JHEP05(2015)115 Springer April 7, 2015 May 22, 2015 April 27, 2015 : : : December 1, 2014 : Revised Published Accepted 10.1007/JHEP05(2015)115 Received doi: c Published for SISSA by [email protected] , and P. Ruiz-Femen´ıa a,b . 3 1411.6924 The Authors. C. Hellmann Cosmology of Theories beyond the SM, Supersymmetric Standard Model, c

This paper concludes the presentation of the non-relativistic effective field a,b , [email protected] E-mail: James-Franck-Straße, D-85748 Garching, Germany Institut f¨urTheoretische Teilchenphysik und Kosmologie,D-52056 RWTH Aachen Aachen, University, Germany Instituto de Corpuscular F´ısica (IFIC), CSIC-UniversitatApdo. de Correos 22085, Val`encia, E-46071 Valencia, Spain Physik Department T31, Technische Universit¨atM¨unchen, b c a Open Access Article funded by SCOAP Keywords: Nonperturbative Effects ArXiv ePrint: of the multi-state Schr¨odingerequation thatated is by free large from mass the splittingscomputation numerical between instabilities of the gener- the scattering states. MSSMrates neutralino Our in results dark the allow matter present for Universe, relic a when abundance precise Sommerfeld and enhancements pair-annihilation are important. minimal supersymmetric standard modelcomputation (MSSM). of While papers the Ihere tree-level and we annihilation II describe rates focused inenhanced detail on that long-distance the the feed corrections. method into This toactions includes the obtain in the the the short-distance computation Sommerfeld MSSM, part, of factors which the are that potential provided contain inter- in the compact analytic form, and a novel solution Abstract: theory formalism designed toannihilation cross calculate sections the of radiative slowly corrections moving neutralinos that and enhance charginos the within pair- the general M. Beneke, Non-relativistic pair annihilation ofdegenerate nearly neutralinos mass and charginosComputation III. of the Sommerfeld enhancements JHEP05(2015)115 ], although it 1 30 27 50 48 10 41 when the DM mass is much larger than 2 g , such that an additional particle exchange W – 1 – /M DM is needed in order to calculate the annihilation cross 8 2 39 M g 2 6 g 26 5 6 36 15 18 1 46 2.1.1 Lagrangian 2.1.2 Annihilation2.1.3 matrices Sommerfeld-corrected cross section 4.3 Improved method 4.4 Second-derivative operators 4.5 Approximate treatment of heavy channels 4.1 NR matrix4.2 elements and the Solution Schr¨odingerequation of the multi-channel Schr¨odingerequation 2.1 Effective theory approach 2.2 Relic abundance is not suppressed bythe the mediator electroweak mass. coupling Themation Sommerfeld of effect diagrams is to all non-perturbativesection. orders in in The the relevance sense of thatcross the a section Sommerfeld resum- of effect (wino- was first or pointed higgsino-like) out neutralino for DM the into annihilation two photons [ electroweak nature of theorders interaction, of magnitude. and This even so-calledbetween exceed Sommerfeld the effect the arises non-relativistic lowest-order when DM an crossthat attractive particles they section interaction have significantly by a distorts larger probabilitythe their to effect wave undergo arises function, annihilation. from such Inticles, the terms exchange which of of Feynman contributes diagrams the a electroweak factor gauge bosons between the DM par- 1 Introduction For heavy neutralino dark matterlevel pair (DM) annihilation there cross is section, class which of can become radiative larger corrections than to naively the expected tree- by the A Explicit expressions for the MSSMB potential Equivalence between method-1 and method-2 5 Summary 3 MSSM potentials 4 Sommerfeld enhancement Contents 1 Introduction 2 Theoretical framework JHEP05(2015)115 3 ], e → χ 10 2 4 – e e ] (pa- 3 χ χ 1 15 e χ → ) enhance c is obtained 3 B j − X χ i A 10 χ X ∼ → · · · → j → v χ 2 i e χ χ 1 e χ ]. 13 – 11 denotes a pair of SM or light Higgs parti- B X A – 2 – X ] (from now on referred to as paper I) and [ 14 , responsible for the Sommerfeld effect, in analogy to to other two-particle states through electroweak gauge W where j j χ /m i χ i χ ) can yield an annihilation rate in accordance with the observed c ) or 1 . . . χ 3 2 . ]. For these models, the larger velocities around DM freeze-out in g )) from the long-distance interactions characterized by the Bohr 0 2 → 3 ∼ e LSP LSP v χ m 4 ( e /m / χ (1 O → B X A annihilation matrix entries in the generalpers MSSM I were obtained and in II. analytic The form off-diagonalthe in annihilation numerical pa- terms codes cannot that be be computeconsequently obtained not the directly considered MSSM from previously tree-level except annihilation in rates, the and strict were wino and higgsino limits. X cles. The transitions of boson exchange prior toare the accounted short-distance for annihilation by off-diagonalvided terms in in analytic the form potentialtwo-particle states in interactions, are required which this to are describe paper. pro- the short-distance Likewise, part. off-diagonal The (off)-diagonal entries in the space of The total annihilation cross sectionfrom for the a imaginary given part co-annihilating of state the forward scattering amplitude Here, and in our previous papers [ • neutral sector of higgsino-likeextends scenarios. previous Further approaches aspects are: where the formalism we present regions where mass degeneraciesscattering are states (channels) generic. requires an Dealing extensionis with of provided the many in conventional nearly NRQCD this setup, mass-degenerate Previous work which works to on accommodate the anyof Sommerfeld number the enhancement of MSSM focused neutralino with on and at the chargino most species. wino three and neutralino-chargino two-particle higgsino states limits in the case of the annihilating particles and separatesat the distances short-distance annihilationradius process of (taking order place 1 the NRQCD treatment of quarkonium annihilation.effects However, of in the the LSP MSSM with co-annihilation heavier neutralino and chargino species have to be accounted for in DM by including theparticle (LSP) Sommerfeld to corrections. be anframework arbitrary By can admixture allowing of be the the used electroweakmerfeld lightest gauginos to correction and supersymmetric may study higgsinos, no the regions longerradiative of be correction. the an MSSM order Our one parameter method effect, space but builds where still upon the constitute the the Som- dominant non-relativistic nature of the pair of Sommerfeld enhancements in cosmichave been ray discussed signatures extensively andbut for in also relevant for the MSSM generic scenarios thermal multi-state with relic dark neutralino matter abundance DM models [ [ per II), we develop asection formalism in that improves the the general calculation minimal of the supersymmetric DM standard annihilation model cross (MSSM) with neutralino Sommerfeld enhanced DM modelsDM attracted annihilation more rates attention [ asthe a early Universe mechanism ( to boostrelic the density, whereas thethe much annihilation smaller rate neutralino into electrons velocities and today positrons ( observed as cosmic rays. Since then the was not until 2008, when an anomalous positron excess was measured by PAMELA, that JHEP05(2015)115 , b DM (1.1) /M 2 W M ]. The general 16 can scatter to heav- j χ i χ . 2 rel b v + is thus needed, which is automat- a b denotes the relative velocity of the | ∼ 2 ~v 3 e χ − 4 e 1 -wave annihilations, the subleading term χ ~v | S → B = – 3 – X A rel v X → 2 e χ 1 e χ Γ ), where -wave contributions, that get multiplied by different Sommer- 2 rel P v ( O - and S , respectively, and presented the complete results for the corresponding Wilson rel v from different non-relativistic operator matrix elements. While the leading ordercontains describes both only feld factors when building theseparation full of annihilation the cross two partial-wave section componentsically from of achieved the in rates our above. EFT framework, A since the Sommerfeld factors in question arise The short-distance annihilation rates are obtaineding in corrections the of non-relativistic limit includ- annihilating particles in their center-of-mass frame, i.e. directly for the entriesinstead of of the solving for relevant the matrix wave functions.time that Leaving provides needed aside the limitations to related Sommerfeld solve toallows factors, the a CPU to system compute of thetwo-particle many Sommerfeld channels coupled factors are differential present. reliably equations, also this when method many non-degenerate ier two-particle states that(cms) are energy. kinematically Eventually, when closed some fornumerical mass-splittings the instabilities become in available larger the center-of-mass than solution oftering the wave for matrix-Schr¨odingerequation functions the scat- prevents onefeld from factors. obtaining We accurate circumvent results this for problem the by Sommer- reformulating the Schr¨odingerproblem Our multi-channel approach implies that an incoming state In the EFT approach the short-distance contributions to the neutralino and chargino • • MSSM parameter-space point andgation the and corresponding discussion relic abundance. ofsome Sommerfeld A popular enhancements detailed MSSM in investi- scenarios thephenomenological is the relic study subject abundance of of calculation Sommerfeldpostponed an in for accompanying enhancements paper a in future [ the publication. MSSM parameter space is the annihilation process and provideof all the the matrix technical elements detailsthe required of other for model-specific the the ingredient, four-fermion the computation operators. non-relativisticgauge potentials boson generated Apart by and from electroweak Higgs thealytic boson annihilation form exchange rates, in in thisfull the work. Sommerfeld-enhanced MSSM, The neutralino are contents and also of chargino provided papers co-annihilation in I-III compact rates therefore an- for allow an us arbitrary to compute the ators gives the fullI annihilation and rates, II we includingthat have the mediate written the Sommerfeld down short-distance corrections.order the annihilation in dimension-6 rates In at and papers leadingcoefficients dimension-8 in and four-fermion the next-to-next-to operators general leading MSSM. In the present paper we turn to the long-distance part of pair-annihilation processes are encodedoperators, into whereas the the Wilson absorptive coefficients part of of the local matrix four-fermion element of these four-fermion oper- JHEP05(2015)115 4.2 . In the 4.4 . The standard 2.1 . As in NRQCD, the ]. More important to 19 . The computation of the -channel resonance, since 4.3 – s ], the case of Sommerfeld 17 2.2 20 . In the first part of this section we relate 4 to outline the details. The computation of 3 , and we show the equivalence between the two – 4 – . Finally, we collect the analytic expressions for A 5 . We discuss the structure of the EFT Lagrangian and derive the 2 we give an approximation to the treatment of heavier channels which ]. For the general case, this is left for future work. 4 11 , ), and therefore a member of an approximate electroweak multiplet. On 4 W m  The contents of the paper are the following. The theoretical framework is summarized A number of limitations needs to be mentioned. The formalism in its present form LSP m can be used toSchr¨odingerequation, reduce and the thus number the of timeof channels required our to for main be its results treatedthe numerical is potentials exactly solution. in given in the A in the MSSMdifferent summary section multi-channel basis in of appendix two-particle states that can be used to evaluate the Sommerfeld factors in numerical instabilities that aremethod caused that solves by this kinematicallymatrix problem closed element is channels. of then explained second-derivativethough The in operators here improved subsection can it begenerated becomes related by more massive to gauge involved the bosons;final due leading part the to of exact order section relation the ones, is presence derived in of subsection Yukawa-like interactions of this paper, andthe we Sommerfeld factors have is devoted the section the subject four-fermion of matrix section elementsby with solving the a scattering wave multi-channeldescribes functions Schr¨odingerequation the that with are standard MSSM determined method potentials. to Subsection solve the Schr¨odingerequation and points out to the at some length in section master formula for the Sommerfeld-correctedformalism cross to section obtain in the subsection thermal relicpotentials abundance for is reviewed neutralino in and chargino scattering in the MSSM is one of the main results general the thermal correctionsenhancements is to special, DM since freeze-outthe they small are depend mass tiny sensitively splittings [ limits on between the some the co-annihilating of range particles. the ofenhancement above In the [ the effects potential wino have and and already Higgsino been studied in the context of Sommerfeld indirect detection signals of DM.up EFT the methods Sudakov have been logarithmsrelic-density proposed computations in recently is multi-TeV for the DM summing temperatures fact annihilation that where for the [ heavy temperature-dependencecan dark of be matter the the relevant, electroweak as freeze-out gauge-bosons occurs can masses at be the temperature-dependence of the gaugino masses. While in ios require theless accidental generic coincidence than of neutralino-chargino( some co-annihilations, MSSM especially parameters whenthe and the technical are LSP side, in we is presentlyas this heavy neglect well sense the as scale electroweak dependenceonly double of important logarithms the when of electroweak the the couplings Sudakov formalism type. is applied The to latter, exclusive of final course, states, are as is relevant to cannot be applied toin DM this annihilation case through the annihilation a processthe narrow-width is pair no longer annihilation short-distance. andand Furthermore, potentials we sfermions, computed of which excludes neutralinos MSSMis and scenarios important. charginos, where Both, but neutralino-sfermion resonant not co-annihilation annihilation of and neutralinos neutralino-sfermion co-annihilation scenar- JHEP05(2015)115 , Y + U(1) sector of × , . . . , n L ± χ = 1 , j 0 , χ j ± χ among other required SUSY µ relic abundance. Our formalism 0 1 co-annihilation processes in a given χ and 2 chargino-states ± χ 2 ]. ≤ , 0 M 14 + χ , n 1 ], the formalism that we present in this work . The description of our EFT framework is M 0 1 χ 15 and – 5 – , 0 14 co-annihilations in the most general MSSM models. 4 result as combinations of the four SU(2) ± , . . . , n χ , ,..., ], but have not yet been implemented in public spectrum 0 = 1 2 are a mixture of the charged wino and higgsino electroweak χ , 27 i – = 1 , , starting with the discussion of the Lagrangian in the effec- 0 i i 24 = 1 χ , , respectively, the higgsino states are associated with the mass- ]. j 0 i 2 2.1 , χ 16 ± j M χ ], where the parameters DR-parameters that are provided by most spectrum calculators. Fur- co-annihilation rates as well as the 23 and – ± 1 χ 21 in the underlying SUSY Lagrangian. , M 0 . As mentioned above, the present papers concentrates on technical aspects of µ χ B 4 neutralino-states We perform our calculation in the framework of an effective field theory (EFT) of A rigorous analysis of Sommerfeld-enhanced A MSSM scenario can be obtained with publicly available MSSM spectrum generators, ≤ 0 non-relativistic neutralinos and charginos, whichn generically covers the case ofthat models with are nearly mass-degeneratethe with purpose the of section instead of to the thermore, the mass splittingsthe between precise the calculation of co-annihilating Sommerfeld statesized enhancements, play masses requiring in an one-loop some on-shell essential cases. renormal- the role MSSM in Results are on the availablegenerators. one-loop [ on-shell renormalized requires positive mass parametersmatically of account the for neutralino bymatrices, and an as chargino explained appropriate states, in rotation which appendix of we A the auto- of paper neutralino Imodel and [ should chargino refer mixing to the on-shell mass spectrum of the neutralino and chargino states, Sommerfeld enhancements in Generically we require for ourmass calculations a parameters, (SLHA mixing formatted) MSSM matricesspectrum spectrum and including calculator. angles, From typicallyenhanced this provided spectrum as output we determine of a the MSSM corresponding Sommerfeld- for example [ parameter inputs have tomodels be with grand specified. unification ofparameters gauge In are couplings, constrained assumed. certain MSSM relations Our among scenarios, setup the as input is SUSY not for restricted instance to such cases, but allows to analyze eigenstates bino, the electricallyThe chargino neutral states wino andeigenstates. the two While electrically the neutralparameters bino higgsinos. and the winosparameter are related to the soft SUSY breaking mass- Together with previous results given inallows [ to describe Sommerfeld-enhancedwithin neutralino generic R-parity and conserving charginoflavor MSSM mixing pair-annihilation in scenarios, rates the including squarkfour the and neutralino most slepton states general sector. form After of electroweak symmetry breaking the the framework, which is ratherwith general electroweak and interactions. applicable toaccompanying any paper A model [ non-technical of heavy discussion dark of matter results is2 presented in an Theoretical framework appendix JHEP05(2015)115 that relic (2.2) (2.1) j 0 1 χ , ζ j η = ]. It consists j are important ), respectively. 14 ψ , and second, it 1 + j . τ χ j LSP ψ or ˜ and m i 1 ! ξ ˜ t and relic abundance calcu- 0 1 − j 2 LSP χ χ is the mass of the lightest ~ ∂ i m ξ 2 ..., ! LSP + ) + m 2 LSP 2 non-relativistic chargino species, ~ ∂ ann LSP m ≤ co-annihilations are important in the L 2 ], that has been employed for comput- m δ + ± 29 n − + . Finally the ) + , where /χ j 2 0 ) pot m χ v ) and charginos ( LSP ( we recall the notation for the four-fermion L 0 i 2.1.3 m χ − + – 6 – LSP t − m i i∂ 2.1.2 kin

L m ( † j = . Our EFT framework does not cover scenarios where ψ − t + =1 n j i∂ X 2.2

η,ζ † i . In section = ξ X NRMSSM ψ 0 L =1 n i + X 2.1.1 ]. The relevant formulae for the Sommerfeld-enhanced annihilation = its small velocity. The framework allows us to compute the inclusive 15 kin , L v 14 4 non-relativistic neutralino species and ≤ ] we have introduced an effective field theory (EFT), the non-relativistic MSSM 0 is given by contains the bilinear terms in the two-component spinor fields The structure of the EFT Lagrangian has already been discussed in [ n 14 relic abundance calculation. kin kin 0 1 not required for theL present calculation ofrepresent the the Sommerfeld-enhanced non-relativistic annihilation neutralinosFor rates. ( L and not just Coulomb potentials. of three parts: where the dots stand for terms of higher order in the non-relativistic expansion, that are ing heavy quarkonium properties and radiativeremarkable corrections accuracy, to and heavy can quark productionFirst, be with the considered NRMSSM as can an accountneutralinos extension for and several of charginos non-relativistic the whose particle latterincludes masses species, potential in are namely interactions two nearly those generated aspects. degenerate by massive with gauge bosons (i.e. Yukawa potentials) off-shell by an amountneutralino of and the order ofannihilation rates ( of pairs oftematic charginos expansion and in neutralinos the movingsimilarities at coupling with small constant the velocities and non-relativistic in the EFT a of velocity. sys- QCD The [ NRMSSM shares many 2.1 Effective theory approach 2.1.1 Lagrangian In [ (NRMSSM), designed to describe the dynamics of charginos and neutralinos which are co-annihilations with nearly mass-degeneratein sfermion the such relic as abundance the EFT calculation. setup These but cases are requirescenarios beyond a from the straightforward our extension scope analysis, of of whereχ our this other than work. For the time being we exclude MSSM operators that reproduce theintroduced short-distance in annihilation [ reactions andmatrix that elements, were which already determineabundance the calculation, co-annihilation are cross given sectionslation in entering is the section summarized in section tive theory in section JHEP05(2015)115 ) | ~r , the (2.3) i 0 e ≡ | ; within χ . r m ( ) ~r . The par- ~x and 1 + − m 0 field. Redundant + − ~x t, ~x i χ χ ( ± e 2 = χ + − e χ χ ~r χ ), as compared to the ) → → represents a field t, ~x i LSP + − ( e 1 χ χ χ e /m + − χ χ χ ) (1 , which are local in space-time ~r O + for the case of a ann double-charged reactions ) are formally considered of order neutral, single-charged and double- L + δ t, ~x 3 ( e LSP 3 † e with non-relativistic relative velocity. In χ m 4 χ 3 e , . . . , n ) e 0 − χ + χ χ i χ 4 ) (whichever is smaller) of the non-relativistic = 1 t, ~x e 0 − ( i → m W χ χ χ 4 e – 7 – † e 2 e χ → → /m χ ) and 0 1 r (1 + , which have been summarized in table e ( 2 χ i χ } e χ ± e O 0 − 3 scattering reactions classified according to the total charge. χ e χ χ ) participating in the reaction is indicated by the label 1 4 , whereas , χ 3 e e 0 i ± e ) or 0 e χ . χχ }{ χ χ v 2 χ 4 → e single-charged reactions e 1 are suppressed in the above table. If or χ e = χχ LSP i . In the same way, the hard modes associated to the SM and { , . . . , n e 0 i 2.1.2 e → χ χ + 0 /m 0 + χ ~rV = 1 2 χ χ 3 χ χ e i (1 0 0 − d ) are not explicitly written. e χ − χ accounts for the exchange of SM gauge bosons and Higgs particles 0 1 O χ χ ], a straightforward extension of the EFT covers the case where the non-relativistic χ e χ NRMSSM Z χ 0 14 → L → → χ ) is taken over all pot → χχ 0 + L 0 + of the same order as the time-derivative and kinetic-energy term in the La- → → χ χ 2.3 X χ χ 0 on the fields 1 0 − χχ − − χ ], i χ χ χ χ e − neutral reactions + 14 [ χ . Possible = can range over 2 i , which are relevant for this work, and the details on the associated Wilson coefficients v e The term pot As discussed in [ ann 1 L LSP L charged reactions with ticular particle species ( particle species are nearly mass-degenerate with respect to two well-separated mass scales in the two-body system: The sum in ( this modified set-up pair annihilations of a set of hydrogen-like two-particle states can be also considered. between the two-particle states the non-relativistic limit, suchand interactions are become described instantaneous but inYukawa- and spatially the Coulomb non-local, EFT potentials by depending four-fermion on operators the whose relative matching distance coefficients are the Wilson coefficients ofbecause four-fermion the operators annihilation takes in placecharacteristic range at short-distances of interactions between the charginos andδ neutralinos. The explicitare form given of below the in operators section in grangian. This impliesSUSY that particles heavier and neutralinos higherfective and mass theory, charginos Higgs) and (as are their wellthe not as virtual operators among further effects in the heavy can degreeslight only of Higgs-particle produced appear freedom in of as neutralino the short-distance and ef- corrections chargino pair-annihilations to are encoded in In order to havetween a two-particle consistent states power-counting formed in fromEFT the the amplitudes we neutralino describing and need transitions charginom that be- species included the in mass the differences ( Table 1 The labels label reactions where the typereaction of particle in position 1 and 2 and/or 3 and 4 are interchanged (e.g. the JHEP05(2015)115 . . ) J is 2 π as L v 4 ann ~ ( 2 S / (2.4) +1 L 2 ~x i gauge O δ S g has, at 2 Y φ = and i ), and con- m α 1 , U(1) ~x r φ 2.12 ⊗ r m L by its eigenvalue − e 3 e ), where  2 i χ 2 ) are generated by re- 4 α α ), which are contracted e 3 ( χ 2.3 ,α , O 2.3 1 2 , corresponding to particle e α i ). Since we shall decompose 4 e . Details on the calculation χ ~σ ζ α 1
e pot ⊗ 2 . The total spin operator χ or i ~ L S 1 e i e χ 3 + e 4 1 , η , and are of e i omitted in ( 2 e are the generic SU(2) ⊗ B e ξ i 1 i ~σ X e α g A B 2 ( are given by dimension-6 four-fermion op- with SM and light Higgs intermediate states X / + one-loop scattering diagrams with two SM or 1 3 e 2 1, we can drop the spin indices in the potentials undergoing potential interactions into 3 ann α – 8 – χ e ≡ , , the contributions to the Wilson coefficients arise 3 4 L 3 i χ e 2 e α δ 4 g δ / χ e χ = 0 ) effects in the (co-)annihilation of neutralino and 1 1 -wave neutralino and chargino scattering processes 4 χ operator acting on 2 α α e S S , since the exchange of gauge and Higgs bosons before v 4 3 → 0 χ 2 ( . α α e → ~ δ S ~σ O 3 χ X 3 2 1 e e e α 4 χ and χ 4 e → 1 2 α by the field symbols 2 e e 2 δ e 1 e i χ e e χ χ + 1 1 χ A 1. 2 ) in the MSSM, where e in all possible ways compatible with charge-conservation in the e ,  α 2 2 χ χ i 3 g + e α ( δ χ = 0 ) = O 2 r S / ( 1 } 3 and α e 4 for 4 i α e − e ~σ S χ χχ }{ 2 , → = in the potentials. The explicit form of the terms in ( e i 1 0 e 2 e χχ { α χ 3 χχ V ,α 1 + 1) = 2 The leading-order contributions to In this work we account for → α 4 S ( α ~ Note that the annihilation ratesthe include space the of two absorptive particle partsthe states of short-distance off-diagonal annihilation amplitudes may in transform one two-particle state intoerators, another. that describe leading-order rates, only the absorptive part ofsequently these the matching Wilson can coefficients be are done required,order for see in the ( the absorptive electroweak part gauge of couplings thefrom amplitude the only. absorptive At part lowest of light Higgs particles in the intermediate state, The short-distance annihilation of the chargino andfinal neutralino states pairs into is SM reproduced and lightThe in Higgs Wilson the coefficients EFT of by theseplitudes local operators for the four-fermion can process be operators determined containedto by in the matching tree-level the matrix MSSMoutgoing am- element states. of For the the four-fermion computation operators of the for neutralino the and same chargino incoming inclusive annihilation and of the potentials at couplings, will be given in section 2.1.2 Annihilation matrices S chargino pairs coming fromcontributions from the the short-distance long-range part. part Consequently,Yukawa of only potential the the leading-order interactions Coulomb- annihilation and need but to ignore be considered in with the corresponding indicesbuilt from in the the spin fieldS operators operators acting onthe the two-particle particles states interactingpartial-wave at states with points defined spin in what follows and replace the leading order, the generic form where we have written explicitly the spin indices placing the generic fields species reaction. At leading order inthe the total non-relativistic spin expansion, of thecontribution the potentials from two-particle depend states, the only which exchange on is of thus a conserved. gauge boson An individual or potential Higgs particle with mass χχ JHEP05(2015)115 ], → ann 14 L 2 [ (2.8) (2.5) (2.6) (2.7) e δ 4 χ e 1 e transitions ) particle χ 4 ↔ . 1 e , , correspond-
, have been 3 P χ S 3

e
ann S J S S − L S S δ S 1 +1 and S S +1 +1 3 3
+1 2 )-suppressed potential e S S J S v 2 2 in analytic form can χ 2 } and/or ( P ( 1 -wave operators with 3 and O e ˆ 2 } B f 2 } S 2 ]. The remaining next- 1 e 1 e 1 e e } e X P 2 χ 1 2 15 }{ 1 . e A e e ↔ 3 χχ 2 1 e }{ − e χχ e X 1 }{ χχ 4 , respectively), while in the 3 →
3 1 e e e 3 χ and }{ → e χχ 1 , → { i c e 4 S 3 4 χχ i 1 e ) contributions to 3 e 1 P e σ χ → e χχ { v 4 e χχ 1 , { 2 4 † ( Q e χ 0 c e † e χ O χχ { P
O 2, are the same as the dimension-6 P 2 † χ } χ pair is changed. They are not considered S 3 c e , 1 O

e 3 S χ S c e 2 − S
e χχ S 3 0 χ S +1 = 1 J c e and i S }{ S χχ 2 i P 1 3 χ σ : +1 2 +1 1 3 e 2, contain one derivative acting on each , 4 4 → S e S 4 † e † e ,
} 2 2 e χ ann 1 3 } S χ χ χχ { 2 † e , 3 L c e } S 4 e } O 3 e = = δ 4 3 χ e e e – 9 – +1 4

}{
4 e = 0 S 2 e 0 1 }{ χχ 1 χχ e ). A master formula and necessary ingredients to 2 2 S S }{ χχ from individual states P 1 e 2 i }{ → χχ → J 2 1 3 e 1 1 2 i e α → {
e e → , 1 χχ ( χχ i { 1 J Q } } e }
e f χχ h 1 1 { 3 χχ S O { J e e e 2 f g 2 2 2 4 P e e +1 e 1 3 ] 1 4 2 M 1 S M }{ }{ }{ χχ χχ χχ 4 2 transitions are ignored. 3 3 2 1 4 2 14 1 M , e e , e → → → O 1 are the corresponding Wilson coefficients, which will be often 4 4 1 ˆ 4 f 2 e e e , =1 D X =0 χχ χχ χχ { {
{ . The explicit form of the dimension-6 X i 1 3 1 , including redundant ones where the particle species at the first , S
, f J 1 O O , J and S 1 X → =0 2 =0 χχ X S =0 J s 1 X
+1 s 1 → 1 S M X +1 S 3 -wave operators 4 ) is taken over all neutralino and chargino scattering reactions P 2 χχ χχ S χχ χχ 1 2 S → → → } χχ X X 2.5 = , which we neglect. For the same reason, dimension-8 four-fermion operators in X . They read [ ] in the MSSM at and 3 f 3 e χχ χχ O → χχ 1 e 4 X 14 pot e =6 P χ + + + χχ d ann 1 L ) in the non-relativistic expansion in momenta and mass differences, dimen- 4 operators the two derivatives act either on the initial or in the final state. The }{ χχ 2 e L 2 = v →
e δ → χ ( 1 S 1 ) below. The absorptive parts of the Wilson coefficients, e =8 χχ P { O S d ann specified in table → 3 1 is f L , 2 3 +1 δ 2.11 e e At S In a parity-violating theory such as the MSSM, there exist also χ χ 2 2 4 1 = 0 e e ing to dimension-7 four-fermionwhere operators the which spin describe and/orhere the because orbital-angular they momentum contribute ofinteractions to the in the annihilation ratescausing only in conjunction with The operators of the initial andP final bilinear operatorsexplicit ( form of theseto-next-to operators leading can be read off from table 1 in [ sion-8 four-fermion operators contribute to see ( calculated in [ obtain the contributions to be found therein. χ and second and/or thirdthat and several operators fourth describe position one specificin are process the interchanged. with a initial This (final)certain redundancy state, symmetry implies and relations as under a the consequence exchange of their the respective labels Wilson coefficients obey S The first sum in ( where abbreviated as χ JHEP05(2015)115 - ) 2 e =8 P χ d ann 1 mul- (2.9) L e m M (2.11) (2.10) i δ χ δ h , and , 3 → e g , χ 2 4 e f e -wave Wilson ). χ ]. For the χ 1 S e or larger should = 1 and ( 15 2.8 χ → , , i 2

e LSP J J χ In the non-relativistic L L 1 m e . ) and ( 3 +1 +1 χ ]. Therefore, in scenarios 2 ∼ S S e . These cuts are automatically 2 2 2.5 14 ij m δm } } 2 3 3 − in ( e e ) in the case 3 below. 4 4 i e e e h 1 }{ }{ m χχ χχ 2 2 e e ) effects [ → → , 1 1 δm M 2 and e e = k v χχ χχ { { e ( k k m O m f, g are process-specific quantities, their value L L + + ) there are symmetry relations under the annihilation cross section summed over all 4 =1 S S m k X j δ 2.8 1) 1) ) refers to a scattering energy eigenstate of the non- χ , | 1 2 i j − − – 10 – χ χ i δm , δ = χ = ( = ( h 1 ( e

) up to a factor ( i J J M or of any other two-particle state in the shaded blob representing m j ) below, which in turn guarantees that the operator matrix element L L 4 χ 2.7 2 i e − = 0 for diagonal annihilation reactions 3 are only relevant for the computation of the off-diagonal χ particles involved in the reaction +1 +1 4 | e 2.17 e S S
k 2 2 m e or S m δ χ 2 S } } e . Particles with masses such that 3 4 = 1 = e e +1 e 4 3 by virtue of unitarity, see figure S e e ) follows. LSP 2 j δm written in ( }{ }{ χχ χχ δm m 1 2 χ i i e e
→ → 2.14 2 1 S Q χ and mass differences e e  χχ χχ { { S k k → M +1 ), where the state δm ]. We reproduce them here for later reference: j S 2 -wave Wilson coefficients in ( χ i 15 S 2.12 χ O the masses of the ) 2 refers to any of the Wilson coefficients, k v e ( k m = 2, where O Analytic results for the absorptive parts of the Wilson coefficients appearing in i We note that the forward amplitude shown in the second line of this figure has further cuts whenever the 3 mass of the two-particle state ladder diagrams is lessexcluded than the in centre-of-mass ( energyrelativistic of Hamiltonian the initial rather state thancomplex conjugate to wave-function in an ( in-stateis (out-state). hermitian such This that ( also explains the appearance of the 2.1.3 Sommerfeld-corrected crossThe section spin-averaged center-of-mass frame accessible light final statesamplitude is given by the corresponding cuts of the forward-scattering where in the general MSSMand can beexchange extracted of from the the particlecoefficients expressions [ labels given in analogous [ to those for the leading-order particle species whose long-distanceare interactions limited are to described withinbe the decoupled EFT formalism explicitlydistance and coefficients integrated in the out effectivecomputation. — Lagrangian, which they are cause not relevant small to the modifications relic of density short- (where the absorptive partcross of section), the amplitudes the isrates. related to The the convergence correspondingquires of annihilation that the the mass non-relativistic differences expansionwhere are considered the for as off-diagonal these annihilation off-diagonal terms can terms be re- relevant, the non-degeneracies among the The mass scale being determined by the curlytiplying bracket labels them. of the Since short-distance coefficients for with operators JHEP05(2015)115 ) ) 3 e χ . In 4 ∗ 2.12 2.12 e  (2.13) (2.14) (2.15) (2.12) ) ) which χ i J ], which p , ) for the L , 0 ) − i 14 J ~p +1 [ and j f S L p , ˆ 2 χ 2 ( − f i ( i e , 4 } i  j +1 ~p χ δ 2 χ i j | ( 4 S e i χ i 1 j ) i 1 2 ) χ j e j e i ( χ π ) instead of the cor- J (3) χ χ i χ χ } J | χ i }{ δ i χχ 3 L | χ 3 ) L 3 e | χ e χ → ) 1 ) | 4 | 4 ) 1 +1 e +1 e ) S π ) 0 χχ { S S S 0 3 ˆ 0 }{ χχ f 2 2 S 3 ( 2 ( S ( P  1 ( e → 1 appear in an expression a 3 ( P 1 ( ( = (2 e | O i } | O χχ | P { j e 1 ) = i ˆ j j | O f 0 e χ J j pair. In what follows we will χ | O 2 χ i ~p i L i e j χ | ) and χ j i χ ~p χ χ h i J }{ +1 χχ = 2 h χ i h h χ j 3 i L S h ) e χ  2 → ) χ  χ 4 h ( ) J i 1 +1 e  2 } ∗ 1 χχ S { χ L S ) 3   | 2 S e ) 3 0 ( ) 4 M 3 ( 2 e +1 S ( | O J O ˆ annihilation reactions where the states g 1 P S 2 j }{ χχ ann L ( , 3 2 2 ˆ h 3 χ 2 ( ( e + → i L e ˆ ˆ 1 ˆ h f +1 f ann δ e χ χ i m χχ i S { h L j 4 | M ˆ j δ 2 f e = δ m j ( χ M χ δ i χ }
χ i – 11 – i 2 o 0 χ ) + 5 χ e | i ) + χ 1 | that S 1 j 1 ) h → e 1 ) ) + P ˆ 1 χ f S 0 0 ) corrections, this observable is obtained as 3 i }{ χχ 3 2 P ). ( 2 } S 3 S e ( χ ˆ 1 3 e 1 f → | 1 ~p 1 ( e χ 4 ). This means that, for instance, the first term in ( ( ( ( 2 Im ) 4 e ˆ h 1 1 2.12 χχ e , and when repeated indices { e J m O  ˆ h | O f | P δ χ }{ χχ L j j j χχ  2 M ) + 3 δm χ e ,s → χ 0 → i i +1 1 i M δm X − s e P S χ χχ χ and { 2 χχ ) 3 h ˆ 1 4 h ) + f ( ( J ) 1 ˆ ) ) the sum symbol over the intermediate states f  ) + P 1 L 0 S 0 2 2  | O χχ 3 P S j S ( 1 +1 2 1 = → 1 M ˆ M ( 2.12 χ ( S f X 1 ( the relative velocity of the annihilating particles in the center-of-mass i ˆ M, δm 2 ˆ g f ˆ M χχ f  | ( χ rel j j } h 3 v ~v + + + + ,s ) e i have the same charge as the incoming X J 4   s − e j L ) the matrix elements of operators 3 1 8 i light }{ e ,s χχ ~v i 2 | +1 X χ s e → → 4 S 1 2.12 j e 1 8 2 e = χ χχ ( { and assuming the non-relativistic normalization i f 5 χ f , χ rel 4 = σ 2  n v e i To make contact with commonly used notation in quarkonium annihilation, we abuse notation when It follows directly from the definition of χ 5 4 Im 1 − e writing in ( responding forward scattering amplitudes;should the not former be contain included an in additional the relation factor ( (2 papers I and II thisabsorptive relation parts; was we incorrectly take written the for opportunity the to Wilson correct coefficients it themselves here. and not for their following from the definition ofimplies the absorptive part of the Wilson coefficients always omit the symbol summation over the particlewe species have will used be that implied. To obtain the last equality in ( and the sumχ extends over all incoming chargino and neutralino one-particle states.we In have order omitted to in make the ( that notation simpler undergo annihilation andas the in labels the in quantities thein Wilson full coefficients form and reads operators (as well with frame, effective theory, including up to JHEP05(2015)115 j j B χ ∗ i , ξ X i  χ A , and ξ A B (2.17) (2.16) j X χ X X ) can be accounts 0 ) S , evaluated 1 and 4 3 S ( L,S e e i ( 3 ij , e χ ~ ψ χ
4 e , ij χ i 1 0) j e , → 2 χ 2 (0 to yield the exclusive e i satisfy e . ψ χ χ i j | ) B 1  e 4 J 1 ann + X χ e ,e L and total spin L 3 A χ X O i j e δ 2 † ). L , ij +1 c e X 2 S 0) χ e , ) account for the long-distance 2 | 1 ( (0 e 0 2.12 } 4 ψ   i h e 2.12 A B 3 0 i | e state with all possible intermediate X X i 3 4 3 ξ }{ c e e e j † 2 c χ j e χ 4 ξ 1 i † e 1 2 e h † χ e e { χ A B ∗ | i O ) short-distance corrections. X X j
2 i χ 1 2 α i , ij e e ( ) = 4 – 12 – χ 0) e , h J O 3 (0 e L = ψ i j i ) in the last line in ( +1 j 2 0 + S with , orbital quantum number 2 χ ,e P )), since infrared divergences are absent at that order. s i ( 1 3 2 i X , ij } e ( rel χ √ 3 i j 1 α | 0) v e , e ( O ) 4 ) can be expressed in terms of non-relativistic wave functions 4 2  (0 e 0 e B O S ψ X }{ ,...e 1 3 X A -component of the scattering wave function for an incoming 1 2.12 ( e i e 2 AB } ) denote the Pauli spinor of the incoming particles that projects onto the Fock space with no neutralino and chargino 4 i X e 1 e ξ | e { → χ †  0 2 c j j 1 e O e ξ χ dPS 2.17 i h i ℑ h }{ χ χχ ] (i.e. at χ 0 3 h | Z e σ → 14 4 e = χχ { 2, with that of ) also applies separately for every final state = = 2 , is the | O j ) 2.12 , ij . For instance, the matrix element of the operator = 1 | χ 2 i j e . Diagrammatic picture for the relation among the annihilation amplitude and the ab- J 1 L,S ~v χ pot ( e h L ), ψ − stands for the trace in spin space. The multi-component wave function J i + It is well-known from quarkonium physics that matrix-elements of four-fermion opera- The matrix-elements of four-fermion operators in ( i ~v P | 3 ( ij kin ... σ two-body chargino and neutralino states with the same charge and identical spin and state with center-of-mass energy for zero relative distance andin normalized the to second the line free of scattering ( h solution. The symbols for the potential interactions of the incoming where and their derivatives evaluated atinsert the the operator origin. Thisstates relation into becomes the clear, four-fermion operators, ifL which we is explicitly exact atwritten the level as of terms included here in at the tree-level [ Therefore ( annihilation rates tors analogous to those in ( Note also that weO have related the spin-average of theinteractions matrix between element the of annihilating pair, theparticles is while operators described the by short-distance the Wilson annihilationpart coefficients. of into the light The Wilson matching coefficients calculation can of be the performed absorptive for exclusive two-particle final states and the fact that the adjoint of the four-fermion operators in Figure 1 sorptive part of the correspondinginteractions. forward scattering amplitude in presence of long-range potential JHEP05(2015)115 , )+  and ij 0 3 ) ex- e (2.19) (2.20) (2.21) (2.22) (2.18) L 0 4 M ,e − 3 2.22 e s , ij 4 2 1) e e , √ i 1 κ ( (0 e j . The deriva- 0 2 ij , ij . e χ ψ ). The precise 2 i 0 1 µ 0) e , A , k` are generated by  χ ,e 1 in their center-of- | δ 4.1 ) 2 . (0 e 1 e 1 ) , ij j 0 2 = 2 ψ 1 e 2 e S S = e 1) 2 χ e , 0 1 2 3 ij δ  S e 1 ∗ , ), where -boson exchange. The ( ,e ) (1 ~p e i i a 2 κ S 0 i + e +1 ) ψ χ g φ ξ S S 1 a S 0 3 ` ( e 1 e 2 and  c σ ( 0 4 ( ) +1 † 2 κ 2 ) + ˆ a ,e S c j e g 1 3 φ 2 of appendix ξ P | O 1 e ( S 3 j 4 e m ( 3 2 e i h P χ ˆ ( i ) + ˆ δ f i 2 ξ 0 0 2 a χ k 5 3 ˆ h e S h X 0 1 σ 1 ) † ( that we consider only leading-order 2 ,e arising from m c j S 2 2 ) and M δ α ) + ξ e 0 2 S ˆ S h 2 1 h 1 e e ∗ e S j P χ 1 ) for the Wilson coefficients, and the spin +1 κ m 3 e ,s 0 1 ) + 2.1.1 M i S ( e δ +1  1 µ X s 2 ˆ S f χ . Let us just mention here that the lowest- ( ) S 2 2.11 κ 1 2 3 S ( 4 ( g ) + + 2 → S 1 O 0 – 13 – ) + 0 2 1 ˆ h 0 S e . It is interesting to note that when we reduce = ˆ e , ij +1 1 0 1 2 ; for an operator with quantum numbers P S 0) i χ ( e , , are given in table , 1 4 ,e 3 2 j 1 1 1 M 0 2 e 2 ( ( 2 δm e (1 e e e χ ˆ ˆ h } between the two components, see ( χ f 0 1 i 1 χ 3 ψ 2 † e M e ,e = 1 χ c e 1 3 S δ ) 4 | 2 2 ) + is determined by the particle labels of the short-distance M e 1 ∗ χ e ) δm + 2 ) 1 2 i 1  a e L P 2 i S }{ ( e S 1 2 1 2 2 ξ c e 3 S M ( e 1) † 1 ( M ~p ) + ˆ c 1 j , which cannot change the spin and orbital angular momentum ˆ M f e , as can be easily checked by explicit computation. 0 f − ξ +1 { j are the total and reduced mass, respectively, of the two-particle = S S ∗ ∗ b  ˆ g pot 1 2 i h e i i 0 2 i ∗ ( ( ij δ e L ξ ˆ i 0 1 i f µ , ij , ij † a c 3 3 j ,e | P ) = 0) 1)  e , ij e e , , 2 ξ j 3 δ S 4 4 e ∗ 1) is postponed to section h e rel , (1 (1 e e χ 1 i S j 4 i and e v ) (0 ψ ψ e ,s → χ κ h h i , ij +1 ψ h ij X s 3 0) L,S S h e , ) the value of 2 2 ) ) ( ij ij 2 , ij light ij 4 M 1 2 b ( S ~p ~p (0 e ~ e ψ } ) is postponed to section ) takes the form → S a L,S ψ 0 3 j 2 ( e e 2.22 + 3 + + h χ 0 4 i +1 ψ e M χ S 2.20 2.12 = 2 ), ( }{ σ ( 0 2 e ˆ g ), where 0 1 e In terms of the non-relativistic wave functions the spin-averaged annihilation cross 4 ij 2.21 { ~p κ ( , there is a relative sign ( ˆ g In ( coefficient in the numeratortends over of all the potential interactions fraction.coefficients The of sum the in potentials, tion the of second ( term in ( and system. In addition we have used the relation between the matrix-elements of operators For a given two-particle state, themass frame relative is momentum related of to particles theO center-of-mass energy of the collision by where we have used thesums symmetry relations ( order perturbative result for thereplacing matrix-elements of four-fermion operators is obtained by section ( potential interactions in of the two-particle states. Boththe wave-function matrix-element components ofS operator definition of partial-wave configuration. Recall from section JHEP05(2015)115 2 6 κ = α and J (2.24) (2.25) (2.23) S term in , 2  α ) ) 1 J S the Sommerfeld P 3 . ) 3 ( s 1 ( } } ) S , ij χχ √ ij -wave operators ac- χχ 3 2 ij S ( e → }{ → 1 L,S }{ } , , ( ij e χχ ) ) ij ij χχ { states with spin χχ ˆ ψ { 0 1 f ˆ }{ → f ) S S ). The Sommerfeld factors ij J 1 3  of the incoming state. They χχ { χχ S ( ( . L )] 3 ) g 2 2 S LO 1 ) | ˆ ˆ h h J rel 2 2 ) +1 S v +1 P 3 S P J )] 3 ˆ 3 S m m ( 2 3 M 1 ( M M -suppressed L 2 ( δ δ ( h ˆ ( 2 ˆ } ˆ S f f f ) to one. The tree-level annihilation } 3 [ v 3 +1  e 5 3 ( ij χχ S ij 4 ) + ) + κ ij 2 e S }{ → 0 1 ( g S 2.24 [ˆ }{ } χχ ij S S ) + 2 )], where we have to set the χχ { ij 1 3 χχ ij ) means that only the leading order in 1 e → ˆ ( ( g S 1 S ) + ) + → P 1 1 }{ e . S 0 2 χχ 1 ij 3 { i ˆ ˆ h h ij ˆ ( S χχ with different total angular momentum f { P 2.23 ˆ +1 ˆ 1 ) + f 1 f ∗ ( S 0 χχ /M with center-of-mass energy ( J i M M | – 14 – } 2 δm δm 2 } S ij ( P j ) χχ ij 1 χχ ij ) ~p 3 ) + κ , ij S χ ( 3 0 → g partial-wave Wilson coefficients is sufficient, since i }{ → } }{ e [ˆ S ) + ) + = P χ ij 4 L,S ij χχ ij 0 1 2 χχ ij ( 3 e { χχ { ˆ +1 ( S S ˆ f }{ P → S ψ f ˆ S 1 3 LO 3 f ) should be kept in the denominator. For our purposes, h ij 2 | ( ( )]  χχ { J 1 3 ( ) ˆ ˆ f f 0 ) g S L ) acquires the simple form 1 S 2 S | O 1 )] ˆ and P ) also allows to incorporate the higher-order (hard) radiative corrections )] = +1 ( 0 1 ) = ) = ) = J M S h +1 ( 1 S 0 1 ˆ χχ 2.12 2 ˆ J -wave states f S 1 L f h ( P [ S S 2.23 2 ) reminds us that the knowledge of the spin-weighted sum over ( P ˆ 3 1 3 P  2 ( f ij κ 3 ( ( +1 } ~p ( ij , g S h h S ) between the leading and the [ˆ ˆ ij 0 ˆ ˆ f S 2 as the ratio f f 2.25 χχ ij ( = P }{ ˆ = 3 + S or, equivalently, of the relative velocity f → L i ij [ 2.20 { s  ij χχ κ rel χχ g v S √ | 2 ij 2 ij ) , describing the short-distance annihilation of ~p S M ˆ f light S + → +1 j S χ 2 i ( χ in a convenient way. Indeed, in terms of the Sommerfeld factors, the spin-averaged σ 2. The pure tree-level annihilation rate with no long-distance corrections is readily We define for an incoming state | P j In general, the definition ( , 6 1 χ i , χχ 0 recovered by setting all the Sommerfeldcross factors section in thus ( obtained depends only on theto diagonal the entry short-distance of part the of the Wilson annihilation coefficients into the Sommerfeld factor. The last relation inthe ( three different the leading-order potential interactionsamong while the being three spin-dependent, spin-1 do not discriminate where we have introduced the combinations of Wilson coefficients allow us to parametrizeχ the long-distance corrections toannihilation the cross annihilation section rate ( of the state The subscript “LO” in theof the denominator Wilson of coefficient ( this is only relevant forto the zero, case so of thatare ˆ functions of coefficients, orbital-momentum less, the relation ( quires the simplerh form familiar from the NRQCD applicationsenhancement to factor associated heavy to quarkonium, a generic Wilson coefficient or combination of Wilson the spectrum of two-particle states to just one state and the exchanged bosons are mass- JHEP05(2015)115 , ) S X S → (2.26) +1 S j 2 χ ( ) are the i j ) accurate χ χ ) tree cross 0 i 0 v χ 1 the matrix } 2.26 v ( ( ij → O are in a definite j ) is valid under O }{  χ 2 i i ij g χ { = 0. /v f 2.26 2 ∼ i ) m g v δ = 2.24 ) reproduces the tree-level -parity conservation implies δm R , ) 2.24 2 eq ) for the Yukawa potential. This n χ − , either directly or indirectly through 1 /m 2 )], which multiply the n ) resummed cross section, which would n 1 2 ( EW S i v . Of course, when 3 ( ) corrections to the potentials, hence the v ( , the sum of the particle number densities of 2 i i O /v h eff ˆ 2 i v, M f n σ [ g v, g 20 the typical temperature where the departure ij ( =1 / −h – 15 – N i S 0 O χ = P ) accuracy. Thus, ( ) therefore receives relative corrections formally of m 0 = v ( ∼ Hn )] and n 2.24 0 ) accuracy, and dresses each term by an f O 2 S T + 3 v 1 taking part in the relevant annihilation reactions ( . Note that we have written in ( ( j i h in the present Universe because dt , the sum of the equilibrium number densities of each particle ˆ O dn f χ χ 1 [ i with eq ) because for the diagonal entries ] have n ij χ S n f of the species involved in those reactions. Note that one can S S T ... [ 1 +1 . ij χ and S S 2 T ( H ) accuracy. When the Sommerfeld effect is very large, the Sommerfeld- j 2 χ to obtain } i v must decay into the lightest one by today. In this work the supersymmetric ) is derived using Maxwell-Boltzmann statistics for all species in thermal ( χ n ij i is effectively replaced by max ( → O χ }{ j v considered in the relic density calculation are the neutralinos and charginos. ij 2.26 χ i { i ) for χ h χ f ). This is of the same order as generic, non-enhanced electroweak radiative correc- 2 i , whose specific form is given below. The other quantities entering ( 2.26 i , g . Eq. ( , where We close this subsection with a comment on the parametric accuracy of our treat- v 2 v eq eff /v ( i, 2 i σ equilibrium instead of Fermi-Diracvery for good fermions approximation and for Bose-Einsteinheavy the dark for temperatures matter bosons, relevant ( for which thefrom is relic equilibrium a density takes calculation place). of the Furthermore, assumption the that evolution the equation particle’s ( phase-space distributions are proportional to those in co-annihilation effects with thehere neutralino, an to extension a ofenter larger the the EFT set Boltzmann framework of equation presented h through two-particle the states thermal would averageHubble be of parameter the needed. effective crossn The section, co-annihilation rates the later decaysolve into ( that all other particles For SUSY models where other particle species (staus for example) may have important that describes the time evolution of all supersymmetric particles which change the lightest neutralino number density element will be close to its tree-level2.2 value. Relic abundance The thermal relic abundanceBoltzmann equation of neutralino dark matter can be obtained by solving the tions. We also note that contraryrelation, to the bound-state two calculations, quantities where arestates independent with parameters variable here, energy. sinceg we The consider non-relativistic scattering matrixresummation elements can sum be all done terms for of any order value of annihilation cross section to Sommerfeld factor. This is notrequire the the same evaluation as of an section, with corrected cross section givenO by ( and not ment. As mentioned above,order we NRMSSM compute Lagrangian, the neglecting Sommerfeld operator factors matrix elements with the leading- corresponding to channel JHEP05(2015)115 with (2.28) (2.29) (2.27) In the /T ) in the 7 ) . 1 j into stan- rel due to the ]. It reads m v χ j s 2 36 χ − , i j ].
χ light m and and 34 + i → – i n j χ s/T χ m i 31 ( √ χ − σ e 1 but not on the energy. K T ij σ , which for neutralinos and , defined as the ratio of the i 2 j the four-momentum and energy of , and gets scaled out from the , χ i m n/s eq in any frame where the two particles E 2 i Hs j, 3 j ≡ m n 2 eq − s and for the Maxwell-Boltzmann equilib- − n Y eq i and √ = 2 written in terms of Lorentz invariants p i, 2 ) i /T i j n i v p E i · ) (there denoted as − ij eff i e ij ds/dt v ) with σ p v j h ( ij = E ij σ i ) corrections. Note that the thermal average of h i particles are maintained in kinetic equilibrium σ – 16 – ds E f 2 rel ( 2 i =1 / v , ) 1. N 2 j ( j χ X
i,j m m O  for ( 2 i + = /T i i m is computed as a function of the temperature rather than i ∞ m /T v ( m i − 1 Z v 2 2.1.3 the modified Bessel function of the second kind of order eff 2 ) m T σ j eff 2 is the so-called Møller velocity of particles K h n p σ j 4 2 i ) evaluated in the center-of-mass frame allow us to use the · h g K ij i π i m p v 4 g i ( that feeds into the equation for the dark matter relic density reads g 2.28 i q ] and further generalized to include co-annihilations in [ =1 s/T > v N N =1 X = i i,j X √ 35 eff , which arises from the asymptotic expansion of the Bessel function ij 2 σ i v h 2 eq π ], this is true if the 1 T can be calculated using 11 n 2 = 2, and v 30 i , with a factor of proportionality that depends on is the thermal average of the co-annihilation cross section of i i = = 11 ij g i f σ v i ij h eq v ij v n ) for large σ eff ij h the internal degrees of freedom of particle species , and is equal to the relative velocity of particles σ σ i h h i is defined by s/T g ij The efficiency of dark matter production and annihilation processes to maintain chem- The quantity √ v ( 7 1 . The integrand in ( expansion of the Universe is theBoltzmann equation. same, namely Since particle move collinearly. ical equilibrium with the plasmathe in Boltzmann an equation expanding Universe is canparticle written be density in better to terms understood the of entropy when suming density that yield the in entropy the per comoving comoving volume cosmic is frame. conserved, the This change of is because, as- n expressions obtained in section non-relativistic limit including up to a heavy co-annihilation channelrespect is typically to suppressed byK a factor with charginos is rium distributions. A generalwas expression first for derived in [ where dard model particles and cosmic comoving frame, i.e. theaverage frame where the plasma is at rest as a whole, the thermal scattering processes stop toin be the effective context and of the Sommerfeld-enhancement have consequences been of investigated early in kinetic [ decoupling As argued inthrough [ elastic scattering withthermal the plasma much during more abundant allIn standard of this model their work particles evolution, indecoupling we even the sufficiently after assume long they that until leave dark freeze-out chemical is matter equilibrium. completed. is The kept possibility in that kinetic the equilibrium elastic after chemical equilibrium, JHEP05(2015)115 . . 1 Y ] as T − 35 eq given (2.34) (2.33) (2.30) (2.31) (2.32) Y , where Mpc eq 2 = 1 n h − Y = 1, where one ) takes the form x 2.26 , . ) above. ) 3 at high temperatures (low 2 eq T Y eq 2.33 2 , and chemical decoupling of , can be finally written as [ in the evolution equation by , Y π ) 45 − eq t 2 ), is known, the neutralino relic  2 Y eq Y 2 0 ) crit 400 MeV), where there is a sharp Y is defined as x eff Y T ( of the energy and entropy densities: ( ( /ρ − 2 dT ) numerically from − i 0 / Y dh eff v 1 eff ∗ 2 Y g h 0 h ≡ eff Y 150 eff s 2.31 ( T σ 1 0 h = i is the current photon temperature in the h the entropy density of the present universe, ∼ 3 Y v m and 1 0 0 s T eff T = m , the Boltzmann equation ( 2 σ eff 1 + can be obtained using the formula for 2 h g x / – 17 –  , s 1 ∗ /T crit eq ) to its energy density. In a radiation-dominated 2 ds dx 1 g 4 / Y eff /ρ 1 eff H T m was initially close to h g G H 1 2 , where π 3 = DM 0 0 30 π Y 45 = ρ ) x = 2 /T r T / ). It is customary to provide the relic density as Ω 1 as the temperature decreases. At a given temperature the ( constant) is not large enough to provide the change in 1 ∗ − g m dx Ω = 2 eff dY eq / g = 2.33 1 ∗ Y = g = 0 dx dY ρ x the critical density and ), we see that the change in the yield tries to compensate for deviations (take . The final step requires the Friedmann equation that relates the ex- 2 πG 8 /s 2.31 /x / i 2 ]. The equilibrium yield eq v n H 37 eff , = σ h = 3 35 eq the gravitational constant. The parameter ) and the parametrization of the entropy density in ( Y crit G ρ In order to obtain the relic density we solve ( In the form ( 2.29 is the value of the Hubble parameter determined at present in units of 100 km s ) it will continue to track with that can be calculated from ( h can safely assume thatinitial the condition, yield up still to Universe. tracks Once the the equilibrium presentdensity value value today and of in the take units yield, of the critical density is determined by the relation needed to keep upthe massive with particle the from rapid theuniverse plasma cools exponential takes down drop place. (phenomenon of cross The known section yield is as then further “sudden remains increasedthat constant freeze-out”) for is as lower or, the the temperatures if case the when the yield the continues annihilation annihilation to be rates reduced; get enhanced by Sommerfeld corrections. in ( from the equilibrium, suchx that if prefactor The effective degrees ofat freedom the are QCD quark-hadron slowly-varyingincrease phase functions [ transition of ( the temperature except with in terms of the effective degrees of freedom universe at early times, the equation for the evolution of In terms of the yield, and defining where pansion of the Universe (Hubble rate time, it is also convenient to trade the independent variable JHEP05(2015)115 . 3 v = 2; e ]. / χ eff LSP ) 4 41 σ D 3 e ] and m p χ below. 0 ), the ~ 38 [  − 4 s, ) we have 4 √ p T LSP ( eff = ( = ( /m h ). The potential Mathematica 2 0 0 P 2 p ~p DarkSUSY = ∼ j has been obtained by ) and 0 p , both taken from [ 0 T 0 3 p ( + Y 2 − / i 1 ∗ ], pair, i.e. p g 3 16 e , as is shown in section χ into several pieces and adapting 4 pair, e GeV cm EW 8 j 2 χ . Arrows in this picture indicate the α j χ h i 5 χ i χ − 4 3 χ = 10 e e p 10 χ χ − x ′ × states, which contributes to the set of radiative p χχ p p – 18 – + − 05368 = 1 to 2 1 . e e 2 2 P P x χ χ = 1 crit ρ running inside a box graph with internal fermions ], which can be found conveniently tabulated as a function 0 35 p pair before and after the interaction (see figure i j p p j i χ χ χχ . For the results that we present in [ ]. Other numerical values needed for the computation of the relic 2 7255 K and . bv 40 , + = 2 39 [ a 0 ) using numerical routines for differential equations built in T ), and fermion and boson propagators can be expanded accordingly. To leading p . Box subgraph of a characteristic diagram with multiple ladder-like exchanges of 2.26 In the center-of-mass system of the incoming In the presence of Sommerfeld corrections the effective annihilation cross section the momentum carried by therelative exchanged momenta boson in the is then equalloop-momentum to routed in the this difference way between is(idem the characterized for by the scaling order in the potential-region expansion, the denominator of the boson propagator, box graphs, which have toA be summed necessary to input all to ordersby perform in electroweak- and such Higgs-boson resummation exchange, are which the are potential given interactions inpotential generated this section. loop momentum can be chosen to be the relative momentum of the Sommerfeld enhancement occurs when thesignificantly interaction distorts between the their incoming two-bodyterms DM wave of particles Feynman function diagrams the away effect froma arises lesser from the extent, the Higgs) exchange initial of bosons plane electroweak betweenlimit gauge the wave. (and the neutralinos to and dominant In charginos. contribution arises In from the non-relativistic the potential loop momentum region in ladder micrOMEGAs density are 3 MSSM potentials Due to the stiffnessis of done the by evolution splittingthe equation, the starting the region and numerical from maximum integrationused step of the sizes the values in equation derivedof in each temperature [ of them. among the For package files of the automated programs acquires a complicated velocitysimple dependence form that cansolving no ( longer be parameterized in the Figure 2 vector and Higgs bosonscorrections among to intermediate thedirection annihilation of of the labelled the momenta. incoming pair JHEP05(2015)115 . = ∼ ij ) † rel (3.2) (3.3) (3.4) (3.1) . + ), the m v , m j H + e 2 is the A 0 m p χ / and thus LSP A i ) + e j 1 m p χ In the latter − ] , and ( ∼ 2 − j 8 0 0 e i -odd ( ij ). eigenstates. For χ M p ) j i † 0 e pairs, that in the 0 e ~p,~p + + m χ CP -field in the interaction χ 4 i = ( e H . We note that the 2 + e + CP ) s χ = χ ij W 3 ~p pot p j e . e = , L for on-shell charginos or 2) or χ † − χ , ] j 2 i Z 0 3 e e + e µ ~p χ χ i χ ) has to be considered as well, , or a charged Higgs boson i and = 1 [( ξM e W -boson exchange, we illustrate 4 j , [ e + e − χ − 2 Z j − 3.3 e V µ χ ), where χ + e i 2 -boson propagator, , m χ 2 A = + e χ v 1 q j Z 0 ] m φ , e χ → e 5 j ν χ , 2 χ H Z 2 e pot γ q j LSP ] e 0 e µ µ χ 5 D M ) interaction vertices of charginos and χ m q ] χ γ 1 γ ( a i 5 e 0 e ij µ -boson (then + γ ) χ χ γ couplings are given in the appendix † Z ij -even ( flow. For consistency, the  ij p ∼ O W ij a ν + = 2 a Z q bosons. χ + CP µ + , p – 19 – -field (which sets j M q LSP e ij ij + ( µ W + s χ γ µ -boson exchange is obtained at leading order in the [ i − , s /m e W i ij γ ) has to be added to describe the charge-conjugated . Note, however, that our results apply as well to the Z e ij v χ 2 ij j ij µν 0 e -field annihilates positive charginos. Therefore, the arrow on the [ χ p ) g 3.1 i † + χ 2 , a e i ), axial-vector (  χ g + e ij W χ v . By means of an example, 2 v , which we neglect consistently given that we are dealing v Z χ 2 2 a with g 3 ) [( ), or the M i = 1 3 j i + e 0 e j − , in the particle spinors and − m e stands for either the χ χ 2 i χ 2 rel 2 can be any of the + e i q − V µ v g e χ φ 4 A χ = m becomes energy independent, LSP j 1 m e − χ ] i ∼ 2 e 0 χ 0 2), then M , term dropped in the boson propagator has a term proportional to the mass , p − 0 2 ) must annihilate positively charged ) p 2 = 1 0 ) p 3.1 p , m − and The amplitude of figure Let us denote the vector ( The leading-order potential interactions in the EFT can be obtained by expanding − -violating MSSM where the neutral Higgs particles are no longer We adopt the convention that the ± 0 0 m 8 Z 2) Higgs bosons, in which case 0 p , H p fermion line for a chargino refersvertex to ( the direction of which introduces the hermitian conjugatesExplicit of the expressions coupling for matrices, all ( the non-relativistic expansion by taking the limitM of small relative momenta, where the scalar field 1 ( CP charged Higgs-boson exchange the hermitian conjugate of ( Likewise, the interactiongeneric vertices form of charginos/neutralinos with Higgs particles has the where the spin-1 field photon field ( case, the hermitian conjugateinteraction, of i.e. ( scattering particles, see figure in the following the essential steps to perform thisneutralinos tree-level with matching. gauge bosons in the MSSM generically as assumed to be atincoming most relative of momentum. order of the full-theory tree-level scatteringneutralinos amplitude exchanging a gaugedirection or of Higgs the fermion boson. flow have Three to tree be considered diagrams depending which on differ the in particle nature the of the represents an instantaneous interactionnon-relativistic between the EFT is taken( into account bydifference the squared potentials ( in with a set of nearly mass-degenerate with the neutralino LSP and mass differences are [( JHEP05(2015)115 . 2 2 rel α v 3  (3.5) (3.6) α i δ σ -boson LSP 1 α ) = 1, is Z m ⊗ 4 p i α ( a. Written ), is equiv- δ ∼ σ − 2 u 3 ) 2 0 p χ ≡ 0 e ( p − 3 ( u Z e χ 1 † , p µ a 0 u 1 ⊗ ˜ Γ p ) can be obtained → e ) 1 4 3 − Z e p 3.5 χ a ( and , − u and i − χ Z ) ¯ σ , 2 1 1 ) -boson. For simplicity we α p M c 1 3 ⊗ ( Z ⊗ i α i u ∼ ⊗ , 1 σ µ σ 1 0 1 2 1 rel e α )Γ ) , -dependent term of the 3 4 v 4 2 Z e ⊗ ξ i → α 0 e p v → − σ ( 1 5 1 m LSP e u 5 2 → 4 ≡ e qγ / γ Z e m − 3 5 − i µ a, ¯ − − Z v e γ 3 χ ⊗ γ σ v χ χ ∼ e µ 3  + 0 1 5 + 0 ) γ 2 e ⊗ χ ⊗ χ m 3 e χ 4 i Z 3 qγ / e 5 p ⊗ )( e σ v → γ → 1 ~p,~p → µ e µ − 2 q , / − − 0 2 e Z δm 2 χ m 3 χ χ – 20 – , γ p ⊗ e 1 0 0 + M e µ χ χ − 5 χ 4 ) arises from the = γ e , γ 2 4 Z δm )( ) we thus obtain e qγ / 1 1 b 1 ⊗ -channel boson exchange that generate the leading-order 3.6 p M , e t δm m 5 4 3.5 5 ⊗ ( processes. e γ − 1 µ ). The relation in the third line of ( qγ 4 / + γ 1 + p δm χ ⊗ → − → 3.5 + 2 p Z = q / q /   µ χ a when the exchanged particle is a q − γ 4 3 ′ 2 ⊗ 2 Z 2 3 ξM + e e → 0 + p ⊗ q / χ 2 2 ig χ χ χ χ M + µ + + 0 0 = ig -gauge. At leading-order, the non-relativistic expansion of the prod- 2 χ γ χ χ χ + ξ q − ~q + 2 R → → → χ ~q + 0 + + χ χ χ = ). Gauge invariance requires that this term is cancelled against the contribu- 0 2 1 0 + e e and χ Z χ χ χ χ 3.4 0 )( χ a 0 ( . Tree-level diagrams with χ → The term in the second line of ( 0 (figure 3a) χ 0 of the amplitude in figure have considered the caseχ where the couplings in both vertices are equal,propagator which ( applies to for the leading-order term in the expansion in in components, the spin operatorsThe above read use of theimplied non-relativistic in normalization the for replacements theusing ( relativistic the spinors, equation of motion forscaling the of relativistic this spinors; we term comment below. on the Using non-relativistic ( in the full-theory amplitude, wherestood the as right-hand-side of the these matricesneutralinos relations and acting should charginos be on at under- the the upper two-component and Pauli lower interaction spinors vertices of of the figure non-relativistic alent to the replacements ( potential among non-relativistic neutralinoslines and indicate charginos. the The fermion arrows flow,written in such below. the that neutralino/chargino each diagram contributes only to the scatteringhere processes defined in the uct of Dirac bilinears in the amplitude of diagram Figure 3 JHEP05(2015)115 , . 2 − Z e 0 i 3 . If and 0 at G e  M (3.7) ) and i m s refers 9 ( ) typi- 1 σ → LSP e  0 EW ij 4 Z ij 5 ⊗ G e 2 3.6 v ), change M γ s i v /m b 2 2 . The can- δm σ g ⊗ − 2 Z ( 3 5 to at most EW - and neutral e LSP O 2 Z Z e a ξM , γ m M is a 1 1 LSP ∼ e ij 4 ⊗ Z e 5 Z /m δm a , γ + 2 /M EW 5 ij 1 γ . M ) one obtains the condition to go from ⊗ ⊗ 1 3 can then be easily obtained δm ∼ e = 0 (3.8) , 1 1 3.6 2 ⊗ 3 e 2 ij 3 1 e 0 e and their diagonalization prop- 1 3 X 2 ) holds provided that G e can be large, but is multiplied by 2 Z e ⊗ exchange by expanding the scalar δm e s A v ) to ( , which has a mass 3 1 1 Z → 1 3.8 0 φ φ e . If the mass difference e Z 0 e ). 3 4 s 4 2 G e G e 1 ) together with the pseudo-Goldstone Z e e 2 in figure /M e s 3 3.6 Z v ) because e ξM 4 e ≡ c ij v (1) due to suppressed vector couplings. φ e  3.6 + → ∞ 1 s 0 2 X = 2 3.7 e O 2 e 4 A 2 φ δm e 0 Z 3 e 3 and e G v Z e ) in the electroweak doublet, but much smaller, m LSP 2 2 v b 2 term in ( m 1 2 e Z m + δm ( ig e LSP 3 2 – 21 – Z 4 e 1 2 0 Z e M e ~q v 4 /m G /M e 2 Z δm 2 2 ). Since the axial couplings go to zero in the decou- e = e 2 = 1 EW 3 3 M : e e δm e φ 4 b 3.8 M φ a with scalar-boson e ( δm ) are suppressed at least as 2 3 O Z δm 1 1 + e δm 1 4 M 3.6 e ) for e 4 -dependent terms. Eq. ( 2 Z e   ξ is of 3.7 δm 2 2 Z 2 ξM δm (figure 3a) ij ig M + 2 2 ), and the Dirac bilinear in the non-relativistic limit; the result reads δm 2 2 φ ig + ~q 2 m ) are given in this limit by the pseudo-Goldstone couplings ~q − , a relation which can be proven using the explicit definition of the cou- − , as follows from ( belong to different multiplets, 0 ). For a very heavy LSP, where we could nevertheless have 3.6 2 = 2 ij ij G q j v ( Z is ), in the triplet. In the latter case, the radiative correction to i/ ± /δm LSP and 3 LSP ): for every line where the fermion flow has been reversed we need to interchange = i m ( EW Z /m 3.6 M 4 The corresponding results for diagrams The term proportional to the mass differences in the first line of the potential ( EW /M ∼ O The mass splitting ∼ (figure 3b) ) 9 M j ( Z ij suppressed contribution for the O dominates over the tree-level mass splitting in the LSP mass region of interest. However, it still yields a the antiparticle spinors asdue the to charge-conjugate Wick ordering. ofHiggs-boson particle For exchange ones), the for diagram sake and of add clarity, a let global us sign write the results for pling limit, it follows thatthe first the line gauge-independent of off-diagonal ( terms in square brackets in from ( the labels in thethe vertex sign couplings of (for the instance, scalar, pseudo-scalar and axial-vector couplings (which arises when writing To see this, considerto the particles decoupling within limit thethe same electroweak mass-difference multiplet terms then inparticles ( v contribution to the potential. cally yields a very small contribution,δm since in the EFT we treatthe only size those species of for which this term would still be at most of for the cancellation ofm the plings in terms of theerties. mixing matrices We given can in therefore appendix drop the second line in ( The pseudoscalar interactions do not survivethe in leading ( order. Adding ( cellation holds because certain conditions amongtralinos the and vector charginos and are scalar fulfilled couplings inbution of the corresponding the MSSM. neu- to To see diagram this,propagator, write the leading-order contri- tion from the exchange of the Goldstone boson JHEP05(2015)115 2 1 ~ S ⊗ part (3.9) 1 (3.11) (3.13) (3.10) (3.12) µν g , r X ) gives the Coulomb r m , , − ) . For these amplitudes, ) e 2 0 3.11 1 ~q χ ) , among the incoming and ( 3 ⊗ ± 3 e X e 4 χ , 1 χχ 4 e m e 2 → 2 e 3 ( e 1 → mr 1 e πr the spin operators acting on the χχ − e b − 4 ± and charged Higgs-boson exchange e ) 2 2 , χ i T 1 S ~ 0 S = ~x ~s 4 W · χ 2 i~q − e m ) = 2 3 (3 1 and 2 ) 2 ~q + 3 ~s π 1 in the potentials. Since the operator − 2 ∓ d , ), with 3 ~q . (2 − χ e ~σ – 22 – 4 2 ~x ± 1 1 · e = 0 Z ~s ⊗ χ 2 i~q e ⊗ e 1 S 1 − e 1 3 → ) for charginos and neutralinos. Working in the basis 2 ) = a ) ~q 3 + 1 r e 0 3 ~ π S ( 2 d 1 χ } φ e ⊗ ) for (2 0 3 s e 1 ) = ( 1 1 ⊗ χ 4 stands for the momentum-space amplitude as given above e r = 2( . Applied to the massless case, ( e Z 4 ( ⊗ ~σ 4 ( i φ e 3 } χχ }{ m e / s 3 σ 1 2 χχ 4 e → 2 ( ( e e in terms of the total spin operator as 4 2 φ 1 / → 2 ⊗ e e i S e χχ 1 i { m 1 χχ 2 σ }{ 2 χχ e σ V ) = 3 2 → e + 0 for the photon propagator. T ig ≡ 1 ⊗ 1 2 e i χχ ) = 2 2 ). From the identity { ⊗ ~q → σ ~s 1 V . The potential from charged boson exchange can be obtained from the 1 Z 3.9 = − + ⊗ , and | χ φ 1 M pairs. For leading-order scalar boson and photon exchange, the coefficient 1 ), ( Before we write the result for the potentials in coordinate space, let us write ~s ~x prescription in the full-theory gauge boson propagators provides the necessary regularization + 1) ( 3.7 χχ = 10 s i ≡ | ( = 0 case. + 1) ( s ~ S r ) with ), ( m S ( The amplitudes above provide the potentials in momentum space. However, the The potentials generated by photon exchange can be obtained from the result for the by or a final 3.4 3.6 The + (figure 3b) S 2 , 10 -boson case by plugging the corresponding vector coupling and keeping only the + 2 1 for the case ofoutgoing the exchange of a vector boson withfor mass the by is the identity operator, wetotal find spin that and a in leading-order coordinate potential space contribution thus in has the the basis form of where particles at the upper and lower vertices in theof scattering eigenstates of diagram, total and spin we for have the replaced neutralino and chargino pairs, we can as well replace we immediately obtain the well-known Yukawa-likeof potential for a amplitudes with force exchange potential. carrier of mass the spin operator where in ( and neutralino pairs acquirespotentials a are obtained simpler by form taking in the coordinate Fourier transform space. The coordinate-space we have to hermitian conjugateχ the coupling matrices at verticesneutral involving boson either amplitudes an initial with trivial replacements in the couplingsSchr¨odingerequation and which the sums boson the mass. all-order exchange of bosons among the chargino Z of ( are relevant for the processes ~s JHEP05(2015)115 , ˆ f } ) 3 e into (the 4 χχ e ( 1 (i.e. (3.14) (3.15) (3.16) 2 4 , e }{ e → ) times ± 1 2 ) } χ e e and sum 4 3 ) 1 e χχ charginos, e = e ( 1 3 { . If all four 3.13 e χ e χχ V + 2 + , Q χ χχ , provided we }{ n 2 n = 24 states in 2 2 e → → e e 0 1 χ = ∓ 2 2 e χχ e { ≤ N χ 2 χ V . The two potential ) are accounted for 1 ± 2 1 N 4 e − 2 S e and e χ + χ , χ χ have identical particle 2 2. Therefore, the total L e ∓ 1 and = 1 ± 6= χ χ 1) with e 1 , 8 3 + ± 2 e and χ 2 − = e χ e 2 n χ , e 3 χ 0 χ 0 4 e , χ Q χ − 1 + ( 1 n . neutralinos and χ χ ∓ 2 e = 4 states in the singly-charged χ } 4 ± 2 0 χ χ 3 that denote a two-particle state e are interchanged. At this point, in the diagrams, integrated over 2 e n different states with = 2 ± 1 = to denote a state in the new basis, χ ± 2 4 3 } N and , χ e n 1 e χ 7 2 is the total number of channels for + ... ± 2 χ χχ , χ 2 }{ χ ± 2 ) N → n | e 2 , e χ , χ ∓ 1 m → 0 ). e 1 Q 0 4 4 + 2 χ 2 | 1 χ χ e n e , χ 1 e χχ e χ χχ e { { ± 1 χ N ± 1 3.9 χ χ V 1 χ = e , χ ± 2 6 = 16 and ), ( χ = 0 4 , . . . , χ χ 1 states of charge χ } , χ , ) ± 2 , whereas 3 0 4 3.6 N – 23 – e ± 2 + 1 ± χ 2 + 4 χχ 0 1 , where e χ χ ( χ , χ n | . The potential entries for method-2, 0 3 ± 1 ± }{ Q 3 → | = 2 ) , χ χ e χ 2. For the case of e 0 1 + 2 0 4 5 , 1 , χ χχ χ e 2 0 2 + 1 ( ≤ χ ± 1 { neutral states, 2 , ± 1 n n 4 -boson exchange the spin-dependent part of the potential V χ ,...N e ∓ ± 1 = , χ W χ = 0 χ 0 , . . . , χ ± 1 4 and ± 0 1 | with ladder-like exchanges in the non-relativistic limit, that = 1 χ N and χ χ by analysing the perturbative expansion of the annihilation 0 1 Q ), and : 0 2 | B 0 χ 2 m + 2 4 ,... - and e 3 χ X n , χ e Z ± 0 2 A ≤ χ = 1 χ 6= X 1 0 1 i 4 e and the one where particles or e and 2 , χ → 2 = 2), then we have to consider the interactions among for 0 1 0 ± i 0 j χ χ χ + and 0 1 0 0 χ χ n i χ 2 χ χ χ e = 0 2 i n vanishes. For 6= , where we have introduced the notation ( χ to a single label 3 1 3 e e e 2 4 4 e e e An alternative basis of two-particle states with a smaller number of states can be used, We show in section = 4 and ) 2 χ e 0 1 1 e e χχ n in method-2 by a( single potential entryand assumed describing that the scatteringare obtained of from state those ( of method-1 as content. The new basis, thatfrom we the shall basis refer just to described as above,identify “method-2 is basis” built in only order by to states contributions distinguish of it method-1 correspondingso-called to “crossed” contribution, different from the former if The ordering of the states in each sector iswhich of takes course into a matter account of the convention. fact that states with 24 different states in total, whereas in the charge-1 sector weand have 16 just channels, 4 in the charge-2 sector, neutralinos and the two charginos are( relevant for the long-distance partthe of the neutral annihilation sector, as welland as doubly-charged those sectors, among respectively. Forneutralinos instance, and in two the charginos, charge-0 the sector single with label all runs four over states χ each total charge sector, we have of the type number of neutral states is corresponding loop momenta,with and the finally appropriate quantum multipliedas numbers. by for No the the additional short-distance combinatoric internalover factors coefficient states all arise we of as consider them. long graph as like different Diagrammatically figure the this states it means is for convenient instance to that switch we from have the to two consider indices a arises from the axial-vector coupling, see ( amplitude the amplitude can bethe two-particle described propagator as of a the product internal of pairs potential interactions like ( b JHEP05(2015)115 3 ], = = e 4). 4 e 15 and , 2 ) even light ˜ 1 N is odd (3.17) 16 . The S m, n L , χχ → 3 n S ( − 1 + ] and [ χ → → L 14 ) arises from + 1 2 1, and χ e ) = (24 m 1 2 ± e 3.17 ) = 0 otherwise. ,N = charginos, contains id 1 χχ 2 if the two-particle n ) by dropping states + , Q ,N n 0 = 1 3.16 N . An additional rule has to , , id light particles. In method-1 L ), ( n } } 4 3 1) e e . → 4 4 − e e } 3.15 4 11 χχ χχ e }{ }{ states with ) 4 2 1 → → e e e ), ( − (i.e. with quantum numbers 1 1 + , where in the case of method-1, and χχ χ e e }{ χχ χχ n { { 1 id | S neutralinos and + 1) comes from Wick ordering, and the → e 0 n V V 3.14 Q 1 0 ) and the selection rule just mentioned χ | + n − e ( 2 2 χχ n 2) { L √ √ V √ = → 3.17 / , whereas for the annihilation matrices, the = = = 1 in the spin wave function and a change of sign n ,...N 11 ˜ } } } ) N ) ) ) 4 3 4 0 – 24 – e e e +1 → 4 4 4 χ = 1 χχ χχ χχ S e e e ( ( ( 0 2. The reduction in the number of states in method- 1) m 3), whereas for method-1 ( }{ }{ }{ χ partial-wave configuration is given by the Wilson co- → → → . In what follows, quantities with indices referring 2 1 1 ) ) ) , , where the ( ± e e e − J 8 1 1 1 m, n χχ χχ χχ , e e e L +1 ( ( ( = { { { S V V V +1 Q are formed of identical particles, and i > j 1) S 2 3 − e ( 4 ) multiplied by (1 ), with ) = (14 e neutral states, J 2 r ) × : : : ( ˜ L N where L 2 + 3 3 3 , χχ n mn e e e j 1 +1 1) ˜ V S χ under exchange of the particle labels, see (8) and (6) in [ N i 2 − = 6= = , ( ( ˆ χ 0 f 4 4 4 } 2 + ) is the absorptive part of the one-loop amplitude for 3 ˜ e e e produces a factor ( / × N e J 4 4 ) and e L or/and ( e 1) J light, which cancel each other by virtue of the symmetry properties of the }{ χχ in front of the crossed diagram amplitude in the first line of ( 2 and and and describes the scattering − + 1) +1 2 L e ↔ e → 2 2 2 1 S 2 states with charge S 0 1 → e 2 e e e / 3 for method-2. We shall also adopt the convention that the potential matrix +1 e + ) n ( mn χχ e { S ( L ˆ | + 1 V = = 6= 2 f 0 ( Q χ mn χχ 1) | n 1 1 1 ˆ +1) f − 1 ˜ e e e − N + mn χ The basis of states in method-2 for the case of Likewise, the annihilation matrices built from the Wilson coefficients of the four- ˆ in the case at hand), the entries involving such states for the potential when = n f ( 1 0 + ,... P ˜ and the potentials, willi.e. be written using the1 single-label notation withelement lower-case letters, entry 2 as compared tothe method-1 non-relativistic EFT is framework. more For significantconsidered instance, as if then all more ( neutralino particle andThe chargino species species list are are of considered method-2 in that states differ can only be read in to off two-particle from states, such ( as the Wilson coefficients that built the annihilation matrices states ( N n fermion operators must beNamely, the supplemented absorptive with part additionalinvolving of method-2 factors states the in in one-loop a theefficient annihilation method-2 amplitude basis. ( and Wilson coefficients respectively. We note thatimply the that the the relations potentialat ( in the method-2 leading-order depends in the on non-relativistic the expansion. orbital angular momentum 1 have to be set to zero.for This prevents a that method-2 forbidden yields atransition state non-zero to annihilation of an amplitude allowed two state, such identicalthis as transition ( neutralinos is automatically or zero because charginos there are through two annihilation an terms, intermediate The ( the product ( exchange in the relative momentum which translatesbe into accounted the for factor when ( buildingparticles the cannot potential form matrix a for 2-particle method-2: state with since odd identical spin-1/2 JHEP05(2015)115 . ± 1 χ and 2 -wave . The (3.23) (3.22) (3.18) (3.19) (3.20) (3.21) i S 1 + δ χ 1 − = , , , χ ], and can be 1 1 − , ,         15 χ N i r 1 1 + Z     Z 1 derives from the . The annihilation − r , r M − 1 , χ Z r 1 1 − r 1 0 0 0 0 1 χ r e M W , − χ W = 0 + 1 − r 1 0 r   M e ) ) together with the rule M 2 W 0 00 0 1 0 − χ S 2 c 2 − , χ e 2 χ e 2 W 0 1 2 0 L,S state:     2 c ( ca m 2 χ α πα 3.17 , 0 1 2 2 2 α ψ 0 1 2 χ α χ α ) − χ − 2 0 1   m 25 24 πα J LO r α χ | − r √ L ) Z 1 r − α J − 25 24 = r M +1 P L ) using this notation reads − − 1 S e 2 = (2) ( +1 r or S )] Z 2 W cb 2.23 2 ˆ r 1 r c 1 f M ( (1) S r 2 − W ∗ S )] 3 e r aa 3 i α M W 1 ˆ ( ) f r ˆ − f S M 2 W e [ − 3 − c L,S 0 ( e 0 2 , and its mass-degenerate chargino partners r – 25 – α ( ba ˆ 2 f χ α 2 [ , ψ α − ) can be obtained from the analytic formulae given α 2 h m 3) correspond to channels − 2 − rel √ , , , and the same equation holds in both methods. In v 2 r α   00 0 − a ( 2 refer now to channels , )] = , 2 − O 1 3 2 J     √     L = 1 3 2 3 2 = 1 2 after plugging in the values of the neutralino and chargino 1 1 r r +1 ) = ) = √ 3 2 3 2 r r S W W A ( ( 2 r r , with mass 2 1 1 1 1 m, n   M M ( S S 0 1 m, n ˆ − − f 2 2 + + 0 χ e e [ 2 χ     L L a 2 2 2 2 m 2 χ πα S α α odd m , πα even − − = , =0     = =0 (2) (2) Q (2) Q )] V V (1) 0 ) = )] S r : 0 1 we demonstrate explicitly for a simplified example that method-1 and method-2 ( 1 ( i S ]. In the pure-wino limit, the explicit expressions for the absorptive parts of ˆ δ B f 1 =0 [ ( (1) Q ˆ f 15 = [ V , To provide an example, let us give both the potential and the annihilation matrices in 1 14 i ± in method-1, and read off from (147) andannihilation tables matrices 5-7 read therein. In the neutral sector, the leading order where the matrix indices matrices for any MSSM modelin to [ the Wilson coefficients have been worked out in detail in appendix C of [ where the matrix indicespotential ( for method-2 thenthat follows sets from to the zero entries rules involving given a in forbidden ( formulae given in appendix mixing matrix entries relevantZ to the calculation in the pure-wino, i.e. yield the same Sommerfeld factors. the well-known wino limit ofwino the lightest neutralino MSSM. The relevantIn particles the in neutral this sector, limit the are potential the matrix pure- of method-1 for both for an incoming two-particleappendix state formula for the Sommerfeld enhancement factor ( JHEP05(2015)115 ) , ij 2 e (4.1) ), we 1 L,S -wave ( e (3.24) (3.25) ψ P , which S 2.17 )] defined + . . J L , L    1) ) +1 , ij     2 3 2 − 1 1 S . They receive e √ 3 2 3 2 2
2 L,S ( ( e J ˆ 2 1 3 2 3 2 f 1 [ ψ L √ 2 1 1 1 1 ij S +1 S +   S     L 2 2 χ denotes the relative mo- 2 2 2 2 4 χ 1) } m ~p 1 m πα − πα 2 e 2 . e 7 3 7 3 L ]. Let us also write the + ( ). With this definition }{ χχ 3 3 = = e ) → , ij 4 4.3 2 e e (1) (2) χχ { 1 L,S   ( e O ) ) ψ is a polynomial in relative momentum J J 2 2  ] P P K 3 3 M M ~p ( ( [ ˆ ˆ f f K   i i ξ – 26 – Γ , † c j ξ     h 1 , = − i 1 1 j   ) have been given before in [ χ − as in or out state are equal up to a factor ( i 1, respectively, and 0 1 0 0 0 0 1 0 0 0 χ , | + 1 particle as defined below in ( 1 3.23 χ       e j − 1 χ 2 2 2 2 = 0 4 χ 2 χ χ  χ S m m πα πα ∂ ←→ ↔ ], respectively). The potential and annihilation matrices for the other 1 6 1 6 i and 2 by computing the left-hand side of for − 1 i = = 15 − ~σ χ ) χ , , ij 2  , + 1 e 2 (2) (1) 14 1 χ L,S K (   e × 2 ) ) Γ ψ 1 1 2 † 2 2 c e P P χ 1 1 ). Within the non-relativistic MSSM the Sommerfeld enhancement arises from the | M M ( ( 0 ˆ ˆ f f h In the first part of this section, we establish the relation between the non-relativistic   2.23 problem. This isgeneral our MSSM main parameter points. result, since Inis practice, time-consuming it the and allows solution often not with to relevant, theon since compute full the the the number heavier final of channels Sommerfeld result. have channels only factorsto In a the for the small treatment effect final of part heavier channels, of which this is section accurate and we fast therefore for describe most an practical purposes. approximation (NR) matrix element above, diagrammatic resummation andSchr¨odingerequation. the We solution of then a describe multi-channel We the shall find actual that solution the of standardcases method the with fails kinematically Schr¨odingerequation. to closed provide channels an and accurate result derive in an many alternative relevant method that solves this Here Γ = 1 (derivatives) corresponding to amentum given of angular the momentum is normalized to one,evaluated when in the the Sommerfeld tree effect approximation. is neglected, and the matrix element is large quantum corrections, which have todetermine be summed to all orders. Generalizing ( 4 Sommerfeld enhancement In this section we discuss thein computation ( of the Sommerfeld factors matrix elements of annihilation operators such as replacement arises as a consequenceand of (2.4) the of exchange [ ofcharge labels sectors in can the be Wilson obtained coefficients similarly. (see eqs. (2.7) Recall that the entries in the annihilation matrices of method-1 which differ only in the annihilation matrices in both methods: in method-2. The results ( JHEP05(2015)115 , µ (4.4) (4.2) (4.3) . The |  2 a k ~ m  | + 1 0 a k m ≡ a i by assumption, both A B , 2 M + X X v ij 2 2 . The masses of the two- a 2 µ ~p 2 the reduced mass. Since 2 LSP v m , and 2 ij m 2 − ~p, a a µ LSP 2 E ≡ m − ) m m 2 a 1 2 , ~ e e k P . 1 a i Γ 2 ij + ) , and scale as m m a µ j a j k E, ~ µ p χ ∼ E ∼ 2 + i = − ~p + LSP χ ( ( E to the solution of a Schr¨odingerequation j m 4 − 2 i LSP ] − + m j 2 1 i – 27 – LSP a 2 m ~p, ~p ], that no other region requires resummation, and − m m a + 1 term can be dropped in the propagator of the ex- 2 a + k i , which measures energy from the common reference a E ≡ 42 2 k 1 − 2 a a − ~ E + m ) P m + 2 + of the exchanged virtual particle satisfies m 0 j 2 a ) m j i j p ij k a k p k M k µ m − m [ + + = − i i i p E ~p ( m 0 a = = π 2 dk s i j √ i j Z p p m m ). in a given ladder rung are denoted by 4.1 , the center-of-mass energy of the annihilation process is a . Since mass differences ij µ LSP . Diagrammatic representation of the leading (ladder) contributions to the non-relativistic is the reduced mass of the two non-relativistic particles. It can be shown by power- m |  µ ~p | The first consists of kinematic simplifications of the propagators in the potential region. , | ~ P convenient to introduce the variable value 2 energy variables have the non-relativistic scaling particle state loop momentum energy-component integralsemploying in each ladder rung can then be performed | up to higher-order terms in the non-relativistic expansion. For later purposes it proves cles. The pair of heavy-particle propagators in each ladder rung can also be simplified. Let be the momenta of the on-shell external state ing the sum of ladderinvolves diagrams a shown number in of figure steps, which we now sketch. We already mentioned thatchanged the particle, so ( that the interaction corresponds to a potential between the heavy parti- The enhancement of non-relativistic scatteringtum region, originates when from the the momentum potentialwhere loop momen- counting arguments, see for instancethat [ the enhancement from the potential region is only present in ladder diagrams. Relat- Figure 4 matrix element ( 4.1 NR matrix elements and the Schr¨odingerequation JHEP05(2015)115 , ) 2 on v = 0 4.6 (4.6) (4.7) (4.5) ) , ij S LSP 2 e ) which 1 ), which L,S m , 1 ( e e ).  ψ . 2 . ) 4.5 e ) ˆ ) is required 4.5 2 E ( b E ) n 2 1 ; µ 1 k ~q ~ 2 a 2 k ~ 4.5 e µ ] in ( + 1 2 − ~p,~q ~p − ~q e ( ( [ ~p ] ¯ e i K − − + 1)-loop ladder dia- ˜ G ] LSP , could be equivalently ~q integral in ( n ) develops a singularity S ( a m b ~q + 1 µ E , the potentials for LSP 2 n L ; ) a ~σ 1 m ~q − should be of order ˆ 1) V are small. Note that within 2 b ~p,~q or j − − 2 ( 1 ) − 2 a M ~p n m n [ ab 1 a × ( ) to the state k ~ ) for the ( a m 2 ˜ µ e G − + − ) + ( 2 (¯ (3) ~p M 2 − ( ˆ 4.7 δ [ E e a E 3 is defined through 2 ; ) ) a − − m ˜ π in ( ] G E m ~p,~q E , while and ; (2 n ( ) ) contain the propagator of the exchanged E LSP and ie a 1 2 k ij µ 1 ~p,~q ˜ k i ~ ( ~p and G m 2 ( = ( 2  i 1 ab m ab ˆ ] i − E ˆ aa V − i m – 28 – ~q ] − = 1. The limiting procedure in ( ξ [ ˆ iH n = 0.) The first term on the right-hand side of ( V 1 − Γ a L a K † 0 a # 2 j LSP c j µ k ~ 3 3 M ~p m i ξ 2 [ ) ) m ~q m ~ h k ab for . The function 3 π π 2 3 δ − − d d = ~q and ] (2 (2 − i E a a j = n Z =1 The factors χ LSP i Y = i M ... )] vanishes for [ a, b, . . . " 2, or any other two-particle state reduced mass. K  m χ 0 ) 11 1 ij | / . Depending on whether Γ = 1 2 1 a − 1 ij i . Note that using the on-shell condition for the external particles 2 µ Z k e ~ ξ µ − ~p 3 E χ (2 LSP Γ 2 − =0 a † ∞  / c X n − j m 2 2 − ξ i M ∂ k ~ ~p [ ←→ ( = 0 and ˆ or E i 2 2 − − a ) =  L ˆ 1 ij ) = E − . The angular-momentum dependence arises through = 0 we set a E E µ ; 1) E S ˆ  V n ; + − + 1 exchanges. K ( × ~p,~q n Γ ( E ~p,~q 2 † ( = 1 for c e → ab refers to the initial two-particle state lim ˆ ab χ ˜ E G | i K 0 H = 1 must be used in the above equations, which causes the dependence of h × With these simplifications to the ladder-diagram sum, the annihilation matrix element, The loop integration of the last loop before the annihilation vertex is the S 11 the total spin reads since the factor [ that corresponds to scattering states with relative-momentum kinetic-energy particle and coupling factors fromtials the discussed two in vertices. section These areand the non-relativistic momentum-space approximations, poten- the Dirac matricesto from the the tree numerator structure can be reduced or (For the term accounts for the tree diagram,gram a with term with given and where annihilates into light particles. Below westates employ this by compact compound notation indices to label two-particle including the tree diagram with no exchange, can be written as expansion. As discussedheavy before, particle we line must are requirebut small, that we that the do is mass not differencesthese have approximations along to the assume denominator a of that given thesubstituted kinetic by energy term, To arrive at this result we systematically neglected higher-order terms in the non-relativistic JHEP05(2015)115 (4.8) (4.13) (4.11) (4.12) (4.10) . ∗ ei )] ) in terms of ) (4.9) , ~q 0 ) ( ~r 4.5 ~q with eigenvalue E ˜ ψ − − H ][ ~r ~p ( ~q ( . [ (3) i i ) } (3) K δ i ˆ δ E 3 ~q,e ) to compute 3 ; ab | − ) ~q p a ~p, ) ) δ | 3 . π ˆ ˆ E π a E d 1 4.5 (  {z 2 ~p,~q (2 µ ( (2 G ~p | − 2 ) = ie ), ( LSP 0 ab Z ~p,i E ˜ ) is in fact the momentum-space h δ G | = ; H m 0 4.1 ] = E i 2 ; ~q i ) [ ) = − r ~r ~r, ) = E ( p a ~p, K E a ~p,~q | ˆ ; q e ~q, E ; ( cb 0 | 3 ( M ) ) ~q 0 ab G  ˆ 3 π ~p,~q ) E ˜ k, ~q ~ ( G d ( r ac (2 ( δ ab − G ac ˜ ~p G } Z ,H (  – 29 – V i ) + ,G  ˆ cb ] E  r , a ) is the Green function for the Schr¨odingerequation ˜ ( i ) . Accordingly, the eigenstates carry a compound G ij 2 ˆ ) + − 2 E + E ) ac µ ( a LSP − ~p ; ~p E 1 0 ˆ 2 | k ij ~ V χ 2 ; ˆ − 0 m ( 1 0 E a 1 µ {z 2 ~p a G − µ 2 ac | ~p r ~r ~r, − χ 2 . The Lippmann-Schwinger equation can be used to derive ) = ˆ ( ˆ + 2 V  ~p,i r ~r ~r, E i r h ( | a rather than the full Green function. Let us define the Green ab H 3 ( =  by including the potential in the unperturbed Lagrangian. = , b ) ab k ~ G i ac M 3 2 π ~p,i E + a are the exact stationary scattering states of 1) G are plane waves. We may represent the former in the plane-wave h | d ) ~p V , a ) = | i − (2 − i ] ! ( ˆ + E ~q χχ , a E E [ ( ~p E [ Z | q a ~q, + p a ~p, → G . With these preparations we use ( h K | − ~p − ˆ i H + E | 3 a = 2 a ) ~q 2 are matrix-valued operators with dimensionality related to the number = lim µ 3 µ π ~p ~ p a ~p, ∇ 2 ab 2 d 0 | (2 1 ) )] , ij G −  2 − 0 ~q ), while e ,

( Z a 1 L,S ( e G E µ E GG ˜ ψ ψ ] (2 → that refers to the component of the wave-function proportional to the two-particle = / ˆ ~p . The states E [ 2 a i a ~p The third and final step uses elementary scattering theory to express ( The second step consists of verifying that K = lim , a = + ~p state E basis by [ | Note that of two-particle states ( index for the interacting and free Hamiltoniansmomentum of eigenstates the and Schr¨odingerproblem the corresponding Note that this result couldLagrangian be given obtained in directly section as an equation of motion from theeigenfunctions effective with energy operators with the coordinate-space potential hence the Fourier-transform Green function of a certainthe Schr¨odingeroperator. Indeed, Lippmann-Schwinger one equation easily checks that it satisfies JHEP05(2015)115 ) ) a,ij )] 4.13 4.13 . Let (4.17) (4.14) (4.18) (4.15) ~r ij ( E in ( ψ ei )] , refers to the ~q , ij . ( ij 2  e E 1 ∗ e ψ ) in the cms frame LSP m (0)] LSP , 2 E m ψ − 2 ~ 12 = 0 (4.16) a ∇ = 0 [ − ). i | bi M i  b,ij i − )] are identical and the correct result , a M . The wave-function [ )] ~r ( ab 4.16 2 in the kinetic term, since the ∗ − = + ( ~r δ / ) )] , ij ( ~p E − ˆ 2 ~r E E e ( , ij ih ) ψ 1 L,S [ 2 E LSP E ψ ) + ( e − e ,S [ , a r ψ 1 ψ ! m ( a (1 e 2 ) ! + µ ~p ) r ab 2 ~p ≈ ( | r ~pψ ˆ V ( ab a 3 ) and [ ) ab ~p µ V ~r 3 π ( V ) = d = 1, we obtain E + r (2 – 30 – ( + ψ L ab ab and Z that defines ab δ , specifies that only the component proportional to refers to the fact that this equation should be solved δ V 1 # 2 for a e # e X E ij χ 1 ~q E ]. Although, eventually, we will not use this method, we 2 † e e c with kinetic energy − , ij − 43 χ 2 = ) = i ] = e ˆ 2 a 2 1 e E and ~q ∗ e LSP µ ( [ in the Schr¨odingerequation ( ~ p i ~p, ~ ∇ ∇ 2 | G m K ab − (0)] − E V of the wave-function at the origin for this incoming state is picked E ). The label ψ " " e + = [ 4.10 = 0 and correction, which we consistently neglect in the long-distance part. The LSP , ij ) 2 L 2 e ,S m v 1 instead of (0 e ). Using the opposite convention we would end up with [ ψ = 2 ba s 4.6 V = 1 for √ K For a symmetric potential the solutions for 12 and we recall the definitions would be obtained eventaken if care the of. conventions However, for in the the Green MSSM function the and potential matrix the is potential complex-hermitian were in not general. consistently Schr¨odingerequation now reads We first solve theclosely the matrix-Schr¨odingerequation method for described the indiscuss [ Sommerfeld it to factors set by upfor the following the framework for mass the splittings, subsequentdifference improvement. we is Under can a our approximate assumption because of the conventionfunction used for ( the leftbut and with right states in the definition of4.2 the Green Solution of the multi-channel Schr¨odingerequation with the potential ( with the initial condition correspondingus to finally a mention particular that incoming the two-particle complex state conjugated scattering wave-function appears in ( of the annihilation. Thethe first, two-particle state out by the annihilation operator can be obtained directly from the matrix-Schr¨odingerequation in terms of coordinate-spacecarries scattering two wave-functions compound at indices the referringincoming origin. to two-particle two-particle Note state states. that The the second, result of the Green operatorsince (ignoring the bound states not relevant to our discussion). Finally, The same result can also be derived from the spectral representation JHEP05(2015)115 . - is = z a LSP th , the ) does v r (4.19) (4.22) (4.20) (4.21) m 2 4.21 − . This holds a ) is completely potentials arise , channel as a superpo- M r ( → ∞ ,a is imaginary. ai /r = V r = 0 (4.23) inf )] a ,a bi as k ~r ( )] inf /r r E V ( , containing the Coulomb ψ E > V L u [ α/r , a ) does not depend on the  c . ) ) θ θ r ( r + a ( r ai ik ,a f ab e , (cos ) inf ) V L V ,a θ P ( ), when written in the two-particle mass- inf r LSP ai bi ( ) should be matched to the behaviour V ), and in the outgoing scattered wave, one has f . A r m V ))). Since it can be shown that the subsequent ( r − 2 + a V + ab v 4.21 z k i )] a with diagonal entries ) in the following. ab r r ik + δ LSP ( e L  inf ) ln(2 m 4.21 pair annihilations. For Coulomb potentials, ( E ) should be replaced by the incoming wave-function in a u ai V ( – 31 – E [ k δ ≡ χχ (2 4.21 L LSP LSP E X = α/ of the radial Schr¨odingerequation →∞ component of ) then always approaches a diagonal matrix (the non-diagonal m a m r r c defined by = ( ab aa = − ai V v )] ai LSP 2 a )] , r )] r k ( m ~r . With the above redefinitions the dependence on the initial ( ~r L 2 ( + ) factor in ( + 1) 2 u E / z r r E 1 a ψ is real. Otherwise the channel is closed and L a ) [ ψ ( ) applies to radial potentials vanishing faster than 1 k [ ik a ( i L k , but not in the equation itself. We note that the MSSM matrix LSP 4.21 bi + )] 2 /m 2 Since the potential is non-diagonal, the scattered spherical wave is not ) ~r . The asymptotic behaviour ( d ( dr ai E 13 ) by exp ( δ − LSP r ψ ) denotes the Legendre polynomials. This expresses [ a m θ  ik 2 of [ − of basis solutions [ i (cos j L bi m P A + The scattering solution describing an incoming plane wave propagating along the i The asymptotic form ( 13 m potential contributions as well asCoulomb constant potentials, mass the splitting terms. exppresence ( of Accounting the for Coulomb the potential presence into the of replace long-range exp ( derivation of the Sommerfeldanalogous factor to including the Coulomb short-range potentialswill potential for on case, brevity the refer and diagonals to leads of the to asymptotic the behaviour same ( result for the enhancement factor, we for Yukawa potentials, arising from massivefrom mediator exchange, photon which exchange is — —not besides the the relevant apply. Coulomb case interaction for However,exclusively photon-exchange in does the not diagonaleigenstate change basis. entries the of For neutralino the largeYukawa state, potentials values potential of so being matrix the exponentially 1 suppressed) with entries where sition proportional to of the general solution where, due toazimuthal the angle. azimuthal symmetry of the problem which determine the asymptotic behaviourkinematically of open the and solutions. When direction and an outgoing scattered spherical wave takes the asymptotic form subscript potential is hermitian,matrix but can in be general complexpotential numbers. not approaches The symmetric, the potential since diagonal isWe matrix the spherically therefore symmetric. entries define the of For wave large the numbers potential This implies that the threshold(( for co-annihilation channels occurs atscattering finite velocities state appears only in the initial condition for the solution as indicated by the We also use the velocity variable JHEP05(2015)115 ) 4.25 (4.28) (4.29) (4.26) (4.27) (4.24) th component a , each of which i ) into account, the . r ( bi ] V 1 . i linearly independent − k  . ) (4.25) N ai M i [ δ ai ) δ r ... ab  a L i k . + ai k δ Matching the asymptotic be- ) assumes short-range potentials. In (0)] is given by 13 +1 ln(2 + i L 14 +1) α r 4.24 . + 1) a L 2 ( L c ai + 1)!! Lπ a ai L , the relevant cases in practice being δ u k [ . L 2 L (no sum over (2 − LSP ai (0)] L (2 r m i iδ a k − L + + 1) +1) e r L solutions are irregular at the origin, hence  ) + 1)! bi ( L L matrix, there exist 2 2 ] ai θ Lπ u 1 -wave annihilation). The leading term in the [ L i (2 n independent initial conditions N sin ( − k N − L P – 32 – i ≡ r . ai N (cos M a × ) as the ratio of the annihilation matrix element [ a L n L k ai 1 r + 1)! P  ) N 4.1 = 1 ( M θ 0 L + 1)-th derivative at the origin. Hence using ( = ( →∞ sin + 1) = → L r r L is a ai L = (cos ai n ai ) for the Sommerfeld factors now require the wave function L (2 and scattering phases )] ai V )] around the origin is given by P r L )] regular ones. The asymptotic behaviour of the i ( = ~r →∞ ai 0 ), we obtain r ( r ai L 4.15 ( n = = → N u )] L ai r [ r free E u bi )] 4.22 ( [ r ai ψ is defined in ( L A ( [ )] L u in the Schr¨odingerequation. The free Schr¨odingerequation can be u ~r [ ) , ij ( , but we do not indicate this explicitly. Eq. ( denoting the ( 2 E inf e L ) corresponding to the ) and ( 1 L,S V ψ ai ( e [ ψ 4.23 by 4.21 (0)] V -wave annihilation) and +1) L -component vector with index S ), we have ( L u N The basic ingredients ( 4.22 Both depend on = 0 ( 14 the presence of Coulombasymptotic potentials behaviour in of the the diagonal regular entries basis of solutions the reads potential matrix The modifications match those of the scattering solution, see footnote replacing solved exactly, and one finds The quantity to the same matrix element evaluated in the tree approximation, which corresponds to with [ in ( and its derivatives at thesuch origin. that We it suppose overlaps that onlyL the with annihilation a operator single partial is wave Taylor constructed expansion of [ with with constant coefficients haviours of ( solutions to ( is an restricting us to the set of of the regular linear independent solution for initial state for each partial wave. When JHEP05(2015)115 (4.34) (4.35) (4.36) (4.30) (4.31) (4.32) (4.33) ) can now . 2.23 aj , M i 0 e ia T T linearly independent ) a , , LO k J appeared in the definition r | N a ) L aj L a J ik by making use of the fact v )] X ∗ bi ]. There the potential matrix +1 L − ] ai r bi S e 1 L ( T 43 2 +1 i +1 − L i ( L M ai i S ] k u be the k 2 [ † , χχ M ( L [ = →∞ ia T i ai ∗ → L r i ei χχ 0 )] ∗ eb )] k T ) looks more natural even in the case of r χχ ij r ee → ( ˆ ( L f = [ )] + 1 →∞ † 0 L L (0)] ]. Note, in compound-index notation, χχ − r ii r + 1)! 4.32 ie ˆ i v v ( ] f L [ † 43 L L ai 2 considered in [ +1) ( behaviour of the singular solutions. This T − )] L [ W 1)!! L ( L [ r r = . ( u aj 2 W [ − } L )] bi  χχ ij v ] r L [ 1 ( – 33 – → }{ Inserting the asymptotic behaviour of the regular , 0 L − ij 1)!! χχ u { ij [ + 1)!! 15 ˆ L + 1)! i = (2 M f − [ k ia , L L ) L = )] (0)] ( , ij ab L ai (2 2 r r e δ (2 ( 1 L,S χχ +1) † L ( e = 0 (0)]  -independent. Let [ L v → ( L ψ r = → r u ) , ij +1) χχ [ ii 2 = [ ˆ L f e )] = with hermitian conjugates ( ai . The Wronskian is defined as ( L 1 L,S ij J ( e u )] T ] L [ in terms of the large- r L ψ L + 1 ) gives ( W + 1)! and ) gives L +1 L W ) , ij [ } v L S 2 2 [ 3 ( 2 e e 4.30 ( 1 L,S 4.28 4 0 ˆ ( e e f [ ψ = → i }{ χχ r 2 S e → from the regular solutions by the following three steps: ij 1 . e )] χχ { ) is a generalization of the expression L T ˆ r f ( L = 4.32 W [ , instead of the hermitian conjugate. Due to the generic hermiticity property of the Schr¨odinger χχ In practice, the matrix-Schr¨odingerequation must be solved numerically. We obtain We can avoid the computation of the phase-shift matrix L → Eq. ( 0 W 15 χχ ee ˆ the matrix was assumed to beof real-symmetric, such thatequation, the the transpose definition ofreal-symmetric of the potentials. matrix which is equivalent tof the result first derived in [ which expresses can be obtained fromdefined the through regular the solutions square asbe of described evaluated the below. in annihilation the The form amplitude. Sommerfeld factor The is definition ( Plugging this into ( Both expressions must be equal due to the constancy of the Wronskian, which implies where the prime denotes the derivative. and singular solutions, we obtain that the Wronskian matrix is singular basis solutions with asymptotic behaviours which defines the matrix for general Comparison with ( JHEP05(2015)115 is = = i T ai (4.37) (4.38) (4.39) (4.40) 61 GeV (0)] . for ) follows until ai +1) . L ∞ )] ( L 4.36 , r ij r u δ ( ai and works well, = 2749 L δ = u )] as function of L ) of the irregular 0 + 1 0 ˆ r , χ ij 012, slightly below S . )] 1
m ( r 4.31 ˆ ( aj f + 1 [ + 1 L = 0 )] 0 1 L L χ W v ∞ 2 0 1 r appearing in ( χ ( 4 GeV, = S L . T u ai [ Mathematica 50 stays at the constant value ≡ a )] ij 0 ik δ > r S ( ) is shown as the light-grey (red), ! − = 2749 = 0 L two-particle states in the Schr¨odinger ∞ ∞ . u 0 1 [ aj x x aj 1 − 1 χ ( )] U − χ S m ∞ ia + 1 U r T χ ( , = = 0 L = ai u state to the problem, so that the Schr¨odinger T . After a rapid variation with a peak structure, δ [ and – 34 – ij LSP ] 5 0 2 0 1 +1 L χ m ∞ χ L 0 0 1 r 0 1 ˆ W r a [ χ χ . The normalization is chosen such that [ ik 7 e + 1 − 1 L ) = 2 state. The result ∞ = 10 , such that the asymptotic behaviour ( r = 0 ( − 1 ∞ r by solving the radial with Schr¨odingerequation initial conditions χ ai aj r within a certain target accuracy. linearly independent regular solution vectors [ 3 matrix. Since the new state is 200 GeV heavier and moreover + 1 )] L U 0 χ × ∞ r N + 1), in which case the Wronskian equals [ r ( known from step 1. Hence the matrix L L u and use ˆ [ (2 25 GeV. Further details of the models are not relevant for the following aj . by keeping only the / )] , check the stability of the result by varying (and increasing) vr is close to zero. In practice, we work with the dimensionless, scaled variable for given r r ∞ ( 59. Next, we add the 0 . states included in the multi-channel Schr¨odingerequation are degenerate to a L r LSP vr + 1)! u m N = 2950 199 L ( LSP 0 2 = ,...,N ai ≈ χ m all ˆ from matrix inversion, value of independent of with and [ δ solution applies, and the Wronskian evaluates to 1 where ˆ r ) Since in practice the Schr¨odingerequation can only be solved up to some finite Pick a large value Determine the m For the purpose of illustration we consider a MSSM parameter point where the LSP The procedure described here has been implemented in = ∞ (3) (2) (1) ( ∞ the threshold fordashed the curve in the upperthe panel Sommerfeld of figure factorS reaches a plateausystem and is for now for a 3 is wino-like. Theand relevant masses are discussion. We firstx compute the Sommerfeldequation and factor the annihilation process. The velocity is chosen other states not relatedignored. to However, the our wino aim is ora to Higgsino larger compute electroweak the part Sommerfeld-enhanced multiplet of radiative are thein corrections decoupled in MSSM the and parameter wino space, or whennumerical Higgsino the problems, limit. mass as splittings we In now become describe, this larger and than case fails the to method provide outlined the above desired solution. encounters severe when high degree. This isment the is case most in effective, MSSM such parameter as regions the where wino the or Sommerfeld Higgsino enhance- limit for the neutralino, and when the JHEP05(2015)115 v )] J P 3 ( ]). In ˆ 2 f [ 50. The v 0 1 χ 0 1 > LSP χ S ∞ m x + refers to the two- -wave Sommerfeld LSP P a m [2 − 100 b 100 M ( )] (upper plot) and 0 state is included. In this case, the S LSP 1 0 2 ( χ ˆ m f 10 0 1 [ 0 1 χ = χ 0 1 2 b χ 10 κ S ¥ two-particle states in the Schr¨odingerequation ¥ x x . ≡ − 1 χ 5 1 S is rather generic. To quantify the issue suppose + 1 for the closed channel involves an exponentially – 35 – b where χ ) is a closed channel, while bi r b )] state. When the mass splittings become larger or κ and x 4.23 < M e 0 2 ( 1 0 1 2 χ L χ 0 1 v u 0 1 χ 0.1 χ LSP m + is slightly large than 1, as seen from the dark-grey (blue) curve in 2 and no reliable result is obtained. LSP 0.1 0 0.01 0 50 5 > m

∞

10 200 150 100 20 15 300 250

012 in a wino-like model as described in the text. The light-grey (red) dashed L H L H ¥ ¥ x S S x x -wave Sommerfeld factors . ∞ P x ), when only the = 0 ∞ v x ( - and S S channel. The solution [ . 0 1 , which drops to 0 after a few spikes. It is not possible to reach the plateau, where χ in the Schr¨odingerequation ( 0 1 ) stabilizes. A similar behaviour impedes the calculation of the 5 b χ The numerical instability originates from the presence of kinematically closed two- ∞ x ( particle state channels, here the small, the situation 2 that LSP growing component proportional to should have little effect on thesolution value fails of the when Sommerfeld factor.figure However, now the numerical S factor, as shown in the lower panel of figure and the annihilation processdark-grey (blue) are solid kept curve andevaluation fails shows the for the asymptotic result regime when is the reached for rather weakly coupled to the two lowest, nearly degenerate wino states, we expect that it Figure 5 (lower plot) for curve shows JHEP05(2015)115 ]. ai ] 44 L u (4.42) (4.41) (4.44) ) reads ), such defined summed, x ( satisfies U ˆ inherit the 4.23 V b not i ai + ) )] x ) can no longer x a inf . ( ˆ k ). (4.43) V L ( a 1 2 u 4.40 = 0. The method →∞ channel, exponential = 0 + x ) = L 0 1 bi ) x χ and dimensionless wave iJ ( )] x 0 1 ( x V χ − ( ˆ vr should not be summed. V ) L = 0 (4.45) . a x u [ ai a LSP ˆ k  )] 2 , so ( m EW ) LSP x . The radial equation ( 1 2 x ( ai 2 M m ( = + / L E 1 L 16 ab u ] x [ > Y ) (no sum over V ] h  x 2 /E ( ) v − 2 ai πx β ab 2 + 1) ) LSP . The matrix inversion ( LSP δ x ) that couple the different channels decrease r x L a r m ( m (  ( − 2 a L – 36 – g ab + − a V − − ) = 2 + 1) of the functions of the right-hand side of the ansatz. Further- 2 x a ) x M LSP ( L ˆ . k x ( ) we seek a method where the inverse of L a ( (  m g 5 − L ai [2 + i α 4.40 − ) − becomes ill-conditioned with exponentially growing entries for the dark matter scenarios of interest, this condition is ai x 1 b )] ( ) ), the left-hand side has indices ai a x x M ( f  U a EW is the mass scale set by the electroweak gauge boson exchange large enough such that the asymptotic regime is reached, which 00 L ˆ k 4.45 ) = [1+ ( + = M u [ ∞ 1 2 v EW r ai ai +  )] )] L M LSP x x J ( ( m L 00 L ( LSP 2 u πx / u [ [ m a ] starts with the variable phase ansatz k r 44 , where = /r a ) = r contains the mass splittings, and is diagonal, while ˆ k x ( ) follows directly from the solution of a differential equation system. This can indeed EW a inf f M V We work with the dimensionless radial variable − We do not indicate the dependence on 4.39 e 16 more, in subsequent equationssince we this do should not beas write clear well from explicitly as the when later index an ones, structure index such of appearing as the ( twice equation. is For instance, in the following equation are linearly independent Bessel function solutions of the free Schr¨odingerequations that proposed in [ Here numbers We separate the asymptotic value of the potential by defining 4.3 Improved method To avoid the matrixin ( inversion ( be found by an adaptation of the reformulation of the described Schr¨odingerproblem in [ degenerate and the matrix in the rows correspondingbe to done open in channels practice for causes the instability seen in figure Since typically easily satisfied unless all two-particle states includedwithin in a the few computation GeV are very or degenerate less. In consequence the formally linearly independent solutions [ oscillating. The off-diagonal potentials as that mediates the channel mixing.exponential growth Hence, from the the open-channel closed solutionsgrowth occurs channels. [ when For at the least two-LSP one of the included kinematically closed channels the absence of channel mixing, the solutions for the kinematically open channels would be JHEP05(2015)115 , ai ) to α , de- (4.55) (4.52) (4.54) (4.49) (4.51) (4.53) (4.46) (4.48) (4.50) α , ˜ 4.42 ) and the N ). x ( x ( ai ai β β ai ]. ˜ α . x ) and 44 a cb x ) and . Specifically, for the ˆ k ( i x x N e . ( ai ai α ac ai E ˆ δ α V = 0 = 1 = , →∞ L 0 x + x L L db . ab ai N . Up to this point we essentially δ 7 , )] b ) = 0 a a cd 0 g − x ) = 0 (4.47) E ai g g ˆ ( V x ] ) = 1 x ab ) ( L ( L x  x O ac u ( ai and the above definitions, one derives u 0 a ai [ a [ ai 0 a ˜ α ˆ ≡ − . k β N g g i )] 0 a a  ) ai ) vanishes for large ] 1 g g a x a ai x ab − − 0 x ( L − ˆ second-order differential equations ( ( k ) is asymptotically trivially related to ˜ g ( a α Z i a L u + a ab ˆ ab k g , (1 u i f δ N [ − = N ai x a ab 4.39 a − – 37 – g 0 − a a − α δ ˆ k  ai − ) a g g i ) 0 b b a ˜ α f x = ˜ x , and g g  ( − ( /g 0 bi + 1 = a 0 ai ai with x 0 a x + ] α ai g from ( a g α L      ) ˆ 0 L k ab 0 a a ) )] i 2 ab x u g g ai x e x [ ( N = ( ( O 0 a U  a a 0 L + bi f f ) = ˆ k = u ˜ α + i [ 0 ai x  ˜ ab ai α ) translate into ab − ( β δ Z x x 0 a a ab a a g g ˆ ˆ k k i i = = 4.37 N e e 1, 0 0 ai ab , = = ˜ α N ai = 0 coupled differential equations of first order for U ]. The condition reduces the satisfy the first-order differential equations L N 44 α [ is a number close to zero, which we set to 10 defined through ai 0 and ˜ ab x O N The crucial observation is now that The important quantities in the present approach are the matrix-functions The last equality follows,relevant cases because since which implies that the matrix The initial conditions ( where reviewed the modification of the variable phase method proposed in [ that with From the original Schr¨odingerequation for [ for every a system of 2 fined by The ansatz doubles the setadditional of freedom unknown can functions by be introducing eliminated by imposing the condition The Wronskian of the free solutions is JHEP05(2015)115 , ) is ≈ ) T 4.2 4.52 -wave ∞ (4.56) , such , such ( S x ∞ S with the x -wave case ) that ∞ P x ∞ x state cannot be ) are computed 50 with 0 2 -wave Sommerfeld χ > 4.36 0 1 P -wave case there is a χ ∞ , that is, the S x 5 ) (4.57) . = 0. For the for large enough ∞ ia L x δ 1 ( 1 − L ) with initial conditions ( − ia − 0 U ˜ α x 4.50 ∞ -wave Sommerfeld factor with the 72. x . ) = a P ˆ k 0 i x . state (dark grey/blue curve) cannot be 199 − ( e 1 ∞ 0 2 ≈ − ia x χ ) ˜ 0 1 α , but contrary to the ) = χ ∞ ∞ ∞ ( x x S – 38 – ( , ). ia ] matrix in the standard and new method agree only ba 1 Z and check (by varying and increasing − U 1 59 to 4.51 the same result as in figure . − ib U ∞ and hence the Sommerfeld factors ( ˜ α 6 x 199 . Since there is no numerical restriction on − T -independent. This is now easily accomplished, since us- ) = [ )] in the wino-LSP model described at the end of section shows the corresponding results for a 2 ∞ x ≈ = 0 )]. Once again the two-state solution (light-grey/red curve ∞ 6 0 ) S /x . J x 1 1 10. The difference arises from the approximation in the second ( ) can be turned into the differential equation ( given by ( P ∞ − ) for sufficiently large ia ˆ within a certain target accuracy. ( 3 f ia < ˜ α [ ( S Z T 0 1 ˆ f ∞ -wave case. 4.51 χ [ ∞ 4.56 ), and then x 0 1 0 1 S x ,...,N ) for several χ χ the first-order differential equations ( 0 1 S χ 4.51 = 1 ab ) with S = 0, ( 4.57 b N in general. The “approximation” is exact only for 4.56 /dx ), which implies that the ) ∞ ˜ α 2, as in the x 1 given in ( 4.54 − ≈ α independent of for every Z (˜ Evaluate ( Solve for Solve ( The lower plot of figure To illustrate the improved performance of the method developed in this section, we To summarize, the matrix We recall that our task is to compute the inverse ∞ d x 31. We also show the two-state solution with the standard method (light-grey/red dashed (3) (1) (2) . the difference vanishes asnew 1 method, the difference canNote always be that made it sufficiently would smallstandard as method not seen for be in the the possible three-stateat lower problem, to plot. since compute the the matrix inversion becomes unstable distinguished on the scale4 of the plot.curve). The plateau All is results reached agreevisible for for difference large for enough line of ( for large Sommerfeld enhancement in this particularby model a — tiny the amount enhancement from factor changes only factor, here for [invisible]) and the one including the heavier but now using the newto method. obtain The with solution thethat for the standard the three-state Sommerfeld method case factorreferring can that to can was now the impossible be be two-state extracted solutiondistinguished evolved and without on to the the difficulty. sufficiently one scale including large In of the fact, the heavier plot. the As two expected, curves the heavier state has no effect on the show in theSommerfeld upper factor plot of figure from the solution to ( by the following three steps: that the result ising practically with JHEP05(2015)115 )] J and P 3 -wave 0 1 ( (4.58) ˆ χ P f [ 0 1 0 1 , χ using the χ -wave op- 0 1 # χ P 1 e S 4.2 χ 2 † c e χ - and 3 S c e χ 2  proportional to mass dif- ), when only the ∂ ←→ ] 100 } ∞ i 2 1 x 15 e 100 -wave operators by the rela- ( 2 − e S )] (upper plot) and S 0 }{  χχ S 3 4 1 e † e → ( 4 ˆ χ e f 10 [ χχ { 0 1 + χ Q 0 1 1 state is included. In both plots the two cases e χ 10 0 2 S χ ¥ ¥ 2 χ x x 0 1 ≡  χ 1 S ∂ ←→ – 39 – i 2 − , we further require the matrix element of the second-  1 2 2 † c e v 0.1 χ . For spin-0 it is defined as [ } 3 1 c e e χ 2 e 4 † e }{ χχ χ 3 e " → 4 e 0.01 1 2 0.1 χχ { 0

0 50 5 012 in a wino-like model as described at the end of section

and the associated operators

. P 10 200 150 100 20 15 300 250

L H L H ¥ ¥ S S x x -wave Sommerfeld factors } ) = 1 0 P e = 0 2 S can be obtained from the non-derivative e 1 v ( } }{ χχ } 1 3 1 e e - and → e 2 4 2 S e e e χχ { . }{ χχ }{ χχ 3 O 3 e → e → 4 two-particle states in the Schr¨odingerequation and the annihilation process are kept. The 4 e e χχ { χχ 1 − { P χ P 1 + with obvious generalization toof spin-1. We show in this section that the matrix element The above method allowserators us to calculateferences. the matrix elements In of orderannihilation the cross to section fully at order describederivative operators the Sommerfeld enhancement of the short-distance χ dark-grey (blue) curve shows the resultare when now the indistinguishable. TheSommerfeld lower factor plot with shows the in “old” addition method (light-grey for (red) the dashed) two-state problem. 4.4 the Second-derivative operators Figure 6 (lower plot) for improved solution method. The light-grey (red) curve shows JHEP05(2015)115 k ~ ). 4 e + ~p ~q ( (4.63) (4.64) (4.65) (4.59) (4.60) (4.61) O . } → ¯ e  ~q )+ ) (4.62) ) → ~q E ( k ~ e + ¯ ∗ ei { ] e ) − ), and use that S E ~q ~q M ˜ ( ( ψ 1) i and the expression [ 0 ∗ ei , . Shifting − ] ∗ e − 4.65 S κ ] i E j A E 1) ˜ ψ χ ˜ LSP + ( [ ψ i − [ 3 χ # m ∗ | ) ) ∗ ~q ) the SU(2) gauge coupling 0 1 0 ) , )] 3 (2 π ~q e π a . e d ( ) + ( ( ee k  ~ χ i 0 ) (2 0 c ( µ 0 (4 ~q a 0 Γ ∗ e a , ( ( ee / ¯ ∗ ee ] 0 2 † φ ), we obtain φ ee c 2 π 2 c e a Z ∗ ei κ E a ] g 4 = 2 2 φ ˆ m ) m χ ˜ V φ S ψ of appendix | [ E [ 2 0 ) e m E 0 − ˜ = ) of the matrix 4.61 a ψ 2 3 m a 1) ~p h 3 ( ee [ ) 4 2 k X ) ~ 0 c ~q , + + − 3 π  3 a e α 2 π → = ∗ ee 2 e 2 d 4.60 d d X α κ (2 k ) leads to ~ (2 ~q e + ( M 2 2 φ 3 0 µ 3 ) = ~q α a ) Z − ~q m e ∗ ee 3 π i X 4.16 in the second term of ( 3 π d j κ 1 i µ 2 + 2 d i + (2 0  χ (2 ξ e i ¯ e 2 LSP ) 0 – 40 – Γ πα e χ k ~ ~q Z + 2 † | ) reads m δ Z ( c j 0 1 1 e i ) → i 0 e ξ k (2 ~ − µ i ee ∗ e h E 1 e ] d 0 χ ξ 2 δ 2.22 ) 2 ) = 4 e − µ E Γ 2 = e + d π − ˜ † k  ~ ψ ~p c d j i ), which for the specific case at hand imply e ( [ 0 (2 j ξ ∂ 3 h ←→ = M χ ee ) ~q i i ) = 2 0 2 Z 3 ˆ π 4.13 = V χ − ~q ee d | ( i − 1 (2 κ j e ∗ ei ] χ  χ LSP i Z 2 E Γ χ ˜ m 2 † i ψ |  ) and ( i c e [ 1 ξ (2 e χ 2 ∂ | e ←→ Γ 4.5 χ ~q 0 † µ i 2 2 c h j 2 3 ). To this end, we show that ξ  ) ~q " h − 3 π ∂ d ←→ = ×  2.22 (2 i 2 † 2 c e χ Z − ), ( | is the mass of the exchanged gauge boson, 0  the coefficients of the potentials given in table φ h ) factorizes the two integrations. Employing dimensional regularization gives the Γ 0 ) 2.20 m 2 a † ( ee c e c We start from ( χ 4.62 | 0 h where the last equality follows fromfor the the definition relative ( momentum ofWe the two-particle can state rename the dummy index finite result for the linearly divergent integral. Inserting this into ( where and in ( The momentum-space potential isbosons, a sum of terms from the exchange of gauge and Higgs The momentum-space Schr¨odingerequation ( where in compound index notation ( tions ( JHEP05(2015)115 , 2 1 s v . can κ (4.66) LSP m 100 GeV. , ∼ < ) correction  ). 2 ) v H E ~q ). Since heavy is only present ( ( i m 2.22 0 involve the same κ O ∼ ¯ ∗ e ] 0 1 4.64 e E ˜ χ ) of ψ LSP 0 2 [ e S m χ 5 unless 4.60 . 1) − 0 → − e 1 > e M χ )+( 2 e ~q LSP ( χ i potential interaction, the simplest Sommerfeld factor in the charge- 0 ∗ e /m 2 ] 0 ) H E as has been done in ( S and χ ˜ 1 ψ † m κ [ 0 2 χ g e Since even for a single MSSM parameter  χ 3 0 1 17 ) ~q e 3 π χ d (2 14 problem is only necessary if all 14 states are → – 41 – Z × 2 i e i ξ χ 1 leads to an enhancement of the radiative correction, Γ e † c j χ ξ ) as h LSP 0 . A general MSSM parameter point, however, may also 2 m ee ∗ 4.65 v κ  14 matrix problem. = LSP i φ × j , the computation of the m m χ i E χ ∼ | 1 e ). Note that the second term in the definition ( χ EW 2 are parametrically of the same order, since  . After this decoupling, the generated local interaction is a 4.59 vm 3 ∂ κ ←→ ∼ i 2 (since the potentials for 0 φ − ee m  κ 2 2 † c e α = However, an exact solution of the 14 In parameter-space regions where the Sommerfeld effect is important and the exchange χ The numbers strongly depend on the model and velocity but provide an indication of the increase in | 0 ¯ e 0 17 ¯ e h the heavier two-particle statesof should the have lighter little states,ladder effect since diagram on the series the propagator is Sommerfeld of off-shell enhancement the and heavier does state not inCPU lead time one to as of the an the number enhancement. of loops channels Yet increases. of there the is the set, many Sommerfeld factors mustthe be computed thermal for average a requires singlerestrict scattering an the energy, and integral number further of over channels, the for scattering which energy, the Schr¨odingerequation CPU isnearly considerations solved degenerate, exactly. which is rarely the case. As soon as the mass splitting becomes large, and scattering energy neutral channel for a velocitywhen close only to two out the of threshold the of totalincreases of a to 14 nearly channels 14 are s degenerate included state for inthe takes four the solution 0 computation, channels, of the 5 CPU the min time full for 14 8 channels and reaches nearly three hours for For a generic MSSM parameter spacerange point and the not two-particle states be willsolution allows span degenerate. us a to certain While cover mass thestill the non-degenerate practical case improved without limitations method numerical related instabilities for to therehence the the are the increasing dimensionality Schr¨odinger-equation CPU of time, the as matrix the increases. number For of example, states, if and for a given MSSM model solution is to decouple the heavytial Higgs of bosons section by not including themto into the the MSSM long-distance poten- part,Higgs-exchange which from we the potential consistently for neglect. Higgs masses In practice, we4.5 simply eliminate Approximate treatment of heavy channels and lead to heavy Higgsa bosons, which large though contribution strongly topower-counting suppressed breaking the in contribution is last the the potential, termHiggs-boson linearly may exchange in divergent give causes integral square ( a suppressed, brackets. local ( The origin of this unphysical for the exchange of massivebe particles, absorbed and not into for an the effective Coulomb annihilation potential. matrix The ˆ factor of a particle withall mass terms in coupling structure), to write ( which proves ( κ JHEP05(2015)115 ) ). ] for 4.68 2.13 ~q (4.68) (4.67) [ K ) 0 ˆ , E eff le ; ˆ , see ( f   k e ~ heavy states, h 2 µ ~p, k ~ n 2 ( il − − ˜ G ] refers to one of the  N h 2 µ . } ij ) ~q 2 LSP 2 2 h µ ~q ~p h 2 ( m 1 − 1 ∗ li 2 ] ] h − { E − ˆ ˜ is the Green function of the E ψ h LSP [ =  il 3 m M when a light channel is replaced 1 ) [ h ˜ ~q 2 G 1) 3 π a − d − − ) and perform the integration over )] (2 ( h E E ) Z LSP M → 4.63 [ ~q lim i ˆ E i m ξ − 2 − i i Γ k ~ ξ † E − ) to the integral involving the wave-function ( c j Γ ) h ξ † lh Suppose that h k c ~ j ˆ – 42 – M ξ V 4.67 ( h ] − 18 = 2 k ~q ~ ) we obtain = [ ( E/ i i j lh of two-particle states (14 in method-2 in the charge- K j 4.5 χ 3 problem for the light-states. The potential interaction ˆ χ V i and summed over all two-particle channels ) i k ~ ] N χ 3 n π } | χ ~q | d 3 1 [ 1 e (2 e × 4 h K χ e , n χ 2 for the contribution in the last loop before the annihilation,  Z 3 } }{ χχ  / ) 0 k 2 ~ ¯ h into the sum over light and the sum over heavy channels and ∂ e 3 → π ←→ ∂ he 1 ←→ d ˆ i e e 2 f (2 = 1 χχ → i { 2 ) runs only over the light (heavy) channels, and the equality in ( ˆ light states, which will be treated exactly, and f a h h − ( Z − ] + { n  ≡ l ~q 3  S [ 0 ) ~q K + 3 K π ee K Γ L ˆ d 0 f Γ ), we find 2 † (2 c e 2 le † 1) ˆ c h f χ | − χ Z   | 4.13 0 2 when the heavy state appears inside the ladder, which justifies the omission. h 0 / × 0 + ( h × refers to the light channels summed over, and ee ˆ l f = 3 converts the light channel into the annihilating heavy channel Our aim is to write the square bracket as an effective annihilation matrix The annihilation cross sections matrix elements such as above are multiplied by annihi- We divide the total number a We also include them as external states, but only at tree-level. Heavy external states are not relevant for lh 18 ˆ the light channels. We write the potential asthe relic-density in calculations, ( since they areannihilation Boltzmann-suppressed, of except for the special heavy casesBoltzmann state, such suppression. as in resonant which the resonant enhancement of the cross section compensates the where the sum over holds when heavy channelsperturbative contribute expansion only of in the the matrix last element. loop before the annihilation of the similar to ( V lation matrices Dividing the sum over proceeding for the heavy channels from ( where Sch¨odingeroperator for the heavy states, then by extension of ( but In the following, weabsorbed describe into how an the effective effect annihilation of matrix the for heavy the channels lighterneutral in channels. sector) the into last loopwhich can we be include only in the last loop. lighter states and henceallow effectively the enhances heavy the channels annihilation toelsewhere rate. appear in in the To ladder the cover diagrams. last thisthere loop Moreover, case, the is before non-relativistic we a the power-counting shows suppression annihilation that factor vertex,by but of a not order heavy one, [ with possibility that a heavy state couples more strongly to the annihilation process than the JHEP05(2015)115 ), in ~q ( L li I ] (4.74) (4.71) (4.72) (4.73) (4.69) (4.70) , hence defined E 2 ˜ ψ ~q S,P . Since the I . . E ) ] and also in our 2 2 2 ~q 45 . , ~q y ~q p ) to the long-distance a 2 2 φ ) , p v m to run over all rele- ( m 2 m 2 , which has also been O , m h ~q l + ) + = 0 in coordinate space. ) + 2 y ~q 0 ) + 2 y i p y 2 2 r √ hh p m ~q − √ ~q ˆ f 2 ] 0 3( − + h E 0 ∗ l y y ) + 2 ) + 2 I → + 2 2 , − − lh ) ~q 0 ~q I 2 2 2 hl LSP ˆ − + m f + m , we can expand the integrals ( ( y m y i i E lh lh ˆ I [2 f − − ln y, m, ~q h ) represents only “one half” of the four- 2 2 + 0 LSP − ( ∗ 19 l ) . 0 1 I m h 2 m m 1 ll I , ( ( ˆ i f 0 ~q 4.67 i + M ) taking channel index 2 I ] keeps in addition terms proportional to mass differences ( ) ∼ 0 / ) – 43 – + = h | 3 ln ~q 45 hl ] = 1) ) 2 ˆ µ ~q y f 2 | 2 4.69 ~q ~q [ (2 π m ~q ~q − lh ] = right ’s inside the loop that originate from the numerator of the full i L I K , 2 χ ~q 2( I + p 1 0 [ ]. It can be checked that the loop integrals ) eff + 2 ll m y ) long-distance corrections. ˆ h ( 0 K f 2 is large compared to the scattering energy 45 i ll v √ µ = 0, ˆ ( f 2 ) = + O L 2 − → LSP = ( = 1, ) m above are not yet useful, since they define functions of ) 0 m 2 2 a L eff ll 2 ( lh L ˆ − ~q f c I terms arise as well from other sources, for instance from the anti-particle pole ) terms in the Wilson coefficients of the annihilation operators. We note that 2 y − ) would require the full momentum-space wave-function [ 2 y, m, ~q v a h ( √ ( X ’s and the virtual 0 LSP ). Keeping only the leading term in this expansion, we approximate to obtain y, m, ~q I 2 χ M 4.68 -integration of the full amplitude, which have been neglected in [ ( 0 α 0 /m ∼ O k ~ ) = q ) I LSP 2 = m -wave case ( LSP LSP 2 lh m I P /m − ) − h y, m, ~q i ( LSP 1 m M I ( m -wave matrix elements ( / We wish to note that the result ( The expressions for 2 − S The non-relativistic expansion performed in [ i ~q 19 m between the incoming 1-loop amplitude. In ourpart, approach, which the latter we( can do correspond not to consider, correctionssimilar of as ( well ascontribution in to the theapproach, short-distance since annihilation, they are where a we class of have kept all the evaluation of ( which is not available, andHowever, not we just would the like valuethe and to mass derivative make at splitting usetypical of relativep these momentum expressions scales only as for heavy channels when vant two-particle states regardless ofapproximation their mass to provides the the 1-loop leading-orderobtained non-relativistic by annihilation direct amplitude expansion in ofin [ the the latter state reference, are equal to (2 left then gives for the effective annihilation matrixin in the the last space loop of of light the states ladder. including the heavy states The superscript “right”fermion reminds operator us that that accountsfor ( for the the fermion square bilinear of matrix the annihilation element amplitude. that multiplies Accounting the annihilation matrix from the for and for the with loop momentum JHEP05(2015)115 ) n ). 2.9 4.73 (4.77) (4.78) (4.79) (4.75) (4.76) , )] of the 0  . 0 S 2 . The spec- 1 hh ! ( = 200), when ) M ˆ ˆ g f a 4.2 [ ( lh ∞ 0 1 c =0 χ x a 0 1 #! ( L GeV, the chargino | φ χ 0 h S 2 } h S 0 µ m ∗ l k ~ 2 I 54 14 matrix problem by . ≡ 2 a − X × 2 S h ) + lh,D a matrix, as defined in ( a 3083 I φ M , φ a g φ + m − m 0 98 2 . m ) + hh + E a a M ( lh ˆ g + h φ h 4]. c y + 2 h y / 3062 m y 2 ) √ , √ ,D ) a 0 2 ( lh √ + 2 LSP h 25 c 3( 0 h . ∗ l h ) m I a y m a 2 X ( lh √ " c =0 ) + 2950 L h | 1 , − a a µ h 4 lh ). The result is that the effective annihilation X X . 2 h I – 44 – 012 shown in the upper plot approaches the value 212.18 -wave operator is somewhat more complicated, m 2 2 . − M + S α α + GeV. We consider the charge-neutral two-particle )( 4.65 2 h h 2749 − = 0 h lh 2 } ), from which the effective annihilation matrix ( µ µ l { ˆ g M v 2 2 ) is built from the particle species that enter the two- i E M 26 m heavier channels are treated approximately by including 2 . − − = − + − + n 0 ] = = 4.78 h,D χ 1 0 E − l ∗ E l 3074 I ). This is done by writing m LSP lightest states are included in the solution of the Schr¨odinger , m + =0 =1 + + m L L [( n -wave, spin-0 configuration on the number of states included in = 200) for | | 61 0 2 2 / . S 4.68 lh lh terms in the potential for all 14 two-particle states, not only for hl LSP LSP ∞

lh I I ˆ g M ) gives x increases from 1 to 14, the dashed curve shows ˆ g h ( m m 2 in ( µ S 2749 n ≡ [2 2 (2 Z,W { 2 the dependence of the Sommerfeld factor 4.70 2

− lh,D k ~ = 2 I 2 7 = /M h  2 1 α /M 20 e + M h ] = k ~ in each term in ( 1 2 4 lh ( χ e state in the µ k ~ g h [ + 0 1 m 0 µ M lightest states are included and the remaining ones are ignored. The solid curve δm K χ = 2 1 0 1 κ,ll e n 2 treated exactly, such that the potential is independent of the number of states = 2 χ g 4 e h n = ˆ y 0 lh,D To illustrate the quality of approximation to the heavy channels described in this sec- The case of the second-derivative δm I This is explains why eff κ,ll 20 ˆ g them in the last loopthe before the annihilation.those For the purpose oftreated this exactly. comparison we keep lightest the computation. As only the shows the result, when the equation, and the remaining 14 tion, we employ the same wino-LSPtrum model of as neutralino discussed at masses themasses is end are of section state sector with methodmass. 2, We in study whichplotting the case in approach there figure to are the 14 full two-particle states treatment of of increasing the 14 The factor particle states specified byFor the instance indices ˆ of the accompanying where which leads to thematrix same for factor the as second-derivative operators in reads ( with follows. since one has to applythe the factor equation-of-motion relation discussed in the previous section to Inserting this into ( JHEP05(2015)115 = ∞ x state, ( S − 1 χ + 1 χ 14 14 = 2 on the dashed curve. n 12 12 )] in the charge-neutral sector 0 S 1 ( 15 (lower plot). The figure shows increases from 1 to 14, the dashed ˆ . f [ 10 10 0 1 n χ 012 (upper plot), slightly below the . 0 1 = 0 χ v S lightest states are included in the solution heavier channels are treated approximately = 0 8 8 n ≡ n v n n S − 59, which corresponds to . 6 6 – 45 – 199 state, and lightest states are included and the remaining ones are terms of the potential, which here are included for all states. ≈ − 1 2 ) n χ ) 4 ∞ 4 + 1 ( EW χ S 2 δm/M 2 when adding states 11, 13 and 14, which suggests that these states are 7 0 0

in figure

1.60 1.58 1.56 1.54 1.52 1.50 205 200 195 1.64 1.62 215 210

L H L H ¥ ¥ 200 S 200 S x x = = S state exactly, as is of course expected, since the bulk of the enhancement = 200), when only the − 1 ∞ χ x ( + 1 012 (upper plot), slightly below the threshold for the nearly degenerate . S χ 06, hence their effect is small. This is expected, since the Sommerfeld enhancement is mainly 15 (lower plot). . . . Dependence of the Sommerfeld factor = 0 v = 1 is always a poor approximation. This says that it is necessary to treat the nearly 206 = 0 n ≈ We show the result for two velocity points, v rather than the previouslyIf, quoted one value the other200) hand the mass-difference termsgenerated in by the the potential two arenegligible. lowest, neglected nearly entirely, The we degenerate rise obtain states, of comparatively for more important which than the the lower mass onesof difference (except salt, the terms since first it and are second), originates completely should from be the taken ( with a grain threshold for the nearly degenerate that degenerate of the Schr¨odingerequation, and theby remaining including 14 them inpoints, the last loopand before the annihilation. We show the result for two velocity Figure 7 on the number ofcurve shows states included inignored. the The computation. solid curve As shows the result, when the JHEP05(2015)115 = 14 value. n - and next-to-next-to-leading P . The calculated enhancements thus sum ). The present paper describes all the nec- 2 rel = 2 is already very close to the LSP v ( n O – 46 – /m ) corrections in the short-distance part has been Z,W 2 rel v ( state with the charged state in models where the LSP M 0 1 O , quantum corrections to the tree-level annihilation rates ∼ χ 1, the dependence of the Sommerfeld enhancement on the 0 1 rel χ ,... v 2 , n > ] focused on the short-distance part of the annihilation process, ∼ 15 2 = 14 than the ones that leave out the heavier channels completely. = 1 α n n , that includes n ) j χ rel ] and II [ i χ /v 14 2 α ( O -wave effects, i.e. up corrections of S A general formula for thechannel Sommerfeld-corrected annihilation cross-section for a given Papers I [ While this shows that the one-loop approximation for heavy states is often a good • counting, which treats up large to all orders. The main results of this this work are: providing analytic results forlation the Wilson rates coefficients of that neutralinosorder reproduce and the charginos tree-level including annihi- essary up ingredients to to account forenhancements the in long-range the interactions annihilation that rates produce at the the Sommerfeld leading order in the non-relativistic power- the long-distance matrix elements; theseThe contains framework the so-called developed Sommerfeld here enhancements. arbitrary applies admixture of to the general electroweak gauginos,all MSSM and the accounts scenarios, for neutralino where the and co-annihilations the charginos, with LSP which are is nearly an mass degenerate with the LSP. neutralinos and charginos atis small similar relative to velocity. thefeatures non-relativistic of The EFT the construction for NRMSSM ofsive heavy are mediators the quarkonium the exchanged NRMSSM annihilation. presence among of the(rather Characteristic more heavy than than particles, colour one which Coulomb) generate heavy potentials; particle electroweak and Yukawa species; weak mas- couplings that allow one to compute 5 Summary With this paper we conclude theformalism, presentation designed of to the calculate non-relativistic the MSSM effective enhanced theory radiative corrections to pair-annihilation of mation is a reasonable approximation toapproximation the in true the loop corrections, potential sinceexact region the is non-relativistic loop not integral expectedthe for to Sommerfeld be very factor a heavy anyway, goodprocedure channels. in approximation described practice, to in Since we the this simply section. such include states them have by little the influence approximate on loop before the annihilationthe process, two but cases shown, to it neglect isstates them essentially approximately, as sufficient in to the the treat solid Schr¨odingerequation. only curve In two for states exactly and allapproximation other to the full,one-loop resummed or result, full resummed it treatment does of very not heavy prove channels that in the either non-relativistic of approxi- the two, the is almost pure wino.remaining Once states is no longerthe large, results resulting that in include only the another heavyfull channels 5% result approximately attained increase. have for a We smoother observeThis approach that demonstrates to that the it is a good approximation to include the heavy channels only in last comes from the mixing of the JHEP05(2015)115 ) as a 2.24 ], which further 16 has been decomposed in ( rel v j χ i χ σ – 47 – . Since the Sommerfeld factors depend on the partial-wave ). The formula is a generalization of the tree-level approximation 2 rel b v 2.24 + a = rel v j χ i χ demonstrated that including heavy channelsprovides only a in the good last approximation loopnumber of before to channels annihilation the to full be treated resummed exactly. result, and thus reduces the factor instead of solvingment for the yields scattering accurate wave functions.channels numerical are This solutions included, important up when improve- to many CPU non-degenerate time limitations. An two-particle approximate treatment of heavy channels,annihilation which matrix can for includes the them Schr¨odingerequation of in a an lower-dimensional effective problem. We A novel methodmethod to does solve provide the anically multi-channel accurate closed Schr¨odingerequation. result channels for areponential the The included growth Sommerfeld due standard of factors to some whenreformulating numerical of the kinemat- instabilities the Schr¨odingerequation for solutions. caused the by The quantity the relevant new for ex- method the solves Sommerfeld this problem by tralino pairs for a generalSchr¨odingerequation that MSSM has parameter to point, be whichtors. feed solved into to Transitions the among obtain multi-channel different the states Sommerfeldin are correction the accounted fac- potential. for by the off-diagonal entries σ configuration of the annihilating state, sum over partial waves, where thecomputed annihilation in coefficients papers for I each and partial II. wave were Compact analytic results for the potential interactions among the chargino and neu- written down in ( While this paper, as well as papers I and II, have been dedicated to the technical details • • • forschungsbereich/Transregio 9 “Computergest¨utzteTheoretische Teilchenphysik”.work of P. The R. has beenthe supported EU in Commission part [Grants by No. the SpanishCPAN)] FPA2011-23778, Government and No. and ERDF by CSD2007-00042 funds Generalitat (Consolider from Project Valencianaalso under thanks Grant the No. “Excellence Cluster PROMETEOII/2013/007. Universe” at P. TU R. Munich for its hospitality and support Acknowledgments The work ofregio M.B. 9 was supported “Computergest¨utzteTheoretischeTelekom Teilchenphysik”. in Stiftung” for part its C.H. support. by thanks ThisLeibniz the work programme the is DFG of supported “Deutsche the in Sonderforschungsbereich/Trans- Deutsche part Forschungsgemeinschaft by (DFG) the and Gottfried the Wilhelm DFG Sonder- describing the computation ofinvestigation of the the Sommerfeld-enhanced effect annihilationheavy on cross the neutralino sections, relic LSP abundance the isillustrates in the the well-motivated subject general MSSM use scenarios of and with an potential accompanying of publication the framework. [ JHEP05(2015)115 ]. = 48 and i (A.1) (A.2) X + i ( , , χ i X -exchange  i  i j j ± , , 2 2 + + e e  -boson or the Z Z   W i i ) ∗ ∗ 1 1 i i Z − − ∗ , 2 2 j + + e e with a (pseudo-) Z Z 1 N e e Z Z e 0 i Z j j -boson are given by ∓ ± 4 4 N N   χ + ∗ ∗ W ) e e provides the coeffi- i i Z Z j θ ), the 2 2 + + 1 W 2 2 N 2 e e 0 m 1 1 Z Z e Z √ √ ∗ ∗ , j j ,A W 1 1  − − − + θ 0 + tan e e m ij Z Z i i δ ∗ 2 2 − −   H . A similar cancellation occurs j ) and part of the , an outgoing chargino e e 2 Z Z N 0 j m m 3 ∗ ∗ 2 W 2 2 e H R ± χ Z j j 2 s + tan ( 3 3 N N Z Z ∗ H j e e i . We have used the abbreviations Z Z − 2 N 2 + + 0 + 2 2 e e Z and an outgoing = 1 1 Z or an incoming G 2 W   ( √ √ j j c i 2 0 j 1 1 ± + 2 + + − 1 χ ≡ √ e e e + + Z Z Z G H , i i ∗ ∗ 0 2 2 2 2 + 2( − − i i +  1 1 1 + + e e or A j ∗ Z Z √ ∗ e e 1 j i + Z Z , 1 1 + − + j j e ∓ ± Z 0 − 2 2 – 48 – e e N N Z Z ∗ G ∗ ∗ i i i generated by the exchange of boson e e A i i ∗ Z Z 1 − denoting the Weinberg angle. The coupling factors 1 1 + − j 1 1 + + e 4 e e N Z e e Z Z + − Z Z ≡ j e W /r Z ∗ ∗ i i 3 N r θ j j + − 1 1 − − 0 1 i e  ]. 2 2 − − , and the mixing matrices are defined as in ref. [ Z e e j ∗ X Z Z ) does not have to be considered, since it cancels against A e e j 1 + Z Z + m  ∗ ∗ 0 2 47 m 1 − 2 [ j j e H   Z G 2 2 A e − N N m Z ∗ Z m m e 1 i i e e H , with Z Z 1 1 1 1 R H and + − ≡ Z e e   W Z Z  Z Z , 0 θ h h   ± 1 2 1 2 1 2 + 2 h ij 2 2 δ − W W H i 1 1 1 , c √ √ c ≡ W 4 2 2 4 ) for each process. The complete potential is obtained as the sum of = = s = cos 0 2 ± ) = Jaxodraw − − − − and W W ij ij + H m W v a = = 0 = = c ,H H ij , + 0 1 p 0 ) = ) = -exchange potential, as explained in section Z γ ij ij γ ij ij Z ( H 0 ] and m 0 m v v a a H Z ,A + m and A ij H ij ), whose explicit expressions are given next. 0 46 m p H ij p ≡ [ ≡ ( ( s W 3.3 θ + 0 0 0 m 1 m 1 A ij H ij H H , γ, H s ), ( s ± The (axial-) vector and (pseudo-) scalar couplings, that encode chargino interactions = sin 3.1 W where Finally, three-point interactions of an incoming where s for the three-point interaction ofeither an an incoming incoming charged neutralino Higgs particle photon read between the charged pseudo-Goldstonepotential. bosons The coefficientsin have ( been written in terms of thewith coupling the factors neutral as scalar defined and pseudo-scalar Higgs particles ( cients of theZ,W Yukawa terms the coefficients multiplied by the correspondingpseudoscalar potential Goldstone terms. boson The ( contributiona from piece the of the Axodraw A Explicit expressions forThe the leading-order MSSM potential non-relativisticchargino potential two-particle interactions states among are MSSM given neutralino in and this appendix. Table during completion of this work. Feynman diagrams have been drawn with the packages JHEP05(2015)115 ). 2 3 ∗ ∗ 2 3 − e e r + + e e + + 3 2 χ 3 2 H e H e + H e H e + s s H s s χ ∗ ∗ r 1 1 1 1 M 0 0 0 0 0 0 e e e e + + + + − 4 4 4 4 → e H e H e H e H e 2 s s s s + α χ − − − − − 2 3 2 3 2 3 χ ,φ e e e e e e 3 2 r 3 2 3 2 φ e φ e (0) e φ e φ e φ e φ s s s s s s m 1 1 4 r 1 1 1 ,φ ,φ − ,φ e e e e e e e 4 4 1 4 4 4 . φ e φ e φ e 2 (0) e (0) e (0) e s s s α s s s − − − − − − 2 Z/W 3 e 2 2 /M 2 e e 2 γ e 3 3 e γ γ e e v 3 2 v v 1 e r 0 0 0 0 0 0 0 0 e 1 4 α 4 e e has to be summed over the “physical” γ e 4 1 δm ) or are vanishing (like γ γ e e v φ 1 0 v v e − χ 4 e + 2 2 2 2 3 3 χ ∗ ∗ ∗ e e 3 3 e e e e δm ∗ 3 3 e e 3 3 2 2 W W e e 2 2 W e W e W e W e r → W W e e v v a a a a 0 v v 1 1 W 1 1 1 1 ∗ ∗ ∗ e e 1 1 e e e e χ ∗ M 4 4 e e 4 4 4 4 r 0 0 0 0 0 0 0 0 0 W W = 1 + e e + 4 4 W e W e W e W e − – 49 – W W e e v v e χ a a a a v v 2 S S S S W W α or W W λ λ λ λ Z/W λ λ λ λ + λ + + + + − − χ . The expressions obtained from the table correspond to the − 0 2 3 χ 3 2 2 2 3 2 ,Z e e e 2 e e e e e 2 2 3 ,Z 3 2 ) and 2 e 3 3 3 2 ,A 3 e e e Z e Z e Z e 3 → (0) e Z e Z e Z e 0 Z e 3 3 2 S r Z e a a v (0) Z Z e e e a a a a v 4 v Z + 1 1 4 v v v , h 1 1 1 1 ,Z ,Z ,Z 1 e e e e ,Z 0 1 4 χ M e e − e 1 e ,Z 4 4 1 1 4 e e r ,Z 0 0 0 0 4 4 4 e − 4 − Z e Z e Z e Z e e 4 1 (0) e (0) e H (0) e 4 e (0) e Z Z 4 e e (3 a a a v χ (0) e a a v (0) e 2 a v v v Z = S S S v α Z S S ≡ Z Z S Z λ λ λ λ Z λ λ λ φ λ λ λ S λ λ − + + + − λ + + + 3 3 3 3 3 3 3 3 3 3 0 e − e − e e + + e − e 0 e 0 e + e − e χ χ χ χ χ χ χ χ χ χ 4 4 4 4 4 4 4 4 4 4 0 e + e + e e − 0 e 0 e + e − e + e − e χ χ χ χ χ χ χ χ χ χ → → → → → → → → → → . Potentials describing the non-relativistic interactions among chargino and neutralino pairs 2 2 2 2 2 2 2 2 2 2 0 e − e 0 e 0 e + e − e + e e − + e − e χ χ χ χ χ χ χ χ χ χ 1 1 1 1 1 1 1 1 1 1 0 e + e 0 e 0 e 0 e 0 e 0 e e 0 + e − e χ χ χ χ χ χ χ χ χ χ neutral Higgs bosons, potential entries in method-1.interchanging indices The (like potentials for theWe have channels introduced not shown are obtained trivially by Table 2 in the MSSM. The potential from neutral scalar exchange JHEP05(2015)115 . 2 ∗ χ ba V (B.1) (A.3) (A.4) c.c. , c.c. , = and ± ± 1 ab i χ i V ), since the ) ) j j 22 V ↔ ), which are all ↔ i = i A.3 33 + ( + ( V   , respectively. We can ∗ ∗ , . 1 j j 1 N 1 χ N ji ji e ) and ( 2 e Z a p Z χ , − W W A.1 θ θ = = (as well as and , . ∗ ij ∗ ij ) = 0 tan 2 tan  23 3         χ V u − ) − 1 j 12 23 22 ∗ 3 matrix where the generic potential ∗ χ + j = j V V V 2 N , 2 N 2 ↔ × 1 e 32 e Z u Z 12 22 23 i χ ( V ( V V V 1   12 χ   ∓ . ∗ ∗ 11 21 21 V i i ∗ , named as method-1 and method-2, gives the 3 N 12 V V V 3 j N ) e 4 + N 3 e Z – 50 – Z e         1 Z i m χχ u m 4 1 N H 1 R e Z Z 11 ) = Z V r − ( − − and ( + ∗ ∗ ∗ i j , a (1) , p i 1 4 N 3 11 N 4 N ) in method-1 when applied to a generic multi-component -boson involve the following (axial-) vector or (pseudo-) ) V ji ji e e e Z Z Z v s Z i ) reads 3 D u N m χχ 3 m 2 4.23 e H 2 R Z Z , u = = Z  2 21, corresponding to 3) reads , ∗ ij ∗ ij W , h  h  , u s v 1 c 2 1 12 i 4 4 1 4 , , u as two different neutralinos (or two different charginos), and we note 2 = 1 ) = ) = ) = χ = 11 0 0 m m ,Z a ,A a, b ,H (0) ij ( and a (0) ij (0) ij ( 1 p p (1) ( ( χ ab ,Z V 0 m 0 m (0) ij ,A v ,H The equation Schr¨odinger in method-1 is a 3 Let us consider the simple case of a system which in method-1 consist of three states (0) ij (0) ij s s amplitudes that produce thesevertices potential of terms the only associatedThe differ diagram. Schr¨odingerequation by ( Note a that relabeling hermiticitywave function of requires Ψ the as = well internal ( that and we have used that thesince potential the for exchange scattering of of state statein 11 12 the to by diagram, either 21, gives 12 which the11. or corresponds 21 same In to is amplitude addition the crossing due we same, the have to lines taken the into of identical account particle that nature of the state think of that only state 11built is from composed only of two identical states, particles. ( Conversely, the basismatrix of method-2 is We wish to show inof this two-particle appendix states by introduced means in ofsame section an outcome example, for that the using Sommerfeld any of enhancement the factors. two basis labelled with as a consequence of the hermiticity of the underlyingB SUSY Lagrangian. Equivalence between method-1 and method-2 The (axial-) vector and (pseudo-)related scalar to coupling interactions factors with in neutral ( SM and Higgs particles, satisfy scalar coupling factors: scalar Higgs particle or the JHEP05(2015)115
), J L (B.5) (B.6) (B.2) (B.3) (B.4) 4.26 +1 ( reduce S and the 2 2 , ab . 1 T } ), in which M 3         e 23 4 , e u 23 23 }{ −         and 0 2 M , e M 12 1 − 23 ab 22 ˆ e f ˆ f { ˆ , u 12 f L L (0)] + ), and acquire the form 22 22 + L ) applied to Ψ M 22 S 3 ˆ S f + u 2 1) M M 2 ), whose specific form is . On the other hand, the , , +1) = (0 S 1) − L B.2 √ ∗ 12 ( L 3 , . 1) ˆ − a f u ] ( . We have to seek for three and = 0 = 0 1+( 4.23 − 11 r L 2 i i = 21 21 . = 0 = 0 2 2 u M u 12 = ( u u 22 ˆ 3 3 in ( 2 M M f 21 ˆ f ) ˆ u u
f = [ 12 √ L = 0 J L 23 ab + V 3 23 22 + δ L V 22 S         23 2 V V ˆ S = even, and up to the global factor f 1) 2 u + +1 √ = , since such an state cannot be formed + + 1) − ) S S S 2 2 2 22 − + 23 ( u u V 1+( i V + + } 1 3 22 23 u e − L L 11 21 21 ,M + ( V V 4 ˆ ˆ ˆ f f f e i 11 – 51 – ). Let us first choose two solutions of the form 22 1 + + V         L L L }{ V u ) from the matrices [ 1 ) are symmetric in + + + 1 1 e , to write the annihilation matrix entries involving + S S S 23 B.2 u u 2 4 21 23 ), we can readily compute the Sommerfeld factors i ) is chosen as Ψ e 0 + ( U 1) 1) 1) 2 2 2 e V 1 { B.2 U 21 21 4.34 ˆ − − − f − 2 23 V V 2. Then the set of equations ( B.6 B.2 ↔ , √ D u 1+( 1+( 1+( 12 + + 3 D u U 2 3 e 22 22 + on the independent variable         ). The entries of the annihilation matrix in method-1 read i = 1 2 U U 2 i = √ ab D u D u V ). For the case 3, suggested above these matrices must have the structure 4.36 ], namely D u 11 , (1) 2 U 21 21 )] , 15 4.36 , J 2 U U built from ( and ), with L i √ 14 a 2 = 1 T u +1 i         , u S , i 2 i 2 ( ˆ f , u [ i 1 (0)] = is determined following ( u 2. A third solution to ( , 2 derived in [ T +1) 2 √ L e L ( = 1 u [ i = ( stands for the differential operator multiplying ↔ i 1 With the matrix for method-1 using ( which are both builtthe from solutions the Ψ independent solutions of the Schr¨odingerequations. Given and likewise for thestate exchange 21 in termsrates of involving those channel of 11out 12. vanish of for two These identical odd samematrix spin-1/2 relations particles. also Note tell also us that that the annihilation where we havee used the symmetry relations under the exchange of the particle labels In order to computeannihilation the matrix, Sommerfeld see enhancement ( factors we need the matrix for case only one equation remains: Ψ to two independent ones (since eqs. ( D irrelevant for the currentof discussion, the and quantities we alsoindependent omit solutions writing of explicitly the the system ( dependence JHEP05(2015)115 . ∗ 12 ˆ 23 f V (B.9) (B.7) (B.8) = − (B.10) 21 22 ˆ f ) and the V , , .  ) = 3.17  . Coming back r     12 11 2 ( ˆ ˆ 12 22 f f , we find ˆ ˆ χ f f 23 for the wave func- 11 22 L 22 11 ∗ ˆ ˆ odd V f f ˆ ˆ ∗ ) , f f ) (see ( + 2 and 2 12 −

(2)

even 12 2 S 1 , U V U 22 22 22 χ 12 22 V (2) 21 U M 11 U M , . V 0 0 0 M M 11 11 2 21 11 | 2     − ), at the expense of using two | M M − = 0 = 0 M M 22 22 11 2 2 − − 11 − − B.8 ˜ M u u ) = U M U r ) ˜ 11 21 ( 21 21 11 12 22 11 23 U 12 V U M . M M V M odd 2 M M 12 2 , 22 12

12 − + )( √ − )( M (2) M M 23 22 M 23 21 22 +

22 21

U , sector we deal with just a single annihilating U V 1 M U 1 2 M

˜ M u S 21 ,V 11 12 + ( + 2 + = 0 = 12 11 + . In method-1 this is taken into account auto- – 52 – M 1 2 2 M V M

    L ˜ | u M 2 the identical-particle state 11 does not exist, and − | = 0, as found in ( − odd odd + 22 22 23 11 12 12 22 21 S 21 1 12 V S S V 21 U M U M ˜ u V U odd + 2 11 U + 2 11 11 21 11 S D 22 √ L 12 22 √ M M M M M V + M ( − − − − ( 2 12 could give a non-vanishing annihilation amplitude for an  21 ˜ u  21 21 11 21 V 11 , we obtain D → U U 2 2 V M M ] 2 Re 2 Re √ 12 22 12. Applying the rules described in section 12 12 √ , +1 √ M M     L a M M k

+ 2 + = odd = 11 ) = = = S r ) of method-2 read a ( 2 + 1)! + ˜ u ) in a more compact form. even even , 12 11 L L 1 even ( S S , u / B.7 (2) V e + 1)!! Ψ = (˜ The Schr¨odingerequations which derive from the potential Let us now turn to method-2. As mentioned above in this case the basis is given by L tion incoming state 11, asmatically argued and one in gets section thedifferent states result (12 and 21) to describeto the method-2, physical we state then built see from thatstate. in We the can odd therefore switch to a one-dimensional problem with In particular, note thatthe for entries of odd the potentiala matrix involving potential that transition state 11 have to be set to zero, to avoid that the two states discussion following that equation), we canto build method-2 the potential from matrix those entries corresponding of method-1. Distinguishing even and odd The Sommerfeld factors forshould the be the state case 21 sinceto are both write states found ( are to physically be equivalent. We equal have used to that those of state 12; this whereas for [(2 JHEP05(2015)115 . 2 , 3 T 1 (0)] , of Ψ (B.14) (B.11) (B.12) (B.13) +1) L ( L 23 u 2 for each u ) as linear √ 22 / (2005) 063528 ) by inserting , u that match the . 12 u 4.36 S     D 71 : + 12 22 . = ( 2 L M M 22 2 ˆ f A theory of dark matter e Ψ 11 21 . = M M ]. 23 Phys. Rev. odd     M , , , ) and . The (one-dimensional) matrices = = = even. The matrices [ (2) 21 S SPIRE ˆ f = 0 S Non-perturbative effect on dark matter IN , u + odd even 2 ][ + , 11 L u u ) ˜ L     ), satisfied by the component 23 obtained for the even and the odd cases ,M ,M V 12 = ( ˆ f B.4 23 22 − ˆ     1 M 2 f U 1 e – 53 – √ Ψ 22 12 22 = V U U follow immediately: 21 ˆ 11 f 2 ˆ , f 11 21 + ( odd 2 1 1 2 1 2 U U arXiv:0810.0713 √ e Ψ ˜ u (0)] and [ are simply given by (0)]     ). This straightforward computation yields results for     ), which permits any use, distribution and reproduction in D S as solution for odd +1) = +1) ]. = L + ( L L 23 ( L B.14 u ) obtained previously for the components of the solutions Ψ u L u even [ , there is just one equation, even , S B.3 SPIRE ) derived above using method-1. (0)] e IN (2) Ψ = + ˆ ), respectively, it is immediate to compute the corresponding f CC-BY 4.0 (2009) 015014 B.8 ][ L +1) for odd L This article is distributed under the terms of the Creative Commons ( L B.13 u [ M D 79 ) and ( ) agree with ( B.7 ) and ( that derive from solutions (0)] and B.10 hep-ph/0412403 annihilation and gamma ray[ signature from galactic center Phys. Rev. M B.11 +1) J. Hisano, S. Matsumoto, M.M. Nojiri and O. Saito, N. Arkani-Hamed, D.P. Finkbeiner, T.R. Slatyer and N. Weiner, L ( L [1] [2] u Open Access. Attribution License ( any medium, provided the original author(s) and source areReferences credited. in ( matrices. Thethe Sommerfeld annihilation factors matrices forthe ( method-2 Sommerfeld then factorformulae follow of ( from states ( 11 and 12 for even and odd From the expressions for [ Finally, we requirefactors. the annihilation These are matrices obtainedidentical-particle in state from involved method-2 those in of to the method-1 entry, compute as by explained the adding in a Sommerfeld section factor of 1 which is the sameHence, we differential can equation choose as[ ( For the case of odd of method-1.independent Therefore solutions we for can method-2and choose in the case Eqs. ( JHEP05(2015)115 , ]. , 10 ] , ] SPIRE IN ]. JCAP ][ , ]. Consistent , , (2014) SPIRE IN ]. ]. SPIRE ][ IN ]. ][ hep-ph/0610249 [ SPIRE SPIRE ]. arXiv:1411.6930 IN IN (2012) 137] ][ SPIRE ][ ][ arXiv:1005.1779 IN ]. 06 Non-perturbative effect on [ SPIRE ][ IN (2007) 34 TUM-HEP-955-14 , ]. Indirect detection analysis: wino ][ SPIRE (2015) 162 arXiv:1102.4295 IN [ arXiv:1210.7928 ][ SPIRE B 646 03 Wino dark matter under siege ]. ]. IN ][ (2010) 023503 Erratum ibid. Soft collinear effective theory for heavy Non-relativistic pair annihilation of nearly Heavy neutralino relic abundance with Heavy dark matter annihilation from effective [ arXiv:1401.6212 arXiv:0911.3795 On the annihilation rate of WIMPs [ [ JHEP SPIRE SPIRE arXiv:1409.7392 (2011) 271 Cosmology and astrophysics of minimal dark IN IN D 82 [ – 54 – Phys. Lett. ][ ][ ]. (2013) 224] [ ]. arXiv:0706.4071 , [ Potentially large one-loop corrections to WIMP 10 (2012) 163 B 699 Non-relativistic pair annihilation of nearly mass Relic densities including Sommerfeld enhancements in the arXiv:1303.0200 (2014) 031 SPIRE SPIRE [ 01 IN IN (2015) 099 [ 07 (2010) 105004 Phys. Rev. arXiv:1010.2172 ][ , [ ]. 01 (2007) 152 ]. 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