Electro-Weak Dark Matter: non-perturbative effect confronting indirect detections
KKIAS-P15030CETUP2015-008
Electro-Weak Dark Matter:non-perturbative effect confronting indirect detections
Eung Jin Chun (1)a and Jong-Chul Park (2)b(1)
Korea Institute for Advanced Study, Seoul 130-722, Korea (2)
Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea
Abstract
We update indirect constraints on Electro-Weak Dark Matter (EWDM) considering theSommerfeld-Ramsauer-Townsend (SRT) effect for its annihilations into a pair of standard modelgauge bosons assuming that EWDM accounts for the observed dark matter (DM) relic density fora given DM mass and mass gaps among the multiplet components. For the radiative or smallermass splitting, the hypercharged triplet and higher multiplet EWDMs are ruled out up to theDM mass ≈
10 – 20 TeV by the combination of the most recent data from AMS-02 (antiproton),Fermi-LAT (gamma-ray), and HESS (gamma-line). The Majorana triplet (wino-like) EWDM canevade all the indirect constraints only around Ramsauer-Townsend dips which can occur for a tinymass splitting of order 10 MeV or less. In the case of the doublet (Higgsino-like) EWDM, a widerange of its mass (cid:38)
500 GeV is allowed except Sommerfeld peak regions. Such a stringent limit onthe triplet DM can be evaded by employing a larger mass gap of the order of 10 GeV which allowsits mass larger than about 1 TeV. However, the future CTA experiment will be able to cover mostof the unconstrained parameter space. a email: [email protected] b email: [email protected] a r X i v : . [ h e p - ph ] J u l . INTRODUCTION The most simplistic candidate for DM would be a neutral component of an SU(2) L × U(1) Y multiplet added to the Standard Model (SM) [1, 2], which is dubbed as EWDM. In the case ofa fermionic EWDM, its physical properties are completely determined by the gauge charge,dark matter mass, and mass differences between the multiplet components. As is well-known,the non-perturbative effect plays a crucial role when the DM is slowly moving and the DMmass is larger than the force-carrying boson mass [1, 3–7]. Furthermore, this effect includesnot only the usual Sommerfeld enhancement but also the Ramsauer-Townsend suppressionwhich are more apparent for larger DM mass or smaller mass gaps [7].In this paper, we revisit the non-perturbative effect of EWDM which is strongly con-strained by the recent indirect detection data on anti-proton flux by AMS-02 [8], gamma-raymeasurement from Milky Way (MW) satellite dwarf galaxies by Fermi Large Area Tele-scope (Fermi-LAT) [9], and gamma-ray line searches by Fermi-LAT [10] and High EnergySpectroscopic System (HESS) [11]. The first two measurements put bounds on the leading-order annihilation processes: χ χ → W W/ZZ , and the last two on the loop-induced ones: χ χ → γγ/γZ . Remark that the updated Fermi-LAT search on gamma-ray lines covers thedark matter mass up to around 500 GeV complimenting the previous HESS search range of500 GeV – 25 TeV. Related studies have been made for the case of the wino or wino-Higgsinodark matter in a supersymmetric theory [12–17].The annihilation of two neutral particles into γ + X is a radiative process and maybe subject to a large correction due to the resummation of electroweak Sudakov logarithms α log ( m /m W ). In Refs. [18, 19], the authors studied the Sudakov resummation effect forthe annihilation rate of the exclusive process χ χ → γγ/γZ and found that the exclusiverate is reduced by up to a factor of 2 – 3 compared to the tree-level plus Sommerfeldenhancement calculation. On the other hand, Refs. [20] treated systematically the semi-inclusive annihilation rate into the final state γ + X within the resolution of the experimentto find that the effect of higher order correction is very limited as the semi-inclusive ratechanges by only O (1%) at m DM = 3 TeV. Thus, we will simply use the leading-orderannihilation cross section of EWDM including only the SRT effect.For our study, we will consider three specific examples of fermionic EWDM in the lowest-lying multiplets: a vector-like (Dirac) doublet with a hypercharge Y = ± / Y = 0 (wino-like), and a vector-like (Dirac) triplet with Y = ±
1. Note that a certain symmetry like Z has to be imposed for the stability ofthese EWDM candidates. They are assumed to form 100 % of the observed DM density inwide ranges of the DM mass and mass gaps among the multiplet components. As shownin Ref. [7], the feature of Sommerfeld peaks and Ramsauer-Townsend dips in the non-perturbative annihilation cross section depends sensitively on those mass parameters, whichleads to an interesting impact on the indirect detection constraints.This paper is organized as follows. In Section II, we review the non-perturbative effect,the SRT effect for annihilations of EWDM summarizing the results in Ref. [7]. In SectionIII, we discuss the existing indirect constrains on annihilations of EWDM into SM gaugebosons based on various cosmic-ray measurements, then conclude in Section IV. Finally, inAppendix A, we present the explicit forms of the potential and scattering matrices describedin Section II. II. EWDM AND SOMMERFELD-RAMSAUER-TOWNSEND EFFECT
Let us first make a brief summary of the non-perturbative effect on the EWDM annihi-lation [7]. The doublet EWDM (Higgsino-like) consists of a Dirac fermion pair denoted by χ u = ( χ + u , χ u ) and χ d = ( χ d , χ − d ) in the chiral representation. The lighter linear combinationof χ u and χ d is the DM candidate and the mass splitting can come from (effective) non-renormalizable operators after the electroweak symmetry breaking. The wino-like EWDMmultiplet, a triplet with Y = 0 is denoted by χ = ( χ + , χ , χ − ) having only one Majorananeutral component. Finally, the triplet EWDM multiplet with Y = ± χ u = ( χ ++ u , χ + u , χ u ) and χ d = ( χ d , χ − d , χ −− d ). Recall that the electroweak one-loopcorrection generates a mass splitting of order 0.1 GeV between the multiplet components[2]. Together with the above-mentioned (tree-level) contribution, this one-loop correctioncan make arbitrary mass gaps as assumed in our analysis.The non-perturbative effect in the EWDM annihilation arises from the exchange of theelectroweak gauge bosons which mixes together the two-body states of the multiplet com-ponents. In the case of the doublet EWDM, there are three states formed by the charged(Dirac) component and two neutral (Majorana) components: χ + u χ − d , χ χ , and χ χ , where χ denotes the dark matter component. For the wino-like EWDM, there are two two-body3tates: χ + χ − and χ χ . The triplet EWDM with Y = ± χ ++ u χ −− d , χ + u χ − d , χ χ , and χ χ .For the calculation of the non-perturbative effect, we apply the analysis of Refs. [1, 3, 4]in which the Green’s functions g ij corresponding to the transition between the two-bodystates i and j are shown to follow the Schr¨odinger equation: − m DM ∂ g ij ( r ) ∂r + V ik ( r ) g kj ( r ) = Kg ij ( r ) , (1)with the boundary condition g ij (0) = δ ij and ∂g ij ( ∞ ) /∂r = i (cid:112) m DM ( K − V ii ( ∞ )) g ij ( ∞ ).Here K = m DM β is the total kinetic energy of the two initial dark matter particles in theannihilation process, where β is the DM velocity. Then, the non-perturbative annihilationcross section of the dark matter χ is σv ( χ χ → AB ) = 2 d i d ∗ j Γ ABij , (2)where d j = g j ( ∞ ) and v = 2 β is the relative velocity between the two incident DMparticles. Here A and B run over the gauge bosons ( W + , W − , Z, γ ), and the gauge bosonfinal states AB can be W + W − , ZZ , γZ , or γγ . The explicit forms of the potential matrix V ij in Eq. (1) and the scattering matrix Γ ABij in Eq. (2) are collected in Appendix A.The non-perturbative calculation of the EWDM annihilation exhibits not only the usualSommerfeld enhancement with resonance peaks, but also a vanishing cross section realizingthe Ramsauer-Townsend effect for particular choices of the model parameters. These effectsare particularly sensitive to the the mass splittings between the dark matter and the chargedcomponents of the multiplet, and can appear even for the DM mass below the TeV scalewhen the mass splitting is reduced to O (10) MeV or less. One can find a detailed analysison non-perturbative effects on the annihilation cross section of the EWDM in Ref. [7]. Theappearance of the Ramsauer-Townsend dips is of a particular interest as it can allow theEWDM to evade strong constraints from various indirect detections. III. CONSTRAINTS ON EWDM FROM INDIRECT DM SEARCHESA. Annihilation into
W W & ZZ final states We first present constraints on EWDM from indirect signals by its annihilation into
W W/ZZ final states. Annihilations of EWDM are expected to produce
W/Z bosons plen-4ifully since EWDM is charged under the SU(2) symmetry of the SM. Fragmentation of theproduced
W/Z bosons leads to sizable contributions to the antiproton flux and the contin-uum photon spectrum which are detectable in cosmic-ray measurement experiments such asAMS-02 and Fermi-LAT.
1. Constraints from AMS-02 antiproton flux measurements
Antiproton production from DM annihilations into
W W and ZZ channels is constrainedby the precise measurements on antiproton flux by AMS-02 [8]. The antiproton contributionfrom DM annihilation should be summed to the secondary antiprotons, produced by colli-sions of energetic cosmic-rays with the interstellar medium, which account for the bulk of theobserved flux. Ref. [21] provided constraints on DM annihilation cross sections for variousfinal states including W W based on the recent antiproton to proton ratio measurements byAMS-02 [8]. The authors of Ref. [21] assumed the Einasto DM halo profile and the MEDpropagation parameter set proposed in Ref. [22]. They also showed that for m DM > ∼ − In our study, we use the 2 σ exclusion bound on the W W channel for the Einasto profile with the MED propagation model in Ref. [21] as the boundon the total cross section σv W W + σv ZZ ≡ σv χ χ → W + W − + σv χ χ → ZZ since the antiprotonyields per annihilation from two final states W W and ZZ are almost undistinguishable. TheAMS-02 antiproton flux limit on σv W W + σv ZZ is shown as a red-thin dotted line in Figures1–4.
2. Fermi-LAT continuum photon constraints: Milky Way satellite dwarf galaxies
The dwarf spheroidal satellite galaxies of the Milky Way are some of the most promisingtargets for the DM indirect detection via gamma-rays since they are highly DM domi-nated objects with a relatively short distance from the Earth. In EWDM annihilation,the continuum photons originate mostly from fragmentation of hadronic final states in the χ χ → W W/ZZ processes which are strongly constrained by the gamma-ray measure- In a number of recent papers based on Galactic synchrotron emission [23–26] and cosmic-ray positrons [24,27, 28], it has been pointed out that the thin halo model in the MIN propagation scheme is seriouslydisfavored. σv W W (+ σv ZZ ) at 2 σ level for the MW satellite dwarf galaxiesfrom Ref. [9] which is shown as a red-thin dashed line in Figures 1–4. B. Annihilation into γγ & γZ final states In this subsection, we provide indirect constraints on EWDM by its annihilation into γγ/γZ final states. Monochromatic photons arise from the one-loop processes χ χ → γγ/γZ . Such a line signature would be quite easily distinguished from astrophysical photonsources since in most cases they produce continuous spectra. Now, the annihilation crosssections for the processes χ χ → γγ/γZ are already in tension with Fermi-LAT and HESSsearches for line-like spectral features in the photon spectrum.
1. Constraints from Fermi-LAT photon line searches
Very recently, the Fermi-LAT collaboration has reported a constraint on a DM annihila-tion cross section based on updated searches for spectral line signatures in the energy range200 MeV – 500 GeV from around the Galactic Center (GC) using 5.8 years of data repro-cessed with the Pass 8 event-level analysis [10]. In the analysis, they searched spectral linesexpected from dark matter annihilation for four signal regions of interest (ROIs), selected tooptimize sensitivity to different dark matter halo profiles: NFW with γ = 1 or 1.3, Einasto,and isothermal. In Ref. [10], it has been shown that the bound for m DM (cid:38)
100 GeV is just2 – 4 times weaker even for the isothermal profile compared to the bound for the Einastoprofile since the Fermi-LAT has measured gamma-rays from all the sky and thus can findthe corresponding optimized ROI for each DM profile. In this study, we use the 2 σ upperlimit on σv γγ for the Einasto profile with the local DM density ρ = 0 . as the6ound on the total cross section σv γγ + σv γZ ≡ σv χ χ → γγ + σv χ χ → γZ , weighted by thenumber of photons for each final state. In Figures 1–4, we plot the constraint for the Einastoprofile on σv γγ + σv γZ as a blue-thick dotted line.
2. Constraints from HESS photon line searches
In Ref. [11], upper limits on line-like gamma-ray signatures in the energy range 500GeV – 25 TeV are provided using the data collected by the HESS, which complementthe limits obtained by the Fermi-LAT at lower energies [10]. In the analysis, the HESScollaboration assumed the Einasto DM halo profile with ρ = 0 . . However, theHESS collaboration has searched gamma-ray lines for only one ROI, a 1 ◦ radius circle aroundthe GC, compared to the Fermi-LAT. Thus, the HESS limit for the Einasto profile can beweakened by about two orders of magnitude for a more cored profile, the isothermal profile.In our analysis, we use the 2 σ limit from the HESS spectral line search for the Einasto profileas a representative constraint on σv γγ + σv γZ which is depicted as a blue-thick dashed linein Figures 1–4. C. Summary of indirect constraints on EWDM
We plot DM annihilation cross sections σv γγ + σv γZ and σv W W + σv ZZ as blue-thick andred-thin solid lines for the doublet, Majorana triplet, and hypercharged triplet ( Y = ± σv γγ + σv γZ and σv W W + σv ZZ as a function of DM mass m DM including the non-perturbativeeffect for two representative values of δm + : the typical mass splitting of O (0 .
1) GeV arisingfrom the electroweak one-loop correction (top-panel) and O (10) MeV (bottom-panel). Forlarger mass gap, the SRT effect becomes much weaker for a given DM mass. In order tosee this effect, we additionally show the results for the Y = ± σv γγ + σv γZ and σv W W + σv ZZ are shown by blue-thick and red-thin curves, respectively. • The limit from AMS-02 antiproton flux measurements : We plot the 2 σ limiton σv W W + σv ZZ from AMS-02 antiproton flux measurements as a red-thin dottedline. 7 IG. 1. Constraints on σv χ χ → W + W − + σv χ χ → ZZ (red-thin curves with the scale on the right)and σv χ χ → γγ + σv χ χ → γZ (blue-thick curves with the scale on the left) for the Higgsino-likeEWDM with δm + = 341 MeV (top) and 8 MeV (bottom) for δm N = 0 . W W/ZZ [33]and γX [34, 35] channels are presented as red-thin and blue-thick dot-dashed curves, respectively. IG. 2. Constraints for the wino-like EWDM with δm + = 166 MeV (top) and 6 MeV (bottom).Each line is the same as Figure1. • The limit from Fermi-LAT continuum photon searches : For σv W W + σv ZZ , theupper region of the red-thin dashed curve is excluded by the limit from the Fermi-LATcontinuum gamma-ray searches for the Milky Way satellite dwarf galaxies at the 2 σ level. 9 IG. 3. Constraints for the triplet EWDM with Y = ± δm + = 525 MeV (top-panel) and 15MeV (bottom-panel) for δm ++ = 1 . δm N = 0 . • The limit from Fermi-LAT photon line searches : For σv γγ + σv γZ , the 2 σ exclusion limit from Fermi-LAT spectral line searches for around the GC is shown asa blue-thick dotted curve. • The limit from HESS photon line searches : The HESS gamma-ray line signaturesearch limit on σv γγ + σv γZ is plotted as a blue-thick dashed curve.10 IG. 4. Constraints for the triplet EWDM with Y = ± δm ++ = 20 GeV, δm + = 10 GeV,and δm N = 0 . • The future sensitivity of CTA : In near future, the remaining parameter regionsfor each EWDM will be probed by various upcoming cosmic-ray observations such asCTA [36] and GAMMA-400 [37]. The CTA sensitivity with a 500 h time exposure on σv W W + σv ZZ [33] is plotted as a red-thin dot-dashed line. For 5 h of GC observationwith CTA, the upper limit on σv γγ + σv γZ [34, 35] is depicted as a blue-thick dot-dashed curve.As stated in Introduction, each EWDM is assumed to account for 100 % of the observedDM relic abundance in wide ranges of the DM mass and mass gaps. In the case of the doublet(Higgsino-like) EWDM with a radiative mass splitting δm + = 341 MeV, the DM mass largerthan about 500 GeV is allowed except a narrow Sommerfeld peak region at 7 TeV. For asmaller mass gap δm + = 8 MeV, the first Sommerfeld peak moves down to about 1 TeV andmore peaks appear at lower DM masses. These peak regions are excluded by either the Fermi-LAT gamma-ray data from the dwarf galaxies or HESS gamma-line data. The Majoranatriplet (wino-like) EWDM is stringently constrained for the whole range of masses up to ∼
10 TeV for the typical mass splitting of 166 MeV by the AMS-02, Fermi-LAT, and HESSresults. However, the indirect constraints on the wino-like EWDM could be evaded aroundRamsauer-Townsend dips which can occur for a very small mass splitting, O (10) MeV or11ess. For the case of the triplet EWDM with Y = ± the indirect constraints become muchmore stringent to rule out for masses up to ∼
20 TeV by a combination of AMS-02, Fermi-LAT, and HESS limits. It is also difficult to dodge the indirect constraints by arrangingRamsauer-Townsend dips with smaller mass splittings as, contrary to the Sommerfeld peaks,the Ramsauer-Townsend dips for σv W W + σv ZZ and σv γγ + σv γZ do not coincide with eachother as shown in the lower panel of Figure 3. This behavior is more apparent in the case ofthe quintuplet EWDM which exhibits non-overlapping narrow dips even with the radiativemass splitting [4], and thus is completely ruled out by the combination of all the indirectconstraints. As shown in Figure 4 taking a large mass splitting of O (10) GeV, the tripletEWDM with Y = ± m DM (cid:38) a couple of TeV except around the Sommerfeld peaks. Similarly, the Wino-likeEWDM can escape from the current indirect limits for m DM (cid:38) O (10) GeV. IV. CONCLUSION
In this paper, we have investigated indirect constraints on EWDM considering the SRTeffect for its annihilations into a pair of SM gauge bosons which are sensitive to the size ofmass splitting among the multiplet components. Assuming that EWDM accounts for theobserved DM relic density for a given DM mass and the radiative (or smaller) mass splitting,we found that the triplet with Y = ± m DM ≈
10 – 20 TeV by the combination of current limits from AMS-02 (antiproton), Fermi-LAT (gamma-ray), and HESS (gamma-ray line) measurements, disregarding a potentiallystrong DM halo profile dependence of the HESS limit. In the case of the Majorana triplet(wino-like) EWDM, there is a chance to dodge the indirect constraints around Ramsauer-Townsend dips only with a tiny mass splitting δm + (cid:46) O (10) MeV. On the other hand,the Higgsino-like EWDM is excluded just for DM masses less than ∼
500 GeV and aroundSommerfeld resonance peaks. Such a stringent limit can be weakened significantly if a largemass splitting of O (10) GeV is employed. The indirect constraints could be evaded even for It is advocated as the unique candidate for asymmetric EWDM in Ref. [38], which is however stringentlyconstrained by DM indirect detection limits as shown in this study. Y = ± m DM (cid:38) O (TeV) exceptaround the Sommerfeld peaks which occur at larger DM masses. However, the unconstrainedparameter regions will be mostly searched by various future cosmic-ray measurements suchas CTA [36] and GAMMA-400 [37]. ACKNOWLEDGMENTS
JCP is supported by Basic Science Research Program through the National ResearchFoundation of Korea funded by the Ministry of Education (NRF-2013R1A1A2061561), andappreciate CETUP* (Center for Theoretical Underground Physics and Related Areas) forits hospitality during completion of this work. We thank Shigeki Matsumoto for usefuldiscussions on the HESS bound.
Appendix A: The potential and tree-level annihilation matrices for EWDMs
Considering the covariant derivative D µ = ∂ µ + ig A µ T A for each gauge boson A = W ± , Z, γ , the potential matrix in Eq. (1) and the tree-level scattering matrix Γ ij take thefollowing general forms: V ij ( r ) = 2 δm i δ ij − α N i N j (cid:88) A (cid:2) T Aij (cid:3) e − m A r r with δm i = m χ i − m χ ; (A1)Γ ABij = πα δ AB ) m f ( x A , x B ) N i N j (cid:8) T A , T B (cid:9) ii (cid:8) T A , T B (cid:9) jj , (A2)where f ( x A , x B ) ≡ (cid:0) − x A + x B (cid:1)(cid:0) − x A + x B (cid:1) (cid:114) − x A + x B x A − x B )
16 with x A = m A m . Here the normalization factor N i is 1 or √ • Doublet (Higgsino-like) EWDMThe potential matrix V and the normalized tree-level scattering matrix Γ AB normalizedby ( πα /m ) f ( x A , x B ) in the basis of the three states ( χ + χ − , χ χ , χ χ ) are given13s follows: V = δm + − s W α r − (1 − s W ) α c W e − mZr r − α √ e − mW r r − α √ e − mW r r − α √ e − mW r r − α c W e − mZr r − α √ e − mW r r − α c W e − mZr r δm N ; (A3)Γ W W = 116 √ √ √ √ , (A4)Γ ZZ = 1128 c W − s W ) √ − s W ) √ − s W ) √ − s W ) √ − s W ) , Γ γZ = s W (1 − s W ) c W , Γ γγ = s W . • Majorana triplet (Wino-like) EWDMThe potential matrix V and the tree-level scattering matrix Γ AB normalized by( πα /m ) f ( x A , x B ) in the basis of the two states ( χ + χ − , χ χ ) are given as follows: V = δm + − s W α r − c W α e − mZr r − √ α e − mW r r − √ α e − mW r r ; (A5)Γ W W = 12 √ √ , Γ ZZ = c W
00 0 , (A6)Γ γZ = s W c W
00 0 , Γ γγ = s W
00 0 . Let us note that there are a few discrepancies in factors of the scattering matrices(A4) and (A6) compared with the previous results in Ref. [3].14
Hypercharged triplet EWDMThe potential matrix V and the tree-level scattering matrix Γ AB normalized by( πα /m ) f ( x A , x B ) in the basis of the four states ( χ ++ χ −− , χ + χ − , χ χ , χ χ ) aregiven as follows: V = δm ++ − s W α r − (1 − s W ) α c W e − mZr r − α e − mW r r − α e − mW r r δm + − s W α r − s W α c W e − mZr r − α √ e − mW r r − α √ e − mW r r α √ e − mZr r α c W e − mZr r − α √ e − mZr r − α c W e − mZr r δm N ;(A7)Γ W W = 14 √ √
24 8 2 √ √ √ √ √ √ , (A8)Γ ZZ = 18 c W − s W ) − s W ) s W √ − s W ) √ − s W ) − s W ) s W s W √ s W √ s W √ − s W ) √ s W √ − s W ) √ s W , Γ γZ = 2 s W c W − s W ) − − s W ) s W − − s W ) s W s W , Γ γγ = s W
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