) unitarity constraints for the SM extendedby a real triplet. Here, we show the maximally allowed valueof tan α max for a given heavy Higgs mass m H and three dif-ferent values of the triplet VEV v T . m H + is chosen to obtain λ T = 0 . In the second row the ratio of tan α max for the fullcalculation and the large s approximation is shown. and without explicit s dependence. However, we alsosee that for smaller m H a large difference exists betweenboth calculations: the value allowed for tan α based onthe point interactions only is about a factor of 3 largerthan the correct one by including all contributions. Thisdifference becomes smaller for increasing m H . The ratiofor tan α max for both calculations is nearly independentof the chosen value of v T as can be seen in the secondrow of Fig. 3.Finally, we want to remark that we were only con-cerned with the improved unitarity constraints in thismodel. In addition to our findings, a recent study foundthat also the impact of the modified Veltman conditionscan place very severe limits on this model, see Ref. [32]for details. IV. THE COMPLEX TRIPLET EXTENSION OFTHE SMA. Model description
The general scalar potential for the SM extended by acomplex SU (2) L triplet with hypercharge Y = 1 can bewritten as V = m H | H | + m T Tr ( T † T ) + (cid:0) κH T T † H + h.c (cid:1) + 12 λ H | H | + 12 λ T Tr ( T † T T † T ) − λ HT | H | Tr ( T T † )+ 12 λ HT (cid:48) H † T † T H + 12 λ T (cid:48) Tr ( T † T ) . (17)The matrix form for the triplet is given by T = (cid:18) T + / √ − T ++ T − T + / √ (cid:19) . (18)The scalar spectrum in this model consists oftwo neutral CP even, one neutral CP odd, twocharged and one doubly-charged Higgs with masses m h , m H , m A , m H + , m H ++ . The relations between themasses and the Lagrangian parameters are λ H = m h + m H t α ( t α + 1) v , (19) λ HT = − m A v + 4 v T − t α ( m h − m H )( m h + m H )( t α + 1) vv T , (20) κ = √ m A v T v + 4 v T , (21) λ T (cid:48) = m A v v v T + 4 v T + 8 m H + v + 2 v T (22) + m h t α + m H − (4 m H + − m H ++ ) (cid:0) t α + 1 (cid:1) ( t α + 1) v T ,λ HT (cid:48) = 8 m H + v + 2 v T − m A v + 4 v T , (23) λ T = − (cid:0) m A − m H + + m H ++ (cid:1) v T + 8 m A v + 4 v T − m H + v + 2 v T , (24)The mass terms m H and m T are fixed by the minimiza-tion conditions of the scalar potential. Here and in the following we use the convention Q em = T L + Y . B. Results
The unitarity constraints in the large s limit are givenbyMax (cid:110) | λ H | , | λ HT | , | λ HT − λ HT (cid:48) | , | λ HT − λ HT (cid:48) | , | λ HT + λ HT (cid:48) | , | λ T − λ T (cid:48) | , | λ T (cid:48) | , | λ T + λ T (cid:48) | , (cid:12)(cid:12) ± (cid:112) − λ H + 3 λ T + 4 λ T (cid:48) ) + 6( λ HT (cid:48) − λ HT ) + 6 λ H + 6 λ T + 8 λ T (cid:48) (cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) ± (cid:113) ( − λ H + 2 λ T + λ T (cid:48) ) + λ HT (cid:48) + λ H + 2 λ T + λ T (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) (cid:111) < π . (25)We see that in this version it is also not possible to havea large cubic interaction which is not proportional to aquartic coupling, because κ is always smaller than λ HT v .Nevertheless, there is an important difference comparedto the case of the real triplet. While it was not possiblein the real case to have a sizeable mass splitting betweenthe additional neutral and charged Higgs states, this isno longer the case here. If we fix m A and m H ++ bythe conditions λ T = λ T (cid:48) = 0 , and assume λ H to besmall compared to λ HT , λ HT (cid:48) , the condition to preserveunitarity becomes (cid:114) (cid:12)(cid:12)(cid:12)(cid:12) m H t α v − m H + v T v v T (cid:12)(cid:12)(cid:12)(cid:12) < π . (26)Thus, m H and m H + do not have to be close in mass tofulfil this condition. In contrast, m H + > m H is evenpreferred for t α v > v T . On the other hand, the process hh → HH is in this limit calculated to a ( hh → HH ) = m H ( t α v − v T )32 π √ sv v T (cid:112) s ( s − m H ) × (cid:104) m H (2 v T − t α v ) log (cid:32) − m h − (cid:112) s ( s − m H ) + s − m h + (cid:112) s ( s − m H ) + s (cid:33) + v T (cid:113) s ( s − m H ) (cid:105) , (27)which is completely independent of m H + and growsquickly with m H . As long as we have a scattering en-ergy √ s < m H + , also no interferences can occur, i.e. wecan expect a different behaviour compared to the large s limit. This is confirmed in Fig. 4, where we find ahard limit for large m H which is nearly independent of m H + if the charged Higgs is heavier than the neutralone. While the constraints according to Eq. (25) rule outpart of the considered parameter space, the full numericalresults including the trilinear interactions constrain themodel more severely. Here, the scattering energies whichplace the strongest constraints range from ∼ GeV to ∼ TeV.
FIG. 4. Exclusion limits using the unitarity constraints inthe large s limit (dashed line) and the full calculation (fulllines) in the ( m H , m H + − m H ) plane. Parameter space tothe right of the contour lines is excluded by the respectiveconditions. The colour code shows the logarithm of the ratioof the maximal eigenvalues of the scattering matrix for thetwo calculations. In the first row, we use tan α = 0 . , and inthe second tan α = 0 . . The masses m A and m H ++ are fixedby the conditions λ T = λ T (cid:48) = 0 . V. THE GEORGI-MACHACEK MODELA. Model description
The GM model extends the SM by two scalar SU (2) L triplets, one real triplet η with hypercharge Y = 0 and acomplex one χ with Y = − , therefore also featuring adoubly-charged component. Both triplets are required toactively participate in EWSB and therefore to assume aVEV v η,φ . In order not to be in conflict with electroweakprecision measurements, and in particular the ρ param-eter, a global SU (2) L × SU (2) R symmetry is imposed,leading to v η = v φ and a simplification of the scalar po- tential. In order to write the latter in a very compactway, it is convenient to express the two triplets as a bi-triplet and the Higgs doublet Φ as a bi-doublet under SU (2) L × SU (2) R : Φ = (cid:18) φ ∗ φ + φ − φ (cid:19) , ∆ = χ ∗ η + χ ++ χ − η χ + χ −− η − χ . (28)The scalar potential can then be written as V (Φ , ∆) = µ † Φ + µ † ∆ + λ (cid:2) TrΦ † Φ (cid:3) + λ TrΦ † Φ Tr∆ † ∆ + λ Tr∆ † ∆∆ † ∆ + λ (cid:2) Tr∆ † ∆ (cid:3) − λ Tr (cid:0) Φ † σ a Φ σ b (cid:1) Tr (cid:0) ∆ † t a ∆ t b (cid:1) − M Tr (cid:0) Φ † τ a Φ τ b (cid:1) ( U ∆ U † ) ab − M Tr (cid:0) ∆ † t a ∆ t b (cid:1) ( U ∆ U † ) ab , (29)with τ a and t a being the SU (2) generators for the doubletand triplet representations; U is e.g. given in Ref. [6].The electroweak VEV v SM can then be re-expressed as v = v φ + 8 v χ (cid:39)
246 GeV , (30)while the relative sizes of the VEVs are parametrised byan angle θ H : s H ≡ sin θ H = 2 √ v χ v SM , c H ≡ cos θ H = v φ v SM . (31)The scalar mass spectrum of the model consists ofseven physical states: three CP-even neutral scalars, onephysical CP-odd scalar (i.e. pseudo-scalar), two (com-plex) physical singly-charged scalars and one (complex)doubly-charged Higgs. Due to the custodial symmetry,one CP-even scalar, one singly- and the doubly-chargedHiggs can be combined into a custodial fiveplet witha common tree-level mass m while the other chargedscalar combines with the pseudo-scalar to a triplet withtree-level mass m . Those masses are given by m = v SM (cid:16) s H (3 √ M + s H λ v SM )+ c H ( M √ s H + 32 λ v SM ) (cid:17) , (32) m = v SM M √ s H + 12 λ v . (33)The remaining two neutral Higgs eigenstates are denoted h and H , where the former corresponds to the measuredSM-like eigenstate. The mixing angle which rotatesbetween these two mass eigenstates is denoted α . Formore details on the model we refer the reader e.g. toRef. [6]. h H H H H H H h H H H H H H h H H H H H H FIG. 5. Diagram contributing to the scalar scattering matrixat finite s . B. Unitarity constraints
The unitarity constraints in the large s approximationfor this model are given by [5, 6]: (cid:113) (6 λ − λ − λ ) + 36 λ + (cid:12)(cid:12) λ + 7 λ + 11 λ (cid:12)(cid:12) < π , (cid:113) (2 λ + λ − λ ) + λ + | λ − λ + 2 λ | < π , | λ + λ | < π , | λ − λ | < π . (34)In the following, we perform a brief analytical estimate ofthe additional terms appearing for small s which can beof similar size – or even larger. The full × scatteringmatrix has dimension × , i.e. it is highly unlikely tolearn anything from this matrix. Therefore, we concen-trate on single scattering channels like the ones depictedin Fig. 5. We find for λ i → ( i = 2 , , , ) that the sumof all diagrams is given by a s ( HH → HH ) = − (cid:112) s ( s − m )216 πss H ( s H −
1) ( t α + 1) v × (cid:34) m h t α v SM (cid:18) − (cid:113) − s H s H t α + 2 √ s H − √ (cid:19) + m H v SM (cid:18) − √ s H t α − (cid:113) − s H s H + 2 √ t α (cid:19) (cid:35) × (cid:34) t α t α + 1 (cid:16) m h − m + m h + s (cid:17) m − s + 1 m h − s + 2 m − s +2 log (cid:16) m H − m + m H + s (cid:17) ( t α + 1) (4 m − s ) − (cid:16) m s − m (cid:17) s − m + 1( t α + 1) ( m H − s ) (cid:35) (35)Here, the remaining Lagrangian parameters appearing inthe vertices have been re-expressed in terms of massesand mixing angles. For the relations see e.g. Ref. [7]. This can be further simplified if we assume s H , t α (cid:28) and m (cid:39) m H a s ( HH → HH ) = − (cid:16) m H s − m H (cid:17) (cid:0) m h t α + m H ( s H − t α ) (cid:1) πs H ( t α + 1) v (cid:112) s ( s − m H ) (36)Under the same assumptions, the maximal eigenvalue ofEq. (34) can be expressed as a max ,s →∞ = 3 (cid:0) m h + m H t α (cid:1) π ( t α + 1) v (37)Thus, if we go close to the kinematic threshold, but stilloff-resonance, and choose s = 5 m (cid:39) m H , we canexpect that the ratio of the two expressions very roughlyis a s ( HH → HH ) a max ,s →∞ = 8 m H log(5)27 √ s H ( m h + m H t α ) (38)This ratio can become huge for small s H and/or large m H . What is the origin of this behaviour? The vertex c H H h involved in the SM-like Higgs exchange is givenby c H H h = − √ M (cid:113) t α + 1 (cid:39) m H (cid:0) √ s H − √ t α (cid:1) s H (cid:113) t α + 3 v SM (39)Thus, the huge scattering amplitudes are a consequenceof large trilinear couplings which are triggered by largevalues of M (and M ). Since M and M do not enterthe unitarity constraints so far used in the literature, thiseffect has been missed entirely. C. Results
In order to validate our rough analytical understand-ing and to further explore the impact of the new uni-tarity constraints, we now use the numerical machineryavailable with the recent update of
SARAH . The imple-mentation of the GM model in
SARAH was discussed inRef. [33]. There are many possibilities for what to use asinput parameters. Naively using the Lagrangian parame-ters λ i , M and M will hardly produce points which arein agreement with the Higgs mass measurements. There-fore, we trade λ , M and M for m h , m H and α . Withthat choice, the full set of input parameters is m h , m H , α, λ , λ , λ , λ , sin θ H . (40)
1. Dependence on the scattering energy
As first step, we show in Fig. 6 the dependence of thefull scattering matrix on the scattering energy √ s for FIG. 6. The maximal scattering eigenvalue as a function ofthe scattering energy √ s for different values of M H . The otherparameters were set to λ = 0 . , λ = 0 . , λ = − . , λ =0 . , α = 20 ◦ , sin θ H = 0 . . The dashed purple line gives theresults using the old constraints. different choices of M H . The other parameters are set to λ = 0 . , λ = 0 . , λ = − . , λ = 0 . ,α = 20 ◦ , sin θ H = 0 . . (41)We can see that for √ s in the TeV range, the unitar-ity limits are clearly violated for m H above 250 GeV.The value of √ s at which this happens is shifted withincreasing m H , but at most 2.5 TeV for m H = 400 GeV.In contrast, agreement with the large s approximation isonly found for much larger values of √ s . The differencebetween our full calculation and the old approximatedone in the maximal scattering element from Eq. (37) canbe as large as a factor of 10 for m H = 400 GeV. Even for m H = 250 GeV a factor of 3 difference is visible.
2. Comparison between old and new unitarity constraints
As next step, we want to make a more exhaustive com-parison between the old and new results. For this pur-pose, we consider the (sin θ H , m H ) plane for the samevalues of α and λ i as in Eq. (41). In the full calculationincluding the propagator diagrams, we scan the scatter-ing energy √ s between 250 and 2500 GeV to find themaximal eigenvalue of the scattering matrix. We com-pare this value with the one obtained by using the oldconstraints which only depend on the quartic couplings.The outcome is summarised in Fig. 7. While the oldconstraints are passed in the entire plane, the improvedcalculation cuts out significant regions. This is not onlythe case for small sin θ H < . as one could expect fromthe previous discussion, but also for sin θ H > . . The
150 250 500800 1500200 400750 10001250
50 50 200 5005001000 100020005000 7500
FIG. 7. First row: Logarithm of the ratio of the new andold results. The red line indicates the region ruled out bythe new constraints, while the old calculation would allowthe entire plane. The dashed contours give the values of m (blue) and m (black). The other parameters were set to λ = 0 . , λ = 0 . , λ = − . , λ = 0 . , α = 20 ◦ . Here wescanned √ s between 250 GeV and 2500 TeV in order to obtainthe tightest constraint. This ‘optimal’ scattering energy isshown in the second row. The black contours in this plotshow the values of | M | in GeV. reason is that m in this parameter region scales as m ∼ m H (cid:115) ( − s H (cid:112) − s H + √ t α − √ s H t α ) s H (cid:112) − s H (1 + t α ) (42) ∼ m H (cid:115) − s H + (cid:112) − s H s H (43)where we have used t α (cid:39) . This equation turns to zerofor s H = 1 / √ (cid:39) . , leading to tachyonic five-plets forlarger s H and very small m for s H values slightly belowthis root. As consequence, the t -channel diagrams witha five-plet exchange become large. In general, we findthat the maximal eigenvalue of the scattering matrix canchange even by a factor of 100 for large M H and verysmall sin θ H . The reason is that in this region the calcu-lated value of M becomes huge and is in the multi-TeVrange. It is worth mentioning that this region still hasa stable vacuum, i.e. the new unitarity conditions reallymake the difference between ‘allowed’ or ‘forbidden’ . Inthe second row of Fig. 7 we also show the value for the‘optimal’ scattering energy, i.e. the energy at which thelargest scattering eigenvalue becomes maximal, exclud-ing resonances. In the regions which are affected mostby the new constraints, this energy is moderately smalland well below our largest chosen value of 2.5 TeV.
3. Comparison with extended perturbativity constraints
In Ref. [7], a set of conditions was proposed which in-dicate if problems with the expansion of the perturbativeseries might exist. Those constraints check the relativeand absolute size of the counter-terms (CTs) when impos-ing an on-shell calculation of the scalar sector. A thirdcondition makes use of the one-loop corrected couplingsin the unitarity constraints on the quartic couplings afterthe breaking of the custodial symmetry. The fourth con-dition checks the finite corrections to the scalar massesif a MS scheme is applied instead of an OS one. We willcompare the new unitarity constraints with these con-straints, namely:1. A parameter point is considered problematic if theCT to at least one Lagrangian parameter is largerthan the tree-level value of this parameter timessome constant value v , i.e. (cid:12)(cid:12)(cid:12)(cid:12) δxx (cid:12)(cid:12)(cid:12)(cid:12) > v . (44)2. A parameter point is considered to violate pertur-bativity if the CT of at least one quartic couplingbecomes larger than some fixed value, i.e. | δx | > c · π , (45)with c within 1 and 4.3. A parameter point is considered to violate pertur-bation theory if the unitarity constraints on thequartic couplings (i.e. in the limit s → ∞ ) are vi-olated when inserting the renormalised couplings,i.e. |M ( λ Nx → λ N + δλ Nx ) | > π . (46) This statement holds until one also includes the perturbativityconstraints which we discuss in the next subsection. Since the custodial symmetry is broken at the loop level, thisdemands to calculate 17 independent CTs.
4. A parameter point is considered to violate pertur-bation theory if the two-loop corrections to at leastone scalar mass, calculated in the MS scheme, arelarger than the one-loop corrections, i.e. | ( m φ ) Tree − ( m φ ) | < | ( m φ ) − ( m φ ) | . (47)We make this comparison for different parameter rangeswhich were discussed in Ref. [7], cf. Figs. 3, 6 and 7 inthis reference. The results are shown in Fig. 8. In theupper row, we vary sin θ H for two different heavy Higgsmasses m H = 300 , GeV. In the lower row we showthe dependence on both m H and λ . For a discussion ofthe choices of varying parameters we refer the interestedreader to Ref. [7]. All other input parameters are listedin the caption of each figure. We show the maximal scat-tering eigenvalue for different intervals of the scatteringenergy √ s . All parameter regions which we show here areallowed by Higgs data, vacuum stability and the old uni-tarity constraints. However, in Ref. [7] it was found thatperturbation theory is not trustworthy in some rangesof the varied parameters. The large loop corrections arealso caused by large (effective) trilinear couplings. Sincethe same couplings can also enhance the scattering am-plitudes, we find that there is actually a nice agreementbetween the perturbativity constraints and the improvedunitarity constraints discussed here. The reason for thevery different scaling of the old and new unitarity con-straints as a function of λ are diagrams with effectivetrilinear couplings ∼ λ v SM . One finds that the domi-nant contributions scale as | a max0 | ∼ − λ v log (cid:16) m H − m H + s (cid:17) π (cid:112) s ( − m H + s ) s H (cid:39) π (cid:18) λ v SM m H s H (cid:19) < (48)while the old constraints impose λ π < , ie. there can bea strong enhancement of λ v m H s H for small s H and nottoo large m H .
4. RGE effects
Finally, we want to estimate the effect of higher-ordercorrections which we are still missing by performing apure tree-level calculation. A full one-loop calculationof the entire → scattering process is well beyondthe scope of this paper. Therefore, we use the one-loopRGEs and calculate the process with running couplingsand masses at the scattering energy √ s . The RGEs forthe GM model have been calculated in Ref. [34] usingthe generalised version of the Lagrangian parameters. Wehave implemented this model in SARAH and cross-checkedthe β -functions. While we agree with the expressions forall scalar parameters, we found discrepancies for g andthe Yukawa couplings. We find for the β functions ofthese parameters in the limit of Y τ → and no flavour0 FIG. 8. Comparison between the new unitarity constraints and the perturbativity constraints proposed in Ref. [7]. The maximaleigenvalue of the scattering matrix calculated for different intervals of the scatter energy √ s are shown. The vertical lines showwhere the different perturbativity constraints, summarised in Eqs. (44) to (47), are violated. The purple dashed line gives theresult using the old calculation with the large s approximation. Upper row: | a max0 | as a function of sin θ H for the input masses m H = 300 GeV (left-hand plot) and 800 GeV (right-hand plot).The other parameters were set to λ = 0 . , λ = 0 . , λ = − . , λ = 0 . , α = 20 ◦ . Lower row: | a max0 | as a function of m H using λ = λ = λ = 0 . , λ = − . , α = 20 ◦ , sin θ H = 0 . (left plane), and as afunction of λ using λ = 0 . , λ = 0 . , λ = − . , α = 20 ◦ , sin θ H = 0 . , m H = 750 (right plane). mixing πβ g = − g , (49) πβ Y t = 9 Y t Y b Y t − Y t (cid:18) g
20 + 9 g g (cid:19) , (50) πβ Y b = 92 Y b + 32 Y t Y b − Y b (cid:18) g + 94 g + 8 g (cid:19) . (51)Having the model and the RGEs at hand, we can checkthe impact of the running. Since we are here only inter-ested in an estimate of the size of this effect, we use thesimplified – but common – approach of tree-level match-ing combined with one-loop running. Higher order cor-rections will be important especially in the presence oflarge quartic couplings [35]. We show in Fig. 9 the max-imal eigenvalue of the scattering matrix as a function ofthe largest considered scattering energy √ s max . For each √ s max , we check for the best scattering energy between s which is in between s min and s max including the RGErunning of all couplings up to √ s . For comparison wealso show a max0 without RGE running. We do so for theheavy Higgs masses m H = 500 GeV and
GeV. Theother parameters were set to λ = 0 , λ = − λ = λ = x with x = { . , } α = 20 ◦ , sin θ H = 0 . . (52)For comparison, we include also the maximal eigenvalueusing the old calculation which includes only the con-tributions from point interactions. For those, we alsoinclude the RGE running up to a scale Q = √ s max . This Per construction, the slope of the obtained curve can never benegative if no RGEs are considered. Small variations are only dueto discrete steps in the scanning of the scattering amplitudes upto √ s max . FIG. 9. The maximal eigenvalue of the scattering matrix de-pending on the highest scattering energy √ s max up to whichwe have computed the processes. We present the results with(dashed line) and without (solid line) the inclusion of RGErunning of the parameters. In the first row we use small quar-tic couplings of ± . , while in the second row the couplingsare ± . might look a bit strange: these constraints use alreadythe large s limit since they neglect the phase space fac-tor (cid:112) − m /s ; however this is a common procedure inliterature and shall only serve for illustration purposeshere. We see that for small quartic couplings, the effectof the RGEs is moderately small. This is not surprisingbecause also the running of the trilinear parameters M and M is proportional to the quartic couplings as wellas gauge and Yukawa couplings squared. Therefore, evenif we run the quartic couplings to TeV, the contribu-tions from the trilinear couplings dominate the unitar-ity constraints. If we go to larger values of the quartics(lower row in Fig. 9), the running becomes stronger be-cause of the λ dependence of the respective β functions.Thus, the scale dependence of the old constraints is quitestrong and to some extent also much stronger than theone of the new constraints. At large √ s max , when thequartic couplings have grown even more through RGEevolution, their point interactions also dominate over the → scattering including propagators – leading to anagreement between the old and new constraints indepen- dently of the chosen value of m H . D. Impact on benchmark scenarios
Before we conclude, we want to comment briefly on theimpact of the new constraints on benchmark scenariosstudied in literature. One widely used benchmark planeis the so-called ( m , s H ) where by construction the tri-linear couplings are small [36, 37]. Therefore, the changein the scattering elements is only moderate. In contrast,very recently Ref. [38] has proposed six benchmark pointswhich cover also larger values of M and M . Therefore,we find quite significant changes. In particular BP2, BP4and BP6 are clearly ruled out. For these three points, themaximal scattering eigenvalue changes by a factor 10 to100. VI. CONCLUSION
In this paper we have computed the full scalar → scattering amplitudes for models with scalar SU (2) L triplets. Those amplitudes are needed for determiningthe bounds from imposing perturbative unitarity. Sofar, these bounds have been computed using the limitof large scattering energy, neglecting all diagrams withinternal propagators. Here, instead, we include the ef-fects stemming from finite energies √ s . We find that thefull calculation is necessary in the presence of large trilin-ear scalar interactions, be it from electroweak symmetrybreaking effects or from trilinear Lagrangian parameters.We showed this at the example of three models: (i) theSM extended by a real triplet with zero hypercharge, (ii)the SM extended by a complex triplet with Y = 1 , and(iii) the Georgi-Machacek model. In all examples we findsizeable regions of parameter space which are excludedby the constraints obtained from the full calculation butwhich would have been regarded as allowed using the oldprocedure. For the last model, we find good agreementof the new unitarity constraints with the recently pro-posed loop-level perturbativity checks of Ref. [7]. Sinceour study makes use of tree-level relations, we have fi-nally estimated the effects of loop corrections by includ-ing renormalization group running of the parameters tothe scattering energy.A more thorough estimate of the loop effects is be-yond the scope of this paper and we leave the inclusionof higher-order corrections to the calculation of the scat-tering amplitudes, which can be important in particu-lar in non-supersymmetric models, to future work. Theresults for other very popular models like the singlet-extended SM and two-Higgs doublet models are discussedelsewhere [14, 39].2 ACKNOWLEDGEMENTS
We thank Mark Goodsell for very fruitful discus-sions and collaboration on related work, and SophieWilliamson for collaboration in the early stages of this work. FS is supported by ERC Recognition Award ERC-RA-0008 of the Helmholtz Association. MEK is sup-ported by the DFG Research Unit 2239 “New Physicsat the LHC”. We further thank the LPTHE in Paris fortheir hospitality while this work triggered. [1] J. Schechter and J. W. F. Valle, Phys. Rev.
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