#### 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
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