Heavy dark matter through the dilaton portal
Benjamin Fuks, Mark D. Goodsell, Dong Woo Kang, Pyungwon Ko, Seung J. Lee, Manuel Utsch
HHeavy dark matter through the dilaton portal
Benjamin Fuks,
1, 2, ∗ Mark D. Goodsell, † Dong Woo Kang, ‡ Pyungwon Ko, § Seung J. Lee, ¶ and Manuel Utsch ∗∗ Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes ´Energies, LPTHE, F-75005 Paris, France Institut Universitaire de France, 103 boulevard Saint-Michel, 75005 Paris, France School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea Department of Physics, Korea University, Seoul 136-713, Korea (Dated: July 20, 2020)We re-examine current and future constraints on a heavy dilaton coupled to a simple dark sectorconsisting of a Majorana fermion or a St¨uckelberg vector field. We include three different treatmentsof dilaton-Higgs mixing, paying particular attention to a gauge-invariant formulation of the model.Moreover, we also invite readers to re-examine effective field theories of vector dark matter, whichwe show are missing important terms. Along with the latest Higgs coupling data, heavy scalarsearch results, and dark matter density/direct detection constraints, we study the LHC bounds onthe model and estimate the prospects of dark matter production at the future HL-LHC and 100 TeVFCC colliders. We additionally compute novel perturbative unitarity constraints involving vectordark matter, dilaton and gluon scattering.
I. INTRODUCTION
Any theory can be made scale invariant by coupling itto a dilaton. The scale invariance can then be softly bro-ken, giving the dilaton a mass and self-interactions, andthis becomes a popular proposal [1–14] for solving the hi-erarchy problem of the Standard Model (SM). Such SMplus dilaton theories can either be thought of as funda-mental, or as the low-energy limit of composite theories,where the dilaton becomes the pseudo-Goldstone bosonassociated with the spontaneous breaking of scale invari-ance. It therefore couples to the SM fields through thetrace of the energy-momentum tensor.The dilaton portal is also an extremely economic wayof coupling the SM to a dark matter particle: a mas-sive dark matter field automatically couples to the dila-ton, so that there is no need to add any additional in-teractions with the SM. Such models are very economi-cal in terms of new parameters: we have effectively justthe dilaton mass, dark matter mass, and the dilaton de-cay constant/symmetry breaking scale as extra degreesof freedom relative to the SM.Models of dilaton portal dark matter have also beenwell studied in the literature [9, 15–18]. In this paper,we will study both the fermionic and vector dark mat-ter cases in detail, pointing out along the way the con-nection between the dilaton portal and models of vectordark matter based on effective field theory. We shallconsider both the vanilla dilaton scenario without mix-ing with the Higgs boson, and also (in section II.2) twodifferent formulations of the theory once mixing is in- ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] cluded: the theory is not uniquely defined, and the sim-plest way to include mixing is not gauge invariant; alongthe way we also write a gauge-invariant formulation ofthe Lagrangian. Our aim is to revisit and update theconstraints on the main parameters of the model, viadark matter constraints, the latest Higgs-like particlesearches for the dilaton itself, collider constraints fromdirect searches for the dark matter particle at the LHC,and perturbative unitarity of scattering amplitudes. Wethen provide projections for future colliders. In partic-ular, we shall restrict our attention to a heavy dilatonabove about 300 GeV (see ref. [14] for a detailed recentexamination of constraints on the dilaton in the low-masswindow) where diboson searches push the dilaton to largemasses and weak couplings. However, a vector dark mat-ter candidate is produced much more copiously at theLHC than a fermion candidate, and thus this model ismore promising for future searches.The rest of this paper is organised as follows. In sec-tion II, we introduce our theoretical framework, discussthe problematics of the Higgs-dilaton mixing and ad-dress the issue of perturbative unitarity. Section III isdedicated to the estimation of the constraints on theHiggs-dilaton mixing and section IV to the bounds thatcan be imposed from dark matter direct detection andLHC searches. Section V finally focuses on future col-lider prospects. We conclude and summarise our workin section VI. In addition, more extensive details on per-turbative unitarity constraints are presented in the ap-pendix. II. THE DILATON PORTAL TO DARKMATTERII.1. Theoretical framework
We consider an extension of the SM that contains adark matter candidate, taken to be a Majorana fermion a r X i v : . [ h e p - ph ] J u l or a neutral vector boson, without any tree-level interac-tion with the Standard Model particles. Assuming spon-taneously broken scale invariance at a high scale f , theparticle content of the model includes a light dilaton σ of mass m σ that couples to all other fields through thetrace of the energy-momentum tensor [1–3, 7, 8, 11, 12].As pointed out in the introduction, the dark matter can-didate couples to the dilaton via its mass term, and thenthe dilaton acts as a mediator for dark matter interac-tions with the Standard Model particles. The dilaton,its self-interactions and its interactions with the SM sec-tor arise from a procedure that converts a theory into ascale-invariant one. We briefly summarise this procedurebelow, in the context of the model studied in this work.We consider a generic Lagrangian L that we write interms of operators O i of classical scaling dimension d i =[ O i ] and coefficients g i depending on the renormalisationscale µ , L = (cid:88) i g i ( µ ) O i ( x ) . (1)Scale-invariance requires that the above Lagrangian con-tains only dimensionless, scale-independent couplings, orequivalently that it is invariant under a scale transforma-tion of parameter λ , x µ → e λ x µ , O i ( x ) → e λd i O i ( e λ x ) , µ → e − λ µ . (2)However, those rules yield a non-vanishing Lagrangianvariation δ L equal to the trace of the energy-momentumtensor T µν , δ L = (cid:88) i (cid:104) ( d i − O i − µ ∂g i ∂µ (cid:105) = T µµ . (3)Scale invariance can then be established by introducinga conformal compensator field χ ( x ), which is a scalar fieldof dimension [ χ ] = 1, and by enforcing the replacementof the couplings g i [11, 12] g i ( µ ) → g i (cid:18) µ χf (cid:19) (cid:18) χf (cid:19) − d i . (4)Such an introduction of the appropriate powers of thecompensator field allows us to retrieve scale invarianceafter imposing the scale transformation law of the com-pensator χ ( x ) → e λ χ ( e λ x ) . (5)Along with the additional field χ , the replacement ineq. (4) also induces a new parameter, the cut-off scale f = (cid:104) χ (cid:105) . This is an artifact of the breaking of the scale For a detailed explanation see, for example, the work of ref. [8]. symmetry, i.e. the vacuum expectation value of χ . There-fore, an appropriate parametrisation involving the associ-ated Goldstone boson σ , namely the dilaton field, wouldbe based on the field redefinition [8, 16] χ ( x ) = f + σ ( x ) . (6)Inserting these modifications into the Lagrangian leadsto L = (cid:88) i g i (cid:16) µ (1 + σ ( x ) /f ) (cid:17) (cid:16) σ ( x ) /f (cid:17) − d i O i ( x )= L + (cid:88) i σ ( x ) f (cid:104) g i ( µ )(4 − d i ) O i ( x )+ β ( g i ) O i ( x ) (cid:105) + σ ( x )2 f (cid:88) i (cid:104) (4 − d i )(3 − d i ) g i ( µ ) O i ( x ) (cid:105) + . . . , (7)where we restricted ourselves to the leading terms in σ / f explicitly, and the ellipsis stands for any (potentially-relevant) higher-order contributions.Considering the broken electroweak phase, the inter-action terms of the dilaton with the SM sector are given,including all the Lagrangian terms of dimension 6 or be-low, by L σ = σf (cid:20) m W W + µ W − µ + m Z Z µ Z µ − m h h − (cid:88) ψ m ψ ψψ − m h v (cid:104) h + hG G + 2 hG + G − (cid:105) + gv (cid:104) ∂ µ G − W + µ + ∂ µ G + W − µ + 1 c W ∂ µ G Z µ (cid:105) + gm W hW + µ W µ − + g c W m Z hZ µ Z µ + ig (cid:48) m W (cid:0) G − W + µ − G + W − µ (cid:1)(cid:0) c W A µ − s W Z µ (cid:1) + 11 α EM π F µν F µν − α s π G aµν G aµν (cid:21) + σ f (cid:104) m W W + µ W − µ + m Z Z µ Z µ − m h h (cid:105) , (8)where the summation over ψ refers to all the SMfermionic mass-eigenstates of mass m ψ , and m h , m W and m Z stand for the Higgs boson, W -boson and Z -bosonmasses. Moreover, v denotes the vacuum expectationvalue of the Standard Model Higgs field, c W and s W thecosine and sine of the electroweak mixing angle, and g , g (cid:48) and g s the weak, hypercharge and strong coupling con-stants. In our expressions, we have included the inter-actions between the physical Higgs ( h ) and electroweak( A µ , W µ , Z µ ) bosons and the three Goldstone bosons( G , G ± ). For the corresponding expression in the un-broken electroweak phase, we refer to ref. [15] that addi-tionally includes a complete Higgs-dilaton mixing analy-sis.The interactions of the dilaton with the two considereddark matter candidates, namely a Majorana fermion Ψ X and a real vector field X µ , read L DM σ = (cid:40) − σ f m Ψ Ψ X Ψ X (Majorana fermion) , (cid:16) σf + σ f (cid:17) m V X µ X µ (vector boson) , (9)where we denote the mass of the dark matter state by m Ψ and m V in the fermion and vector case respectively. Inorder to ensure the stability of the dark matter particle,our setup assumes a Z symmetry.At this level, the only purpose of introducing the dila-ton was to make the existing terms of the Lagrangianscale-invariant. We should however also include a kineticand a mass term for the dilaton, as well as its poten-tial self-interaction terms. We rely on a dilaton poten-tial that is constructed under the assumption that theconformally-invariant field theory, for which our modelrepresents an effective theory, is explicitly broken dueto the addition of an operator with a scaling dimension∆ O (cid:54) = 4. This yields a potential V ( χ ) [4, 8], V ( χ ) = χ ∞ (cid:88) n =0 c n (∆ O ) (cid:18) χf (cid:19) n (∆ O − , (10)that we add to the effective Lagrangian. At the minimumof the potential at which (cid:104) χ (cid:105) = f , the coefficients c n can be related to the the parameters of the underlyingconformal field theory . This potential is supposed to beat the origin of the dilaton mass m σ , which translatesinto the condition m σ = d V ( χ )d χ (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) χ (cid:105) = f > . (11)With the assumption that | ∆ O − | (cid:28)
1, it is possibleto expand the potential in | ∆ O − | such that the ex-plicit ∆ O -dependence of the different coefficients c n dis-appears [8], V ( χ ) = 116 m f χ (cid:20) χf − (cid:21) + O ( | ∆ O − | ) . (12)By applying the parametrisation of eq. (6), one obtains,after adding the dilaton kinetic term [16], L self σ = 12 ∂ µ σ∂ µ σ − m σ σ − m σ f σ − m σ f σ + . . . , (13)where the dots stand for higher-dimensional interactions. II.2. Higgs-Dilaton mixing
When two physical neutral scalars are present in thetheory (the Higgs field h and the dilaton field σ ), they Note that the exact details of the underlying theory is beyondthe scope of this paper, and in our analysis we take an effectivetheory approach. could in principle mix, unless it is forbidden by somesymmetry. If the mixing is allowed, its origins can befound through a UV completion of our effective setup.This option was first studied in ref. [17], and will be re-investigated in the light of the most recent experimentaldata in section III.Keeping in mind the effective approach adopted in thiswork, we will not speculate about any UV-physics thatdrives the source, and hence the amount, of mixing. In-stead, we will study the possibility of a non-vanishingHiggs-dilaton mixing by introducing a mixing angle α as an additional parameter. We will relabel our originalflavour states as h , σ with mass parameters m h, , m σ, and relate them to the new mass eigenstates h, σ via therotation (cid:18) hσ (cid:19) = (cid:18) cos α sin α − sin α cos α (cid:19) (cid:18) h σ (cid:19) ≡ (cid:18) c α s α − s α c α (cid:19) (cid:18) h σ (cid:19) . (14)We then assume that the (lighter, by assumption) scalar h can be identified as the experimentally confirmed scalarof mass of about 125 GeV, so that it is mostly compatiblewith the SM Higgs boson. In contrast, the heavier scalarfield σ is mostly dilaton-like.The impact of the non-zero mixing of the Higgs bo-son and the dilaton can be inferred from the similar-ity between the couplings of the h and σ states withthe remaining particle content in the zero-mixing case.Without mixing and in unitary gauge ( i.e. ignoring Gold-stone bosons), there is a corresponding dilaton couplingfor every Higgs boson coupling. This is not surprisingsince these dilaton couplings originate from the presenceof a dimensionful coupling in the SM Lagrangian afterelectroweak symmetry breaking, where the dimensionfulquantity in the coupling constants is actually the Higgsvacuum expectation value v . In all of these cases, thecoupling constants appearing in the dilaton interactionvertices therefore differ by a factor of r f = v/f for eachdilaton participating in the interaction, when comparedwith the corresponding Higgs-boson interaction. In prin-ciple, dilaton couplings involving the Higgs or Goldstonebosons (leaving the unitary gauge) should be discussedas well, but they cannot be related to any SM counter-part by factors of v/f . We refer to the discussion belowfor what concerns those multi-scalar interactions.For the dilaton and Higgs Yukawa couplings to the SMfermions, this yields the interaction Lagrangian L ψσ , L ψσ = (cid:88) ψ m ψ v (cid:16) h + r f σ (cid:17) ψψ = (cid:88) ψ m ψ v (cid:104) ( c α + r f s α ) h + ( r f c α − s α ) σ (cid:105) ψψ . (15)Similarly, the massive gauge boson interactions read, atleading order in the scalar fields, L Vσ = (cid:104) ( c α + r f s α ) h + ( r f c α − s α ) σ (cid:105) × (cid:20) m W v W + µ W − µ + m Z v Z µ Z µ (cid:21) . (16)The W - and Z -boson couplings to a pair of scalar fields h and/or σ are obtained analogously.These couplings are potentially relevant for processesaddressing both scalar production and decays. For ex-ample, in scalar production at hadron colliders throughgluon fusion, the leading-order contribution involves tri-angle diagrams featuring a loop of quarks (the top quarkone being the most relevant by virtue of its largest mass),so that the associated predictions are affected by themodifications of eq. (15). In case of the light scalar, afactor of ( c α + r f s α ) is introduced into the amplitudecompared to the zero-mixing case. On the contrary, forthe production of the heavy scalar, the extra factor isgiven by ( − s α + r f c α ). This feature is obviously alsopresent for any other production mode at colliders, likeassociated production ( V h or V σ ) or vector-boson fu-sion, that involve the coupling of the scalar to the W -and Z -bosons.On the other hand, the dark matter mass term is at theorigin of a dilaton coupling which obviously does not havean analogue for the Higgs boson due to the absence of thedark matter in the SM and of a Higgs portal relating thedark and the visible sector. Depending on the type ofdark matter, the Lagrangian in terms of the scalars h and σ gives L DM σ = − m Ψ f s α h Ψ X Ψ X − m Ψ f c α σ Ψ X Ψ X (17)in the Majorana case and L DM σ = m V f ( s α h + c α σ ) X µ X µ + m V f ( s α h + c α σ ) X µ X µ (18)in the vector case. In the latter case, the trilinear cou-plings are the most relevant ones for Higgs measurementsand heavy scalar searches, but the quartic couplings arevery important for dark matter searches when m V (cid:29) m σ .Consequently, mixing leads to a coupling of the SM-likeHiggs state to the dark-matter candidate that would notexist without mixing.The same applies to the tree-level couplings of the dila-ton to the massless gauge-bosons, which are not inducedby dimensionful couplings but by the scale-dependenceof the electromagnetic and the strong coupling. Here thecontribution to the Lagrangian in the mass eigenbasis L V V S reads L V V S = 11 α EM πv (cid:16) r f s α h + r f c α σ (cid:17) F µν F µν − α s πv (cid:16) r f s α h + r f c α σ (cid:17) G aµν G µνa . (19)Without mixing, the dilaton coupling to photons or glu-ons must take into account both a tree-level and a one-loop contribution at leading order in the electromagneticor strong coupling, while the Higgs boson interacts withthese bosons only beyond tree-level. Eq. (19) shows thatin the mixing case, the tree level contributions enter intothe amplitudes for both the light and the heavy scalarwith a factor of sin α or cos α respectively. Furthermore, there are also couplings which involvesome combination of the two scalars. These result fromthe Higgs and the dilaton potentials, where couplings be-tween the two scalars h and σ are only present whenthe mixing is non-zero. On the other hand, the dimen-sionful couplings of the Higgs potential also give riseto interaction terms involving both the dilaton and theHiggs boson, which are present even in the zero-mixingcase. These trilinear couplings are phenomenologicallyvery important, because they allow the dilaton to de-cay into a pair of Higgs bosons. Unfortunately, in ourmodel they are not uniquely defined . Starting from theLagrangian introduced in the previous subsection, thecubic couplings come from the terms L φ = − v m h, h − ξ m σ, f σ − m h f h σ . (20)Here ξ is a model-dependent dilaton self-coupling, thatwe have fixed to 5/6 in eq. (13). The trilinear couplingsbetween the two mass eigenstates are then found by sub-stituting, in the above equation, the mixing relation ofeq. (14).Note that the parameter m h, ≡ √ λ SM v is not equalto the mass of the lightest scalar. The physical masses areinstead determined by the diagonalisation of the dilaton-Higgs mass matrix extracted from the bilinear terms ofthe scalar potential, L φ = − m h, h − m σ, σ − m hσ h σ (21)where m h, = m h c α + m σ s α , m σ, = m h s α + m σ c α ,m hσ = − c α s α ( m σ − m h ) . (22)Note also that in the work of ref. [17] the param-eter m hσ is not explicitly discussed. However, it isclear that if it is present, then the full Lagrangian ofthe theory cannot be written in a gauge-invariant way:we will have to give up precision calculations for themodel and it is harder to make a connection with a UV-completion. On the other hand, if we want to write thelowest-dimension effective operator which can yield sucha mass and preserve the gauge symmetry, then we mustwrite the SM degrees of freedom in terms of a doublet H ⊃ H = √ [( v + h ) + iG ] , and it follows that thephysics that generates a dilaton mass should also gener-ate a new coupling L ⊃ − m hσ v σ | H | ⊃ − m hσ h σ − m hσ v h σ . (23)This new trilinear coupling has dramatic consequencesfor the phenomenology: it allows for unsuppressed de-cays of the dilaton into two Higgs bosons. Thereforewe will investigate both scenarios: the “minimal mixingscenario” of eq. (21) and the “gauge invariant mixingscenario” of eq. (23). II.3. Fermionic and vector dark matter throughthe dilaton portal
The dilaton portal is particularly interesting and sim-ple for dark matter models because the dark matter cou-pling is unique (being determined only by the dark mat-ter mass). The most commonly considered and perhapssimplest dark matter candidate in a dilaton portal modelis a Majorana fermion Ψ or a real scalar S , with a Z sym-metry. Note that in ref. [17] a Dirac fermion was con-sidered, which has instead a continuous unbroken global U (1) symmetry (although this was not explicitly stated).The phenomenology is however very similar up to somefactors of 2.There is little to add to the analysis of dark matter con-straints performed in this earlier work (and for the caseof zero mixing, in the works of ref. [9, 15, 16]). However,we update those results by providing limits from recentmonojet and multijet searches in the fermionic case insection IV.1, and projections for a future 100 TeV col-lider in section V.A vector boson as the dark matter particle has at-tracted some interest as an alternative to the more com-mon fermion or scalar candidates. The particular chal-lenge for the vector case is to allow it to remain stable;a Z symmetry forbids conventional gauge interactions.We must either accept this [9, 19–22] or we can stabilisethe vector through another symmetry such as a custodialone [23]. Here we shall be considering the minimal model,which can arise from a St¨uckelberg U (1) field coupled tothe dilaton. This automatically features a Z symmetryprovided there is no matter charged under it; and thesymmetry also forbids any kinetic mixing term involvingthe dark vector field, through which it could decay.Since our dilaton must be rather heavy and couples toall the SM particles including the Higgs boson, we couldconsider it as generating effective Higgs portal interac-tions. Indeed, integrating it out generates the effectiveLagrangian L eff , L eff = 12 f m σ (cid:20) (cid:88) ψ m ψ ψψ − m V X µ X µ − m Z Z µ Z µ − m W W + µ W − µ + m h h (cid:21) , (24)after neglecting any higher-dimensional operators. Thiscan be compared with the standard Higgs-portal La-grangian for vector dark matter L VHPDM (which has be-come a popular benchmark scenario) [21, 22], L VHPDM = λ hv h X µ X µ + 12 m V X µ X µ + λ X µ X µ ) , (25)that exhibits the same classes of terms. These types ofvector dark matter effective Lagragians including a massterm generated by the St¨uckelberg mechanism exhibit aviolation of unitarity, in particular at high energy collid-ers and even for the standard Higgs portal. The Higgs invisible decay width hence diverges when m V →
0. Onesimple way to cure these problems consists of introduc-ing a Higgs mechanism in the dark sector [24]. Then,the Higgs invisible decay width becomes finite [25] andunitarity is restored [26–28]. In the following, we will notsuffer this complication.In the dilaton case, the generic coefficients λ hv , m V and λ are expressed in terms of m V and f . While in L VHPDM the vector self-coupling λ is rather unimpor-tant, L eff instead features additional dimensionless quar-tic terms ( X µ X µ )( Z µ Z µ ) and ( X µ X µ )( W + µ W − µ ) thatare crucial for the phenomenology of the model. Theyindeed provide the principal annihilation channels for thedark matter. We therefore see no reason why these cou-plings should be neglected in a generic vector dark mattereffective field theory, and invite the reader to reconsiderbenchmark scenarios that omit them.Another interesting feature of a vector coupling to thedilaton is that a heavy dilaton will predominantly decayinto vectors. The partial width of the dilaton into darkmatter becomesΓ( σ → XX ) = m σ πf (cid:20) − m V m σ + 3 m V m σ (cid:21)(cid:115) − m V m σ → m σ (cid:29) m V m σ πf , (26)which is independent of the vector mass. When we fullyconsider the decay channels into Z and W bosons, wefind that the total width of the dilaton can be well ap-proximated by Γ (cid:39) m σ (cid:29) m V ,m Z m σ πf , (27)and the branching ratio of the dilaton into dark matterthen becomes roughly 0.25. This is much larger than forthe scalar or fermionic cases, where the dilaton branchingratio into dark matter tends to zero for small dark mattermasses. Consequently, vector dark matter productionat colliders could be potentially enhanced via a dilatonresonance.This width also provides a limit on the size of the dila-ton mass from the requirement that it be a narrow state.However, as we shall see, this is actually a weaker con-straint than perturbative unitarity of scattering ampli-tudes. II.4. Perturbative Partial Wave Unitarity
Since our theory contains non-renormalisable opera-tors, it must have a cutoff comparable to the scale f . Weexpect this to be manifest, even at tree level, as perturba-tive unitarity constraints on two-body scattering ampli-tudes. In particular, there are some couplings that couldin principle be large compared to f since they come withadditional massive factors – such as the masses of vec-tor bosons, and particularly the dark matter (in the vec-tor dark matter scenario). We have therefore calculatedthe constraints originating from imposing the unitarity oftwo-particle scattering when vector and dilaton fields areinvolved, as well as the one stemming from the scatter-ing of gluons into heavy vectors via a dilaton exchange.The details are given in appendix A that also includesthe derivation of useful approximate formulae, such asan upper bound on the dilaton mass from self-scatteringin eq. (A9) of m σ (cid:46) f , (28)which gives Γ /m σ (cid:46) / π .In particular, the scattering of gluons into vectors viaa dilaton places a constraint on the maximum scatteringenergy permissible in our theory. This will be relevant forthe LHC and future collider constraints; and also for allother unitarity constraints (which necessarily also involvechoosing a scattering momentum).As shown in appendix 4, unitarity is violated if s > π f α s . (29)In principle, when the process centre-of-mass energy s isabove this value (30 TeV for f = 3 TeV), the cross sectioncalculation in our theory is not reliable. However, incollider processes, this is of course not the centre-of-massenergy of the proton-proton collision that is relevant, butthe partonic one ˆ s . We have insteadˆ s ≡ x x s < π f α s ≡ ˆ s unitarity , (30)or equivalently x x < . (cid:18) f (cid:19) (100 TeV) s . (31)Here x , are the momentum fractions carried by theintial-state gluons, the relevant processes being inducedby gluon fusion. So provided that the gluon parton dis-tribution function (PDF) is negligible for x i > O (10 − ),calculations for a 100 TeV collider are safe. We considerprocesses where a single dilaton is resonantly produced.The relevant scale therefore consists of its mass m σ , andtaking into account the dilaton mass range investigated inthis work, the typical gluon-gluon parton collision scale is x x ∼ − − − . This therefore guarantees the sup-pression of any growth in the cross section coming fromhigh-energy subprocesses.On the other hand, the above constraint is very impor-tant when considering the unitarity of (vector) dark mat-ter scattering. While we typically want to take the high-energy limit for unitarity calculations to simplify mattersand avoid resonances, we have seen that it is not possi-ble for our model. In particular, the scattering of dilaton pairs into longitudinal vectors exhibits (only) a logarith-mic growth with energy, so that the cutoff of eq. (29) alsoreduces the power of other unitarity constraints. To besafe for f = 2 TeV we should take √ s <
20 TeV and for f = 1 TeV we should take √ s <
10 TeV.Throughout the rest of the paper we shall use the abovelimit as a guide for fixing the cutoff on our theory. Thisonly appears in the other unitarity constraints, arisingfrom vector and dilaton scattering, which we scan all mo-mentum values up to a maximum vector centre-of-massmomentum p V, max where p V = (cid:112) s/ − m V . Since theseother constraints grow logarithmically with p V, max this isboth necessary (to give finite results) and conservative. III. CONSTRAINTS ON THE HIGGS-DILATONMIXING
In order to assess the viable regions of the model pa-rameter space, we use input from experimental Higgs-boson measurements and heavy scalar searches, as well asconstraints on the magnitude of the new physics contri-butions to the Peskin-Takeuchi parameters S and T [29].For the Higgs and heavy scalar set of constraints, weenforce that predictions for the signal strengths of vari-ous scattering processes ( i.e. the ratio of total rates totheir Standard Model counterparts) is consistent withobservations (within uncertainties). In this context, itis assumed that the light or heavy scalar is resonantlyproduced. It is therefore allowed to factorise the scatter-ing cross sections into the scalar production cross sectiontimes the branching ratio of the considered decay mode.The Higgs and heavy scalar constraints are determinedusing Lilith -2.2.0 [30] and
HiggsSignals -2.4.0 [31] forthe SM-like Higgs measurements, and
HiggsBounds -5.5.0 [32] for the heavy scalar searches. The input forthese codes is generated using a specially written mainprogram for
MicrOMEGAs -5.0.8 [33] using model filesgenerated with
FeynRules -2.3.36 [34]. The Higgs andheavy scalar search limits require the computation of ra-tios of scalar couplings to the ones of a SM-like Higgsparticle of the same mass, both for the lighter (observed)state and the heavier one. These are estimated usinga modified version of the routines embedded into
Hig-gsBounds , that calculate tree-level scalar decays intoquarks with their masses evaluated at a fixed runningscale of 100 GeV, as well as quark-loop-induced couplingsto photons and gluons (including some higher-order QCDfactors) which we supplement with the higher-order op-erators given by eq. (19). We have verified that in thelimit of small dilaton masses, the loop-induced operatorsexactly cancel against the dilatonic ones for the dilatonstate, as they should.Predictions for the electroweak parameters S and T are obtained by rescaling Standard Model results, whichwe compare with the experimental values extracted fromthe electroweak precision fits [35] according to formulætaken from ref. [36]. For the calculation of the contribu-tions from the h and σ scalars ( δS and δT ), we use theSM Higgs-boson contribution X SM (for X = S, T ) [37],after replacing the Higgs-boson mass by the correspond-ing scalar mass and modifying the couplings as explainedin section II.2. This leads to δX = (cid:2) ( r f s α + c α ) − (cid:3) X SM ( m h )+ ( r f c α − s α ) X SM ( m σ ) . (32) III.1. Constraints from the light SM-like Higgsboson
Since a mass-mixing term between the dilaton and theSM-like Higgs scalar must violate the electroweak sym-metry, it is natural to assume that this mixing shouldbe small. However, in the absence of complete top-downconstraints, we should consider all possible bottom-upvalues for the mixing angle α . The allowed amount ofmixing, depending on the parameter f (or r f ), was ex-amined in ref. [17] for heavy scalar masses of 200, 600and 900 GeV with the Dirac dark matter mass beingfixed to 300 GeV. While the bounds from heavy scalarsearches obviously are strongly dependent on the massof the heavy scalar (having a rather weak dependenceon the dark matter mass with some provisos), the con-straints on the light Higgs couplings are independentof both the dilaton and the dark matter masses (pro-vided that the dark matter is not lighter than half theHiggs-boson mass, opening up invisible decays). Theconstraints from the S and T parameters are also onlyrelatively weakly dependent on the dilaton mass and havehence not drastically altered since 2014. The possible val-ues are therefore restricted to | sin α | (cid:28) r f (cid:28) | sin α | and r f .Figure 1 shows the preferred parameter space regionsin the (sin α, r f ) plane, after considering 99% exclu-sion bounds stemming from the most recent experimen-tal input from Higgs measurements (as implemented in Lilith [30] and
HiggsSignals [31]), as well as valuesfor the S and T parameters extracted from the most up-to-date electroweak fits [35]. The dark matter mass hasbeen chosen sufficiently high such that none of the scalarscan decay invisibly, in order to obtain bounds on sin α and r f which are not influenced by dark matter. Thebounds originating from the electroweak precision fit arepresented for heavy scalar masses of 500 and 3000 GeV.We have found out that a large amount of mixing isstill permitted by data. Larger mixing however modi-fies the constraints related to both heavy Higgs searches(see below) and the dark matter sector. For the latter,the effect is to more strongly couple the dark matter tothe Higgs boson, which greatly strengthens constraintsfrom direct detection since the Higgs couples much morestrongly to light quarks than the dilaton does.Our results additionally allow for a comparison be- sin v / f FIG. 1. Constraints on the mixing angle between the Higgsand dilaton against the ratio v/f , computed using
Lilith (solid green contour) and
HiggsSignals (blue dot-dashedcontour). We also show the allowed regions from constraintsoriginating from the electroweak S and T parameters for dila-ton masses of 3000 GeV (inner, light brown shaded region)and 500 GeV (outer, light red shaded region). tween the Lilith and
HiggsSignals programs, the for-mer requiring slightly more input (the decays needing tobe specified) while the latter calculates the decay tablebased on coupling ratios. The results agree at an ex-cellent level and do not substantially differ from olderbounds [17].
III.2. Heavy scalar searches
Heavy scalar searches provide rather powerful comple-mentary information, and indeed have been substantiallystrengthened during the second run of the LHC. Theseconstitute some of the main new results of this paper.Here we probe the allowed parameter space for f and m σ for fixed mixing angles. The results depend cruciallyon the treatment of the dilaton-Higgs mixing term, as wedescribe below.We consider a heavy dilaton with a mass larger than300 GeV. Throughout most of the parameter space themain constraint therefore comes from diboson searchessince the dilaton predominantly decays to heavy vectorbosons. However, there are two notable special cases.The first is when the dilaton decay constant is not largecompared to the dilaton mass. As we have seen, the dila-ton can become a wide resonance, for large m σ . Eventhough unitarity constrains Γ /m σ (cid:46) / π , the widthmay be large enough that the standard narrow resonancesearches do not apply. We shall avoid this issue by re-stricting our attention to a narrow dilaton.The second special case, as already pointed out in f [GeV]50010001500200025003000 m σ [ G e V ] sin α =0.0 f [GeV]50010001500200025003000 m σ [ G e V ] sin α =0.04 f [GeV]50010001500200025003000 m σ [ G e V ] sin α =0.11 f [GeV]50010001500200025003000 m σ [ G e V ] sin α =0.13 FIG. 2. Excluded regions of the model parameter space, presented in the ( f, m σ ) plane for a selection of mixing angles, withthe ‘minimal mixing term’ treatment of the Higgs-dilaton mixing. We include 99% confidence level exclusions from Higgsmeasurements (blue) and the electroweak S and T parameters (red/purple), as well as 95% confidence level bounds from heavyscalar searches (green). The parameter scans were done using Lilith and
HiggsBounds . Our findings demonstrate that forsufficiently low positive mixing angles, a gap in the exclusion emerges around the value f that satisfies v / f = tan α . In this gap,the couplings of the heavy scalar to the Standard Model fermions and massive gauge bosons are close or equal to zero, so thatsearches for heavy scalars turn out to be insensitive and arbitrary heavy scalar masses are allowed. ref. [17], is when f and α fulfill the condition( − s α + r f c α ) = 0 ⇔ r f = tan α . (33)Here the couplings of the heavy scalar σ to fermions andheavy vector bosons in eqs. (15) and (16) vanish, inde-pendently of m σ . This does not mean that the heavyscalar completely decouples from the Standard Model:the tree-level couplings to photons and especially gluonsdo not carry the factor ( − s α + r f c α ) and are thereforenon-vanishing (although the contribution from fermionand vector loops vanishes). Moreover, since the darkmatter particle does not obtain its mass from the Higgsmechanism, its coupling to the heavy scalar is also non-vanishing at this point. Therefore, in a ‘magic window’around r f = tan α the most stringent constraints on theheavy scalar disappear, yet it can still be produced by gluon fusion and decay significantly to dark matter.There are still constraints even exactly at this magicvalue, notably from diphoton and di-Higgs decays. Thedetails depend crucially on the treatment of the mixingmass term, as described in section II.2. If we take theminimal approach to the dilaton-Higgs mixing, and donot introduce additional interactions, then for small val-ues of the mixing angle we do see this ‘magic window’appear. The width of this window in terms of values of f depends on the mixing and mass m σ , and so does the depth : for sufficiently large mixing the window actuallydisappears. The appearance of a wide window at smallsin α = 0 .
04 and large f = 6 TeV, and its eventual disap-pearance around sin α = 0 .
13 are illustrated in figure 2.At larger mixing angles, the constraints from the lightHiggs measurements and the
S/T parameters moreoverdominate.On the other hand, if we introduce an additional termin the Lagrangian to restore gauge invariance, then ourresults are dramatically modified. In particular, the‘magic window’ around v/f = tan α that was found withthe minimal treatment of the mixing disappears. In the‘minimal mixing’ case and for large dilaton mass, the tri-linear σh coupling λ σhh becomes λ σhh −→ v/f = s α /c α m σ v c α s α ( ξ − − m h v c α s α + ... (34)where the ellipsis denotes additional terms suppressed bypowers of s α . On the other hand, in the ‘gauge invariantmixing’ case it becomes λ σhh −→ v/f = s α /c α − c α s α m σ v + ... (35)which is dramatically enhanced compared to the previousvalue, by a factor of order m σ /m h . This leads to dilatondecays into two Higgs fields dominating for much of theparameter space, and has the effect of completely erasingthe ‘magic window’ . We show the constraints in this casein figure 3, which shows current heavy scalar searcheswiping out all of the interesting parameter space. IV. UNDERGROUND SEARCHES FOR DARKMATTER
We now turn to considering the constraints on our dila-ton dark matter model originating from the relic density,direct detection, and the LHC. Combined with unitarityconstraints, we will show which part of the parameterspace remains viable, and to what extent the differentsearches are complementary.
IV.1. Collider constraints
In order to assess the constraints originating from darkmatter searches at the LHC, we used the implementa-tion in the
FeynRules package [34] of the Lagrangianin eq. (8) described in section III, to generate a UFO li-brary [39]. We then employ
MG5 aMC [40] to generatehard-scattering events relevant for the production of apair of dark matter particles together with jets, pp → XXj , (36)where X generically denotes the dark matter particle (aMajorana fermion or a vector boson). In our simulations,we convolute leading-order (LO) matrix elements withthe LO set of NNPDF 3.0 parton densities [41]. More-over, for every dilaton/dark matter mass configuration,we evaluate the dilaton width with the MadSpin [42] and
MadWidth [43] packages and make sure that the dila-ton is narrow (see eq. (27)). In other words, we enforcethe m σ /f ratio to be small enough. The above hard process gives rise to a monojet or amultijet plus missing transverse energy (MET) collidersignature, that is targeted by numerous dark mattersearches undertaken by the ATLAS and CMS collabo-rations [38, 44]. As those searches usually select eventsfeaturing at least one highly-energetic central jet, we im-pose, at the event generator level, that the transversemomentum of the jet satisfies p T >
100 GeV and that itspseudo-rapidity fulfills | η | < Pythia − of LHC data at a centre-of-massenergy of 13 TeV. This analysis targets multijet eventsfeaturing a monojet-like topology, i.e. it requires eventsto exhibit a large amount of missing transverse energy, alarge number of jets with at least one of them being veryhard. As a consequence, it gives a great handle on themodel considered in this work and dark matter modelsin general.Starting from Monte Carlo simulations of the dilaton-induced dark matter signal, we make use of the Mad-Analysis
Delphes
FastJet software [49] for event reconstruction on thebasis of the anti- k T jet algorithm [50]. We then esti-mate, still within the MadAnalysis s method [52].In figure 4, we present the obtained constraints for var-ious dark matter and dilaton mass configurations, bothfor Majorana fermion (left panel) and vector (right panel)dark matter. For each mass configuration, we evaluatethe maximum value of the cut-off scale that is excludedby the ATLAS analysis under consideration.This shows that the LHC has no sensitivity to scenar-ios in which the dilaton cannot be produced on-shell andthen decay into a pair of dark matter particles (which cor-responds to the parameter space region lying above theblue line). In contrast, when 2 m X (cid:46) m σ (with m X gener-ically denoting the dark matter mass), cut-off scales f around the TeV scale can be reached, which closes a partof the small light dilaton window visible allowed by Higgsdata (see the upper left panel of figure 2). Moreover, asexpected from the spin nature, constraints are tighter inthe vector dark matter case than in the fermionic one.Enforcing a naive scaling of both the background andthe signal [53], we have verified that the future high-luminosity operation of the LHC will not substantiallyaffect those conclusions, even with 3000 fb − of data.0 f [GeV]50010001500200025003000 m σ [ G e V ] sin α =0.11 f [GeV]50010001500200025003000 m σ [ G e V ] sin α =0.15 FIG. 3. Constraints on the parameter space of f and m σ for sin α = 0 .
11 (left) and sin α = 0 .
15 (right) with a ‘gauge invariant’treatment of the dilaton-Higgs mixing. The colour coding is the same as for figure 2.FIG. 4. Constraints on dilaton-induced dark matter for fermionic dark matter (left) and vector dark matter (right), presentedin the ( m σ , m V ) plane. For each mass configuration, we evaluate the maximum value of the cut-off scale that can be probedby using 140 fb − of LHC data and the ATLAS analysis of ref. [38]. Multijet plus missing transverse energy LHC analysis tar-geting a monojet-like topology within a multijet environ-ment are indeed already limited by the systematics [54]so that mild improvements can only be expected with alarger amount of luminosity. Moreover, in the presentcase, the limit depends on the fourth power of the cut-offscale, so that a noticeable improvement at the level ofthe bounds on f would require a huge improvement atthe analysis level. IV.2. Vector dark matter at zero mixing
The dilaton portal for dark matter is very simple, and,as described above, rather special. Its phenomenology inthe case of no mixing between the Higgs and the dilatonis therefore rather straightforward: we have two regimes, either near the s -channel resonance where m V (cid:39) m σ / m V (cid:29) m σ inwhich there is a combination of t -channel annihilationsof the dark matter to dilaton pairs and s -channel anni-hilation. However, in the previous sections we saw thatheavy Higgs searches force m σ > f < m σ is roughly given by3000 GeV /f. Clearly this will limit the possibilities fordetecting dilaton-induced vector dark matter at collidersor via direct detection.We show in figure 5 the contours of relic density match-ing the Planck results [55] for fixed f values in the( m σ , m V ) plane, for values of f upwards of 3 TeV. Wesee that the sensitivity of heavy scalar searches is not no-ticeably weakened due to the invisible dilaton decays intodark matter, and so the most promising regions for detec-tion corresponds to a resonant configuration. This thus1 m (GeV) m V ( G e V ) f = f = f = f = f = f = f =7000 GeV f =6000 GeV f =5000 GeV f =5000 GeV f =4000 GeV f =3000 GeV FIG. 5. Dilaton and vector dark matter masses that satu-rate the observed dark matter relic density Ω h = 0 .
12, fordiffering values of f and no mixing between the dilaton andthe Higgs boson. The curves are green and solid when thepoints are not excluded by any observations; they are orangeand dot-dashed when excluded by heavy Higgs searches. Thedark matter is underdense (and thus allowed if there is an-other source of dark matter) between the two curves relatedto a given f value, and overdense (hence excluded unless thereis some mechanism to dilute the dark matter density) outsidethem. For the values f = 3000 , m V → ∞ .Unitarity constraints are shown as the shaded grey regions. leads to m σ > f = 4000 GeV, butwhere unfortunately σ proton , spin − independent (cid:39) − cm , well beyond the reach of current and near-future direct-detection experiments. Moreover, the LHC and HL-LHCsearches described in the previous section do not limitthe parameter space in the figure at all. Alternativelywe can say that this model has a large unexcluded viableparameter space.We also present limits from unitarity as described inappendix A, which appear at the edges of figure 5. Theconstraints come from dark-matter scattering at low mo-mentum via a dilaton exchange, and at small dilatonmasses. This could also be interpreted as the regimewhere Sommerfeld enhancement should be taken into ac-count in the calculations. In the top right corner of thefigure, high-energy scattering constraints become visible,where we have taken a maximum centre-of-mass vectormomentum of p V, max = 20 TeV.It is legitimate to ask what happens at smaller val-ues of f : can there remain some viable parameter space?For f = 1 TeV, the constraints on the gluon scatter-ing momentum force us to impose a maximum cutoff of s = 10 TeV, leading to p V, max = 5 TeV. In the upperpanel of figure 6, we see that this excludes m σ < .
500 1000 1500 2000 2500 3000 3500 m (GeV) m V ( G e V ) p V ,max = 5 TeV p V ,max = 20 TeV f = 1 TeV
500 1000 1500 2000 2500 3000 3500 m (GeV) m V ( G e V ) p V ,max =10 TeV p V ,max =20 TeV f = 2 TeV FIG. 6. Dark matter curve as in figure 5, but for f = 1000 GeV (upper) and 2000 GeV (lower); the red portionof the curve corresponds to regions excluded by both heavyHiggs searches and dark matter direct detection experiments.The solid and hatched regions show (current) LHC and (fu-ture) HL-LHC exclusions from dark-matter-inspired collidersearches: the solid blue region is the future exclusion reachat the HL-LHC, after accounting for LO signal cross sections,and the hatched blue region shows the same constraint butwith a signal enhanced by a K -factor of 2. Similarly, the redsolid and hatched regions depict the current LHC exclusionwithout and with a K -factor of 2. Unitarity constraints areagain shown as shaded grey regions. On the upper figure, theentire parameter space of the model is excluded by a combi-nation of heavy Higgs searches and unitarity. On the lowerpanel, the monojet/multijet+MET searches are barely visi-ble, and some viable parameter space exists above the reachof heavy Higgs searches. that figure, we also show the sensitivity of the collidersearches for dark matter, although the entire ( m σ , m V ) plane is excluded by a combination of heavy Higgs searchresults, dark matter and unitarity constraints .On the other hand, as shown in the lower panel of fig-ure 6, the unitarity constraints give us an upper boundon the dark matter mass for f = 2 TeV. In our results,2we naively limit the considered parameter space regionsto m σ > m V ∈ [1 . , .
5] TeV. The reasonis that at m σ = 3 TeV the dilaton is rather wide , withΓ σ (cid:39)
250 GeV. This means that the dark matter den-sity constraint does not longer result in a funnel, as darkmatter is underdense everywhere above the shown curve.Clearly, the choice f = 2 TeV is therefore rather bor-derline in terms of whether we trust the results of thenumerical calculations. IV.3. Collider and vector dark matter constraintsat non-zero mixing
The hope of detecting dark matter greatly improvesonce we allow for dilaton-Higgs boson mixing: the darkmatter acquires a coupling to the Higgs, and so interactsmuch more strongly with nuclei (in principle the modelcan then accommodate a Higgs portal, which we shallnot consider as being very fine-tuned). Moreover, we alsohave the possibility of sitting in the ‘magic window’ where f = v/ tan α , which should also maximise the reach ofdark matter collider searches.In the ‘minimal mixing’ case, this would seem to bethe ideal situation: the dilaton still has barely suppressedgluon and dark vector couplings, but its couplings to SMbosons and fermions vanish. This means that the dila-ton decays only to the dark vector and the SM Higgsboson. Potentially, then, monojet and multijet + METsearches could probe some interesting part of the parame-ter space of the model. Since the collider searches dependso strongly on f , to have the best sensitivity we shouldlook for the lowest possible value. We saw earlier thatin the ‘minimal mixing’ case at the magic window, for m σ >
300 GeV, the minimum value of f that survives allconstraints was for sin α = 0 .
11, giving f = 2 . α = 0 .
11 and sin α = 0 .
15. The relic density anddirect detection cross sections were computed using
Mi-crOMEGAs and compared with the limits summarisedin ref. [56]. The LHC limits and projections were inferredfrom the results in section IV.1 by recomputing the pro-duction cross sections (for the pp → V V j process withthe same cuts on the hard jet) for the mixing case. Dueto the dilaton widths and the relative coupling changes,we could not naively rescale the cross sections. However,the same cutflows/limits on the total rate can be used.We observe that the (HL-)LHC searches do not overlapwith any of the viable regions of the parameter space, sothat for sin α = 0 .
11 we could have a viable dark mattermodel for m σ <
200 400 600 800 1000 1200 m (GeV) m V ( G e V )
200 400 600 800 1000 1200 1400 m (GeV) m V ( G e V ) FIG. 7. Combined dark matter, Higgs and collider constraintson the considered model for sin α = 0 .
11 (upper) and 0.15(lower) in the ‘magic window’ where f = v/ tan α and un-der the assumption of a ‘minimal Higgs-dilaton mixing’ treat-ment. The solid lines show the curves where the dark matterdensity matches the Planck limit of Ω h = 0 .
12 with the cir-cular shading between them showing the underdense regions.The solid line is green and orange for allowed and excludedby heavy Higgs searches, and black when excluded by darkmatter direct detection. The solid and hatched regions showthe current and future exclusions from dark matter collidersearches: the solid blue region is the future exclusion reachat the HL-LHC after accounting for LO signal cross sectionsand the hatched blue region is the same constraint but withthe signal enhanced by 2 σ according to the uncertainty onits total rate. The red solid and hatched region represent thecorresponding constraints at the end of the LHC run 2. therefore present the same results but for the ‘gauge in-variant mixing’ case in figure 8. There is no dark matterparameter space available for m σ >
300 GeV, and theLHC/HL-LHC searches are completely wiped out as theproduction of dark matter greatly diminishes. The ‘gaugeinvariant mixing’ scenario is therefore entirely unappeal-ing phenomenologically, and invites other model-building3
200 400 600 800 1000 1200 m (GeV) m V ( G e V )
200 400 600 800 1000 1200 1400 m (GeV) m V ( G e V ) FIG. 8. Combined dark matter and Higgs constraints forsin α = 0 .
11 (upper panel) and 0.15 (lower panel) for a ‘gaugeinvariant’ treatment of the Higgs-dilaton mixing. The descrip-tion is similar as in figure 7, except that there are no LHC orHL-LHC constraints from monojets or multijet searches, thecross sections being orders of magnitude too small. solutions.
V. FUTURE COLLIDER CONSTRAINTS
We showed in the previous sections that searches fordark matter at the LHC and at its future high-luminosityoperation are not sensitive to our scenario. In this sec-tion, we investigate instead monojet and multijet+METcollider probes at a future 100 TeV collider. We make useof the simulation chain introduced in section IV.1, study-ing the hard-scattering process of eq. (36). At the gen-erator level, we impose that the transverse momentumof the hardest jet satisfies p T > | η | <
5. In addition, we consideras the main backgrounds to our analysis the productionof an invisible Z -boson with jets.
500 1000 1500 2000 2500 3000 3500 40005001000150020002500300035004000 m σ [ GeV ] m ψ [ G e V ] m σ [ GeV ] m V [ G e V ] FIG. 9. Missing energy selection thresholds as a function ofthe dilaton and fermionic (upper panel) and vector (lowerpanel) dark matter masses to achieve the best sensitivity at afuture 100 TeV proton-proton collider. In order to avoid anypotential unitarity issues at high energies, we impose that thethreshold value is smaller than 5 TeV.
In our analysis, we first veto the presence of chargedleptons with a transverse momentum p T >
20 GeV and apseudo-rapidity | η | < . p T >
40 GeV and | η | < .
3. Wenext require that the leading jet is central and very hard,with p T ( j ) > | η ( j ) | < .
4, and allowfor some extra hadronic activity in the selected events.This hadronic activity is associated with the ensembleof non-leading jets whose p T >
30 GeV and | η | < . ϕ ( / p T , j i ) < . , (37)and additionally, the second jet is prevented from being4back-to-back with the leading jet,∆ ϕ ( j , j ) < . . (38)After this preselection, our analysis relies on varioussignal regions to estimate the sensitivity of a future100 TeV proton-proton collider to the dark matter mono-jet/multijet+MET signal predicted in our model. Eachsignal region is defined by a different missing transversemomentum selection, /E T > /E thr .T with /E thr .T ∈ [2 −
5] TeV , (39)so that any considered dark matter and dilaton massconfigurations could be optimally covered. In the abovesetup, we restrict all missing transverse energy thresh-olds to be smaller than 5 TeV, which guarantees to avoidany potential unitarity issues. The best MET thresholdvalue for a given mass spectrum depends on both masses,as depicted in the upper and lower panels of figure 9 forthe fermionic and vector dark matter cases respectively.For dark matter masses much larger than half the dila-ton mass ( i.e. far from any resonant configuration), theoptimal selection enforces the missing energy to be largerthan 1–3 times the dark matter mass for the two classes ofmodels, the MET spectrum being in general flat enoughto guarantee a large signal selection efficiency and a goodrejection of the Z +jets background. In contrast, whendark matter production is enhanced by the existence of adilaton resonance ( i.e. for m Ψ < m σ / m V < m σ / − of proton-proton collisions at 100 TeV are suffi-cient to probe cutoff scales lying in the multi-TeV regimefor both the fermion (upper panel) and vector (lowerpanel) dark matter cases. The results are presented inthe ( m σ , m Ψ ,V ) mass plane respectively, and the valuesof the f scale that are reachable for each mass spectrumare indicated by a colour code. For each configurationfor which the dark matter can be produced from the de-cay of a resonantly produced dilaton (below the red line),scales larger than 4 TeV can be probed, providing hencecomplementary constraints to models allowed by cosmo-logical considerations.In contrast, for configurations in which the dilaton can-not decay into a pair of dark matter particles, the signalcross sections are smaller. This results in a loss of sensi-tivity, in particular in regions favoured by cosmology. VI. CONCLUSIONS
We have presented a comprehensive and up-to-date setof current and future constraints on the most interesting(heavy) dilaton-portal dark matter models. While heavyscalar and unitarity constraints push the model to largemasses and weak couplings, to the extent that the direct
500 1000 1500 2000 2500 3000 3500 40005001000150020002500300035004000 m σ [ GeV ] m ψ [ G e V ] m ψ = m σ
500 1000 1500 2000 2500 3000 3500 40005001000150020002500300035004000 m σ [ GeV ] m V [ G e V ] m V = m σ FIG. 10. Sensitivity of a 100 TeV future proton-proton col-lider to the dilaton portal dark matter model considered inthis work, for the case of Majorana (upper panel) and vector(lower panel) dark matter. The results are presented in the( m σ , m Ψ ,V ) plane and we indicate, through the colour coding,the expected reach on the theory cutoff scale f . Our findingscorrespond to integrated luminosity of 3 ab − . dark matter production at the LHC can only probe alight dilaton, and not reach any viable parameter spaceabove 300 GeV, a future collider would potentially besensitive with the same searches. It would be interestingto examine future projections for heavy scalar and darkmatter searches, to see whether these will be complemen-tary. On the other hand, the (HL) LHC reach could beenhanced if the experimental collaborations extend theirpublished diboson limits above 3 TeV.We discussed the fact that vector dark matter is morepromising for collider searches thanks to the dilaton’smuch larger branching ratio into vectors compared tofermions or scalars. Allowing mixing of the dilaton withthe Higgs boson then apparently leads to a way to weakenor evade heavy scalar searches (and increase the coupling5of the dark matter to the visible sector via the Higgs por-tal) via the opening of a ‘magic window’. We showed thatthis could allow the dark matter and dilaton below before electroweak symmetry break-ing , as performed in ref. [15]: it would be interestingto revisit our constraints and searches in this alternative(and inequivalent) formulation of the theory. ACKNOWLEDGMENTS
MDG acknowledges support from the grant“HiggsAutomator” of the Agence Nationale de laRecherche (ANR) (ANR-15-CE31-0002). He thankshis children for many stimulating conversations duringthe preparation of this draft. DWK is supported by aKIAS Individual Grant (Grant No. PG076201) at KoreaInstitute for Advanced Study. P.K. is supported in partby KIAS Individual Grant (Grant No. PG021403) atKorea Institute for Advanced Study and by NationalResearch Foundation of Korea (NRF) Grant No. NRF-2019R1A2C3005009, funded by the Korea government(MSIT). S.J.L. acknowledge support by the SamsungScience and Technology Foundation under ProjectNumber SSTF-BA1601-07.
Appendix A: Perturbative unitarity constraints
Since our theory contains non-renormalisable opera-tors, it must have a cutoff comparable to the scale f .We expect this to be manifest, even at tree level, inperturbative unitarity constraints on two-body scatter-ing amplitudes. In particular, there are some couplingsthat could in principle be large compared to f since theyare enhanced by additional massive factors involving themasses of the theory vector bosons (especially in the darkmatter case). In this appendix, we describe the calcula-tion of the unitarity constraints involving massive vec-tors. For simplicity, we restrict to the case of no mix-ing between the dilaton and the Higgs, and neglect theHiggs boson entirely. For references on the calculation ofunitarity constraints, and in particular including vectorbosons, see refs. [57–61].
1. Scattering to dark matter
We first consider the scattering of vector bosonsamongst themselves and into pairs of dilatons, as wella dilaton self-scattering. The relevant terms in the La-grangian read L σV = m V f σV µ V µ + m V f σ V µ V µ − ξ m σ f σ − ζ m σ f σ , (A1)where ξ and ζ are model-dependent and usually taken tobe 5/6 and 11 respectively.Since the model has a Z symmetry, states with oddnumbers of vectors can only scatter to states with oddnumbers of vectors, so that we only need to consider the M σσ → σσ , M σσ → V V , M V V → σσ and M V V → V V scatter-ing amplitudes. These are given by i M σσ → σσ = − iξ m σ f (cid:20) s − m σ + 1 t − m σ + 1 u − m σ (cid:21) − ζi m σ f ,i M V V → σσ = i m V f (cid:15) · (cid:15) + 6 iξ(cid:15) · (cid:15) m V m σ f ( s − m σ )+ i (cid:18) m V f (cid:19) (cid:15) µ (cid:15) ν (cid:20) t − m V (cid:18) η µν + k µ k ν m V (cid:19) + 1 u − m V (cid:18) η µν + k µ k ν m V (cid:19)(cid:21) ,i M V V → V V = − i m V f (cid:15) µ (cid:15) ν ˜ (cid:15) ρ ˜ (cid:15) κ (cid:20) η µν η ρκ s − m V + η µρ η νκ t − m V + η µκ η νρ u − m V (cid:21) , (A2)where (cid:15) , denote the incoming polarisation vectors and˜ (cid:15) , the outgoing ones. Note that in all of the amplitudes s/t/u channel poles are not possible, and so we can searchover all scattering momenta up to some potential cutoffwithout needing to excise singular regions or submatricesas required in the general case [59, 61].Since there are three initial polarisations possible foreach vector, the scattering matrix (including the dilaton-dilaton state) is in principle of rank 10. However, itbreaks into irreducible blocks under the Lorentz algebra,and in particular we can separate off the symmetric andantisymmetric states of (cid:15) µi (cid:15) νj for i (cid:54) = j .Typically, only the high-energy limit is considered,where we retain only longitudinal gauge bosons. We find,however, that even the transverse components can con-tribute in this limit. In fact, in the low-energy regimerelevant for dark matter scattering, it is the contribu-tion of the transverse components that dominates. So,we can take a suitable basis of polarisation vectors, suchas (0 , , , , , , p V /m V , , , E V /m V ) for a vec-tor aligned along the third spatial component and insert6them in the above amplitudes. We then extract the ze-roth moment of the scattering matrix T ij = = 164 π (cid:114) | p in | | p out | s (cid:90) − d (cos θ ) M ij (A3)where we have included appropriate symmetry factors forour incoming and outgoing states to be identical pairs,and p in and p out are the three-momenta in the centre-of-mass frame for the incoming and outgoing states respec-tively.To find the limit from unitarity, and since the incomingand outgoing states may be related by a unitary rotationwhich we are not interested in, we take the square root ofthe eigenvalues of the T matrix and compare the maxi-mum of these to 1 / | a max0 | ≡ (cid:115) max (cid:20) eigenvalues (cid:18) T ( T ) † (cid:19)(cid:21) < . (A4)For convenience we shall define T ij ≡ πf ˜ T ij . (A5)The full expressions are too cumbersome to list here, butcan be made available in Mathematica format or in a c program; we shall here identify different regions of pa-rameter space of interest and give approximate formulaefor those cases.
2. Scattering at high energy
In the limit of large s , we find the following for thescattering of longitudinal gauge bosons and the dilaton,˜ T V L V L )( V L V L ) → − m σ , ˜ T V L V L )( σσ ) → (cid:20) m V + (6 ξ − m σ − m V log sm V (cid:21) , ˜ T σσ )( σσ ) → − m σ . (A6)At this level, scattering involving the transverse modesis also relevant, and the scattering matrix can be rotatedinto˜ T = −√ m V √ m V −√ m V ˜ T V L V L )( V L V L ) ˜ T V L V L )( σσ ) √ m V ˜ T σσ )( V L V L ) ˜ T σσ )( σσ ) . (A7)In the limit that m V is small and the scattering energyis large, the amplitude is dominated by dilaton-dilatonscattering. We find a max0 = m σ πf (cid:113)
261 + 28 √ . (A8) f (TeV) p V, max (TeV) Approx. max( m V ) (TeV)1 5 1.91 20 1.42 10 3.82 20 3.1TABLE I. Bounds on the vector dark matter mass m V forgiven illustrative cutoff values f and typical maximum centre-of-mass vector momentum p V, max . This is essentially the constraint originating from dilatonself-scattering, 11 m σ πf (cid:46) → m σ (cid:46) f, (A9)that is considerably stronger than the one from pure lon-gitudinal vector scattering.We can use the above result in the limit that s (cid:29) m V ,so that the logarithmic term appearing in the second re-lation of eq. (A6) is large. For reasonable values, wefind that the obtained constraint on m σ is comparable orslightly stronger than the above. For the limit m V (cid:29) m σ ,18 πf m V log sm V < / , (A10)which bounds m V for a given cutoff. We can directlycompare this with the values shown in figure 6 and thatwe report in table I. These approximate values agree withthe maximum values of m V found numerically, althoughthe high-energy limit is generally found not to be a verygood approximation for the full scattering matrix.Finally, while we have computed separate constraintsfrom the symmetric/antisymmetric scattering of thetransverse and longitudinal vector modes, in the largemomentum limit these all reduce to √ m V πf < / . (A11)
3. Scattering at low energy
When m V (cid:29) m σ the amplitude is largest at low mo-menta, and dominated by the scattering of transverse vectors of the type that obey (cid:15) · k , = 0. We find thatthe other modes reduce the scattering amplitude and mixlittle with the other states. The largest eigenvalue is wellapproximated by the (11) ↔ (11) scattering, and largestwhen p V = m σ , a = m V πf E V (cid:20) p V log (cid:18) p V m σ (cid:19) − p V m V + 4 p V − m σ (cid:21) (cid:39) m V πm σ f log 5 . (A12)7This places a constraint on the minimum dilaton mass, m σ (cid:38) m V πf log 5 . (A13)This can correspondingly be considered the point atwhich Sommerfeld enhancement of the amplitudes is sig-nificant, as clearly seen in figure 5.
4. Scattering to gluons
In this work, we are interested in vector dark matterpartly because it can be more copiously produced thanfermions or scalars via the dilaton portal. It should notbe a surprise then that a unitarity limit on our theoryarises from the scattering of gluons into vector bosonsvia the dilaton. The corresponding scattering amplitudeis given by i M gg → V V = (cid:18) i α s b πf (cid:15) g · (cid:15) g s (cid:19) is − m σ (cid:18) i m V f ˜ (cid:15) · ˜ (cid:15) (cid:19) , (A14)where (cid:15) g , are the gluon polarisation vectors and b denotes the QCD beta function. For the longitudinalmodes, ˜ (cid:15) · ˜ (cid:15) = 1 m V ( p + E ) = s m V , (A15)so that in the s (cid:29) m V limit, we have i M gg → V V → − i α s b πf s s − m S . (A16) Now | T gg ) , ( V L V L ) | = | M | π , but there are 2 incoming spins,and 8 incoming pairs of colours that give non-zero results.So the scattering matrix looks like T = T gg ) , ( V L V L ) · · · T gg ) , ( V L V L ) , (A17)so T ( T ) † = | T gg ) , ( V L V L ) | · · · . (A18)Therefore, we finally obtain a max0 = √ × α s b π f s. (A19)However, this is only for scattering into one vector bosonspecies. After accounting for the contributions of the Z and W bosons (that act as the equivalent of threeindividual vectors), we obtain an additional factor of twoin the limit, √ × √ × α s b π f s < / → s < π f α s . (A20)Similar constraints on the maximal scattering energy viagluon fusion were found long ago in ref. [58]. [1] W. A. Bardeen, C. N. Leung, and S. T. Love, Phys. Rev.Lett. , 1230 (1986).[2] W. Buchmuller and N. Dragon, Phys. Lett. B , 417(1987).[3] W. Buchmuller and N. Dragon, Nucl. Phys. B , 207(1989).[4] R. Rattazzi and A. Zaffaroni, JHEP , 021 (2001),arXiv:hep-th/0012248 [hep-th].[5] C. Csaki, M. L. Graesser, and G. D. Kribs, Phys. Rev.D , 065002 (2001), arXiv:hep-th/0008151.[6] D. Dominici, B. Grzadkowski, J. F. Gunion, andM. Toharia, Nucl. Phys. B , 243 (2003), arXiv:hep-ph/0206192.[7] C. Csaki, J. Hubisz, and S. J. Lee, Phys. Rev. D76 ,125015 (2007), arXiv:0705.3844 [hep-ph].[8] W. D. Goldberger, B. Grinstein, and W. Skiba, Phys.Rev. Lett. , 111802 (2008), arXiv:0708.1463 [hep-ph].[9] Y. Bai, M. Carena, and J. Lykken, Phys. Rev. Lett. ,261803 (2009), arXiv:0909.1319 [hep-ph].[10] B. Grzadkowski, J. F. Gunion, and M. Toharia, Phys.Lett. B , 70 (2012), arXiv:1202.5017 [hep-ph]. [11] Z. Chacko and R. K. Mishra, Phys. Rev.
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