OOn the Validity of the Effective Field Theory for DarkMatter Searches at the LHC
Giorgio Busoni ∗ SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136 Trieste, ItalyE-mail: [email protected]
Andrea De Simone
SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136 Trieste, ItalyE-mail: [email protected]
Johanna Gramling
Section de Physique, Université de Genève,24 quai E. Ansermet, CH-1211 Geneva, SwitzerlandE-mail: [email protected]
Enrico Morgante
Section de Physique, Université de Genève,24 quai E. Ansermet, CH-1211 Geneva, SwitzerlandE-mail: [email protected]
Antonio Riotto
Section de Physique, Université de Genève,24 quai E. Ansermet, CH-1211 Geneva, SwitzerlandE-mail: [email protected]
We generalize in several directions our recent analysis of the limitations to the use of the effectivefield theory approach to study dark matter at the LHC. Firstly, we study the full list of operatorsconnecting fermion DM to quarks and gluons, corresponding to integrating out a heavy mediatorin the s -channel; secondly, we provide analytical results for the validity of the EFT descriptionfor both √ s = XXII. International Workshop on Deep-Inelastic Scattering and Related Subjects,28 April - 2 May 2014Warsaw, Poland ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - ph ] N ov FT for Dark Matter Searched at LHC
Giorgio Busoni
1. Introduction
While there are many cosmological and astrophysical evidences that our universe contains asizable amount of dark Matter (DM), i.e. a component which clusters at small scales, its nature isstill a mystery. Currently, there are several ways to search for such DM candidates. DM particles(if they are light enough) might reveal themselves in particle colliders, namely at the LHC. ManyLHC searches for DM are based on the idea of looking at events with missing energy plus a singlejet or photon, emitted from the initial state in pp collisions pp → χ + χ + jet , (1.1)where χ indicates the DM particle. Several results are already available from two LHC collabora-tions [2–4]. In order to avoid the overwhelming model-dependence introduced by the plethora ofDM models discussed in the literature, DM searches at the LHC have made use of the EffectiveField Theory (EFT) [5, 6]. However, as far as collider searches are concerned, with the LHC beingsuch a powerful machine, it is not guaranteed that the events used to constrain an effective inter-action are not occurring at an energy scale larger than the cutoff scale of the effective description.The question about the validity of the EFT for collider searches of DM has become pressing (seealso Refs. [6, 7]), 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 quarksand DM are coupled with couplings g q and g χ , respectively. The EFT is a good approximationonly at low energies. Indeed, it is possible at low energies to integrate out the heavy mediatorfrom 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 DMproduction 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 thecouplings g q , χ < π , implies Λ > Q tr √ g q g χ > Q tr π . (1.2)It is clear that the details of condition (1.2) 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 UVtheory of O ( ) . So, we will deal with the condition (but we will discuss also the impact of takingcouplings larger than 1) Q tr (cid:46) Λ . (1.3)In Ref. [7] we have started the discussion of the limitations to the use of the EFT approach forDM 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 effectiveoperators to describe processes with very high momentum transfer. Our criteria indicated up towhat cutoff energy scale, and with what precision, the effective description is valid, depending onthe DM mass and couplings. 2 FT for Dark Matter Searched at LHC
Giorgio Busoni
2. Validity of the EFT: analytical approach
The starting point of our analysis is the list of the 18 operators listed in [1] which are commonlyused in the literature [6]. We have considered not only the operators connecting the DM fermionto quarks (D1-D10), but also those involving gluon field strengths (D11-D14). Furthermore, theoperators 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 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 initialstate f + ¯ f → χ + ¯ χ + g , where f is either a quark (for operators D1-D10), or a gluon (for operatorsD11-D14).In order to get the cross sections initiated by the colliding protons one needs to average overthe PDFs. We have performed the analytical calculation only for the emission of an initial stategluon (identified with the final jet observed experimentally). The extension to include also thesmaller contribution coming from initial radiation of quarks ( qg → χ χ + q ) is done numerically inSection 3. In what regions of the parameter space ( Λ , m DM ) is the effective description accurate andreliable? The truncation to the lowest-dimensional operator of the EFT expansion is accurate onlyif the momentum transfer is smaller than an energy scale of the order of Λ , see Eqs.(1.3). Thereforewe 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. R tot Λ ≡ σ | Q tr < Λ σ = (cid:82) p maxT p minT d p T (cid:82) − d η d σ d p T d η (cid:12)(cid:12)(cid:12)(cid:12) Q tr < Λ (cid:82) p maxT p minT d p T (cid:82) − d η d σ d p T d η . (2.1)The results are shown in Fig.1. We show only results for representative operators D (cid:48) , D , D
9. Thisratio R tot Λ gets closer to unity for large values of Λ , as in this case the effect of the cutoff becomesnegligible. The ratio drops for large m DM because the momentum transfer increases in this regime.This confirms our precedent analysis of Ref. [7], that the EFT works better for large Λ and small m DM . Notice also that, going from √ s = √ s = 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 Fig.2 for √ s = Λ requires such a cutoff scale to be above ∼ √ s = ∼ √ s =
14 TeV.To close this section let us comment on another question one may ask: what is the differencebetween interpreting data with an effective operator and with its simplest UV completion? This3
FT for Dark Matter Searched at LHC
Giorgio Busoni (cid:76) (cid:64)
TeV (cid:68) R (cid:76) t o t D1' SolidD5 DashedD9 Dotted s (cid:61) (cid:163) p T (cid:163) (cid:200) Η (cid:200) (cid:163) m DM (cid:61)
10 GeV m DM (cid:61)
500 GeV m DM (cid:61) (cid:76) (cid:64) TeV (cid:68) R (cid:76) t o t D1' SolidD5 DashedD9 Dotted s (cid:61)
14 TeV500 GeV (cid:163) p T (cid:163) (cid:200) Η (cid:200) (cid:163) m DM (cid:61)
10 GeV m DM (cid:61) m DM (cid:61)
10 10 m DM (cid:64) GeV (cid:68) R (cid:76) t o t D1' Solid , D5 Dashed , D9 Dotted s (cid:61) (cid:163) p T (cid:163) (cid:200) Η (cid:200) (cid:163) (cid:76)(cid:61) (cid:76)(cid:61)
10 10 m DM (cid:64) GeV (cid:68) R (cid:76) t o t D1' Solid , D5 Dashed , D9 Dotted s (cid:61)
14 TeV500 GeV (cid:163) p T (cid:163) (cid:200) Η (cid:200) (cid:163) (cid:76)(cid:61) (cid:76)(cid:61) Figure 1:
The ratio R tot Λ defined in Eq.(2.1) for operators D (cid:48) (solid lines), D (dashed lines) and D (dottedlines) as a function of Λ and m DM , for √ s = TeV (left panel) and TeV (right panel). m DM (cid:64) GeV (cid:68) (cid:76) (cid:64) G e V (cid:68) D5 s (cid:61) (cid:163) p T (cid:163) (cid:200) Η (cid:200) (cid:163) R (cid:76) tot (cid:61) (cid:37) R (cid:76) tot (cid:61) (cid:37) R (cid:76) tot (cid:61) (cid:37) R (cid:76) tot (cid:61) (cid:37) (cid:76) (cid:60) m DM m DM (cid:64) GeV (cid:68) (cid:76) (cid:64) G e V (cid:68) D11 s (cid:61) (cid:163) p T (cid:163) (cid:200) Η (cid:200) (cid:163) R (cid:76) tot (cid:61) (cid:37) R (cid:76) tot (cid:61) (cid:37) R (cid:76) tot (cid:61) (cid:37) R (cid:76) tot (cid:61) (cid:37) (cid:76) (cid:60) m DM Figure 2:
Contours for the ratio R tot Λ , defined in Eq.(2.1), on the plane ( m DM , Λ ) , for the different operators.We set √ s = , | η | ≤ and
500 GeV < p T < . question has already been addressed in Ref. [7] for the operator D (cid:48) , by studying the ratio ofthe cross sections obtained with the UV theory and with the effective operator. For each of theoperators listed in [1] one can write a simple UV-complete Lagrangian. The very same analysiscan be repeated for all the other operators and we checked that the same qualitative conclusionscan be drawn. In particular, if Λ is not larger than a few TeV, interpreting the experimental data interms of EFT or in terms of a simplified model with a mediator can make a significant difference.4 FT for Dark Matter Searched at LHC
Giorgio Busoni (GeV) DM m ( G e V ) L DM < 2m L (GeV) DM m ( G e V ) L h g, 500 GeV, | |<2 h j, 500 GeV, | |<2 h j, 350 GeV, | |<2 h j(j), 350 GeV, | |<4.5 h j(j), 350 GeV, | |<4.5 h j(j), 500 GeV, | DM < 2m L Figure 3: (Left Panel)Comparison of the contour R tot Λ = for the analytical calculation (dashed line)and the simulation (solid line) for the different operators D (cid:48) , D (cid:48) , D , D and D . The results agree withinless than 7 %. (Right Panel) The changes of the contour of R tot Λ = are shown for several variationsfrom the analytically calculated scenario to a scenario close to the cuts used in the ATLAS monojet analysisexemplarily for the operator D at √ s = TeV. In the legend, “g” means only gluon radiation, “j” standsfor either quark- or gluon-initiated jets, “j(j)” means a second jet is allowed.
3. Comparison with MonteCarlo Simulations
In order to perform an alternative check of our analytical results and to be able to compareto the experimental limits as close as possible, we present in this section the results of numericalevent simulations.We made use of M AD G RAPH pp collisions at √ s = √ s =
14 TeV.For details about this procedure, see [1]. According to the event kinematics we have evaluatedwhether or not the conditions of validity discussed in Section 2 are fulfilled. Specifically, we havechecked if Eqs.(1.2) are fulfilled, that is, if the following condition is satisfied Λ > Q tr √ g q g χ > m DM √ g q g χ . (3.1)From the simulated samples the fraction of events fulfilling Λ > Q tr / √ g q g χ for each pair ofDM mass and cutoff scale can be evaluated, if one assumes a certain value for the couplings √ g χ g q .In order to confirm that analytical and numerical results are in agreement, Figure 3 shows acomparison for the operators D (cid:48) , D (cid:48) , D D D
9. The contours of R tot Λ =
50% from analyticaland numerical evaluation agree within less than 7 %. The remaining differences could be due to theupper jet p T cut not imposed during event simulation but needed for the analytical calculation, andthe details of the fitting procedures. Next, we vary the kinematical constraints step by step from thescenario considered in the analytical calculations, to a scenario closest to the analysis cuts appliedin the ATLAS monojet analysis [4]. The effect of the variation of the cuts can be seen in Figure 3.Moving to the scenario closer to the experimental analysis leads to contours that are at most ∼ Λ . After having extracted R tot Λ for each WIMP and mediator mass, a curve can be fittedthrough the points obtained in the plane of R tot Λ and Λ . See [1] for more details.5 FT for Dark Matter Searched at LHC
Giorgio Busoni (GeV) DM m ( G e V ) L L R = 75% Lp R = 50% L R = 50% Lp R = 25% L R = 25% Lp R DM < 2m L p DM
2m < L (GeV) DM m ( G e V ) L L R = 75% Lp R = 50% L R = 50% Lp R = 25% L R = 25% Lp R DM < 2m L p DM
2m < L Figure 4: tot Λ , compared to the experimental limits from AT-LAS [4] (blue line). Also indicated are the contours of R tot Λ in the extreme case when setting the couplings √ g q g χ = π (dashed lines). Results are shown for different operators: D5 (left panel) and D11 (right panel).
4. Implications of the limited validity of EFT in DM searches at LHC
Figure 4 shows the experimental limits obtained from the ATLAS monojet analysis [4] in theplane ( Λ , m DM ), for the opearators D5 and D11. The contours of R tot Λ for 25%, 50% and 75% aresuperimposed. The experimental limits are placed in a region where only about 30% of the eventscan be expected to fulfill the EFT conditions. Especially the limit on the gluon operator D
11 seemsquestionnable. For comparison, dashed lines show the contours of R tot Λ for the case of √ g q g χ = π ,presenting the limiting case for which the theory is still considered perturbative.In Fig.5 we show the new limits for the operators D5 and D11, for the conditions Q tr < Λ , Λ , π Λ , corresponding different choices of the UV couplings: √ g q g χ = , , π , respectively.The ATLAS bound reported is the 90%CL observed limit. The functions R tot Λ used are taken fromthe fitting functions described in [1], which include both quark and gluon jets, and the same cutsas the “Signal Region 3” used by ATLAS. As expected, the weaker is the condition on Q tr , themore the new limits approach the ATLAS bound. In the case of extreme couplings √ g q g χ = π ,while for D5 the new limit is indistinguishable from the ATLAS one, for D11 the bound at largeDM masses still need to be corrected. In general, for couplings of order one, the limits which aresafe from the EFT point of view are appreciably weaker than those reported. We encourage theexperimental collaborations to take this point into account when publishing their limits.
5. Conclusions
Following Ref. [7], we have studied the quantity R tot Λ (see Eq.(2.1), which quantifies the errormade when using EFT to describe processes with very high momentum transfer. Our criterionindicates up to what cutoff energy scale the EFT is valid, depending on the DM mass and cou-plings. We have performed the analysis for the full list of EFT operators, connecting fermion DMparticles and quarks or gluons, used by the ATLAS and CMS collaborations and originated fromthe exchange of heavy mediators in the s -channel. We have also extended our analysis to the caseof √ s =
14 TeV. Furthermore, we have validated our analytical results by performing numerical6
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Giorgio Busoni
10 20 50 100 200 500 10002004006008001000 m DM (cid:64) GeV (cid:68) (cid:76) (cid:64) G e V (cid:68) ATLAS limit D5 Q tr (cid:60) Π (cid:76) Q tr (cid:60) (cid:76) Q tr (cid:60)(cid:76) (cid:76) (cid:60) m D M (cid:76) (cid:60) m D M (cid:76) (cid:60) m D M (cid:144) (cid:72) Π (cid:76)
10 20 50 100 200 500 10000100200300400500 m DM (cid:64) GeV (cid:68) (cid:76) (cid:64) G e V (cid:68) ATLAS limit D11 Q tr (cid:60) Π (cid:76) Q tr (cid:60) (cid:76) Q tr (cid:60)(cid:76) (cid:76) (cid:60) m D M (cid:76) (cid:60) m D M (cid:76) (cid:60) m D M (cid:144) (cid:72) Π (cid:76) Figure 5:
The experimental limits by ATLAS [4] on the suppression scale Λ are shown as solid blue lines.The updated limits taking into account EFT validity are shown as dashed black lines, for Q tr < Λ , Λ , π Λ ,corresponding to different choices of the UV couplings: √ g q g χ = , , π , respectively. The correspondingkinematical constraints (Eq.(3.1)) are denoted by gray bands. The different plots refer to different operators:D5 (left panel) and D11 (right panel). event simulations which reproduce the experimental situation in the closest possible way. Our re-sults 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 [9] 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 ofthe EFT, most possibly through identifying a handful of classes of models (able to reproduce theEFT operators in the heavy mediator limit); this would allow a consistent analysis of the currentand future LHC data by consistently taking into account the role played by the mediator. On thepositive side, keep working with the EFT allows to avoid the overwhelming model-dependencegenerated 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. References [1] G. Busoni, A. De Simone, J. Gramling, E. Morgante and A. Riotto, [arXiv:1402.1275] [hep-ph].[2] G. Aad et al. [ATLAS Collaboration], JHEP , 075 (2013) [arXiv:1210.4491].[3] S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. Lett. , 201804 (2011) [arXiv:1106.4775].[4] ATLAS-CONF-2012-147.[5] Y. Bai, P. J. Fox and R. Harnik, JHEP , 048 (2010) [arXiv:1005.3797].[6] J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and H. -B. Yu, Phys. Rev. D ,116010 (2010) [arXiv:1008.1783].[7] G. Busoni, A. De Simone, E. Morgante and A. Riotto, Physics Letters B 728C (2014)[arXiv:1307.2253].[8] J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, T. Stelzer, JHEP 1106(2011)128 [arXiv:1106.0522].[9] G. Busoni, A. De Simone, T. Jacques, E. Morgante and A. Riotto, [arXiv:1405.3101] [hep-ph].,116010 (2010) [arXiv:1008.1783].[7] G. Busoni, A. De Simone, E. Morgante and A. Riotto, Physics Letters B 728C (2014)[arXiv:1307.2253].[8] J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, T. Stelzer, JHEP 1106(2011)128 [arXiv:1106.0522].[9] G. Busoni, A. De Simone, T. Jacques, E. Morgante and A. Riotto, [arXiv:1405.3101] [hep-ph].