On the validity of the effective field theory for dark matter searches at the LHC, part II: complete analysis for the s-channel

We generalize in several directions our recent analysis of the limitations to the use of the effective field theory approach to study dark matter at the LHC. Firstly, we study the full list of operators connecting fermion DM to quarks and gluons, corresponding to integrating out a heavy mediator in the s-channel; secondly, we provide analytical results for the validity of the EFT description for both √s = 8 TeV and 14 TeV; thirdly, we make use of a MonteCarlo event generator approach to assess the validity of our analytical conclusions. We apply our results to revisit the current collider bounds on the ultraviolet cut-off scale of the effective field theory and show that these bounds are weakened once the validity conditions of the effective field theory are imposed.


Introduction
While there are many cosmological and astrophysical evidences that our universe contains a sizable amount of dark Matter (DM), i.e. a component which clusters at small scales, its nature is still a mystery. Various considerations point towards the possibility that DM is made of neutral particles whose mass and interactions are dictated by physics in the electroweak energy range. If so, the DM relic density of these particles, assuming they were in thermal equilibrium during the evolution of the universe, turns out to be Ω DM h 2 0.110 ≈ 3 × 10 −26 cm 3 /sec σv ann , (1.1) where σv ann is the (thermally-averaged) annihilation cross section. A weak interaction strength provides the abundance in the right range measured by the Planck collaboration: Ω DM = 0.315 ± 0.0175 [1]. This numerical coincidence represents the main reason why it is generically believed that DM is made of weakly-interacting particles with a mass in the range (10 2 − 10 4 ) GeV.
Currently, there are several ways to search for such DM candidates. Apart from the indirect [2] and direct [3] searches, DM particles (if they are light enough) might reveal themselves in particle colliders, namely at the LHC. Many LHC searches for DM are based on the idea of looking at events with missing energy plus a single jet or photon, emitted from the initial state in pp collisions (for alternative kinds of DM searches at the LHC see e.g. refs. [4][5][6][7][8][9]) pp → χ + χ + jet/photon, (1.2) where χ indicates the DM particle. Several results are already available from two LHC collaborations [10][11][12][13][14][15][16][17].

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In order to avoid the overwhelming model-dependence introduced by the plethora of DM models discussed in the literature, DM searches at the LHC have made use of the Effective Field Theory (EFT) [18][19][20][21][22][23][24][25][26][27][28]. This approach is a very powerful and economical way to grasp the main features of a physical process, only in terms of the degrees of freedom which are excited at the scale of the process. EFT techniques are successfully applied in many branches of physics, and in particular they have become a standard way to present experimental results for DM searches.
However, as far as collider searches are concerned, with the LHC being such a powerful machine, it is not guaranteed that the events used to constrain an effective interaction are not occurring at an energy scale larger than the cutoff scale of the effective description. In other words, some (or many) events of DM production may occur with such a high momentum transfer that the EFT is not a good description anymore. The question about the validity of the EFT for collider searches of DM has become pressing (see also refs. [7,21,[29][30][31][32][33][34][35]), especially in the perspective of analysing the data from the future LHC run at (13)(14) TeV.
Let us consider a simple model where there is a heavy mediator of mass M , to which the quarks and DM are coupled with couplings g q and g χ , respectively. The EFT is a good approximation only at low energies. Indeed, it is possible at low energies to integrate out the heavy mediator from the theory and obtain a tower of operators. The matching condition of the ultra-violet (UV) theory with the mediator and its low-energy effective counterpart implies Λ = M/ √ g q g χ . A DM production event occurs at an energy at which the EFT is reliable as long as Q tr < M , where Q tr is the momentum transfer in the process; this, together with the condition of perturbativity of the couplings g q,χ < 4π, implies If, in addition, one assumes the momentum transfer to occur in the s-channel, then kinematics imposes Q tr > 2m DM , so eq. (1.3) becomes This is a very minimal requirement which is refined event-by-event by the stronger condition eq. (1.3), which depends on m DM through Q tr . It is clear that the details of condition (1.3) depend on the values of the couplings in the UV theory. In the following, for definiteness, we will mostly identify the mass of the new degrees of freedom M with the suppression scale of the operator Λ. This is equivalent to consider couplings in the UV theory of O(1). So, we will deal with the condition (but we will discuss also the impact of taking couplings larger than 1) In ref. [29] we have started the discussion of the limitations to the use of the EFT approach for DM searches at the LHC by adopting a toy model where the heavy mediator is exchanged in the s-channel and by introducing a few quantities which quantify the error made when using effective operators to describe processes with very high momentum transfer. Our criteria indicated up to what cutoff energy scale, and with what precision, the effective description is valid, depending on the DM mass and couplings. In this paper we significantly extend our previous work along four different directions: 1. we consider the full list of operators connecting fermion DM to quarks and corresponding to integrating out the heavy mediator in the s-channel; we provide analytical results for the validity of the EFT description for both  √  s = 8  TeV and 14 TeV;   3. we follow a MonteCarlo approach to assess the validity of the EFT and compare this  fully numerical results with the analytical calculations; 4. we apply our results to revisit the current experimental bounds on the effective operator scale; by requiring that only the events which are "safe" from the EFT point of view should be considered, the bounds get weakened.
The rest of the present paper is organized as follows. In section 2 we present and discuss the results of our analytical approach to assess the validity of EFT. In section 3, the fully numerical approach is described and the results are compared with the analytical calculations.
In section 4 we analyze the impact of the limitation of the validity of the EFT for the current limits from the LHC searches. Finally, we draw our conclusions in section 5. The details of the analytical results can be found in the appendix A.
2 Validity of the EFT: analytical approach

Operators and cross sections
The starting point of our analysis is the list of the 18 operators reported in table 1 which are commonly used in the literature [18]. We have considered not only the operators connecting the DM fermion to quarks (D1-D10), but also those involving gluon field strengths (D11-D14). Furthermore, the operators can originate from heavy mediators exchange in the s-channel. For instance, the D1' (D5) operators may be originated by the tree-level s-channel exchange of a very heavy scalar (vector) boson S (V µ ), with lagrangians Notice the presence of the "primed" operators D1 -D4 , very similar to the ones often considered D1-D4, respectively, but with a different normalization, independent of the quark masses. In fact, they may arise from integrating out heavy scalars which do not take a vacuum expectation value and therefore do not give rise to quark masses. 1 We have computed the tree-level differential cross sections in the transverse momentum p T and rapidity η of the final jet for the hard scattering process with gluon radiation from the initial state f (p 1 ) +f (p 2 ) → χ(p 3 ) + χ(p 4 ) + g(k), where f is either a quark (for operators where Λ is an energy scale of the order some TeV related to the Electroweak Symmetry Breaking and m t,b , λ t,b are the mass and the Yukawa coupling with the Higgs of the top/bottom quark, depending on whether the quark q is up-like or down-like. In the present work, we will be agnostic about this point, and we'll keep both the primed and unprimed operators into account on the same footing as all others.

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Name Operator Coefficient D1χχqq Table 1. Operators used throughout this work. The nomenclature is mostly taken from ref. [21]. D1-D10), or a gluon (for operators D11-D14). The results are conveniently written in terms of the momentum transfer in the s-channel where x 1 , x 2 are the fractions of momentum carried by initial partons and η, p T are the pseudo-rapidity and the transverse momentum of the final state gluon, respectively. The expressions are of course valid for all admitted values of the parameters. It's only when integrated numerically over the PDFs and over η, p T that the dependence on these values comes in. We obtain The reader can find the details of the derivation of eqs. (2.5)-(2.13) in appendix A. As for the other operators, we get in the limit of massless light quarks. The operators D1-D4 are simply related to D1 -D4 by a straightforward rescaling We checked that the differences between the cross sections for D1 -D4 computed for m q = 0 and those reported above assuming m q = 0 are at the per-mille level, so the approximation m q = 0 which we used in all our analytical calculations is justified. The cross sections for the UV completions of dim-6 operators, with s-channel exchange of a mediator of mass M med , are simply obtained by the replacement 1/Λ 4 → g 2 -5 -

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In order to get the cross sections initiated by the colliding protons one needs to average over the PDFs. For example, for processes with initial state quarks We have performed the analytical calculation only for the emission of an initial state gluon (identified with the final jet observed experimentally). The extension to include also the smaller contribution coming from initial radiation of quarks (qg → χχ+q) is done numerically in section 3.

Results and discussion
In what regions of the parameter space (Λ, m DM ) is the effective description accurate and reliable? The truncation to the lowest-dimensional operator of the EFT expansion is accurate only if the momentum transfer is smaller than an energy scale of the order of Λ, see eqs. (1.5). Therefore we want to compute the fraction of events with momentum transfer lower than the EFT cutoff scale. To this end we define the ratio of the cross section obtained in the EFT with the requirement Q tr < Λ on the PDF integration domain, over the total cross section obtained in the EFT. (2.18) To sum over the possible p T , η of the jets, we integrate the differential cross sections over values typically considered in the experimental searches. We consider p min T = 500 GeV (as used in the signal region SR4 of [12]), |η| < 2 and the two cases with center-of-mass energies √ s = 8 TeV and 14 TeV. For p max T we used 1, 2 TeV for √ s = 8, 14 TeV, respectively. The sum over quark flavours is performed only considering u, d, c, s quarks.
We first study the behavior of the ratio R tot Λ , as a function of Λ and m DM The results are shown in figure 1. We show only results for representative operators D1 , D5, D9. This ratio R tot Λ gets closer to unity for large values of Λ, as in this case the effect of the cutoff becomes negligible. The ratio drops for large m DM because the momentum transfer increases in this regime. This confirms our precedent analysis of ref. [29], that the EFT works better for large Λ and small m DM . Notice also that, going from √ s = 8TeV to √ s = 14TeV, the results scale almost linearly with the energy, so for the same value of the ratio m DM /Λ one obtains nearly the same R tot Λ . Next, we turn to study the contours of constant values of the quantity R tot Λ , in the plane (m DM , Λ). These contour curves for the different operators are shown in figure 2 for √ s = 8 TeV and in figure 3 for √ s = 14 TeV. The requirement that at least 50% of the events occur with momentum transfer below the cutoff scale Λ requires such a cutoff scale to be above ∼ 1TeV for √ s = 8 TeV, or above ∼ 2TeV for √ s = 14 TeV. Note also that the contours for D1-D4 differ by the corresponding contours for D1 -D4 by O(1) factors, due to the different weighting of the quarks' PDFs. On the other hand, the experimental bounds on the scale of the operators D1-D4 are much lower (of the order of tens of GeV), as such operators experience an additional suppression of m q /Λ. This means that the bounds on D1-D4 are not reliable from the point of view of EFT validity.
We stress once again that the precise definition of a cutoff scale for an EFT is only possible when the details of the UV completion are known. The most conservative regime is  when the couplings of the UV theory reach their maximal values allowed by perturbativity. In such a situation, the requirement on the momentum transfer becomes Q tr < 4πΛ. We show the effect of varying the cutoff scale in figure 4, for the representative contour R tot Λ = 50% of D5. As it should be clear, the variation of the cutoff scale is equivalent to a change of the unknown couplings of the UV theory. All the operators have very similar results, as the contours scale linearly with the cutoff. As a comparison, we show as a shaded area the region Λ > m DM /(2π) often used as a benchmark for the validity of the EFT (see eq. (1.4)). The 50% contour is above such a region, meaning that the parameter space regions of validity of the effective operator approach is smaller than commonly considered.
To close this section let us comment on another question one may ask: what is the difference between interpreting data with an effective operator and with its simplest UV completion? This question has already been addressed in ref. [29] for the operator D1 , by studying the ratio of the cross sections obtained with the UV theory and with the effective operator. For each of the operators in table 1 one can write a simple UV-complete Lagrangian, see e.g. eqs. (2.1)-(2.2). The very same analysis can be repeated for all the other operators and we checked that the same qualitative conclusions can be drawn. In particular, if Λ is not larger than a few TeV, interpreting the experimental data in terms of EFT or in terms of a simplified model with a mediator can make a significant difference.

Comparison with MonteCarlo simulations
In order to perform an alternative check of our analytical results and to be able to compare to the experimental limits as close as possible, we present in this section the results of numerical event simulations.

Simulation and analysis description
We made use of MadGraph 5 [38] to simulate pp collisions at √ s = 8 TeV and √ s = 14 TeV. Both PDF sets CTEQ6L1 and MSTW2008LO (discussed in refs. [39,40]) are employed. The PDF choice affects the cross section, but only minimally the acceptance. Hence, the change in contours of R tot Λ is negligible. Since MSTW2008LO is used for the analytical calculations, this set is also used where direct comparisons between simulation and calculation are shown. For the comparison to the experimental results, CTEQ6L1 is used instead. Only u, d, c, s quarks were considered, both in the initial and in the final state.
According to the event kinematics we have evaluated whether or not the conditions of validity discussed in section 2 are fulfilled. Specifically, we have checked if eqs. (1.3) and (1.4) are fulfilled, that is, if the following condition is satisfied From the simulated samples the fraction of events fulfilling Λ > Q tr / √ g q g χ for each pair of DM mass and cutoff scale can be evaluated, if one assumes a certain value for the couplings √ g χ g q connecting the cutoff scale Λ and the mediator mass M via Λ = M/ √ g q g χ .
As above, g q g χ was assumed to be 1.

Results
In order to confirm that analytical and numerical results are in agreement, figure 5 shows a comparison for the operators D1 , D4 , D5, D8 and D9. The results were obtained for the scenario of one radiated gluon jet above 500 GeV within |η| < 2. The contours of R tot Λ = 50% from analytical and numerical evaluation agree within less than 7 %. The remaining differences could be due to the upper jet p T cut not imposed during event simulation but needed for the analytical calculation, and the details of the fitting procedures.   Figure 6. The changes of the contour of R tot Λ = 50% are shown for several variations from the analytically calculated scenario to a scenario close to the cuts used in the ATLAS monojet analysis exemplarily for the operator D5 at √ s = 8 TeV. In the legend, "g" means only gluon radiation, "j" stands for either quark-or gluon-initiated jets, "j(j)" means a second jet is allowed.
Next, we vary the kinematical constraints step by step from the scenario considered in the analytical calculations, namely one radiated gluon jet above 500 GeV within |η| < 2, to a scenario closest to the analysis cuts applied in the ATLAS monojet analysis [12]. More specifically, the leading jet is allowed to come from either a gluon or a quark being radiated, the leading jet p T cut is changed from 500 GeV to 350 GeV, a second jet is allowed and its range in η is enlarged to |η| < 4.5. No further cuts are applied at simulation level.
The effect of the variation of the cuts can be seen in figure 6. Allowing not only for a gluon jet but also taking into account the possibility of a quark jet changes the R tot Λ contours appreciably. The change from lowering the p T of the leading jet has a smaller effect. Allowing for a second jet and enhancing its rapidity range barely changes the R tot Λ contour, especially at large m DM values.   Table 2. Coefficient for the fitting functions for R tot Λ in eq. (3.2), in the cases √ s = 8 and 14 TeV. The fitting functions describe processes where quarks and/or gluons are radiated, the final state contains 1 or 2 jets, where the leading jet has minimum p T of 350 GeV while the second jet is allowed to be within |η| < 4.5. See text for further details.
If the collision energy is augmented to √ s = 14 TeV, all the R tot Λ contours increase. As seen for √ s = 8 TeV, moving to the scenario closer to the experimental analysis leads to contours that are at most ∼ 30% lower in Λ.
After having extracted R tot Λ for each WIMP and mediator mass, a curve can be fitted through the points obtained in the plane of R tot Λ and Λ. The following functional form is used for this purpose Further, the parameters are fitted for each DM mass separately. From these fits, the points denoting a cutoff scale where R tot Λ equals e.g. 50% can be extracted for each DM mass, and the lines of constant R tot Λ can be plotted in the usual limit-setting plane Λ vs. m DM . Table 2 collects the values of the fitting parameters for all operators except D12-D14, for which no experimental analysis exists. Figure 7 shows the experimental limits obtained from the ATLAS monojet analysis [12] in the plane (Λ, m DM ), for the opearators D5, D8 and D11. The contours of R tot Λ for 25%, 50% and 75% are superimposed. The experimental limits are placed in a region where about 30% of the events can be expected to fulfill the EFT conditions -the exact number depends on the operator considered. Especially the limit on the gluon operator D11 seems questionnable. For comparison, dashed lines show the contours of R tot Λ for the extreme case of couplings √ g q g χ = 4π, presenting the limiting case for which the theory is still considered perturbative.

Implications of the limited validity of EFT in DM searches at LHC
Unfortuntately, there is no possibility to measure Q tr in data, on an event-by-event basis. So the information on what is the fraction of the events to cut out comes from analytical computations or a numerical simulation, as we explained in this paper. To assess the impact of the limited validity of the EFT on the current collider bounds, we adopt the procedure that relies on the assumption that the p T (or MET) distributions with the Q tr cut are simply a rescaling of those without the cut. A more refined study should account for  Now, if we do not consider any information about the shapes of the p T or MET distributions, the experimental bound only comes from the total number of events passing given cuts. The fact that a fraction of the events involve a transfer momentum exceeding the cutoff scale of the EFT means that the number of signal events for placing a limit gets reduced by a factor R tot Λ . Therefore, actually N signal (Λ, m DM ) → R tot Λ (m DM )N signal (Λ, m DM ), so the new limit is found by solving the implicit equation and it turns out to be weaker than Λ expt . In figure 8 we show the new limits for the dim-6 operators D5, D8 and the dim-7 operator D11, for the conditions Q tr < Λ, 2Λ, 4πΛ, corresponding different choices of the UV couplings: √ g q g χ = 1, 2, 4π, respectively. The curves are obtained solving eq. (4.2) with R tot Λ , R tot 2Λ , R tot 4πΛ respectively. The ATLAS bound reported is the 90%CL observed limit. The functions R tot Λ used are taken from the fitting functions described in table 2, which include both quark and gluon jets, and the same cuts as the "Signal Region 3" used by ATLAS. As expected, the weaker is the condition on Q tr , the more the new limits approach the ATLAS bound. In the case of extreme couplings √ g q g χ = 4π, the condition on the momentum transfer is very conservative Q tr < 4πΛ. For D5 and D8, the new limit is indisinguishable from the ATLAS one, meaning that the experimental results are safe from the EFT point of view, in this limiting situation. For D11, even for extreme values of the couplings, the bound at large DM masses must be corrected. In general, for couplings of order one, the limits which are safe from the EFT point of view are appreciably weaker than those reported. We encourage the experimental collaborations to take this point into account when publishing their limits.

Conclusions
The search for DM is one of the main targets of LHC analyses. In this paper we have continued our previous investigation to assess the validity of the EFT commonly used in interpreting -14 -

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such searches. Following ref. [29], we have studied the quantity R tot Λ (see eq. (2.18), which quantifies the error made when using effective operators to describe processes with very high momentum transfer. Our criterion indicates up to what cutoff energy scale the effective description is valid, depending on the DM mass and couplings. We have performed the analysis for the full list of EFT operators, connecting fermion DM particles and quarks or gluons, used by the ATLAS and CMS collaborations and originated from the exchange of heavy mediators in the s-channel. We have also extended our analysis to the case of √ s = 14 TeV. Furthermore, we have validated our analytical results by performing numerical event simulations which reproduce the experimental situation in the closest possible way. Our results indicate that the range of validity of the EFT is significantly limited in the parameter space (Λ, m DM ). While our findings are valid for the s-channel, a similar analysis is under way for the t-channel [42] where similar results are obtained.
Does it mean that the EFT is not the best tool to interpret the current LHC data of DM searches? The answer is yes and no. On the negative side, our results clearly cry out for an overcoming of the EFT, most possibly through identifying a handful of classes of models (able to reproduce the EFT operators in the heavy mediator limit); this would allow a consistent analysis of the current and future LHC data by consistently taking into account the role played by the mediator. On the positive side, keep working with the EFT allows to avoid the overwhelming model-dependence generated by the many DM models proposed so far. Nonetheless, as we have shown in section 4, the price to pay is a deterioration of the limits presented so far.

A.1 Generalities
In this appendix we show the details of the calculations of the tree-level cross sections for the hard scattering process f (p 1 ) +f (p 2 ) → χ(p 3 ) + χ(p 4 ) + g(k), where f is either a quark (operators D1-D10) or a gluon (D11-D14), and the final gluon is emitted from the initial state.