W/Z Bremsstrahlung as the Dominant Annihilation Channel for Dark Matter
Nicole F. Bell, James B. Dent, Thomas D. Jacques, Thomas J. Weiler
aa r X i v : . [ h e p - ph ] S e p ERRATUM
Due to an error in a Fierz identity published elsewhere [43], some of the results presented in this paper are also inerror. Here we outline which results of this paper are correct and which are incorrect, and then briefly discuss theconsequences for the latter. Further details can be found in [44].The various Fierz identities we presented in this paper (Eqns. (3,4,7) and all the equations of Appendix A) are allcorrect. However, due to parallel-processing of our efforts, our explicit cross section calculation was performed usingthe Fierz identity given in Okun’s textbook. This identity should correctly read F il G mk = 14 X A ∆ A ( F γ A G ) ml ( γ A ) ik , (1)in correspondence with our Eq. (A7), but does not. In the published expression of Ref [43], the indices { k, i } areincorrectly interchanged with { l, m } on one side of the equation, which is equivalent to exchanging F and G on oneside of Eq. (1). This transposition is thus not an issue for the usual application of the Fierz identity to 2 → F = G . However, it becomes an issue for 2 → M η ≫ M χ . Since the four Fermi limit gives a zero s-wave result, there are no interestingnew figures to be presented in this erratum. However, if the four Fermi limit is not adopted, then the cancellation isnot complete and an unsuppressed s-wave amplitude results. This means that the interesting parameter space is thatwhere M η and M χ are comparable. In this regime, one should consider not only the 4 diagrams of Fig. 1, but alsothe two additional diagrams in which the W/Z boson is radiated from the η propagator. This in turn means that theleading s-wave amplitude is one power of ( M χ /M η ) higher than implied by our errant calculation. In Ref. [44], withtwo additional authors, we present the higher order calculation of the unsuppressed s-wave cross section.Many of the important qualitative conclusions of this paper still hold:(i) W / Z bremsstrahlung can lift helicity suppressions and thus be the dominant annihilation model, albeit for aregion of parameter space smaller than we originally proposed.(ii) antiprotons produced by the decays of the W and Z gauge bosons can prove lethal for models attempting toproduce positrons without overproducing antiprotons.We thank Ahmad Galea for his signifiant help in identifying the errant formulas. /Z Bremsstrahlung as the Dominant Annihilation Channel for Dark Matter Nicole F. Bell, James B. Dent, Thomas D. Jacques, and Thomas J. Weiler School of Physics, The University of Melbourne, Victoria 3010, Australia Department of Physics and School of Earth and Space Exploration,Arizona State University, Tempe, AZ 85287-1404, USA Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA (Dated: September 23, 2011)Dark matter annihilation to leptons, χχ → ℓℓ , is necessarily accompanied by electroweak radiativecorrections, in which a W or Z boson is radiated from a final state particle. Given that the W and Z gauge bosons decay dominantly via hadronic channels, it is thus impossible to produce finalstate leptons without accompanying protons, antiprotons, and gamma rays. Significantly, whilemany dark matter models feature a helicity suppressed annihilation rate to fermions, radiating amassive gauge boson from a final state fermion removes this helicity suppression, such that thebranching ratios Br( ℓνW ), Br( ℓ + ℓ − Z ), and Br( ν ¯ νZ ) dominate over Br( ℓℓ ). W/Z -bremsstrahlungthus allows indirect detection of many WIMP models that would otherwise be helicity-suppressed, or v suppressed. Antiprotons and even anti-deuterons become consequential final state particles. Thisis an important result for future DM searches. We discuss the implications of W/Z -bremsstrahlungfor “leptonic” DM models which aim to fit recent cosmic ray positron and antiproton data.
PACS numbers: 95.35.+d, 12.15.Lk, 95.85.Ry
I. INTRODUCTION
An abundance of cosmological and astrophysical evidenceattests to the existence of dark matter (DM), whose pres-ence is inferred via its gravitational influence [1–3]. How-ever, the fundamental particle properties of DM remainessentially unknown. One important means of probingDM’s particle nature is via indirect detection, wherebywe search for products of DM annihilation (or decay) em-anating from regions of DM concentration in the Universetoday.The dark matter annihilation cross section is oftenparametrized as h vσ A i = a + bv + · · · , where h vσ A i isthe thermally-averaged annihilation cross section. Theconstant a comes from s-wave annihilation, while the ve-locity suppressed bv term receives both s-wave and p-wave contributions; the L th partial wave contribution tothe annihilation rate is suppressed as v L . Given that v ∼ − c in galactic halos, even the p-wave contributionis highly suppressed and thus only the s-wave contribu-tion is expected to be significant in the Universe today.However, in many DM models the s-wave annihilationinto a fermion pair χχ → ¯ f f is helicity suppressed by afactor ( m f /M χ ) (only → ¯ tt modes remain of interest,and then only for a certain range of χ mass).When computing DM annihilation signals, it is nor-mally assumed that only the lowest order tree-level pro-cesses make a significant contribution. However, thereare important exceptions to this statement. Dark matterannihilation into charged particles, χχ → ¯ f f , is neces-sarily accompanied by the internal bremsstrahlung pro-cess χχ → ¯ f f γ , where the photon may be radiated fromone of the external particle legs (final state radiation,FSR) or, possibly, from a virtual propagator (virtualinternal bremsstrahlung, VIB). On the face of it, theradiative rate is down by the usual QED coupling fac- tor of α/π ∼ s -wave process [4], which more than compensates for theextra coupling factor. Such a striking enhancement canarise when a symmetry of the initial state χχ is satis-fied by the three body final state ¯ f f γ , but not by thetwo body final state ¯ f f . For bremsstrahlung of photons,only VIB is effective in lifting the helicity suppression,as FSR is dominated by soft or collinear photons (suchthat the two and three body final states have the samesymmetry properties) as discussed in Ref. [5].In this paper we examine electroweakbremsstrahlung [6–12], i.e., bremsstrahlung of Z or W ± electroweak gauge bosons to produce ¯ f f Z and¯ ℓνW final states. The virtue for W/Z bremsstrahlungto lift initial-state velocity and final-state helicity sup-pressions, alluded to in [8, 11], has not been previouslyexplored. We show that
W/Z -bremsstrahlung can alsolift suppression and become the dominant annihilationchannel. Thus,
W/Z bremsstrahlung allows indirectdetection of many WIMP models that would otherwisebe helicity-suppressed, or v suppressed. This is animportant result for future DM searches.There are a number of important distinctions be-tween electromagnetic (EM) and electroweak (EW)bremsstrahlung. An obvious one is that EMbremsstrahlung produces just photons, whereas EWbremsstrahlung and subsequent decay of the gaugebosons leads to leptons, hadrons and gamma rays, of-fering correlated “multi-messenger” signals for indirectdark matter searches. Another distinction is that W/Z -bremsstrahlung from final state particles (FSR) is suf-ficient to lift a suppression. This is due to the nonzerogauge boson masses, and the coupling of the gauge bosonsto the non-conserved axial current which leads to a differ-ent form for the polarization sum than in the case of thephoton (or a gluon in the similar QCD process). In con-trast, for the EM process, VIB is required for the photonto lift a suppression. Because an additional propagatorappears for VIB, suppression-lifting EM bremsstrahlungis itself suppressed by an additional factor of M χ /M η rel-ative to electroweak’s FSR, where M η is the mass of theinternal exchange-particle. Only in the event of a near-degeneracy M χ ∼ M η is this relative suppression of EMbremsstrahlung negligible.DM annihilation to charged leptons has been the sub-ject of much recent attention, due to recently measuredcosmic ray anomalies which point to an excess of cos-mic ray positrons above those that may be attributed toconventional astrophysical processes. PAMELA has ob-served a sharp excess in the e + / ( e − + e + ) fraction at ener-gies beyond approximately 10 GeV [13], without a corre-sponding excess in the antiproton/proton data [14, 15],while Fermi and HESS have reported more modest ex-cesses in the ( e − + e + ) flux at energies of order 1 TeV [16].These signals have led to a re-examination of positronproduction in nearby pulsars [17], emission from super-nova remnants [18], acceleration of e + e − in cosmic raysources [19], and propagation in conventional cosmic raymodels [20]. As an alternative to these astrophysicalmechanisms, it has also been proposed that the excess e + and e − are produced via dark matter annihilationin the Galactic halo, with an abundance of DM modelsproposed to accomplish this end. A recent overview of e ± -excess data and possible interpretations is availablein [21].However, some of the most popular models suffer fromhelicity or v -suppression. A prototypical example ofsuppressed production of Standard Model (SM) fermionpairs is provided by supersymmetry: Majorana neutrali-nos annihilate into a pair of SM fermions via t - and u -channel exchange of SU (2)-doublet sfermions. To over-come the suppression, proponents of these models haveinvoked large “boost” factors. These boost factors maybe astrophysical in origin, as with postulated local over-densities of dark matter, or they may arise from particlephysics, as with the Sommerfeld enhancement that arisesfrom light scalar exchange between dark matter particles.Although not ruled out, these factors do seem to be a con-trivance designed to overcome the innate suppression.A further problem with suppressed models is theoverproduction of antiprotons from unsuppressed W/Z bremsstrahlung. Given that hadronic decay modes ofthe W and Z bosons will lead to significant numbers ofboth antiprotons and gamma rays, this will impact theviability of models that might otherwise have explainedthe observed positron excess. Even in models which donot feature a suppression, the W/Z-bremsstrahlung hasimportant phenomenological consequences, as the decayproducts of the gauge bosons make a pure leptonic e + e − signal impossible [11].In Section II we discuss the circumstances under whichdark matter annihilation may be suppressed, and in Sec-tion III explain how W/Z bremsstrahlung is able to cir- cumvent such a suppression. In Section IV we considera representative model, and explicitly calculate the crosssections for both the lowest order annihilation process,and for the
W/Z bremsstrahlung process. We discussimplications of these results in Section V. Calculationaldetails are collected in five Appendices.
II. UNDERSTANDING SUPPRESSION USINGFIERZ TRANSFORMATIONS
In this section we describe the origin of v and helicitysuppressions. We shall make use of Fierz transformationand partial wave decomposition to determine under whatcircumstances these suppressions will or will not arise.Dark matter candidates may be scalar, fermionic, orvector in nature; if fermionic, they may be either Diracor Majorana. Permissible annihilation models include s -, t -, and u -channel exchanges of a new particle, and thevarious possibilities are listed in Refs. [22–24]. In everycase, it is useful to classify the partial waves available tothe decay process, and to analyze the dependence on themass of the SM particle-pair in the final state. In thisarticle, we focus on fermionic Majorana dark matter.For fermionic dark matter, the natural projection of2 → χχ → ¯ f f , andexplain the use of Fierz transforms to convert the matrixelements for t / u -channel annihilation, which are of theform ( χ Γ A l )(¯ l Γ B χ ), to a sum of s -channel amplitudesof the form ( χ Γ χ )(¯ l Γ l ). In the following subsectionwe then categorize the Fierzed s -channel amplitudes intopartial waves and fermion-pair spin states, which deter-mines whether the amplitudes are velocity suppressed,mass-suppressed, or unsuppressed. In the third and finalsubsection, we put our findings together to determinewhich class of models will have a suppressed 2 → → W/Z -bremsstrahlung processis unsuppressed, and in fact dominant for 2 M χ > M W .We will find in Section III that a generalization of theFierz transformation offers useful insight into the non-suppression of the 2 → A. Fierz Transformations in the Chiral Basis
Helicity projection operators are essential in chiralgauge theories, so it is worth considering the reformu-lation of Fierz transformations in the chiral basis [25].(A discussion of standard Fierz transformations may befound in, e.g. Ref. [26].) We place hats above the gener-alized Dirac matrices constituting the chiral basis. Thesematrices are { ˆΓ B } = { P R , P L , P R γ µ , P L γ µ , σ µν } , and { ˆΓ B } = { P R , P L , P L γ µ , P R γ µ , σ µν } , (2)where P R ≡ (1 + γ ) and P L ≡ (1 − γ ) are the usualhelicity projectors. Notice that the dual of P R γ µ is P L γ µ ,and the dual of P L γ µ is P R γ µ . The tensor matrices in thisbasis contain factors of : ˆΓ T = σ µν and ˆΓ T = σ µν .These facts result from the orthogonality and normaliza-tion properties of the chiral basis and its dual, as ex- plained in detail in Appendix A.Using completeness of the basis (see Appendix A), onearrives at a master formula which expands the outerproduct of two chiral matrices in terms of their Fierzedforms:(ˆΓ D ) [ˆΓ E ] = 14 T r [ˆΓ D ˆΓ C ˆΓ E ˆΓ B ] (ˆΓ B ] [ˆΓ C ) , (3)where the parentheses symbols are a convenient short-hand for matrix indices [27] (see the appendix for de-tails). Evaluating the trace in Eq. (3) leads to the Fierztransformation matrix in the chiral-basis: ( P R ) [ P R ]( P L ) [ P L ]( ˆ T ) [ ˆ T ]( γ ˆ T ) [ ˆ T ]( P R ) [ P L ]( P R γ µ ) [ P L γ µ ]( P L ) [ P R ]( P L γ µ ) [ P R γ µ ]( P R γ µ ) [ P R γ µ ]( P L γ µ ) [ P L γ µ ] = 14 −
16 6 − − − − ( P R ] [ P R )( P L ] [ P L )( ˆ T ] [ ˆ T )( γ ˆ T ] [ ˆ T )( P R ] [ P L )( P R γ µ ] [ P L γ µ )( P L ] [ P R )( P L γ µ ] [ P R γ µ )( P R γ µ ] [ P R γ µ )( P L γ µ ] [ P L γ µ ) . (4)Non-explicit matrix elements in (4) are zero, and we haveintroduced a shorthand ˆ T for either ˆΓ T = σ µν or ˆΓ T = σ µν .The importance of this transformation for us is thatit converts t -channel and u -channel exchange graphs into s -channel form, for which it is straightforward to eval-uate the partial waves. The block-diagonal structures,delineated with horizontal and vertical lines, show that“mixing” occurs only within the subsets { P R ⊗ P R , P L ⊗ P L , ˆ T ⊗ ˆ T , γ ˆ T ⊗ ˆ T } , and { P R ⊗ P L , P R γ µ ⊗ P L γ µ } . TheFierz transform matrix is idempotent, meaning its squareis equal to the identity matrix. This follows from thefact that two Fierz rearrangements return the process toits initial ordering. A consequence of the block-diagonalform is that each sub-block is itself idempotent.In Eq. (4) we have included one non-member of the ba-sis set, namely γ ˆ T ; it is connected to ˆ T via the relation γ σ µν = i ǫ µναβ σ αβ . (5)Explicit use of γ ˆ T in Eq. (4) is an efficient way to expressthe chiral Fierz transformation.So far we have not used the qualifier in the assumption,that the dark matter is Majorana. Majorana particlesare invariants under charge conjugation C , which impliesthat vector and tensor bilinears are disallowed. Anotherway of understanding this is to note that interchangingthe two identical Majorana particles in a t -channel dia- gram generates an accompanying u -channel diagram witha relative minus sign (from fermion anticommutation).When Fierzed, these two amplitudes cancel for V and T couplings (exactly so in the Four-Fermi limit wherethe differing momenta in the t - and u -channel propaga-tors can be ignored – refer to Appendix B for details).We must thus drop V and T couplings appearing in theFierzed bilinears of the χ -current. B. Origin of v and Helicity Suppressions One can use partial wave decomposition (see. e.g.,the textbooks [28–30], or the convenient summary in theAppendix of [8]) to expand the scattering amplitudes asa sum of angular momentum components. Partial wavesdo not interfere, and the L th partial wave contributionto the total cross section σv is proportional to v L . Theannihilating χ particles are very non-relativistic today, soan unsuppressed s-wave ( L = 0), if present, will dominatethe annihilation cross section. The DM virial velocitywithin our Galaxy is about 10 − (in units of c ), leadingto a suppression of v ∼ − for p -wave processes.On the other hand, the SM fermions produced in the2 → t ¯ t production). For many annihilation channelsthe spin state of the fermion pair gives rise to a helic-ity suppression by a factor of ( m l /M χ ) , where m l is thefermion mass.Unfortunately, many popular models for annihilationof Majorana dark matter to charged leptons are sub-ject to one or more of these two suppressions, the v and/or ( m ℓ /M χ ) suppressions. This includes some ofthe models proposed to accommodate the positron and e + e − excesses observed in PAMELA, Fermi-LAT, andHESS data. In Section III, we show that in the class ofmodels which have suppressed rates for χχ → ℓ + ℓ − , the2 → W ± or Z tothe final state particles of the 2 → W ’s and Z ’s will decay to, amongother particles, antiprotons. Since an excess generationof antiprotons is not observed by PAMELA, this class ofmodels is ruled out by the present work.Consider products of s-channel bilinears of the form( χ Γ χ )(¯ l Γ l ). To further address the question of whichproducts of currents are suppressed and which are not, wemay set v to zero in the χ -current, and m ℓ to zero in thelepton current, and ask whether the product of currentsis suppressed. If the product of currents is non-zero inthis limit, the corresponding amplitude is unsuppressed.In Table I we give the results for the product of all stan-dard Dirac bilinears. (The derivation of these results isoutlined in Appendix C.) Suppressed bilinears enter thistable as zeroes. One can read across rows of this table to discover thatthe only unsuppressed s -channel products of bilinears forthe 2 → C. Class of Models for which χχ → ℓℓ Annihilationis Suppressed
We now put the results of the previous two subsec-tions together to explain which class of models have a v and/or ( m ℓ /M χ ) suppressed 2 → s -channel annihilationwith a P coupling is unsuppressed, while S and A con-tributions are suppressed (and V and T forbidden). Letus now consider t -channel or u -channel processes.Any t -channel or u -channel diagram that Fierz’s to an s -channel form containing a pseudoscalar coupling will It is seen that the only bilinears in the table without velocity-suppression are those of the pseudo-scalar, the three-vector partof the vector, the zero th component of the axial vector, and thetime-space part of the tensor (or equivalently, the space-spacepart of the pseudotensor). It is also seen that the only bilinearswithout fermion mass-suppression are the scalar, pseudoscalar,three-vector parts of the vector and axial vector, and the tensor. have an unsuppressed L = 0 s -wave amplitude. Fromthe matrix in Eq. (4), one deduces that such will be thecase for any t - or u -channel current product on the leftside which finds a contribution in the 1 st , 2 nd , 5 th , or7 th columns of the right side. This constitutes the t -or u -channel tensor, same-chirality scalar, and oppositechirality vector products (rows 1 through 4, and 6 and8 on the left). On the other hand, the t - or u -channelopposite chirality scalars or same-chirality vectors (rows5, 7, 9, and 10 on the left) do not contain a pseudoscalarcoupling after Fierzing to s -channel form. Rather, it isthe suppressed axial-vector and vector (Dirac fermionsonly) that appears.Interestingly, a class of the most popular models forfermionic dark matter annihilation to charged leptons,fall into this latter, suppressed, category. It is preciselythe opposite-chirality t - or u -channel scalar exchange thatappears in these models, an explicit example of which willbe discussed below. Thus it is rows 5 and 7 in Eq. (4)that categorize the model we will analyze. After Fierzingto s -channel form, it is seen that the Dirac bilinears areopposite-chirality vectors (i.e., V or A ). Dropping thevector term from the χ -current we see that the 2 → χ -current to a relativis-tic SM fermion-current which is an equal mixture of A and V . Accordingly, this model has an s -wave amplitudeoccurring only in the L = 0, J = 1, S = 1 channel, withthe spin flip from S = 0 to S = 1 (or equivalently, themismatch between zero net chirality and one unit of he-licity) costing a fermion mass-insertion and a ( m f /M χ ) suppression in the rate.Let us pause to explain why this t - or u -channel scalarexchange with opposite fermion chiralities at the verticesis so common. It follows from a single popular assump-tion, namely that the dark matter is a gauge-singlet Ma-jorana fermion. As a consequence of this assumption,annihilation to SM fermions, which are SU (2) doubletsor singlets, requires either an s -channel singlet boson or a t - or u -channel singlet or doublet scalar that couples to χ - f . In the first instance, there is no symmetry to forbid anew force between SM fermions, a disfavored possibility.In the second instance, unitarity fixes the second vertexas the hermitian adjoint of the first. Since the fermions ofthe SM are left-chiral doublets and right-chiral singlets,one gets chiral-opposites for the two vertices of the t - or u -channel.Supersymmetry provides an analog of such a model. Inthis case the dark matter consists of Majorana neutrali-nos, which annihilate to SM fermions via the exchangeof (“right”- and “left”-handed) SU (2)-doublet sleptonfields. In fact, the implementation in 1983 of supersym-metric photinos as dark matter provided the first explicitcalculation of s -wave suppressed Majorana dark mat-ter [31]. However, the class of models described aboveis more general than the class of supersymmetric models.To illustrate our arguments, we choose a simple ex-ample of the class of model under discussion. This isprovided by the leptophilic model proposed in Ref. [32] s-channel bilinear ¯Ψ Γ D Ψ v = 0 limit M = 0 limitparallel spinors antiparallel spinors parallel spinors antiparallel spinorsscalar ¯Ψ Ψ 0 0 √ s iγ Ψ − iM − i √ s γ γ Ψ 2 M γ γ j Ψ 0 0 0 √ s ( ± δ j − iδ j )vector ¯Ψ γ Ψ 0 0 0 0¯Ψ γ j Ψ ∓ M δ j − M ( δ j ∓ iδ j ) 0 −√ s ( δ j ∓ iδ j )tensor ¯Ψ σ j Ψ ∓ iM δ j − iM ( δ j ± δ j ) − i √ s δ j σ jk Ψ 0 0 ±√ s δ j δ k γ σ j Ψ 0 0 ± i √ s δ j γ σ jk Ψ ∓ M δ j δ k − M ( δ j δ k ∓ iδ j δ k ) −√ s δ j δ k v = 0 limit” columns, and the lepton bilinear must have a non-zero term in the appropriate cell of the“ M = 0 limit” columns. Otherwise, the term is suppressed. (The tensor and pseudo-tensor are not independent, but ratherare related by γ σ µν = i ǫ µναβ σ αβ .) We recall that antiparallel spinors correspond to parallel particle spins (and antiparallelparticle helicities for the M = 0 current), and vice versa. Amplitudes are shown for u Γ D v = [ v Γ D u ] ∗ . The two-fold ± ambiguities reflect the two-fold spin assignments for parallel spins, and separately for antiparallel spins. by Cao, Ma and Shaughnessy. Here the DM consists ofa gauge-singlet Majorana fermion χ which annihilates toleptons via the SU (2)-invariant interaction term f (cid:0) ν ℓ − (cid:1) L ε η + η ! χ + h.c. = f ( ν L η − ℓ − L η + ) χ + h.c. (6)where f is a coupling constant, ε is the 2 × η + , η ) form the new SU (2) doubletscalar which mediates the annihilation. (This model wasoriginally discussed in Ref. [33], and an expanded discus-sion of its cosmology may be found in Ref. [34].)As discussed above, the u - and t -channel ampli-tudes for DM annihilation to leptons, of the form( χP L l ) (¯ lP R χ ), become pure ( χP L γ µ χ ) (¯ lP R γ µ l ) underthe chiral Fierz transformation. The product of the Ma-jorana and fermion bilinears then leads to an AA termand an AV term. However, reference to Table I showsthat neither of these terms leads to an unsuppressed am-plitude: in all cases, either the lepton bilinear is sup-pressed by m ℓ , the DM bilinear by v , or both are sup-pressed. Thus, Majorana DM annihilation to a leptonpair is suppressed in this model, in accordance with theexplicit calculation in Ref. [32]. III. LIFTING THE SUPPRESSION
Allowing the lepton bilinear to radiate a W or Z bo-son (as shown in Fig. (1)) does yield an unsuppressedamplitude. In the rate, there will be the usual radia-tive suppression factor of α π ∼ − . But, this willbe partially compensated by a 3-body phase space fac- tor ∼ ( M χ /M W ) / π relative to 2-body massless phasespace, which exceeds unity for dark matter masses ex-ceeding ∼ TeV. More importantly, the v suppressionfor Majorana annihilation to 2-body final states will belifted by the 3-body W -bremsstrahlung process. In Sec-tion IV we show, by explicit calculation, that the 2 → M χ that allows the W to be produced on-shell, i.e.,for 2 M χ > M W .The next inevitable question is “Why is the radiative2 → → N processes, N ≥
3. The relevant equation,derived in Appendix A states that any 4 ×
1) [ B Y ] [ Γ B )= 14 T r [X Γ B Y Γ C ] (Γ C ] [Γ B ) , (7)where the Dirac matrices here are taken in the standardbasis defined in Eq. (A1).From Table (I) we see that setting Γ C to γ γ , theonly structure available to a non-relativistic Majorana When M χ ≫ M W , the rate for single W production is dominatedby infrared and collinear divergences, leading to a suppressed fac-tor ln (cid:18) s M W (cid:19) − (cid:18) s M W (cid:19) [10, 35] instead of our (cid:18) s M W (cid:19) .Moreover, the rate for multiple production of W ’s becomes solarge that resummation techniques are necessary. FIG. 1. t -channel (A and C) and u -channel (B and D) contributions to χχ → e + νW − . Emission from the scalar propagatoris not included, as it is suppressed by 1 /M η . Note that all fermion momenta flow with the arrow except p , so q = p + Q , q = − p − Q . current other than the pseudoscalar, and Γ B to either γ j or γ γ j , provides an unsuppressed product of the Majo-rana dark matter bilinear and the charged lepton bilinear.Moreover, for the W/Z -bremsstrahlung process, X and Y in the general Fierz equation are the un-Fierzed cou-plings P L and q − P R /q P L /ǫ , respectively. So we will haveshown that the radiative process is unsuppressed if wecan show that q − T r [ P L ( γ j or γ γ j ) P R /q P L /ǫ γ γ ] isunsuppressed. This trace reduces to q − T r [ P R γ γ j /q /ǫ ].The expansion of this trace as scalar products containsterms such as q · ǫ j and ( ~ǫ × ~q ) j , which are nonzero andunsuppressed by fermion masses. Thus, the 2 → s -wave amplitude.Physically, the un-suppression works because thegauge boson carries away a unit of angular momentum,allowing a fermion spin-flip such that there is no longer amismatch between the chirality of the leptons and theirallowed two-particle spin state.One may ask why emission of a gamma-ray ratherthan a W/Z boson is less effectual. It has been knownfor some time [4, 5] that gamma-ray emission in the fi-nal state does produce an unsuppressed s -wave contribu-tion, but at second order rather than lowest order in theinverse mass-squared M − η of the t - and u -channel ex-change particle(s). The reason is that gamma-ray emis-sion from the final state fermions (FSR) is dominatedby infra-red and collinear singularities, each of whichputs the intermediate lepton on-shell (virtuality q → q − from the squared propagator in thephase space integral (see Eq. (21)), one gets the factor R sM V dq q ( s − q ) ( q − M V ), where M V is the mass of theradiated boson (photon or W or Z ). For a gamma-ray,with M V = 0, one readily sees the infra-red and collinearsingularities in R dq q . An on-shell particle is observable,so the spin states of the q → q →
0, the trace for gammaemission,
T r [ γ γ j /q /ǫ ] = T r [ γ γ j ( P R + P L ) /q /ǫ ] goesover to T r [ γ γ j P R ] T r [ P R /q /ǫ ] + T r [ γ γ j P L ] T r [ P L /q /ǫ ].The first trace in each term of this sum vanishes. Con-sequently, the gamma-emission amplitude remains sup-pressed at order M − η . However, at order M − η , thegamma-ray may be emitted from the internal particle η (VIB). For VIB, phase space does not favor q = 0,and an unsuppressed amplitude results.The emission of a massive W (or Z ) boson contrastssignificantly from the emission of a massless photon.With the W emission, the relevant phase space integralover virtuality q is R sM W dq q ( s − q ) ( q − M W ). The min-imum virtuality of the intermediate fermion is q = M W ,and the mean virtuality for s ≫ M W is greater again bythe factor 2 ln( s/M W ). With no infra-red or collinear sin-gularities for W/Z -emission, an unsuppressed amplituderesults already at order M − η .Before looking at an explicit example in which elec-troweak bremsstrahlung is seen to lift a suppression, wepause to summarize some important facts for the 2 → • Fierz transformation is used to re-express t - and u -channel amplitudes of the form ( χ ˆΓ A l )(¯ l ˆΓ B χ ) as asum of s -channel (not to be confused with s -wave)amplitudes of the form ( χ ˆΓ C χ )(¯ l ˆΓ D l ). • For Majorana dark matter, only S , P , and A s -channel bilinears are allowed, with the V and T bilinears forbidden by the self-conjugate propertiesof Majorana particles. • Considering the product of an s -channel χ -currentwith an s -channel fermion-current, we find that thepseudo-scalar is the only member of the set ( S , P , A ) which is unsuppressed. The other combi-nations are either helicity ( m ℓ /m χ ) or velocity ( v )suppressed. • The annihilation process χχ → ℓℓ via t - and u -channel exchange of a scalar is suppressed. Im-portantly, electroweak bremsstrahlung lifts thissuppression at lowest order in the propagatormass-squared ( M − η in amplitude), whereas photonbremsstrahlung lifts the suppression at the next or-der ( M − η in amplitude).Amplification of the latter remark is the purpose ofthis paper. IV. EXPLICIT CALCULATION OFSUPPRESSION-LIFTING WITHELECTROWEAK BREMSSTRAHLUNG
To explicitly demonstrate that emission of a W ± or Z boson does lift helicity suppression, we calculate thecross section for χχ → e ∓ ( − ) ν W ± below in the leptophilicmodel of Ref [32]. The interaction term for this model isthat given above in Eq. (6). A. Example of Helicity-Suppressed Rate
In the model of ref. [32], the cross section for the 2 → χχ → e + e − or ν ¯ ν with Majorana DM is given as v σ = f v r π M χ (1 − r + 2 r ) , (8)where m l ≃ M η ± = M η have been assumed, and r = M χ / ( M η + M χ ). The suppressions discussed in Sec-tion II are apparent in Eq. (8). The helicity suppressed s -wave term is absent in the m l = 0 limit, and thus onlythe v -suppressed term remains.This 2 → t -channel exchange of η andthe associated u -channel exchange obtained by crossingthe Majorana particles. The relative sign between thegraphs is negative, due to the fermion exchange. Sum-ming and squaring, one has three terms including the interference term. Alternatively, one may Fierz trans-form the fermion bilinears in the two contributing am-plitudes. The relative minus sign is compensated by thespecial Majorana minus sign described in Eq. (B2). Ref-erence to Eq. (4) then shows that one gets ( P L ) [ P R ] → ( P L γ µ ] [ P R γ µ ) ×
2, where the final factor of 2 counts thetwo contributing amplitudes, which are identical in thefour-fermi limit M η ≫ t and u . We are left with just oneamplitude, f M η [ v ( k )( γ ) v ( p )] [ u ( p ) P L γ µ v ( p )]. Thesurviving Dirac structure for the Majorana current ispure axial vector, since the vector (and tensor) part of aMajorana current vanishes. With just a single productof bilinears, the remaining part of the 2 → v σ = f M χ π M η (cid:20) m l s + 23 v + O ( v ) (cid:21) , (9)in agreement with the four-fermi, m ℓ = 0 limit of Eq. (8).Here, the helicity suppression of the s -wave amplitude,proportional to a helicity flip, in turn proportional to amass insertion, is manifest. B. W Emission and Unsuppressed S -wave We now turn to the calculation of the cross sectionfor the process χχ → e + νW − (equal to that for χχ → e − ¯ νW + ). The four contributing Feynman diagrams areshown in Fig. 1. Note that we consider bremsstrahlungonly from the final state particles (FSR), and neglectemission from the virtual scalar (VIB). Strictly speak-ing, the distinction between FSR and VIB is somewhatartificial in the sense that the partition depends uponthe choice of gauge. However, we shall work in unitarygauge, in which emission from the internal line is sup-pressed by a further power of M η due to the additionalscalar propagator; consequently, we expect our results tobe valid to order M − η in amplitude, i.e. order M − η inrate.We retain the assumptions m l ≃ M η ± = M η .The matrix element for the top-left diagram is M A = igf √ q t − M η (cid:16) ¯ v ( k ) P L v ( p ) (cid:17) × (cid:16) ¯ u ( p ) γ µ P L /q u ( k ) (cid:17) ǫ Qµ . (10)where i g √ γ µ P L is the coupling at the ℓνW vertex, and t , t , u , u are the standard Mandelstam variables, t = ( k − q ) = ( p − k ) t = ( k − p ) = ( − q − k ) u = ( k − q ) = ( p − k ) u = ( k − p ) = ( − q − k ) . (11)Upon applying Eq (7) to Fierz transform the matrix el-ement, we obtain M A = igf √ q t − M η ǫ Qµ h(cid:16) ¯ v ( k ) u ( k ) (cid:17)(cid:16) ¯ u ( p ) P L γ µ P L /q v ( p ) (cid:17) + (cid:16) ¯ v ( k ) γ u ( k ) (cid:17)(cid:16) ¯ u ( p ) P L γ γ µ P L /q v ( p ) (cid:17) + (cid:16) ¯ v ( k ) γ γ α u ( k ) (cid:17)(cid:16) ¯ u ( p ) γ α γ µ P L /q v ( p ) (cid:17)i = igf √ q t − M η ǫ Qµ × (cid:16) ¯ v ( k ) γ γ α u ( k ) (cid:17)(cid:16) ¯ u ( p ) P L γ α γ µ /q v ( p ) (cid:17) . (12)The first two terms after the first equality are zero dueto the helicity projection operators, leaving only an axialvector term. (Vector and tensor χ -bilinears have beenomitted, as they will cancel between u and t channel di-agrams in the heavy M η limit, as discussed above.) Notethat although this matrix element resembles that of an s -channel annihilation process, the γ matrices in the lepton bilinear would be in a different order for a true s -channelannihilation process involving W/Z -bremsstrahlung fromone of the final state leptons.Similarly, the matrix element for the top-right diagramcan be written as M B = − igf √ q u − M η (cid:16) ¯ v ( k ) γ γ α u ( k ) (cid:17) × (cid:16) ¯ u ( p ) P L γ α γ µ /q v ( p ) (cid:17) ǫ Qµ , (13)and those for the bottom diagrams, M C = − igf √ q t − M η (cid:16) ¯ v ( k ) γ γ α u ( k ) (cid:17) × (cid:16) ¯ u ( p ) P L /q γ µ γ α v ( p ) (cid:17) ǫ Qµ , (14) M D = igf √ q u − M η (cid:16) ¯ v ( k ) γ γ α u ( k ) (cid:17) × (cid:16) ¯ u ( p ) P L /q γ µ γ α v ( p ) (cid:17) ǫ Qµ . (15)Performing the sum over spins and polarizations, we find X spin, pol. |M| = X spin, pol. | ( M A + M C ) − ( M B + M D ) | = (cid:18) gf √ (cid:19)
116 Tr [( /k + M χ ) γ α ( /k + M χ ) γ β ] (cid:18) g µν − Q µ Q ν M W (cid:19) q (cid:18) t − M η + 1 u − M η (cid:19) Tr h /p γ α γ µ /q /p /q γ ν γ β P R i − q q (cid:18) t − M η + 1 u − M η (cid:19) (cid:18) t − M η + 1 u − M η (cid:19) (cid:18) Tr h /p γ α γ µ /q /p γ β γ ν /q P R i + Tr h /p /q γ µ γ α /p /q γ ν γ β P R i(cid:19) + 1 q (cid:18) t − M η + 1 u − M η (cid:19) Tr h /p /q γ µ γ α /p γ β γ ν /q P R i! (16)We evaluate this in terms of scalar products using thestandard Dirac Algebra, leading to a result too lengthyto record here.The thermally-averaged rate is given by v dσ = 12 s Z X spin, pol. |M| dLips (17)where the arises from averaging over the spins of theinitial χ pair, and v = q − M χ s is the mean dark matterrelative velocity, as well as the dark matter single-particlevelocity in the center of mass frame . Informative discussions of the meaning of v are given in [36], and,including thermal averaging, in [37]. The three-body Lorentz Invariant Phase Space is dLips = (2 π ) d ~p E d ~p E d ~Q E W δ ( P − p − p − Q )(2 π ) (18)and P = k + k . This factorizes into the product oftwo two-body phase space integrals, convolved with anintegral over the fermion propagator momentum, dLips = R sM W dq π (cid:18) d ~q E q d ~p E δ ( P − q − p )(2 π ) (cid:19) × d ~p E d ~Q E W δ ( q − Q − p )(2 π ) ! = R sM W dq π dLips ( P , q , p ) dLips ( q , Q , p ) . (19)Evaluating the two-body phase space factors in their re-spective center of momentum frames, and using p =0 - M Χ H TeV L R FIG. 2. The ratio R = v σ ( χχ → e + νW − ) /v σ ( χχ → e + e − )for the example model [32], with M η ≫ M χ . We have used v = 10 − c , appropriate for the Galactic halo. x W = E W (cid:144) M Χ d N (cid:144) d x W FIG. 3. The W spectrum per χχ → eνW event for the ex-ample model, with M χ = 300 GeV and M η ≫ M χ . p = 0, we have dLips ( x , y ,
0) = x − y πx d ¯Ω4 π . (20)This allow us to write the three-body phase space as dLips = 12 (2 π ) Z sM W dq (21) × ( s − q )( q − Q ) sq dφ d cos θ P d cos θ q , where φ is the angle of intersection of the plane definedby χχ → ee ∗ with that defined by eνW , and θ P and θ q are defined in P (CoM) and q rest frames respectively.We evaluate the scalar products that arise from Eq.(16)in terms of the invariants q , Q = M W , s , t , and u ,and the angles θ P , θ q , and φ . We then use Eq. (17) toevaluate the cross section. As we have neglected dia-grams suppressed by M − η relative to those in Fig. 1, wepresent our results to leading order in M − η (i.e., we take M η ≫ t , t , u , u ). To leading order in powers of M χ and M W in the numerator, we find v σ = g f M W M η π ( M χ
13 ln " M χ M W − ! + M χ M W ln " M χ M W (cid:20) M W M χ M W + 4 M χ (cid:21)(cid:27) − " M χ M W + 4 M χ − Li (cid:20) M W M W + 4 M χ (cid:21) ! + O ( M W ) ) . (22)The Spence function (or “dilogarithm”) is defined asLi ( z ) ≡ − R z dζζ ln | − ζ | = P ∞ k =1 z k k . The full expres-sion (retaining sub-leading terms in M χ in the numer-ator) is specified in Appendix D. Clearly, the leadingterms are neither helicity nor velocity suppressed.The effectiveness of the W -strahlung processes in lift-ing suppression of the annihilation rate can be seenFig. 2, where we plot the ratio of the W -strahlungcross section to that of the lowest order process, R W = v σ ( χχ → e + νW − ) /v σ ( χχ → e + e − ). We see that the W -strahlung rate rises with DM mass, to quickly dom-inate over the lowest order annihilation process. The W -bremsstrahlung rate rises approximately as M χ . As M χ increases, eventually phase space allows multi- W/Z radiative production, with such a large rate that resum-mation techniques become necessary. The onset of multi-
W/Z dominance has been discussed in [6–8].To obtain the energy spectrum of the W , we computethe differential cross section in terms of E W by makingthe transformation d cos( θ q ) → − √ sq ( s − q )( q − M W ) dE W . (23)We find [38], again to leading order M − η , v dσdE W = g f E W M W M η π × ( E W q E W − M W (cid:0) M W − E W + 8 E W M χ − M χ (cid:1) + (cid:16) E W − E W M χ + (2 E W − M W ) (cid:0) M χ + M W (cid:1) (cid:17) × ln " E W + p E W − M W E W − p E W − M W . (24)The W spectrum per χχ → eνW event is given in Fig. 3.We use the scaling variable x W ≡ E W /M χ , and plot dN/dx W ≡ ( σ e + νW − ) dσ e + νW − dx W . The kinematic range of x w is [ M W M χ , (1+ M W M χ )], with the lower limit correspondingto a W produced at rest, and the upper limit correspond-ing to parallel lepton momenta balancing the opposite Wmomentum. As evident in Fig. 3, the W boson spectrum1 x l = E l (cid:144) M Χ d N (cid:144) d x l FIG. 4. The primary lepton spectrum per χχ → eνW for theexample model, with M χ = 300 GeV and M η ≫ M χ . x l = E l (cid:144) M Χ d N (cid:144) d x l FIG. 5. The secondary lepton spectrum (i.e., from W → ν ℓ ℓ )per W for the example model, with M χ = 300 GeV and M η ≫ M χ . (The branching ratio for W → νl , 11% per flavor, is notincluded here.) has a broad energy distribution, including a significantcomponent at high energy E W ∼ M χ .The energy spectrum of the the primary leptons is cal-culated in similar fashion. We present the analytic resultin Appendix D (along with more detailed expressions for v σ and v dσ/dE W ). Here the range of the scaling vari-able x ℓ ≡ E ℓ /M χ is [ 0 , − M W M χ ]. Both limits arise whenone lepton has zero energy and the other is producedback-to-back with the W . The lepton spectrum is shownin Fig. 4. Note that this lepton spectrum is valid for ei-ther e + or ν from the annihilation χχ → e + νW − , andfor either e − or ¯ ν from the annihilation χχ → e − ¯ νW + .The primary lepton spectrum in Fig. 4 features a sharpcut off near E ℓ = M χ , and a dip in the spectrum that isdue to an absorptive interference effect.To obtain the full lepton spectrum, the contributionsfrom the subsequent decays of the gauge bosons to lep-tons must be included. (The contribution from the lowestorder 2 → χχ → e + e − or ν ¯ ν is negligible. We also neglect final state leptons resulting from µ decay andfrom the τ decay chain. These leptons are softer thanthose we consider.) For leptons from W -decay, the rangeof the scaling variable x ℓ is [ M W M χ , W decay may be calculatedin a simple but approximate way, as we describe in Ap-pendix E leading to Eq. (E8). The resulting secondarylepton spectrum is shown in Fig (5). Unsurprisingly, thespectrum of secondary leptons is softer than the spec-trum of primary leptons.When combining the primary lepton and secondarylepton spectra, the relative weights are model dependent.For example, the primary ℓ -spectrum is weighted by BR ( χχ → W νℓ ) + 2 BR ( χχ → Zℓℓ ), while the secondary ℓ -spectrum is weighted by BR ( χχ → W + X ) × BR ( W → νℓ ) + BR ( χχ → Z + X ) × BR ( Z → ℓℓ ).We note that the final charged-lepton spectra will bymodified by cosmic propagation effects. The injected e ± will suffer rapid energy losses from synchrotron and in-verse Compton processes on the Universe’s backgroundmagnetic and radiation fields (see, e.g., Ref. [39] for arecent analysis). On the other hand, the injected neutri-nos do not interact with the environment, and so theirspectra remain unmodified. C. Unsuppressed Z Emission
Consider the process producing the ¯ ννZ final state.The cross sections for the Z-strahlung processes are re-lated to those for W-strahlung in a simple way: The am-plitudes producing ¯ ννZ arise from the same four graphsof Fig. (1), where e , W and η + are replaced everywhereby ν and Z and η , respectively. The calculation ofthe amplitudes, and their interferences, thus proceedsin an identical fashion. After making the replacement M W → M Z , the cross section for the annihilation pro-cess χχ → ν ¯ νZ differs from that for χχ → e + νW − byonly an overall normalization factor, v σ ν ¯ νZ = 1(2 cos θ W ) × v σ e + νW − (cid:12)(cid:12)(cid:12)(cid:12) M W → M Z ≃ . × v σ e + νW − (cid:12)(cid:12)(cid:12) M W → M Z . (25)Consider now the e + e − Z final state. Again, the am-plitudes arise from the same four basic graphs of Fig. (1).Since only the left-handed leptons couple to the darkmatter via the SU(2) doublet η , only the left handedcomponent of e − participates in the interaction with the Z . Therefore, the couplings of the charged leptons to Z and W take the same form, up to a normalization con-stant. We thus find v σ e + e − Z = 2 (cid:0) sin θ W − (cid:1) cos θ W × v σ e + νW − (cid:12)(cid:12)(cid:12) M W → M Z ≃ . × v σ e + νW − (cid:12)(cid:12)(cid:12) M W → M Z . (26)2 V. CONCLUSIONS
In an attempt to explain recent anomalies in cosmic raydata in a dark matter framework, various non-standardproperties have been invoked such as dominant annihila-tion to leptons in so-called leptophilic models. When thedark matter is Majorana in nature, such annihilationsinvariably are confronted by suppressions of such pro-cesses via either p-wave velocity suppression or helicitysuppression. With the aid of Fierz transformation tech-nology, which we have presented in some detail, we haveelucidated the general circumstances where suppressionsmay be encountered.It has been known for some time that photonbremsstrahlung may have a dramatic effect on such sup-pressions. We have shown that once one considers theinclusion of three body final states due to electroweakbremsstrahlung, one may also lift these suppressions andobtain rates which may be several orders of magnitudebeyond those without such radiative corrections. Infact, barring an unexpected mass-degeneracy, the EW-bremsstrahlung lifts the suppression at one order lowerin a certain small ratio of squared masses than does EM-bremsstrahlung, as explained in the text.Such radiative processes may be lethal for modelsattempting to produce positrons without overproduc-ing antiprotons due to the subsequent hadronizationof the radiated gauge bosons. Given that electroweakbremsstrahlung is the dominant annihilation channel forthe DM models under consideration, and both W and Z decay to hadrons with a branching ratio of approximately70%, a large hadronic component is unavoidable. Impor-tantly in the context of recent cosmic ray data, therewill be sizable antiproton production. We also note thatdark matter searches triggering on anti-deuterons willfind a sample in the W - and Z -bremsstrahlung processes.The Aleph experiment has measured an anti-deuteronproduction rate of 5 . ± . × − anti-deuterons perhadronic decay of the Z [40]. We expect the rate foranti-deuteron production in W -decay to be similar.Even for models which do not suffer a suppression ofthe lowest order process, we see that it is impossibleto have purely leptonic annihilation products, including“leptophilic” models in which the dark matter has directcouplings only to leptons. In a broader context the re-sults presented here show the importance that may beplayed by electroweak bremsstrahlung in future searchesof indirect dark matter detection. For any DM model forwhich electroweak bremsstrahlung makes an importantcontribution to observable fluxes, there will be large, cor-related fluxes of e ± , neutrinos, hadrons and gamma rays.We will explore the detection of these signals in a futurearticle [41]. ACKNOWLEDGEMENTS
We thank Sheldon Campbell, Bhaskar Dutta, SourishDutta, Haim Goldberg, Lawrence Krauss, Danny Mar-fatia, Yudi Santoso and Nick Setzer for helpful discus-sions. NFB was supported by the Australian ResearchCouncil, TDJ was supported by the Commonwealth ofAustralia, and TJW and JBD were supported in partby U.S. DoE grant DE–FG05–85ER40226. TJW ben-efited from a AvHumboldt Senior Research Award andhospitality at MPIK (Heidelberg), MPIH (Munich), U.Dortmund and the Aspen Center for Physics.
Appendix A: Fundamentals of Fierzing
In this paper we have made use of standard Fierz trans-formations, helicity-basis Fierz transformations, and gen-eralizations of the two. In this Appendix, we derive thesetransformations. The procedure for standard Fierz trans-formation can be found in, e.g., [26], while more generalFierz transformations are laid down in [25]. The startingpoint is to define a basis { Γ B } and a dual basis { Γ B } ,each spanning 4 × C , such that an orthogonality relation holds. Thestandard Fierz transformation uses the “hermitian” bases { Γ B } = { , iγ , γ µ , γ γ µ , σ µν } , and { Γ B } = { , ( − iγ ) , γ µ , ( − γ γ µ ) , σ µν } , (A1)respectively. Because of their Lorentz and parity trans-formation properties, these basis matrices and their dualsare often labeled as S and ˜ S (scalars), P and ˜ P (pseu-doscalars), V and ˜ V (vectors, four for V , four for ˜ V ), A and ˜ A (axial vector, four for A , four for ˜ A ), and T and ˜ T (antisymmetric tensor, six for T , six for ˜ T ). Asusual, spacetime indices are lowered with the Minkowskimetric, γ = γ = iγ γ γ γ , σ µν ≡ i [ γ µ , γ ν ], (and γ σ µν = i ǫ µναβ σ αβ ). Note the change of sign betweenthe the basis and dual for the P and A matrices. Thebases are “hermitian” in that γ Γ † B γ = Γ B , so that theassociated Dirac bilinears satisfy [ ¯Ψ Γ B Ψ ] † = ¯Ψ Γ B Ψ and [ ¯Ψ Γ B Ψ ] † = ¯Ψ Γ B Ψ . Importantly, we have Γ B =(Γ B ) − in the sense of the accompanying orthogonalityrelation T r [Γ C Γ B ] = 4 δ BC , B, C = 1 , . . . , . (A2)Note that the factor of in the definition of ˜ T = σ µν (but not in T = σ µν ) provides the normalization requiredby Eq. (A2): T r [Γ B Γ B ] (nosum) = X C T r [Γ C Γ B ] = 4 . (A3)The orthogonality relation allows us to expand any 4 ×
34 complex matrix X in terms of the basis asX = X B Γ B = X B Γ B , withX B = 14 T r [ X Γ B ] , and X B = 14 T r [ X Γ B ] , i . e ., X = 14
T r [X Γ B ] Γ B = 14 T r [X Γ B ] Γ B . (A4)One readily finds that the particular matrix element(X) ab satisfies(X) cd δ db δ ac = 14 [(X) cd (Γ B ) dc ] (Γ B ) ab . (A5)Since each element (X) cd is arbitrary, Eq. (A5) is equiv-alent to a completeness relation(
1) [
1] = 14 (Γ B ] [Γ B ) = 14 (Γ B ] [Γ B ) , (A6)where we have adopted Takahashi’s notation [27] wherematrix indices are replaced by parentheses ( · · · ) andbrackets [ · · · ], in an obvious way. Thus, any 4 ×
1) [ B Y ] [ Γ B )= 14 T r [X Γ B Y Γ C ] (Γ C ] [Γ B ) . (A7)This equation is presented as Eq. (7) in the main text.Alternatively, we may express any 4 ×
1) [ Y
1] = 14 (X Γ B ] [ Y Γ B ) (A8)= 14 T r [X Γ B Γ C ] T r [Y Γ B Γ D ] (Γ C ] [Γ D ) . The RHS’s of Eqs. (A7) and (A8) offer two useful op-tions for Fierzing matrices. The first option sandwichesboth LHS matrices into one of the two spinor bilinears,and ultimately into a single long trace. The second optionsandwiches each LHS matrix into a separate spinor bilin-ear, and ultimately into separate trace factors. Eq. (A7)seems to be more useful than (A8). One use we will makeof Eq. (A7) will be to express chiral vertices in terms ofFierzed standard vertices. But first we reproduce thestandard Fierz transformation rules by setting X = Γ D and Y = Γ E in Eq. (A7), to wit:(Γ D ) [Γ E ] = 14 T r [Γ D Γ B Γ E Γ C ] (Γ C ] [Γ B ) . (A9)(An additional minus sign arises if the matrices are sand-wiched between anticommuting field operators, ratherthan between Dirac spinors.) Evaluation of the tracein Eq. (A9) for the various choices of ( B, C ) leads to theoft-quoted result [26] ( S ) [ ˜ S ]( V ) [ ˜ V ]( T ) [ ˜ T ]( A ) [ ˜ A ]( P ) [ ˜ P ] = 14 − −
46 0 − − − − − ( S ] [ ˜ S )( V ] [ ˜ V )( T ] [ ˜ T )( A ] [ ˜ A )( P ] [ ˜ P ) . (A10) TABLE II. Fierz-invariant combinations in the standard ba-sis. Fierz-invariant combination eigenvalue3 ( S ⊗ ˜ S + P ⊗ ˜ P ) + T ⊗ ˜ T +12 ( S ⊗ ˜ S − P ⊗ ˜ P ) + ( V ⊗ ˜ V + A ⊗ ˜ A ) +1 V ⊗ ˜ V − A ⊗ ˜ A − S ⊗ ˜ S + P ⊗ ˜ P − T ⊗ ˜ T −
12 ( S ⊗ ˜ S − P ⊗ ˜ P ) − ( V ⊗ ˜ V + A ⊗ ˜ A ) − More relevant for us, as will be seen, is the ordering
P, S, A, V, T , which leads to a Fierz matrix obtainedfrom the one above with the swapping of matrix indices1 → → → → →
1. The result is ( P ) [ ˜ P ]( S ) [ ˜ S ]( A ) [ ˜ A ]( V ) [ ˜ V ]( T ) [ ˜ T ] = 14 − − − − − − − ( P ] [ ˜ P )( S ] [ ˜ S )( A ] [ ˜ A )( V ] [ ˜ V )( T ] [ ˜ T ) . (A11)(The zeroes make it clear that Fierzing induces no cou-pling between tensor interactions and vector and axialvector interactions.) As an example of how to read thismatrix,( A ) [ ˜ A ] = − ( P ] [ ˜ P ) + ( S ] [ ˜ S ) −
12 ( A ] [ ˜ A ) + 12 ( V ] [ ˜ V ) , (A12)or, multiplying by spinors and giving the explicit formsof the gamma-matrices,( uγ γ µ u ) ( v ( − γ γ µ ) v )= − ( uiγ v ) ( v ( − iγ ) u ) + ( uv ) ( vu ) (A13) −
12 ( uγ γ µ v ) ( v ( − γ γ µ ) u ) + 12 ( uγ µ v ) ( vγ µ u ) . The Fierz matrix M for the standard basis is nonsin-gular, and hence has five nonzero eigenvalues λ j . Sincetwo swaps of Dirac indices returns the indices to theiroriginal order, the matrix is idempotent, with M = M − = M . Accordingly, the five eigen-values satisfy λ j = 1, so individual eigenvalues must be λ j = ±
1. Also, the corresponding eigenvectors are in-variant under the interchange of two Dirac indices. InTable (II) we list the eigenvalues and “Fierz-invariant”eigenvectors.Helicity projection operators are often present in the-ories where the DM couple to the SU (2) lepton doublet,so it is worth considering Fierz transformations in themore convenient chiral basis.One derivation of chiral Fierz transformations utilizes4the following chiral bases (hatted) [25]: { ˆΓ B } = { P R , P L , P R γ µ , P L γ µ , σ µν } , and { ˆΓ B } = { P R , P L , P L γ µ , P R γ µ , σ µν } , (A14)where P R ≡ (1 + γ ) and P L ≡ (1 − γ ) are the usualhelicity projectors. The orthogonality property betweenthe chiral basis and its dual is T r [ˆΓ C ˆΓ B ] = 2 δ BC , B, C = 1 , . . . , , (A15)which implies the normalization T r [ˆΓ B ˆΓ B ] (no sum) = X C T r [ˆΓ C ˆΓ B ] = 2 . (A16)Notice that because { γ , γ µ } = 0, the dual of P R γ µ is P L γ µ , and the dual of P L γ µ is P R γ µ . Notice also thatthe normalization for the chiral bases necessitates factorsof in both ˆ T = σ µν and ˜ˆ T = σ µν , in contrast to thetensor elements of the standard bases, given in Eq. (A1).In the chiral basis, one is led to a general expansionX = 12 T r [X ˆΓ B ] ˆΓ B = 12 T r [X ˆΓ B ] ˆΓ B , (A17)and to a completeness relation(
1) [
1] = 12 (ˆΓ B ] [ˆΓ B ) = 12 (ˆΓ B ] [ˆΓ B ) . (A18)Thus, any 4 ×
1) [ B Y ] [ ˆΓ B )= 14 T r [X ˆΓ C Y ˆΓ B ] (ˆΓ B ] [ˆΓ C ) . (A19)Substituting X = ˆΓ D and Y = ˆΓ E into Eq. (A19), onegets(ˆΓ D ) [ˆΓ E ] = 14 T r [ˆΓ D ˆΓ C ˆΓ E ˆΓ B ] (ˆΓ B ] [ˆΓ C ) . (A20)Evaluating the trace in Eq. (3) leads to the chiral-basisanalog of (A10) or (A11), presented in Eq. (4) of themain text. As a check, we note that the matrix M in Eq. (4) isidempotent, M =
1, as it must be. The eigenvalues aretherefore ±
1. Eigenvalues and Fierz-invariant eigenvec-tors for the chiral basis are given in Table (III). The finaltwo eigenvectors in the Table simply express again the in-variance of V ± A interactions under Fierz-transpositionof Dirac indices. This invariance is also evident in thediagonal nature of the bottom two rows of the matrixEq. (4).One may instead want the Fierz transformation thattakes chiral bilinears to standard bilinears. Since mod-els are typically formulated in terms of chiral fermions,a projection onto standard s -channel bilinears would be TABLE III. Fierz-invariant combinations in the chiral basis.Fierz-invariant combination eigenvalue3 ( P R ⊗ P R + P L ⊗ P L ) + ˆ T ⊗ ˜ˆ T +12 P R ⊗ P L + P R γ µ ⊗ P L γ µ +12 P L ⊗ P R + P L γ µ ⊗ P R γ µ +1 P R ⊗ P R + P L ⊗ P L − ˆ T ⊗ ˜ˆ T − P R ⊗ P L − P R γ µ ⊗ P L γ µ − P L ⊗ P R − P L γ µ ⊗ P R γ µ − P R γ µ ⊗ P R γ µ − P L γ µ ⊗ P L γ µ − well- suited for a partial wave analysis. Because differ-ent partial waves do not interfere with one another, thecalculation simplifies in terms of s -channel partial waves.Setting X = ˆΓ D and Y = ˆΓ E in Eq. (A7), we readilyget(ˆΓ D ) [ˆΓ E ] = 14 T r [ˆΓ D Γ B ˆΓ E Γ C ] (Γ C ] [Γ B ) . (A21)We (should) get the same result by resolving the RHSvector in Eq. (4) into standard basis matrices. The resultis5 ( P R ) [ P R ]( P L ) [ P L ]( P R γ µ ) [ P L γ µ ]( P L γ µ ) [ P R γ µ ]( ˆ T ) [ ˆ T ]( γ ˆ T ) [ ˆ T ]( P R ) [ P L ]( P L ) [ P R ]( P R γ µ ) [ P R γ µ ]( P L γ µ ) [ P L γ µ ] = 18 − − − − − − − − − −
10 0 0 0 0 0 1 − − − − − −
20 0 0 0 0 0 − − (
1] [ γ ] [ γ )( γ ] [
1] [ γ )( T ] [ ˜ T )( γ T ] [ ˜ T )( γ µ ] [ γ µ )( γ γ µ ] [ γ γ µ )( γ γ µ ] [ γ µ )( γ µ ] [ γ γ µ ) (A22)All relations are invariant under the simultaneous inter-changes P R ↔ P L and γ → − γ . The matrix in (A22),relating two different bases, is not idempotent. In fact,it is singular. Appendix B: Cancellation of Vector and TensorAmplitudes for Majorana Fermions
Majorana particles are invariants under charge conju-gation C . Accordingly, the Majorana field creates andannihilates the same particle. This implies that foreach t -channel diagram, there is an accompanying u -channel diagram, obtained by interchanging the momen-tum and spin of the two Majorana fermions. The rela-tive sign between the t - and u -channel amplitudes is − s -channel) bilinear for χ -annihilation:¯ v ( k , s )Γ B u ( k , s ). The associated Fierzed bilinearfrom the ( k ↔ k )-exchange graph, with its relativeminus sign, is − ¯ v ( k , s )Γ B u ( k , s ). Constraints relat-ing the four-component Dirac spinors to their underly-ing two-component Majorana spinors must be imposed.These constraints, any one of which implies the otherthree, are u ( p, s ) = C ¯ v T ( p, s ) , ¯ u ( p, s ) = − v T ( p, s ) C − ,v ( p, s ) = C ¯ u T ( p, s ) , ¯ v ( p, s ) = − u T ( p, s ) C − . (B1)Here, C is the charge conjugation matrix. These Ma-jorana conditions on the spinors allow us to rewrite theexchange bilinear as (suppressing spin labels for brevityof notation) − ¯ v ( k )Γ B u ( k )) = u T ( k ) C − Γ B C ¯ v T ( k )= (cid:2) ¯ v ( k )( C − Γ B C ) T u ( k ) (cid:3) T = ¯ v ( k )( η B Γ B ) u ( k ) . (B2)For the final equality, we have used (i) the fact that thetranspose symbol can be dropped from a number, and (ii)the identity ( C − Γ B C ) T = ( η B (Γ B ) T ) T = η B Γ B , where η B = +1 for Γ = scalar, pseudoscalar, axial vector, and η B = − t - and u -channel propaga-tors can be ignored, one obtains an elegant simplification.Subtracting the u-channel amplitude from the t-channelamplitude, one arrives at the weighting factor (1 + η B ),which is two for S, P, and A couplings, and zero for V and T couplings. Thus, we must drop V and T couplingsappearing in the Fierzed bilinears of the χ -current. Whatthis means for the model under discussion is that afterFierzing, only the axial vector coupling of the χ -currentremains, and the factor of 1 + η A = 2 is multiplied by the(7-8)-element = in the Fierz matrix of Eq. (4) to givea net weight of 1. Appendix C: Non-Relativistic andExtreme-Relativistic Limits of Fermion Bilinears
We work in the chiral representation of the Dirac alge-bra, and we follow the notation of [28]. Accordingly, γ = ! , ~γ = ~σ − ~σ ! , γ = − ! . (C1)The rest-frame four-spinor is u ( ~p = 0) = √ M ξξ ! , (C2)where ξ is a two-dimensional spinor. The spinor witharbitrary momentum is obtained by boosting. One gets u ( p ) = √ p · σ ξ √ p · σ ξ , (C3)where σ ≡ (1 , ~σ ) and σ ≡ (1 , − ~σ ).In a standard fashion, we choose the up and down spineigenstates of σ as the basis for the two-spinors. Thesebasis two-spinors are ξ + ≡ ! , ξ − ≡ ! . (C4)6In terms of the chosen basis, we have for the NR u -spinors, u ± NR −→ √ M ξ ± ξ ± ! . (C5)We get the ER limit of the u -spinors from Eq. (C3). Aftera bit of algebra, one finds u + ER −→ √ E ξ + , u − ER −→ √ E ξ − . (C6)The arbitrary v -spinor is given by v ( p ) = √ p · σ η −√ p · σ η . (C7)In the Dirac bilinear the two-spinor η is independent ofthe two-spinor xi , and so it is given an independent name, η . However, the basis η ± remains ξ ± as defined above.It is the minus sign in the lower components of v relativeto the upper components that distinguishes v in eq. (C7)from u in eq. (C3) in a fundamental way. After a smallamount of algebra, one finds the limits v ± NR −→ v ± ( ~p = 0) = √ M η ± − η ± ! , (C8)and v + ER −→ √ E − η + , v − ER −→ √ E η − . (C9)Finally, we apply the above to determine the values ofDirac bilinears in the NR and ER limits. The ¯ u ≡ u † γ and ¯ v ≡ v † γ conjugate spinors are are easily found from the u and v spinors. We let Γ denote any of the hermitianbasis Dirac-matrices { , i γ , γ µ , γ γ µ , σ µν } . Then, theNR limit of ¯ u ( p ) Γ v ( p ) is just¯ u ( p ) Γ v ( p ) NR −→ M " ( ξ , ξ ) Γ η − η ! . (C10)Non-relativistic results for the various choices of basis Γ’sand spin combinations are listed in Table I of the text.To give a succinct formula for the ER limit of¯ u ( p ) Γ v ( p ), we take ˆ p = − ˆ p = ˆ3, i.e. we work ina frame where ˆ p and ˆ p are collinear, and we quantizethe spin along this collinear axis. The result is¯ u ( p ) Γ v ( p ) ER −→ p E E " ξ (Λ + , Λ − ) Γ Λ + − Λ − ! η , (C11)where the matrices Λ ± are just up and down spin pro-jectors along the quantization axis ˆ3: λ + = ! , Λ − = ! . (C12)Extreme-relativistic results for the various choices of ba-sis Γ’s and spin combinations are listed in Table I of thetext. Appendix D: Full Cross Section Results
We present here the full results of the cross sectioncalculations for the process χχ → e ∓ ( − ) ν W ± , includingterms of all orders in M χ . In Section IV we presentedonly the leading order terms, which dominate in the large M χ limit. For M χ not too much heavier than M W , it isimportant to retain sub-leading terms.The total cross section for χχ → e ∓ ( − ) ν W ± is given by v σ e + νW − = g f M W M η π ( (cid:18) M W M χ − M W M χ + 4 M W − M W M χ − M χ (cid:19) + ln " M χ M W M W M χ + 43 M W M χ − M W + 16 M W M χ + 163 M χ + 8 M W (cid:0) M W + 2 M χ (cid:1) ln (cid:20) M W M χ M W + 4 M χ (cid:21) (cid:19) + 8 M W (cid:0) M W + 2 M χ (cid:1) Li " M χ M W + 4 M χ − Li (cid:20) M W M W + 4 M χ (cid:21)! + O ( v , M − η , m ℓ ) ) . (D1)7The W energy spectrum is v dσ e + νW − dE W = g f E W M W M η π ( E W q E W − M W (cid:0) M W − E W + 8 E W M χ − M χ (cid:1) (D2)+ (cid:16) E W − E W M χ + (2 E W − M W ) (cid:0) M χ + M W (cid:1) (cid:17) ln " E W + p E W − M W E W − p E W − M W + O ( v , M − η , m ℓ ) ) , while the lepton spectrum (for either the charged lepton or the neutrino) is v dσ e + νW − dE ℓ = g f M η ( M χ − E e ) π ( E e (cid:0) M χ − E e ) M χ − M W (cid:1) M W M χ ( M W + 4 E e M χ ) ( M χ − E e ) × (cid:18) × E e M W + 2 E e M χ (cid:18) M W − M χ (cid:19) − E e M χ (cid:18) M W + 1463 M χ M W − M χ (cid:19) − E e M χ (cid:0) M W − M W M χ − M χ M W + 2704 M χ (cid:1) + E e (cid:18) − M W + 1643 M W M χ − M W M χ − M χ M W + 191 × M χ (cid:19) + 8 E e M χ (cid:18) M W − M W M χ + 116 M W M χ + 13763 M χ M W − M χ (cid:19) − E e M χ (cid:0) M W M χ − M W M χ + 400 M W M χ + 960 M χ M W − M χ (cid:1) + 2 M χ (cid:0) M W + 2 M χ (cid:1) (cid:19) + 2 (cid:18) M χ + M W − E e ( M χ − E e ) M W (cid:19) ln (cid:20) M W M χ ( M χ − E e ) ( M W + 4 E e M χ ) (cid:21) ) + O ( v , M − η , m ℓ ) . (D3) Appendix E: Approximate Spectrum for Boosted W Decay Products
If any possible polarization of the produced W is ne-glected, then a simple calculation results for the spectraof the finals state particles from W decay. The lab framespectra of the decay product (of type or “flavor” F ) de-pends on a one-dimensional convolution of the isotropicspectrum in the W rest frame (RF energy E ′ ), dN F dE ′ F , , withthe W spectrum in the lab frame, dNdE W . We now developthis convolution.Given the energy distribution dN W /dγ of produced W ’s (with γ = E W /M W ), and the energy distribution dN F /dE ′ F of decay particle F in the W rest frame,normalized to the multiplicity of F per W decay (i.e.,there is a branching ratio W → F multiplier implicit in dN F /dE ′ F ) and assumed to be isotropic, one gets thespectrum dN F /dE F of particle F in the lab via: dN F ( E ) dE = Z − d cos θ ′ Z dγ dN W dγ (E1) × Z dE ′ dN F dE ′ δ ( E − [ γE ′ + βγp ′ cos θ ′ ]) , If the W polarization is not neglected, then the W decay am-plitude includes Wigner functions d µ i µ f ( θ ), which introduce alinear cos θ or sin θ term into Eq. (E1). with p ′ = p E ′ − m F , βγ = p γ −
1. The cos θ ′ inte-gral is easily done, and one gets dN F ( E ) dE = 12 Z ∞ dγ p γ − dN W dγ Z E ′ + E ′− dE ′ p ′ dN F dE ′ , (E2)with E ′± = γE ± βγp . Equivalently, we get dN F ( E ) dE = 12 Z ∞ m F dE ′ p ′ dN F dE ′ , Z γ + γ − dγ p γ − dN W dγ , (E3)with γ ± = ( EE ′ ± pp ′ ) /m F and p = p E − m F . Thisformulation neglects interferences between identical par-ticles produced in both the primary and secondary chan-nels, if any.As given, Eq. (E3) applies to any particle type in the W ’s final state. For example, it could be used to cal-culate the antiproton or antineutron spectrum from W production and decay, if the fragmentation functions for W → ¯ p or ¯ n , i.e. f ( x ¯ B ≡ E ¯ B /M W ) were input.Here we perform a the convolution for the especiallysimple case of W decay to two massless particles, say ν e and e . For massless leptons, we have dN ν dE ′ = dN e dE ′ = BR ( W → ν e) δ ( E ′ − M W ) , (E4)with γ + = ( E W /M W ) max = ( s + M W ) / √ sM W ≈ (4 M χ + M W ) / M χ M W , and γ − = (4 E + M W ) / EM W .8The spectrum in the lab is given by Eq. (E3) becomesjust dN ν dE = dN e dE = ( BR ) M W Z γ + γ − dγ p γ − dN W dγ . (E5)The W -spectrum shown in Fig. (3) is approximately halfof an ellipse, suggesting the fit ln (cid:16) dNdx W (cid:17) − ln 0 . . − ln 0 . + (cid:18) x W − . .
50 (1 . − . (cid:19) = 1 , (E6)valid for 0 . . x W . .
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