Continuum-Mediated Self-Interacting Dark Matter
UUCR-TR-2021-FLIP-MSE-6
Continuum-MediatedSelf-Interacting Dark Matter
Ian Chaffey a , Sylvain Fichet b , and Philip Tanedo a [email protected], [email protected], [email protected] a Department of Physics & Astronomy, University of California, Riverside, CA b ICTP SAIFR & IFT-UNESP
R. Dr. Bento Teobaldo Ferraz 271, S˜ao Paulo, Brazil
Abstract
Dark matter may self-interact through a continuum of low-mass states. This hap-pens if dark matter couples to a strongly-coupled nearly-conformal hidden sector. Thistype of theory is holographically described by brane-localized dark matter interactingwith bulk fields in a slice of anti-de Sitter space. The long-range potential in thisscenario depends on a non-integer power of the spatial separation, in contrast to theYukawa potential generated by the exchange of a single mediator. The resultingself-interaction cross section scales like a non-integer power of velocity. We identifythe Born, classical and resonant regimes and investigate them using state-of-the-artnumerical methods. We demonstrate the viability of our continuum-mediated frame-work to address the astrophysical small-scale structure anomalies. Investigating thecontinuum-mediated Sommerfeld enhancement, we demonstrate that a pattern of res-onances can occur depending on the non-integer power. We conclude that continuummediators introduce novel power-law scalings which open new possibilities for darkmatter self-interaction phenomenology. a r X i v : . [ h e p - ph ] F e b CR-TR-2021-FLIP-MSE-6
Contents α < α = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.5 Validation of Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A AdS/CFT with UV brane
B Derivation of Gapless α = 1 Potential C Validity of the Born Approximation D Classical Transfer Cross Section
E Sommerfeld Enhancement from a /r Potential F Self-Interacting Dark Matter Numerical Method
Introduction
A dark sector is a set of fields that include dark matter and low-mass particles that me-diate interactions of the dark matter [1–6]. If these mediators interact with the StandardModel, their signatures may appear in a suite of laboratory based experiments. Even if theseStandard Model interactions are negligible, the mediators induce long-range potentials be-tween dark matter particles that may be tested astronomically [7, 8]. This self-interactingdark matter framework has been spurred by the observation that it may address poten-tial small-scale structure tensions between simulations of cold dark matter and astronomicalobservations [9, 10].A single mediator typically produces a Yukawa potential between dark matter particles, V ( r ) ∼ − e − m ϕ r /r , where m ϕ is the mass of the mediator. This long-range behavior can bedramatically altered when the single-mediator exchange picture breaks down, for examplewhen the mediator is represented by a continuum of states. Models of continuum darksectors have existed for at least a decade in the form of conformal hidden sectors [11, 12] andclosely related work on unparticle hidden sectors [13–16]. The proposal that such modelsmay lead to novel self-interactions was first identified in Ref. [17] for a spin-0 mediatormodeled in the holographic description of a warped extra dimension. This paper describescontinuum-mediated self-interacting dark matter phenomenology in that benchmark theory.The dynamics of the model generate a long-range potential on the UV brane that scales as anon-integer power of separation, V ( r ) ∼ r (cid:18) r (cid:19) non-integer , (1.1)where Λ is a cutoff scale.The long-range forces between dark matter particles allow energy exchange in dark matterhalos and create a cored density profile compared to standard cold dark matter N -bodysimulations. Observations of small-scale structure anomalies in dwarf spheroidal galaxies areindicative of cored halo profiles and are thus a tantalizing possible signature for dark matterdynamics [9]. Alternative proposals to address these anomalies include baryonic feedback onthe dark matter halo. Future generations of N -body simulations may be able to ultimatelydistinguish between the two scenarios, and it is plausible that nature may even invoke acombination of the two mechanisms. We refer to Ref. [19] for a recent review of the status ofthese anomalies. A key result of our study is that continuum-mediated interactions leads toa non-integer velocity dependence on the dark matter self-scattering cross section, a quantitythat relates the fundamental particle physics parameters of the dark sector to astronomicalobservations. Schematically, σ ( v ) ∼ v non-integer . (1.2) In this work we use continuum to refer to the discrete set of Kaluza–Klein modes. This could be alsoreferred to as a ‘discretuum,’ as opposed to the ‘continuous continuum’ regime in which the KK modesmerge [18] Because a potential is generated by t -channel diagrams, the mediator field carries spacelike four-momentum. This makes it mostly insensitive to whether the spectral distribution is continuous or discreteand no distinction between these scenarios is necessary.
3e proceed as follows. In Section 2 we motivate a class of conformal models that generatenon-integer potentials of the form (1.1) and specifically highlight a dual picture with amass gap. We give a precise definition of the gapped, continuum-mediated self-interactingdark matter model in Section 3. We discuss experimental constraints beyond self-interactionsin Section 4; these constraints can be avoided for the types of parameters needed to addresssmall scale structure puzzles in astronomy. The long-range potential is derived in Section 5using spectral techniques. We present closed form expressions using asymptotic limits thatwe validate numerically. In Section 6 we evaluate the figure of merit for astronomical appli-cations, the self scattering transfer cross section. In the so-called Born and classical regimesof dark matter coupling and velocity, we demonstrate novel scaling in the dark matter ve-locity compared to non-continuum self-interacting models. We confirm the presence of aresonant regime and analyze all regimes numerically. Continuum-mediated self-interactionscan explain small-scale structure observations even when the slope of its potential differssignificantly from a standard Yukawa potential. In Section 7 we show that Sommerfeld en-hancement produces a pattern of resonances that depend on the potential slope and massgap. We conclude in Section 8. The Appendices include a streamlined review of AdS / CFT with a UV brane (Appendix A), a calculation of the approximate transfer cross section inthe non-perturbative classical regime (Appendix D), a proof that there is no Sommerfeldenhancement for a 1 /r potential (Appendix E), and a review of the numerical method usedto solve for the transfer cross section (Appendix F). The simplest assumption for dark matter self-interactions is that dark matter currents, J DM ,interact by exchanging spin-0 or spin-1 mediators at tree-level. In momentum space, thematrix elements take schematically the form= J DM ( q ) 1 q − m J DM ( − q ) . (2.1)The corresponding potential between dark matter currents in position space is Yukawa-like, V ( r ) ∼ e − mr /r , or Coulomb-like if m = 0. The mediator mass, m , cuts off the potential inthe infrared and is important for realizing required low-velocity scaling of the dark matterself-scattering cross section for small scale structure anomalies.The exchange of a single, non-derivatively coupled, weakly-interacting field in (2.1) isthe simplest dark matter self-interaction. The resulting r − potential is the longest rangedpotential allowed by the lower bound on the dimension of the exchanged operator set byunitarity, ∆ ≥
1. However, it is also plausible that the leading self-interaction is shorterrange than 1 /r and thus there are a variety of possibilities that have yet to be thoroughlyinvestigated. An extreme example is a zero-range interaction, J DM ( q ) J DM ( − q ), which givecontact-interactions in position space, V ( r ) ∼ δ (3) ( r ). This possibility is too extreme: thecontact interactions produce velocity-independent cross sections that are tightly constrainedby the upper bound on dark matter self-scattering at high velocities from observations ofgalaxy cluster collisions like the Bullet Cluster. Other short range possibilities include tree-level exchange of a pseudoscalar (see e.g. [20]) and loop-level
4n this work we explore intermediate possibilities where the self-interaction potential hasfinite range that is shorter than the Yukawa/Coulomb limit. The simplest possibility amountsto a matrix element = J DM ( q ) 1 (cid:16)(cid:112) − q (cid:17) − J DM ( − q ) . (2.2)The parameter ∆ satisfies ∆ ≥
1, where ∆ = 1 recovers the Coulomb case. The position-space potential scales as V ( r ) ∼ r − and becomes steeper near the origin for ∆ > CFT );this sector may be a gauge theory with large ’t Hooft coupling. Currents of elementary darkmatter, J DM , interact with CFT operators. Even though the mediator sector is strongly-interacting, conformal symmetry constrains the
CFT correlation functions and provides awell-controlled framework for calculations. The
CFT two-point function has a continuousspectral representation and so we refer to this scenario as continuum-mediated self-interactingdark matter . An analogous description of dark matter–nucleon scattering is used in Ref. [23].A purely conformal hidden sector does not have a mass gap. This prevents an infraredcutoff that is usually set by the mediator mass. In order to restore the desired exponentialdamping at long distances, we assume an infrared ( IR ) mass gap in a slightly more evolvedmodel that is most simply described holographically in five dimensional anti-de Sitter ( AdS )space. In this scenario, a field Φ propagates in the bulk and interacts with the brane-localized dark matter currents, J DM .The AdS dual of the ungapped amplitude (2.2) is schematically:= (2.3)see Appendix A for relevant details from the
AdS/CFT correspondence. In the descrip-tion of continuum-mediated self-interacting dark matter, dark matter itself is a degreeof freedom localized on the UV brane near the AdS boundary. This is identified with anelementary degree of freedom that probes the
CFT sector. The mediator continuum is a bulkfield coupled to the fields on the boundary. The mass gap in the
AdS description is encodedby an infrared ( IR ) brane localized further away from the AdS boundary:= . (2.4) mediated processes [21, 22], which induce potentials going as ∝ /r n with n integer and ≥
5n the description, the mass gap follows from the bulk field having two boundary conditionsat finite distance. The exact CFT limit (2.3) is recovered when the IR brane is decoupled bysending it to spatial infinity. The model is shown in Figure 1 and is described preciselyin the following section. Figure 1: Schematic description of the continuum-mediated self-interacting dark matter scenario.
We detail a model in
5D AdS space that realizes the continuum-mediated self-interactingdark matter scenario; the choices of parameters are discussed in the following section. Themodel is based on the warped dark sector framework [17], which is itself closely related tothe Randall–Sundrum 2 model of a warped extra dimension [24].
Geometry.
The metric for the
AdS spacetime in conformal coordinates is ds = (cid:18) kz (cid:19) (cid:0) η µν dx µ dx ν − dz (cid:1) (3.1)where k is AdS curvature. We restrict to a slice of this
AdS space and place UV and IR branesat the endpoints, z UV ≤ z ≤ z IR z UV = 1 k z IR = 1 µ . (3.2)In our model, the scale µ characterizes the mass gap of the mediator sector; we take µ (cid:28) k .We assume that some stabilization mechanism prevents the two branes from falling intoone another; though we may remain agnostic about the specific choice as the details are notcrucial to our study. For concreteness, one may assume the Goldberger–Wise mechanism [25].6e ignore gravitational backreaction effects near the IR brane and approximate the metricto be exactly AdS over the entire space.The action for the theory includes bulk and brane-localized quadratic terms for the realscalar mediator Φ, UV brane-localized quadratic terms for the dark matter χ , and interactionsbetween dark matter and mediator: S = (cid:90) z IR z UV (cid:90) d x √ g L Φ + √ ¯ g (cid:0) L χ + L int + L UVΦ (cid:1) δ ( z − z UV ) + √ ¯ g L IRΦ δ ( z − z IR ) , (3.3)where ¯ g is the induced metric on the brane, with √ ¯ g = ( kz ) − . Additional terms that do notplay a role in the self-interaction phenomenology are the Einstein–Hilbert term, the Standard Model action localized to the UV brane, and possible Standard Model interactionswith the mediator. The dark matter Lagrangian terms encode a mass m χ and Yukawacoupling to the bulk mediator: L χ = ¯ χγ µ ∂ µ χ − m χ ¯ χχ L int = λ √ k Φ ¯ χχ . (3.4)Writing Lorentz indices M , the bulk mediator Lagrangian is L Φ = 12 (cid:2) ( ∂ M Φ)( ∂ M Φ) − M Φ (cid:3) , (3.5)where the bulk mass M Φ is tied to the dimension ∆ of the operator exchanged between darkmatter particles in the CFT picture. The brane-localized Lagrangian terms for the bulk scalarencode mass and kinetic terms: L UVΦ = 12 k Φ B UV [ ∂ ]Φ L IRΦ = 12 k Φ B IR [ ∂ ]Φ B i [ ∂ ] = m i + c i ∂ + . . . . (3.6)The B i [ ∂ ] are polynomials in the Laplacian ∂ = ∂ µ ∂ µ ; the constant term is the brane-localized masses m i . Higher order terms are typically small and irrelevant for our study.We remark that the low-energy effective theory also contains a radion that is identifiedwith the dilaton in the theory. This mode is light, but localized on the IR brane andhence has negligible contributions to the dark matter dynamics on the UV brane. We thusdo not include it in our analysis as it would produce only a minor shift in the long-rangepotential. interacting theories are non-renormalizable and are understood to be low-energy effectivefield theory ( EFT ) valid up to a cutoff, Λ. The cutoff is tied to the strongest interaction—either gravity or another interaction in the theory. na¨ıve dimensional analysis ( NDA ) [26–30], in turn, relates the cutoff to the
AdS curvature [18],Λ (cid:38) (cid:96) (cid:96) k ∼ πk , (3.7)where the and loop factors are (cid:96) = 16 π and (cid:96) = 24 π , respectively.7n our dark sector model, the cutoff sets the dark matter–mediator Yukawa coupling λ .Thus
5D NDA bounds the Yukawa coupling by λ (cid:46) (cid:114) (cid:96) k Λ (cid:46) π , (3.8)where we have used (3.7) in the second inequality.While the theory is valid below Λ, the AdS / CFT dictionary is valid only up to a cutoffscale on the order of k <
Λ. From the perspective, a CFT coupled to gravity has a cutoffparametrically smaller than M Pl because of the large degrees of freedom of the CFT . Thiscutoff turns out to be k , for example by using the species scale conjecture (see e.g. [31]).Our
5D EFT contains isolated degrees of freedom localized on a brane. In a realistic theorywith gravity, localized fields are special modes from bulk fields and are necessarilyaccompanied by a spectrum of KK modes [32]. We assume an appropriate limit where theobservable effects of these modes are negligible. For the purposes of studying novel, continuum-mediated dark matter self-interactions, werestrict the parameters presented in the model in Section 3.1. The AdS curvature, k ,corresponds to the cutoff of the theory, as described in Section 3.2. To ensure that the cutoffof the theory is beyond the experimental reach of the Large Hadron Collider to detect, e.g.,Kaluza–Klein gravitons, we set k to be k = 10 TeV . (3.9)This sets the position of the UV brane z UV = k − and the upper bound on all other dimen-sionful parameters in the theory. The AdS curvature is much smaller than the Planck scale,in the spirit of ‘little Randall–Sundrum’ models [33].The mediator mass, M Φ is related to the dimension ∆ of the continuum mediator operatorand is conveniently described by the dimensionless parameter α , α ≡ M k = (2 − ∆) . (3.10)The range of α corresponds to the ∆ − branch of AdS / CFT (see details in Appendix A) andcan be established as follows. Unitarity of
CFT operators requires ∆ ≥
1, implying α ≤ AdS implies α ≥ α ≥ ≤ α ≤ V ( r ) ∼ r − for α = 1 and V ( r ) ∼ r − for α = 0. For potentials more singular than r − , solving for thephenomenology becomes computationally intractable and, furthermore, the theory is unlikelyto produce the effects relevant for small scale structure anomalies. We thus further restrictthe range to α ≥ / / ≤ α ≤ (cid:46) µ (cid:28) k Early universeDark matter mass µ (cid:46) m χ (cid:46) k Nonlocal potential,
EFT validityYukawa coupling λ ≤ π EFT perturbativity
Table 1: Range of parameters in our model. The
AdS curvature is set to k = 10 TeV; larger valuesgenerically suppress self-interaction effects. The dimensionless brane-localized masses and kineticterms defined in (3.6) and (3.11) are assumed to be O (1), with the exception of b UV which is tunedto zero to reproduce the long-range behavior, (2.2). The early universe bound on µ is described inSection 4. Our theory includes brane-localized masses m i Φ( x, z i ) and kinetic terms c i [ ∂ Φ( x, z i )] for the mediator. It is convenient to parameterize the former into dimensionless variables, b IR ≡ m k + (2 − α ) b UV ≡ m k + (2 − α ) . (3.11)The IR parameters b IR and c IR generically have O (1) values so we set them all to one. Theseonly have a mild impact on the self-interaction phenomenology. Conversely, we tune b UV = 0as required to reproduce the CFT behavior in (2.2) since b UV corresponds to a double tracedeformation in the conformal theory. The UV brane kinetic coefficient c UV is assumed to be O (1), though it is only significant in the limiting case α = 1.With these benchmark values in place, the theory is described by the parameters inTable 1. The IR scale µ defines the mass gap of the theory by setting the scale of the lightestKaluza–Klein mode and its lower bound is set by dark radiation constraints, described inSection 4. It is convenient to work in position space for the z -direction but momentum space along Minkowski slices. The mediator field is decomposed asΦ p ( z ) = (cid:90) d x e ip · x Φ( x µ , z ) p · x = p µ x µ . (3.12)The norm p = √ η µν p µ p ν is real for timelike p µ and imaginary for spacelike p µ . In thesecoordinates, the free scalar propagator is the two-point Green’s function, see e.g. [32], G p ( z, z (cid:48) ) = i πk ( zz (cid:48) ) (cid:104) (cid:101) Y UV α J α ( pz < ) − (cid:101) J UV α Y α ( pz < ) (cid:105) (cid:104) (cid:101) Y IR α J α ( pz > ) − (cid:101) J IR α Y α ( pz > ) (cid:105)(cid:101) J UV α (cid:101) Y IR α − (cid:101) Y UV α (cid:101) J IR α , (3.13)where z <,> is the lesser/greater of the endpoints z and z (cid:48) . The quantities (cid:101) J UV,IR are (cid:101) J UV α = pk J α − (cid:16) pk (cid:17) + B UV ( p ) J α (cid:16) pk (cid:17) (cid:101) J IR α = pµ J α − (cid:18) pµ (cid:19) + B IR ( p ) J α (cid:18) pµ (cid:19) , (3.14)9ith similar definitions for (cid:101) Y UV,IR . The boundary functions B i ( p ) encode brane-localizedoperators and are defined in (3.6). We refer to (3.13) as the canonical representation of thepropagator.The propagator has an infinite series of isolated poles set by the zeros of (cid:101) J UV α (cid:101) Y IR α − (cid:101) Y UV α (cid:101) J IR α and referred to as Kaluza–Klein ( KK ) modes. The free propagator can thus equivalently bewritten as a series G p ( z, z (cid:48) ) = i (cid:88) n f n ( z ) f n ( z (cid:48) ) p − m n + i(cid:15) , (3.15)we refer to this particular momentum-space spectral representation as the KK representationof the propagator. Depending on the context, either the canonical or KK representation maybe more convenient. Assuming that the UV brane mass parameter is zero, b UV = 0, and thatthe other brane parameters have O (1) coefficients, then the KK spectrum for p (cid:29) µ is m n ≈ (cid:18) n − α (cid:19) πµ n > , (3.16)as can be seen from identifying the poles in the limiting form of the propagator in (5.10).The mass of the lightest mode m depends on the brane-localized parameters and is detailedin Section 5.2. The
AdS / CFT correspondence describes the equivalence between a quantum field theory on
AdS d +1 space and a conformal gauge theory with large ’t Hooft coupling and large- N in flat d -dimensional space (for initial works see [36–43], for some reviews see [44–47]). AdS bulkfields correspond to
CFT operators in a way that is exact (to the best of our knowledge) inthe full
AdS spacetime and in the presence of a UV brane.Fields localized on the UV brane are understood to be external fields probing the CFT ;these are equivalently called elementary states in contrast to
CFT degrees of freedom. Inthe context of our model, dark matter and Standard Model particles are elementary fields.We require that dark matter couples to a scalar operator of the mediator
CFT sector; thisscalar operator corresponds to the bulk mediator field Φ. The mediator CFT two-pointcorrelation function gives the self-interaction amplitude in (2.3).The understanding of the dual theory is only qualitative in the presence of IR branecutting off large z values. The IR brane is interpreted as a spontaneous breaking of theconformal symmetry analogous to confinement in a strongly-interacting gauge theory [48,49].The theory is thus only approximately conformal at scales much larger than µ , however wefollow the common colloquiual practice of referring to the theory as a CFT . The scale µ = z − is naturally associated to the mass gap characterizing conformal symmetry breaking,similar to the QCD confinement scale. KK modes are identified with composite states thatare allowed when conformal invariance is broken. In the simplest realization, the compositestates are glueballs of adjoint gauge fields.Either the AdS or CFT description of the theory may be more convenient depending onthe context. We primarily focus on the description where the model is concretely defined.10he qualitative behavior of this theory is general and captures what is expected for a purely near-conformal mediator; one may view the construction as a simple quantitative toolto describe such a theory. We briefly comment on implications of our model beyond the dark matter self-interactionphenomenology that is our primary focus.
Models of near-conformal dark sectors necessarily introduce large numbers of degrees offreedom. Many of these may be relativistic in the early universe and are thus constrained bybig bang nucleosynthesis (
BBN ) and the cosmic microwave background (
CMB ). There are atleast three ways to avoid the tight constraints on the effective number of relativistic degreesof freedom, N eff :1. The theory may have a sufficiently large mass gap, O (MeV), so that all states arenon-relativistic at the relevant times. In this case there is no dark radiation.2. The relativistic states decay quickly enough that they do not affect BBN or the
CMB [23].3. The dark sector may be much colder than the Standard Model so that the density ofstates is suppressed compared to visible matter. This is a natural possibility and hasbeen studied in the context of gravitational interactions in
AdS [50–52]. Dark radiationfrom a bulk scalar will be studied in an upcoming work [53].With these features in mind, we focus on µ (cid:38) O (MeV), but allow for µ the possibility oflower scales subject to additional model building to accommodate N eff limits. Bulk graviton exchange leads to deviations from the Newtonian gravitational potential of theform [24, 54] V N ( r ) = − G N r (cid:20) O (cid:18) k r (cid:19)(cid:21) . (4.1)Constraint from fifth force searches set k (cid:38) k − (cid:46) µ m and hence can be ignored;see e.g. [55] for a recent measurement, [56] for a review of r − constraints. Standard Model fields are assumed to be localized on the UV brane. For the purposesof dark matter self interaction phenomenology, we neglect any direct UV -brane interactionsbetween the dark matter and Standard Model and assume that the mediator–Standard Modelcouplings are negligible. These couplings are phenomenologically relevant, for example indark matter direct detection experiments [23] or in searches for novel forces between Standard11odel particles [17, 22], but are not directly related to the small scale structure anomaliesthat are the primary phenomenological focus of this paper.In principle the brane-localized fields are limits of fields with heavy KK modes [32]. Themost significant effects of these modes are deviations in the Standard Model gauge sector:gauge bosons can scatter off gravitons and the gauge couplings pick up an anomalouslogarithmic running above the IR scale µ . Both of these effects are small enough to beundetected with current data in the limit where Λ is sufficiently close to k . Since we alreadyassume this in (3.7), the model is safe from these effects. The potential V between two particles is obtained from the t -channel scattering amplitudewith the external legs taken to the appropriate non-relativistic limit, i M ≡ − im χ (cid:101) V ( | q | ) = − λ k G | q | ( z UV , z UV ) , (5.1)with t ≈ −| q | where q is the three-dimensional momentum transfer. On the right-handside we insert the expression from the exchange of a t -channel bulk mediator between darkmatter currents. The position-space potential is related by a Fourier transform V ( r ) = (cid:90) d q (2 π ) (cid:101) V ( | q | ) e i q · r , (5.2)with r = | r | . Even though our effective theory has a cutoff, one may integrate (5.2) to infinite | q | under the assumption of a smooth cutoff, as shown in the Appendix B of Ref. [22].Simply inserting the exact propagator (3.13) is analytically challenging. We proceed byusing a spectral representation where the discontinuity of the two-point function is evaluatedin the appropriate asymptotic limits from Section 5.2. The spectral representation for the bulk propagator is [58] G p ( z, z (cid:48) ) = 12 πi (cid:90) ∞ dρ Disc ρ (cid:2) G √ ρ ( z, z (cid:48) ) (cid:3) ρ − p , (5.3)where Disc ρ [ g ( ρ )] is the discontinuity of g ( ρ ) across the branch cut along the real line, ρ ∈ R + :Disc ρ [ g ( ρ )] = lim (cid:15) → g ( ρ + i(cid:15) ) − g ( ρ − i(cid:15) ) (cid:15) > . (5.4)We compute the non-relativistic potential using this spectral representation of the propagator.Performing the d q integral yields a general representation of the long-range potential: V ( r ) = − π λ k (cid:90) ∞ dρ Disc ρ (cid:2) G √ ρ ( z UV , z UV ) (cid:3) e −√ ρr r . (5.5) In principle u -channel diagrams contribute when the scattering particles are identical. This is an O (few)effect [57, App. C]. We neglect the u -channel contribution for simplicity and ease of direct comparison toRef. [9]. aluza–Klein representation. One may use the KK representation of the free propagator(3.15) in the spectral representation of the potential (5.5); this amounts to identifying theexchange of a bulk scalar with the sum of t -channel diagrams with each KK mode:= + + + · · · (5.6)The spectral distribution is Disc ρ (cid:2) G √ ρ ( z, z (cid:48) ) (cid:3) = (cid:80) n f n ( z ) f n ( z (cid:48) )2 πδ ( p − m n ), so that thepotential is an infinite sum of Yukawa potentials from each KK mode: V ( r ) = − π λ k (cid:88) n f n ( z ) e − m n r r . (5.7)While this KK representation of V is exact, it requires knowledge of the entire spectrum of KK masses and wavefunctions. Canonical representation.
One may alternatively use the canonical representation ofthe propagator (3.13) in the spectral representation of the potential (5.5). In this case, onemay apply the closed-form asymptotic expressions derived in the following section. Theseasymptotic expressions carry the same poles as the KK representation. The momentumflowing through the propagator is necessarily spacelike in diagrams that contribute to thepotential. Thus we may readily use the asymptotic expressions for large | p | that are validaway from the poles, (5.11) for α < α = 1. We numerically validate thisapproximation in Section 5.5. We present the limits of the bulk propagator G p for Minkowski momenta p much smaller andlarger than the mass gap, µ . We focus on propagation to and from the UV brane where thedark matter currents are localized. These limits illuminate the properties of the theory andyield simplifications for the self-interaction potential.We treat the α < α = 1 cases separately; the asymptotic behavior of Besselfunctions with near integer order have an extra contribution that is neglected for non-integerorder. As a result, one typically cannot obtain the α = 1 asymptotic behavior as the α → α < α = 1 case is a meaningful benchmark as it isequivalent to the exchange of a single mediator. This is due to the expression for the Bessel function of the second kind with integer index α → n , Y n ( z ) = 1 π ∂J α ( z ) ∂α (cid:12)(cid:12)(cid:12)(cid:12) α = n + ( − ) n π ∂J α ( z ) ∂α (cid:12)(cid:12)(cid:12)(cid:12) α = − n . .2.1 Propagator Asymptotics for < α < Small momentum asymptotic, | p | (cid:28) µ . For Minkowski momenta much less than themass gap we find a single pole: G p ( z UV , z UV ) = i k (1 − α ) (2 α + b IR ) α (2 + b IR ) p − α (1 − α ) b IR µ (cid:16) µk (cid:17) − α . (5.8)All other poles are be heavier than O ( µ ). For b IR (cid:46) O (1), the light 4D mode mass is m = 4(1 − α ) b IR b IR µ . (5.9) Large momentum asymptotic, | p | (cid:29) µ . For momenta much larger than the mass gap, G p ( z UV , z UV ) = i k Γ ( α )Γ ( − α + 1) (cid:18) k p (cid:19) α S α ( p ) S α ( p ) = sin (cid:16) pµ − π (1 − α ) (cid:17) sin (cid:16) pµ − π (1 + 2 α ) (cid:17) . (5.10)The tower of KK poles are encoded in S α ( p ). The propagator further simplifies when themomentum has an imaginary part Im( p/µ ) (cid:38) G p ( z UV , z UV ) = i k Γ ( α )Γ ( − α + 1) (cid:18) k − p (cid:19) α , (5.11)where we have used S α ≈ ( − α in this limit. This includes the case of spacelike momen-tum. In this limit the conformal scaling appears: recalling that α = 2 − ∆, the propagatorreproduces the scaling of the amplitude (2.2). Observe that the UV brane kinetic term doesnot appear in this expression. This reflects the fact that none of the modes are localized nearthe UV brane. α = 1 Small momentum asymptotic, | p | (cid:28) µ . For Minkowski momenta much less than themass gap, we find G p ( z UV , z UV ) = (2 + b IR )2 ikp [(2 + b IR )(2 c UV k + log( k /µ )) − b IR ] − b IR µ . (5.12)This carries a single pole. The mass of this light mode is m = 4 b IR µ (2 + b IR ) [2 c UV k + log( k /µ )] − b IR . (5.13)This mass is suppressed by c UV + log( k/µ ), where c UV is the coefficient of the UV brane-localized kinetic term and log( k/µ ) describes the bulk volume. One may understand (5.13)as a dressing of the zero mode with an IR brane-localized mass. Loops from bulk interactions cause heavy KK modes to acquire large widths and give an effective imaginarypart to timelike four-momentum in the bulk propagator [18,59,60]. This physical imaginary part is importantfor timelike processes but is not for spacelike processes, hence it is irrelevant for the potential. arge momentum asymptotic, | p | (cid:29) µ . For momenta much larger than the mass gap, G p ( z UV , z UV ) = 2 ikp (cid:104) c UV − π cot (cid:16) pµ + π (cid:17) − log (cid:16) p k (cid:17) − γ (cid:105) . (5.14)When Im( p/µ ) (cid:38) − i and the propagator simplifies, G p ( z UV , z UV ) = 2 ikp (cid:104) c UV − log (cid:16) − p k (cid:17) − γ (cid:105) . (5.15)In contrast to the α < UV brane kinetic term is not negligible. Thispropagator describes a mode with a logarithmic running of its wavefunction. It is similarto the well known case of a bulk gauge field in AdS . We can absorb a large logarithm byredefining the brane wavefunction coefficient c UV at a physical scale p :ˆ c UV = c UV + [log ( k/p ) − γ ] G p ( z UV , z UV ) = 2 ikp (cid:104) c UV − log (cid:16) − p p (cid:17)(cid:105) . (5.16)For the astrophysical applications of self-interacting dark matter, the energy transfer rangesover only a few orders of magnitude and the logarithmic running is thus negligible. The α = 1 case thus reproduces the standard single-mediator self-interacting dark matter modeland serves as a useful benchmark. α < For bulk masses in the range 0 < α < IR brane mass parameter b IR ∼ O (1),the lightest excitations have mass on the order of µ ; see (5.9). Since there is no light modeto contribute to non-analyticities of G p for | p | < µ , we may apply the | p | (cid:29) µ approximationof the propagator to the spectral integral (5.5). The lower limit of the spectral integral isformally the mass of the lightest KK mode, V ( r ) = − π λ k (cid:90) ∞ m dρ Disc ρ (cid:2) G √ ρ ( z UV , z UV ) (cid:3) e −√ ρr r . (5.17)However, because m = O ( µ ), by using the ρ (cid:29) µ approximation for the propagator (5.11),we introduce some uncertainty in the lower bound of the spectral integral. We verify thevalidity of this approximation in Section 5.5.The discontinuity across the branch cut along ρ > ρ (cid:2) ∆ √ ρ ( z , z ) (cid:3) = 1 k (cid:18) k ρ (cid:19) α Γ( α )Γ(1 − α ) sin( πα ) , (5.18)where we have used (5.11). This is valid for Im( p/µ ) (cid:38)
1, which we assume because p is spacelike. Evaluating the integral across the discontinuity using the Γ reflection andduplication formulas gives the main expression we use in our analysis: V ( r ) = − λ π / Γ(3 / − α )Γ(1 − α ) 1 r (cid:18) kr (cid:19) − α Q (2 − α, m r ) , (5.19) Namely: Γ (1 − z ) Γ ( z ) = π/ sin ( πz ) and Γ (2 z ) = π − / z − Γ ( z ) Γ ( z + 1 / Q (2 − α, m r ) is the regularized incomplete Γ function, Q ( p, z ) = 1Γ ( p ) (cid:90) ∞ z dx x p − e − x . (5.20)For r (cid:29) m − , the potential is exponentially suppressed at long distances, V ( r ) ∝ − (cid:16) m k (cid:17) − α kr e − m r . (5.21)We see that Q (2 − α, r ) takes the place of the e − mr Yukawa factor that encodes the massgap in the single-mediator scenario. In turn, this mass gap is a key ingredient for cutting offunwanted long-range dark forces.It is illustrative to check the behavior in the gapless limit µ →
0. The large, spacelikemomentum approximation of the propagator (5.15) is exact in this limit and potential canbe evaluated exactly. We recover the gapless limit in (5.19) the gapless limit is recovered bytaking m →
0, giving V gapless ( r ) = − λ π / Γ(3 / − α )Γ(1 − α ) 1 r (cid:18) kr (cid:19) − α , (5.22)which matches the result from [17]. The power law behavior obtained matches the proposedscaling in (2.2) with the AdS / CFT identification ∆ = 2 − α . α = 1 For bulk mass parameter α = 1 and with generic IR brane mass parameter b IR ∼ O (1),there is a mode lighter than the scale µ . The suppression relative to µ is the kinetic factor( c UV + log( k/µ )) / in (5.13). This is in contrast to the α < G √ ρ must thus take into account this pole in the ρ (cid:28) µ regime inaddition non-analyticities in the ρ (cid:29) µ regime. We separate the potential into two piecesaccordingly, V = V light + V KK . Light mode contribution.
The light mode contributes a simple Yukawa potential: V light = − λ πk f ( z UV ) e − m r r (5.23)where the profile evaluated on the UV brane is f ( z UV ) = (2 + b IR )2 k (2 + b IR ) (cid:104) c UV + log (cid:16) k µ (cid:17)(cid:105) − b IR ≈ k ˆ c UV + log (cid:16) p µ (cid:17) + γ . (5.24)as can be derived from the pole of the small momentum transfer limit of the propagator(5.12). On the right-hand side we use the assumption that b IR ∼ O (1), apply the µ (cid:28) k limit, insert the renormalized brane kinetic term coefficient ˆ c UV defined at the scale p from(5.16). 16 K mode contribution.
The KK mode contribution uses the | p | (cid:29) µ asymptotic of thebulk α = 1 propagator (5.16) applied to the large-momentum spectral integral, (5.17). Toobtain an analytically tractable expression we take the limit ˆ c UV (cid:29) log( ρ/p ) over the range ρ ∈ [ m , r − ]; the upper bound comes from the exp (cid:0) −√ ρr (cid:1) factor in the spectral integral.The resulting propagator is G p ( z UV , z UV ) = ikp ˆ c UV (cid:20) − p /p )2ˆ c UV + O (cid:18) c (cid:19)(cid:21) . (5.25)The discontinuity in the spectral intergal isDisc ρ (cid:2) G √ ρ ( z UV , z UV ) (cid:3) = 2 πk ˆ c UV δ ( ρ ) + k ˆ c πρ + O (cid:18) c (cid:19) . (5.26)The singular δ ( ρ ) term is outside the range of integration and does not contribute. Theleading contribution comes from the O (cid:0) ˆ c − (cid:1) term and evaluates to V KK ( r ) = − πr λ ˆ c Γ(0 , m r ) . (5.27) The α = 1 Potential and Limits.
Since we have used the ˆ c UV (cid:29) log( ρ/p ) limit in the KK potential, we may apply the same approximation to the light mode contribution. Thisproduces the full α = 1 potential V ( r ) = − λ πr (cid:20) c UV (cid:18) − log( p /µ ) + γ ˆ c UV (cid:19) e − m r + Γ(0 , m r )ˆ c (cid:21) + O (cid:18) c (cid:19) . (5.28)At long distances, r (cid:29) m − , Γ(0 , m r )ˆ c → c e − m r m r . (5.29)One can explicitly see the exponential suppression from both the light mode and KK modemass gaps. In the short distance r (cid:28) m − limit, the incomplete Γ function is Γ(0 , x ) ≈− (log x + γ ) and one has e − m r ∼
1. Since m r (cid:28)
1, we obtain V ( r ) = − λ ˆ c UV πr (cid:20) − c UV log (cid:18) rr (cid:19)(cid:21) + O (cid:18) c (cid:19) , (5.30)where we introduce the scale r log r = log p + 2 γ + log (cid:18) m µ (cid:19) (5.31)to absorb O (1) coefficients. The explicit µ dependence vanishes because the log µ from thelight mode and the log m = log µ + O (1) from the KK modes cancel.While (5.30) could be understood as the µ → α = 1 potential, the ˆ c UV (cid:29) log( ρ/p ) assumption we used to evaluate the spatial potential formally does not hold in this17 igure 2: Absolute potential | V ( r ) | plotted to validate the continuum-mediated potential with amass gap (black) against a sum over n max Kaluza–Klein modes (colored). The potential with n max KK modes is valid for separations larger than r (cid:38) m − n max . The disagreement at long separationsbetween the blue and black lines represents our numerical error and does not change the quantitativebehavior of integrals over the potential. Also shown: the non-integer power law limit (dashed gray)that is realized in the gap-less limit m → limit. Instead the full log n r series would need to be resummed. Nevertheless, we verify thatthe Fourier transform of the propagator (5.25) matches the potential (5.30). Interestingly,in this limit the contribution from the light mode is replaced by the δ ( ρ ) contribution in thediscontinuity across the propagator, (5.26), which is otherwise cut off at finite µ . Details ofthis calculation are presented in Appendix B.The expressions in this section show that the KK mode contribution tends to be smallwith respect to the light mode for both large and small r . This logarithmic correction isnegligible in our self-interacting dark matter calculations and thus the α = 1 case matchesthe standard single mediator scenario. It can thus be used as a benchmark comparing to α (cid:54) = 1 phenomenology. In this study we use the asymptotic approximation of the gapped continuum-mediated po-tential (5.19). In order to quantify its validity, we compare our approximation to an explicitsum over Kaluza–Klein mediated Yukawa potentials (5.7). This is a meaningful check sincea sum over n max KK modes is a valid approximation to the full sum on scales longer thanthe inverse mass of the heaviest mode, r (cid:38) m − n max . We thus test for agreement of the gappedcontinuum-mediated potential with the sum over a large number of KK in the regime wherethe latter is valid.We present our validation in Figure 2. The key comparison is between sum over n max =10 KK modes (blue) and the continuum-mediated potential (black). For values of α (cid:46) . n max KK modes agrees with the continuum potential in the regime where thefinite KK sum is valid, r (cid:38) m − n max . However, at distances longer than the inverse massgap, r (cid:38) m − , the curves diverge slightly while maintaining the same qualitative gappedbehavior. This discrepancy is caused by the | p | (cid:29) µ limit assumed in the derivation of thecontinuum-mediated potential (5.19). This discrepancy grows when α ≈
1; see Footnote 4.18 igure 3: Regimes of self-interacting dark matter. The horizontal axis measures whether the ladderof mediator exchanges can be approximated by a single mediator exchange. The vertical axis is ameasure of the velocity. The figures of merit are scaled by the ratio of the dark matter mass tothe mediator mass (or mass gap) so that the regimes are limits relative to unity. The perturbativeregime is described by the Born approximation over the range of all velocities, whereas the non-perturbative regime is separated into a classical regime at high velocities and a resonant regime atlow velocities. Blue: asymptotic velocity scaling of the transfer cross section σ T in the continuum-mediated scenario. No simple scaling exists in the resonant regime. The standard case of a single mediator corresponds to α = 1. Practically, we restrict the continuum-mediated potential for α (cid:46) .
95. In this range, thelarge- α discrepancy does not change the qualitative behavior of the continuum-mediatedpotential, nor the quantitative behavior of integrals of this potential. For larger values of α , the potential reproduces the well-known case of a single mediator, as described inSection 5.4.Figure 2 also demonstrates how a sum of Yukawa potentials can reproduce a potentialthat goes like a non-integer power of the separation, (1.1). The lightest KK mass sets a long-range length scale, m − . In the regime m − n max (cid:46) r (cid:28) m , the sum over Yukawa potentialsfrom n max KK modes produces a total potential that matches the power law of (5.22). We apply our continuum-mediated model to the phenomenology of self-interacting dark mat-ter for small-scale structure. The quantity that connects particle physics parameters to as-tronomy is the transfer cross section. We demonstrate the dependence of this cross sectionon our model parameters and provide representative fits.
We summarize key results of self-interacting dark matter phenomenology; see Ref. [10] fora detailed review. Long-range dark matter self-interactions affect halo density profiles bythermalizing the inner halo and reducing the central density. The effect of dark matter19elf-interactions on halos depends on the scattering rate, σv ( ρ χ /m χ ). Since the dark matterdensity ρ χ and the relative velocity v are known for the relevant astrophysical systems,the figure of merit is the ratio of the cross section to the dark matter mass, σ/m χ . Dwarfspheroidal galaxies have low relative velocities ( v ∼
10 km/s) and exhibit small-scale structureanomalies that could be explained by sufficient self-interactions [9,61,62]. On the other hand,galaxy clusters have large relative velocities ( v ∼ (cid:18) σm χ (cid:19) dwarf ∼ g (cid:18) σm χ (cid:19) cluster (cid:46) . g . (6.1)The small-scale target and large-scale upper limit are simultaneously satisfied in self-interactingdark matter models due to the velocity dependence of the cross section. In fact, a more rel-evant quantity for fitting to astronomical observations is the transfer cross section , which isweighted by the amount of transverse momentum transferred between dark matter particles: σ T = (cid:90) d Ω dσd Ω (1 − cos θ ) . (6.2)This accounts for the fact that back-to-back scattering does not change the distribution ofenergy between halo dark matter particles. The transfer cross section is the figure of meritfor determining the effect of self-interactions on the dark matter halo profile. The behavioris classified according to regimes along two axes: perturbativity and relative velocity, seeFigure 3.
Perturbativity.
The horizontal axis of Figure 3 distinguishes whether the transfer crosssection is accurately described by the exchange of a single mediator (perturbative) or other-wise requires a sum over ladder diagrams (non-perturbative). In the former case, one mayuse the Born approximation. For a dark sector with a single mediator of mass m φ andcorresponding potential V ∼ α χ e − m φ r /r , these regimes correspond toBorn: α χ m χ m φ (cid:28) α χ m χ m φ (cid:29) . (6.3)The weighted coupling, α χ m χ /m φ , measures whether the Hamiltonian eigenstates are dis-torted from the non-interacting case [63, (7.2.13)]. The sum over ladder diagrams in thenon-perturbative regime reproduces the distortions of the asymptotic states relative to thenon-interacting eigenstates. Velocity.
The horizontal axis of Figure 3 distinguishes whether the dark matter relativevelocity (kinetic energy) is large enough to ignore the effect of the mediator mass. Whenthe theory is perturbative, the Born approximation may be applied across the entire rangeof velocities. On the other hand the velocity separates the non-perturbative case into two A more symmetric treatment is to use the viscosity cross section, σ V = (cid:82) d Ω sin θdσ/d Ω . In order to mapto the standard self-interacting dark matter literature, we use σ T which differs from σ V by at most an O (1)factor [10]. m χ v ) − iscomparable to the screening length (inverse mediator mass), m − φ :resonant: m χ vm φ (cid:28) m χ vm φ (cid:29) . (6.4)The classical regime is the case where the zeroth-order WKB approximation is valid; thiscorresponds to the (cid:126) → dark sector with a single mediator of mass m φ , theclassical regime is the case where the mediator mass is negligible and the theory reproducesthe case of Rutherford/Coulomb scattering. In contrast, in the resonant regime the Yukawafactor deforms the potential away from the Coulomb limit enough to support quasi-boundstates. In this regime, one must numerically solve the Schr¨odinger equation in a partial waveexpansion to determine the transfer cross section [9].Figure 3 shows that v < α χ is a necessary condition for the existence of resonances oversome range of v . Conversely, v > α χ is a sufficient condition for having no resonance for anyvalue of v . The transfer cross section from a continuum-mediated potential can be mapped onto theself-interacting dark matter regimes described above and pictured in Figure 3.
Effective coupling.
The condition for perturbativity depends on the dark fine structureconstant, which is α χ = g χ / π for a single mediator. We can identify an effective finestructure constant α eff χ for our continuum mediator. For bulk mass parameters 1 / < α < α eff χ = λ m πk (cid:88) n f n ( z UV ) m n ≈ λ π (cid:20) α − − α ) (cid:21) (cid:16) m k (cid:17) − α . (6.5)This follows from applying the Born approximation condition (6.3) to the sum of Kaluza–Klein potentials (5.7). On the right-hand side we use the spectral representation (5.3) toevaluate the sum. This calculation is detailed in Appendix C, where we also discuss thelimiting cases where the bulk masses satisfy α = 1 / α = 1. We note that the factorof ( m /k ) − α ∼ ( µ/k ) − α in (6.5) suppresses the effective coupling compared to a na¨ıveestimate λ / π . Transfer cross section regimes.
The self-interaction regimes in Figure 3 are mappedto the continuum-mediated scenario by identifying the mediator mass with the lightest KK mode mass (the mass gap), m φ → m . We find that the effective coupling α eff χ replaces α χ inthe demarcation of the perturbative (Born) and non-perturbative regimes,Born: α eff χ m χ m (cid:28) α eff χ m χ m (cid:29) . (6.6)We can likewise divide the non-perturbative regime into the classical and resonant regimes:Classical: m χ vm (cid:29) m χ vm (cid:28) . (6.7)21 igure 4: Velocity dependence of the transfer cross section in the Born regime. Comparison betweenthe Born approximation and (blue/solid) and the numerical result from a sum of partial waves(orange/dashed). The results asymptotically scale like v − α at large velocity (green). Unlike the case of a Yukawa potential, there are no analytic expressions for the transfer crosssection in the entire non-perturbative classical regime. We show the scaling of the transfercross section for the classical regime in the small mass gap/high velocity limit and give aclosed form result in the low velocity regime below, see Appendix D.
Continuum-mediated Born regime.
In the Born regime, the transfer cross section com-puted perturbatively from the 1 / < α < (cid:18) dσd Ω (cid:19) Born = (cid:0) α eff χ (cid:1) m χ m (2 α − F (cid:0) , α ; 1 + α ; −| q | /m (cid:1) , (6.8)where F (1 , α ; 1 + α ; −| q | /m ) is the hypergeometric function that encodes the mass gap.The transferred three-momentum, q , satisfies | q | = m χ v (1 − cos θ ) where θ is the scatteringangle in the center of mass frame. We compute the angular integral numerically.We may examine (6.8) in the limits of large and small transferred three-momentum. Fora transfer momentum much larger than the mass gap, | q | (cid:29) m , the transfer cross section is σ BornT ≈ λ m χ πk (1 − α ) (cid:20) Γ( α )Γ(1 − α ) (cid:21) (cid:18) km χ v (cid:19) α | q | (cid:29) m . (6.9)This matches the result from the gapless potential, (5.22). In the opposite limit, | q | (cid:28) m ,the transfer cross section approaches a constant: σ BornT ≈ λ m χ π α k Γ(1 − α ) (cid:18) km (cid:19) α | q | (cid:29) m . (6.10)Figure 4 compares these asymptotic behaviors to a numerical solution.Early astrophysical simulations of self-interacting dark matter assumed a constant σ T and found that the cross sections required to address small-scale structure anomalies were See Ref. [64] for a discussion of scattering in the limit of no mass gap. α controls the velocity-scaling in the high-velocity Born limit.This parametric control is not possible for the exchange of a single mediator. Continuum-mediated classical regime.
Unlike in the Born regime, in the classicalregime closed form results for the transfer cross section do not follow from straightforwardcalculation. While in the case of a Yukawa potential closed form expressions can be deter-mined for the entire non-perturbative classical regime, see e.g. Ref. [9, eqn. (7)], analyticexpressions for the continuum mediated transfer cross section are harder to come by. In thelimit of a small mass gap/large velocity, one can determine its velocity dependence. In theopposite low velocity limit, one finds a closed form expression. The calculations are detailedin Appendix D.One can write the transfer cross section in this regime as an integral over the impactparameter ρ . It is convenient to introduce the dimensionless quantities ξ = ρ/ρ , where ρ isa characteristic length scale defined from the potential (5.19), σ classicalT = 2 πρ (cid:90) ∞ [1 − cos θ ( ξ, m ρ )] ξdξ ρ ≡ (cid:20) λ π / m χ v k − α Γ(3 / − α )Γ(1 − α ) (cid:21) − α . (6.11)When m ρ (cid:28)
1, corresponding to the small mass gap/high velocity limit, the scatteringangle θ is a function of the ratio ξ only [64]. In this case, the transfer cross section dependson a non-integer power of the relative velocity, − / (3 − α ). A finite mass gap inducescorrections to this scaling.While this scaling holds in the small mass gap/high velocity limit of the classical regime,an approximate closed form solution for the transfer cross section can be computed for lowervelocities. Following the methodology of Ref. [65], we calculate the transfer cross section interms of the parameter β = 2 α eff χ m v m χ (2 α − . (6.12)In the limit β (cid:29)
1, the transfer cross section is found to approximately be σ classicalT ≈ πm (cid:18) β log β (cid:19) − (2 α − β + (cid:0) α − (cid:1) log (cid:16) β log β (cid:17) . (6.13)See Appendix D.2 for details. Our analytical result is shown to be in good agreement withthe numerical solution to the Schr¨odinger equation, see Figure 5.23 ummary of Velocity Scaling We summarize the velocity scaling in the different regimes: σ T ∼ v Born (low velocity) v − α Born (high velocity) v − / (3 − α ) Classicalno simple scaling Resonant (6.14)The dependence on the bulk mass parameter α is a key difference from the standard ,single mediator case. The scenario corresponds to α = 1. To make quantitative statements about the transfer cross section that extend to the classicaland resonant regimes, we numerically solve the Schr¨odinger equation using a partial waveexpansion, σ T = 4 π ( m χ v/ (cid:88) (cid:96) ( (cid:96) + 1) sin ( δ (cid:96) +1 − δ (cid:96) ) , (6.15)where δ (cid:96) is the scattering phase shifts partial wave (cid:96) . We follow the methodology of Ref. [9]with a more relaxed numerical algorithm described in Appendix F.For bulk mass parameters α ≤ /
2, the potential dominates over the repulsive centrifugalbarrier for r →
0. In this case one must place a short distance cutoff on r that encodesdata from the UV completion. Practically, the partial wave expansion converges poorly andbecomes numerically intractable for potentials more singular than r − . As such, we restrictthe bulk mass parameter to the range 1 / < α <
1, where the upper limit is the theoreticalupper limit established in Section 3.3.
Realization of the transfer cross section regimes.
The scattering rate density relevantfor thermalizing the cores of dark matter halos is the transfer cross section times the darkmatter number density, σ T n χ ∼ σ T ρ χ /m χ . The dark matter density ρ χ is a measured input,so a useful figure of merit is the ratio σ T /m χ , for which the typical value required for small-scale structure is σ T /m χ ∼ O (1).To demonstrate the self-interacting dark matter regimes discussed in this section, Figure 5scans the ratio σ T /m χ over the mass gap µ ∼ m for different values of the bulk massparameter α . These one-dimensional plots are slices of the transfer cross section over the two-parameter space of regimes in Figure 3. For each of these plots, large values of µ correspond tothe low-velocity Born regime. Figure 5 confirms the agreement with the Born approximationin this limit. As one decreases µ , one moves upward and to the right in Figure 3, crosses theresonant regime with pronounced peaks in the cross section, and finally enters the classicalregime. Figure 5 confirms that our approximate analytical results in the classical regimeagree with the numerical solution to the Schr¨odinger equation. For smaller values of α , ourapproximation for the transfer cross section in the classical regime breaks down as expected.24 igure 5: Comparison of the numerically calculated transfer cross section to the analytic approx-imations introduced in Figure 3. The general behavior displays distinct regimes, similar to thatof a single mediator, see e.g. Ref. [9, Fig. 2]. The blue line is the numerical solution. Orange(dashed)/green (dotted) lines correspond to analytic Born/classical approximations valid in theirrespective regimes; (6.8) and (6.13) . Resonances and the bulk mass parameter.
The resonance structure of transfer crosssection can be very sensitive to the bulk mass parameter α . This parameter has no analogin self-interacting dark matter models with a single mediator and represents a new modeldegree of freedom to affect phenomenology. The bulk mass feeds into both the overall effectivecoupling α eff χ (6.5) and the slope of the potential at short distances (5.19). We demonstratethe α -sensitivity of the transfer cross section with a set of benchmark parameters in Figure 6.The two plots scan over both α and the relative velocity v to highlight the interplay in theresonance structure.We remark that Figure 6 plots σ T m χ to make it straightforward to use scaling relationsto connect results to different parameters. The partial wave expansion (6.15) makes it clearthat σ T ∼ m − χ . The additional m χ dependence of the phase shifts δ (cid:96) depend only on theratios m χ /m and m χ /k ; see Appendix F. Thus the plots are unchanged by the followingrescaling of parameters by η : m χ → ηm χ k → ηk µ → ηµ . (6.16)This extends the scaling arguments in Ref. [9] to the case of a continuum mediator.25 igure 6: Transfer cross section as a function of relative dark matter velocity v (left) and bulkmass parameter α (right). The plots demonstrate the presence of resonances and anti-resonances.Vertical markers identify parameters used in the opposite plot. The scattering rate, σ T v ( ρ χ /m χ ), determines the energy transfer in dark matter halos. Fig-ure 7 plots the figure of merit σ T v/m χ for a set of benchmark parameters compared to theastronomical data points presented in Ref. [9]. The plot includes a Yukawa potential to rep-resent the single mediator case. These benchmarks correspond to a range of bulk massparameters α . The other parameters are set to give fits of comparable χ to the Yukawapotential. We remark that this is not a scan to minimize χ and is only meant to demon-strate the range of parameter possibilities that can fit the data. The ultimate cause for thedark matter halo density profile observations may partially (or wholly) include contributionsfrom baryonic feedback, see Ref. [19] for a recent status report. Thus one may conservativelyinterpret the data in Figure 7 as upper limits on the transfer cross section for a viable model.The mass hierarchy between the dark matter and lightest KK mass is comparable to thatof the benchmark self-interacting dark matter theory, m χ /m φ ∼ O (10 ). While λ can varyover a few orders of magnitude, the effective coupling α eff χ remains approximately constant forthe benchmarks in Figure 7. In the extreme case α = 0 .
55, the effective coupling α eff χ is smallcompared to the other benchmarks. This is compensated by a small dark matter mass. Thisinterplay between α ( i.e. the bulk mass) and the dark matter–mediator coupling λ may beused, for example, to maintain the fit to data in Figure 7 while adjusting a mediator–StandardModel coupling λ SM to realize other phenomenology.We remark that while we restrict to the range of bulk mass parameters 1 / < α < α ≈ .
55, reproducing the desired (cid:104) σ T v (cid:105) /m χ behavior requires sub-GeV dark matter and a Kaluza–Klein scale of O (10 keV),which may cause tension with cosmological constraints [66]. On the other hand, for largevalues of α →
1, one must take care to use the appropriate limiting form of the bulk prop-agator, as discussed in Section 5.2. Since the α = 1 case essentially describes a single mediator, this limit approaches that of ordinary self-interacting dark matter models.Beyond simply describing the model parameters that reproduce astrophysical data, it isalso illustrative to plot a range of model parameters to see how they distort the (cid:104) σ T v (cid:105) /m χ igure 7: Velocity dependence of the thermally averaged transfer cross section. The parameters arechosen to be reasonably fit astronomical data. A benchmark self-interacting dark matter modelwith a scalar mediator is shown for comparison. The data points for velocities v ∼ −
200 km / sare determined from the observed rotation curves of dwarf (red) and low-surface brightness galaxies(blue) respectively; points for velocities v (cid:38) km / s correspond to galaxy clusters (green) and aredetermined from stellar line-of-sight velocity dispersion data [61]. behavior from the ideal case. Figure 8 presents such a scan over the mass gap µ and m χ withother parameters fixed.Varying the mass gap µ primarily affects the behavior at low velocities (low momentumtransfer), though it leads to an overall rescaling because it is a multiplicative factor in theeffective coupling α eff χ (6.5). Thus for a set of parameters that fit the cluster data well, onecan tune the mass gap to help fit the low-velocity data.The dependence on the dark matter mass m χ , on the other hand, is highly nontrivial.One can see this because the phase shifts in (6.15) depend on the dimensionless combinations m χ /m and m χ /k , as described in Appendix F. Varying m χ thus affects two independentquantities in the numerical solution of the partial waves. The purpose of this study is to demonstrate the distinctive self-interaction phenomenology ofour model and we have remained agnostic about whether or not dark matter is a relic fromthermal freeze out. Thus we have not restricted the dark matter mass m χ and bulk coupling λ to fit that of a thermal relic, even though such a restriction would itself be an interestingbenchmark. Indeed, one of the constraints on typical self-interacting dark matter modelsis that the required self-interactions for small scale structure are generally too large for darkmatter to be a thermal relic in the simplest cosmological scenarios. Recent work has shownthat in the presence of bulk self-interactions, the high KK -number states of the scalar arenot valid asymptotic states due to the breakdown of the narrow width approximation [18].As a result, the production of KK modes is heavily suppressed by phase space. This canlead to a tantalizing mechanism to suppress the annihilation rate: by increasing the bulk27 igure 8: Velocity dependence of the thermally averaged transfer cross section, analogous to Fig-ure 7, for a range of µ and m χ choices to demonstrate the behavior with respect to these parameters. scalar self-interaction—a new parameter in the theory—one may control the total number ofeffectively allowed final states. We leave this topic for future work. The same dynamics that generate dark matter self-interactions also lead to Sommerfeldenhancements. Sommerfeld enhancements encode the effect of the long-range force on a short-distance process (annihilation) and so depend on the solution to the two-body Schr¨odingerequation at the origin, Ψ(0) [67–73]. In contrast, the dark matter self-interactions that arethe main focus of this paper are intrinsically long-ranged. Diagrammatically both processesinvolve a ladder of exchanged force mediators between the dark matter initial states. Whenthe potential has a mass gap, the potential supports resonances at large enough coupling.We investigate the Sommerfeld effect in our continuum-mediated model. The continuum-mediated potentials we consider are shorter-ranged than the 1 /r factor in Yukawa potentials.Since the Sommerfeld enhancement is a long range effect, one may expect that the continuum-mediated Sommerfeld effect is suppressed as compared to the Coulomb case. However, thepossibility of resonances may compensate for this and a detailed quantitative analysis isrequired.Analytical results for Sommerfeld enhancement are only available for Coulomb potentials.More generally, one must use numerical methods to solve for the enhancement from moregeneral potentials, see e.g. Ref. [74]. This method is valid for potentials that scale like r − to r − , corresponding to bulk masses 1 / ≤ α ≤ r − require a separate treatment because the potential termdominates the centrifugal term at small distances. These potentials require a short-distancecutoff as expected from a low-energy effective theory.We numerically explore the Sommerfeld enhancement over the range of bulk masses1 / ≤ α < α (cid:46) .
9. Figure 9 shows the Sommerfeldenhancement as a function of α and µ/m χ for a benchmark coupling λ = 10. The key result isthat Sommerfeld enhancement occur even when α < igure 9: Sommerfeld enhancement of the (cid:96) = 0 partial wave for a range of α and the ratio µ/m χ .Our approximation of the potential breaks down near α = 1 and hence this region is removed. dark matter mass, µ/m χ . For example for µ/m χ = 10 − we find S ∼
10 for α ∼ . α – µ plane. With the assumptions inFigure 9, the theory exhibits resonant behavior occurring for bulk masses as low as α ∼ . µ . The large coupling case λ = 10exhibits resonances, while a smaller coupling λ = 1 does not. In this case the Sommerfeldeffect is found to quickly decrease with α .The Sommerfeld enhancement decreases quickly with α and eventually vanishes near α = 1 /
2, corresponding to a r − potential. We remark that the enhancement for an ungapped V ( r ) ∝ r − potential can be solved exactly. In this case, the centrifugal term has the samescaling as the potential so that the (cid:96) = 0 solution is singular and dependent on the EFT cutoff. To the best of our knowledge, Sommerfeld enhancement for this case has not beendiscussed in the literature. We present details of this calculation in Appendix E. We find S = 1 whenever the dark matter mass is much smaller than the EFT cutoff. In other words,the 1 /r potential is too short-ranged to induce any Sommerfeld enhancement, confirmingthe numerical result in Figure 9. 29 igure 10: Sommerfeld enhancement of the (cid:96) = 0 partial wave as a function of α . We propose a model where dark matter self-interacts through a continuum of mediators.This generalizes work on self-interacting dark matter that has otherwise focused on the caseof a single massive mediator producing a Yukawa potential. A continuum mediator may arisein a strongly-coupled gauge sector. We assume that this mediator sector is nearly conformalso that its features are dictated by symmetry. Applications of the self-interacting dark matterparadigm to small-scale structure anomalies require a mass gap to cut off the potential atlong distances. A natural choice to realize this mass gap is to assume that the strongly-coupled sector has a large number of colors so that the theory is described holographicallyby a brane-localized dark matter interacting with a bulk field in a slice of
5D AdS space.We present a concrete realization where the continuum mediator is a scalar. Weaddress aspects of effective field theory and constraints from experiments and cosmology.The key parameter that characterizes the hallmark features of our model is α , which encodesthe scalar field’s bulk mass and maps onto the conformal dimension ∆ of the dual scalaroperator.We evaluate the non-relativistic potential induced by a continuum mediator with a massgap using the spectral representation and asymptotic expressions for the propagator. Weobtain simple closed-form expressions for the α < α = 1 cases and validate themnumerically. The α = 1 case corresponds to a Yukawa potential. At long distances, thepotential scales like a non-integer power, V ∼ r α − . We focus on the range 1 / < α ≤ CFT unitarity.The astronomical phenomenology of dark matter self-interactions depends on the transfercross section, σ T . We calculate this quantity in the continuum-mediated scenario and demar-cate three types of qualitative behavior—the Born, resonant, and classical regimes. Theseregimes are qualitatively similar to those of a single mediator, but in the continuum-mediated model the regimes depend on α in addition to to the strength of the dark mattercoupling and mass gap.The velocity-dependence of the transfer cross section allows a self-interacting dark mattermodel to explain small-scale structure anomalies while avoiding cluster-scale constraints. Incontrast to the single mediator, the transfer cross section in the continuum-mediatedmodel exhibits non-integer velocity scaling. For example, in the perturbative Born regime,30 T ∼ v − α for large velocities. In the non-perturbative classical regime, σ T ∼ v − / (3 − α ) inthe small mass gap limit. In contrast, a Yukawa potential in both of these regimes has atransfer cross section scaling of σ T ∼ v − .We present benchmark fits of the transfer cross section to astrophysical data. In theextreme case of bulk mass parameters α ∼ .
55, fits typically require sub-GeV dark matterand sub-MeV mass gaps, which may be cosmologically challenging. Larger values of α permithigher mass scales so long as the ratio of the dark matter mass to the mass gap is m χ /µ ∼ .Larger bulk masses cause KK mode profiles to localize away from the UV brane that containsdark matter. Thus larger bulk masses typically require larger dark matter–mediator couplingsbetween the dark matter and mediator.Our model necessarily leads to continuum-mediated Sommerfeld enhancement. We demon-strate the pattern of resonances that occur in the ( µ, α ) plane. The enhancements vanish as α → /
2, consistent with our analytical results for a 1 /r potential.We conclude that models of dark matter with continuum mediators introduce novel power-law scalings in self-interaction effects. The bulk mass parameter, α , has no analog in standard self-interacting dark matter models and is a new way to control the phenomenology. Sincethe bulk mass controls the localization of the mediator, it naturally plays a role in possibleeffective couplings to the Standard Model. These observations open new possibilities for darkmatter phenomenology. Acknowledgments
We thank Gerardo Alvarez, Lexi Costantino, and Hai-Bo Yu for useful comments and discus-sions. p.t. thanks the Aspen Center for Physics (NSF grant p.t. is supported by the DOE grant de-sc /0008541. p.t. and i.c. thank the Physics 40B (Winter 2021, Section 001) studentsof UC Riverside for their patience with grade postings while this manuscript was being com-pleted.
A AdS/CFT with UV brane
The
AdS / CFT correspondence states that boundary correlators of quantum field theory in
AdS d +1 spacetime are equivalent to correlators of a conformal field theory in d -dimensionalspacetime [44–47]. For a given bulk field in AdS , the corresponding
CFT operator arisesthrough the asymptotic behavior of the field near the
AdS boundary. In this appendix werevisit and streamline the two branches of the correspondence in the presence of a UV brane. A.1 The Two Branches
A scalar bulk field Φ in
AdS corresponds to a scalar operator O of a CFT . The conformaldimension of O is denoted ∆. An analysis of the boundary asymptotics shows that therelation between AdS bulk mass and ∆ is given by [36–43]∆(∆ + d ) k = M , (A.1)31r equivalently (3.10). We recall that M = ( α − k and α ≥ ± = 2 ± α . (A.2)These two roots indicate that the correspondence has two branches; for a given AdS fieldthere can be two
CFT duals. Unitarity of the operator implies ∆ ≥
1. It follows that the ∆ + branch exists for α ∈ R + , but the ∆ − branch exists only for 0 ≤ α ≤ AdS boundary Φ ≡ Φ( X M → boundary). Starting from the AdS partition function,one integrates over the bulk degrees of freedom while holding Φ constant. This defines theboundary effective action (cid:90) Φ D Φ e iS AdS [Φ] = e i Γ AdS [Φ ] . (A.3)The two branches of the correspondence are then formulated as follows.In the ∆ + branch, the dual CFT is defined by the correspondenceΓ
AdS [Φ ] ≡ W CFT [Φ ] (A.4)with W CFT [ J ] the generating functional of connected correlators of a CFT where J is thesource of the operator O (with [ O ] = ∆ + ), Z CFT [ J ] = (cid:90) D φ CFT e iS CFT [ φ CFT ]+ (cid:82) d x O J = e iW CFT [ J ] . (A.5)In this branch we can observe that the Φ variable corresponds to the source of the O operator.In the ∆ − branch the dual CFT is defined by the correspondenceΓ
AdS [ O ] ≡ Σ CFT [ O ] (A.6)with Σ CFT [ O cl ] the Legendre transform of W CFT [ J ],Σ CFT [ O ] = W CFT [ J ] − (cid:90) dx µ O J . (A.7)Σ is constructed similarly to an effective action. Its argument is understood to be an expec-tation value, e.g. O cl , this is left implicit here. In the ∆ − branch we can observe that Φ corresponds to the expectation value of the O operator itself. A.2 The Two Branches with a UV brane
One can truncate
AdS with a UV brane and identify Φ = Φ( X M → UV brane). The above
AdS / CFT relations from full
AdS remain structurally the same, however fields on a braneaway from the boundary can be dynamical, hence the UV brane has a localized action S UV .In particular the Φ variable is in general dynamical instead of being static as in the full AdS case. Thus Φ is now a field, external to the CFT .32he
AdS partition function is (cid:90) D Φ e iS UV [Φ ] (cid:90) Φ D Φ e iS AdS [Φ] = (cid:90) D Φ e iS UV [Φ ]+ i Γ AdS [Φ ] . (A.8)To formulate the theory in terms of a generating functional of connected correlators, onewould have to introduce new static sources coupled to Φ and O . Instead we can Legendretransform and describe the theory directly in terms of an effective action Γ . We introduceΓ UV , the effective action generated by S UV .Consider the ∆ + branch. The theory is identified as in (A.4), appending S UV on bothsides. The W CFT is substituted by its Legendre transform using Eq. (A.7), where the J sourceis localized on the UV brane and can now be dynamical. It follows that the effective actionof the theory is given byΓ UV [Φ ] + Γ AdS [Φ ] ≡ Γ UV [Φ ] + Σ CFT [ O ] + (cid:90) dx µ O Φ = Γ [Φ , O ] . (A.9)To illustrate the theory defined by (A.9), consider a dynamical UV brane-localized current J UV coupled to Φ as S UV = (cid:82) d xJ UV Φ , and evaluate the (cid:104) J UV J UV (cid:105) correlator. One findsthat the J UV currents exchange a propagator of Φ , which is itself dressed by the two-pointfunction of O .In the ∆ − branch the effective action of the theory is identified asΓ UV [ O ] + Γ AdS [ O ] ≡ Γ UV [ O ] + Σ CFT [ O ] = Γ [ O ] . (A.10)Consider again the (cid:104) J UV J UV (cid:105) correlator from the S UV = (cid:82) d xJ UV Φ interaction. What weobtain is that the J UV currents exchange a two-point correlator of O . This ∆ − branch of theduality is the one used for our model. Identifying the J UV current as J DM , the (cid:104) J J (cid:105) correlatordiscussed here describes formally the relation given in (2.3).
B Derivation of Gapless α = 1 Potential
In this appendix we show how to evaluate the Fourier transform of (5.25). The first term isa simple pole at the origin and thus gives a Coulomb potential. The next-to-leading termgoes as log( q ) /q . To evaluate its Fourier transform we uselog q q n = − ∂ α q α (cid:12)(cid:12)(cid:12)(cid:12) α → n (B.1)with n = 1. The Fourier transform of q − α is1(2 π ) (cid:90) d q e i qr q α = 1(2 π ) α ) (cid:90) d q e i qr (cid:90) dtt t α − e − tq = 1(4 π ) / Γ(3 / − α )Γ( α ) (cid:18) r (cid:19) / − α . (B.2)We then evaluate the α derivative and set α = 1, which gives1(4 π ) / ∂ α (cid:32) Γ(3 / − α )Γ( α ) (cid:0) r (cid:1) / − α (cid:33) α → = 12 π ( γ + log r ) . (B.3)Combining these identities gives (5.30). 33 Validity of the Born Approximation
In order to determine the validity of the Born approximation, consider the wave function fora dark matter particle scattering off of a potential V ( x ), ψ ( x ) ∼ e i p · x − m χ (cid:90) d x (cid:48) e ip | x − x (cid:48) | π | x − x (cid:48) | V ( x (cid:48) ) e i p · x (cid:48) , (C.1)where | p | = m χ v/ p · x (cid:48) = pr (cid:48) cos θ . Near the origin | x − x (cid:48) | ≈ r (cid:48) , thus the conditionfor when the Born approximation is valid is (cid:12)(cid:12)(cid:12)(cid:12) m χ π (cid:90) d x (cid:48) e ipr (cid:48) r (cid:48) V ( x (cid:48) ) e i p · x (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) . (C.2)For a Yukawa potential V ( r ) = α χ e − m φ r /r this condition is simply α χ m χ /m (cid:28)
1. At lowenergies we can replace the exponentials by 1. For a central potential in spherical coordinatesthe angular integral is trivial. Evaluating (C.2) for a Yukawa potential gives the condition α χ m χ /m φ (cid:28)
1. This bound can be equivalently determined by considering the typicalmomentum flowing through a ladder diagram is of order α χ m χ [76, 77]. We evaluate (C.2)for (5.7) and arrive at the result λ πk (cid:88) n f n ( z UV ) m n (cid:28) . (C.3)In order to make the connection to the Yukawa case more explicit, we define the effectivecoupling α eff χ = λ m πk (cid:88) n f n ( z UV ) m n , (C.4)such that the condition for when then Born approximation is valid becomes α eff χ m χ m (cid:28) , (C.5)analogous to the Yukawa case.Recalling that the bulk profiles depend on the bulk mass parameter α , we note that thesum over KK modes in α χ, eff diverges for α ≤ /
2. This is consistent with the Schr¨odingerequation in which, near the origin, the continuum mediated potential (5.19) dominates overthe centrifugal barrier for α ≤ /
2. In order to achieve finite results in the case when α ≤ / α eff χ = λ m πk (cid:88) n f n ( z UV ) m n −→ α eff χ (Λ) = λ m πk (cid:88) n f n ( z UV ) m n e − m n / Λ , (C.6)where Λ − is the short distance cutoff. We evaluate the KK sum using the spectral represen-tation of the propagator (5.3) and using the large-momentum asymptotics (5.11). We arriveat the result α eff χ = λ π Γ(1 − α ) m Λ (cid:18) k Λ (cid:19) α − Γ (cid:16) − α, m Λ (cid:17) . (C.7)34hen α > /
2, the limit Λ → ∞ is finite and cutoff independent, α eff χ (cid:12)(cid:12) α> / = λ π (cid:20) α − − α ) (cid:21) (cid:16) m k (cid:17) − α . (C.8)This result is identical to evaluating (C.2) for the continuum mediated potential (5.19).For the special case of a bulk mass parameter α = 1 /
2, we find that the effective couplingfor the Born approximation is α eff χ = λ m π k log (cid:18) Λ e − γ m (cid:19) . (C.9)The other limit, α → α < α → α = 1, scattering is governed by the Yukawapotential (5.23) and we can directly apply (6.3) so that α eff χ = λ m πk f ( z UV ) (C.10)where f ( z UV ) is given by (5.24).The accuracy of the Born approximation improves at higher energies. This can also beshown from (C.2) by computing the angular integral for a general central potential, D Classical Transfer Cross Section
We calculate the transfer cross section in the classical regime and observe its velocity depen-dence. The angle by which a particle in a central potential is deflected is θ ( ρ ) = | π − ϕ ( ρ ) | where [78] ϕ ( ρ ) = ρ (cid:90) ∞ r min drr (cid:112) − ρ /r − V ( r ) /m χ v (D.1)and ρ is the impact parameter. The lower limit of integration r min is the largest root of thedenominator of (D.1). In contrast to the quantum case, the classical cross section is typicallygiven in terms of the impact parameter rather than the angular variables. In the classicallimit, integration over the deflection angle can be troublesome since the solution to (D.1) for ϕ ( ρ ) and thus θ ( ρ ) takes values greater than π for cases other than a 1 /r potential. On theother hand the impact parameter always ranges between zero and infinity.The differential cross section is dσ = 2 πρ dρ . The transfer cross section is thus σ classicalT = 2 π (cid:90) ∞ [1 − cos θ ( ρ )] ρdρ . (D.2)To connect to the deflection angle, we note that (cid:18) dσd Ω (cid:19) classical = ρ ( χ )sin θ (cid:12)(cid:12)(cid:12)(cid:12) dρdθ (cid:12)(cid:12)(cid:12)(cid:12) (D.3)where in these variables d Ω = 2 π d cos θ . We present calculations for the velocity scaling inthe small mass gap/high velocity limit and an analytical result in the low velocity region ofthe non-perturbative classical regime. 35 .1 Velocity Scaling in the Small Mass Gap/High Velocity Limit For the sake of this calculation we assume the gapless limit where the potential is (5.22) V ( r ) = − λ π / Γ(3 / − α )Γ(1 − α ) 1 r ( kr ) − α . (D.4)This approximation also accounts for the high velocity limit where the particle momentumis much greater than the mass gap. Given our potential we can define a characteristic lengthscale ρ ≡ (cid:20) λ π / m χ v k − α Γ(3 / − α )Γ(1 − α ) (cid:21) − α (D.5)so that after making the change of variables r = ρ/x , (D.1) becomes [64] ϕ ( ρ ) = (cid:90) x max dx (cid:112) − x + 2( ρ x/ρ ) − α (D.6)where the limit of integration x max is the smallest positive root of the denominator. Observethat ϕ (and by extension χ ) and as x max are functions of the dimensionless combination ρ/ρ and not ρ independently. Making the change of variables ρ = ρ ξ , the transfer cross sectionis σ classicalT = 2 πρ (cid:90) ∞ [1 − cos χ ( ξ )] ξdξ . (D.7)Because χ is a function of ξ and α only, the integral (D.7) only depends on the bulk massparameter. We can thus conclude from (D.5) that the velocity dependence of the transfercross section in the classical regime is σ classicalT ∼ v − / (3 − α ) .The presence of the mass gap spoils the velocity dependence derived in (D.7). For thegapped potential (5.19), after changing variables, the deflection angle depends on the quantity m ρ as well. Thus a small but nonzero m induces corrections to (D.7). D.2 Low Velocity Classical Regime
We present a closed form result for the transfer cross section in the low velocity region ofthe non-perturbative classical regime. Following the method of Ref. [65], the transfer crosssection is a function of a single unique parameter, β = 2 α eff χ m m χ v (2 α − . (D.8)The transfer cross section is σ T ≈ πρ ∗ where ρ ∗ is found by solving the set of equations (cid:101) V eff ( r max , ρ ∗ ) = 1 d (cid:101) V eff ( r, ρ ∗ ) dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r max = 0 (D.9)36here (cid:101) V eff is the effective potential (cid:101) V eff ( r, ρ ) = ρ r + 4 m χ v V ( r ) . (D.10)These conditions correspond to the maximum of the effective potential (cid:101) V eff .We find for β (cid:29) σ classicalT ≈ πm (cid:20) (cid:18) β log β (cid:19) − (2 α −
1) log − β + (cid:18) α − (cid:19) log − (cid:18) β log β (cid:19)(cid:21) . (D.11)The accuracy of (D.11) is confirmed in Figure 5. E Sommerfeld Enhancement from a /r Potential
The Sommerfeld effect amounts to the enhancement of the particle wavefunction at thepoint where the local annihilation process happens. It comes from the dressing from ladderdiagrams generated by a potential V ( r ). The dressed wavefunction is determined by directlysolving the Schr¨odinger equation. The Sommerfeld enhancement factor is defined as σ = S ( p ) σ with σ the undressed cross section. The method to evaluate the Sommerfeld effectis well known, here we follow [68] (see also [70]).The Schr¨odinger equation is − M ∆Ψ( r ) + V ( r )Ψ( r ) = p M Ψ( r ) . (E.1)In any solution of the Schr¨odinger equation with rotational invariance around z , the solutionscan be expanded as Ψ = (cid:88) a l P (cid:96) (cos θ ) R (cid:96) ( r ) . (E.2)The radial wavefunction satisfies − M r ddr (cid:18) r dR (cid:96) dr (cid:19) + (cid:18) V ( r ) + (cid:96) ( (cid:96) + 1)2 M r (cid:19) R (cid:96) ( r ) = p M R (cid:96) ( r ) . (E.3)In the standard approach one uses the fact that angular momentum with (cid:96) > R (cid:96) ∼ r (cid:96) at small r , which implies that the (cid:96) > (cid:96) = 0 angular momentum.For our continuum-mediated potential V ( r ) ∝ r α − , the vanishing of (cid:96) > α ≥ /
2. For α > /
2, the (cid:96) = 0 mode gives R (cid:96) ∼ constant at small r . But for α = 1 /
2, which is the V ( r ) ∝ /r potential of our interest, the (cid:96) = 0 component divergesat small r . This feature is not an inconsistency. We work in a low-energy EFT so the r coordinate cannot be zero, it is rather cut at a small value corresponding to the UV cutoff, r = r . In our AdS model the cutoff is at r ∼ k − . Of course, the subsequent results maybe cutoff dependent, but this is not a conceptual problem, this simply reflects that an EFT prediction can depend on the unknown UV physics.37ere we parametrize the α = 1 / V ( r ) = − κ r . (E.4)The matching to the physical couplings from the AdS model is κ = λ π k .Introducing χ (cid:96) ( r ) = rR (cid:96) ( r ) the Schr¨odinger equation becomes − M ∂ r χ ( r ) + V ( r ) χ ( r ) = p M χ ( r ) . (E.5)From this equation, various equivalent methods lead to the Sommerfeld factor, which differsby the boundary conditions chosen for χ [68]. We use the following. χ (cid:96) is chosen to satisfy ∂ r χ (cid:96) = ipχ (cid:96) at r = ∞ . Using this solution, the Sommerfeld factor is S = (cid:12)(cid:12)(cid:12)(cid:12) χ ( r = ∞ ) χ ( r = r ) (cid:12)(cid:12)(cid:12)(cid:12) . (E.6)Notice that since we are in an EFT with have replaced the r = 0 by r = r .The solution satisfying the condition at r = ∞ is found to be χ ( r ) (cid:96) ∝ √ rH (1) η ( pr ) , η = (cid:114) − M κ . (E.7)The dimensionful κ coupling is of order of the inverse cutoff of the EFT . The
EFT wouldbreak at
M κ ∼
1, we are rather interested in
M κ (cid:28)
1, i.e. the dark matter mass is muchlower than the cutoff k .Expanding in the small parameter M κ we find χ ( r ) ∝ i + O ( pr ) . (E.8)We have that pr is necessarily (cid:28) r is the inverse cutoff k − , and because thenon-relativistic approximation requires p < M and the EFT validity requires
M < k .It follows that within the range of validity of the
EFT , we can simply take η ≈ /
2. TheHankel simplifies to H (1) ( z ) ∝ z − / e iz , thus χ ( r ) ∝ e ipr for any pr . The Sommerfeld factoris then exactly S = 1 for any p . F Self-Interacting Dark Matter Numerical Method
We summarize the methodology for determining the dark matter self-interaction cross sec-tion. We closely follow the procedure in Ref. [9] however we employ a slightly more relaxedalgorithm. The relevant quantity is the transfer cross section, σ T = (cid:90) d Ω (1 − cos θ ) dσd Ω , (F.1)which characterizes interaction cross section weighted by momentum transfer. This regulatesthe cos θ → (cid:96) th partialwave phase shift, δ (cid:96) , by σ T = 4 πp ∞ (cid:88) (cid:96) =0 ( (cid:96) + 1) sin ( δ (cid:96) +1 − δ (cid:96) ) . (F.2)The δ (cid:96) are, in turn, obtained by solving the radial Schr¨odinger equation (E.3) taking M = m χ / p = m X v/ v is the relative velocity of the two-particle dark matter system. δ l is found by comparing with the asymptotic solution for R (cid:96) :lim r →∞ R (cid:96) ( r ) ∝ cos δ (cid:96) j (cid:96) ( pr ) − sin δ (cid:96) n (cid:96) ( pr ) , (F.3)where j (cid:96) ( n (cid:96) ) is the spherical Bessel (Neumann) function of the (cid:96) th order. We again definethe function χ (cid:96) ≡ rR (cid:96) along with the dimensionless variables x ≡ α X m X r a = v α X b = α X m X m c = α X m X k , (F.4)so that we can rewrite (E.3) as [79] (cid:20) d dx + a − (cid:96) ( (cid:96) + 1) x ± π / x (cid:16) cx (cid:17) − α Γ(3 / − α )Γ(1 − α ) Q (2 − α, x/b ) (cid:21) χ (cid:96) ( x ) = 0 . (F.5)Near the origin for α > /
2, the angular momentum term dominates over the potential.This implies that χ (cid:96) ∝ x (cid:96) +1 close to x = 0. When α ≤ / χ (cid:96) ( x ) = 1 and χ (cid:48) (cid:96) ( x ) = ( (cid:96) + 1) /x where x is a point close to the origin chosen to satisfy x (cid:28) b and x (cid:28) ( (cid:96) + 1) /a . We take x as thelower limit for the range in which we numerically solve the Schr¨odinger equation. Similarly,to define the upper limit, we pick a point x m satisfying the condition a (cid:29) π / x (cid:16) cx (cid:17) − α Γ(3 / − α )Γ(1 − α ) Q (2 − α, x/b ) . (F.6)When x m satisfies this condition, the potential term is negligible compared to the kineticterm and the solution approaches χ (cid:96) ( x ) ∝ xe iδ (cid:96) (cos δ (cid:96) j (cid:96) ( ax ) − sin δ (cid:96) n (cid:96) ( ax )) . (F.7)The phase shift is thentan δ (cid:96) = ax m j (cid:48) (cid:96) ( ax m ) − β (cid:96) j (cid:96) ( ax m ) ax m n (cid:48) (cid:96) ( ax m ) − β (cid:96) n (cid:96) ( ax m ) where β (cid:96) = x m χ (cid:48) (cid:96) ( x m ) χ (cid:96) ( x m ) − . (F.8)39or an initial guess of the range ( x , x m ) and the maximum number of partial waves requiredfor convergence, (cid:96) max , we calculate δ (cid:96) from (F.8). In Ref. [9] x m and x are increased anddecreased respectively, recalculating δ (cid:96) until the differences of successive iterations convergeto be within 1%. This condition can be quite cumbersome numerically and is not strictlyrequired unless one wishes to do a fine grained scan over the parameter space. Instead, wetake the value of δ (cid:96) given by our initial guess. This method is sufficient to reproduce thebenchmark results in Ref. [61].We then sum (F.2) from (cid:96) = 0 to (cid:96) = (cid:96) max to obtain an estimate for σ T . Next we increment (cid:96) max → (cid:96) max + 1 and repeat the procedure until successive values of σ T converge to be within1% and δ (cid:96) max < .
01. Ref. [9] iterates (cid:96) max until σ T converged and δ (cid:96) max < .
01 ten consecutivetimes. We have found that the “StiffenessSwitching” method from the
NDSolveUtilities package in
Mathematica to be particularly useful.We employ this method to calculate the Sommerfeld enhancements as well. The enhance-ment factor is [70, 74] S = (cid:20) (2 (cid:96) + 1)!! C (cid:21) (F.9)where C is C = (cid:0) χ (cid:96) ( x ) − χ (cid:96) ( x − π/ a ) (cid:1) x →∞ . (F.10) References [1] M. Pospelov, “Secluded U(1) Below the WeakScale,”
Phys. Rev.
D80 (2009) 095002, arXiv:0811.1030 [hep-ph] .[2] M. Pospelov and A. Ritz, “AstrophysicalSignatures of Secluded Dark Matter,”
Phys.Lett.
B671 (2009) 391–397, arXiv:0810.1502[hep-ph] .[3] M. Pospelov, A. Ritz, and M. B. Voloshin,“Secluded WIMP Dark Matter,”
Phys. Lett.
B662 (2008) 53–61, arXiv:0711.4866[hep-ph] .[4] R. Essig et al. , “Working Group Report: NewLight Weakly Coupled Particles,” in
Proceedings, 2013 Community Summer Studyon the Future of U.S. Particle Physics:Snowmass on the Mississippi (Cs S .2013. arXiv:1311.0029 [hep-ph] . .[5] J. Alexander et al. , “Dark Sectors 2016Workshop: Community Report,” 2016. arXiv:1608.08632 [hep-ph] . http://lss.fnal.gov/archive/2016/conf/fermilab-conf-16-421.pdf .[6] M. Battaglieri et al. , “Us Cosmic Visions: NewIdeas in Dark Matter 2017: CommunityReport,” in U.S. Cosmic Visions: New Ideas inDark Matter College Park, Md, Usa, March23-25, 2017 . 2017. arXiv:1707.04591[hep-ph] . http://lss.fnal.gov/archive/2017/conf/fermilab-conf-17-282-ae-ppd-t.pdf .[7] E. D. Carlson, M. E. Machacek, and L. J. Hall,“Self-Interacting Dark Matter,” Astrophys. J. (1992) 43–52.[8] D. N. Spergel and P. J. Steinhardt,“Observational Evidence for SelfinteractingCold Dark Matter,”
Phys. Rev. Lett. (2000)3760–3763, arXiv:astro-ph/9909386 .[9] S. Tulin, H.-B. Yu, and K. M. Zurek, “BeyondCollisionless Dark Matter: Particle PhysicsDynamics for Dark Matter Halo Structure,” Phys. Rev. D no. 11, (2013) 115007, arXiv:1302.3898 [hep-ph] .
10] S. Tulin and H.-B. Yu, “Dark MatterSelf-Interactions and Small Scale Structure,”
Phys. Rept. (2018) 1–57, arXiv:1705.02358 [hep-ph] .[11] T. Gherghetta and B. von Harling, “A WarpedModel of Dark Matter,”
JHEP (2010) 039, arXiv:1002.2967 [hep-ph] .[12] B. von Harling and K. L. McDonald, “SecludedDark Matter Coupled to a Hidden CFT,” JHEP (2012) 048, arXiv:1203.6646 [hep-ph] .[13] M. J. Strassler, “Why Unparticle Models withMass Gaps are Examples of Hidden Valleys,” arXiv:0801.0629 [hep-ph] .[14] C.-H. Chen and C. Kim, “SommerfeldEnhancement from Unparticle Exchange forDark Matter Annihilation,” Phys. Lett. B (2010) 232–235, arXiv:0909.1878 [hep-ph] .[15] A. Friedland, M. Giannotti, and M. Graesser,“On the R S Realization of Unparticles,”
Phys.Lett. B (2009) 149–155, arXiv:0902.3676[hep-th] .[16] A. Friedland, M. Giannotti, and M. L.Graesser, “Vector Bosons in theRandall-Sundrum 2 and Lykken-RandallModels and Unparticles,”
JHEP (2009) 033, arXiv:0905.2607 [hep-th] .[17] P. Brax, S. Fichet, and P. Tanedo, “TheWarped Dark Sector,” Phys. Lett. B (2019) 135012, arXiv:1906.02199 [hep-ph] .[18] A. Costantino, S. Fichet, and P. Tanedo,“Effective Field Theory in AdS: ContinuumRegime, Soft Bombs, and IR Emergence,” arXiv:2002.12335 [hep-th] .[19] J. S. Bullock and M. Boylan-Kolchin,“Small-Scale Challenges to the ΛCDMParadigm,”
Ann. Rev. Astron. Astrophys. (2017) 343–387, arXiv:1707.04256[astro-ph.CO] .[20] P. Fadeev, Y. V. Stadnik, F. Ficek, M. G.Kozlov, V. V. Flambaum, and D. Budker,“Revisiting spin-dependent forces mediated bynew bosons: Potentials in the coordinate-spacerepresentation for macroscopic- andatomic-scale experiments,” Phys. Rev. A no. 2, (2019) 022113, arXiv:1810.10364[hep-ph] .[21] S. Fichet, “Quantum Forces from Dark Matterand Where to Find Them,” Phys. Rev. Lett. no. 13, (2018) 131801, arXiv:1705.10331[hep-ph] . [22] A. Costantino, S. Fichet, and P. Tanedo,“Exotic Spin-Dependent Forces from a HiddenSector,”
JHEP (2020) 148, arXiv:1910.02972 [hep-ph] .[23] A. Katz, M. Reece, and A. Sajjad,“Continuum-mediated dark matter–baryonscattering,” Phys. Dark Univ. (2016) 24–36, arXiv:1509.03628 [hep-ph] .[24] L. Randall and R. Sundrum, “An Alternative toCompactification,” Phys. Rev. Lett. (1999)4690–4693, arXiv:hep-th/9906064 [hep-th] .[25] W. D. Goldberger and M. B. Wise, “ModulusStabilization with Bulk Fields,” Phys. Rev.Lett. (1999) 4922–4925, arXiv:hep-ph/9907447 .[26] A. Manohar and H. Georgi, “Chiral Quarks andthe Nonrelativistic Quark Model,” Nucl. Phys.B (1984) 189–212.[27] H. Georgi and L. Randall, “Flavor ConservingCP Violation in Invisible Axion Models,”
Nucl.Phys. B (1986) 241–252.[28] H. Georgi, “Generalized Dimensional Analysis,”
Phys. Lett. B (1993) 187–189, arXiv:hep-ph/9207278 .[29] M. A. Luty, “Naive Dimensional Analysis andSupersymmetry,”
Phys. Rev. D (1998)1531–1538, arXiv:hep-ph/9706235 .[30] E. E. Jenkins, A. V. Manohar, and M. Trott,“Naive Dimensional Analysis Counting ofGauge Theory Amplitudes and AnomalousDimensions,” Phys. Lett. B (2013)697–702, arXiv:1309.0819 [hep-ph] .[31] G. Dvali and C. Gomez, “Quantum Informationand Gravity Cutoff in Theories with Species,”
Phys. Lett. B (2009) 303–307, arXiv:0812.1940 [hep-th] .[32] S. Fichet, “Braneworld Effective Field Theories— Holography, Consistency and ConformalEffects,”
JHEP (2020) 016, arXiv:1912.12316 [hep-th] .[33] H. Davoudiasl, G. Perez, and A. Soni, “TheLittle Randall-Sundrum Model at the LargeHadron Collider,” Phys. Lett. B (2008)67–71, arXiv:0802.0203 [hep-ph] .[34] P. Breitenlohner and D. Z. Freedman, “PositiveEnergy in Anti-de Sitter Backgrounds andGauged Extended Supergravity,”
Phys. Lett. B (1982) 197–201.
35] P. Breitenlohner and D. Z. Freedman,“Stability in Gauged Extended Supergravity,”
Annals Phys. (1982) 249.[36] J. M. Maldacena, “The Large N limit ofsuperconformal field theories and supergravity,”
Int. J. Theor. Phys. (1999) 1113–1133, arXiv:hep-th/9711200 [hep-th] . [Adv.Theor. Math. Phys.2,231(1998)].[37] S. S. Gubser, I. R. Klebanov, and A. M.Polyakov, “Gauge theory correlators fromnoncritical string theory,” Phys. Lett.
B428 (1998) 105–114, arXiv:hep-th/9802109[hep-th] .[38] E. Witten, “Anti-de Sitter space andholography,”
Adv. Theor. Math. Phys. (1998)253–291, arXiv:hep-th/9802150 [hep-th] .[39] D. Z. Freedman, S. D. Mathur, A. Matusis, andL. Rastelli, “Comments on 4 point functions inthe CFT / AdS correspondence,” Phys. Lett. B (1999) 61–68, arXiv:hep-th/9808006 .[40] H. Liu and A. A. Tseytlin, “On four pointfunctions in the CFT / AdS correspondence,”
Phys. Rev. D (1999) 086002, arXiv:hep-th/9807097 .[41] D. Z. Freedman, S. D. Mathur, A. Matusis, andL. Rastelli, “Correlation functions in theCFT(d) / AdS(d+1) correspondence,” Nucl.Phys. B (1999) 96–118, arXiv:hep-th/9804058 .[42] E. D’Hoker, D. Z. Freedman, and L. Rastelli,“AdS / CFT four point functions: How tosucceed at z integrals without really trying,”
Nucl. Phys. B (1999) 395–411, arXiv:hep-th/9905049 .[43] E. D’Hoker, D. Z. Freedman, S. D. Mathur,A. Matusis, and L. Rastelli, “Gravitonexchange and complete four point functions inthe AdS / CFT correspondence,”
Nucl. Phys. B (1999) 353–394, arXiv:hep-th/9903196 .[44] O. Aharony, S. S. Gubser, J. M. Maldacena,H. Ooguri, and Y. Oz, “Large N field theories,string theory and gravity,”
Phys. Rept. (2000) 183–386, arXiv:hep-th/9905111[hep-th] .[45] A. Zaffaroni, “Introduction to the AdS-CFTcorrespondence,”
Class. Quant. Grav. (2000) 3571–3597.[46] H. Nastase, “Introduction to AdS-CFT,” arXiv:0712.0689 [hep-th] . [47] J. Kaplan, “Lectures on AdS/CFT from theBottom Up.”.[48] N. Arkani-Hamed, M. Porrati, and L. Randall,“Holography and phenomenology,” JHEP (2001) 017, arXiv:hep-th/0012148 [hep-th] .[49] P. Creminelli, A. Nicolis, and R. Rattazzi,“Holography and the Electroweak PhaseTransition,” JHEP (2002) 051, arXiv:hep-th/0107141 .[50] A. Hebecker and J. March-Russell,“Randall-Sundrum II Cosmology, AdS / CFT,and the Bulk Black Hole,” Nucl. Phys.
B608 (2001) 375–393, arXiv:hep-ph/0103214[hep-ph] .[51] D. Langlois, L. Sorbo, andM. Rodriguez-Martinez, “Cosmology of a braneradiating gravitons into the extra dimension,”
Phys. Rev. Lett. (2002) 171301, arXiv:hep-th/0206146 .[52] D. Langlois and L. Sorbo, “Bulk gravitons froma cosmological brane,” Phys. Rev. D (2003)084006, arXiv:hep-th/0306281 .[53] A. Costantino, S. Fichet, and F. Tanedo,“Work in progress.”.[54] S. B. Giddings, E. Katz, and L. Randall,“Linearized gravity in brane backgrounds,” JHEP (2000) 023, arXiv:hep-th/0002091 .[55] J. G. Lee, E. G. Adelberger, T. S. Cook, S. M.Fleischer, and B. R. Heckel, “New Test of theGravitational 1 /r Law at Separations down to52 µ m,” Phys. Rev. Lett. no. 10, (2020)101101, arXiv:2002.11761 [hep-ex] .[56] P. Brax, S. Fichet, and G. Pignol, “BoundingQuantum Dark Forces,”
Phys. Rev. D no. 11, (2018) 115034, arXiv:1710.00850[hep-ph] .[57] F. Kahlhoefer, K. Schmidt-Hoberg, andS. Wild, “Dark Matter Self-Interactions from aGeneral Spin-0 Mediator,” JCAP (2017)003, arXiv:1704.02149 [hep-ph] .[58] R. Zwicky, “A Brief Introduction to DispersionRelations and Analyticity,” in Quantum FieldTheory at the Limits: From Strong Fields toHeavy Quarks , pp. 93–120. 2017. arXiv:1610.06090 [hep-ph] .[59] S. Fichet, “Opacity and effective field theory inanti–de Sitter backgrounds,”
Phys. Rev. D no. 9, (2019) 095002, arXiv:1905.05779[hep-th] .
60] A. Costantino and S. Fichet, “Opacity fromLoops in AdS,” arXiv:2011.06603 [hep-th] .[61] M. Kaplinghat, S. Tulin, and H.-B. Yu, “DarkMatter Halos as Particle Colliders: UnifiedSolution to Small-Scale Structure Puzzles fromDwarfs to Clusters,”
Phys. Rev. Lett. no. 4, (2016) 041302, arXiv:1508.03339[astro-ph.CO] .[62] R. Dave, D. N. Spergel, P. J. Steinhardt, andB. D. Wandelt, “Halo properties incosmological simulations of selfinteracting colddark matter,”
Astrophys. J. (2001)574–589, arXiv:astro-ph/0006218 .[63] J. J. Sakurai,
Modern quantum mechanics; rev.ed.
Addison-Wesley, Reading, MA, 1994. https://cds.cern.ch/record/1167961 .[64] D. Chiron and B. Marcos, “Classical particlescattering for power-law two-body potentials,” arXiv:1601.00064 [cond-mat.stat-mech] .[65] S. A. Khrapak, A. V. Ivlev, G. E. Morfill, andS. K. Zhdanov, “Scattering in the AttractiveYukawa Potential in the Limit of StrongInteraction,”
Phys. Rev. Lett. no. 22, (2003)225002.[66] R. H. Cyburt, B. D. Fields, K. A. Olive, andT.-H. Yeh, “Big Bang Nucleosynthesis: 2015,” Rev. Mod. Phys. (2016) 015004, arXiv:1505.01076 [astro-ph.CO] .[67] J. Hisano, S. Matsumoto, M. M. Nojiri, andO. Saito, “Non-Perturbative Effect on DarkMatter Annihilation and Gamma RaySignature from Galactic Center,” Phys. Rev. D (2005) 063528, arXiv:hep-ph/0412403 .[68] N. Arkani-Hamed, D. P. Finkbeiner, T. R.Slatyer, and N. Weiner, “A Theory of DarkMatter,” Phys. Rev. D (2009) 015014, arXiv:0810.0713 [hep-ph] .[69] M. Lattanzi and J. I. Silk, “Can the WIMPannihilation boost factor be boosted by theSommerfeld enhancement?,” Phys. Rev. D (2009) 083523, arXiv:0812.0360 [astro-ph] . [70] R. Iengo, “Sommerfeld Enhancement: GeneralResults from Field Theory Diagrams,” JHEP (2009) 024, arXiv:0902.0688 [hep-ph] .[71] R. Iengo, “Sommerfeld enhancement for aYukawa potential,” arXiv:0903.0317[hep-ph] .[72] S. Cassel, “Sommerfeld factor for arbitrarypartial wave processes,” J. Phys. G (2010)105009, arXiv:0903.5307 [hep-ph] .[73] S. Hannestad and T. Tram, “SommerfeldEnhancement of DM Annihilation: ResonanceStructure, Freeze-Out and CMB SpectralBound,” JCAP (2011) 016, arXiv:1008.1511 [astro-ph.CO] .[74] B. Bellazzini, M. Cliche, and P. Tanedo,“Effective theory of self-interacting darkmatter,” Phys. Rev. D no. 8, (2013) 083506, arXiv:1307.1129 [hep-ph] .[75] I. R. Klebanov and E. Witten, “AdS / CFTcorrespondence and symmetry breaking,” Nucl.Phys. B (1999) 89–114, arXiv:hep-th/9905104 .[76] F. Gross,
Relativistic Quantum Mechanics andField Theory . A Wiley-Intersience publication.Wiley, 1999.[77] K. Petraki, M. Postma, and J. de Vries,“Radiative bound-state-formation cross-sectionsfor dark matter interacting via a Yukawapotential,”
JHEP (2017) 077, arXiv:1611.01394 [hep-ph] .[78] L. D. Landau and E. M. Lifshitz, Mechanics,Third Edition: Volume 1 (Course of TheoreticalPhysics) . Butterworth-Heinemann, 3 ed., Jan.,1976. .[79] M. R. Buckley and P. J. Fox, “Dark MatterSelf-Interactions and Light Force Carriers,”
Phys. Rev.
D81 (2010) 083522, arXiv:0911.3898 [hep-ph] ..