Collisions of Jets of Particles from Active Galactic Nuclei with Neutralino Dark Matter
PPrepared for submission to JCAP
Collisions of Jets of Particles from Active GalacticNuclei with Neutralino Dark Matter
Jinrui Huang, Arvind Rajaraman, and Tim M.P. Tait
Department of Physics and Astronomy,University of California, Irvine, California 92697
E-mail: [email protected], [email protected], [email protected]
Abstract.
We examine the possibility that energetic Standard Model particles containedin the jets produced by active galactic nuclei (AGN) may scatter off of the dark matterhalo which is expected to surround the AGN. In particular, if there are nearby states in thedark sector which can appear resonantly in the scattering, the cross section can be enhancedand a distinctive edge feature in the energy spectrum may appear. We examine bounds onsupersymmetric models which may be obtained from the Fermi Gamma-ray Space Telescopeobservation of the nearby AGN Centaurus A.
Keywords: dark matter theory, particle physics-cosmology connection, supersymmetry andcosmology, active galactic nuclei a r X i v : . [ h e p - ph ] J a n ontents e − ˜ χ → e − ˜ χγ
11B Proof of Gauge Invariance 14
Active galactic nuclei (AGN) [1–7] are among the most powerful natural accelerators in theUniverse, producing jets of charged particles which travel macroscopic distances and can evenbe imaged from the Earth through the subsequent production of high energy photons. Whilethe nature of the jet constituents (proton or electron) remains a subject of some controversy,there is no doubt that they are among the highest energy and most interesting objects in thecosmos, with energies reaching into tens of GeV and with huge luminosities.The supermassive black holes which power these jets are typically hosted in very largegalaxies, where one would expect the largest concentrations of dark matter (DM) also occur.Taken together, the combination of dense dark matter with high energy particles represent anew opportunity to study the interactions of dark matter and Standard Model (SM) particlesat high energies. Indeed, while particle accelerators on the Earth reach higher energies andwith (much) better understood beams, the huge density of dark matter expected close toAGNs implies that they represent a unique opportunity to learn about dark matter interac-tions with high energy SM particles.If there is some other particle in the “dark sector” which can be produced as a resonancein the scattering of dark matter with a jet particle, the rate for scattering in the high energytail of the jet may be enhanced. Beam particles with energies E greater than, E ≥ m e − m χ m ˜ χ (1.1)(where m ˜ χ is the mass of the dark matter and m ˜ e the mass of the resonance) have sufficientenergy to produce the resonance on-shell, leading to an enhancement of the cross section.Since the beam particles are presumably charged (so that they can be efficiently acceleratedby the AGN), the resonant state should also be charged. As a result, its decays can produce– 1 –amma rays which can stick out from the background produced by the AGN and jet them-selves. Such a situation is common in the most popular theories of dark matter, such as insupersymmetric theories in which a neutralino plays the role of dark matter [8], and whichcan be up-scattered by an energetic beam electron into a selectron and was first proposedby [9] and recently studied in [13]. Such a degeneracy may even be central to realizing aneutralino WIMP as dark matter, to allow for efficient enough coannihilation to produce theobserved density from thermal freeze-out. In fact, a very degenerate selectron is challengingto detect at the LHC, because its visible decay products have energies controlled by m ˜ e − m ˜ χ ,and are typically very soft. Thus, the signal from AGNs is enhanced precisely where the LHChas difficulty, and can help study motivated but otherwise experimentally difficult regions ofparameter space [14].Since the resonant state must be charged, its subsequent de-excitation includes higherorder processes in which a photon is produced in addition to the primary SM and darkmatter decay products. While suppressed by α , such processes may be enhanced by collinearlogarithms that rise when the photon is emitted along the same direction as the final statecharged SM particle. Unlike charged particles, these radiated photons have very long meanfree paths, and could be detected by detectors on the Earth as an indicator that the jet-DMscatterings are taking place. For dark matter masses on the order of a few hundred GeV,these photons fall precisely in the range for which the Fermi Gamma-ray Space Telescope [15]is sensitive, and thus it is precisely the right time to explore such signals. In fact, as pointedout in Ref. [13], the rise in cross section as beam particles cross the ˜ e threshold leads to avery interesting feature in the spectrum of gamma rays, a rise in the number of high energyphotons followed by a sharp edge. This feature provides an interesting discriminant from thebackground of gamma rays emitted from the jet itself, one which conventional astrophysicalsources are expected to have difficulty faking. Inspired by Ref. [13], we study the dark matterand AGN electron jet scattering process in SUSY models, presenting the complete scatteringamplitudes and performing an explicit check of gauge invariance. We scan the parameterspace of neutralino and selectron masses to define precise constraints from the Fermi-LATdata.This article is organized as follows. In Section 2, we discuss the astrophysical inputs tothe computation, namely the density profile of dark matter around the AGN and the intensityand energy spectrum of the beam particles. In Section 3, we revisit the computation of thecross sections for scattering in supersymmetric theories. In Section 4 we examine the boundsone can place for different astrophysical assumptions. We conclude in Section 5. In this section, we discuss the astrophysical inputs to our estimates. These boil down intothe properties of particles inside the AGN jet, and the density of dark matter surroundingit. In both cases, we follow closely the discussion of Ref. [13], which contains a more detaileddiscussion, and study of the impact varying these parameters has on the signal. We restrictourselves to looking at Centaurus A, which offers a good balance of being relatively close by( D ∼ . Similarly, in theories with Universal Extra Dimensions [10], the dark matter particle may be a Kaluza-Klein excitation of the hypercharge vector boson [11, 12], which may up-scatter into a Kaluza-Klein mode ofthe electron [13]. – 2 – .1 Dark Matter Distribution
The impact of the dark matter distribution is through its integral along the “line of sight”of the jet particles, δ DM ≡ (cid:90) r max r min dr ρ ( r ) (2.1)where ρ ( r ) is the density of dark matter at a radial distance r from the center of the AGN, r min characterizes the minimum distance of interest (the base of the jet), and r max the distance atwhich the jet loses cohesion and becomes irrelevant. In practice, results depend sensitivelyon r min and are rather insensitive to r max [13] due to the fact that dark matter densitydistribution ρ ( r ) drops steeply as the the radius r increases.The dark matter density distribution surrounding a compact object was modeled inRef. [17] in the collisionless limit and under the assumption that the central black holegrew adiabatically by accretion. The resulting steady-state distribution is characterizedby a central spike of dark matter, which gets depleted in the core by pair annihilation.The resulting distribution is thus sensitive to the underlying particle physics model of darkmatter, with the central density enhanced for WIMPs with very low (or in the case of apurely asymmetric dark matter halo, zero) annihilation cross sections. Ref [17] begins with acusped density profile ρ i ( r ) = ρ ( r/r ) − γ , and determines a final profile (close to the center)of the form, ρ ( r ) = ρ (cid:48) ( r ) ρ core ρ (cid:48) ( r ) + ρ core , (2.2)where the core density is given in terms of the velocity-averaged annihilation rate (cid:104) σv (cid:105) andthe age of the black hole t BH as, ρ core (cid:39) m ˜ χ (cid:104) σv (cid:105) t BH . (2.3)The dependence on r is contained in the function, ρ (cid:48) ( r ) = ρ (cid:18) R sp r (cid:19) − γ (cid:18) − R S r (cid:19) (cid:18) R sp r (cid:19) γ sp , (2.4)where γ sp is slope of the density profile in the dark matter spike, R sp its radius, and R S isthe Schwarzschild radius. Below 4 R S , the dark matter is accreted into the black hole, andhas ρ = 0. The spike radius and power are given by, R sp = α γ r (cid:18) M BH ρ r (cid:19) − γ and γ sp = 9 − γ − γ , (2.5)where α γ ∼ . γ / for γ (cid:28) M BH = (5 . ± . × M (cid:12) (2.6)– 3 –hich corresponds to R S (cid:39) × − pc. Given a choice of γ , we determine ρ by requiring,3 × M (cid:12) ≥ (cid:90) R S R S dr πr ρ ( r ) , (2.7)where 10 R S characterizes the distances at which the stellar populations are measured.Ultimately, the factor δ DM is very sensitive to the product (cid:104) σv (cid:105) × t BH , which determinesthe density within the spike. For black hole ages between 10 to 10 years and cross sectionsranging from 10 − cm s − (appropriate for a thermal relic) to 10 − cm s − (typical forthe “coannihilation” region of the MSSM, in which the superpartners of the charged leptonsplay an active role in freeze-out), one finds δ DM ranging from, δ DM ∼ to 10 M (cid:12) / pc (2.8)with rather mild dependence on the choices of γ , r min , and r ∼ r max ∼
15 kpc [13].
For our purposes, the details of the AGN jet geometry are not very important, while modelingthe energy distribution of particles in the jet is crucial. We assume most of the particles inthe Centaurus A jet are electrons. Based on observation of emitted gamma rays by the FermiLAT experiment [20], we assume the distribution in the electron boost is a broken power law, d Φ (AGN) e dγ (cid:48) ( γ (cid:48) ) = 12 k e γ (cid:48) − s (cid:20) γ (cid:48) γ (cid:48) br ) ( s − s ) (cid:21) − (cid:0) γ (cid:48) min < γ (cid:48) < γ (cid:48) max (cid:1) . (2.9)Here the primed variables refer to the “blob frame”, in which the electrons move isotropically.The parameters s , s , γ (cid:48) br , γ (cid:48) min and γ (cid:48) max are fixed to [20] s = 1 . , s = 3 . , γ (cid:48) br = 4 × , γ (cid:48) min = 8 × , γ (cid:48) max = 10 . (2.10)The normalization k e can be determined from the jet power in electrons, which is defined inthe black hole frame as, L e = (cid:90) − dµ Γ B (1 − β B µ ) (cid:90) γ max γ min dγ ( m e γ ) d Φ AGNe dγ (cid:18) γ (Γ B (1 − β B µ )) (cid:19) , (2.11)(2.12)where µ = cos θ ( θ is the polar angle with respect to the jet axis) and γ = (cid:15)/m e where (cid:15) isthe energy of the electron. Similarly, the primed variables in the blob frame are µ (cid:48) = cos θ (cid:48) and γ (cid:48) = (cid:15) (cid:48) /m e . The quantities in the two frames are related by the blob velocity β B andboost Γ B = 1 / (cid:113) − β B ( ∼ µ (cid:48) = µ − β B − β B µ , γ (cid:48) = γ Γ B (1 − β B µ ) , (2.13)or equivalently, µ = µ (cid:48) + β B β B µ (cid:48) , γ = γ (cid:48) Γ B (1 − β B µ ) . (2.14)– 4 – χ ( p ) e ( p ) ˜ χ ( p ′ ) e ( p ′ ) γ ( r )˜ e ( p + p − r ) ˜ χ ( p ) e ( p ) ˜ χ ( p ′ ) e ( p ′ ) γ ( r )˜ e ( p + p ) ˜ χ ( p ) e ( p ) ˜ χ ( p ′ ) e ( p ′ ) γ ( r ) ( p + p ) ( p + p − r ) ˜ e ˜ χ ( p ) e ( p ′ ) e ( p ) ˜ χ ( p ′ ) ˜ e ( p − p ′ ) γ ( r ) ˜ χ ( p ) e ( p ′ ) e ( p ) ˜ χ ( p ′ ) ˜ e ( p − p ′ ) γ ( r ) γ ( r )˜ χ ( p ) e ( p ′ ) e ( p ) ˜ χ ( p ′ ) ˜ e ( p − p ′ )( p − p ′ − r ) Figure 1 : Feynman diagrams for the process e − ˜ χ → e − ˜ χ γ through ˜ e exchange. Theamplitudes corresponding to these diagrams are, sequentially, M − M .The limits on the γ integral are given by, γ min = γ (cid:48) min Γ B (1 − β B µ ) , γ max = γ (cid:48) max Γ B (1 − β B µ ) , (2.15)whereas the limits on µ in the black hole frame correspond to 0 . −
1, leading to a highlycollimated jet of electrons.The function d Φ (AGN) e /dγ can be written explicitly as, d Φ e dγ (AGN) (cid:18) γ (Γ B (1 − β B µ )) (cid:19) = 12 k e (cid:20) γ Γ B (1 − β B µ ) (cid:21) − s (cid:20) (cid:18) γ (Γ B (1 − β B µ )) /γ (cid:48) br (cid:19) s − s (cid:21) − . (2.16)Saturating the upper limit of L e ≤ erg s − [13], we arrive at12 k e = 4 . × s − . (2.17) The total cross section for the process e ˜ χ → e ˜ χ γ may be written, dσ = (2 π ) δ ( p + p − p (cid:48) − p (cid:48) − r ) 14 m ˜ χ E d p (cid:48) (2 π ) E (cid:48) d p (cid:48) (2 π ) E (cid:48) d r (2 π ) E γ (cid:88) |M| , (3.1)where the (cid:80) |M| is the sum of all of the matrix elements squared. In addition to the s -channel contributions included in [13]), the u -channel ˜ e exchange diagrams are also included,as shown in the Fig. 1. We ignore t -channel contributions mediated by the SM Z boson asit is non-resonant, and thus negligible. The amplitudes corresponding to these diagrams arelabelled M − M . – 5 –he experimental observable is the distribution of photon energies, νS ν ≡ E γ d Φ γ dE γ at afixed angle relative to the jet axis ( ∼ ◦ for the Centaurus A). For fixed incoming electronenergy, we integrate over the phase space of the final state neutralino and electron, resultingin d σ/dE γ d Ω γ . We work in the rest frame of the initial neutralino dark matter particle,which is expected to be moving non-relativistically ( ∼
300 km/s [21]) in the rest frame ofthe Earth. d σ/dE γ d Ω γ can be expressed, d σdE γ d Ω γ = 1(2 π ) m ˜ χ E E (cid:48) (cid:90) d Ω p (cid:48) (cid:18) E (cid:48) E γ | J | (cid:88) |M| (cid:19) , (3.2)where J is a Jacobian factor from the integration of the function δ ( p + p − p (cid:48) − p (cid:48) − r )over dE (cid:48) and can be explicitly written as J = E (cid:48) E (cid:48) − ( (cid:126)p + (cid:126)p − (cid:126)r ) · (cid:126)p (cid:48) E (cid:48) E (cid:48) (3.3)= E (cid:48) E (cid:48) − ( p x − r x ) p (cid:48) x + ( p y − r y ) p (cid:48) y + ( p z − r z ) p (cid:48) z E (cid:48) E (cid:48) . (3.4)The full expression for the amplitude squared may be found in Appendix A. The crosssection is enhanced in three kinematic configurations: • M − are resonantly enhanced when the incoming electron energy is such that theintermediate selectron is approximately on-shell [9]. • M and M have a collinear enhancement when the emitted photon lines up with thefinal state electron [13]. • M and M have a soft enhancement for low energy photon emission.Note that because the jet axis makes a fixed angle with respect to the line of sight fromthe Earth, there is no possibility of a collinear enhancement from the initial electron in M and M . In practice, the soft enhancement (the infinity of which is formally matched by theone-loop QED correction to e − ˜ χ → e − ˜ χ scattering) is not observationally interesting. Thedominant contribution comes from the on-shell selectron, collinear photon region of M : | M | = α eff ( p (cid:48) · r )( p · p ) | Σ s | (cid:18) ( p (cid:48) · r ) − m e (cid:19) , (3.5)where α eff = 4 πα EM ( a L + a R ) is the effective neutralino-selectron-electron coupling. We assumethe left- and right-handed selectrons are degenerate in mass, and a mostly bino neutralino,for which, a R = 2 g w tan θ w , (3.6) a L = a R / . (3.7)The factor 1 / | Σ s | is the selectron propagator, which has been modified to maintain gaugeinvariance , using the prescription of Ref. [23]. To avoid the divergence from soft radiation,( p (cid:48) · r ) in the propagator has been shifted by m e / Proof of gauge invariance (correcting Ref [13]) can be found in Appendix B. – 6 –
GeV) e E ) ( G e V γ Ω d γ / d E σ d = 10GeV γ E = 60GeV χ∼ m = 100GeV e~ m (GeV) e E ) ( G e V γ Ω d γ / d E σ d = 30GeV γ E = 60GeV χ∼ m = 100GeV e~ m (GeV) e E ) ( G e V γ Ω d γ / d E σ d = 50GeV γ E = 60GeV χ∼ m = 100GeV e~ m Figure 2 : Differential cross sections, d σ/dE γ d Ω γ for the process e − ˜ χ → e − ˜ χγ , for aneutralino mass of 60 GeV and selectron mass of 100 GeV, and fixed photon energies (topto bottom) of 10 GeV, 30 GeV, and 50 GeV.The differential cross sections d σ/dE e d Ω γ are shown as a function of the incomingelectron energy in Fig. 2, for fixed photon energies of 10 GeV, 30 GeV and 50 GeV. As abenchmark point, we have chosen the mass of the neutralino to be m ˜ χ = 60 GeV and themass of the selectron to be m ˜ e = 100 GeV. Notice that for a photon energy of 10 GeV or30 GeV, there are two peaks which arise from the (shifted) resonant behavior of M and M separately. However, as the photon energy increases, the incoming electron energy mustbe large enough to produce such a hard photon in the scattering process, closing off thepossibility for scattering (for fixed E γ ) below a certain threshold. Once the threshold risesabove ∼ m ˜ e /
2, it becomes impossible to produce a final state photon from an on-shell ˜ e decay, and the M resonance effectively disappears. Putting the astrophysical inputs together with the cross section, we have: d Φ γ dE γ = (cid:18) δ DM (cid:19) × (cid:90) dE e (cid:18) D d Φ AGN e dE e (cid:19) × (cid:18) m ˜ χ d σdE γ d Ω γ (cid:19) . (4.1)– 7 – GeV) g E10 ) - c m - ( e r g s n S n -15 -14 -13 -12 -11 -10 -2 pc M = 10 DM d -2 pc M = 10 DM d n S n Figure 3 : The photon energy distributions for δ DM = 10 M (cid:12) pc − (lower curve) and δ DM = 10 M (cid:12) pc − (upper curve) exhibiting a drop-off around E γ ∼
35 GeV.where the three factors δ DM , d Φ AGN e /dE e and d σ/dE γ d Ω γ have been discussed in Sec-tions 2.1, 2.2 and 3, respectively. We adopt two choices of δ DM = 10 M (cid:12) pc − and δ DM = 10 M (cid:12) pc − and to begin with, the benchmark point with a bino neutralino of mass m ˜ χ = 60 GeV and degenerate left- and right-handed sleptons of masses m ˜ e = 100 GeV. InFigure 3, we plot the distribution of photon energies for the benchmark point, for both valuesof δ DM discussed above. We note the drop at energies around E γ ∼
35 GeV, where, as dis-cussed above, the required photon energy is so large that it is no longer produced efficientlyfrom on-shell intermediate selectrons.Using the existing energy sprectrum data from Fermi LAT observations of CentaurusA, we can put bounds on the coannihilation region of SUSY parameter space. Ideally, onecould do a shape-based analysis, as advocated in Ref. [13], looking for a significant drop inthe energy spectrum, which occurs as one demands the photon energy be high enough thatkinematics forbid it from arising from an on-shell selectron decay. However, with the limitedstatistics of the current data, it is more feasible to consider a counting experiment analysisin each energy bin. We place 95% confidence limits on regions of MSSM parameter spacewhere a contribution larger than the expected 95% fluctuations (based on the error bars)in the spectrum is predicted. We use the quoted uncertainties on the Fermi measurementpoints, √ B ∼ . × − erg s − cm − , (4.2)roughly independent of energy [20]. These bounds are subject to our assumptions concerning δ DM and the beam composition and energy spectrum.In Figure 4, we plot the 95% CL bounds in the plane of m ˜ χ and ∆ m , where∆ m ≡ m ˜ e − m ˜ χ (4.3)is the mass splitting between the neutralino and selectron. The left panel corresponds to δ DM = 10 M (cid:12) pc − , and the right to δ DM = 10 M (cid:12) pc − . These values correspond to a– 8 – GeV) χ∼ m0 50 100 150 200 250 m ( G e V ) ∆ pc M = 10 DM δ = [2, 5]GeV γ E = [5, 10]GeV γ E = [10, 30]GeV γ E = [30, 50]GeV γ E (GeV) χ∼ m0 50 100 150 200 250 m ( G e V ) ∆ pc M = 10 DM δ = [2, 5]GeV γ E = [5, 10]GeV γ E = [10, 30]GeV γ E = [30, 50]GeV γ E Figure 4 : Contours of 95% CL constraint on the m ˜ χ − ∆ m plane based on one year ofobservation of Centaurus A by the Fermi LAT. The colored regions indicate constraints fromeach energy bin reported by Fermi, and the black dashed line indicates m ˜ e = 100 GeV. Thetwo panels assume different δ DM , as indicated.neutralino annihilation cross section of (cid:104) σv (cid:105) ∼ − cm s − (appropriate for a coannihilationscenario) and a black hole lifetime of t BH ∼ or 10 years, respectively. We continue toassume a bino neutralino and selectrons which are degenerate in mass. Since the optimalenergy bin for the search varies with m ˜ χ and m ˜ e , we derive the ruled out region from eachenergy bin independently, as indicated on the figure. We find that the largest resolvingpower comes from the highest energy bins, but nevertheless the lower energy bins provideinteresting constraints, and a combined shape analysis would probably do a little better thantreating each bin independently. The black dashed line on each panel indicates the line of m ˜ e ≥
100 GeV, roughly the bound from LEP-II null searches [24]. There are no current LHCbounds on slepton masses and what they turn out to be will ultimately depend on the LHCperformance. For comparison, the theoretical study in Ref. [14] concluded that the LHCat √ s = 14 TeV and with ∼
100 fb − has sensitivity for slepton masses around 100 GeV,provided the mass splitting is larger than about ∆ m = 20 GeV. Active galactic nuclei are among the most energetic natural accelerators in the Universe. Theyare also located in regions rich with dark matter, and provide a great opportunity to studythe high energy interactions of WIMPs with SM particles. In this paper, we have estimatedthe regions of MSSM parameter space which can be probed from Fermi observations. Wefind that there is greatest sensitivity for models with low annihilation cross sections andnearly degenerate sleptons – as occurs in coannihilation regions of the MSSM, where sleptonsare active during freeze-out. Indeed, sleptons are very challenging to discover at the LHCunder ideal circumstances, and almost impossible when nearly degenerate with the lightestsupersymmetry particle. Observations of the gamma rays from AGNs can help cover aninteresting and important gap in LHC coverage of MSSM parameter space. Analogously, onecan study the scattering process between dark matter and AGN jets consisting of protons,although the rates at the highest energies will be suppressed by the parton distributionfunctions; we leave this direction for future work.– 9 –ltimately, the limiting factors in utilizing these naturally occurring accelerators asprobes of the property of dark matter are the uncertainties in the properties of the AGNsthemselves and the dark matter environment around them. These uncertainties can bemitigated by further observations, to help pin down the underlying properties of the jet,and observation of the dynamics of the surrounding stars and gas to better constrain thegravitational dynamics. The MSSM signal considered here also leads to a distinctive featurein the gamma ray spectrum, which could be exploited with more statistics to minimize thedependence on the unknown background emission. Ultimately, if the particle properties ofdark matter can be pinned down more precisely using data from the LHC or direct or indirectdark matter detection experiments, one could even turn the process around and use collisionsbetween jets and the WIMPs to learn about the astrophysics of these fascinating systems.
T. Tait is glad to acknowledge conversations with L. Costamente, S. Profumo, and L. Ubaldi,and the hospitality of the SLAC theory group, for their generosity during his many visits.The work of AR and JH is supported in part by NSF grants PHY-0653656 and PHY-0709742.The work of TMPT is supported in part by NSF grant PHY-0970171.– 10 –
Matrix Elements for e − ˜ χ → e − ˜ χγ In this appendix, we summarize the full expression for the amplitude squared for the process e − ˜ χ → e − ˜ χγ . We begin with some notations, α eff = 4 πα EM ( a L + a R ) ; s = ( p + p ) ; s (cid:48) = ( p (cid:48) + p (cid:48) ) ; u = ( p − p (cid:48) ) ; u (cid:48) = ( p − p (cid:48) ) ; E s = ( s + m χ − m e ) / (2 √ s ) ; E s = √ s − E s ; E s (cid:48) = ( s (cid:48) + m χ − m e ) / (2 √ s (cid:48) ) ; E s (cid:48) = √ s (cid:48) − E s (cid:48) ; r s = (cid:113) ( s − ( m e + m ˜ χ ) )( s − ( m ˜ χ − m e ) ) / (2 √ s ) ; r s (cid:48) = (cid:113) ( s (cid:48) − ( m e + m ˜ χ ) )( s (cid:48) − ( m ˜ χ − m e ) ) / (2 √ s (cid:48) ) ;Γ = (( a L + a R )( E s E s + r s ) r s ) / (4 πm ˜ e √ s );Γ = (( a L + a R )( E s (cid:48) E s (cid:48) + r s (cid:48) ) r s (cid:48) ) / (4 πm ˜ e √ s (cid:48) ) ;Σ s = s − m e − i m ˜ e Γ ;Σ s (cid:48) = s (cid:48) − m e − i m ˜ e Γ ;Σ u = u − m e ;Σ u (cid:48) = u (cid:48) − m e ; f = 1 + i ( m ˜ e (Γ − Γ )) / (2(( p + p ) · r )) ; | Σ s | = ( s − m e ) + ( m ˜ e Γ ) ; | Σ s (cid:48) | = ( s (cid:48) − m e ) + ( m ˜ e Γ ) . (A.1)Here p , p (cid:48) , p , p (cid:48) , r are the four-vectors for the momenta of the incoming and outgoing darkmatter particles (neutralino), incoming and outgoing electrons and photon. E s , E s , E s (cid:48) ,and E s (cid:48) are the energies of the initial and final neutralino and electron in the lab frame wherethe initial neutralino is at rest. The s and s (cid:48) subscripts correspond to the s and s (cid:48) diagrams,respectively. Γ i are the momentum dependent decay widths and Σ s , Σ s (cid:48) and Σ u , Σ u (cid:48) are thepropagators in the s - and u -channels. f is the modified vertex factor in amplitude M whichinsures gauge invariance, see appendix B. The squared amplitudes, including the interference– 11 –erms, are: | M | = [ α eff ( p · r )( p (cid:48) · p (cid:48) )] / [ | Σ s (cid:48) | ( p · r )] ; | M | = [ α eff ( p (cid:48) · r )( p · p )] / [ | Σ s | (( p (cid:48) · r ) − m e )] ; | M | = [ − α eff ( p · p )( p (cid:48) · p (cid:48) )(2( p · p ) − ( p · r ) − ( p · r ) + m χ ) | f | ] / [ | Σ s | | Σ s (cid:48) | ] ;2 Re ( M M † ) = Re (cid:20) − α eff Σ s (cid:48) Σ ∗ s ( p · r )(( p (cid:48) · r ) − m e / (cid:18) ( p · p (cid:48) )( p · r )( p · r ) − ( p · p (cid:48) )( p · r )( p · p )+ ( p · p (cid:48) )( p · r ) − p · p (cid:48) )( p · r )( p · p ) + 2( p · p (cid:48) )( p · p ) − ( p · p (cid:48) )( p · r ) − ( p · p (cid:48) )( p · r )( p · r ) + ( p · p (cid:48) )( p · r )( p · p ) − ( p · p (cid:48) )( p · r )( p · p ) − ( p · r ) ( p · p ) − p · r )( p · r )( p · p )+ 3( p · r )( p · p ) − ( p · r ) ( p · p ) + 3( p · r )( p · p ) − p · p ) (cid:19)(cid:21) ;2 Re ( M M † ) = Re (cid:20) α eff f | Σ s (cid:48) | Σ ∗ s ( p · r ) (cid:18) p · r ) ( p · p ) + 3( p · r )( p · r )( p · p ) − ( p · r )( p · r ) m χ − p · r )( p · p ) + ( p · r ) ( p · p ) − ( p · r ) m χ − p · r )( p · p ) + ( p · r )( p · p ) m χ + 2( p · p ) (cid:19)(cid:21) ;2 Re ( M M † ) = Re (cid:20) − α eff f | Σ s | Σ ∗ s (cid:48) (( p (cid:48) · r ) − m e / (cid:18) ( p · p (cid:48) )( p · r )( p · p ) + ( p · p (cid:48) )( p · r )( p · p ) − ( p · p (cid:48) )( p · p ) − ( p · p (cid:48) )( p · p ) m χ + ( p · p (cid:48) )( p · r )( p · p )+ ( p · p (cid:48) )( p · r )( p · p ) − ( p · p (cid:48) )( p · p ) − ( p · p (cid:48) )( p · p ) m χ + ( p · r ) ( p · p ) + 2( p · r )( p · r )( p · p ) − p · r )( p · p ) − ( p · r )( p · p ) m χ + ( p · r ) ( p · p ) − p · r )( p · p ) − ( p · r )( p · p ) m χ + 3( p · p ) + ( p · p ) m χ (cid:19)(cid:21) ; (A.2)– 12 – M | = [ α eff ( p (cid:48) · r )( p · p (cid:48) )] / [Σ u ( p · r )] ; | M | = [ α eff ( p · r )( p (cid:48) · p )] / [Σ u (cid:48) ( p (cid:48) · r − m e / | M | = [4 α eff ( p · p (cid:48) )( p · p (cid:48) )(( p · p (cid:48) ) + ( p · p (cid:48) ) − m χ )] / [Σ u Σ u (cid:48) ]2 Re ( M M † ) = Re (cid:20) α eff Σ u Σ u (cid:48) ( p · r )(( p (cid:48) · r ) − m e / (cid:18) p · p (cid:48) ) ( p · p (cid:48) ) + ( p · p (cid:48) ) ( p · r ) − ( p · p (cid:48) ) ( p · p ) + ( p · p (cid:48) )( p · p (cid:48) )( p · r ) + ( p · p ) − ( p · p (cid:48) )( p · p (cid:48) )( p · r ) − ( p · p (cid:48) )( p · r )( p · p ) − ( p · p (cid:48) )( p · r ) + ( p · p (cid:48) )( p · r )( p · p ) − ( p · p (cid:48) ) ( p · r ) + ( p · p (cid:48) ) ( p · p ) − ( p · p (cid:48) )( p · r )( p · r ) + 2( p · p (cid:48) )( p · r )( p · p ) + 2( p · p (cid:48) )( p · r )( p · p ) − p · p (cid:48) )( p · p ) + ( p · r )( p · r )( p · p ) − ( p · r )( p · p ) + ( p · r ) ( p · p ) − p · r )( p · p ) (cid:19)(cid:21) ;2 Re ( M M † ) = Re (cid:20) α eff Σ u Σ u (cid:48) ( p · r ) (cid:18) p · p (cid:48) ) ( p · p (cid:48) ) + 2( p · p (cid:48) ) ( p · r ) − p · p (cid:48) ) ( p · p ) − ( p · p (cid:48) )( p · p (cid:48) )( p · r ) − ( p · p (cid:48) )( p · r ) + ( p · p (cid:48) )( p · r )( p · p ) − ( p · p (cid:48) )( p · r ) m χ (cid:19)(cid:21) ;2 Re ( M M † ) = Re (cid:20) − α eff Σ u Σ u (cid:48) (( p (cid:48) · r ) − m e / (cid:18) ( p · p (cid:48) ) ( p · p (cid:48) ) + ( p · p (cid:48) ) ( p · r ) − ( p · p (cid:48) ) ( p · p ) − ( p · p (cid:48) )( p · p (cid:48) ) + ( p · p (cid:48) )( p · p (cid:48) )( p · r ) − p · p (cid:48) )( p · p (cid:48) )( p · r ) + 2( p · p (cid:48) )( p · p (cid:48) )( p · p ) − ( p · p (cid:48) )( p · p (cid:48) ) m χ + ( p · p (cid:48) )( p · r )( p · r ) − ( p · p (cid:48) )( p · r )( p · p ) − ( p · p (cid:48) )( p · r ) + 2( p · p (cid:48) )( p · r )( p · p ) − ( p · p (cid:48) )( p · r ) m χ − ( p · p (cid:48) )( p · p ) + ( p · p (cid:48) )( p · p ) m χ − ( p · p (cid:48) ) m χ − ( p · p (cid:48) )( p · r ) m χ − p · p (cid:48) )( p · r ) m χ + 2( p · p (cid:48) )( p · p ) m χ − ( p · r )( p · r ) m χ + ( p · r )( p · p ) m χ − ( p · r ) m χ + 2( p · r )( p · p ) m χ − ( p · p ) m χ (cid:19)(cid:21) ;(A.3)– 13 – Re ( M M † ) = Re (cid:20) α eff m χ Σ s (cid:48) Σ u ( p · r ) (cid:18) ( p · p (cid:48) ) + ( p · p (cid:48) ) + ( p · r ) + ( p · r ) − ( p · p ) (cid:19)(cid:21) ;2 Re ( M M † ) = Re (cid:20) − α eff m χ Σ s (cid:48) Σ u (cid:48) ( p · r )(( p (cid:48) · r ) − m e / (cid:18) ( p · p (cid:48) )( p · p (cid:48) ) + ( p · p (cid:48) )( p · r ) − ( p · p (cid:48) )( p · p ) (cid:19)(cid:21) ;2 Re ( M M † ) = Re (cid:20) − α eff m χ Σ s (cid:48) Σ u (cid:48) Σ u ( p · r ) (cid:18) p · p (cid:48) )( p · p (cid:48) ) + ( p · p (cid:48) )( p · r ) − ( p · p (cid:48) )( p · p ) + ( p · p (cid:48) )( p · r ) + ( p · p (cid:48) )( p · r ) − ( p · p (cid:48) )( p · p ) − ( p · r )( p · p ) − ( p · r )( p · p ) + ( p · p ) (cid:19)(cid:21) ;2 Re ( M M † ) = Re (cid:20) − α eff m χ Σ u Σ ∗ s ( p · r )(( p (cid:48) · r ) − m e / (cid:18) ( p · p (cid:48) )( p · p (cid:48) ) + ( p · p (cid:48) )( p · r ) − ( p · p (cid:48) )( p · p ) (cid:19)(cid:21) ;2 Re ( M M † ) = Re (cid:20) [ α eff m χ ( p · r )] / [Σ s Σ u (cid:48) (( p (cid:48) · r ) − m e / (cid:21) ;2 Re ( M M † ) = Re (cid:20) α eff m χ Σ s Σ u Σ u (cid:48) (( p (cid:48) · r ) − m e / (cid:18) ( p · p (cid:48) )( p · p (cid:48) ) + ( p · p (cid:48) )( p · r ) − ( p · p (cid:48) )( p · p ) − ( p · p (cid:48) ) − ( p · p (cid:48) )( p · r ) − ( p · r )( p · p ) − ( p · r )( p · p ) + ( p · p ) (cid:19)(cid:21) ;2 Re ( M M † ) = Re (cid:20) α eff m χ f Σ u Σ ∗ s Σ ∗ s (cid:48) ( p · r ) (cid:18) ( p · p (cid:48) )( p · r ) − ( p · p (cid:48) )( p · p ) − ( p · p (cid:48) )( p · r ) − ( p · p (cid:48) )( p · r ) + ( p · p (cid:48) )( p · p ) − ( p · r )( p · p ) − ( p · r )( p · p ) + ( p · p ) (cid:19)(cid:21) ;2 Re ( M M † ) = Re (cid:20) − α eff m χ f Σ u (cid:48) Σ ∗ s Σ ∗ s (cid:48) (( p (cid:48) · r ) − m e / (cid:18) ( p · p (cid:48) )( p · p (cid:48) ) + ( p · p (cid:48) )( p · r ) − ( p · p (cid:48) )( p · p ) + ( p · p (cid:48) ) + ( p · p (cid:48) )( p · r ) − ( p · r )( p · p ) − ( p · r )( p · p ) + ( p · p ) (cid:19)(cid:21) ;2 Re ( M M † ) = Re (cid:20) − α eff m χ f ∗ Σ s Σ s (cid:48) Σ u Σ u (cid:48) (cid:18) ( p · p (cid:48) )( p · p (cid:48) ) + ( p · p (cid:48) ) + ( p · p (cid:48) )( p · r ) − ( p · p (cid:48) )( p · p ) − p · p (cid:48) ) m χ (cid:19)(cid:21) . (A.4) B Proof of Gauge Invariance
In this appendix, we verify the invariance of the amplitude under U (1) EM gauge transforma-tions. We begin with the set of u -channel Feynman graphs. The amplitudes of M , M and– 14 – χ ( p ) e ( p ) ˜ e qq − p − p ˜ χ ( p ′ ) e ( p ′ ) p + p Figure 5 : A general “bubble” diagram containing no photon radiation. M can be written as, M = (cid:20) ˜ χ ( p (cid:48) )( a L P L + a R P R ) iγ. ( p − r ) − m e ( − ieγ µ ) e ( p ) (cid:21)(cid:20) ¯ e ( p (cid:48) )( a L P L + a R P R ) χ ( p ) (cid:21)(cid:20) i ( p − p (cid:48) − r ) − m e (cid:21) ,M = (cid:20) ˜ χ ( p (cid:48) )( a L P L + a R P R ) e ( p ) (cid:21)(cid:20) i ( p − p (cid:48) ) − m e (cid:21)(cid:20) ¯ e ( p (cid:48) )( − ieγ µ ) iγ. ( p (cid:48) + r ) − m e ( a L P L + a R P R ) χ ( p ) (cid:21) ,M = (cid:20) ˜ χ ( p (cid:48) )( a L P L + a R P R ) e ( p ) (cid:21)(cid:20) ¯ e ( p (cid:48) )( a L P L + a R P R ) χ ( p ) (cid:21)(cid:20) i ( p − p (cid:48) ) − m e ( − ie )(2 p − p (cid:48) − r ) µ i ( p − p (cid:48) − r ) − m e (cid:21) . (B.1)Applying the Ward identity to M - M individually gives rise to M = e (cid:20) ˜ χ ( p (cid:48) )( a L P L + a R P R ) e ( p ) (cid:21)(cid:20) ¯ e ( p (cid:48) )( a L P L + a R P R ) ˜ χ ( p ) (cid:21)(cid:20) i ( p − p (cid:48) − r ) − m e (cid:21) ,M = − e (cid:20) ˜ χ ( p (cid:48) )( a L P L + a R P R ) e ( p ) (cid:21)(cid:20) ¯ e ( p (cid:48) )( a L P L + a R P R ) ˜ χ ( p ) (cid:21)(cid:20) i ( p − p (cid:48) ) − m e (cid:21) ,M = e (cid:20) ˜ χ ( p (cid:48) )( a L P L + a R P R ) e ( p ) (cid:21)(cid:20) ¯ e ( p (cid:48) )( a L P L + a R P R ) ˜ χ ( p ) (cid:21)(cid:20) i ( p − p (cid:48) ) − m e − i ( p − p (cid:48) − r ) − m e (cid:21) , (B.2)and they sum to zero as expected.The set of s -channel diagrams is somewhat more complicated. To begin with, we con-sider the process with no final state photon, but a sum of bubble diagrams as shown in Fig. 5,which will be useful later. The key is the imaginary part of the loop which produces the widthof the selectron. Writting the sum of diagrams with all numbers of bubble insertions as, ip − m e + ip − m e ( − i Σ( p )) ip − m e + ... = ip − m e − Σ( p ) (B.3)we have − i Σ( p ) = − (cid:90) d q (2 π ) Tr (cid:20) iγ.q − m e ( a L P L + a R P R ) iγ. ( q − p ) − m ˜ χ ( a L P L + a R P R ) (cid:21) , (B.4)– 15 –here the width is determined by m Γ = Im (Σ( p )) at p = m e . The diagram with no photonradiation is further simplified to, (cid:20) ˜ χ ( p ) ( a L P L + a R P R ) e ( p ) (cid:21)(cid:20) ip − m e − im ˜ e Γ (cid:21)(cid:20) ¯ e ( p (cid:48) ) ( a L P L + a R P R ) ˜ χ ( p (cid:48) ) (cid:21) (B.5)with p = p + p .When the photon line is added, we get new diagrams by attaching the photon in allpossible ways. The first possibility is ˜ χ ( p ) e ( p ) ˜ e qq − p − p + r ˜ χ ( p ′ ) e ( p ′ ) p + p − r γ ( r ) M = (cid:20) ˜ χ ( p )( a L P L + a R P R ) iγ. ( p − r ) − m e ( − ie ) γ µ e ( p ) (cid:21)(cid:20) i ( p (cid:48) + p (cid:48) ) − m e − im ˜ e Γ( p + p − r ) (cid:21)(cid:20) ¯ e ( p (cid:48) )( a L P L + a R P R ) ˜ χ ( p (cid:48) ) (cid:21) (B.6)We have taken the momentum of the photon (with polarization µ ) to be r going out from thediagram. This graph corresponds to M . We also have the diagram corresponding to M , ˜ χ ( p ) e ( p ) ˜ e qq − p − p ˜ χ ( p ′ ) e ( p ′ ) p + p p ′ + r γ ( r ) M = (cid:20) ˜ χ ( p )( a L P L + a R P R ) e ( p ) (cid:21)(cid:20) i ( p + p ) − m e − im ˜ e Γ( p + p ) (cid:21)(cid:20) ¯ e ( p (cid:48) )( − ie ) γ µ iγ. ( p (cid:48) + r ) ( a L P L + a R P R ) ˜ χ ( p (cid:48) ) (cid:21) (B.7)While for M , the photon attaches to the selectron; there are two diagrams where one is M , = (cid:20) ˜ χ ( p )( a L P L + a R P R ) e ( p ) (cid:21)(cid:20) ¯ e ( p (cid:48) )( a L P L + a R P R ) ˜ χ ( p (cid:48) ) (cid:21)(cid:20) ( − ie )(2 p + 2 p − r ) µ (cid:21)(cid:20) i ( p + p ) − m e − im ˜ e Γ( p + p ) i ( p + p − r ) − m e − im ˜ e Γ( p + p − r ) (cid:21) (B.8)– 16 – χ ( p ) e ( p ) ˜ e qq − p − p χ ( p ′ ) e ( p ′ ) p + p p + p − r γ ( r )˜ χ ( p ) e ( p ) ˜ e qq − p − p ˜ χ ( p ′ ) e ( p ′ ) p + p − r γ ( r ) q − r and the other one has the photon attached to an internal loop, M , = (cid:20) ˜ χ ( p )( a L P L + a R P R ) e ( p ) (cid:21)(cid:20) ¯ e ( p (cid:48) )( a L P L + a R P R ) ˜ χ ( p (cid:48) ) (cid:21)(cid:20) − i Σ µ ( p + p , − r ) (cid:21)(cid:20) i ( p + p ) − m e − im ˜ e Γ( p + p ) i ( p + p − r ) − m e − im ˜ e Γ( p + p − r ) (cid:21) . (B.9)Now we need to extract the imaginary part of − i Σ µ ( p, − r ) in this vertex, − i Σ µ ( p, − r ) = (cid:90) d q (2 π ) Tr (cid:20) iγ. ( q − r ) − m e ( − ie ) γ µ iγ.q − m e ( a L P L + a R P R ) iγ. ( q − p ) − m ˜ χ ( a L P L + a R P R ) (cid:21) = − (cid:90) d q (2 π ) Tr (cid:20) iγ. ( q − r ) − m e ( − ie ) γ. ( − r ) iγ.q − m e ( a L P L + a R P R ) iγ. ( q − p ) − m ˜ χ ( a L P L + a R P R ) (cid:21) = − ( − e ) (cid:90) d q (2 π ) Tr (cid:20) ( iγ. ( q − r ) − m e − iγ.q − m e )( a L P L + a R P R ) iγ. ( q − p ) − m ˜ χ ( a L P L + a R P R ) (cid:21) = ( − e )( − i Σ( p − r ) + i Σ( p )) . (B.10)We can write generally, − i Σ µ ( p, − r ) = A ( p + p (cid:48) ) µ + B r µ . The equation above tells us that A ( p + p (cid:48) ) . ( − r ) = ( − e )( − i Σ( p − r ) + i Σ( p )) A = ( − e )( − i Σ( p − r ) + i Σ( p )) − p · r . (B.11)The coefficient B is irrelevant since r · (cid:15) = 0, and the vertex is effectively − i Σ µ ( p, − r ) =( − ie )( p + p (cid:48) ) µ ( − Σ( p − r )+Σ( p )) − p · r which can be thought of as a correction to the tree level vertex.– 17 –ince we are only interested in the imaginary piece of the correction,( − ie )( p + p (cid:48) ) µ ( − Γ( p − r ) + Γ( p )) − p · r , (B.12)we can combine the last two diagrams, arriving at the expression, M = (cid:20) ˜ χ ( p )( a L P L + a R P R ) e ( p ) (cid:21)(cid:20) ¯ e ( p (cid:48) )( a L P L + a R P R ) ˜ χ ( p (cid:48) ) (cid:21)(cid:20) im ˜ e Γ( p + p ) − Γ( p + p − r )2( p + p ) · ( − r ) (cid:21) i ( p + p ) − m e − im ˜ e Γ( p + p ) ( − ie )( p + p − r ) µ i ( p + p − r ) − m e − im ˜ e Γ( p + p − r )= (cid:20) ˜ χ ( p )( a L P L + a R P R ) e ( p ) (cid:21)(cid:20) ¯ e ( p (cid:48) )( a L P L + a R P R ) ˜ χ ( p (cid:48) )( − ie )( p + p − r ) µ (cid:21)(cid:20) p + p ) · ( − r ) (cid:21)(cid:20) p + p ) − m e − im ˜ e Γ( p + p ) − p + p − r ) − m e − im ˜ e Γ( p + p − r ) (cid:21) . 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