Shining light on polarizable dark particles

Journal of High Energy Physics, Apr 2017

We investigate the possibilities of searching for a self-conjugate polarizable particle in the self-interactions of light. We first observe that polarizability can arise either from the exchange of mediator states or as a consequence of the inner structure of the particle. To exemplify this second possibility we calculate the polarizability of a neutral bosonic open string, and find it is described only by dimension-8 operators. Focussing on the spin-0 case, we calculate the light-by-light scattering amplitudes induced by the dimension-6 and 8 polarizability operators. Performing a simulation of exclusive diphoton production with proton tagging at the LHC, we find that the imprint of the polarizable dark particle can be potentially detected at 5σ significance for mass and cutoff reaching values above the TeV scale, for \( \sqrt{s}=13 \) TeV and 300 fb−1 of integrated luminosity. If the polarizable dark particle is stable, it can be a dark matter candidate, in which case we argue this exclusive diphoton search may complement the existing LHC searches for polarizable dark matter.

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Shining light on polarizable dark particles

Received: December Shining light on polarizable dark particles Sylvain Fichet 0 1 2 3 4 5 0 Rua Dr. Bento Teobaldo Ferraz 271, Bloco 2 , Barra Funda , Brazil 1 Instituto de Fisica Teorica, Sao Paulo State University 2 ICTP South American Institute for Fundamental Research 3 Open Access , c The Authors 4 signi cance for mass and cuto 5 nd that the imprint of the polarizable We investigate the possibilities of searching for a self-conjugate polarizable particle in the self-interactions of light. We rst observe that polarizability can arise either from the exchange of mediator states or as a consequence of the inner structure of the particle. To exemplify this second possibility we calculate the polarizability of a neutral bosonic open string, and nd it is described only by dimension-8 operators. Focussing on the spin-0 case, we calculate the light-by-light scattering amplitudes induced by the dimension-6 and 8 polarizability operators. Performing a simulation of exclusive diphoton production with proton tagging at the LHC, we dark particle can be potentially detected at 5 values above the TeV scale, for p s = 13 TeV and 300 fb 1 of integrated luminosity. If the polarizable dark particle is stable, it can be a dark matter candidate, in which case we argue this exclusive diphoton search may complement the existing LHC searches for polarizable dark matter. Forward physics; Dark matter; Hadron-Hadron scattering (experiments) 1 Introduction 2 Polarizability operators 2.1 Microscopic origin Mediated polarizability Intrinsic polarizability Consistency of the approach Consistency of the calculation Helicity amplitudes Light-by-light scattering as a probe for polarizable dark particles Sensitivity at 13 TeV and L = 300 fb 1 5.2 Interplay with other searches for a stable dark particle Conclusions A Neutral open string in an electromagnetic background B Four-photon amplitude calculations Polarizability of the neutral bosonic string Four-photon amplitudes from polarizable dark particles Introduction Among the speculations about what lies beyond the Standard Model (SM) of particles, there is the intriguing possibility of particles that are electrically neutral but can still slightly interact with photons. The existence of such \almost-dark" particles is theoretically well-motivated. These could for example be the hadrons created by a hidden strongly-interacting gauge force, binding together electrically charged constituents (for recent scenarios featuring such bound states, see for instance Stealth dark matter [1] and Vectorlike con nement [2]). Models where the dark particle has an electromagnetic coupling have been investigated in the scope of explaining Dark Matter (DM) of the universe. When interpreted as DM, the dark particle is assumed to be stable. In the present work this assumption of stability will not be needed. In general, a particle with no electric charge may still interact with one photon through a dipole operator and/or a charge radius operator | in which case the particle couples directly to the photon eld strength F . Such scenarios for dark particles have been investigated in [3{20] in the context of dark matter. However, if the neutral particle can be described as a self-conjugate eld in a low-energy theory, these operators vanish.1 The main interaction of the dark particle with light is then controlled by its polarizability, i.e. its tendency to interact with two photons. Such scenarios have also been investigated [4, 16, 21{28], still in the context of dark matter. Such polarizable dark particles are the topic of the present paper. The interactions of our focus are bilinear in both the dark particle and in F the case of a scalar, a linear coupling of the form (F )2 may also exist in principle. We will assume this coupling is either negligible or forbidden by a symmetry.2 By electroweak (EW) gauge invariance, a dark particle with electromagnetic polarizability should also be polarizable with respect to the W and Z bosons. This aspect will play little role in our analysis, but is relevant when it comes to comparisons with the literature. To the best of our knowledge of the literature, most of the searches for a dark particle polarizable by EW gauge bosons are done within the assumption that the dark particle is stable. This is true by de nition for direct and indirect detection, and is also the case for collider searches [22, 23, 26, 28{37], where the search strategies always involve detection of large missing energy. If this kind of searches turned out to be successful, it would provide a striking signature for the existence of dark matter. In this paper we would like to adopt a slightly di erent strategy. Instead of readily testing the existence of a stable dark particle, we propose to rather test the existence of a dark particle, whether it is stable or not. In such approach, the assessment of stability is postponed to the post-discovery era, together with the characterization of the other properties of the new particle such as spin and mass. A consequence of this approach is that observing a large missing energy is not required anymore. Rather, one can set up a search which is independent of the hypothesis of stability. Adopting this slightly di erent viewpoint naturally leads to consider searches for the e ects of virtual polarizable dark particles. From a theoretical viewpoint, one advantage of looking for virtual dark particles is that, if a full dark sector is present, all the dark polarizable particles contribute to the signal, and not only the stable ones. This implies that the signal is enhanced with respect to searches only focussed on stable dark particles (e.g. usual DM searches). In the following we will consider the case of a single dark particle, unless stated otherwise. As a rst step, we will classify the CP-even polarizability operators up to dimension-8 and discuss their possible microscopic origin in section 2. As an example, the intrinsic, dimension-8 polarizability of the neutral bosonic open string case is calculated in section 3. These preliminary studies are needed to establish under which conditions the virtual search we propose is relevant. The virtual process we will focus on in this paper is photon-photon scattering. The amplitudes in the spin-0 case are given in section 4. Moreover we will 1By self-conjugate we mean a eld transforming in a real representation: real scalar, real vector, Majorana fermion. . . . 2The presence of a sizeable (F )2 term would considerably change the phenomenological prospects for the dark particle. In particular, the dark particle could be constrained by resonant production at colliders, and by Casimir force experiments if its mass is below the keV scale. This scenario lies outside the scope of consider the exclusive channel, where outgoing protons remain intact and are detected. The simulation and its results are presented in section 5, and section 6 contains our conclusions. Polarizability operators We use a low-energy e ective eld theory (EFT) approach. Here the set of CP-even polarizability operators up to dimension-8 is classi ed. One writes down the operators featuring two photon eld strengths and two dark particles of a given spin. It will be claimed below that the dimension-6 operators can be vanishing depending on the UV origin of polarizability, hence the dimension-8 operators can potentially be the dominant ones. The cuto scale is denoted , and validity of the e ective description of polarizability by local operators requires that the dark particle mass and the energy owing through the polarizability vertices be smaller than (see also section 4.1). The e ective Lagrangian describing the spin-s polarizable dark particle has the form Ls = Lkin + Ls6 + Ls7 + Ls8 + O s One introduces the dual electromagnetic eld strength = 12 (F )2 = F F , and one de nes = F F ; (F F~) = F = F O81=a~2 = i (F~:F~) , O81a~ = X X (F~:F~) table 1, as they decompose as The coe cients of the operators of eq. (2.1) should be understood as given at the EFT matching scale . These coe cients should be in general written as ci( ). However, only the coe cients de ned at the scale will e ectively appear in our results, hence we will simply refer to them as ci in the following. be denoted m. The operators allowed by EW gauge invariance and inequivalent under eld rede nitions and integration by parts are classi ed in table 1. We do not include operators that induce a coupling to gauge bosons after EW breaking, which would arise from Higgs coare not independent of the ones given in O8sa~ = O8sa + and are thus not included. When truncating the EFT expansion at dimension 8, higher-order contributions to the coe cients of operators with lower dimension should be kept up to dimension 8 (see [38] for related discussions). We will actually encounter such situation, with dimension-6 and 7 operators coming respectively with a prefactor m2 2 and m . These operators are of dimension-8 in the sense they come with a 4 factor, and will be referred to as The coe cients of the O^6si operators will be written c^s6i, and similarly for O^7si. 3We use a metric with (+; ; ; ) signature, except in section 3. Spin 1=2 X X (F:F ) (X:X) (F:F ) (X:F ) (X:F ) notations for the eld strength contractions of X are the same as for F . Finally, we remark that the dimension-6 and 7 operators can be naturally suppressed with respect to the dimension-8 ones if the dark particle has an approximate shift-symmetry. When this happens, the dark particle mass should be suppressed similarly. parametrize the explicit breaking of the shift-symmetry using the dark particle mass, and thus be identi ed as the hatted operators of eq. (2.4). This situation occurs for instance if the dark particle is the Nambu-Goldstone particle of a spontaneously broken approximate global symmetry, for example a U(n) symmetry or supersymmetry, respectively giving a Nambu-Goldstone scalar and a Nambu-Goldstini (see [39, 40] for a related analysis in the context of dark matter). Microscopic origin Even though we simply listed the polarizablity operators in an e ective theory approach, some aspects of the UV origin of these operators can already be deduced. We identify two mechanisms. Either polarizability could arise from the exchange of heavy virtual particles, referred to as \mediator". Or the polarizable particle may actually be an extended object in the UV, and polarizability could then originate from the inner structure of the particle. We shall refer to these two scenarios as mediated polarizability and intrinsic polarizability. Here we consider the case of polarizability induced by heavy mediators. First, we notice that no operator in table 1 can be generated via the tree-level exchange of a particle in a fully renormalizable theory. It may seem possible in the case of the O61b operator, starting from a dipole operator and integrating out the heavy spin-1 mediator Y . However, renormalisability requires X and Y to arise as massive gauge elds of a spontaneously broken gauge symmetry G arise from the kinetic term of the G gauge eld. Inspecting the broken sector (see [41]), it turns out that OXY is controlled by the broken constant structures f a^^bc, where the hatted 1 (unhatted) indexes label the broken (unbroken) generators. These same constant structures determine the coupling of the massive gauge elds to the electroweak gauge elds. One concludes that the X and Y 1 elds have to be charged in order for OXY to be non-zero. This is in contradiction with the hypothesis of a self-conjugate X, therefore polarizability of X cannot be induced by tree-level exchange of a spin-1 mediator in a renormalizable theory. Possibilities for tree-level exchange of heavy mediators arise in be generated together if a Majorana fermion shares a dipole operator with another, heavier Majorana fermion [21]. A similar possibility is that the components of be part of a single Dirac fermion, with a mass splitting induced by a Majorana 1=2 mass [42]. Finally, the O80a, O8a , O81a can be generated by a spin-2 mediator (such as a 1=2 Kaluza-Klein graviton), together with O80b, O8b , O81b terms coming from tracelessness of the spin-2 representation. All of the operators of table 1 could in principle be generated at loop-level, in particular by loops of charged mediators. In such case, the coe cient cis of the polarizability operators must come with a factor e2=16 2, and is identi ed with the mass of the particle in the loop. Intrinsic polarizability Here, we consider the possibility that polarizability arises from the inner structure of the dark particle. Let us consider a generic 4-point amplitude with two dark particles and two photons in external legs. We focus on the scalar case for concreteness, but the same reasoning applies to spin 1/2 and 1 similarly. The scattering amplitude has the form M = is a function of the momenta and of the intrinsic scale of the dark particle . Using Ward identities and the fact that the photon does not couple to the dark particle through covariant derivatives by de nition, we readily know that the V where one introduces the projector R (p) = p g p g . The dimensionless tensor F a a b b is the general form factor of the dark particle, that encodes the information about its inner structure. In the low-energy domain s; t; u; m2 < 2, where s; t; u are the Mandelstam variables, the lower order Lorentz structures can then be written as F a a b b = F0(s; t; u; )g a b g a b + 2 F1(s; t; u; )p1a p2b g a b + O We assume that a massless polarizable dark particle can exist, and thus ask for the amplitude to remain nite in the massless limit. This implies that the form factors should decrease at least as F0;1 m2= 2 at small m= in order to compensate the m 2 in eq. (2.7). This can also be checked taking the massless limit of the amplitudes of section 4.3. Expanding the form factors for large and using the symmetries of the diaF~0(s; t; u; ) = A + (Bp1:p2 + Cm2)= 2 + O( 4), F~1(s; t; u; ) = D + O( 2) where A, B, C, D are constants. The general form factor reads F a a b b = Bp1:p2 + Cm2 All the terms vanish in the pointlike limit ! 1, as expected from e ects arising from compositeness. The A, B, C, D constants are in direct correspondence with the spin-0 e ective operators of table 1. Identifying the Lorentz structures, one has simply A = c60a B = c80b C = c^60a D = c80a : We can deduce some physical features of the polarizability operators by studying the non-relativistic limit (pi)2 m2, where pi is the three-momentum. Here we limit our Let us rst remark that in the non-relativistic limit the operators satisfy where Ei, Bi are the standard electric and magnetic elds. One also has The (Ei)2 and (Bi)2 term that appear in the non-relativistic Lagrangian correspond respectively to the static electric and magnetic susceptibilities of the inner structure of the dark s + t + u = 2m2. 5The case of a massive non-relativistic spin-1 particle is not straigthforward to analyze, for example one particle. We can see that the O8a with purely electric (respectively magnetic) origin. These properties can in turn be used to infer some features of the polarizability operators for a given object. In the case of dark hadrons, made of electrically charged fermions glued by a hidden strong interaction, we certainly expect an electric polarizability, as these constituents form an electronic density that can be deformed by an external electric eld. Also, as the constituents carry intrinsic spin, a magnetic polarizability should exist, however both theoretical arguments [43] and observations [44] suggest that it is suppressed with respect 2 0 to the electric one, thus one may expect c08b + c^60a + m2 c6a c18=a2. The case of a neutral string with non-zero charges at endpoints is also interesting. In that case one should consider the e ective operators associated with the quantum states of the string.6 An electric polarizability should exist as one has two charges binded together. In contrast, as no intrinsic spin is attached to any point of the string, the string cannot have Finally, one may wonder how electro-magnetic duality applies to the arguments above. From a macroscopic viewpoint, electromagnetic duality exchange the suscepti B, and thus exchanges the E 6= 0; M = 0 case with E = 0; M 6= 0 in the string case. Microscopically, such object would be a sort of open string with magnetic monopoles attached at endpoints. Such objects, called D-strings, do exist in string theories, and are related by S-duality to the original strings (see [46]). From a low energy point of view, the polarizability of such objects should be expected to be described by the O8a~ Polarizability of the neutral bosonic string To give a concrete example of an object with intrinsic polarizability, we work out the case q1 = q). For the sake of describing polarizability of the string states, there is no need to assume that spacetime has critical dimension. In fact, being ultimately interested in the 4D case, the string we consider cannot be considered as a fundamental one. Instead, it may for example be taken as a QCD-like string, i.e. an e ective description of the binding between a quark and an antiquark arising in a gauge theory with large number of colors. A mostly-plus signature ( ; +; +; +) is used for g in this section. The action of an open string with length scale ls ground is given by p 0 in an electromagnetic backS = A X_ where A is the canonically normalized electromagnetic eld. Propagation of a bosonic open string in an abelian background gauge eld has been worked out in ref. [47], and 6The dipole operators associated with the quantum states of the open bosonic and super strings have been evaluated in ref. [45] and are vanishing if the string is neutral. canonical quantization is done in details in ref. [48]. Our calculation follows closely [48], details are given in appendix A. Computing the solutions of the equation of motion, de ning an orthogonal basis for the oscillator and zero modes, and asking for canonical commutators between the position, momentum and Fourier operators x ; p ; a(ny), the string decomposition over orthonormal = x + 2ls2 g It turns out that the background eld does not a ect the oscillator modes, only the zero mode gets deformed.7 The L0 Virasoro operator of the open string is then given by L0 = where N = P1 n = pn ayn. The states of the string are built from a ground state j0i using creation operators, 1 1::: s (x ) = The L0 operator satis es the condition = 0 ; where a is a constant from normal ordering which is left unspeci ed and is irrelevant regarding the property of polarizability.8 Equation (3.5) gives the equation of motion for the string states, 1 2::: s = 0 ; of ls gives 1 2::: s = 0 : This equation of motion describes the polarizability of a string state of any integer spin s. Going back to the mostly-minus metric used for the e ective Lagrangian of eq. (2.1), we can deduce the Lagrangian giving rise to the equation of motion eq. (3.7) in case of spin 0 7There is a freedom in normalising the x and p operators inside the zero mode. It is convenient to let the position operator unchanged and to incorporate all the e ect of the background eld into the 8We will assume a 0 whenever discussing the spin-0 state, otherwise it is tachyonic. and 1. We conclude that polarizability of the spin-0 state and spin-1 state is respectively described by the operators O80a, O81a. Identifying the operator coe cients are (2 )2 q2, so that the e ective Lagrangian is with the inverse string length, 4 O80a + 4 O81a : = ls 1, Establishing the consistent Lagrangian for a neutral polarizable state of higher spin is probably more challenging conceptually and technically, and lies outside the scope of this study. In particular, the electromagnetic interactions of the auxiliary elds present in the higher-spin Lagrangian would have to be determined.9 Four-photon amplitudes from polarizable dark particles Polarizable dark particles automatically induce loops with four external photon legs (see gure 1). Following our strategy of focussing on virtual processes (see section 1), we propose to use such anomalous photon couplings as a probe for the existence of a dark particle. For a rst analysis of this proposal, we focus on the case of a dark particle of spin-0. The spin-1/2 and spin-1 cases would deserve to be treated similarly, but lie outside the scope of this paper. Consistency of the approach A necessary condition for our proposal to make sense is that the dark particle produces the main contribution to the four-photon coupling. While in principle the complete UV picture is needed to answer this question, the considerations on the microscopic origin of polarizability made in section 2 already provide a useful constraint. Indeed, in the case where polarizability is induced by mediators, the mediators themselves can form diagrams with four external photons. The contributions from dark particles are expected to be smaller than the ones from mediators by at least a loop factor. This happens in both the cases of loop and tree diagrams, induced respectively by charged mediators and mediators with non-renormalizable couplings. The four-photon search then essentially probes the existence of these mediators. The sensitivity for such particles has already been estimated [49, 50], irrespective of the existence of a dark particle. In contrast, if polarizability originates from the inner structure of the dark particle, the dark particle loop can in principle be the dominant contribution to the anomalous four-photon vertex. Some consistency constraints also come from the validity of the EFT approach. The validity of the low-energy expansion requires that otherwise the form factor from UV physics becomes important, and the description of polarizability of the dark particle by local operators is not valid anymore. The partonic center-of-mass energy for exclusive photon scattering is typically of ps 1 TeV at the 9These aspects might be treated in a further work. 13 TeV LHC. Moreover, tree-level unitary of photon - dark particle scattering imposes the jc80ajs2= 4 < 16 ; jc60ajs= 2 < 8 ; , the bound translates as ci < 8 . It is worth noticing that for are allowed to be larger than 8 . In our estimations of LHC sensitivity of section 5 we will to requiring perturbativity of the e ective interactions in the EFT. Constraints similar to those of eq. (4.2) can be obtained by requiring that a diagram with n + 1 loops be smaller or of same order of magnitude than a diagram with n loops (when using dimensional Consistency of the calculation An important subtlety is that the four-photon loop diagrams we consider come from higherdimensional operators and are thus more divergent than the four-photon diagrams from the UV theory. This implies that four-photon local operators (i.e. counter-terms) are also present in the e ective Lagrangian to cancel the divergences which are not present in the UV theory. The nite contribution from these local operators is xed by the UV theory at the matching scale, and is expected to be of same order as the coe cient of the log in the amplitude by naive dimensional analysis (this situation is analog to renormalisation of the non-linear sigma model, see ref. [51]). This implies that the amplitudes obtained from calculating the loop graphs should only be considered as estimates of the complete amplitudes, the latter being determined only once the UV theory is speci ed. Concretely, for four-photon interactions induced by loops with dimension-8 operators, local four-photon operators of dimension-12 are present in the Lagrangian. Four-photon interactions induced by loops of dimension-6 operators imply the presence of dimension-8 operators, corresponding to the two Lorentz structures shown in eq. (4.19). Cuto regularisation in an e ective theory is very di cult because it breaks the expansion with respect to 1, as loops from operators of arbitrarily high dimension contribute at same order to the amplitudes (see [51]). A much simpler scheme is dimensional regularisation, in which case power-counting is respected and it is thus consistent to include only operators of lower dimension (up to dimension-8 in our case). The matching of the e ective theory with the UV theory being done at the scale , we can readily identify the log( = 2) ; 2 log( = 2) : As a nal remark, we note that in the limit of heavy mass, m2 s; t; u, the loops reduce to local e ective interactions. The amplitudes from these local interactions are given in [50], and have the Lorentz structure check for our loop calculations, we observe that all our amplitudes reproduce the structure of eq. (4.6) at dimension-8 operators corresponding to each loop can also be deduced from eq. (4.6), and will be given below. Helicity amplitudes Focussing on the case of a spin-0 dark particle, we calculate the four-photon amplitudes induced by the dimension-8 polarizability operators O80a, O80b, O^60a, which are theoretically well-motivated as discussed in section 4.1. We limit ourselves to cases where one of these operators is dominant and do not calculate diagrams involving two di erent operators. Helicity amplitudes are given under the form M a b 1 2 (s; t; u), where a;b = notes the polarization of two ingoing photons and 1;2 denotes the polarization of two out+(s; t; u) = M++++(t; s; u), only the M++++, M++ con gurations have to be calculated (see ref. [54]). Full amplitudes and details of the calculation are given in appendix B. The amplitude is found to be exactly zero in all cases. Here below we display only the helicity amplitudes in the high energy limit m2 s; t; u and in the low-energy limit m2, where in both cases s; t; u; m2 < divergent integrals as (see [52, 53])10 10The running of the ci0( ) coe cients is taken into account at leading-log order with this method. + 175 s4 log s2 + t4 log t2 + u4 log u2 : M++++ = 3 (c80b)2 s2m4 log m22 ; 2 2 8 + s2 log s2 + t2 log t2 + u2 log u2 : O^60a operator Finally, in the m2 s;t;u case, it is well-known that four-photon interactions can be represented by two independent dimension-8 operators and the helicity amplitudes as a function of the b1;2 coe cients have been given in ref. [49].11 Matching these amplitudes to the low-energy limit of the ones from loops of polarizable particles eqs. (4.9), (4.10), (4.13), (4.14), (4.17), (4.18) gives b1 = ; b2 = from the O80a operator, from the O^60a operator. b1 = b1 = ; b2 = 0 ; b2 = 0 Light-by-light scattering as a probe for polarizable dark particles We propose to focus on photon-photon scattering in the exclusive channel, where the two protons remain intact after the collision, These intact protons can be detected and characterised using forward proton detectors along the beam pipe, that are scheduled by both ATLAS [55] and CMS/TOTEM [56] collaborations. The interest of the exclusive diphoton channel with proton characterization is that there is enough kinematic information to eliminate most of the background. The sensitivity of this measurement to new physics has been studied in details in [49, 50, 57], where the residual background rate after all cuts has been estimated to 3 10 4 fb. This background comes from inclusive diphoton events occuring simultaneously with the tagging of two intact protons from pileup.12 Sensitivity at 13 TeV and L = 300 fb 1 In order to obtain a realistic estimation for the discovery potential of the dark particle, we implemented the four-photon amplitudes induced by dark particles in the Forward Physics Monte Carlo generator (FPMC [68]). The model of photon ux of ref. [69] is assumed. We reproduce the acceptance of the forward detectors by constraining the fractional momentum loss of both protons to be13 11To adapt the amplitudes given in the conventions of [49] to the ones of the present paper, the amplitudes in [49] have to be multiplied by a factor 12Other studies using proton-tagging at the LHC for New Physics searches can be found in refs. [41, 58{66]. We refer to [67] for a study of light-by-light scattering at the LHC without proton tagging. 13For CMS, these expectations have recently been updated to be 0:037 < < 0:15 [70]. We checked that our results are essentially the same with this new range, the sensitivity regions decrease only slightly. O80a polarizability O80b polarizability EFT EFT 1000 1500 2000 2500 3000 2000 2500 3000 plane, and assuming sensitivity in presence of N = 5 copies of the dark particle. The two dotted lines corresponds to the 5 sensitivities for the spin-0 = ls 1. We set a cut of jpT j > 150 GeV on the transverse momentum of each photon. Like in ref. [50], the main impact on the signal rates is expected to come from these cuts. We include the e ect of the other cuts on the signal with a global e ciency of s = 90%. The average sensitivities for a signal induced by the O80a, O80b, O60a operators are shown in gures 2 and 3. We simply set that 3 statistical signi cance for the existence analysis give similar conclusions. The uncertainty on the cross-section is expected to blow up when approaching the m = limit. In fact, the cross-sections used for the gures are probably under-estimated in this region because we did not included the local 4 operators small (see also discussion in section 4.2). We observe that for the chosen values of ci0, the sensitivity regions can go above the TeV. It turns out that the sensitivity for the O80b operator is better than for the O^60a, have sensibly di erent shapes. In particular, a sensitivity remains at low mass for the O80a, O80b operators, while it vanishes for the O^60a operator. These estimations are for a single self-conjugate scalar. An important point to keep in mind is that the search we propose is multiplicity-sensitive: the more the dark sector is populated by polarizable dark particles, the more the sensitivity regions improve. For illustration we show how the regions grow photon cross-section is enhanced by a N 2 factor. O^60a polarizability O60a polarizability EFT Beyond EFT 2000 2500 3000 2000 2500 3000 Although our present study is limited to the case of one operator turned on at a time, some conclusions can already be drawn regarding some realizations of the spin-0 dark polarizability, for which the photon-photon search is the less sensitive. However, if one charge of q = 1, the sensitivity reaches m 3 TeV, as shown in gure 2. Regarding the dark spin-0 baryon of the Stealth DM scenario [27], only a polarizability of O8a has been considered. However, to the best of our understanding, the O80b, O^60a 0 operators do not need to be zero, provided that the sum of their coe cients is small (see section 2). This may make an important di erence in the prospects for the diphoton search, as the sensitivity to O^60a and particularly O80b is much better than for the O80a coe cient. Finally, for a pNGB dark particle, we expect all of the three operators to be non-zero.14 The present study, as a proof of principle, is limited to the spin-0 case and to turning on one operator at a time. Given these encouraging rst results, it would be worthwhile to go further by computing the loops in presence of all operators at a time. Also, it would be certainly interesting to similarly analyze the spin-1/2 and spin-1 cases. Interplay with other searches for a stable dark particle Here we brie y discuss the case where the dark particle is stable and identi ed as dark matter. We recall that, compared to DM searches, a general drawback of the diphoton search is that it does not detect stability, while a general advantage is its sensitivity to the entire spectrum of polarizable dark particles. 14For the pNGB dark particle, it is not clear to us if the operators should enter in a speci c combination. Comparison with collider searches. A quantitative comparison with the reach of missing-energy searches obtained in the literature (see e.g. [22, 26]) would require to take into account the nature of the dark particle, the assumed luminosity and center-of-mass energy, assumptions on the couplings, normalization of the operators and statistical criteria. Here we will remain at a qualitative level. We observe that in the prospects for missing-energy based searches at the 13/14 TeV LHC, the sensitivity drops quickly above m > 1 TeV. While in our case, one can see from gures 2, 3 that the sensitivity goes over regions with masses above 1 TeV. This can be understood from the kinematics of the two kinds of process: the cross-section for producing two on-shell dark particles plus other states drops faster with the center-of-mass energy than for a photon-photon nal state. There is thus a complementarity between the two kind of searches. One can also notice that if the stable dark particle has a multiplicity N , the diphoton cross-section grows with N 2, but the cross-section for pair production grows only with N . Thus a large multiplicity for the stable dark particle favours the diphoton search, as the photon-photon production is enhanced by N with respect to pair-production.15 For these reasons we conclude that, qualitatively, the proposed diphoton search seems to compete with and sometimes complement missing-energy searches at the LHC. Comment on indirect detection. A strong constraint on stable polarizable dark particles naturally comes from indirect detection bounds on photons. If the annihilation rate is not velocity-suppressed, these bounds are expected (see [26] and references therein) to dominate over collider and direct searches. As velocity-suppression annihilation is a crucial izabilities. None of these operators alone lead to a suppressed annihilation rate. However, it turns out that the full squared matrix element takes the form of a complete square 4(c8b + c^6a) Thus there exists a combination of coe cients for which the annihilation rate is velocitycorresponds precisely to coupling the traceless part of F Such operator appears in particular when integrating out a heavy spin-2 particle, like a KK graviton. It would be interesting to further investigate this e ective scenario of a \spin-2 portal". From the point of view of the diphoton search, the spin-2 particle is a mediator, thus the loop of the polarizable scalar is subdominant with respect to the spin-2 induced four-photon loop. It would be interesting to investigate whether the combination of eq. (5.4) can vanish in a scenario with intrinsic polarizability. 15A roughly similar conclusion is expected for N particles which are non-degenerate, as the decay chains of unstable particles end up with the stable one and thus contribute to missing energy signatures. 16This is consistent with the velocity-suppressed rate found in [39], table 4. We propose to test the existence of a self-conjugate polarizable particle by searching for the virtual e ects it induces. We focus on the process of photon-photon scattering, occuring via loops of this \almost dark" particle. The method does not depend on whether the particle is stable. Thus if there is a dark sector with many polarizable dark particles, the search is sensitive to the cumulative e ect of the whole spectrum. As a preliminary step we classi ed the CP-even polarizability operators up to dimension microscopic nature of polarizability: mediated and intrinsic polarizability. We illustrate intrinsic polarizability in the case of a neutral bosonic open string and nd it is described by dimension-8 operators. The scenario of a dark particle with intrinsic polarizability is the relevant one for the search we propose. Focussing on the spin-0 case, we evaluate the four-photon helicity amplitudes induced by the dimension-8 polarizability operators. The matching of this e ective interaction onto local four-photon operators for s m2 is also provided. We then evaluate the prospects of a pp ! pp search at the 13 TeV LHC using forward detectors to characterize the intact protons. This channel is known for being sensitive to new physics searches. For operator coe cients equal to 10, it turns out that the sensitivity in mass and cuto can go beyond the TeV. For the string with unit charge, mass and inverse string length can be probed up to roughly 1:5 TeV. The center-of-mass energy of the process is typically of 1 TeV, hence the EFT expansion is roughly valid unless the coe cients of the operators get too small. In case the dark particle is stable, it is a DM candidate. In this context we qualitatively compare DM collider searches with our diphoton search. It turns out that these two methods are fairly complementary, as the diphoton search tends to have a sensitivity to higher masses and is multiplicity-enhanced. The annihilation rate of two dark particles into photons is found to be suppressed if the c08a 4(c08b + c^60a) combination vanishes. This happens in case of mediated polarizability from a spin-2 particle, and it would be interesting to nd a UV completion of intrinsic polarizability in which this cancellation occurs. We emphasize that the present study of the spin-0 case should be taken as a proof of concept, used to get a rough idea of the sensitivities that can be reached. As the rst conclusions seem encouraging, it would be interesting to further analyze the spin-0 case, Acknowledgments I would like to thank E. Ponton, G. von Gersdor , R. Mattheus and A. Ballon for useful discussions and M. Saimpert for clari cations on FPMC. This work was supported by the S~ao Paulo Research Foundation (FAPESP) under grants #2011/11973 and #2014/21477-2. Neutral open string in an electromagnetic background following from eq. (3.1) are given by X00 = 0 ; X0 = qF X with q0 = 2 2 block diagonal form by orthogonal transformations, and it is thus enough to focus on two dimensions, taken to be space dimensions with = 1; 2. One has n = i mn sgn(n) : The boundary conditions become simply The oscillator modes are according to the inner product h mj ni = X+ = p (X1 + iX2) ; = p (X1 iX2) : X+0 = iqf X_ + if = 1; 2 : Is is further convenient to rotate space coordinates as P = 1 X_ + + qA+( ( ) To go further, one uses the approximation that the background eld is constant. The potential is then linear in X1;2, and one can make the following gauge choice as in [48], which reproduces well the background eld eq. (A.3) when using the de nition = @ A @ A . This provides the canonical momentum A = f P = which gives eq. (3.3) using We have then everything to express the operators x , p , a(ny), in terms of X P , using the inner product and eq. (A.10). Using the canonical equal-time commutators [Xu( ; ); Xv( ; 0)] = 0 ; [Pu( ; ); Pv( ; 0)] = 0 ; [Xu( ; ); Pv( ; 0)] = i uv ( we can check that all the operators satisfy well canonical commutation relations. Finally, the L0 operator of the Virasao algebra is given by L0 = 1 X2 (X_ + X0 )2 = (X_ + + X+0)(X_ after rotating back to X1;2 coordinates and putting together all block matrices to restore all dimensions of spacetime. Four-photon amplitude calculations We de ne = m2 x)q2. After loop integration, all the x-dependence of the numerators appears via powers of x(1 x) after combination of all terms. Thus we introduce a basis of loop functions fn(q2; m; ) = over which all amplitudes decompose. One further introduces the combinations A(q2; m; ) = (m4f0 2m2q2f1 + q4f2) ; X(q2; m; ) = (3m4 + 2m2q2)f0 (30m2q2 + 2q4)f1 + 28q4f2 ; C(q2; m; ) = (12m4 + 2q2m2)f0 (32m2q2 + 2q4)f1 + 24q4f2 : The helicity amplitudes are then given by M++++ = = 0 : s2(X(s; m; ) + A(t; m; ) + A(u; m; )) ; 32 2 8 (s2 X(s; m; ) + t2 X(t; m; ) + u2 X(u; m; )) ; M++++ = = 0 : M++++ = = 0 : 8 2 8 (s2C(s; m; ) + t2C(t; m; ) + u2C(u; m; )) ; 2 2 4 (s2f0(s; m; ) + t2f0(t; m; ) + u2f0(u; m; )) ; The unpolarized cross-section is given by Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. baryon dark matter on the lattice: SU(4) baryon spectrum and the e ective Higgs interaction, Phys. Rev. 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Shining light on polarizable dark particles, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP04(2017)088