The Bactrian Effect: Multiple Resonances and Light Dirac Dark Matter
SSLAC-PUB-17576February 9, 2021
The Bactrian Effect: Multiple Resonances and Light Dirac DarkMatter
Thomas G. Rizzo † SLAC National Accelerator Laboratory 2575 Sand Hill Rd., Menlo Park, CA, 94025 USA
Abstract
The possibility of light dark matter (DM) annihilating through a dark photon (DP) which kineticallymixes (KM) with the Standard Model (SM) hypercharge field is a very attractive scenario. For DMin the interesting mass range below ∼ s -wave annihilation process for DM in this mass range as would bethe case, e.g. , if the DM were a Dirac fermion. In an extra-dimensional setup explored previously,it was found that the s -channel exchange of multiple gauge bosons could simultaneously encompassa suppressed annihilation cross section during the CMB era while also producing a sufficiently largeannihilation rate during freeze-out to recover the DM relic density. In this paper, we analyze moreglobally the necessary requirements for this mechanism to work successfully and then realize themwithin the context of a simple model with two ‘dark’ gauge bosons having masses of a similar magni-tude and whose contributions to the annihilation amplitude destructively interfere. We show that ifthe DM mass threshold lies appropriately in the saddle region of this destructive interference betweenthe two resonance humps it then becomes possible to satisfy these requirements simultaneously pro-vided several ancillary conditions are met. The multiple constraints on the parameter space of thissetup are then explored in detail to identify the phenomenologically successful regions. † [email protected] a r X i v : . [ h e p - ph ] F e b Introduction
Although dark matter (DM) is known to exist at multiple scales in the universe we don’t yet know whatit is or if it interacts with the particles of the Standard Model (SM) through any forces other than viagravity. However, in order to obtain the observed relic density as measured by Planck [1] it is more thanlikely that some sort of non-gravitational interactions are responsible. The traditional DM candidates,Weakly Interacting Massive Particles (WIMPs) [2,3] and the familiar axion [4–6], either assume the usualStandard Model (SM) electroweak interactions or some new high scale physics is responsible for obtainingthe relic density. While such theories remain very interesting, the lack of any observational signatures atthe LHC or in either direct or indirect detection searches [7–10] has resulted in a slowly shrinking allowedparameter space for these models. This has led to the construction of a plethora of new DM scenariosbased on the introduction of non-SM interactions to reproduce the observed relic abundance [11,12] withvery wide ranges in both the possible DM masses and coupling strengths [13–15]. Many of these potentialnew interactions can be described via a set of ‘portals’ which link DM, and possibly other ‘dark’ sectorfields, with those of the SM, only a few of which can result from renormalizable, dimension-4 terms inthe Lagrangian.Perhaps the most attractive of these ideas, and one that has received much attention in the recentliterature, is the vector boson/kinetic mixing (KM) portal [16,17] which will be the subject of the analysisthat follows below. The main ingredients of this setup in its basic incarnation can be deceptively simple:DM is assumed to be a SM singlet but instead carries a charge under a new ‘dark’ gauge interaction, e.g. , U (1) D , with a corresponding gauge coupling g D . The associated gauge field is thus termed the‘dark photon’ (DP) [18] which has a mass that can be generated by the dark analog of the usual Higgsmechanism, i.e. , via the ‘dark Higgs’. The coupling of the DM and other dark sector fields to the SMis then generated by the KM of the U (1) D DP with the SM U (1) Y hypercharge gauge boson whichcan be accomplished at 1-loop via a set of ‘portal matter’ fields that are charged under both gaugegroups [19–23]. Once all the fields are canonically normalized to remove the effects of this KM and boththe U (1) D and SM gauge symmetries are spontaneously broken, one finds that the the DP has pickedup a small loop-induced coupling to the SM fields. For the range of DP masses below ∼ Z mass-squared ratio, one finds thewell-known result that this coupling can be very well approximated as (cid:15)eQ em , where (cid:15) is a dimensionlessparameter, here assumed to roughly lie in the interval ∼ − − − , that describes the magnitude ofthis loop-suppressed KM.When both the DM and the DP are both light and have somewhat comparable masses, < ∼ U (1) D or inKM models with extra dimensions where the compactification radius sets the common scale for particlemasses [24–27]. In this low mass regime, there are several constraints on the model parameters: first,there is the required annihilation cross section necessary to obtain the observed relic density during freezeout, e.g. , < σv rel > F O (cid:39) . × − cm s − for an s -wave annihilating Dirac fermion DM [11, 12], whichis the case that we will consider below. For such a light mass, we will assume in what follows that pairannihilation of DM via virtual spin-1 exchanges is responsible for this and that it results in a SM finalstate consisting of pairs of electrons, muons, or light charged hadrons. Second, a lower bound on theDM mass exists arising from Big Bang Nucleosynthesis considerations of roughly ∼
10 MeV (which wetake from Ref. [28]). Lastly, in this same DM mass range of ∼ − z ∼ )constraints from Planck [1] tell us that at that time the DM annihilation cross section into light SMcharged states, e.g. , e + e − , must be substantially suppressed [29–32] thus avoiding the possible injectionof any additional electromagnetic energy into the SM plasma. A recent analysis [33] of this constraintinforms us that it lies roughly at the level of ∼ × − ( m DM /
100 MeV) cm s − , noting that it dependsapproximately linearly on the DM mass, but is, in any case, roughly three orders of magnitude below thatneeded at freeze out to recover the observed relic density. However, as the DM get heavier, this constraintbecomes quite weak and can be essentially ignorable for DM masses above roughly > ∼ −
20 GeV. Wefurther note that this constraint from the CMB is not expected to strengthen by more than a factor of1 s -channel exchange of spin-1 mediators like the DP, e.g. , if DM is aDirac fermion (as will be considered here), this annihilation process is dominantly s -wave assuming vectorcouplings. In such a case, since the reaction rate is generally not very sensitive to the relative velocityof the annihilating DM, v rel , the cross sections at freeze-out and during the CMB are not expected tobe much different thus conflicting with the requirements above. Does this imply that light Dirac fermionDM in the KM setup and annihilating to the SM as described above is excluded in this mass range?In the simple canonical DP scenario – without any ‘tweaking’ – as discussed earlier the answer in ‘yes’.However, modifications of this basic idea may allow for this possibility and several more or less successfulbut diverging paths might be followed, one of which we will consider here. In recent work [26] on the 5-Dextension of this usual 4-D KM setup with Dirac DM, it was found in a random scan that certain regionsof the model parameter space simultaneously satisfied the CMB bound while still leading to the desiredDM annihilation cross section (via multiple s -channel Kaluza-Klein DP exchanges) at freeze out. Whilethe exact mechanism at work in this case was speculated upon and the necessary ingredients for thissuccess never fully identified, it was clear that the existence of more than one particle exchange and withthe proper interference structure were clearly necessary ingredients. In this paper, we will further examinethis issue in some detail and then construct a simpler, more tractable and transparent 4-D scenario whichsatisfies all of the necessary conditions. To this end we will employ a modified version of the dark sectormodel considered in Ref. [20] based on a SM-like, but fully broken, SU (2) I × U (1) Y I dark gauge group,naturally having two diagonally coupled gauge bosons with only a few adjustable parameters.The outline of this paper is as follows: in Section 2, based on our previous work, we consider andoutline in detail the necessary inputs and constraints on a model of Dirac fermion DM which interactswith the fields of the SM through (at least) a pair of two spin-1 mediators, Z i , whose couplings aregenerated by KM, thus generalizing the conventional DP setup. We then construct a simple but realisticmodel that satisfies all of these requirements. In Section 3, we discuss the phenomenological implicationsof the model we construct based on the requirements arrived at in the previous Section and then weexplore how their interplay impacts the model’s parameter surviving space. Our results and conclusionsare then summarized in Section 4. In this Section, we will discuss the essential requirements for and the set of constraints imposed uponmodels that may realize the expectations described above as well as the reasoning behind them. A simple,prototypical – but potentially physically realistic – proof of principle model of this kind with the desiredproperties will then be presented and examined in some detail.
The mechanism envisioned here has several important distinct components – some of which were super-ficially touched upon in our earlier work [26]. In this subsection we will clarify what these are and whattheir interplay is with one another. Based on these observations we will make a number of model buildingassumptions in what follows and then explore how they can be realized.( i ) We imagine that light Dirac fermionic DM, χ , with a mass in the in the 10 to 1000 MeV range,realizes the observed relic density via the usual pair annihilation to SM fields, e.g. , e + e − , via the s -channel exchange of two (or more) new neutral gauge bosons, Z i , which both have masses of comparablemagnitude to the DM. As noted above, this is a rather natural occurrence in, e.g. , ED models of KMwherein the masses of all the low lying states are set by the inverse size of the extra size of the ED, R − [24–27], or in models where the masses are determined by the single vacuum expectation value (vev)of a scalar field. While the couplings of the DM to the Z i will be set by a common overall dark gauge2oupling, g D (modulo Clebsch-Gordon and mixing angle factors as appear in the SM), the Z i couplingsto SM fields will be determined up to similar overall factors via a single kinetic mixing with the SMhypercharge field and so are related to one another but are also, as is usual, suppressed by loop factors.This loose framework is just a rather straightforward generalization of the familiar DP/KM model [16,17].To simplify matters and make things more tractable we will specifically concern ourselves with the caseof only two Z i → Z , in what follows but the arguments we make can be generalized as in the caseof, e.g. , ED that was previously considered as well as to other scenarios with multiple s -channel spin-1exchanges.( ii ) Though axial-vector couplings of the DM to the Z i , a DMi , can be present, and we will return to thispossibility below, we will assume that the DM [and the SM fermions] must at least have [only have] vectorcouplings to the Z i , v DM,SMi (cid:54) = 0, so that in the non-relativistic, low relative velocity limit, v rel →
0, theannihilation process is primarily an s -wave and is also not, e.g. , helicity or threshold suppressed by anysmall SM fermion masses that may appear in the final state. This is, again, just a generalization of thefamiliar DP/KM scenario.Figure 1: Semi-quantitative picture of the Dirac fermion DM annihilation cross section, in random units,when √ s is given in units of m and where m /m = 3 has been assumed for purposes of demonstration.Both the constructive or destructive interference possibilities are shown.( iii ) Now consider the DM annihilation cross section during the CMB era at z ∼ when thetemperature is sufficiently low so that taking the v rel → Z i , a DMi , can be safely ignored as their contributions to theannihilation rate are all v rel suppressed in this case. This implies that in this limit there is effectivelyonly a single ‘vector-vector’ coupling amplitude contributing to the DM annihilation process to a givenfinal fermion state which is made up of the sum of the individual contributions of the various Z i andwhich we can write in the simple familiar form A (cid:39) (cid:88) i =1 , v DMi v SMi s − m i + i Γ i m i , (1)where s is the usual Mandelstam variable and here m i = m Z i with Γ i being the total widths of thesestates (assumed here to be at least somewhat narrow Γ i /m i < a few % or perhaps significantly smaller).Further, we now make the additional assumption that the DM mass is such that 2 m DM lies within thesaddle region, i.e. , m < m DM = √ s < m between the two resonance humps where the equality follows3rom the fact that we have taken v rel →
0; such a situation may be envisioned as that shown qualitativelyin Fig. 1. Trivially, if the product of the DM and SM couplings to the Z , have the same (opposite) signin both cases, then destructive (constructive) interference between the two contributions to the amplitudein the saddle region between the resonances will take place. In the case of destructive interference, inwhich we will be interested, the location of this very deep cross section minimum lies approximately (inthe zero width limit) at the center of mass energy √ s (cid:39) (cid:104) m + X m X (cid:105) / , (2)where X > Z i , i.e. , X = v DM v SM / ( v DM v SM ).For a fixed value of the mass ratio m /m , the value of X determines the proximity of this minimumto the location of either resonance, e.g. , moving closer to m relative to m as X increases. (We willreturn to this relationship below within the context of a specific model.) Now we easily imagine that forDM lying in this mass range this destructive interference is at least partially responsible for the relativelysuppressed annihilation cross section which must hold during the CMB (as well at at present times),provided the value of m DM is properly chosen.While such a deep destructive interference may be possible to achieve if two or more distinct ampli-tude structures of comparable magnitude contributed to the annihilation process, it certainly would besignificantly more difficult to arrange since the precise relative weights of the contributions to the totalamplitude would in general be quite different . This is our reasoning behind the assumption made abovethat the Z i couplings to the SM are solely vector-like; while taking the v rel → e.g. , in the mass range ofinterest to us here it may simply be the e + e − final state. Of course if the DM is sufficiently massivethen other final states such as µ + µ − and/or hadrons may also be kinematically accessible and the totalannihilation cross section is then a weighted sum of these various contributions. It is, of course, a stronglydestructive minimum in this total cross section that we seek here. In such a case, certainly, we will needall of these individual contributions to have destructive minima at the same value of √ s as given bythe expression above and to that end we must require that the ratio v SM /v SM be the same for allaccessible SM final states. Note that this is a weaker requirement than demanding that the separate v SM , individually be the same for all of these final states. This weaker requirement can be easily satisfied if, e.g. , v SMi = c i Q em (or with Q em here replaced by any other fixed combination of gauge group generators),where the c i are final state independent constants. This will indeed be the case in the simple model thatwe will construct below and this requirement occurs relatively naturally if both of these couplings aregenerated via the same KM but result in different corresponding strengths due to mixing angle effects.( iv ) A further constraint on this setup is that we must require (in its weakest form and again some-thing we will return more seriously to below) that m DM < m so that the s -wave, non-KM or velocitysuppressed process ¯ χχ → Z is kinematically forbidden when v rel → e.g. , during freeze-out, will lead to a required strengthening of this bound as can begleaned from the detailed study of Forbidden DM models [39–43]. As we will see below, this also leadsto a further bound on the ratio m /m and thus will also play a rather strict role as a constraint on ourmodel parameter space.( v ) Although we may manage to sufficiently suppress the annihilation rate of Dirac fermion DMduring the CMB via destructive interference, we still need to have a correspondingly large annihilationcross section at freeze-out, < σv rel > F O (cid:39) . · − cm s − ≡ . σ [11,12], for Dirac fermion DM when m DM /T F O (cid:39)
20 or so, to recover the DM relic density as observed by Planck. To do this we rely on thenon-zero temperature effects present during the early universe to insure that v rel (cid:54) = 0 be large enough to For example, while there can be destructive interference of the γ and Z contributions below the SM Z resonance for thefamiliar e + e − → ¯ ff process, the resulting cross section suppression is not extremely large due to the existence of severalcompeting amplitudes. T F O (cid:54) = 0 the DM has a sufficiently enhanced center ofmass energy to feel the influence of the Z resonance hump. For the mass range of interest to us here,this effect must be strong enough so as to enhance the annihilation cross section in comparison to CMBtimes by a factor by roughly K ∼ a few · or so as mentioned above and will be further discussedbelow. However, unlike in the case of ordinary resonant enhancement, the cross section in our case startsout quite suppressed at low temperatures due to the destructive interference implying that these finitetemperature effects may now be potentially much more significant as we saw in our earlier work on ED..Obviously, if m is too large in comparison to 2 m DM the influence of this second resonance will be bereduced unless the coupling ratio X is sufficiently large so as to compensate for this effect. We note thatdue to ( iv ) we cannot arbitrarily increase the value of m DM to bring the DM ‘closer’ to experiencing thesecond hump and, since need to rely only on these thermal effects, m cannot be made arbitrarily largein comparison to m . Thus we might expect that, e.g. , m /m < ∼ To move forward, we consider a simple model of the dark sector gauge interactions a variant of whichwe have analyzed previous [20] in a very different context and which we will realize here in a somewhatdifferent manner. Consider generalizing the familiar the dark gauge group from U (1) D to SU (2) I × U (1) Y I with the gauge couplings g I , g (cid:48) I in analogy with the SM. Unlike in the earlier version of this model, theSM fields themselves will remain singlets under this gauge group. Unlike the SM, however, this gaugegroup must be completely broken at or below the < ∼ g (cid:48) I /g I = tan θ I = t I , e I = g I s I = g (cid:48) I c I , with s I = sin θ I , etc . Note that, again analogous to the SM, wewill define the ‘dark charge’ to which the dark photon will couple as Q D = T I + Y I / U (1) Y hyperchargegauge boson, ˆ B µ , and the analogous U (1) Y I field, ˆ B µI , generated as usual at the 1-loop level through theaction of some portal matter (PM) fields but whose detailed nature is beyond the scope of the presentdiscussion [19–23]. This KM is described in familiar notation by L KM = (cid:15) c w c I ˆ B µν ˆ B µνI , (3)where typically (cid:15) = 10 − (3 − . Here we will always consider (cid:15) to be sufficiently small so that we cangenerally work to linear order in this parameter except where necessary. This KM is removed (to lowestorder in (cid:15) ) via the usual simple field redefinitions: ˆ B → B + (cid:15)c w c I B I and ˆ B I → B I . Now consider allof the gauge fields in the SM plus those in the dark sector in a familiar basis: W ± , Z and A defined asusual and now also W ± I (where here the ± labels the electrically neutral W I ’s dark charge as we will seebelow), ˆ Z I and ˆ A I . In such a basis, after KM has been removed, the SM gauge fields will couple as theyusually do but the hermitian dark sector gauge fields will pick up additional interactions proportional tothe SM hypercharge g I √ T + I W I + h.c. + e I Q D A I + g I c I ( T I − s I Q D ) Z I + (cid:15)g Y c w c I Y c I A I − s I Z I ) , (4)Note that at this point we have only removed the KM and have gone to a somewhat convenient andfamiliar basis; none of the gauge symmetries have yet been broken which is what we need to do next.As usual, we will assume that the SM gauge group is broken by the T L = − Y / / v (cid:39) . e.g. , the W ± it’s usualtree-level mass M W = ( gv/
2) while leaving the SM photon massless. Of course v (cid:54) = 0 also generates theusual diagonal mass term for the Z , M Z = ( gv/ c w ) but, via the KM terms in the couplings, therewill also be both diagonal and off-diagonal terms in the dark sector as well as mixing terms with the Z . We note, however, that at this step dark gauge symmetries remain unbroken. To accomplish thisfurther breaking we first add an SU (2) I doublet, SM singlet scalar field which has Y I / − / Q D = 0 element obtains a vev, v D ∼ W ± I , i.e. , M W I = g I v D / SU (2) I , as well as SM, singlet complex scalarfield with Q D = 1 that also obtains a vev, v S , of a similar (but perhaps slightly smaller) magnitude.Abbreviating the suggestive combinations M Z I = ( g I v D / c I ) and M A I = ( e I v S ) , the full 3 × Z, A I , Z I ) basis M × = M Z − (cid:15)t w M Z (cid:15)t w t I M Z − (cid:15)t w M Z (cid:15) t w t I M Z + M A I − (cid:15) t w t I M Z − t I M A I (cid:15)t w t I M Z − (cid:15) t w t I M Z − t I M A I (cid:15) t w t I M Z + t I M A I + M Z I . (5)Making the small rotations A I → A I − (cid:15)t w Z , Z I → Z I + (cid:15)t w t I Z and Z → Z + (cid:15)t w ( A I − t I Z I ) thenremoves the mixings between the now physical Z and both A I , Z I to this order as well as all of O ( (cid:15) )entries in the lower right 2 × A I , Z I gauge bosons (which are not yet mass eigenstates) will now couple to SM fields inthe combination e(cid:15)Q em ( A I − t I Z I ) and that the physical Z picks up an O ( (cid:15) ) coupling to the dark sectorfields. These results assume that M Z I ,A I << M Z which is certainly true for the parameter choices wehave made so far. We can now decouple the Z and then the remaining neutral gauge boson mixing isseen to lie totally within the dark sector and has significantly simplified to just (now in the A I , Z I basis): M × = (cid:18) M A I − t I M A I − t I M A I t I M A I + M Z I (cid:19) , (6)where we now see very transparently that the Q D = 1 singlet vev, v S , is obviously required for bothof the eigenstates masses to be non-zero. This matrix is easily diagonalized by defining the new masseigenstate fields Z , where A I = Z c φ − Z s φ and Z I = Z c φ + Z s φ with s φ ( c φ ) = sin φ (cos φ ) andwhere the angle φ is given by the expressiontan 2 φ = 2 t I M A I M Z I + ( t I − M A I . (7)In terms of the physical fields Z , , the coupling of these dark gauge bosons with the visible sector SMcan be simply written as L SM − int = e(cid:15) eff Q em ( Z − T Z ) , (8)where we have now defined the combinations T = tan( φ + θ I ) = t φ + t I − t φ t I , (cid:15) eff = (cid:15) ( c φ − t I s φ ) = (cid:15)c φ (1 − t φ t I ) . (9)Note that, within the parameter ranges employed below, it is always true that (cid:15) eff ≤ (cid:15) . Also notethat, trivially, the Z , couplings to the SM are proportional to one another, i.e. , v SM = e(cid:15) eff Q em , v SM = − T v SM in the notation of Eq.(1). The corresponding couplings of the Z i to the dark sector fieldsare given by L DM − int = (cid:104) g I c I ( T I − s I Q D ) s φ + e I Q D c φ (cid:105) Z + (cid:104) g I c I ( T I − s I Q D ) c φ − e I Q D s φ (cid:105) Z . (10)To go further we must posit the transformation of the DM field under SU (2) I × U (1) Y I requiring, trivially,that Q D ( χ ) (cid:54) = 0 and that the DM be the lightest member of the SU (2) I multiplet to which it belongsto insure its stability. The simplest possibility satisfying these requirements is that χ is a Q D = 1 state6hich is also an SU (2) I singlet, i.e. , T I ( χ ) = 0 . Assuming this to be the case, then if we define thecombination g D = e I c φ (1 − t φ t I ) , (11)we obtain that v χ = g D Q D ( χ ) ≡ g D and v χ = − T v χ . Finally, combining both sets of couplings weobserve that v χ v SM v χ v SM = T , (12)where we see that we’ve reproduced the desired result from the discussion in the previous subsectionabove with the identification X → T and, since T = tan( φ + θ I ), 0 ≤ T ≤ ∞ .Next, we need to address the masses of the Z i themselves, m i , and their relationships to the othermodel parameters. Given the discussion in the previous subsection we recall that we will be particularlyinterested in parameter values where the mass ratio λ R = m /m is held fixed. Given the simple formof the mass squared matrix above it is clear that the ratio of its eigenvalues, λ R = λ + /λ − , will dependonly upon the value of t I and the ratio ρ = M A I /M Z I . Explicitly,( M Z I ) − λ ± = 12 (cid:2) ρ (1 + t I ) (cid:3) ± (cid:2) ρ ( t I −
1) + ρ (1 + t i ) (cid:3) / ≡ A ± B , (13)so that λ R = 1 + R − R with R = BA = λ R − λ R + 1 . (14)For a given λ R one can now determine (the physical) value of ρ ( t I ) as the ‘+’ root of quadratic equation R − (cid:2) R (1 + t I ) + 1 − t I (cid:3) ρ + ( R − t I ) ρ = 0 , (15)and requiring this root to be real places an upper bound on t I : t maxI = R (1 − R ) / = λ R − λ R , (16)with ρ ( t maxI ) = [1 + t max I ] − . Using the definition of the angle φ in terms of ρ and t I then leads to ananalogous upper bound on t φ which after some algebra becomes t maxφ = (cid:2) t max I (cid:3) / − t maxI = λ − R , (17)so that, after more algebra and employing the definition of T above, we finally arrive at the simple upperbound T max = λ R . (18)This bound is phenomenologically very important because, as we noted above, we will need to increase T as m /m becomes larger to keep the cross section minimum within the range given by the requirements( iv ) and ( v ) above.To see how this parameter constraint and the other requirements above play out in this setup, weneed to perform a detailed numerical study to which we now turn. This model as constructed has only vectorial couplings for the DM and SM to the Z i and basicallyhas only 3 dimensionless parameters apart from an overall coupling strength and a mass scale; we takethese parameters to be r = 2 m DM /m , λ R = m /m and T . As we saw above, model consistency plusphenomenological constraints impose somewhat sever restrictions on their interrelated allowed values. The dark sector may, of course, contain other additional fields in various multiplets of the dark gauge symmetry all ofwhich are more massive than the DM itself.
7o proceed, we first consider the DM annihilation cross section for the process ¯ χχ → Z ∗ i → ¯ f f , wherethe fermion field, f , is here being used as a placeholder for the SM in generality. This cross section isgiven in the above model by by a simple generalization of the well-known result [44] σv rel = 2 α(cid:15) eff g D s N fc Q f β f − β f − β χ (cid:88) i,j P ij ˜ v χi ˜ v χj ˜ v fi ˜ v fj , (19)where β χ,f = 1 − m χ,f /s . As is also clear, and as previously noted, the β χ terms will essentially vanishat the time of the CMB due to the low temperatures/DM velocities. For simplicity, we will consider thespecific case of f = e in what follows so that N fc = | Q f | = β f = 1 in the kinematic region of interest butthe reader should remember that it may be a factor of a few time larger numerically when additional finalstate channels become kinematically allowed. Note that with this chosen normalization and employingthe results above we find that ˜ v χ = ˜ v f = 1 and ˜ v χ = ˜ v f = − T . The kinematic propagator factorappearing in this expression, P ij , is given as usual by P ij = s ( s − m i )( s − m j ) + Γ i Γ j m i m j [( s − m i ) + (Γ i m i ) ][ i → j ] . (20)Since we are assuming that m < m DM < m as per the above discussion, Z can decay only to SMstates, i.e. , the electron, so that it has a suppressed width, Γ /m = ( e(cid:15) eff ) / π , whereas Z candominantly decay directly to pairs of DM fermions, Γ ( DM ) /m = PS · ( g D T ) / π , where ‘PS’ is asimple phase space factor, i.e. , PS = (1 − m χ /m ) / (1 + 2 m χ /m ). Z can also decay, like Z , intoSM fields but with a partial width that also is highly suppressed, i.e. , Γ ( SM ) /m = ( e(cid:15) eff T ) / π ,which can generally be neglected but will be included here for completeness since we will sometimesapproach the kinematic region where PS →
0. For numerical purposes we can conveniently express thisDM annihilation rate in units of σ = 10 − cm s − which sets the typical scale for that required toobtain the observed relic density (recalling that the required Dirac fermion annihilation rate to achievethis density for DM masses in this mass range of interest is (cid:39) . σ [11, 12]) as σv rel σ ≡ g D e (cid:16) (cid:15) eff − (cid:17) (cid:16)
100 MeV m (cid:17) σσ . (21)As noted above, during the CMB and at present times, temperatures are sufficiently low so thattaking v rel , β χ → s = 4 m DM in suchcircumstances. Consider the sample case with the parameter choices m /m = 2 with g D /e = 1(0 . m = 100 MeV and (cid:15) eff = 10 − which we will typically employ as basic realizations of our setup. Sincethe cross section approximately factorizes as seen above, it is straightforward to obtain the correspondingresults for any other choices of g D /e , m and (cid:15) eff . For such a parameter set we can completely determinethe DM annihilation cross section in the low velocity limit as a function of r = 2 m DM /m assumingdifferent values of the parameter T as input; the results of this calculation are shown in Fig. 2 assumingthat g D /e = 1 for purposes of demonstration. Here we see the presence of the two resonance peaks witha series of destructive minima lying between them; the location of the minimum is seen to move closerto the Z hump as the value of T increases as expected from the discussion above. However, we cannot continually push this minimum to lower values of r since T has a maximum value, i.e. , T max = λ R = 2in the present case, and thus the two furthest left curves in the lower panel are not actually allowed bythis constraint and appear here only for the sake of comparison. We note that the range of parameterscomfortably satisfying this CMB constraint is rather modest (to say the least) when g D /e = 1 is assumed.To further clarify these points, Fig. 3 shows the location of the v rel → T for various values of the Z , mass ratio, m /m ; also shown is the corre-sponding upper bound on T for the same range of values of m /m that we have determined previously The presence of possiblel additional axial couplings of the DM to the Z i can be easily accommodated by letting (in acommon normalization) ˜ v χi ˜ v χj → ˜ v χi ˜ v χj +2 β χ (˜ a χi ˜ a χj ) / (3 − β χ ) in this expression above. However, this does not happen in thepresent simple model realization that we are considering here but if present would generally only make O(1) modificationsto the discussion below at the time of freeze-out but would have no effect during the CMB as noted previously. v rel → m = 100 MeV, (cid:15) eff = 10 − and g D /e = 1 in units of σ = 10 − cm s − , shown as afunction of r = 2 m DM /m . Here it is also assumed that m /m = 2 and also that, for the minimum,from right to left, (Top) T = 0 . , . , . (cid:39) tan θ w ) , . , . T = 1 . , . , . , , , v real → m as a function of T assuming that,from top to bottom, m /m = 7 , , , , ,
2, respectively. The vertical dashed lines show the maximumallowed value of T for, from left to right, the corresponding value m /m = 2 , , , , ,
7, respectively. Ineach case, the region to the right of the dashed line is unphysical and so excluded.above. For a fixed T the location of the minimum will move to larger (smaller) values as m /m increases(decreases) and similarly, for fixed m /m the value of the minimum location will decrease (increase) as T increases (decreases). However, we see that due to the bound on T from above, the location of theallowed physical minimum can never be pushed to a value of r smaller than that given by r min = (cid:16) √ s m (cid:17) min = (cid:104) λ R λ R (cid:105) / , (22)for a given λ R so that, e.g. , for λ R = m /m = 2(3) = T max , r min = √ . √ .
8) and this minimumasymptotes to the value √ λ R → ∞ .Returning now to Fig. 2, we see that, quite generally, the suppressed saddle region between theresonance humps can very easily lead to cross sections of order ∼ a few 10 − σ or larger when we choose g D /e = 1 over a modest mass range given the proper choices of T . However, we recall that in the unitsintroduced here the CMB cross section bound is roughly given by [33] σ CMB /σ < . × − r as alsocan be seen in this Figure. To increase the size of our ‘zone of comfort’ where we quite safely satisfy thisconstraint in the saddle region and for later phenomenological reasons, we will chose to shift our defaultvalue of g D /e downward, i.e. , to g D /e = 0 .
1, so that all of the model predictions displayed in this Figurewill also shift downwards by a factor of 100. This value shift now provides us with a significantly largerregion of parameter space safely satisfying the current (and any near future) CMB constraint discussedabove for this range of DM masses; we will assume this value of g D /e (= 0 .
1) in the discussion thatfollows .We have now obtained annihilation cross sections easily satisfying the CMB constraint as v rel → < σv rel > F O (cid:39) . σ for the same set of input We note that at this point we could have just as easily instead have assumed that (cid:15) eff = 10 − to recover the samereduced cross section as these are both simple overall numerical factors. However, this smaller value of (cid:15) eff is somewhatmore difficult to arrange at the 1-loop level and the benefits of the choice of reducing the coupling ratio g D /e instead willbe made more obvious below. K , as a function of r assuming x F = 20 and λ R = m /m = 2. From right to left the curves correspond to T = 0 . , . , . , . , . , . , . , . , , , g D , (cid:15) eff , m . Note that due to the overall parameter factorization exhibited in Eq.(21), therequired cross section enhancement factor, K , as will be defined below, is independent of the specificallychosen values of g D /e and (cid:15) eff and will instead depend solely upon the values of the kinematic parameters r, λ R and T as well as the temperature at freeze-out, T F O . At freeze-out, after some algebra, the thermalaveraged cross section can be written as (see, e.g. , Refs. [2, 45]) < σv rel > F O = 8 x F K ( x F ) (cid:90) ∞ γ min dγ γ ( γ − K (2 γx F ) σ ¯ χχ → SM , (23)where here the role of ‘SM’ will still be played by the e + e − final state as above, x F = m χ /T F O (cid:39) − K , are the familiar modified Bessel functions and γ = √ s/ m χ with γ min = 1 here; note that it is only σ and not σv rel that appears inside of the integrand in this expression. We now define the ‘enhancementfactor’, K , as the ratio of the annihilation cross section at freeze-out to that obtained during the CMBwhen v rel →
0, discussed above, i.e. , K = < σv rel > F O < σv rel > CMB , (24)where we will require, roughly, that K ∼ a few 10 or so to get the necessary numerics to work outproperly. We gain stress that K itself does not depend on the values of g D , (cid:15) eff or even m to a rathergood approximation since they simply cancel in this ratio but instead depends only upon the two massratios and the value of T . To be specific, let us assume that λ R = 2 and x F = 20; we can then calculate K as a function of r for different values of the parameter T as is shown in Fig. 4 and then search for theregions where K has the desired range of values. Here we see that for roughly the range 0 . < ∼ T < ∼ . K can easily lie within the desired range of ∼ a few 10 or so; this corresponds roughlyto the scaled DM mass range of 1 . < ∼ r < ∼ .
95. For larger values of T , the locations of the v rel → Z resonance hump to obtain anadequate enhancement – especially so if we must also require that T ≤ T increases the width of the Z increases, lowering the peak height, alsoleading to a further suppression of the value of K , although this is not numerically a very importanteffect. For smaller values of T outside the above range, the Z coupling is simply too weak and theproximity of the minimum too close to the Z peak to provide the cross section boost that is needed. Asa further comment on this Figure, we can also see that the values of K obtainable in this setup from theusual [46–48] resonant enhancement mechanism associated with the Z is ∼ −
200 and is clearly fartoo small for our purposes by a factor of ∼ −
30 or more.It is worthwhile to consider a few variations on this calculation while keeping λ R = 2 held fixed; we firstconsider varying out choice of x F = 20 to, e.g. , larger values, i.e. , x F = 25 ,
30. Since x F = m DM /T F O ,an increase in x F lowers the freeze-out temperature and thus the typical values of β χ , v rel occurring inthe DM collision process are also reduced since < β χ > (cid:39) / (3 x F ) and, hence, so is the typical value of √ s . This would imply that for fixed r the DM is less able to feel the influence of the second resonancehump and we thus expect the value of K to decrease with increasing x F . Fig. 5 shows what happenswhen we move to the larger values of x F = 25 or 30 and we see that our expectations are indeed met andthat the range of T over which the value of K is sufficiently large to satisfy our requirements is indeedreduced, but not by a very serious amount. For example, even when x F = 30, we see that the parameterrange 0 . < ∼ T < ∼ K .We briefly consider two other modifications related to the the Z total width since its intrinsic ‘nar-rowness’ as 2 m DM → m does plays a role in the calculation, specifically, how it compares with thethermal ‘doppler-induced’ resonance width. ( i ) One may wonder if the use of ‘running’ decay widths(see, e.g. , [49]), which scale like ∼ √ s , instead of our default use of fixed widths might lead to some-what different results when x F , T and λ R (as well as both g D /e and (cid:15) eff ) are held fixed. The toppanel of Fig. 6 addresses this issue for a particular choice of the parameter set; at least in this casewe can barely see the difference between the two predictions for K and we conclude that this choicelikely makes little difference. ( ii ) Since the width of the Z becomes (cid:15) eff suppressed in the limit when2 m DM → m , one might ask how any additional decays of the Z , into, e.g. , other possible dark sectorfields, might influence our results due to the increased Z width. We recall that in the current setup12igure 5: Same as the previous Figure but now assuming that (Top) x F = 25 or (Bottom) x F = 30.13 /m (cid:39) Γ ( DM ) /m = PS ( g D T ) / π (cid:39) . × − PS ( g D /e ) T , where PS is just the phase spacefactor introduced above PS = (1 − m χ /m ) / (1 + 2 m χ /m ), which is generally rather narrow evenwhen g D /e = 1. Clearly as this width increases, the height of the Z resonance hump decreases leadingto a suppression of the enhancement of the value of K which is obtainable when all other parametervalues are held fixed. A priori , we don’t expect that these contributions can be very large since whateverthese additional dark fields into which the Z can decay may be, they must be heavier than m DM (bydefinition) so the window for their kinematic accessibility is quite small. The lower panel of Fig. 6 showsthe effect of adding these potential ad hoc contributions to the Z width with all of the other parametersheld fixed. Clearly, if these contributions could become large then there can be a significant reduction inthe possible values of K by over an order or magnitude. However, as noted, since m is not that muchlarger than 2 m DM when λ R = 2, there is very not much of a window for such a large suppression to takeplace. Of course as λ R increases the possibility of such significant contributions can also increase due tothe opening up of the allowed phase space. However, as we will see below, such scenarios already faceother more significant issues.So far, we have not spent much time concerning ourselves with the model building constraint ( iv )above, i.e. , that we need to avoid a potentially sizable s -wave ¯ χχ → Z process cross section, otherthan by requiring that m χ = m DM < m so that, at least when v rel → √ s sufficiently so that this process becomes kinematicallyallowed although still remaining somewhat suppressed by Boltzmann factors. In the current setup, thisprocess occurs through t − and u − channel χ exchange similar to the familiar e + e − pair annihilationprocess in QED. Interestingly, if at least part of the fermion DM’s mass were to be generated by one ormore of the dark Higgs field vevs (which, given our coupling structure, is not the case presently underconsideration here and can more easily occur in the case of scalar DM) then those scalars would alsocontribute to this process as s − channel exchanges. If we want the usual ¯ χχ → Z ∗ i → e + e − reaction toremain the dominant DM annihilation process and we don’t want the ¯ χχ → Z process to reduce theamount of DM from that we observe, then we must require that the corresponding annihilation crosssection for the 2 Z final state satisfy the rough bound < σ (2 Z ) v rel > F O /σ < ∼ . To examine thisreaction in the present context we make use of the cross section expression for this process as given inRef. [50] with only a few modifications. This reaction is, of course independent of the values of both m and (cid:15) eff (which is one reason that it can be so large) but is proportional to ( g D /e ) and will dependon the value of r and, of course, x F , to which we expect some substantial sensitivity since as x F → ∞ this annihilation rate will vanish due to the Boltzmann factors. Recall that the larger the value of x F the lower the average DM velocity is in the thermal bath and thus the lower is the average value of √ s .Based on this Boltzmann suppression, semi-quantitatively, we may expect this cross section to to scaleroughly as [39–43] ∼ e [ − ( m − m χ ) x F /m χ ] = e − (2 /r − x F which gives a fair approximation to the shape ofthe numerical results that we obtain below.Fig. 7 shows the result of this cross section calculation as a function of r , provided we assume that g D /e = 0 . x F = 20 ,
25 or 30. This result was obtained by returning to Eq.(23), adopting thecross section section from Ref. [50], as noted above, and now employing γ min = m /m χ = 2 /r dueto the 2 Z mass threshold. Here we see several important things: ( i ) Simply applying the constraintthat < σ (2 Z ) v rel > F O /σ < ∼ r < ∼ . . , .
76) for x F =20(25 , ii ) If we assume that λ R = 2 as above, then this constraint tells us that we must require that0 . − . < ∼ T , depending on the exact value of x F , to avoid this excluded range of r . Simultaneously, T is also bounded from above if we are to simultaneously obtain a sufficiently large value of K as well asto satisfy the T < ∼ T max = λ R limit. ( iii ) The annihilation rate is seen to be is an exponentially strongfunction of r , reflecting the Boltzmann factor, rising extremely rapidly as r increases. For example, wesee that for values of r only slightly larger than implied by these bounds the annihilation rate is alreadyfound to be more than an order of magnitude greater than σ or possibly larger. ( iv ) This process isalso quite sensitive to g D /e , as noted above, due to its overall g D coupling dependence; this is the mainreason for making the choice g D /e = 0 . . Changes in this parameter will also We expect that the Z is sufficiently massive so that the Z Z and 2 Z final states do not pose any similar problems. This choice also renders us safe from the corresponding process where one of the Z ’s is produced off-shell [51]. K assuming that x F = 20, T = 0 .
54 and m /m = 2 (Top) comparingthe result obtained employing a running Z width (blue) with that from fixed width calculation (green)and (Bottom) showing the impact of a larger, fixed Z total width assuming that, from top to bottom, δ Γ /m = 0 . , . , . , . , . , .
03, respectively.15igure 7: Dirac fermion DM pair annihilation cross section into 2 Z via thermal effects, in units of σ ,taking g D /e = 0 .
1, as a function of r and assuming, from left to right, that x F = 20 , ,
30, respectively.lead to some substantial modifications on the constraints on the value of r and consequently the valueof T as we can see by comparing Fig. 7 and Fig. 4. ( v ) Lastly, we note that as m χ /m → x F converge to a common result for the cross section. This shouldbe no surprise since as r → √ s to exceed2 m shrinks rapidly to zero and so the cross section becomes independent of the temperature.Given these results, we necessarily must focus on a somewhat narrower model parameter space region.To this end, Fig. 8 displays the value of T = T min for the v rel → √ s /m as a function of the mass ratio λ R = m /m ; this is also, very closely,the location where K is maximized. Hence, for example, if we require a maximum value of r to lienear r = 1 . . e.g. , for λ R = 2(2 . , . , T take on values close to 1 . . , . , . . . , . , . K which results will be sufficiently large so as tomeet our needs and to determine that we must perform a detailed calculation as we did for the case of λ R = 2 above. Note that when this constraint from 2 Z production is included only the approximaterange 1 . < ∼ r < ∼ .
70 can now yield a sufficiently large value of K when we assume λ R = 2.Due to the non-abelian structure of our setup, there is a second, similarly kinematically forbiddenprocess that we may also be concerned about, i.e. , ¯ χχ → Z ∗ , → W + I W − I where the Z , exchangesin the s -channel are found to destructively interfere to maintain tree-level unitarity. The corresponding t, u − channel exchanges, familiar from the SM, are absent here as the DM, χ , is an SU (2) I singlet state.One finds, however, that it is always true for the set of parameters considered in the present analysisthat roughly (1 . − . m < ∼ m W I . (This further implies that the decay channel Z → W + I W − I for theon-shell final state will open up once (2 . − . m < ∼ m .) Thus this kinematic suppression coupled withthe destructive interference of the two amplitudes in the s − channel as well as the absence of t, u − channelexchanges renders this process far less important than the 2 Z final state we have already consideredabove when obtaining parameter constraints. This result remains true even if other values of λ R (cid:54) = 2 areconsidered, a subject to which we now turn.Up to this point we have mostly limited our discussion to the case of λ R = 2 and it behooves us to now16igure 8: Required value of T = T min for the cross section minimum to lie at √ s /m = 1 . − . m /m . The dashed line represent themaximum allowed value of T as a function of m /m as described in the text.ask what happens to our results when this value is modified. First, let us consider the case where λ R < λ R decreases the location of the destructiveminimum must move to lower values of r thus making it much easier or even trivial to avoid the constraint r < ∼ . Z production process. However, as was already noted above in our discussionof Fig. 4, it is not advantageous to have r too close to m /m and lowering λ R significantly decreases thepossible range of r over which K can be large. Fig. 9 shows the result of our calculation of K ( r ) as wegradually lower the value of λ R from 1.7 to 1.5 to 1.2 with all the other parameters held fixed. When, e.g. , λ R = 1 .
7, we see that for 0 . < ∼ T < ∼ .
5, corresponding to roughly 1 . < ∼ r < ∼ .
65, a sufficientlylarge value of K is obtained while automatically avoiding a large rate for the 2 Z DM annihilation mode.However, we see that as λ R further decreases, the allowed range of T is somewhat reduced due to therequirement T ≤ λ R , but that for r is drastically reduced, i.e. , 1 . < ∼ r < ∼ .
45 when λ R =1.5 and onlythe narrow window 1 . < ∼ r < ∼ .
17 when λ R = 1 .
2. Thus the λ R < R , our expectation is, since we require that both r < ∼ . Z annihilation cross section) and T < ∼ λ R (from model self-consistency), that the values of K ( r ) will be somewhat reduced as λ R increases when all the other parameters are held fixed. The reasonfor this expectation was noted above: as the Z resonance hump moves away from the value of 2 m DM ,the ability of the the DM to ‘feel’ this resonance sufficiently to increase the annihilation cross sectionat freeze-out is reduced, hence, leading to a lower value of K . Clearly, at some point λ R will becomesufficient large, with r < ∼ .
7, that no region of the parameter space allowed by other constraints producesvalues of K in excess of the required value of a few × and the model again begins to fail.Fig. 10 shows the response of K in our r range of interest to increasing values of λ R to 5 and 6 forthe same default values of the other parameters as considered previously above in Fig. 4. As might beexpected, for a fixed value of r , the peak of the K distribution moves to higher values of T , while forfixed T , the peak moves to larger values or r , more frequently beyond our range on interest as λ R onlyincreases further. Here we see already that for λ R = 5(6), only the range 1 . < ∼ T < ∼ . . < ∼ r < ∼ .
70 ( T ∼ .
20 with r (cid:39) .
70) provides a sufficiently large value of K while also avoiding the17igure 9: Same as in Figure 4 but now assuming that (Top left) λ R = 1 .
7, (Top right) λ R = 1 . λ R = 1 .
2, respectively, but now with reduced ranges of T , still beginning with T = 0 . λ R = m /m = 5(6) in the top(bottom) panel. 19igure 11: Same as the previous Figure but now assuming, from top left to bottom, that λ R = m /m =7 , ,
9, respectively. 20 Z constraint. These conflicting requirements are brought home even more strongly in Fig. 11 whereeven larger values of λ R are considered. For λ R = 7 we see that T is constrained from both directionsto lie near ∼ . − . r (cid:39) .
70 while for even larger values of λ R , no values of K > ∼ seem tobe obtainable and thus no region of parameter space remains tenable given the choices above. From thisanalysis we see that once λ R becomes much larger than ∼
2, the size of the allowed parameter rangesrapidly fall to zero essentially forcing us to consider only the range λ R < ∼ . − . Z i exchanges. As is well-known, in thisDM mass range, χ − e elastic scattering may likely be the most sensitive channel [52–57]. Assuming thatthe DM mass is in the mass range such that µ = m e m χ / ( m e + m χ ) → m e and noting that the momentumtransfer Q << m , , this cross section is given numerically by the expression σ χe (cid:39) . × − cm (cid:16)
100 MeV m (cid:17) (cid:104) g D /e . (cid:15) eff − (cid:105) (1 + T /λ R ) . (25)Note that all of the model dependence that we have been concerned with up to now is quite weak in thiscase and essentially lies completely isolated within the last term appearing here such that, since T ≤ λ R ,we must have (1 + T /λ R ) ≤ O (1) changes in thiscross section from the predictions of the single DP setup with the same input values of g D , (cid:15) and m . The possibility of light dark matter coupling to the SM via the kinetic mixing of a similarly light darkphoton with the familiar Standard Model hypercharge gauge boson is very attractive for numerous rea-sons. Such a scenario can lead to a DM relic density consistent with the Planck measurements via theusual freeze out mechanism in the same parameter range that is accessible to multiple future plannedexperiments. This same accessibility leads to some already significant restrictions on the parameter spaceof this scenario, if realized in its most simple form, from a wide variety of existing experiments. In par-ticular, measurements from the CMB impose rather strong constraints on the DM thermally averagedannihilation cross section at z ∼ , < σv rel > CMB , informing us that this quantity must be suppressedby a factor of K ∼ a few 10 or more, depending upon the light DM mass, in comparison to the analogouscross section at freeze out, < σv rel > F O , that is required to reproduce the observed relic density. Thiswould seem to imply that this reaction must be temperature and/or velocity dependent. Naively, thisexcludes the possibility of DM annihilation being an s -wave process as would be the case, e.g. , of Diracfermion DM annihilating via an s − channel DP exchange into the SM fermions since this type of processis generally temperature/velocity independent. This observation lends support to the possibilities of co-annihilating Majorana DM, which is an s − wave but is Boltzmann suppressed, or p − wave annihilatingcomplex scalar DM, which is velocity-squared suppressed, during the CMB epoch.In this paper, we have fully examined a previously proposed mechanism by which the Dirac DMannihilation process can be made simultaneously consistent with both the relic density and CMB con-straints, albeit within a restricted kinematic range. Semi-quantitatively, this requires the existence of(at least) two dark gauge bosons, Z , , by which the DM can pair annihilate via s -channel exchangeto SM fields – as noted, this being an s − wave process. The Z i couplings must be such that their con-tributions to this annihilation process destructively interfere, in a manner which is independent of theparticular SM final state, when the DM pair threshold lies between the masses of these two resonances, i.e. , m < m DM < m . Requiring that m DM /m < ∼ .
85, to avoid the s -wave, thermally excited DMpair annihilation into 2 Z (which is not suppressed by KM) while also simultaneously keeping 2 m DM nottoo far below m , so that a very strong resonant enhancement from the deep destructive minimum canoccur, greatly restricts the parameter space of any potential concrete model.In order to explore the interplaying roles of these rather restrictive requirements we, constructed anon-abelian, SM-like SU (2) I × U (1) Y I dark sector model but one whose gauge symmetry is completelybroken leading to Z , of comparable masses. The structure of the model’s couplings automatically lead21o the necessary common destructive interference over a significant parameter space region when the m < m DM < m condition is satisfied for all SM final states. While the DM couplings of Z i essentiallyarise from the gauge group structure and the DM representation, here chosen to be a Y I / Q D = 1, SU (2) I isosinglet to help insure it is the lightest dark sector state, the corresponding SM couplings to the Z i are both generated via KM and the various mixing angles required to obtain kinetically normalizedfields in the mass eigenstate basis. Within this setup it was found that all of the constraints could besatisfied for a respectable range of couplings and values of the mass ratios m /m and m DM /m – but ina correlated manner. In particular, it was found that for the ratio of the Z , masses roughly in the range1 . < ∼ λ R = m /m < ∼ . Z mass ratio in the (correlated) range 1 . < ∼ r = 2 m DM /m < ∼ . Z to Z couplings to the SM and DM of O(1). This demonstrates not only proof of principlebut also that realistic models with all of the desired properties can be constructed allowing for light Diracfermion dark matter.Light dark matter with a light mediator below the ∼ Acknowledgements
The author would like to particularly thank J.L. Hewett, D. Rueter and G. Wojcik for very valuablediscussions related to this work. This work was supported by the Department of Energy, ContractDE-AC02-76SF00515.
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