Unitarisation of EFT Amplitudes for Dark Matter Searches at the LHC
Nicole F. Bell, Giorgio Busoni, Archil Kobakhidze, David M. Long, Michael A. Schmidt
UUnitarisation of EFT Amplitudes for Dark MatterSearches at the LHC
Nicole F. Bell a , , Giorgio Busoni a , , Archil Kobakhidze b , , David M. Long c , , Michael A. Schmidt b , , a ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, TheUniversity of Melbourne, Victoria 3010, Australia b ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, TheUniversity of Sydney, NSW 2006, Australia c School of Physics, The University of Sydney, NSW 2006, Australia
Abstract
We propose a new approach to the LHC dark matter search analysis within the effec-tive field theory framework by utilising the K -matrix unitarisation formalism. Thisapproach provides a reasonable estimate of the dark matter production cross section athigh energies, and hence allows reliable bounds to be placed on the cut-off scale of rel-evant operators without running into the problem of perturbative unitarity violation.We exemplify this procedure for the effective operator D5 in monojet dark mattersearches in the collinear approximation. We compare our bounds to those obtainedusing the truncation method and identify a parameter region where the unitarisationprescription leads to more stringent bounds. [email protected] [email protected] [email protected] [email protected] [email protected] a r X i v : . [ h e p - ph ] A ug Introduction
A dedicated search for Dark Matter (DM) at the Large Hadron Collider is currently one of theforemost objectives in particle physics. The most generic search channel is the mono-jet plusmissing transverse energy signal, which searches for a single jet recoiling against the momentum ofthe DM particles which escape the detector unseen [1–6]. In order to make such a search possible,it is necessarily to have a framework in which to describe the interactions of dark matter particleswith SM fields. Given the plethora of possible dark matter models in the literature, it is impracticalto perform a dedicated analysis of each model. It is thus imperative to work with a small numberof models that capture the essential aspects of the physics in some approximate way. Effective fieldtheories (EFTs) achieve this aim, by parameterising the DM interactions with SM particles by asmall set of non-renormalizable operators. For instance, the lowest order operators that describethe interaction of a pair of fermionic DM particles, χ , with a pair of SM fermions, f , are of theform 1Λ ( χ Γ χ χ ) (cid:0) f Γ f f (cid:1) , (1.1)where the Lorentz structure Γ χ,f can be 1 , γ , γ µ , γ γ µ , σ µν . A full set of operators can be foundin [7, 8], where a standard naming convention has been defined. Such operators are not intendedto be complete description of DM interactions, valid at arbitrarily high energy. They would beobtained as a low energy approximation of some more complete theory by integrating out heavydegrees of freedom. The energy scale Λ is related to the parameters of that high energy theory asΛ = g/M , where g is a coupling constant and M is the mass of a heavy mediator.The EFT description will clearly break down at energies comparable to Λ, at which scale weexpect the mediators to be produced on-shell or give rise to cross section resonances. Moreover,while the EFT will provide physically well-behaved cross sections at low energies, they will giverise to bad high energy behaviour if used outside their region of validity. This manifests as aviolation of perturbative unitarity [9–12]. While these issues may be remedied with a SimplifiedModel [13] in which a mediator is explicitly introduced, issues of unitarity violation can persist ifgauge invariance is not respected. The shortcoming of EFTs and Simplified Models that violateSM gauge invariance [14–17] or dark-sector gauge invariance [18, 19] have recently been discussed.Given the usefulness of the EFT and Simplified Model description of DM interactions, theywill continue to be used in collider DM search analyses. Therefore, it is important to limit analysesto parameters that respect perturbative unitarity. One such approach is to use a truncation tech-nique [20–22], which introduces a momentum cutoff equal to the mass of the would-be integrated-out mediator. In this paper we will instead use a procedure known as K -matrix unitarisation[23–28] to enforce unitarisation of all scattering amplitudes. Although this procedure will notcapture the resonance structure of the true high energy theory, it will force scattering amplitudesto be well behaved at high energies, allowing us to derive meaningful limits on EFT models fromLHC collisions with high centre of mass energies.We will use the K -matrix approach to unitarise the 2 to 2 scattering amplitudes, such as qq → χχ . This will allow us to determine unitarised cross sections for the 2 to 3 mono-jet processes suchas qq → χχg , under the assumption that the gluon can be treated with the collinear approximation.We will also compare the results obtained from this unitarisation technique with those obtainedwith truncation. The rest of the paper is organised as follows: in Section 2 we summarise thetheoretical framework for the unitarisation procedure. We illustrate the unitarisation procedure intwo toy models in Section 3 and apply it to the standard vector operator D5 in Section 4. Section 51ontains the conclusions, while in Appendix A we derive the relevant cross sections in the collinearlimit. The K -matrix formalism was first introduced in Ref. [23, 24]. It is a technique to impose unitarityon amplitudes which naively violate unitarity. In the derivation we largely follow the notation andarguments in Refs. [25, 26, 29] . Unitarity of the S -matrix, S = I + 2 iT , (2.1)implies the well-known relation for the T -matrix T − T † = 2 iT T † . (2.2)Note the factor of 2 in the definition of the T -matrix which has been introduced for convenience.Following the seminal work by Jacob and Wick [27], for scattering processes a b → c d we candescribe both the initial and the final state in terms of two-particle helicity states | Ω λ λ (cid:105) whichare characterised by the helicities λ i of the two particles and two angles θ and φ , collectivelydenoted Ω. Choosing the initial state to align with the z -axis, the individual T -matrix element fora process a b → c d with fixed helicities in the initial and final state is given by (cid:104) Ω λ c λ d | T | λ a λ b (cid:105) = 14 π (cid:88) J (2 J + 1) T Jλ (cid:48) λ D J ∗ λλ (cid:48) ( φ, θ, , (2.3)in terms of the partial waves T Jλ (cid:48) λ ≡ (cid:104) J λ c λ d | T | J λ a λ b (cid:105) = (cid:90) d Ω (cid:104) Ω λ c λ d | T | λ a λ b (cid:105) D Jλλ (cid:48) ( φ, θ, , (2.4)the Wigner D -functions D Jλλ (cid:48) with total angular momentum J , and the resultant helicity of thetwo-particle states λ = λ a − λ b and λ (cid:48) = λ c − λ d , where we used the normalisation of the Wigner D -functions in Ref. [26]. Assuming that no three-particle states are kinematically accessible, ananalogous unitarity relation holds for each partial wave T Jλλ (cid:48) separately, T J − T J † = 2 iT J † T J , (2.5)in terms of matrices T J with components T Jλ (cid:48) λ . This condition can be rewritten in terms of (cid:0) K J (cid:1) − ≡ (cid:0) T J (cid:1) − + i I = (cid:16)(cid:0) T J (cid:1) − + i I (cid:17) † , (2.6)which motivates the definition of the K -matrix for the J th partial wave, K J . The K -matrix ishermitean, K J = K J † . If the S -matrix is invariant under time reversal, the K -matrix is symmetricand thus K J and (cid:0) K J (cid:1) − are real. Hence ( K J ) − can be considered as the real part of ( T J ) − and the imaginary part of T J is determined by the term i I in Eq. (2.6). We can invert the relationin Eq. (2.6) to obtain T J ≡ K J ) − − i I . (2.7)The matrix T J is given by the stereographic projection of the K -matrix on the Argand circle as See Ref. [27, 28] for further details. J K J i i T JU Figure 1:
Argand circle and Thales projection. shown in Fig. 1. If perturbative unitarity is violated in any amplitude, it can be enforced byimposing reality on ( K J ) − , i.e. replacing ( K J ) − by Re[( T J ) − ], which leads to the unitarised T -matrix T JU ≡ T J ) − ] − i I . (2.8)Particularly in case the T -matrix quadratically grows with the centre of mass energy, T J ∝ s π Λ ,the unitarised T -matrix asymptotically reaches saturation T JU = 1 π Λ s − i s →∞ −→ i , (2.9)which can be interpreted as a resonance at infinity. Note that the restriction to the real part of (cid:0) T J (cid:1) − can be understood as the Thales projection onto the real axis [29], if the T -matrix T J iscomplex, i.e. points lying on the red dashed circle in Fig. 1 are projected onto the same unitarised T -matrix T JU as K J . All discussed operators in Secs. 3 and 4 lead to a real T -matrix T J in theconsidered scattering processes. Alternatively, following Ref. [31, 32] the hermitean K -matrix canbe considered as an approximation to the scattering amplitude, which can be obtained order byorder in perturbation theory using Eq. (2.6). Using the fact that the K -matrix is the Cayleytransform of the S -matrix [33, 34] S = I + iK I − iK , (2.10)it is possible to reconstruct a unitary S -matrix starting from an approximate K -matrix. The S -matrix defined in Eq. (2.10) restores unitarity, which is lost in the usual expansion of the S -matrix, if only a finite number of terms are taken into account in perturbation theory. The K -matrix formalism can be considered minimal, since it does not introduce new parameters orvisible structures in scattering amplitudes like resonances. However it does not yield a viable UVcompletion of the effective theory. New resonances have to be included by hand. See Refs. [29, 35,36] for a recent discussion in the context of W W scattering.In the following, we will make use of this prescription to obtain unitary amplitudes for DM pairproduction at the LHC. Taking the normalisation of the two-particle states properly into account,the T -matrix is related to the usual Lorentz-invariant matrix element M fi by (cid:104) Ω λ c λ d | T | λ a λ b (cid:105) = 132 π (cid:114) p f p i s M fi , (2.11) Note that this prescription is not analytic at T J = 0 [29]. K -matrix unitarisation does not enforce a consistentanalytic structure [30]. In practice this is not important, because we are interested in studying monojet searches atthe LHC where the amplitudes are large and T J (cid:54) = 0. p i,f / √ s approach unity, simplifying the calculation of the unitarised T -matrixconsiderably. Finally, the differential cross section in terms of the T -matrix element is given by dσ fi d Ω = (4 π ) s s p i |(cid:104) Ω λ c λ d | T | λ a λ b (cid:105)| , (2.12)and thus the total cross section can be conveniently expressed in terms of the partial waves σ fi = 4 πs s p i (cid:88) J (2 J + 1) (cid:12)(cid:12) T Jλ (cid:48) λ (cid:12)(cid:12) = 4 πs − m i (cid:88) J (2 J + 1) (cid:12)(cid:12) T Jλ (cid:48) λ (cid:12)(cid:12) . (2.13)Note that this is the cross section for fixed helicities. The unpolarised and color averaged crosssection is obtained in the usual way by averaging over the initial state helicities and number ofcolours and summing over the final state ones, i.e., σ ( q ¯ q → X ) = 112 (cid:88) helicities σ fi (2.14)for the unpolarised cross section q ¯ q → X with two quarks in the initial state. The unitarised crosssection is obtained by replacing T Jλ (cid:48) λ by the corresponding unitarised T -matrix element T JUλ (cid:48) λ .Thus the cross section is unitarised for each quark color and helicity separately. To illustrate the unitarisation procedure, we will make a simplifying assumption concerning thequark states in the operator and consider two simple models which feature only two channels. Theeffective operator D5 shall then be discussed in the next section.
As we are working in the collinear approximation, in the T -matrix we ought to consider all coupledtwo-particles states to expect the unitarity of the S -matrix to hold. If we consider the SM plusthe DM particle coupled with an EFT operator, this implies the consideration of all possible two-particle states in the standard model with zero charge, baryon and lepton number, in addition to χ ¯ χ . Taking into account color, helicity and flavour, this results in 3 · · · R ¯ R + V ¯ V + B ¯ B √ , (3.1)because all other color combinations decouple from this state and the DM sector. Moreover weassume the same operator suppression scale Λ for all quark flavours. In this case we can alsoconsider just one flavour state: u ¯ u + d ¯ d + s ¯ s + c ¯ c + b ¯ b + t ¯ t √ , (3.2)as, again, all other flavour combinations decouple from this state and the DM sector. Now, if we“turn off” electro-weak interactions, i.e. approximating α EW (cid:28) α s , this state decouples from allother standard model states, and only couples to itself and the DM states.4 .2 EFT Motivated by T-channel Scalar Exchange We now consider a toy model scenario that can be solved analytically. We take the following EFToperator connecting the dark and the visible sector: L = 1Λ qχ ¯ qγ µ P R q ¯ χγ µ P L χ . (3.3)This operator can arise by integrating out a heavy coloured scalar t-channel mediator couplingonly to right-handed quarks and left-handed DM particles. In the limit of massless particles, s (cid:29) m χ , m q , the only non-zero T -matrix elements are (cid:104) χ L ¯ χ R | T | q R ¯ q L (cid:105) and the matrix element (cid:104) q R ¯ q L | T | χ L ¯ χ R (cid:105) related by time-reversal. Thus we are left with a 2 × T -matrix, T = − π s Λ qχ (cid:32) (cid:33) sin θ , (3.4)in the basis of the two helicity 1 two-particle states ( | q R ¯ q L (cid:105) , | χ L ¯ χ R (cid:105) ). We only include the contri-bution of the effective operator and neglect any QCD contribution. The partial wave expansiononly contains the term with total angular momentum J = 1, T = − π s Λ qχ (cid:32) (cid:33) , (3.5)which grows linearly with s and thus is going to violate perturbative unitarity for scales s (cid:38) π Λ qχ . After unitarising the amplitudes using K -matrix unitarisation, the unitarised amplitudeturns out to be T U = 1 s + 144 π Λ qχ (cid:32) is − πs Λ qχ − πs Λ qχ is (cid:33) . (3.6)Note that the unitarisation procedure introduces contributions to the scattering of ¯ qq → ¯ qq and¯ χχ → ¯ χχ . The denominator leads to a smooth cutoff around s ∼ π Λ qχ , indicating that the non-unitarised amplitude strongly violates perturbative unitarity above such energy. When discussingthe validity of the EFT, this in turn means that, unless new states and/or new interactions areintroduced, the EFT breaks at this energy scale. The unitarised T -matrix is well-behaved for large s and converges to i I and it can be thus used to interpret scattering events, like monojet signaturesat the LHC. In fact, the high-energy tail leads to a negligible contribution due to the suppressionof the parton distribution function at high-energy in contrast to the EFT. Generally there might also be operators between two quark currents or two dark matter currents.As second example we consider an effective theory with three operators L = 12Λ qq ¯ qγ µ P R q ¯ qγ µ P R q + 1Λ qχ ¯ qγ µ P R q ¯ χγ µ P R χ + 12Λ χχ ¯ χγ µ P R χ ¯ χγ µ P R χ , (3.7)which might arise from a Simplified Model with a Z (cid:48) gauge boson coupling only to the right-handed quark and DM currents. In such a model the EFT parameters are related to those of the Note that right-handed (left-handed) particles have helicity + ( − ).
5V complete theory according to Λ qq = M Z (cid:48) /g q , Λ qχ = M Z (cid:48) / ( g q g χ ) and Λ χχ = M Z (cid:48) /g χ , where g q,χ are the couplings of the Z (cid:48) to the quarks and the DM, and M Z (cid:48) is the mediator mass. The effectiveoperators lead to four non-vanishing entries in the T -matrix, (cid:104) q R ¯ q L | T | q R ¯ q L (cid:105) , (cid:104) χ R ¯ χ L | T | χ R ¯ χ L (cid:105) , (cid:104) χ R ¯ χ L | T | q R ¯ q L (cid:105) , and (cid:104) q R ¯ q L | T | χ R ¯ χ L (cid:105) , where the latter two are related by time-reversal. The T -matrix in the basis ( | q R ¯ q L (cid:105) , | χ R ¯ χ L (cid:105) ) is then given by T = − π (cid:32) s Λ qq s Λ qχ s Λ qχ s Λ χχ (cid:33) cos θ . (3.8)Gluon s-channel exchange between quark - anti-quark pairs leads to an additional contribution tothe (cid:104) q R ¯ q L | T | q R ¯ q L (cid:105) element. It does not grow with s like the other contributions and thus can beneglected for large s , when perturbative unitarity becomes an issue. The only non-vanishing termin the partial wave expansion has total angular momentum J = 1 reading T = − π (cid:32) s Λ qq s Λ qχ s Λ qχ s Λ χχ (cid:33) . (3.9)The expression for the unitarised T -matrix turns out to be complicated. Assuming an underlyingSimplified Model with a Z (cid:48) mediator, the operator suppression scales are related viaΛ qq Λ χχ = Λ qχ . (3.10)This motivates the definition of the ratio r = Λ qχ Λ χχ = Λ qq Λ qχ . (3.11)In terms of the ratio r , the unitarised T -matrix, T , is T U,r = 1 r s − iπ ( r + 1) s Λ qχ − π r Λ qχ (cid:32) is r + 8 πs Λ qχ πr s Λ qχ πr s Λ qχ is r + 8 πs Λ qχ (cid:33) . (3.12)Note that one can always parameterize new physics using a complete set of EFT operators likethe ones in Eq. (3.7), thus this choice is not model dependent, if one chooses a complete basis.The only model-dependent hypothesis we are using comes from imposing the relation (3.10) basedon the assumption that the chosen EFT operators comes from an integrated-out Z (cid:48) mediator.Even though this choice is model dependent, we will keep this constraint to reduce the number ofparameters of the model. In the following we will restrict ourselves to this relation for simplicityand study the impact of the unitarisation procedure on the cross section using the well-studiedD5 operator and the corresponding four-fermion operators with only quark and dark matter fields,respectively. The K -matrix unitarisation procedure can be applied to any of the studied operators. We will focuson the operator D5 which might arise from a Simplified Model with a Z (cid:48) gauge boson couplingto both the quark and DM vector currents. Besides the operator D5, whose Wilson coefficient The operator D5 belongs to the list of operators presented in Ref. [7], which have been widely used in the LHCmonojet searches reported by the ATLAS and CMS collaborations. See Tab. 1 for the full list of operators.
6e denote by Λ − qχ , we have to consider the two four-fermion operators with only quarks and DMparticles χ , respectively L D = 12Λ qq ¯ qγ µ q ¯ qγ µ q + 1Λ qχ ¯ qγ µ q ¯ χγ µ χ + 12Λ χχ ¯ χγ µ χ ¯ χγ µ χ . (4.1)The explicit expressions for the T -matrix, the partial waves and the unitarised partial waves aresummarised in App. C. Similarly to the second toy model in the previous section, we assume relation(3.10) for simplicity and express the results in terms of the ratio (3.11). K -matrix unitarisationdoes not depend on this assumption, but it considerably simplifies the analysis by constrainingthe parameter space of the three Wilson coefficients to the two-dimensional submanifold definedby Eq. (3.10).Before comparing the result of K -matrix unitarisation with the 8 TeV ATLAS EFT limits forthe operator D5 [3] and the method of truncation, we comment on the validity of the collinearapproximation and the importance of quark jets. The collinear approximation is technically only valid in the limit of small scattering angles, i.e.small transverse momentum p T compared to the centre of mass energy √ s . Thus it is essential toestimate how well the collinear approximation performs for monojet searches, which usually employa high cut on p T to suppress QCD background. The full three-body final state cross section for theeffective operator D5 with an emission of one gluon jet is presented in the appendix of Ref. [21].
100 200 300 400 500 600 700 8000.600.650.700.750.800.850.90 p T , min [ GeV ] σ f u ll / σ c o ll Figure 2:
Ratio of the full cross section to the collinear one as a function of the minimum transverse momentum p T for m DM = 100 GeV. The Blue line refers to beam energy of 13TeV, the red one to 8TeV. Fig. 2 depicts the ratio of the cross section using the analytic result in Ref. [21] over the crosssection obtained in the collinear approximation as a function of the minimum p T,min both for 8TeV (red line) and 13 TeV (blue line). The collinear approximation leads to an enhancement ofless than about 10% of the cross section for a minimum p T,min (cid:39)
100 GeV, which grows to 45%(30%) with p T,min = 800 GeV for 8 TeV (13 TeV) centre of mass energy. The ATLAS 8 TeV Note that the collinear limit for the effective operators D1 and D4 agrees with the exact result. p T >
120 GeV and thus the collinear approximation overestimatesthe cross section by about 13%. The 13 TeV monojet searches plan to require p T,min = 600 GeVleading to about 37% overestimation of the cross section by taking the collinear limit. We expectsimilar results for the cross section in the unitarised EFT, which is suggested by the fact that thecross section in the effective theory can be factorised in the two-body cross section q ¯ q → χ ¯ χ and afunction dependent on the scattering angle of the jet and its rapidity. Consequently we expect theoverestimation by taking the collinear limit to mostly cancel out in the ratio of the cross sections( R U and R Λ , defined below). Hence the ratios calculated with the collinear approximation willbe closer to the values obtained from a full 3-body final state calculation than the result in Fig. 2suggests. Thus the collinear approximation works well, which is also supported by a similar analysisin Ref. [37]. Going beyond the collinear limit requires the inclusion of three-body states in the T -matrix rendering the K -matrix unitarisation procedure more complicated. We will defer ananalysis beyond the collinear limit to a future publication. In the previous subsection we only considered gluon jets, shown in Fig. 3a, and neglected theadditional contribution from quark jets. It originates from diagrams with gluons in the initialstate as shown in Fig. 3b. Quark jets generally lead to a 10% increase in the cross section, as it qq g χχq qq g χχq (a)
Gluon jets gq q χχq qg q χχq (b)
Quark jets
Figure 3:
Initial state radiation leading to monojet signature in DM pair production at the LHC. is suggested by Fig. 6 in Ref. [21]. We included quark jets and show in Fig. 4 the ratio of theunitarised cross section over the cross section using the effective field theory in the collinear limitfor a fixed value of the DM mass, m DM = 100 GeV, R U = σ unitarised , coll . σ EFT , coll . , (4.2)for different values of r = 1 , ,
5. The dotted lines show the ratio R U , if quark jets are neglected,while the solid lines take both contributions into account. The additional contribution of quarkjets generally enhances the unitarised cross section over the EFT cross section. ATLAS performed a monojet analysis with their full 8 TeV dataset of 20.3 fb − . The limits wereinterpreted for different EFT models including the operator D5. Besides the EFT limit, ATLASalso quotes the limit obtained using truncation, where only events are kept, which are consistentwith the EFT interpretation and satisfy the constraintΛ > Q tr √ g q g χ > m DM √ g q g χ , (4.3)i.e. the requirement that the momentum transfer Q tr is always smaller than the mass of themediator M = √ g q g χ Λ, which is expressed in terms of the cutoff scale Λ and the couplings g q,χ
8f the quarks and DM particles χ to the mediator. In case of D5, this could be the mass ofan Z (cid:48) gauge boson, which is exchanged in the s-channel, and the corresponding gauge couplingswith quarks and DM. For gauge couplings, we naively expect the couplings to be of a similarorder of magnitude. We reproduce in Fig. 5 the official ATLAS 8 TeV monojet limit shown inFig. 10b of Ref. [3]. The blue solid line refers to the ATLAS EFT limit, and the green and yellowregions indicate the 1 and 2 σ uncertainty bands. The red dashed line corresponds to the limit Λ [ GeV ] R s =
13 TeV600 GeV ≤ p T ≤ | η | ≤ m DM =
100 GeV R U , r = R U , r = R U , r = Figure 4:
The ratio R U as a function of the cut-off scale Λ, for different values of r for m DM = 100 GeV. The solidlines refer to R U including both quark and gluon jets, the dotted lines refer to R U including only gluon jets. m DM [ GeV ] Λ [ G e V ] Figure 5:
Reinterpretation of ATLAS limit at 8TeV. The blue line refers to the ATLAS limit, the green andyellow band indicating the 1 and 2 sigma uncertainty bands, as in [3]. The red dashed line indicates the limit usingtruncation with maximal couplings, the purple dashed one using truncation with unit couplings. The purple dottedlines refers to our result using the collinear limit for the truncation with unit couplings, while the black lines referto the unitarised amplitude with r = 1 , , , , g q g χ = 4 π and the purple dashed line to the one usingtruncation with couplings g q g χ = 1. The purple dotted line is our result for truncation withunit couplings using the collinear limit. The black solid lines show the limit obtained using theunitarised amplitude with r = 1 , , , , m DM <
100 GeV, because they are derived neglecting the DM mass. In our analysis,we employ the collinear limit and only include the leading jet unlike the ATLAS analysis, whichincluded a second jet. These effects go in the opposite direction and partly cancel each other. Theunitarised amplitude with r ≤ g q g χ = 1. Using the cross section ratio, it is straightforward to apply the same method to a future analysis.The EFT cross section is suppressed by the fourth power of the scale of the effective operatorΛ ≡ Λ qχ . Thus a reduction of the unitarised cross section by a factor R U approximately results ina decrease of the limit on the scale Λ by a factor of R / U . In practice the unitarised limit has to beobtained iteratively [38]. Fig. 6 shows the ratio R U as a function of the cutoff scale Λ for different Λ [ GeV ] R s =
13 TeV600 GeV ≤ p T ≤ | η | ≤ m DM =
100 GeV R U , r = R U , r = R U , r = R Λ , g q g χ = π R Λ , g q g χ = R Λ , g q g χ = R Λ , g q g χ = Figure 6:
Quantities R U , R Λ as a function of the cut-off scale Λ for different values of r = 1 , ,
10 and g q g χ =0 . , , , π for m DM = 100 GeV. The solid lines refer to R U , the dashed lines refer to R Λ . Both gluon and quarkjets were included in both cases. values of r = 1 , ,
10 as solid lines. The dashed lines serve as a comparison to the correspondingratio R Λ = σ truncated , coll . σ EFT , coll . , (4.4)using the truncated amplitudes for different benchmark values of the couplings g q g χ = 0 . , , , π .All ratios have been obtained using the collinear approximation including exactly one jet, whichcan be either a quark or a gluon jet. The centre of mass energy is fixed to √ s = 13TeV and theDM mass to m DM = 100 GeV. The transverse momentum p T is limited to 600 GeV ≤ p T ≤ | η | ≤
2. The ratios R U and R Λ do not change much if the cuton p T is slightly increased to 700 GeV. 10 g q g χ Figure 7:
The light blue shaded region is the region of the parameter space with R U < R Λ . The region covered byblack dashed lines is excluded by dijet search [39] (for mediators below 3TeV). The brown, blue and red lines arecontours where the value of g q g χ is constant, and equal to 1 / The suppression is generally stronger for low cut-off scales Λ, because more events have to bediscarded using the truncation procedure or the amplitude is reduced for smaller center of massenergies √ s using K -matrix unitarisation. The more a value deviates from r = 1, the more theunitarised cross section is suppressed, similar to smaller couplings g q g χ when using truncation.This can be clearly seen in Fig. 6.The values of R U reported in Fig. 6 can be used to rescale EFT limits in the same way as with R Λ . The precise description of the rescaling procedure and its main consequences are outlined inRef. [38].Finally we compare the K -matrix unitarisation to the truncation procedure in the Fig. 7.The solid lines show the lines of constant g q g χ = 0 . , , g q (cid:46) .
25 for mediator masses upto 3 TeV [39]. The light blue shaded region has R U < R Λ , i.e. unitarisation leads to a largersuppression of the cross section than truncation and thus a less stringent limit. Generally thetruncated amplitude is less suppressed for g q g χ (cid:38) g q (cid:46) .
25, we find that the unitarisationmethod leads to a stronger limit, R U > R Λ , for g χ (cid:46) Conclusions
Non-renormalisable operators lead to violation of perturbative unitarity in scattering amplitudesabove the scale of the operator. This particularly poses a problem for the interpretation of monojetsearches at the LHC experiments in terms of EFTs, because the limits on the cut-off scale Λobtained assuming an EFT are lower than the centre of mass energy √ s . Thus there are manyhigh-energy collisions with a centre of mass energy greater than Λ. Although high-energy eventsare penalised by the small values of the parton distribution functions, this is cancelled by theenhanced scattering amplitude, which grows proportional to the centre of mass energy. K -matrix unitarisation allows consistent limits to be obtained within the EFT framework. Weexemplified this for the operator D5 as well as two other simple toy models. It leads to a smoothsuppression of the scattering amplitude. In the limit of large centre of mass energy, √ s → ∞ ,the T -matrix approaches i and thus the off-diagonal elements describing DM pair production atthe LHC vanish. K -matrix unitarisation introduces a dependence on the other T -matrix elementsand thus the cut-off scales of other operators, e.g. four quark operators and operators with fourDM particles. The smallest cut-off scale among all relevant operators determines the scale whenthe suppression due to K -matrix unitarisation sets in. Hence the least suppression of the crosssection in the K -matrix unitarisation framework is obtained if the cut-off scales are of a similarorder of magnitude. This can be clearly seen for the D5 operator: The suppression increase with r = Λ qχ / Λ χχ , since the smallest cut-off scale decreases with r .We recast the ATLAS 8 TeV monojet limit on the operator D5 for five benchmark values of r = 1 , , , , r = 1 which grew to more than50% for r = 5. Given the suppression of the cross section as a function of the cut-off scale Λ, itis straightforward to recast the limit obtained using an EFT to a limit for the unitarised EFT.We provide this ratio for three different choices of r , for a centre of mass energy of √ s = 13TeV,which can be directly used to obtain the unitarised EFT limit given the EFT limit. Note howeverthat all results have been obtained in the collinear approximation and without including a possiblesecond jet. Going beyond these two approximations, and the application of the same procedure tothe other considered operators, will be an interesting extension of the present work. K -matrix unitarisation of EFT amplitudes provides a new way to extract model-independentand theoretically reliable limits on the dark matter production cross section at the LHC. Themethod can be applied to a wide class of scenarios, including other mono-X searches or SimplifiedModels without manifest gauge invariance, providing, in certain cases, more stringent limits thanthe truncation method currently used. Acknowledgements
We thank Lei Wu for collaboration during the initial stages of thisproject. This work was supported in part by the Australian Research Council.
A Collinear Approximation
The collinear approximation allows to drastically simplify the discussion. This appendix containsa detailed derivation of the relevant cross section. Starting by simplifying the three-body phase12pace, we can write dφ body = (2 π ) δ ( (cid:88) i =1 p i − p ) (cid:89) i =1 d p i (2 π ) E i = 12 π d p d p E E E δ ( E + E + E − E ) . (A.1)The phase space is Lorentz invariant, so we are free to evaluate this expression in any referencesystem. After introducing the four-momentum p = p + p with the corresponding energy E = p and invariant four-momentum s = p , it is possible to use the identities1 = ds δ ( s − p ) θ ( p ) (A.2) δ ( E + E + E − E ) = δ ( E + E − E ) δ ( E + E − E ) dE (A.3)to separate the two-body phase space of particles 2 and 3 dφ body = 12 π d p d p E E E ds δ ( s − p ) θ ( p ) δ ( E + E − E ) δ ( E + E − E ) dE (A.4)= 12 π d p E ds δ ( s − p ) θ ( p ) δ ( E + E − E ) dE dφ body , , (A.5)where in the last step we have used the definition of the two-body phase space of the particles 2and 3 dφ body , ≡ π d p E E δ ( E + E − E ) , (A.6)which will be included in the two-body cross section. The remaining part can be further simplifiedby integrating over E dφ body = dφ body , ds π π d p E E δ ( E + E − E ) . (A.7)While we are not interested in simplifying dφ body , further, as its expression in terms of kinematicvariables will be necessary only to calculate the cross section σ q ¯ q → χ ¯ χ , we want to simplify the lastdelta function in dφ body = dφ body , ds π d cos θ E dE E δ ( E + E − E ) , (A.8)which can be evaluated using E = (cid:113) E + s dE dE = E (cid:112) E + s = E E . (A.9)Thus we obtain after the integration with respect to E dφ body = dφ body , ds d cos θ π E E (A.10)The phase space and cross section are simple to evaluate in the centre of mass frame, wheremomentum fraction of the partons equal x = x = x and the following kinematic relations holdˆ s = ( p + p + p ) = sx E = √ sx z s = ( p + p ) = sx (1 − z ) E = √ sx (cid:16) − z (cid:17) (A.12)13he definition of z , θ is given in the following parametrisation of the momenta in the centre ofmass frame p µ = √ sx z , , sin θ , cos θ ) (A.13) p µ = √ sx (cid:18) − y , (cid:112) (1 − y ) − a ˆ p (cid:19) (A.14) p µ = √ sx (cid:18) y − z , (cid:112) (1 + y − z ) − a ˆ p (cid:19) , (A.15)where the angle between p and p is fixed by momentum conservation and the fraction m DM √ sx .Using this parametrisation allows us to write the three-body phase space as dφ body = dφ body , sx z π dz d cos θ (A.16)clearly separating the two-body phase space factor from the variables z and cos θ describing theadditional jet.After the derivation of the convenient form of the three-body phase space factor, we are readyto work with the collinear approximation. The four-momentum of the jet is denoted p , while thefour-momenta of the DM particles are p , . Following the standard discussion of the collinear limit(See e.g. [40]), the monojet cross section with a gluon jet can be written as dσ q ¯ q → χ ¯ χ + j ( g ) = 1 | v q − v ¯ q | E q E ¯ q (cid:20) (cid:88) | M | (cid:21) p q, ¯ q − p ) | M | q ¯ q → χ ¯ χ ( s ) dφ body (A.17)= 12 sx (cid:20) g s p T z (1 − z ) 1 + (1 − z ) z (cid:21) z p T | M | q ¯ q → χ ¯ χ ( s ) dφ body , sx z π dz d cos θ neglecting the color factor. The four-momentum p q, ¯ q denotes the initial state four-momentum ofthe parton radiating off the gluon and the transverse momentum of the gluon is given by p T = √ sx z θ . (A.18)The 2 → q ¯ q → χ ¯ χ , σ q ¯ q → χ ¯ χ ( s ) = 14 | M | q ¯ q → χ ¯ χ ( s )2 sx (1 − z ) dφ body , , (A.19)can be factored out leading to dσ q ¯ q → χ ¯ χ + j ( g ) = σ q ¯ q → χ ¯ χ ( s ) α s π − z ) z z p T sx z dz d cos θ (A.20)= σ q ¯ q → χ ¯ χ ( s ) α s π − z ) z θ dz d cos θ , (A.21)where Eq. (A.18) has been used in the last line. Finally the cross section has to expressed interms of the variables in the lab frame to properly take the detector geometry into account. Thechange from the so-far considered variables in the centre of mass frame ( z , θ ) to the transversemomentum and rapidity of the jet, ( p T , η ), leads to the following Jacobian factor dz d cos θ dp T dη = 4 p T sx x z (A.22) Note that only one of the two diagrams contributes, as only one can be ”collinear”. Consequently also interferenceis negligible. θ = sx x p T z z = p T √ s x e − η + x e η x x . (A.23)Finally the color factors have to be included. For gluon emission it is 1 / T a T a ] = 1 / C F = 4 /
3. Thecross section σ q ¯ q → χ ¯ χ contains the color factor 1 /
3: (1 / for the color average and 3 for the colorsum. Thus the color factor for the full 3body cross section is 4 / x by x x is given by σ q ¯ q → χ ¯ χ + j ( g ) = (cid:88) q (cid:90) dx dx dp T dη ( f q ( x ) f ¯ q ( x ) + f q ( x ) f ¯ q ( x )) dzd cos θ dp T dη σ q ¯ q → χ ¯ χ ( s ) P q → g ( z , θ )(A.24)with the splitting function P q → g ( z , θ ) = 4 α s π − z ) z sin θ . (A.25)This expression is consistent with the expression in Ref. [37]. The factor 2 for the 2 emissions fromthe initial quark and anti-quark lines is already taken into account, because the expression is onlyvalid for θ ∈ (0 , θ max ) for the emission from parton 1 or θ ∈ ( θ max , π ) for the emission from parton2. Each time only one of the 2 diagrams contributes. Extending to the maximum, i.e. θ max = π/ θ , withoutany additional factor of 2.Similarly, the cross section for radiating off a quark-jet is given by σ q ¯ q → χ ¯ χ + j ( q ) = (cid:88) q (cid:90) dx dx dp T dη ( f q ( x ) f g ( x ) + f q ( x ) f g ( x ) + [ q → ¯ q ]) dzd cos θ dp T dη σ q ¯ q → χ ¯ χ ( s ) P g → q ( z , θ ) , (A.26)where the splitting function for a quark-jet with n f different possible quark flavours is P g → q ( z , θ ) = n f α s π z + (1 − z ) sin θ . (A.27) B Convention for Spinors
We explicitly list the helicity spinors used in our calculations to fix the convention of phases. Inthe ultra-relativistic limit and setting the azimuthal angle φ = 0, the helicity spinors take the form u R ( E, θ ) = v L ( E, θ ) = √ E θ i sin θ u L ( E, θ ) = − v R ( E, θ ) = √ E i sin θ cos θ . (B.1)15 K-Matrix Unitarisation of D5
The T -matrix for 2 → T = − π s cos ( θ ) Λ qq s sin ( θ ) Λ qq s cos ( θ ) Λ qχ s sin ( θ ) Λ qχ s Λ qq s Λ qq s sin ( θ ) Λ qq s cos ( θ ) Λ qq s sin ( θ ) Λ qχ s cos ( θ ) Λ qχ s cos ( θ ) Λ qχ s sin ( θ ) Λ qχ s cos ( θ ) Λ χχ s sin ( θ ) Λ χχ s Λ χχ s Λ χχ s sin ( θ ) Λ qχ s cos ( θ ) Λ qχ s sin ( θ ) Λ χχ s cos ( θ ) Λ χχ (C.1)in the basis ( | q L ¯ q R (cid:105) , | q L ¯ q L (cid:105) , | q R ¯ q R (cid:105) , | q R ¯ q L (cid:105) , | χ L ¯ χ R (cid:105) , | χ L ¯ χ L (cid:105) , | χ R ¯ χ R (cid:105) , | χ R ¯ χ L (cid:105) ). The two-particlestates with the same helicity, completely decouple from the other states and can be treated sep-arately. They are pairwise related by parity and they only contribute to the J = 0 term in thepartial wave expansion (cid:10) q L ¯ q L | T | q L ¯ q L (cid:11) = (cid:10) q R ¯ q R | T | q R ¯ q R (cid:11) = − π s Λ qq (C.2) (cid:10) χ L ¯ χ L | T | χ L ¯ χ L (cid:11) = (cid:10) χ R ¯ χ R | T | χ R ¯ χ R (cid:11) = − π s Λ χχ . (C.3)Thus the only non-vanishing elements of the unitarised T -matrix, T , are given by (cid:10) q L ¯ q L | T U | q L ¯ q L (cid:11) = (cid:10) q R ¯ q R | T U | q R ¯ q R (cid:11) = iss − πi Λ qq (C.4) (cid:10) χ L ¯ χ L | T U | χ L ¯ χ L (cid:11) = (cid:10) χ R ¯ χ R | T U | χ R ¯ χ R (cid:11) = iss − πi Λ χχ . (C.5)The remaining states with opposite helicities contribute to the J = 1 term in the partial waveexpansion. The 4 × T -matrix, T , in the basis ( | q L ¯ q R (cid:105) , | q R ¯ q L (cid:105) , | χ L ¯ χ R (cid:105) , | χ R ¯ χ L (cid:105) )is given by T = − π s Λ qq s Λ qq s Λ qχ s Λ qχ s Λ qq s Λ qq s Λ qχ s Λ qχ s Λ qχ s Λ qχ s Λ χχ s Λ χχ s Λ qχ s Λ qχ s Λ χχ s Λ χχ . (C.6)Many of the elements are related by the time-reversal symmetry and parity. [28, 41] There areonly 6 independent matrix elements and we find for the independent elements of the unitarised16 -matrix T U (cid:10) q L ¯ q R | T U | q L ¯ q R (cid:11) = s (cid:0) π Λ qχ r s + ir (cid:0) s − π Λ qχ (cid:1) + 36 π Λ qχ s (cid:1)(cid:0) s − iπ Λ qχ r (cid:1) (cid:0) − iπ Λ qχ r s + r (cid:0) s − π Λ qχ (cid:1) − iπ Λ qχ s (cid:1) (C.7) (cid:10) q L ¯ q R | T U | q R ¯ q L (cid:11) = 12 π Λ qχ r s (cid:0) r s − iπ Λ qχ (cid:1)(cid:0) s − iπ Λ qχ r (cid:1) (cid:0) − iπ Λ qχ r s + r (cid:0) s − π Λ qχ (cid:1) − iπ Λ qχ s (cid:1) (C.8) (cid:10) q L ¯ q R | T U | χ L ¯ χ R (cid:11) = − π Λ qχ r s iπ Λ qχ r s + r (cid:0) π Λ qχ − s (cid:1) + 36 iπ Λ qχ s (C.9) (cid:10) q L ¯ q R | T U | χ R ¯ χ L (cid:11) = (cid:10) q L ¯ q R | T U | χ L ¯ χ R (cid:11) (C.10) (cid:10) χ L ¯ χ R | T U | χ L ¯ χ R (cid:11) = (cid:10) q L ¯ q R | T U | q L ¯ q R (cid:11) (cid:20) r → r (cid:21) (C.11) (cid:10) χ L ¯ χ R | T U | χ R ¯ χ L (cid:11) = (cid:10) q L ¯ q R | T U | q R ¯ q L (cid:11) (cid:20) r → r (cid:21) . (C.12)The fourth equation follows from the interaction being vector-like and the last two equations followfrom the symmetry q ↔ χ . The remaining matrix elements can be obtained from time-reversaland parity symmetry: Time reversal symmetry implies that T U is symmetric, i.e. T U = (cid:0) T U (cid:1) T .Parity conservation implies that matrix elements are invariant under flipping all helicities, i.e. (cid:10) λ (cid:48) λ (cid:48) | T U | λ λ (cid:11) = (cid:10) − λ (cid:48) − λ (cid:48) | T U | − λ − λ (cid:11) . D Effective SM-WIMP Operators
We list the operators coupling the SM to Dirac fermion WIMPs [7] in Tab. 1.Name D1 D2 D3 D4 D5Op. ¯ χχ ¯ qq ¯ χγ χ ¯ qq ¯ χχ ¯ qγ q ¯ χγ χ ¯ qγ q ¯ χγ µ χ ¯ qγ µ q Name D6 D7 D8 D9 D10Op ¯ χγ µ γ χ ¯ qγ µ q ¯ χγ µ χ ¯ qγ µ γ q ¯ χγ µ γ χ ¯ qγ µ γ q ¯ χσ µν χ ¯ qσ µν q ¯ χσ µν γ χ ¯ qσ αβ q Name D11 D12 D13 D14Op ¯ χχG µν G µν ¯ χγ χG µν G µν ¯ χχG µν ˜ G µν ¯ χγ χG µν ˜ G µν Table 1:
Operators coupling SM to WIMPs first shown in Ref. [7].
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