JCAP07(2016)005 2 to a √ / ) hysics ia P + decays into two s s le ic a t ar . s doi:10.1088/1475-7516/2016/07/005 strop A Yann Mambrini a,b , also responsible for generating its mass, allows for the χ , each of which subsequently decays into a pair of collimated a . Content from this work may be used a 3 osmology and and osmology Pradipta Ghosh, C a 1603.05601 dark matter theory, particle physics - cosmology connection We investigate a simple setup in which an excess in the di-photon invariant mass rnal of rnal Article funded byunder SCOAP the terms of the Creative Commons Attribution 3.0 License . ou An IOP and SISSA journal An IOP and Laboratoire de Physique Th´eorique,CNRS, Univ.Universit´eParis-Saclay, 91405 Orsay, Paris-Sud, France Centre de Physique Th´eorique,Ecole polytechnique,Universit´eParis-Saclay, 91128 CNRS, Palaiseau Cedex, France E-mail: [email protected], , [email protected] , [email protected] [email protected] b a Any further distribution ofand this the work must title maintain of attribution the to work, the author(s) journal citation and DOI. fermionic dark matter candidate correct relic density in adirect large or region indirect of detection the experiments.co-exist parameter Moreover, space, with the while a correct not relatively relic being large abundance excluded can width by naturally for the the resonant field Keywords: ArXiv ePrint: Giorgio Arcadi, Re-opening dark matter windows compatible with a diphoton excess Received April 5, 2016 Revised June 2, 2016 Accepted June 17, 2016 Published July 5, 2016 Abstract. J and Mathias Pierre distribution around 750 GeV,through as a seen pair by of the collimatedlight ATLAS photon pseudo-Goldstone and bosons pairs. CMS In collaborations,photons this is framework which a originated are scalar state symmetry misidentified as breaking, a we single show photon. that coupling In a a complex minimal scalar context field of Φ spontaneous = ( JCAP07(2016)005 (1.1) , such that of data and s 1 − , respectively [3]. , σ [3] for the ATLAS 6 . α µν σ G 1 [3] for an excess peaked µν α and 1 < σ σ 9 sG . 4 . –2 Λ GG σ c field strength, respectively. The 8 . [4] and C + σ i µν W µν i and SU(3) sW L Λ – 1 – WW c + , SU(2) for an excess peaked at 750 GeV whereas the CMS, Y µν σ B for example, with gluons and photons which are described µν giving the mass of the resonance, large values of the coupling s sB s m 3 fb2], [ respectively which can be interpreted as a (pseudo)scalar Λ of data at 8 TeV, the largest CMS excess, assuming a narrow width, i.e., BB c 1 ± are the U(1) − = α µν 0 G − L and 3759 GeV 375 GeV 11 of data, has reported a local significance of 2 i µν 1 10% [3,4] with . > 3 fb1] [ and 6 − W − χ χ ± 1 m m µν ∼ B 1 GeV), appears at 750 GeV with a local and global significance of 3 s . (10 fb) and simultaneously a rather large decay width of the resonance, Γ 760 GeV. The global significance is reduced to 2.0 (0 Combining with 19.7 fb 4.1 4.2 4.3 Very4.4 light case: the Dark freeze4.5 matter in detection regime Summary 12 14 16 3.1 Condition3.2 to mimic a Di-jet di-photon3.37 constraint process5 Fitting the di-photon excess8 1 /m 1 s at and CMS, respectively.the These subject announcements [5–160], have tryinginduced initiated to (typically a fit by the new largecouplings data of vector-like amount in the fermions, of resonance, effective with works say frameworks respect on [ 161–168], to featuring the loop standard model (SM)) ∼ O by dimension five operators of the form: 1 Introduction Recently the ATLASphoton [1] invariant and mass CMS distributionlevel of in [2] 10 the collaborations vicinityresonance have of with reported 750 a GeV an (relatively)observed in excess large a their width. in local 13 TeV the significance The data di- of ATLAS at has 3.9 the analyzed 3.2 fb where parameter Λ represents theorder scale of at the which massas new of shown dynamics new appears, in fermions which eq.only entering is (1.1). with in typically gluons the of However,∼ loop and the it O to photons, was generate in quickly anΓ realized order effective that, to Lagrangian in produce the the presence observed of production couplings cross-section 4 The analysis implementing the dark9 sector 5 Conclusion using 3.3 fb 17 Contents 1 Introduction1 2 The model3 3 The LHC analysis5 JCAP07(2016)005 1 . → aa 0 s (1.2) . → s after the GG given the c χ 2 . Thus, the 4 m | ) ia decay process is + aa s is induced after the 1, the limit predicted ( . → process, i.e., via Γ( → | saa s (1 TeV), due to the con- aa 4 are disfavoured, | (300 GeV). Thus, the pres- Φ | → could come from the ∼ O BB can be mimicked by a process , the light dark matter region, s c s ∼ O well below 0 γγ ¯ χχg GG ¯ χχ, → c → χ s m gg representing the mass of the dark matter couplings, as an outgrowth, strongly relax → + χ at the LHC [170, 171], predicting = 40 GeV through GG m pp ¯ χχ c s and, is responsible for generating s gg is a light pseudoscalar decaying into two colli- – 2 – χ 1, of the coupling χ g ) has nevertheless been invoked, although the new . a → 0 = can be achieved by allowing new decay channels for s Br ∼ > 1 s → − L , where gg is primarily determined by = 0 in eq. (1.1). γ couplings, respectively. These associated values, even after s 4 2 appears excluded [175]. / WW s c → BB c m aa coupling to secure Γ ∼ < χ and → m saa s GG c → pp coupling remains proportional to the quartic coupling. The same complex scalar as the relevant Yukawa coupling and 2 TeV. Similarly, high values, saa χ 375 GeV, i.e., ∼ (100 MeV) of the lightest CP-odd Higgs since otherwise its branching ratio into two g A possible solution to increase Γ However, all these studies converged to show that in the said scenario, the direct detec- An interesting alternative has been proposed in refs. [179–182], considering the possi- In this work, we will combine the two preceding approaches in the context of a simple We will show that when Γ would be essential. This possibility, however, contradicts (see ref. [169]) with the current coupling is generated by the spontaneous breaking of a Peccei-Quinn (PQ) symmetry. . This scenario corresponds to 2 ), the di-photon excess can be obtained for values of χ . GG for Λ constraint from the production through photon fusion [172]. the resonance. Anwidth intriguing is possibility provided is byof represented the refs. by decay [47, channel the]173–175 into casefermionic (see dark in (Dirac also matter or which ref. pairs. Majorana) the [ 176]) Along dark additional have this matter considered line, candidate a the as: coupling authors of the resonance with a c observations from di-jet searches with χ tion (in the case ofresonance), a strongly scalar resonance) constraints or thecerned the dark indirect values detection matter of (in mass the case up of to a pseudoscalar bility that the di-photon production process respecting the di-jet [169] anddate photon the fusion observed [172] constraints, di-photonthe appear production constraints naturally cross-section. to from accommo- monojet Combining these searches [177, limits178], further with m of the form mated photons. In this scenario a sizable contribution to Γ decay process. Thissaa possibility has beenHowever, investigated, a for sizable instance, in ref. [182], where the spontaneous symmetry breaking by a classical potential term model featuring spontaneously broken symmetry. The coupling final states have notstudies been have identified been asmodel performed dark (NMSSM) in matter [183–185]. the candidates. In∼ this case model The O of the similar LHC the andphotons signal next-to-minimal can extensive would be supersymmetric be reproduced heavily standard only suppressed for compared masses to the same into SM fermion pairs. field Φ is also coupled to the dark matter rather unnatural since itence of would an require invisible branching a fraction small ( PQ scale derived spontaneous symmetry breaking. aa from the di-jet constraint.the Such constraints low values from of monojetwith and the direct WMAP/PLANCK dark [186, matter187] searches favoured while values of still the remain dark compatible matter relic density. JCAP07(2016)005 3 (2.2) (2.3) (2.4) (2.1) , and Φ = ., ρσ c . can mimic the B γ 4 + h µνρσ  . → α µν 1 2 ) . e G c . aa = µν α α µν , it generates automatically G G χ. → µν + h 2 Φ Φ 5 Φ e µν α λ µ 2 s B ¯ G χγ GG Λ ' (Φ a Φ 100 GeV), (ii) a heavier window → ic 2 Φ 2 Φ 2 2 Φ χ .   λ g c ' Λ √ GG − pp . i c ) through the spontaneous symmetry + + , χ i µν 4 2 Φ Φ χ − | + + h λ µ m vev L f W Φ | i µν ¯ χχ = + µν Φ χχ i s W λ 2 Φ Φ Φ¯ 2 χ W v χ L µν g − i √ g Φ 2 – 3 – + | W − − 0 Φ Φ Λ | WW χ χ L ) with 2 Φ µ µ ic ∂ ∂ µ Λ = , which is then promoted to a pseudo-Goldstone state. ia WW µ µ a − c + + L ¯ ¯ s ∗ χγ χγ + µν i i Φ + e 1 1 2 2 B µ Φ µν ∂ v µν B = = ( Φ 2 B µ χ 1 µν √ Φ ∂ L 2) and (iii) a very light case (keV dark matter). For all of these B / denote the dual field strength, e.g. = Φ s BB Λ Φ m ic α µν Λ BB L c e − G ∼ > can naturally be introduced in this framework through the Lagrangian: , = χ i µν 0 f W −L , 2 is a complex scalar field whose Lagrangian can be expressed as: √ µν ) and giving, at the same time, the correct relic abundance for the dark matter. / e s B ) We can see that eq. (2.3) contains a term that explicitly breaks the U(1) symmetry, Once Φ acquires a vacuum expectation value ( The paper is organized as follows. After a summary of the model in section2, we Regarding the WMAP/PLANCK constraints, three regions of the parameters space are ia in section3. In the next section we introduce the dark matter candidate and compute 400 GeV, i.e., We will work in the framework of Majorana dark matter throughout this analysis. The extension to the s 3 + s & giving mass to the Goldstone boson where A dark matter Dirac case is straightforward. observed di-photon signal. Weobservations, will and then show determine that theof parameter the space di-photon allowed signal by can theits remain relic LHC compatible abundance, evaluating with the adi-photon parameter large excess space and width respecting the at the WMAP/PLANCK same constraints. time Finally the we observed conclude. 2 The model The Lagrangian of eq. (1.1)a can SM be singlet expressed fermion forcomplete (Majorana a complex Lagrangian or scalar can Dirac Φ then like) in be further a with written similar way. eq. as: Adding (1.1) is straightforward. The ( breaking mechanism, Φ = will briefly review under which conditions the process windows, we have checked thatcompatible the with direct a and relatively indirectare large detection summarized width limits in are of the respected the finalconstraints and scalar, plot (e.g., are exchanged where di-jet, as we monojet, have alarge etc.), shown mediator. points Γ observations respecting (di-photon Our production all results the cross-section relevant and LHC ( with now open: (i) the electroweak-scale dark matter ( JCAP07(2016)005 , a Φ /v (2.5) sZZ, sZγ 2 ) 2 a 0. If now one fixes + coupling and hence, 2 → s ( Φ saa 4 λ WW µν c + ) and the dark matter mass e F 4 Φ 3 s v µν s , being the scale of new physics, Φ Φ λ m aF µν α v , Φ e 2 2 W ) = , which preserve the U(1) symmetry. G λ c χ a µν α µν 5 r s saa BB G ¯ λ χγ aG + m C µν a Φ 2 ia G λ GG sa + s Φ C s √ † Φ m m Φ – 4 – ). If one further wants to fit simultaneously, the ¯ + λ χχ χ Φ 2 s 2 √ ( GG λ Λ coupling ( µν m c , after spontaneous breaking of the U(1) symmetry e ) is the field(dual field) strength for the photon field, s χ Φ , F + r µν m a v m e field. µν saa F µν α + m Φ ( 2 a 2 λ , G sF µν a Φ α µν 2 a F r 2 W λ 2 ’s. Choosing logically Λ = c , , m ), the + ii sG Φ W c s BB θ v + BB ¯ 2 m C χχ GG Φ 2 C χ s s Φ λ C , 2 s 2 λ cos Φ m m 2 λ √ √ 2 m 1 GG and re-express the relevant part of the Lagrangian (see eq. (2.2)) as the = √ C + + + , from the requirement of gauge invariance, also generates interactions like Φ = 2 W v s c BB , m C ). Moreover, the parameter Λ in eq. (2.2) can always be freely chosen up to a ii ( c Φ −L ⊃ v s is the pseudoscalar mass and we have worked with χ g Φ = 2 , the di-photon cross-section and the relic abundance, one is left with only two free  , required to be much suppressed in order to generate an experimentally viable 2 s ii 2 √ C GG √ Notice that, sticking to an effective field theory approach, the dimension-5 interaction The main purpose of this paper is to provide a phenomenological explanation of the = , c = The coupling χ 4 a BB m etc. We, however,our do analysis. not explore them in detail since they are observed to have a marginal impact in and thus, the presencestudied of scenario, these as dimension-5c will operators be is clarified ratherproduction subsequently, cross-section natural. the with coefficients collimated Moreover,inclusion photons. of in of In these dimension-6 the this operators, operators, setup, e.g., These e.g., one terms, might with also the consider the choice of Λ = parameters. The model then becomes very predictive. terms in eq. (2.2),symmetry. between the As field alreadybeen Φ mentioned, and introduced an the in explicit SM eq. breaking gauge (2.3) bosons, of in do the order not U(1) to preserve symmetry the have has a U(1) already non-zero mass for the pseudoscalar field width of would produce new effective dimension-5 interactions for the scalar, featuring the same 1 di-photon excess through themain production aspects of collimated of photon dark-mattercouplings pairs physics. of simultaneously various with For operators this the andforwardly reason hence, encapsulated in the through effect our a of analysis redefinition, dimension-6 we operators including will can the freely be contributions vary straight- from the different di- where m the mass of theof five scalar parameters at ( 750 GeV, all the physics is then completely determined by a set the impact ofinteractions including for dimension-6 the pseudoscalar, terms however,can would remain the lead hardly same to affect since couplings only our linear dimension-5 findings. in operators the The relevant redefinition of the couplings the mass of ( one can eliminate suppression associated with the U(1) violatingbetween ones. the This would modify scalar theproduction trilinear component vertex couplings and of decay width Φstudied of framework and the the scalar the decay field. width SM Now, is as gauge will dominated bosons be by explained the and later, tree-level in thus, the will affect the function of masses and couplings as: JCAP07(2016)005 0 = π → γ m a (3.1) (3.2) ( ( 4 0 π /γ → = 10 cm process. 2 m l represents : 3 ss ∼ a γγ 10 cm, might ∼ < BB process remains φ → → a C − [188, 189]. This process remains . remains actually s 10 fb and finally, ¯ 1 gg χχ m 5 2 . / 2 − γγ 1 ∼ 1 and Γ → γγ  l  − → a s 2. → 01 / 2 BB . a a 05 γ 0 a C /m . and the SM gauge bosons. m s 0 , can be expressed only as p (20 mrad) Γ s 0 ∼ < π   . In order to mimic the LHC =  χ values according to the one of s 2 a m = 40 GeV. A further constraint m ∼ O 3 a  process. This plot is made with a 05 βγ s /m . m ∼ < s 0 Γ a γγ a  m m → 4  a 500 MeV 100 MeV region, as reported in ref. [192],  2 . a / 3 a process corresponding to m 2 GeV [179, 190, 191]. At the same time, a very  m – 5 – 1 meter, where γγ 500 MeV . s a   3 → a , corresponding to Γ m m  coupling (see eq. (2.5)) for a Γ Φ λ 750 GeV 1 meter disfavours the gray coloured region corresponding ¯ χχ  s 1 meter gives the following lower bound on a βγ/ 3 ∼ < m − ∼ < l = l l 10 as: 750 GeV , 1 meter from the collision point. A minimal requirement is then which would lead to characteristic signals, e.g., .  × a . This constraint is mostly relevant for values of ◦ 12 [187]. BB a 7 ∼ . . aa 15 . 0 C 2 0005 and of 1 . 3 cm . or ≈ & ∼ 7 2 and = 0 ss h ≈ a BB l ) plane. We also show the C GG m C BB As a consequence, the decay length, for ,C 6 a 200 MeV. For reference we have also plotted the line corresponding to , being the boost factor) of the two photons emitted in the m a . 60 GeV, the di-photon production cross-section in the range of 1 m We note in passing that for this work we consider the width of the resonance in the span Now the requirement of We present in figure.1, for illustration, the allowed The existence of dimension-six interactions, nevertheless, would appear effective for the a , demanding a pair of collimated photons from − 2 This corresponds to One should note that in this regime the invisible branching fraction from / m 5 6 . A detail and subsequent discussion of such signal, however, is beyond the scopes of the s BB γ on the associated decay length in the ( below the angular resolution of the LHC detectors, which is m condition provides an upper bound as 3 The LHC analysis 3.1 Condition toThe mimic decay a di-photon of(dubbed process a as “photon substantially jets”)minimal light condition which to pseudoscalar can realize be can this potentially kind produce misidentified of as highly scenario a is collimated single that photons the photon. opening The angle ∆ of 4 8 current article. the relic density Ω di-photon signal, thewhich decay is should at occur aset distance before on of the the decay electromagneticthe length total calorimeter of decay (ECAL), width for denotes the neutral pionaccessible. mass), such that only the decay process negligible, given the suppression of the high boost typically generates an enhanced decay length for the function of already be distinguishable from a pair of promptC photons originating from fixed value of to pair production of Thus, we will not explicitly refer any further to the presence of dimension-6 operators. We remark that eq. (3.2)for should be example, regarded displaced as photons a from conservative bound. As argued in ref. [181], mensionalities, of the effective couplings between the scalar field JCAP07(2016)005 ) ≡ in → → s γ BB s 4 aa ,C a > σ → m ) h γγ → . We have thus s , above which the 0 1 meter (excluding decays. Given the BB π → C ∼ 0 < 25 to 4 giving Γ m . π pp ( σ process leads to a collimated ≡ γ 600 MeV (based on the similar 2 γγ . As will be clarified in the next σ − is also possible in our model since 7 → GG a γγ C 1 is excluded by the 8 TeV searches of . 0 , ranging from 0 which is duly explained in the text. → Φ & γ and s λ 4 → BB BB C > σ processes, it is useful to adopt those variables C pp γ 2 – 6 – 2 GeV, can be constrained further by considering ) plane when γγ σ ≤ BB → a a ,C m a m ≤ and γγ 2 GeV → . 2 GeV in the numerical computations. 0 π ≤ 001 and for four values of ). . a being the SM Higgs boson, can discriminate, through suitably defined γ m 4 h = 0 ≤ → 0 π , can couple to two photons through the same coupling 3 aa GG , compared to figure1, is now lowered to 500 a ) = 1 irrespective of the values of C a m → : Allowed region in the ( m γγ s processes [193–197]. One should interpret figure1 just as an illustrative plot. = 10 cm. The red coloured region is excluded by constraints reported in ref. [192]. → l → We finally remark that the allowed parameter space can be constrained even further by We have then redetermined, in2 figure the allowed parameter space in the ( It is important to note that the process The allowed range, 0 a In reality in figure1 the region corresponding to process, with ( 7 pp ( γ , just like ZZ, Zγ photon pairs and simultaneously, the associated decay length remains Figure 1 the gray coloured region).with The black coloured solidThe line green represents coloured a region typical corresponds displaced to decay considering the different photon conversion rate, i.e., the probability of the interaction of a s represented (in green colour) in figure1 the region where suitable isolation cuts.4 Indeed, experimental searches of the somewhatsimilarity similar between the to reduce the probability of faking single photon signals [198]. section, this will lead towill a use rather this predictive region scenario.this of work. To the utilize parameter In this space predictive thethe behaviour, for absence region we most of of a the dedicated analytical experimental estimates study, presented we in have nevertheless included (represented by the red colouredof region) a which light remains pseudoscalar constrained in from beam-dump the and direct fixed searches target experiments. calorimeter variables, isolated photons from the ones coming from σ plane, for the span oflimit 4 GeV on to 60 GeV,considerations a by slightly using weaker limit themost of of treatment 800 the MeV illustrated has allowedhadronic in parameter been space found decays ref. in now become [182]. ref. lies accessibleBr [181]). below for The the Interestingly, the threshold upper of pseudoscalar. 3 As a consequence, in this regime, JCAP07(2016)005 8 9 (3.3) channels , ensuring s gg 5 pb, with . represent the 2 and < T eV , events. A detailed ) ) aa 8 γ jj gg 2 GG = 8 TeV, respectively. GG, C 2 GG 001. I s → . → C etc., compared to the di- s √ and 4 s ( 32 = 0 γ and 1 + 32 ZZ Br → s s s 5 pb, at 8 TeV centre-of-mass GG comes from the di-jet searches. . or 2 m m C pp ( Φ Φ GG < σ λ λ are constrained, by the requirement Zγ π π C ) 2 2 GG GG jj for the different values of Γ BB C C C a 2 2 → m pair, between 2 s T eV T eV and − 8 8 e → . The quantities + with s GG e – 7 – GG, GG, pp C I I ( 1. s s BB σ . s s 2 2 0 C π π m m . 8 8 ≈ GG ) = C jj → s ) = 1 regime the corresponding cross-section is given by: → γγ pp ( → σ a ( Br : The variations of coupling = 280, is satisfied for T eV where we have used eq. (2.5) and retained the dependence only on the In the 8 One can, for example,Here see we ref. have [199, considered200] MSTW for 2008 a NLO similar PDF bound [201–203] with and 13 TeV included centre-of-mass the energy. next-to-leading order scal- 8 9 GG, (see next section) in the determination of Γ ing factor. square of thequantities centre-of-mass like energy parton and distribution the functions (PDFs), integral evaluated of at a function involving dimensionless It can be easily seen from eq. (3.3) that the condition Figure 2 photon within the calorimeter toinvestigation produce of a this possibility has been performed in3.2 ref. [179]. Di-jet constraint As already emphasized, that the couplings For this purpose we have imposed that that the pseudoscalar decayspairs before to the mimic ECAL the observed and di-photon produces signal. sufficiently For collimated this photon plot of the relative absence ofphoton additional one. signals, In e.g., our di-jets, scenario the strongest constraintenergy on [170, 171]. I JCAP07(2016)005 ) s and m ) for (3.4) (3.5) (3.6) (3.7) s mode . The  GG m gg GG C χ C 5% m → ∼ s ∼ ( . threshold when (for 2 s BB 0 ) s C π 2 2 GG , m  /m C χ GG GG 001 . 3400 using MSTW 2008 . I C m 0 2  ' 2 GG ∝ )] C (1 + 32 s γγ ) s m π → Φ . 05 /m . λ a s 0 2 ( coupling. Indeed, from eq. (3.7) one 2 coupling GG (Γ C Br ) = 1 )[ aγγ ¯ χχ s are suppressed compared to gg aa 16 fb 2). Thus, collectively one can use eq. (3.5) , respectively, even when / 1 + 32 → , the ' → s Zγ 500 MeV, i.e., below the 3 W 2 s 10 fb) together with a large Γ s ≈ s m θ s ( . BB → ) – 8 – − Γ( 2 Γ m C a ∼ s 2 Br production cross-section depends only on aa (1 , , ) m sin ) χ s γ . We, however, do not consider this possibility in this / m π → gg 2 8 γ GG m ZZ 4 ∼ O s , Φ C 16 ( 2 → GG λ γ W → the 4 4 → s C θ s Br σ 0 4 π , Γ( ) = aa 3 s sin s (1 + 32 γγ 2 m aa → / π m 8 ∼ s < GG 8 → → , I a s s 2 → = W m s π θ γ Γ( 4 4 4 γγ is constrained from the photon fusion process. The invisible decay mode (see figure3) is given by σ ≈ γ cos 4 γ BB / 4 C σ → : Main channel contributing to the di-photon excess observed at the LHC. ) = 1. In this region, combining eq. (3.6) with eq. (3.4) one obtains: aa being the integral over the PDFs. This is estimated to be also suffers suppression, either from the γγ → . The desired signal, as already mentioned, can also be induced through the photon GG s → ¯ χχ At this point one can use eq. (2.5) to compute: Now the processes This equation is especially valid for Interestingly, for I Figure 3 a GG couplings while remains independent of → ( I → Φ upper limit of s or from the phase space factor (when to get with NLO PDF [201–203] andof including the next-to-leading orderfusion [204] scaling process factor inarticle. the definition by factors like 8 λ can see the possibility to obtain Br 3.3 Fitting theThe di-photon production cross-section excess for the studiedgg di-photon signal through the gluon fusion process JCAP07(2016)005 , 1 γ . 4 0 σ (4.1) (3.8) (3.9) , . 10 fb) 4  mode is − GG (1 C γγ s m ∼ O , → 4 , as well as into 65. γ 750 GeV with the physical . s 4 without requiring  0 aa σ  Φ γ . 2 4 λ 2   σ s  and 60 GeV and (iii) correct m BB χ process. We have already ≤ GG 005 C gg . 750 GeV s C m 0 ,  Γ γγ  2 that give a values and check if there exists 100 GeV γγ 4 ) and is given by: does not directly appear in ≤   χ →  4 GG GG  s m 500 MeV with a sizable branching BB s C C and a sizable χ BB 005 C 1, implies . & s ). This feature allows us to trade, in C m 64 05 0 . /m . / 2 a s channel compared to  0 aa  / 100 GeV 1 Γ 8 W BB 2 ) ) m c γ )  C  → 4 aa γγ 2 ) plane the parameter space allowed by the  4 s 2 BB → 27 GG →  . s s s C → C 0 for different γ GG Γ( ( 1 Γ( Γ( 4 – 9 – 10 fb, (ii) 4 GeV C s Φ 05 σ . /m . ≈ ≈ 16 fb λ , s 0 ≤ φ 2 s  Γ GG λ γ )) 4 (1 + 32 30  C and σ ) − 2 γγ s  ≤ 10 GG 05 → 1 /m . × . C BB s 0 a 0 5 ( 60 GeV while still respecting the PLANCK constraints. We couplings. We finally notice that the di-jet limit, C (Γ by demanding a suppressed  ≈ ∼ < 2 χ Br 2 BB , respectively. In reality, however, there exists one more way to s ) 5 fb 2 χ m BB Γ C can also be obtained for 27 m 4 W BB C ≈ 2 GG γ 4 s c ∼ < − 2 the dark matter can annihilate into C 4 4 s γ C 2 channels mostly originate from s-channel exchange of the pseudoscalar. GG / 4 σ 12 [187]. It is evident from figure4 that for any value of the dark matter and 10 s . 2 BB σ πm C 0 πm , consistent with the di-jet constraints. This comes trivially from the fact C m × 2 Φ gg 2 Φ GG λ coupling is responsible for the large Γ ≈ 9 as a function of GG λ . C (1 + 32 16 2 Φ χ into the hadronic channels. In such a case the cross-section is obtained by C . and 375 GeV χ λ s , whether kinematically open (the last two modes are suppressed in the setup a ≈ = = . γγ ZZ gg χ < m γγ i i a m The condition for the dominance of The A good fit for It is important to note that although the coupling σv σv m h and h 4 The analysis implementingOnce one the couples dark the sector darkabundance matter of to the field Φ through eq. (2.4), one can compute the relic multiplying eq. (3.7) with ( The corresponding thermallyestimated averaged as cross-sections, [47, using173, eq.174]: (3.6) and eq. (3.7), can be fraction of together with 4 GeV will study the different regimes in detail4.1 in the following sub-sections. For Zγ considered in this work). three constraints, namely: (i) 1 fb mass between 200 GeV to 700 GeV, there exist values of low values of high values of implies that for all practical purposes Γ used this to derive eq. (3.6). The condition similarly modified and becomes: parameter space compatible withfigure4 thewhere observed we di-photon have excess. shown The in result therelic is ( abundance presented in that a large the analytical expressions considered inobservable this Γ section, the free parameter it is not unconstrained inprocess nature. and In the fact, requirementand the of absence lower a of bound pseudoscalar di-photon signal decaying for from before photon the fusion ECAL provide an upper constrain the coupling JCAP07(2016)005 005, . (4.2) . 4 1 for the 100 GeV.  . 0 = 200 GeV ∼ < χ . s χ 001 and 0 m . m m BB 750 GeV C for .  1 2 −  and s 3 γ 500 MeV, by the FERMI 4 χ cm σ ≥ m a couplings, as 0 30) 005. . m 100 GeV ÷  BB 2 (29 = 0 as depicted in the figure5. The C −  ) plane consistent with the observa- s s BB GG 05 C and /m . C s 0 , Γ Φ GG  λ C 2 ) 2 GG – 10 – C 1 10 fb, as observed at the LHC. The red and magenta − (1 + 32 1 − s 3 . We now clearly see the difficulty to achieve the correct relic cm 1 2 27 − s v − 3 4 s -channel exchange of the scalar 2 Φ 10 [208]. Fixed values for the s λ channel gives rise to a p-wave velocity suppressed cross-section (by CP 2 χ πm × 60 GeV) whereas the blue coloured lines (solid and dashed) represent a di- 3 m aa . 9 384 − 5 ≈ = final states, respectively. Moreover, the gluon channel is also constrained by the aa : The allowed parameter space in the ( i MicrOMEGAs γγ σv h In figure6 we show the result of complete numerical analysis, performed precisely by in the unit of cm Even if the and determining the darkpackage matter annihilation cross-sections and its relic density through the tions of: (a) correcta relic density, relatively (b) large the observed widthwidth di-photon of production region the cross-section (4 and resonance. (c) The gray coloured band corresponds to theand 700 large GeV, respectively. For this plot we consider gg searches of gamma-rays from DSphphoton [205, channel206]. is, instead, On substantiallygamma-ray the excluded signals other [207], by hand, the which the give FERMI a contribution searches limit from of as the monochromatic strong as 10 concerned thermally averaged cross-section can be estimated as: photon production cross-section of 1 di-jet constraint, eq. (3.3), and, to a lesser extent and only for coloured lines correspond to points that respect the PLANCK constraints for Figure 4 abundance through the s-wave channel due to the constraints on arguments), it ismostly the through dominant the annihilation process at the decoupling time. It proceeds JCAP07(2016)005 ss (or aa approaches χ 375 GeV. m . χ m 50 GeV while still fitting ≤ s Γ ) plane. The differently coloured s ≤ Γ , χ , shown in figure7, becomes kinemat- m sa → χχ – 11 – 300 GeV) and for 12 GeV − (200 ∼ O χ 2, the annihilation process / ) as it is the only channel which is not velocity suppressed. The relevant m : Main channel contributing to the relic abundance for ) plane corresponding to different di-photon production cross-sections. We s s s 375 GeV Γ , > m > > m χ : The contour of correct relic density in the ( χ χ m χ m 375 GeV, the correct relic density can be achieved, through the annihilation channel m m Figure 5 2, the width needs to be narrow to avoid the under-abundance due to thermal broadening . / s χ Figure 6 the LHC di-photon data.m We alsoeffect notice [209]. the typical On poleany the indirect effect, other detection hand, i.e., signal being when as velocity it suppressed, will this be channel discussed4.2 cannot in sub-section account4.4. for When ically allowed. Thiswhen channel will dominate the annihilation process compared to horizontal lines correspond to the various values of the di-photon production cross-sections. respectively, are used for thisin figure. the Here we ( have plotted the contour of correct relic density have also used them same methods and toolsof for figure5, figure4. for We notice from6 figure that, when JCAP07(2016)005 . 4  (4.4) (4.3) (4.5) s . 4 one gets: m .  4 , 10 750 GeV  4   s 375 GeV. 4 s m  > s m 750 GeV χ m a  750 GeV m 2 m 750 GeV  5 GeV  . 4  375 GeV is achieved only 0  4  couplings, corresponds to a >  χ 7 a χ  m a m BB m 1 GeV m annihilation channel, 400 GeV 5%, the cross-section exceeds the χ 1 GeV ,C   6 ∼ m  1 GeV γγ 2 .  GG 6 s  0 C  s  /m χ 2 s 05 /m . χ  m 1 GeV s 0 . m Γ 0 1 GeV . BB 005  0  . C 0 1  – 12 – − 2 BB  2 with the Hubble expansion rate, both evaluated at s 2 ) 2 C 3 GG  2 Φ C 2 s GG λ 4 W χ,eq cm 2 Φ 1 c C n , such that the validity of the WIMP paradigm itself λ 05 − i 2 1 1 /m . GG 25 s s 2 0 BB − − − 3 C s σv s Γ C 2 Φ h 3 3 10 2 Φ  cm λ final states through s-channel exchange of the pseudoscalar. = 2 λ 4 s (1 + 32 × cm
34 4 s 6 χ ) cm m 3 − γγ 6 χ 33 4 a m ann 26 2 m GG − 4 a 10 ' m − m 2 C . ). Considering for reference the m 10 3 × χ 4 s 2 χ and (10 π 2 × m m π m . ( 128 1024 3 2 gg 1+32 2 Φ π 8 λ ∼ O ≈ = ≈ = : Main channel responsible for the relic abundance for ∼ O ≈ ' gg γγ ) T (200 MeV), and then the hadronic annihilation channel remains inaccessible. i i on the lower side. ) χ sa χ γ i σv decreases, the relevant annihilation processes occur below the temperature of the QCD phase σv m 4 h ∼ O h 12 GeV. The latter, for the chosen values of m σ χ σv = h = m < Figure 7 T As evident from eq. (4.4) that these cross-sections are very far from the thermally As evident from eq. (4.3) that in the case of Γ ( s T ( As ann H 10 Γ transition, becomes questionable. Thus, wematter apply annihilation the rate conventional rule-of-thumb Γ oftemperature comparing the dark favoured value, 4.3 Very light case:In the this freeze case inannihilations the regime into dominant contribution to the dark matter relic density is given by the thermally averaged cross-section is given by: thermally favoured value and leads,also consequently, reflected to an from underabundant figure6 darkthat matter. the This correct is relicvalue density of for The corresponding thermally averaged cross-sections are: for Γ JCAP07(2016)005 (4.6) (4.8) (4.7) 1% or . . s  (keV), the . . After this O a /m  s s 100 eV, incon- & m a m ∼ ∼ m decay process can 750 GeV 2%. Remembering χ T 1 GeV  m  > 2 ¯ χχ / 2 s 1   → /m s s , ). This requirement can be a s a 5 GeV giving Γ  . m m m 7 χ 1 GeV ( 750 GeV m .  1 eV s   3 ∼ O . As can be seen, the correct dark . ) 02 s /m .  a d s 0 T m Γ ( 100 MeV, compatible with the LHC di- χ eff g ,S m ) → s ∼ < ∗ is the number of internal degrees of freedom 1 keV g χ in turn, kept into equilibrium by its couplings a ¯ 2 χχ 2 GG eff a − m g m C   2 a → 2%, on the contrary, there exists a small window s 10 – 13 – a drops very steeply as long as the dark matter mass m < × 01 /m . Γ( s , with s 6 0 process is rather efficient and keeps the dark matter 1 + 32 . ann H χ 2 χ Γ 9 Γ 2 s m m /m q  a s ¯ χχ ≈ 2 ) (1 keV) (or even below) for Γ , πm / on the smaller side, i.e., m a 2 1 8 3 Φ g ) s ↔ h 2 λ GG a 3 keV ≥ 27 . a rel C ∼ O m 1 ) 1 χ, 10 a χ ) = = Ω & m × m ¯ χ and of the pseudoscalar field ) χχ T ( a = 09 s m (1 + 32 . ,S → m 3 1 T ∗ . g a 10 keV, for which the dark matter cannot get into the thermal equilibrium 0 )( = ) represents the number of relativistic degrees of freedom at the tempera- → − d ≈ = ¯ T χχ 1 T ( ( 2 ∼ H h → ,S ∗ χ FI a g Ω m Γ( below a keV is excluded by structure formation, we conclude that the light dark of decoupling of the dark matter while χ 2 GeV. d m We thus notice that the If the dark matter is not in the thermal equilibrium, the Two examples of the numerical solution of this system are shown in8. figure Here two For the chosen model parameters with where The analytical expressions reported above have been validated by solving a system of T ∼ a also give rise toto freeze-in eq. production. (4.8), However, achieved its with contribution the is replacement suppressed with respect values of the(right). dark In matter the masses lighter have dark matter been case, considered, we notice namely, that 2 the keV dark matter (left) abundance and is 50 always keV in the equilibrium unless that matter is coupled to the primordial thermal bath until the temperature matter relic density requires a Γ is further reduced. The darkthe decay matter process could Γ( nevertheless bewith kept the into gauge thermal bosons, equilibrium at by least down to temperatures of masses, in the Early Universe.by In the this decay of case the the pseudoscalar. dark matter The corresponding is produced relic density through is freeze-in given [210–212] by: coupled Boltzmann equations tracking theones abundance of of the the scalar dark field matter itself as well as the photon signal, the darkstill matter remains annihilations relativistic and become the inefficient. ratio The dark matter, however, translated into the following condition: temperature relativistic decoupling happens with a relic density given by: ture sistently with the conditions stateddark above. matter is For viable overabundant. values For of Γ the mass, i.e., m of the dark matter itself. The correct relic density can be achieved for JCAP07(2016)005 and (4.9) , 2 GG C  , as noted s curve. It is GG 001 . C 0 11  2 ) and then remains  a m ( χ m = 50 keV, the dark matter , coming from the QCD phase ∼ O χ 100 GeV ,S (10). ∗ T m  g 8 ∼ O  x s fields, as determined by the numerical m χ 750 GeV 5 keV monochromatic signal as recently .  and 2 a  , s s – 14 – 05 /m . is due to the change of s 0 , namely, 2 keV (left) and 50 keV (right). As evident Γ x χ  m 2
2 GG C 1 (100 MeV). The latter corresponds to 1 + 32 ), as expected for the freeze-in mechanism, and the numerical value a 2 ∼ O m T ( cm 48 − ∼ O 10 T × 7 : Evolution of the abundance of the . 6 couplings, compared to the direct production from a 750 GeV resonance. As already coupling. The latter also implies an very suppressed monojet production cross-section ≈ The raise of the curve at high values of SI χp 11 GG BB σ transition occurring at which remains well below theC current limit from LUXand [214], thereby, given easily the evades very the small corresponding value of experimental the constraint. Figure 8 solution of a systemThe of two panels coupled refer Boltzmann tofrom two equations values the in of comparison the withline), very the light that dark dark matter for matter equilibrium theequilibrium distribution regime. and lighter it (dotted mass, is orange produced the coloured can through dark freeze-in. however go matter In into the is case the not of thermal capable equilibrium of and get getting decoupled intowell while the below still thermal being the relativistic. equilibrium one, represented by the dotted orange coloured C mentioned in the introduction, such valuesmatter can searches. potentially evade Regarding the direct constraints from detectionare the of the dark the Spin dark Independent matter thealso (SI) most in ones, relevant refs. interactions produced [47, by173–175]. the The exchange SI of cross-section, the in scalar the field studied framework, can be written as: observed in different clusters of galaxies]. [213 4.4 Dark matterOne detection of the maindi-photon characteristics production of cross-section the is proposed scenario obtained is for the relatively fact lower that values here of the the observed produced at of its abundance, asagreement determined with by the the theoreticalchosen solution set prediction of of stated the parameters, system in theOn dark of eq. the matter equations, contrary, (4.8). abundance for is matchesperfectly the in We the tracks heavier also very experimental the value expectation. good notice thermal ofconstant that, equilibrium the confirming function for mass, the until, i.e., the prediction again, 50 ofa a keV, scenario relativistic the decoupling. dark would matter Interesting be signatures abundance of the such interpretation of the 3 JCAP07(2016)005 2. / s > m value. As χ χ m 001) or even . m (0 ) plane from the ∼ O BB values considered for ,C χ increasing ) plane excluded by the BB m values C BB ,C = 4 and 60 GeV, respectively. BB χ values considered in this work, s C s m 400 GeV. The gamma-ray box is narrow > χ gives a characteristic signal represented = 4 GeV) and by dashed green coloured 1 meter as observed in figure2. m s sa depending on the ) plane using the constraints from the . regime, may appear incompatible with the → 1 meter, as depicted in figure1. BB BB a C . m ,C χχ – 15 – χ m = 4 GeV and 60 GeV, are represented with the cyan s values (see figure2), which assure a pseudoscalar decay which can be probed by the CTA [216] in the near future. s 12 500 MeV. . minimum value for a m decreasing = 60 GeV), respectively. One should note that the . The blue and green coloured horizontal dashed lines represent two reference s : Allowed region in the ( s 1 meter for . values for the same two Γ The impact of the indirect detection on the relevant model parameters is represented On the contrary, a potential impact is still retained by the indirect dark matter searches. Concerning detection of the dark matter, an interesting scenario appears for A wide gamma-ray box is actually achieved only when 12 BB in figure9. Here we have shown the excluded regions in the ( smaller. Therequirement latter, of especially a in pseudoscalar decay the length low FERMI/HESS searches for thei.e., minimum 4 GeV and maximum and Γ 60 GeV, respectively. The regions in the ( figure9 always give a pseudoscalar decay length when close to the kinematical threshold and becomes sensitive to the gamma-ray line searches. We have At this moment, the mostsearches effective of constraint the is gamma-ray the one lines. coming These from the searches FERMI can and trace HESS The black and grayvalues coloured of contours Γ represent the correct relic density for the same two FERMI searches, corresponding to Γ Figure 9 C FERMI/HESS searches of theexcluded gamma-ray lines. from the The cyan FERMI/HESS and searches, light correspond green to coloured Γ regimes, length and light green colour, respectively.the The correct black relic and density, as thecan suggested gray extract by coloured a the contours PLANCK. correspond It to is apparent from figure9 that one by a wide gamma-ray box [215] a reference, we also comparerepresented these by limits the with dashed the blueline ones coloured obtained (for line from Γ (for2. figure Γ The latter are In this case the s-wave annihilation process ¯ JCAP07(2016)005 (see (4.10) (4.11) aa 12 GeV) → ∼ < ( s s → , χχ , 12 .  ) plane for the gamma- 0 1 s . 0 Γ , ≈ , , 4 2 χ . This plot is prepared for − h fixed at 750 GeV: m Ω sa s 10 , 2] GeV m × , → 5 2 .  [0 ∈ χχ 1 meter etc.) as well as with the dark 300 GeV) in which the relic abundance 60 GeV ∈ a . BB . ∼ < s χ Γ m ∼ < 20 GeV) is necessary to compensate the velocity – 16 – ,C  & , m 1 . s 0 4 GeV , , 4 abundance of the dark matter in the Universe. (ii) In the − 400 GeV, the correct abundance is obtained through the . 10 . 10 fb 4] over , 1000] GeV × χ ∼ , < 5 25 m  . γ 4 [10 [0 ∈ σ . ∈ ∈ ) over the following ranges, keeping ∼ < χ a GG Φ C m λ , m 1 fb χ , decay length of the pseudoscalar , m Φ ZZ ) plane possessing the correct relic density and corresponds to different val- s , λ : Plot showing the CTA detection prospects in the ( , Γ , BB Zγ χ ,C m Finally, we kept only the points respecting and remain compatible with the relevant accelerator constraints (8 TeV searches of the GG C included the corresponding constraint in the parameter scan described in the sub-section 4.4. Figure 10 ray boxes originating from the annihilation process ¯ ues of thecolours. di-photon This result production( is cross-sections, obtained after as a represented dedicated numerical with scan the on the three set different of parameters the four possible choicescoloured of points represent the the darkscan set matter described of profile, in viable the as model sub-section mentioned points4.4. in which emerged the from plot. theThe parameter The detection orange prospects arecan shown efficiently in probe figure theany10. studied signal, corner can It also of is exclude the evident the parameter from entire space4.5 viable figure and parameter10 thus, space. that in Summary the the CTA We absence of summarized ourthe results ( in figure .11 This plot accommodates a set of points in di-jets, is dominated by thefigure5). p-wave In (velocity this suppressed) case annihilation a channel large width ¯ (Γ matter detection limits discussed in theregions previous in sub-section. figure We can11: clearly distinguish (i) three The low mass region ( suppression effect and to avoid second region, 300 GeV pole region of the same diagram (figure5). In this case, a narrow width of JCAP07(2016)005 . BB C and considered Φ GG 300 GeV, the λ C . χ m ) plane that simultaneously re- s ,Γ χ couplings needed to fit the observed m BB below 12 GeV. The latter corresponds to ,C s GG C – 17 – couplings affects the running of the gauge couplings, has an odd parity, there is no velocity suppression and more precisely, it will become possible to estimate the . In view of this, the higher values of s Φ sa BB 60 GeV, might result in tension with constraints from the λ − ,C 40 GG abundance of the dark matter. (iii) Finally, in the last region C ∼ s (10 GeV)) is needed to respect the WMAP/PLANCK constraint. under ∼ O . As the final state χ : Summary plot representing points in the ( 8 which allows to maintain a safe theoretical framework at least up to an energy scale 400 GeV), the t-channel (s-wave) annihilation depicted in figure7 determines the relic . 0 Before moving to the conclusion we just briefly comment on some theoretical aspects It is also interesting to notice from figure 11 that if the future ATLAS/CMS analysis & . χ Φ m As pointed out, forloop example, level in to refs. [20, generatedriving25, the them182, towards217, theThus,218], non-perturbative too the regime extra much matterwe already high needed then at values at remark energy of the thatLHC scales these the excess, of couplings low compared a remain to valuesalso few of the disfavoured. the TeV. case the theoretical of Similarconsidered a constraints, in to resonance this besides work, ref. decaying a the strong into [182], running impact two is experimental of isolated still retained ones. photons, the by thehere, relax quartic renormalization corresponding In group equation coupling to scenarios Γ theoretical like consistency. the On one the other hand, we remark that except for phenomenological constraints favour values of Γ λ relevant for the phenomenological analysis. 5 Conclusion We have considered in this notethrough the the possibility decay that of the di-photon twodistinguishable signal light from is pseudoscalars the in into reality isolated two produced photons. pairs of We have highly shown collimated that, photons, in in- a spontaneous symmetry relative to our scenario. Ourwe work have is not motivated by referred pure to phenomenological purposes any and particular thus, setup for the origin of the couplings spect the LHC and cosmologicalcross-sections, constraints and as correspond depicted to with different di-photon the production three different colours. is necessary to( avoid abundance of a moderate width ( dark matter mass, consistentconfirmed, with the an cosmological electroweak scale constraint. mass If will the be large favoured. width scenario is Figure 11 determines the value of the width Γ JCAP07(2016)005 , ) , D , of TeV and ] pp 8 1 C 76 − of Phys. Rev. fb 1 , 3 − Confronting . 3 fb 2 . 3 GeV diphoton excess GeV diphoton excess Eur. Phys. J. 750 , PITN-GA-2011- 289442 750 arXiv:1602.07866 , the France-US PICS no. GeV diphoton excesses in a extension of the minimal L − 750 GeV diphoton excess B ]. 750 The U(1) ]. , ATLAS-CONF-2015-081 (2015). (2016) 111702[ SPIRE ]. Addressing the LHC IN FPA2010-17747 [ SPIRE IN interpretation on the [ D 93 Collider tests of (composite) diphoton resonances SPIRE ]. – 18 – IN U(1) ][ × TeV and combined interpretation of searches at SPIRE Faking the diphoton excess by displaced dark photon decays IN Phys. Rev. (“HiggsTools”). , , the contract ][ = 13 SU(2) s Left-right asymmetry and ]. ]. ]. × arXiv:1603.01606 Diphoton resonances in a √ , , CMS-PAS-EXO-15-004 (2015). arXiv:1602.09099 , Search for resonances in diphoton events with the ATLAS detector at Search for resonances decaying to photon pairs in SPIRE SPIRE SPIRE TeV with the ATLAS detector TeV Search for new physics in high mass diphoton eventsSearch in for proton-proton new physics in high mass diphoton events in IN IN IN [ [ [ arXiv:1603.05146 = 13 = 13 s s CSD2009-00064 , ATLAS-CONF-2016-018 (2016). √ √ arXiv:1603.04479 TeV ]. , CMS-PAS-EXO-16-018 (2016). 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Shafi, breaking framework, once onesible couples to the re-open scalarexperimental a sector constraints. to large the range dark of matter, the it dark appearsAcknowledgments matter fea- masses compatible with allY.M. the wants present to thankwork. especially The authors J.B thanks dework Emilian is Vivie Dudas also and whose supported Ulrich help by Ellwangergrant the for was Multi-Dark Spanish fruitful fundamental MICINN’s discussions. throughout Consolider-Ingenio This 2010 our Programme under and the ERC advanceddivision grants for Higgs@LHC the and hospitalitythe MassTeV. during support part G. of from A. the P2IO thanksin completion Excellence the part of Laboratory CERN by this (LABEX). the theory project.Agreement This Research P.G. research Executive acknowledges was Agency also (REA) supported of the EuropeanReferences Union under the Grant 06482 and thethe LIA-TCAP European of Union CNRS. FP7 ITN Y. INVISIBLES M. (Marie Curie and Actions, G. 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