Relativistic and nonrelativistic annihilation of dark matter: a sanity check using an effective field theory approach
aa r X i v : . [ h e p - ph ] M a r Relativistic and nonrelativistic annihilation of dark matter: a sanity checkusing an effective field theory approach
Mirco Cannoni
Departamento de F´ısica Aplicada, Facultad de Ciencias Experimentales, Universidad de Huelva, 21071 Huelva, Spain
We find an exact formula for the thermally averaged cross section times the relative velocity h σv rel i with relativistic Maxwell-Boltzmann statistics. The formula is valid in the effective fieldtheory approach when the masses of the annihilation products can be neglected compared with thedark matter mass and cut-off scale. The expansion at x = m/T ≫ h σv rel i which is usually used to compute the relic abundance for heavy particles that decouplewhen they are nonrelativistic. We compare this expansion with the one obtained by expanding thetotal cross section σ ( s ) in powers of the nonrelativistic relative velocity v r . We show the correctinvariant procedure that gives the nonrelativistic average h σ nr v r i nr coinciding with the large x expansion of h σv rel i in the comoving frame. We explicitly formulate flux, cross section, thermalaverage, collision integral of the Boltzmann equation in an invariant way using the true relativisticrelative v rel , showing the uselessness of the Møller velocity and further elucidating the conceptualand numerical inconsistencies related with its use. I. INTRODUCTION
While there are compelling evidences in astrophysicsand cosmology that most of the mass of the Universe iscomposed by a new form of non baryonic dark matter(DM), there is a lack of evidence for the existence of newphysics at LHC and other particle physics experiments.On the theory side, many specific models with new par-ticles and interactions beyond the standard model havebeen proposed to account for DM.Under these circumstances where no clear indicationsin favour of a particular model are at our disposal, thephenomenology of DM as been studied in a model inde-pendent way using an effective field theory approach, seefor example [1–23].Measurements of the parameters of standard model ofcosmology [24, 25] furnish the present day mass density ofDM, the relic abundance, Ω h ∼ .
11 with an uncertaintyat the level of 1%. Any model that pretends to accountfor DM must reproduce this number, which, on the otherhand, sets strong constraints on the free parameters ofthe model.When the DM particles are weakly interacting massiveparticles that decouple from the primordial plasma at atemperature when they are nonrelativistic, the relativis-tic averaged annihilation rate h σv rel i can be well approx-imated by taking the nonrelativistic average of the firsttwo terms of the expansion of σ in powers of the nonrel-ativistic relative velocity. With v rel we indicate the rel-ativistic relative velocity and with v r the nonrelativisticrelative velocity , as defined in B. To describe collisions ina gas, and in particular in the primordial plasma, the ref-erence frame that matters is the comoving frame (COF)where the observer sees the gas at rest as a whole andthe colliding particles have general velocities v , withoutany further specification of the kinematics.It is thus desirable to formulate cross sections and ratesin a relativistic invariant way, such that all the formulasand nonrelativistic expansions are valid automatically in the COF. Obviously, invariant formulas give the same re-sults in the lab frame (LF), the frame where one massiveparticle is at rest, and in the center of mass frame (CMF)where the total momentum is zero. We will see that thekey for the invariant formulation is v rel .On the contrary, in DM literature [26] instead of v rel it is used the so–called Møller velocity ¯ v , see B. Thatthis is incorrect was already discussed in Ref. [27] butpapers using ¯ v continue to appear. The problem with ¯ v ,which is not the relative velocity, is its non invariant andnonphysical nature, for it can take values larger than c .In this paper we first find an exact formula for h σv rel i as a function of x = m/T calculated with the relativisticMaxwell-Boltzmann statistics. The formula is valid inthe effective field theory framework such that the massesof the annihilation products can be neglected comparedwith the DM and the cut-off scale. For concreteness wework with fermion DM. We find the thermal functionscorresponding to various interactions and in particularthose corresponding to s and p wave scattering in thenonrelativistic limit which is given by the expansion at x ≫
1. This is done in Section II, and A contains somemathematical results needed for the derivation of the ex-act formula and its asymptotic expansions.Then, in Section III, we present the correct invariantmethod for obtaining the same expansion by expandingthe total annihilation cross section σ ( s ) in powers of v r .We then discuss in Section IV the problems with theuse ¯ v , while the numerical impact on the relic abundanceof some incorrect methods employed in literature is eval-uated in Section V.B is preparatory for the whole paper: we remind howrelativistic flux, cross section, rate, collision term of theBoltzmann equation and thermal averaged rate can bedefined in the invariant way in terms of v rel showing theuselessness of the Møller velocity. Figure 1. s and t channel annihilation diagrams reducing tothe effective vertex corresponding to the lagrangian Eq. (1). II. EXACT FORMULA FOR THE THERMALAVERAGE IN THE EFFECTIVE APPROACH
We consider a DM fermion field χ that couples to otherfermion fields ψ through an effective dimension-6 opera-tor of the type L Λ = λ a λ b Λ ( ¯ χ Γ a χ )( ¯ ψ Γ b ψ ) . (1)The DM particles can be of Dirac or Majorana nature andhave mass m , while ψ are the standard model fermionsor new ones. Here λ a,b are dimensionless coupling asso-ciated with the interactions described by combination ofDirac matrices Γ a,b . Λ is the energy scale below whichthe effective field theory is valid. In the exact theory Λcorresponds to the mass of a heavy scalar or vector bosonmediator that appears in the propagators. The ψ massescan be neglected compared to Λ and m . The exchangeof a heavy mediator with mass Λ may take place in the s -channel and/or in t -channel, as depicted in Figure 1,depending on the specific model. A. Exact formula for h σv rel i In all generality, for 2 → s and t , and the squared matrix ele-ment is dimensionless. After integrating over the CMFangle, for example, the only remaining dependence is on s and m . Any amplitude related to the operator (1) givesan integrated squared matrix element |M| summed overthe final spins and averaged over the initial spins that isa simple polynomial of the type w = Z |M| d cos θ = p s + p m s + p m , (2)with p , ..., p depending on Λ and λ a,b . To get the for-mula for h σv rel i in a useful form, it is convenient to definethe reduced cross section σ = 12 m π w, (3)and the effective cross section σ Λ = λ a λ b π m Λ , (4)which contains all the couplings. In terms of the effectivecross section (4), and of the dimensionless variable y = s/ (4 m ), the reduced cross section Eq. (3) becomes σ = σ Λ (cid:16) a y + a y + a (cid:17) , (5)where now a , ..., a are pure numbers. The total unpo-larized cross section then is σ = 2 m s p λ ( s, m , m ) p λ ( s, m , m ) σ . (6)We now set m = m = m , m = m = 0 inEq. (6) and in Eq. (B20), and change variable to y . ThusEq. (B20) becomes h σv rel i = 2 xK ( x ) Z ∞ dy p y − K (2 x √ y ) σ ( y ) . (7)Using the integrals of A, we find h σv rel i = σ Λ
116 [8 a + 2 a + (5 a + 2 a + a ) K ( x ) K ( x )+ 3 a K ( x ) K ( x ) ] . (8)In the case m = m = 0 the pure mass terms do notappear in the cross sections, thus a = 0. Furthermore,we can relate a and a each other by an appropriatemultiplicative factor, a = ka , (9)and express the cross sections as a function of a only.The general formula (8) thus finally becomes h σv rel i = σ Λ a F k ( x ) , (10)with F k ( x ) = 116 (cid:18) k + (5 + 2 k ) K ( x ) K ( x ) + 3 K ( x ) K ( x ) (cid:19) (11)the factored out thermal function.The nonrelativistic thermal average is given by theexpansion at x ≫
1. Using the asymptotic expansionsEq. (A2) we find h σ nr v r i nr = σ Λ a (cid:18) k − kx (cid:19) + O ( x − ) . (12)In the ultrarelativistic limit, x ≪
1, using the expan-sions (A3), the thermal functions behave as 3 /x , thus h σv rel i ur ∼ σ Λ a x = λ a λ b π Λ a T , (13)which is the expected result for massless particles.The exact integration is possible because the effec-tive operator removes the momentum dependence in thepropagators that are reduced to a multiplicative constantand the assumption m = m = 0 allows to simplify thesquare root p λ ( s, m , m ) = s in the cross section (6).For example, with m = m = m ψ , equation (7) becomes h σv rel i = 2 xK ( x ) Z ∞ y dy √ y − ρ p y − K (2 x √ y ) σ ( y, ρ ) . with ρ = m ψ /m and y = 1 if m ≥ m ψ , y = ρ if m < m ψ . In this case the exact integration is not possiblebut nonrelativistic expansions exist also in the case ρ = 1and ρ ≫ B. Applications
In order to show the thermal behaviour of differentinteractions, we calculate the cross sections for variousoperators of the type (1), both for s and t channel anni-hilation. We list the quantity ̟ = Λ / ( λ a λ b ) w and theresulting average Eq. (10).For the s -channel annihilation we find:1) Scalar: ( ¯ χχ )( ¯ ψψ ), ( ¯ χχ )( ¯ ψγ ψ ). ̟ = 2 s ( s − m ) , h σ S v rel i = σ Λ F − ( x ) . (14)2) Pseudo-scalar: ( ¯ χγ χ )( ¯ ψγ ψ ), ( ¯ χγ χ )( ¯ ψψ ): ̟ = 2 s , h σ P S v rel i = σ Λ F ( x ) . (15)3) Chiral: ( ¯ χP L,R χ )( ¯ ψP L,R ψ ). ̟ = 12 s ( s − m ) , h σ C v rel i = σ Λ F − ( x ) . (16)4) Pseudo-vector: ( ¯ χγ µ γ χ )( ¯ ψγ µ γ ψ ), ( ¯ χγ µ γ χ )( ¯ ψγ µ ψ ). ̟ = 83 s ( s − m ) , h σ P V v rel i = σ Λ F − ( x ) (17)5) Vector: ( ¯ χγ µ χ )( ¯ ψγ µ ψ ), ( ¯ χγ µ χ )( ¯ ψγ µ γ ψ ). ̟ = 83 s ( s + 2 m ) , h σ V v rel i = σ Λ F ( x ) . (18)6) Vector-chiral: ( ¯ χγ µ P L,R χ )( ¯ ψγ µ P L,R ψ ). ̟ = 83 s ( s − m ) , h σ V C v rel i = σ Λ F − ( x ) . (19)The tensor interaction σ µν gives the same function asthe vector case and is not reported. In the case of a Ma-jorana χ clearly the vector and tensor interactions areabsent, and the inclusion of a factor 1 / t -channel annihila-tion for operators common to Dirac and Majorana DMannihilation: x F k ( x ) k=2k=0k=-1k=-7/4k=-2k=-16/7k=-14/5k=-43/2x3/x Figure 2. The thermal function (11) for the interactions andannihilation cross sections considered in the text.
1) Scalar, pseudo-scalar: ( ¯ χχ )( ¯ ψψ ), ( ¯ χχ )( ¯ ψγ ψ ),( ¯ χγ χ )( ¯ ψγ ψ ), ( ¯ χγ χ )( ¯ ψψ ). ̟ D = 23 s ( s − m ) , h σ D,tS,P S v rel i = σ Λ F − ( x ) . (20) ̟ M = 13 s (5 s − m ) , h σ M,tS,P S v rel i = σ Λ F − ( x ) . (21)2) Chiral: ( ¯ χP L,R χ )( ¯ ψP L,R ψ ). ̟ D = 16 s ( s − m ) , h σ D,tC v rel i = σ Λ F − ( x ) . (22) ̟ M = 13 s ( s − m ) , h σ M,tC v rel i = σ Λ F − ( x ) . (23)3) Pseudo-vector: ( ¯ χγ µ γ χ )( ¯ ψγ µ γ ψ ), ( ¯ χγ µ γ χ )( ¯ ψγ µ ψ ). ̟ D = 43 s (4 s − m ) , h σ D,tP V v rel i = σ Λ F − ( x ) . (24) ̟ M = 83 s (7 s − m ) , h σ M,tP V v rel i = σ Λ F − ( x ) . (25)The thermal functions corresponding to the previouscases are shown in Figure 2 where the asymptotic be-haviours are clearly seen. In particular we note that F ( x ) = 116 (cid:18) K ( x ) K ( x ) + 3 K ( x ) K ( x ) (cid:19) , (26) F − ( x ) = 316 (cid:18) − K ( x ) K ( x ) + K ( x ) K ( x ) (cid:19) , (27)behave in the nonrelativistic limit as F ( x ) ∼ O ( x − ) , F − ( x ) ∼ x + O ( x − ) . The function F ( x ), which appears in the s -channel an-nihilation through a pseudoscalar interaction, is the onlycase where the term of order O ( x − ) is absent, while F − ( x ), which appears in the scalar and axial-vector s -channel annihilation and in the chiral t -channel Majoranafermion annihilation, is the only case where the constant O ( x ) term is zero. These are the exact temperature de-pendent factors that correspond to the phenomenologicalinterpolating functions proposed in Ref. [28] to model the s -wave and p -wave behaviour in the nonrelativistic limit.For all other interactions both s -wave and p -wave contri-bution are present. The function F − ( x ) can be also readoff from the formulas of Ref. [29] where the t -channel an-nihilation of Majorana fermions with the exchange of ascalar with chiral couplings was considered.We note that although we have concentrated on thecase of fermion DM, the formula is valid for DM scalarand vector candidates as well, with the necessary redefi-nition of σ Λ . III. EXPANSION OF THE CROSS SECTION INPOWERS OF THE RELATIVE VELOCITY
In the general case m = m = m ψ = 0 the exact in-tegration is not possible. If the relative velocity of theannihilating particles is small compared with the veloc-ity of light we can work directly with nonrelativistic for-mulas. The exothermic annihilation cross section in thenonrelativistic limit, to the lowest orders in v r , is usuallyexpanded as σ nr ∼ a/v r + bv r , and multiplying by v r , σ nr v r ∼ a + bv r . (28)Then, using Eq. (B18) and (B19), the nonrelativisticthermal average of Eq. (28) is h σ nr v r i nr ∼ a + 6 bx . (29)In the case of our cross sections, comparing Eq. (29)with Eq. (12), the coefficients are thus a = σ Λ a (cid:18) k (cid:19) , b = − σ Λ a k . (30)We now ask, given σ ( s ), how to perform the expan-sion in terms of the relative velocity to find the coeffi-cients a and b that correspond to the large x expansion This result must coincide with the expansion of Ref. [27, 30].With our notation the expansion is h σ nr v r i nr ∼ σ | y =1 + 3 x (cid:18) − σ | y =1 + 12 σ ′ | y =1 (cid:19) , where the prime indicate derivative respect to the variable y .Comparison with the expansion (29) requires to identify a ≡ σ | y =1 , b ≡ (cid:18) σ ′ | y =1 − σ | y =1 (cid:19) . Using Eq. (5) with a = 0 and a = ka , it is easy to verify thatone obtains again Eq. (12). of the relativistic thermal average in the COF. Combin-ing equations (5), (6), (9), the general total annihilationcross section reads σ = σ Λ a √ s √ s − m (cid:18) s m + k (cid:19) . (31)The correct way to proceed is to use the invariant relationEq. (B5) with m = m = m and to solve it for s as afunction of v rel : s = 2 m p − v ! . (32)This formula is valid in every frame and substituted inEq. (31) gives the exact dependence of the cross sec-tion on the relativistic relative velocity, σ ( v rel ). Then,if v rel ∼ v r ≪
1, we can expand the obtained expressionto the desired order in v r and the nonrelativistic averagetaken using Eq. (B19) will coincide with the expansionof Eq. (10) for x ≫
1, that is the expansion (12).Equivalently, in order to find the expansion (28), wenote that the squared roots in the annihilation cross sec-tion (31) imply that a term of order v r in s will contributeto the order v r in σ . Thus we need to expand s , formula(32), at least to order v r , s ∼ m + m v r + 34 m v r . (33)Substituting Eq. (33) in Eq. (31) and performing the ex-pansion in powers of v r it easy to find σ nr v r ∼ σ Λ a (cid:18) k − k v r (cid:19) , (34)in agreement with (30).In the case of coannihilations [31], for example whena DM particles scatter off another particle with differentmass, the Mandelstam invariant takes the form s = ( m − m ) + 2 m m p − v ! , (35)with the expansion s ∼ ( m − m ) + m m v r + 34 m m v r . (36)This procedure gives the correct expansion in the COFwhere the velocities v , of the colliding particles arespecified in this frame. Clearly, the same expansion withthe same coefficients is obtained in the LF and in theCMF. IV. THE PROBLEMS WITH THE MØLLERVELOCITY
The simple outlined procedure has not been recognizedin DM literature where, incorrectly, the Møller velocity¯ v , Eq. (B22), instead of v rel is considered. As remindedin B, ¯ v is a non-invariant, non-physical velocity. Theexpression of ¯ v in terms of s is thus different in differentframes and the expansion of σ takes different values indifferent frames.Before discussing the problems with the Møller velocitywe note that if we take the limit m f → s is truncated to the lowest orderin v r , s ∼ m + m v r . (37)If we substitute this in Eq. (31) and expand, we find σ nr v r ∼ σ Λ a (cid:18) k k v r (cid:19) , (38)with an incorrect coefficient b . Clearly the same wrongresult is obtained truncating (33) to order v r , whateverthe frame in which v r is specified, CMF, LF or COF.We now go back to the Møller velocity (B22). Evalu-ated in the CMF taking m = m = m reads¯ v ∗ = 2 √ s ∗ p s ∗ − m . (39)We indicate the quantities evaluated in the CMF with a”*”. By inverting Eq. (39) we find s ∗ = 4 m − ¯ v ∗ . (40)This relation is different from (32) and is often incorrectlyidentified as the relation between s and the relative ve-locity in the CMF, see for example [31], [10]. In facts,the expansion to order O ( v r, ∗ ) reads s ∗ ∼ m + m v r, ∗ + m v r, ∗ . (41)When used in (31), it gives the following nonreltivisticexpansion of the cross section σ nr v r, ∗ ∼ σ Λ a (cid:18) k v r, ∗ (cid:19) , (42)which is different from the correct expansion (34).Other authors, [26] and [20–23], perform the expansionwith the Møller velocity evaluated in the rest frame of oneparticle. Indicating with ” ℓ ” the quantities in this frame,Eq. (B22) becomes¯ v ℓ = √ s ℓ √ s ℓ − m s ℓ − m , (43)and by inverting Eq. (43) we obtain s ℓ = 2 m p − ¯ v ℓ ! . (44) This expression is formally identical to Eq. (32), thuswhen ¯ v ℓ ∼ v r,ℓ and s ℓ is expanded up to the order v r,ℓ we obtain the expansion σ nr v r,ℓ which formally coincideswith Eq. (34), with v r,ℓ in place of v r .It should be clear that this is just a mathematicalcoincidence due to the fact that ¯ v reduces to v rel onlywhen one of the two velocities v , is zero as it is evidentfrom the definitions Eq. (B2) and Eq. (B22). In otherwords, the expansion found in Refs. [20–23] are correctbecause the authors have implicitly used the relative ve-locity, Eq. (B5) and (33).We thus emphasize some common statements found inDM literature and why they do not subsist:1) In the relativistic Boltzmann equation the v in σv is ¯ v and h σv i must be calculated in the LF frame. This is not true, as shown in details in Ref. [27] and inB. Using v rel and recognizing the nonphysical nature of¯ v , one works always with invariant quantities and theconsistency of the relativistic and nonrelativistic formu-las and expansions is obtained in the comoving framewithout any further specification of the kinematics. TheLF, also called Møller frame in Ref. [23], cannot be aprivileged frame for the relic abundances calculation alsobecause for massless particles the rest frame does not ex-ist.2) The Møller velocity coincides with relative velocity ina frame where the velocities are collinear.
This not true because, for example, in the CMF wherethe particles have velocities v ∗ , the Møller velocity is 2 v ∗ while the relative velocity is 2 v ∗ / (1 + v ∗ ). Note that thetrue relative velocity is never superluminal. V. IMPACT ON THE RELIC ABUNDANCE
Only in the case k = − a turns out to be al-ways the same, the coefficient b is different in any othercase. To illustrate the impact of b on the value of the relicabundance we consider the case of the s -channel annihila-tion with vector interaction, Eq. (18), and the s -channelannihilation with a pseudoscalar exchange, Eq. (15). Inthe first case k = 2, a = 8 /
3, and the correct coefficients a and b are a V = 4 σ Λ , b V = − σ Λ , (45)while the incorrect coefficient b in (38) and (42) is b V = 76 σ Λ , b V = 23 σ Λ . (46)In the second case, k = 0 and a = 2, thus a P S = 2 σ Λ , b P S = 0 , (47)and the wrong b coefficients are b P S = 34 σ Λ , b P S = 12 σ Λ . (48)We calculate the relic abundance following the exacttheory of freeze out presented in Ref. [32]. We briefly re-call the main points. Let Y = 45 / (4 π )( g χ /g s ) x K ( x )be the initial equilibrium abundance (number densityover the entropy density), with g χ = 2 for spin 1/2fermions and g s the relativistic degrees of freedom as-sociated with the entropy density. The function Y ( x )that gives the abundance up to the point x ∗ where Y ( x ) − Y ( x ) is maximal is Y ( x ) = (1 + δ ( x )) Y ( x ) , (49) δ ( x ) = s − x C h σv rel i Y Y dY dx − , (50)with x ∗ given by the condition − Y ( x ) dY ( x ) dx = 1 δ ( x ) dδ ( x ) dx at x = x ∗ . (51)The abundance at x > x ∗ is found by integrating numer-ically the usual equation dYdx = Cx h σv rel i ( Y − Y ) , (52)with the initial condition ( x ∗ , Y ( x ∗ ) = Y ( x ∗ )). The fac-tor C is defined by C = p π M P m χ √ g ∗ , where M P is thePlank mass and √ g ∗ = g s / √ g ρ (1 + T / d (ln g s ) /dT ) ac-counts for the temperature dependence of the relativisticdegrees of freedom associated with the energy density, g ρ ,and g s [26, 30]. For WIMP masses larger than 10 GeV wecan neglect the temperature dependence of the degrees offreedom [33, 34] and take g s = g ρ = g = 100, √ g ∗ = √ g .In solving numerically (52) and (51) with the exposedmethod, we use the exact formula for h σv rel i , Eq. (10).We compare the previous numerical solution with theone obtained using the nonrelativistic freeze out approx-imation (FOA) that is commonly employed in literature.The FOA consists in integrating equation (52) with aninitial condition ( x f , Y ( x f )) such that the equilibriumterm proportional Y can be neglected. We choose thefreeze out point at the point x where Y ( x ) ≃ Y ( x ) =2 Y ( x ). As shown in Ref. [32], Y ( x ) well approximatesthe true abundance also in the interval x ∗ < x < x . x is the optimal point for the FOA and corresponds to thetemperature where the extent of the inverse creation re-action ψ ¯ ψ → χχ is maximal. The solution in the freezeout approximation is then Y F OA = 2 Y ( x )1 + 2 Y ( x ) Cx ( a + 3 bx ) . (53)The freeze out point x is given by the condition − Y dY dx = 3 Cx h σv rel i Y , which, in terms of the method of Ref. [35]corresponds to c ( c + 2) = 3, that is c = 1. Using thenonrelativistic form of Y , Y = 454 π g χ g s r π x / e − x , (54)
100 200 300 400 500 600 700 800 9001000 m (GeV) b PS b V b PS b V b PS b V
100 200 300 400 500 600 700 800 9001000 m (GeV) Λ =1 TeV Λ =10 TeV( Ω h ) exact /( Ω h ) FOA ( Ω h ) exact /( Ω h ) FOA
Figure 3. Ratio of the relic abundance obtained by solving nu-merically equation (52) over the value given by the freeze outapproximation, for the pseudoscalar and vector interactions.In the bottom blue curves for the FOA the correct coefficients(45) and (47) are used. The red and the black curves showthe effect of the wrong coefficients (46) and (48), respectively. x is given by the root of3 C (cid:18) a + 6 bx (cid:19) r π x − / e − x = 1 . (55)Calling α = 3 aC p π/
2, an accurate analytical approxi-mate solution of Eq. (55) is given by x = ln α −
12 ln(ln α ) + ln(1 + 6 ba (ln α ) − ) . (56)The relic abundance normalized over the critical den-sity is Ω h = 2 . × ( m/ GeV) Y ( ∞ ) for a Majoranafermion and two times that quantity for a Dirac fermionwith the same density of antiparticles. We now comparethe exact relic abundance Ω h with the value (Ω h ) F OA furnished by the nonrelativistic FOA calculated using thecorrect and the wrong expansions. We take the couplings λ a,b = 1 for illustrative purposes and two values of thecut off scale, Λ = 1 ,
10 TeV. The value of the freeze outpoints x ∗ and x varies roughly between 18 and 30 in theparameter space with m < Λ where the effective treat-ment is supposed to be valid. The ratio Ω h / (Ω h ) F OA is shown in Figure 3 as a function of the DM mass forthe chosen examples. The bottom blue curves show thatthe FOA with the correct coefficients (45) and (47) un-derestimates the numerical value by less than 2%, andthat in most part of the parameter space the error is atthe level of 1% or less. This a test of goodness for ourFOA, and confirms what shown in Ref. [32]. The redand the black curves show the effect of the wrong coeffi-cients (46) and (48), respectively. The wrong expansionsunderestimate the relic abundance by a factor between3% and 12% for both interactions for masses larger than10 GeV as shown in the plot. The behaviour is similarfor the other interactions not shown in figure. The er-ror becomes even larger at smaller masses and we haveverified that using for example c = 1 / ACKNOWLEDGMENTS
This work was supported by a grant under theMINECO/FEDER project: SOM: Sabor y origen dela Materia (CPI-14-397). The author acknowledgesRoberto Ruiz de Austri, Nuria Rius and Pilar Hernandezfor hospitality and for useful discussions at the Institutode Fisica Corpuscolar (IFIC) in Valencia where part ofthis work was done.
Appendix A: Integrals and expansions
Equation (7) can be written as h σv rel i = σ Λ xK ( x ) (cid:16) a A + a A + a A (cid:17) . (A1)The integrals are evaluated with methods similar to thosedescribed in Ref. [27] in terms of Bessel functions of thesecond kind: A = Z ∞ dy p y − K (2 x √ y ) = 12 x K ( x ) , A = Z ∞ dy p y − yK (2 x √ y ) = 12 x K ( x ) + K ( x )2 , A = Z ∞ dy p y − y K (2 x √ y )= 12 x
116 [5 K ( x ) + 8 K ( x ) + 3 K ( x )] . The expansions at x ≫ K ( x ) K ( x ) ∼ − x + O ( x − ) , K ( x ) K ( x ) ∼ x + O ( x − ) , (A2)while for x ≪ K ( x ) K ( x ) ∼ x O ( x ) , K ( x ) K ( x ) ∼ x + O ( x ) . (A3) Appendix B: Invariant formulation using v rel In this Appendix we remind, based on the results ofRef. [27], the main points about the relation between therelative velocity, the Møller velocity, flux and thermalaverage which are used in the main text.
1. Invariant relative velocity
The relativistic relative velocity that generalizes thenonrelativistic relative velocity v r = | v − v | , (B1)is given by v rel = q ( v − v ) − ( v × v ) c − v · v c . (B2)We have explicitly written the dependence on the velocityof light c to make manifest that v rel coincide with v r inthe nonrelativistic limit because the scalar and vectorproducts are of order ( v/c ) . In the following we go backto natural units.The relative velocity v rel can be written using the Man-delstam invariant s = ( p + p ) , where p , are the four-momenta, and λ , the Mandelstam triangular function, λ ( s, m , m ) = [ s − ( m + m ) ][ s − ( m − m ) ] , (B3)in a generic frame, v rel = p ( p · p ) − m m p · p (B4)= p λ ( s, m , m ) s − ( m + m ) , (B5)showing its invariant nature.
2. Flux factor
Given two bunches of particles with number densities n , and velocities v , in a generic inertial frame, in non-relativistic physics the flux is F nr = n n v r . To obtainthe relativistic invariant flux that reduces to F nr in thenonrelativistic limit, the easiest way is to consider the4-currents J i = ( n i , n i v i ), thus F = ( J · J ) v rel = n n (1 − v · v ) v rel . (B6)Note that the factor (1 − v · v ) that guarantees theLorentz invariance of the product of the number densitiescan also be written as1 − v · v = γ r γ γ = p · p E E , (B7)where γ r = 1 / p − v is the Lorentz factor associatedwith v rel and γ i the Lorentz factors associated with v i .If the element of Lorentz invariant phase space is de-fined as usual d ˜ p i = d p i (2 π ) E i , (B8)and one particle states for bosons and fermions are nor-malized to 2 E i such that the density per unit volume is2 E i , then, using (B7), the flux (B6) simplifies to F = 4( p · p ) v rel . (B9)Substituting the expression of v rel in the momentum rep-resentation, formula (B4), in Eq. (B9), the scalar product p · p cancels out and the standard explicit form is re-covered F = 4 q ( p · p ) − m m . (B10)
3. Cross section and collision integral
The integrated collision term of the Boltzmann equa-tion, neglecting quantum effects, can be written as, Z Y i =1 d ˜ p i [ f f W (3 , | , − f f W (1 , | , , where W ( ij | kl ) = (2 π ) δ ( P ij − P kl ) P s i ,s f |M ij → kl | ,and f i is the phase space distribution.Using the unitary condition R d ˜ p d ˜ p W (3 , | ,
2) = R d ˜ p d ˜ p W (1 , | ,
4) to write the collision integral onlyin terms of the annihilation rate11 + δ Z Y i =1 d ˜ p i ( f f − f f ) W (1 , | , , (B11)we keep out a statistical factor accounting for the possi-bility of identical particles.By definition, the invariant cross section, using the fluxin the form (B9), is σ = 14( p · p ) v rel Z d ˜ p d ˜ p W (1 , | , g g , (B12)being g i = (2 s i + 1) the spin degrees of freedom.Assuming as usual that the annihilation products aredescribed by the equilibrium phase space distribution atzero chemical potential f ,i , we have f f = f , f , = f , f , , the last equality following from energy conser-vation. Hence g g δ Z Y i =1 d p i (2 π ) E i ( p · p )( f , f , − f f ) σv rel . The equilibrium phase-space distribution f ,i is relatedto the number density n and to the momentum distri-bution f ,p ( p ) by g i / (2 π ) f ,i = n ,i f ,p ( p ). Assumingfurther that the non-equilibrium phase-space function atfinite chemical potential f i remains proportional to theequilibrium momentum distribution by a factor given bythe non-equilibrium number density n i , g i / (2 π ) f i = n i f ,p ( p ), we obtain11 + δ ( n , n , − n n ) h σv rel i . (B13) When the species 1 and 2 are the same, it takes the usualform h σv rel i ( n − n ) with the factor 1/2 cancelled by sto-ichiometric coefficient appearing in the left-hand side ofthe complete kinetic equation, see for example Ref. [32].
4. Averaged thermal rate
In Eq. (B13) the general definition of relativistic ther-mal averaged rate is h σv rel i = Z Y i =1 d p i E i ( p · p ) f ,p ( p ) f ,p ( p ) σv rel . (B14)In the case of the relativistic Maxwell-Boltzmann-Juttnerstatistics, the momentum distribution is f ,p ( p ) = 14 πm T K ( x ) e − √ p + m /T , (B15)and as shown in Ref. [27], the six-dimensional integral onthe right-hand side of Eq. (B14) reduces to h σv rel i = Z dv rel P ( v rel ) σv rel , (B16)where the probability distribution of v rel , for example for m = m = m , is P ( v rel ) = x √ K ( x ) γ r ( γ r − √ γ r + 1 K ( √ x p γ r + 1) . (B17)This is completely analogous to the nonrelativistic casewhere the probability distribution of v r , for m = m = m , is P ( v r ) = r π x / v r e − x v r , (B18)and the thermal average reads h σ nr v r i nr = Z ∞ dv r P ( v r ) σ nr v r . (B19)Given the total annihilation cross section σ the product σv rel will reduce to the nonrelativistic limit σ nr v r and h σv rel i to h σ nr v r i nr in the COF when v rel ∼ v r ≪ s using (B5), we obtainthe usual integral [27, 36] useful for practical calculation h σv rel i = 18 T Q i m i K ( x i ) × Z ∞ M ds λ ( s, m , m ) √ s K ( √ sT ) σ, (B20)with x i = m i /T and M = ( m + m ).We have recently become aware of the paper [37]where, probably for the first time, the thermal average ofrelativistic rates was discussed and it was realized thatwith the relativistic Maxwell-Boltzmann statistics for-mula (B14) reduces to a single integral over the distri-bution over the relative momentum. With some algebraand change of variables it is easy to verify that for exam-ple Eqs. (11b) and (12a) of [37] coincide with Eqs. (29)and (37) of Ref. [27]. In Ref. [37] the cases of collisions oftwo massive particles, two massless particles and a mas-sive with a massless particles are treated separately as ifdifferent definitions of flux and cross sections were neces-sary in each case. Clearly this distinction is unnecessaryfor the formulation we have given is completely generaland valid in any case. We finally note that an integralformula similar to (B20) was also given in Ref. [29].
5. No need for the Møller velocity.
By noting that in Eq. (B6) the factor (1 − v · v ) cancancel the same factor in the denominator of v rel , theinvariant flux can also be written in the form F = n n p ( v − v ) − ( v × v ) . (B21)In the textbook by Landau and Lifschits [38] this formis attributed to Pauli without giving any reference, whileits origin is more generally attributed to Møller [39].It is interesting to look at original paper by Møller [39].With our notation, he wants to prove that the flux given(B21) is invariant. In order to do that he shows that thiscan be written as a product of two invariant quantities:the ratio n n E E and the quantity B = p ( p · p ) − m m and there he stops. The flux factor written in the form (B21) has the samestructure of thee nonrelativistic expression n n v r . Prob-ably for this reason it has been later introduced in theliterature the notion of Møller velocity¯ v = p ( v − v ) − ( v × v ) = p ( p · p ) − m m E E . (B22)It is worth to stress that neither Møller nor Landau andLifschits attribute any particular meaning to Eq. (B22)and do not define it as a particular velocity, even less asrelative velocity. Clearly ¯ v is nothing but the numeratorof the formula defining v rel because ¯ v = (1 − v · v ) v rel ,where the factor (1 − v · v ) comes from the definition ofthe invariant flux (B6). Already this fact indicates that¯ v is not a fundamental physical quantity and overall, itis not the relative velocity, nor when the velocities arecollinear.On the contrary, in DM literature and in textbooks,when defining the flux factor for the relativistic invariantcross section, it is incorrectly asserted that in a framewhere the velocities are collinear the quantity | v − v | isthe relative velocity, while in a generic frame is given by(B22). The form (B21) of the flux is a simple consequenceof the fundamental quantities (B2) and (B6), there is nonew physics or concept in it. For these reasons, and forits noninvariant and nonphysical nature, ¯ v should not beused. [1] A. Kurylov and M. Kamionkowski, “Generalized anal-ysis of weakly interacting massive particle searches,” Phys. Rev. D , 063503 (2004) [hep-ph/0307185].[2] J. Fan, M. Reece and L. T. Wang, “Non-relativisticeffective theory of dark matter direct detection,” JCAP , 042 (2010) [1008.1591].[3] A. L. Fitzpatrick, W. Haxton, E. Katz, N. Lubbers andY. Xu, “The Effective Field Theory of Dark Matter DirectDetection,”
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