text: 2 materialized in laboratories include Bi Se 26,27 (a 3D or glide), and possibly even for the minority of nonsym- 2 3 Z topological insulator),28–31 SnTe32–34 and KHgSb35–37 morphic groups without nonsymmorphic elements; these 2 (topological crystalline insulators).38–48 groups just do not allow for simple bands.3 This work presents rigorous results that apply in any Applications of our no-go statement include: (i) the spatial dimension d and to solids whose magnetic space establishment of a minimal dimension (at each wavevec- groups (denoted G) are symmorphic, i.e., G consists tor) for the tight-binding Hamiltonian of any localizable only of symmetries that are factorizable into products topological insulator with symmorphic symmetry. In of point-group symmetries with lattice translations. A the real-space perspective, the dimension of a tight- case in point is SnTe, whose rocksalt structure has the binding Hamiltonian is the number of orthogonal Wan- symmetry of the symmorphic space group 225; in con- nier functions that span the tight-binding Hilbert space trast, nonsymmorphic KHgSb is symmetric under glide, in one unit cell. The minimal dimension depends on which is the product of a reﬂection with half a lattice whether the localizable topological groundstate is sta- translation. ble, i.e., whether it remains localizable topological upon Our ﬁrst result is a no-go statement: a localizable topo- addition of band representations – stable phases are logical band cannot be a simple band. By ‘simple’, we classiﬁed by topological K-theory,46,55 and unstable (or mean a single band that is nondegenerate (in energy) ‘fragile’564) phases manifest in a ﬁner classiﬁcation of throughout the Brillouin torus. A localizable band that vector bundles.58,59 We ﬁnd that the minimal dimension is unitarily equivalent to locally symmetric Wannier func- is four in the stable case, and three in the fragile case. tions is deﬁned to be a band representation;49–52 colloqui- Additionally, we propose (ii) a sure-ﬁre recipe to design ally, a band representation resembles a lattice of locally- and/or identify bands with nontrivial Chern number, as symmetric atomic orbitals,7 as illustrated in Fig. 1.(c). well as fragile localizable topological bands – this recipe The contrapositive of the no-go statement is that any may be systematized for mass identiﬁcation of topologi- simple band that is not a band representation cannot be cal materials. (i-ii) and other applications are elaborated (cid:54) localizable, i.e., it has nontrivial ﬁrst Chern class (c =0); in Sec. . 1 this is presented pictorially in Fig. 1.(a-b), which com- Sec. generalizes the theorem to half-integer-spin rep- pares the topological categorization of simple bands with resentations of point-group symmetries, with application non-simple bands.2 Our results follow from a theorem to solids with spin-orbit coupling. After establishing a that applies to simple bands occurring as energy eigen- few preliminary results on space groups and their repre- functions of Schr¨odinger-type or tight-binding Hamiltoni- sentations in Sec. , we prove in Sec. that a simple band ans with the symmetry G: for such simple bands, being that is a band representation is topologically trivial, and topologically trivial (in the category of complex vector in Sec. that a topologically trivial simple band is a band bundles) is equivalent to being a band representation of representation. Finally in Sec. , we summarize our results G. from the perspective of establishing rank constraints for The standard notion of topological triviality, as well as bands. its relation to the existence of Wannier functions, is re- viewed in Sec. to further motivate the theorem. A more general and precise statement of the theorem is provided I. EXISTENCE OF WANNIER FUNCTIONS in Sec. . One may question if there is a loss of gener- FOR TOPOLOGICALLY TRIVIAL BANDS ality in our restriction to symmorphic G. Actually the hypothesis of the theorem becomes superﬂuous for all Our theorem may be viewed as a symmetry-reﬁned nonsymmorphic magnetic space groups with at least one analog of known relations between topological nontrivi- nonsymmorphic element (a symmetry that is partially a ality and an obstruction to the existence of Wannier func- translation by a fraction of a lattice vector, e.g., screw tions; these relations will be brieﬂy reviewed in Sec. , and 3 For space groups with at least one nonsymmorphic element, 2 A non-simple band necessarily has rank N>1, but we do not re- bands must be nontrivially connected as a graph due to the quire that a (N>1)-band subspace is connected by band touch- monodromy of symmetry representations, as proven in the sup- ings. Our deﬁnition of unlocalizable and localizable topological plementary material of Ref. 7. Less general proofs exist for solids bands [cf. Fig. 1.(a-b)] should in principle coincide with that without spin-orbit coupling in Ref. 96 and for band representa- of ‘topologically non-trivial bands’ in Ref. 5, which ‘cannot be tions only in Ref. 79. All type I (without time-reversal sym- continued to any atomic limit without either closing a gap or metry) and II (with time-reversal symmetry by itself) nonsym- breaking a symmetry.’ Note that they have not deﬁned topolog- morphic magnetic space groups have nonsymmorphic elements in ≥ ical triviality in the category of complex vector bundles, as we d=2, but not in d 3. For nonsymmorphic space groups without have done in this work. To verify their notion of non-triviality, nonsymmorphic elements, we are not aware of a general proof it is necessary to construct a continuous family of Hamiltoni- that simple bands do not exist; however this seems empirically ans that symmetrically interpolates between the Hamiltonian of to be true for speciﬁc case studies in d=3.24,79,96,97 interest to an atomic-limit Hamiltonian (which is not uniquely 4 ‘Fragility’ is a symmetry-enriched generalization of the notion of deﬁned). In comparison, our deﬁnition oﬀers a diagnostic test ‘stable triviality’ in bundle theory.4,69 Nontriviality of the fragile (for the non-existence of locally-symmetric Wannier functions) topological phases may manifest as a spectral ﬂow in the holon- that can be implemented on a single Hamiltonian of interest. omy over noncontractible Brillouin-zone loops.41,114 ##LTLine## ##LTLine##

text: materialized

text: in

text: laboratories

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text: to

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text: theorem.

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text: EXISTENCE

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text: ings.

text: Our

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text: of

text: unlocalizable

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text: Ref.

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text: proofs

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text: should

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text: that

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text: Ref.

text: and

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text: of

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text: Ref.

text: which

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text: All

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text: d

text: For

text: nonsymmorphic

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text: To

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text: notion

text: of

text: non-triviality,

text: nonsymmorphic

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text: we

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text: a

text: general

text: proof

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text: to

text: construct

text: a

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text: that

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text: exist;

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text: ans

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text: atomic-limit

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text: the

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text: holon-

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text: be

text: implemented

text: on

text: a

text: single

text: Hamiltonian

text: of

text: interest.

text: omy

text: over

text: noncontractible

text: Brillouin-zone

text: ##LTLine##

text: ##LTLine##


text: 2 hilation signals and tightly constrain the DM interpreta- larger regions of the sky. Increasing the radial cut in r tion of the GeV Excess. We verify the robustness of these beyond that used here could potentially improve sensi- results on variations of the Galactic foreground models, tivities to DM by ∼20–30% or more, depending on the and in the presence of a potential DM signal. density proﬁle. Analysis Procedure. We make use of 413 weeks of The expected photon count, µpi (θi), in each pixel Fermi-LAT Pass 8 data collected between August 4, 2008 of the ROI depends on contributions from standard and July 7, 2016. We analyze the subset of photons in the astrophysical sources as well as DM, if present. We ULTRACLEANVETO event class, restricting to the top quar- account for four astrophysical components that trace: ter of photons by quality of point-spread function (PSF) (i) the Galactic diﬀuse emission, as described by the reconstruction. The data is binned in 18 logarithmically- Fermi gll iem v02 P6 V11 DIFFUSE (p6v11) model,2 spaced energy bins between ∼0.8–50 GeV. The recom- (ii) the Fermi bubbles [45], (iii) isotropic emission, and mended quality cuts are applied, corresponding to zenith (iv) Fermi 3FGL point sources [46]. The smooth Galac- ◦ angle less than 90 , LAT CONFIG = 1, and DATA QUAL > tic DM template is modeled using a generalized Navarro- 0.1 Each energy bin is spatially binned into individual Frenk-White (NFW) proﬁle [47]: pixels using HEALPix [37] with nside = 128; the dataset is thus reduced to an array of integers that describes the ρ 0 number of photons in the energy bin, i, and pixel, p. ρNFW(r) = (r/r )γ[1 + (r/r )]3−γ (3) s s Template ﬁtting is a standard astrophysical procedure where the data is described by a set of spatial maps (re- with inner-slope γ = 1, scale radius r = 17 kpc, and ferred to as templates) that are binned in the same way − s local density ρ(r(cid:12)) = 0.4 GeV cm 3 [48, 49] at the So- as the data, which describe the separate components that lar radius r(cid:12) = 8 kpc [50]. All templates are smoothed contribute to the total photon count. Each template is with the energy-dependent PSF of the LAT instrument, associated with a normalization that is treated as a free modeled as a King function.3 model parameter in the ﬁt. The likelihood for a given In our ﬁducial study, we use the p6v11 Galactic diﬀuse energy bin is then a product of the Poisson probabilities associated with the observed counts np in each pixel of emission model, which includes contributions from π0- i decay and Bremsstrahlung emission, as traced by maps the region-of-interest: of gas column-densities, as well as inverse-Compton emis- sion, as predicted using Galprop [51]. The relative nor- − L (d |θ ) = (cid:89) µpi (θi)npi e µpi (θi) , (2) malizations of these separate components are ﬁxed. The i i i np! fact that the p6v11 model should be used with caution p i for energies above ∼50 GeV sets our upper energy cut- oﬀ. To give the p6v11 template more freedom, we divide where di denotes the data in energy bin i, θi represents it into eight radial slices of equal area within our ROI. the set of model parameters and µpi (θi) is the number of The normalization of each slice is then varied separately expected counts in a given pixel and energy bin. The in the ﬁtting procedure. Each slice is roughly ∼440 deg2 total likelihood is simply the product over the individual in area, comparable in size to the regions used in dwarf Li for each energy bin. and galaxy group studies (∼100 and 316 deg2, respec- The region-of-interest (ROI) for this study is chosen to tively) [5, 7]. The additional freedom given to the Galac- maximize the strength of the DM signal while minimizing tic diﬀuse template allows the ﬁt to better account for the eﬀects of◦ foreground mis-modeling. Speciﬁcally, we localized excesses or mis-modeled features in the emis- take |b| > 20 to avoid the Galactic plane, where cosmic- sion. Note that we do not use the gll iem v06 (p8R2) ray emission is particularly bright and there are more◦un- model [52], which is recommended for the Pass 8 dataset. resolved point sources. In addition, we take r < 50 to p8R2 includes large-scale residuals obtained from a ﬁt reduce the possibility of over-subtraction and/or spurious to the Fermi data that have been added back into the excesses obtained from ﬁtting the foreground model over model, and is therefore not appropriate when searching large sky areas. Better modeling of the Galactic diﬀuse for extended DM signals. We also do not use the p7v6 emission is an ongoing eﬀort [38–42], and the use of image diﬀuse model, which contains large-scale structures in- reconstruction and parametric modeling techniques such cluding Loop I and the Fermi bubbles with a ﬁxed nor- as SkyFACT [43] and D3PO [36, 44] could improve mod- malization. eling of the Galactic diﬀuse emission over signiﬁcantly 2 https://fermi.gsfc.nasa.gov/ssc/data/access/lat/ring_ 1 http://fermi.gsfc.nasa.gov/ssc/data/analysis/ for_FSSC_final4.pdf documentation/cicerone/ciceronedataexploration/ 3 https://fermi.gsfc.nasa.gov/ssc/data/analysis/ datapreparation.html. documentation/Cicerone/Cicerone_LAT_IRFs/IRF_PSF.html ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: hilation

text: signals

text: and

text: tightly

text: constrain

text: the

text: DM

text: interpreta-

text: larger

text: regions

text: of

text: the

text: sky.

text: Increasing

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text: datapreparation.html.

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text: 11 Model A Model B Model C Smooth Galactic Halo (Monte Carlo) Smooth Galactic Halo (Monte Carlo) Smooth Galactic Halo (Monte Carlo) Best-ﬁt “excess” parameters (1σ/2σ) Smooth Galactic Halo (Fermi data) Best-ﬁt “excess” parameters (1σ/2σ) 10−24 Smooth Galactic Halo (Fermi data) p6v11 (Fiducial) Smooth Galactic Halo (Fermi data) p6v11 (Fiducial) Fermi dwarfs (2016) p6v11 (Fiducial) Fermi dwarfs (2016) Fermi dwarfs (2016) 10−25 Thermal relic cross section Thermal relic cross section Thermal relic cross section 10−26 Smooth Galactic Halo Smooth Galactic Halo Smooth Galactic Halo 10−27 Fermi-LATPass8Data,bb Fermi-LATPass8Data,bb Fermi-LATPass8Data,bb 101 102 103 101 102 103 101 102 103 m [GeV] m [GeV] m [GeV] χ χ χ FIG. S5. Similar to Fig. 1 of the Letter, except using the Model A, B, and C foreground models (left, middle, and right panel, respectively) as provided by the Fermi-LAT Collaboration [8]. Note that the foreground templates are still divided into eight radial slices, as in the ﬁducial study, but the normalizations of the inverse-Compton and π0+Bremsstrahlung templates are allowed to ﬂoat independently. The ﬁducial limit obtained using the p6v11 foreground model is shown by the dashed black line for comparison. The “excesses” in the Model A and C studies (with signiﬁcances TSmax ∼ 28 and 14, respectively) are well- understood in terms of the source populations included in these models; see text for further discussion. Model B is statistically preferred over Models A and C as a description of the data in our ROI; the diﬀerence in the maximum log-likelihood between Model B and A(C) is ∆ log Lmax = 136(119). procedure, as they are associated with sixteen free parameters (rather than just eight, as in the p6v11 case). This is a very important cross-check of the ﬁducial results, because the relative normalizations of foreground components are ﬁxed in p6v11, with the ratio set by a previous ﬁt to the data. However, because that ﬁt did not include a DM template, one might worry that a potential signal—if present—would be absorbed by the foreground components (particularly the IC component) in the initial ﬁtting procedure. If this were the case, using p6v11 for a Galactic DM search could potentially give artiﬁcially stringent DM limits. Fig. S5 shows the limits obtained using Models A, B, and C. The diﬀerences between the results can be understood in terms of the assumptions going into the separate models, which we now describe in detail: • Model A: Model A is based on the class of Galactic diﬀuse models studied in [74], and is described in detail in [8]. Here, we only highlight the main elements that distinguish it from Models B and C. For Model A, cosmic-ray electrons and nuclei are both sourced by the same population of pulsars, and the cosmic-ray diﬀusion coeﬃcient and re-acceleration strength are held constant. The left panel of Fig. S5 shows the Monte Carlo expectation and data limit when rerunning the ﬁducial analysis using the Model A templates. The recovered data limit is weaker than the Monte Carlo expectation, which suggests that there is excess gamma-ray emission in the ROI that is not captured by the Model A templates. It should be noted that the foreground templates are given considerable freedom in the ﬁtting procedure, as the normalizations of the π0+Brem and IC templates are allowed to ﬂoat separately in each radial slice. Despite this freedom, a large amount of DM emission is still needed to improve the quality of the ﬁt. A DM “excess” with a TS ∼ 28 is observed, with the best-ﬁt 1σ max and 2σ (corresponding to deviations in TS of −2.30 and −6.18 from the global maximum) containment regions as shown in the ﬁgure. The fact that the DM parameter space that is favored is clearly excluded by the dwarf searches strongly suggests that the weakening of the bounds is not due to DM, and is likely of astrophysical origin. • Model B: Model B provides an important counterpoint to Model A [8]. It includes an additional source popu- lation of electrons at the Galactic Center, which contributes to the IC emission. Unlike Model A, which closely reproduces the local cosmic-ray electron spectrum, Model B under-predicts the distribution below ∼20 GV. However, this disparity can be accounted for by contributions from other more local sources. The middle panel of Fig. S5 shows the Monte Carlo expectation and data limit for the Model B study. The limit is comparable to the ﬁducial case at low masses and is somewhat tighter for masses above ∼100 GeV, although still consistent within the Monte Carlo expectation. Clearly, the IC component in Model B absorbs the excess emission that was previously attributed to DM in the Model A study, bringing the data limit in line with the Monte Carlo expectation. Quantitatively, Model B is a better overall ﬁt to the data than Models A and C in our ROI; the diﬀerence in the maximum log-likelihood between Model B and A(C) is ∆ log L = 136(119). The better max overall ﬁt of Model B to the data in this case and the fact that the additional emission is absorbed by the IC ]1−s 3mc[ ivσh ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: because

text: that

text: ﬁt

text: did

text: not

text: include

text: a

text: DM

text: template,

text: one

text: might

text: worry

text: that

text: a

text: potential

text: signal—if

text: present—would

text: be

text: absorbed

text: by

text: the

text: foreground

text: components

text: (particularly

text: the

text: IC

text: component)

text: in

text: the

text: initial

text: ﬁtting

text: procedure.

text: If

text: this

text: were

text: the

text: case,

text: using

text: for

text: a

text: Galactic

text: DM

text: search

text: could

text: potentially

text: give

text: artiﬁcially

text: stringent

text: DM

text: limits.

text: Fig.

text: shows

text: the

text: limits

text: obtained

text: using

text: Models

text: A,

text: B,

text: and

text: C.

text: The

text: diﬀerences

text: between

text: the

text: results

text: can

text: be

text: understood

text: in

text: terms

text: of

text: the

text: assumptions

text: going

text: into

text: the

text: separate

text: models,

text: which

text: we

text: now

text: describe

text: in

text: detail:

text: Model

text: A:

text: Model

text: A

text: is

text: based

text: on

text: the

text: class

text: of

text: Galactic

text: diﬀuse

text: models

text: studied

text: in

text: and

text: is

text: described

text: in

text: detail

text: in

text: Here,

text: we

text: only

text: highlight

text: the

text: main

text: elements

text: that

text: distinguish

text: it

text: from

text: Models

text: B

text: and

text: C.

text: For

text: Model

text: A,

text: cosmic-ray

text: electrons

text: and

text: nuclei

text: are

text: both

text: sourced

text: by

text: the

text: same

text: population

text: of

text: pulsars,

text: and

text: the

text: cosmic-ray

text: diﬀusion

text: coeﬃcient

text: and

text: re-acceleration

text: strength

text: are

text: held

text: constant.

text: The

text: left

text: panel

text: of

text: Fig.

text: shows

text: the

text: Monte

text: Carlo

text: expectation

text: and

text: data

text: limit

text: when

text: rerunning

text: the

text: ﬁducial

text: analysis

text: using

text: the

text: Model

text: A

text: templates.

text: The

text: recovered

text: data

text: limit

text: is

text: weaker

text: than

text: the

text: Monte

text: Carlo

text: expectation,

text: which

text: suggests

text: that

text: there

text: is

text: excess

text: gamma-ray

text: emission

text: in

text: the

text: ROI

text: that

text: is

text: not

text: captured

text: by

text: the

text: Model

text: A

text: templates.

text: It

text: should

text: be

text: noted

text: that

text: the

text: foreground

text: templates

text: are

text: given

text: considerable

text: freedom

text: in

text: the

text: ﬁtting

text: procedure,

text: as

text: the

text: normalizations

text: of

text: the

text: and

text: IC

text: templates

text: are

text: allowed

text: to

text: ﬂoat

text: separately

text: in

text: each

text: radial

text: slice.

text: Despite

text: this

text: freedom,

text: a

text: large

text: amount

text: of

text: DM

text: emission

text: is

text: still

text: needed

text: to

text: improve

text: the

text: quality

text: of

text: the

text: ﬁt.

text: A

text: DM

text: “excess”

text: with

text: a

text: TS

text: is

text: observed,

text: with

text: the

text: best-ﬁt

text: 1σ

text: max

text: and

text: 2σ

text: (corresponding

text: to

text: deviations

text: in

text: TS

text: of

text: and

text: from

text: the

text: global

text: maximum)

text: containment

text: regions

text: as

text: shown

text: in

text: the

text: ﬁgure.

text: The

text: fact

text: that

text: the

text: DM

text: parameter

text: space

text: that

text: is

text: favored

text: is

text: clearly

text: excluded

text: by

text: the

text: dwarf

text: searches

text: strongly

text: suggests

text: that

text: the

text: weakening

text: of

text: the

text: bounds

text: is

text: not

text: due

text: to

text: DM,

text: and

text: is

text: likely

text: of

text: astrophysical

text: origin.

text: Model

text: B:

text: Model

text: B

text: provides

text: an

text: important

text: counterpoint

text: to

text: Model

text: A

text: It

text: includes

text: an

text: additional

text: source

text: popu-

text: lation

text: of

text: electrons

text: at

text: the

text: Galactic

text: Center,

text: which

text: contributes

text: to

text: the

text: IC

text: emission.

text: Unlike

text: Model

text: A,

text: which

text: closely

text: reproduces

text: the

text: local

text: cosmic-ray

text: electron

text: spectrum,

text: Model

text: B

text: under-predicts

text: the

text: distribution

text: below

text: GV.

text: However,

text: this

text: disparity

text: can

text: be

text: accounted

text: for

text: by

text: contributions

text: from

text: other

text: more

text: local

text: sources.

text: The

text: middle

text: panel

text: of

text: Fig.

text: shows

text: the

text: Monte

text: Carlo

text: expectation

text: and

text: data

text: limit

text: for

text: the

text: Model

text: B

text: study.

text: The

text: limit

text: is

text: comparable

text: to

text: the

text: ﬁducial

text: case

text: at

text: low

text: masses

text: and

text: is

text: somewhat

text: tighter

text: for

text: masses

text: above

text: GeV,

text: although

text: still

text: consistent

text: within

text: the

text: Monte

text: Carlo

text: expectation.

text: Clearly,

text: the

text: IC

text: component

text: in

text: Model

text: B

text: absorbs

text: the

text: excess

text: emission

text: that

text: was

text: previously

text: attributed

text: to

text: DM

text: in

text: the

text: Model

text: A

text: study,

text: bringing

text: the

text: data

text: limit

text: in

text: line

text: with

text: the

text: Monte

text: Carlo

text: expectation.

text: Quantitatively,

text: Model

text: B

text: is

text: a

text: better

text: overall

text: ﬁt

text: to

text: the

text: data

text: than

text: Models

text: A

text: and

text: C

text: in

text: our

text: ROI;

text: the

text: diﬀerence

text: in

text: the

text: maximum

text: log-likelihood

text: between

text: Model

text: B

text: and

text: A(C)

text: is

text: log

text: L

text: The

text: better

text: max

text: overall

text: ﬁt

text: of

text: Model

text: B

text: to

text: the

text: data

text: in

text: this

text: case

text: and

text: the

text: fact

text: that

text: the

text: additional

text: emission

text: is

text: absorbed

text: by

text: the

text: IC

text: ivσh

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