Zero-Range Effective Field Theory for Resonant Wino Dark Matter II. Coulomb Resummation
PPrepared for submission to JHEP
Zero-Range Effective Field Theory for Resonant Wino Dark Matter
II. Coulomb Resummation
Eric Braaten, Evan Johnson, and Hong Zhang
Department of Physics, The Ohio State University, Columbus, OH 43210, USA
E-mail: [email protected] , [email protected] , [email protected] Abstract:
Near a critical value of the wino mass where there is a zero-energy S-waveresonance at the neutral-wino-pair threshold, low-energy winos can be described by azero-range effective field theory (ZREFT) in which the winos interact nonperturbativelythrough a contact interaction and charged winos also have electromagnetic interactions.At energies near the wino-pair thresholds, the Coulomb interaction from photon ex-change between charged winos must also be treated nonperturbatively. The parametersof ZREFT can be determined by matching wino-wino scattering amplitudes calculatedby solving the Schr¨odinger equation for winos interacting through a potential due tothe exchange of electroweak gauge bosons. With Coulomb resummation, ZREFT atleading order gives a good description of the low-energy two-body observables for winos.
Keywords:
Dark matter, Effective Field Theories, Renormalization Group, Scatter-ing Amplitudes, Beyond Standard Model a r X i v : . [ h e p - ph ] F e b ontents A.1 Short-distance transition amplitudes 42A.2 Amplitude for creating a charged-wino pair at a point 45– 1 –
Introduction
A weakly interacting massive particle ( wimp ) is one of the best motivated candidatesfor a dark-matter particle that provides most of the mass of the universe. A stableparticle with weak interactions and whose mass is roughly at the electroweak scaleis naturally produced in the early universe with a relic abundance comparable to theobserved mass density of dark matter [1, 2]. If the wimp mass M is in the TeV range,the self-interactions of nonrelativistic wimps are complicated by a nonperturbativeeffect pointed out by Hisano et al. [3]. Weak interactions between low-energy wimpsare nonperturbative in the same sense as Coulomb interactions between low-energycharged particles: the exchange of gauge bosons must be summed to all orders in thegauge coupling constant. There can be critical values of the wimp mass where there isa resonance at the wimp-pair threshold. If the wimp mass is near such a critical mass,the annihilation rate of pairs of wimps into electroweak gauge bosons can be enhancedby orders of magnitude [4, 5]. Wimp-wimp cross sections at low relative velocity canalso be increased by orders of magnitude, which can affect the relic abundance of darkmatter [6, 7].A resonance in an S-wave channel can generally produce a more dramatic enhance-ment over a broader range of M than a resonance in a channel with higher orbitalangular momentum. There is also a qualitative difference between a near-thresholdresonance in an S-wave channel and in a channel for a higher partial wave. The S-waveresonance generates dynamically a length scale that is much larger than the rangeof the interactions. This length scale is the absolute value of the S-wave scatteringlength a , which can be orders of magnitude larger than the range. If there are nopair-annihilation channels, the scattering length can even be infinitely large.In a fundamental quantum field theory, wimps interact through the exchange ofelectroweak gauge bosons to which they couple through local gauge interactions. Theenhancement of low-energy wimp-wimp cross sections can be calculated by summingan infinite set of diagrams in that quantum field theory. The enhancement can becalculated more simply using a nonrelativistic effective field theory (NREFT) in whichthe wimps have instantaneous interactions at a distance through a potential generatedby the exchange of the electroweak gauge bosons. In NREFT, few-body reaction ratesof nonrelativistic wimps can be calculated by the numerical solution of a Schr¨odingerequation [3]. A thorough development of NREFT for nearly degenerate neutralinosand charginos in the MSSM has been presented in ref. [8]. NREFT has recently beenused to calculate the capture rates of two neutral winos into wino-pair bound statesthrough the radiation of a photon [9].In the case of an S-wave resonance near threshold, low-energy wimps can be de-– 2 –cribed more simply using a zero-range effective field theory (ZREFT) in which theweak interactions are replaced by zero-range interactions. ZREFT exploits the largelength scale that is generated dynamically by an S-wave resonance. The S-wave scat-tering length a diverges at critical values of the wimp mass M . ZREFT is applicable if M is close enough to a critical value that | a | is large compared to the range 1 /m W ofthe weak interactions. ZREFT can be used to calculate analytically wimp-wimp crosssections for wimps with relative momentum less than m W . There have been severalprevious applications of zero-range effective field theories to dark matter with resonantS-wave self-interactions. Braaten and Hammer pointed out that the elastic scatteringcross section of the dark-matter particles, their total annihilation cross section, and thebinding energy and width of a dark matter bound state are all determined by the com-plex S-wave scattering length [10]. Laha and Braaten studied the nuclear recoil energyspectrum in dark-matter direct detection experiments due to both elastic scatteringand breakup scattering of an incident dark-matter bound state [11]. Laha extendedthat analysis to the angular recoil spectrum in directional detection experiments [12].In Ref. [13], we developed the ZREFT for wimps that consist of the neutral dark-matter particle w and charged wimps w + and w − with a slightly larger mass. We referto these wimps as winos , because the fundamental theory describing them could bethe minimal supersymmetric standard model (MSSM) in a region of parameter spacewhere the neutral wino is the lightest supersymmetric particle. The ZREFT for winoscan be organized into a systematically improvable effective field theory by expandingaround a renormalization group fixed point. At the RG fixed point, the mass splittingbetween charged winos and neutral winos is zero, the electromagnetic interactions areturned off, the S-wave unitarity bound is saturated in a scattering channel that is alinear combination of w w and w + w − , and there is no scattering in the orthogonalchannel. In Ref. [13], we calculated the wino-wino cross sections analytically in ZREFTwithout electromagnetism at leading order (LO) and at next-to-leading order (NLO)in the ZREFT power counting. The interaction parameters of ZREFT at LO and atNLO were determined by matching numerical results for scattering amplitudes obtainedby solving the Schr¨odinger equation for NREFT. ZREFT at LO gives fairly accuratepredictions for the wino-wino cross sections in the wino-pair threshold region, with theexception of the charged-wino elastic cross section. ZREFT at NLO gives systematicallyimproved predictions for all the wino-wino cross sections. The power of ZREFT wasdemonstrated in Ref. [13] by using it to calculate the formation rate of a wino-pairbound state in the scattering of two neutral winos by a double radiative transition inwhich two photons are emitted.In this paper, we extend the results in Ref. [13] by carrying out the Coulombresummation of diagrams in which photons are exchanged between pairs of charged– 3 –inos. We calculate the wino-wino cross sections analytically in ZREFT at LO. Theinteraction parameters of ZREFT at LO are determined by matching scattering ampli-tudes with numerical results obtained by solving the Schr¨odinger equation for NREFT.We show that ZREFT at LO gives good predictions for the wino-wino cross sectionsin the wino-pair threshold region. In particular, it reproduces the resonances in theneutral-wino elastic cross section just below the charged-wino-pair threshold and thedramatic oscillations in the charged-wino elastic cross section just above the threshold.This paper is organized as follows. We begin in section 2 by summarizing variousquantum field theories that can be used to describe nonrelativistic winos, includingthe fundamental theory, NREFT, and ZREFT. In section 3, we use the Schr¨odingerequation of NREFT to numerically calculate wino-wino cross sections. In section 4, wecalculate wino-wino cross sections with Coulomb resummation analytically in a fieldtheory with zero-range interactions called the Zero-Range Model. In section 5, wepresent analytic results for low-energy two-body observables in ZREFT at LO withCoulomb resummation. We determine the parameters of ZREFT at LO by matchingscattering amplitudes from NREFT. We compare the resulting predictions of ZREFT atLO for wino-wino cross sections and for the binding energy of a wino-pair bound statewith numerical results from solving the Schr¨odinger equation for NREFT. Our resultsare summarized in section 6. In an Appendix, we solve the Lippmann-Schwinger equa-tions for short-distance transition amplitudes in the Zero-Range Model with Coulombresummation. In this Section, we summarize field theories that can be used to describe nonrelativisticwinos, including the fundamental theory and the effective field theories NREFT andZREFT.
We assume the dark-matter particle is the neutral member of an SU (2) triplet ofMajorana fermions with zero hypercharge. The Lorentz-invariant quantum field theorythat provides a fundamental description of these fermions could simply be an extensionof the Standard Model with this additional SU (2) multiplet and with a symmetrythat forbids the decay of the fermion into Standard Model particles. The fundamentaltheory could also be the Minimal Supersymmetric Standard Model (MSSM) in a regionof parameter space where the lightest supersymmetric particle is a wino-like neutralino.In either case, we refer to the particles in the SU (2) multiplet as winos . We denote theneutral wino by w and the charged winos by w + and w − .– 4 – igure 1 . Feynman diagrams in the fundamental theory for wino-wino scattering throughthe exchange of electroweak gauge bosons. The solid lines are neutral winos or charged winos,and the wavy lines are electroweak gauge bosons. If the winos are nonrelativistic, these ladderdiagrams must be summed to all orders. The relic density of the neutral wino is compatible with the observed mass densityof dark matter if the neutral wino mass M is roughly at the electroweak scale [1].We are particularly interested in a mass M at the TeV scale so that effects from theexchange of electroweak gauge bosons between nonrelativistic winos must be summedto all orders. For the neutral wino to be stable, the charged wino must have a largermass M + δ . In the MSSM, the mass splitting δ arises from radiative corrections. Thesplitting from one-loop radiative corrections is determined by M and Standard Modelparameters only [14–16]. As M ranges from 1 TeV to 10 TeV, the one-loop splitting δ remains very close to 174 MeV. The two-loop radiative corrections decrease δ by a fewMeV [17]. We take the wino mass splitting to be δ = 170 MeV.The winos can be represented by a triplet χ i of 4-component Majorana spinor fields,where the neutral-wino field is χ and the charged-wino fields are linear combinations of χ and χ . The most important interactions of the winos are those with the electroweakgauge bosons: the photon, the W ± , and the Z . The Lagrangian for the winos is L wino = (cid:88) i (cid:0) i χ Ti Cγ µ D µ χ i − M χ Ti Cχ i (cid:1) , (2.1)where D µ is the SU (2) gauge-covariant derivative and C is a charge conjugation ma-trix. The mass M of the winos is an adjustable parameter. The splitting δ betweenthe masses of w ± and w arises from electroweak radiative corrections. The relevantStandard Model parameters are the mass m W = 80 . W ± , the mass m Z = 91 . Z , the SU (2) coupling constant α = 1 / .
5, the electro-magnetic coupling constant α = 1 / .
04, and the weak mixing angle, which is givenby sin θ w = 0 . α M is of order m W or larger, loop diagrams in which electroweak gaugebosons are exchanged between nonrelativistic winos are not suppressed [3]. The elec-troweak interactions between a pair of nonrelativistic winos must therefore be treatednonperturbatively by summing ladder diagrams from the exchange of electroweak bosons– 5 –etween the winos to all orders. For wino-wino scattering, the first few diagrams inthe sum are shown in figure 1. The resummation of the ladder diagrams to all orderscan be carried out more easily by solving a Schr¨odinger equation in a nonrelativisticeffective field theory for the winos. Low-energy winos can be described by a nonrelativistic effective field theory in whichthey interact through potentials that arise from the exchange of weak gauge bosons andin which charged winos also have electromagnetic interactions. We call this effectivefield theory
NREFT . In NREFT, the nonrelativistic wino fields are 2-component spinorfields: ζ which annihilates a neutral wino w , η which annihilates a charged wino w − ,and ξ which creates a charged wino w + . The kinetic terms for winos in the Lagrangianare L kinetic = ζ † (cid:18) i∂ + ∇ M (cid:19) ζ + η † (cid:18) iD + D M − δ (cid:19) η + ξ † (cid:18) iD − D M + δ (cid:19) ξ, (2.2)where D and D are electromagnetic covariant derivatives acting on the charged winofields. The neutral and charged winos have the same kinetic mass M , and the winomass splitting δ is taken into account through the rest energy of the charged winos. Theweak interaction terms in the Hamiltonian are instantaneous interactions at a distancethrough a potential produced by the exchange of the W ± and Z gauge bosons: H weak = − (cid:90) d x (cid:90) d y (cid:18) α cos θ W | x − y | e − m Z | x − y | η † ( x ) ξ ( y ) ξ † ( y ) η ( x )+ α | x − y | e − m W | x − y | (cid:2) ζ † ( x ) ζ c ( y ) ξ † ( y ) η ( x ) + ζ c † ( x ) ζ ( y ) η † ( y ) ξ ( x ) (cid:3) (cid:19) , (2.3)where ζ c = − iσ ξ ∗ and σ is a Pauli matrix. The potentials from the exchange of W ± and Z have ranges of order 1 /m W . The amplitudes for wino-wino scattering canbe represented diagrammatically by the same sum of ladder diagrams as in figure 1,except that the wavy lines for the weak bosons W ± and Z should be interpreted asinstantaneous interactions at a distance through the potentials in eq. (2.3).In refs. [4, 5], Hisano, Matsumoto, and Nojiri calculated the nonperturbative effectof the exchange of electroweak gauge bosons between winos on the annihilation rate ofa pair of winos into electroweak gauge bosons by solving a Schr¨odinger equation thatcan be derived from NREFT. A particularly dramatic consequence is the existence of azero-energy resonance at the neutral-wino-pair threshold 2 M at a sequence of criticalvalues of M . Near these resonances, the annihilation rate of a wino pair into a pair of– 6 –lectroweak gauge bosons is increased by orders of magnitude. For δ = 170 MeV, thefirst such resonance is an S-wave resonance at M = 2 .
39 TeV.There are many important momentum scales for nonrelativistic winos. The inverserange of the weak interactions is m W = 80 . α M . The momentum scale below whichelectromagnetic interactions are nonperturbative is the Bohr momentum αM . Anotherimportant momentum scale is the scale √ M δ associated with transitions between aneutral-wino pair and a charged-wino pair. For δ = 170 MeV and the first resonancemass M = 2 .
39 TeV, these momentum scales are α M = 81 . αM = 17 . √ M δ = 28 . There can be a resonance at the neutral-wino-pair threshold in any partial wave. AnS-wave resonance at the threshold is special, because there is a dynamically generatedlength scale that is much larger than the range 1 /m W of the weak interactions [18].This length scale is the absolute value of the neutral-wino scattering length a . Thecorresponding momentum scale γ = 1 /a can be much smaller than any of the othermomentum scales provided by interactions described above. For winos with relativemomenta small compared to m W , the effects of the exchange of weak bosons can bemimicked by zero-range interactions. Thus winos with sufficiently low energy can bedescribed by a nonrelativistic field theory with local interactions and with electromag-netic interactions. This remains true even if there is an S-wave resonance near theneutral-wino-pair threshold. However in this case, the zero-range interactions must benonperturbative, because otherwise they cannot generate the large length scale | a | .A simple nonrelativistic field theory for low-energy winos with local interactionsis the Zero-Range Model introduced in Ref. [13]. The winos are described by nonrela-tivistic two-component spinor fields w , w + , and w − that annihilate w , w + , and w − ,respectively. They can be identified with the fields ζ , ξ † , and η in NREFT, respectively.The kinetic terms for winos in the Lagrangian for zero-range model are L kinetic = w † (cid:18) i∂ + ∇ M (cid:19) w + (cid:88) ± w †± (cid:18) iD + D M − δ (cid:19) w ± . (2.4)The electromagnetic covariant derivatives are D w ± = ( ∂ ± ieA ) w ± , D w ± = ( ∇ ∓ ie A ) w ± . (2.5)The neutral and charged winos have the same kinetic mass M , and the mass splitting δ is taken into account through the rest energy of the charged winos. Since the neutral– 7 – igure 2 . Feynman diagrams for wino-wino scattering in the Zero-Range Model withoutelectromagnetism. The solid lines are neutral winos or charged winos. The bubble diagramsmust be summed to all orders. In the Zero-Range Model with electromagnetism, one mustalso sum ladder diagrams in which photons are exchanged between incoming w + and w − lines, between outgoing w + and w − lines, and between the w + and w − lines in each bubble. wino is a Majorana fermion, a pair of neutral winos can have an S-wave resonanceat threshold only in the spin-singlet channel. That channel is coupled to the spin-singlet channel for charged winos. The Lagrangian for zero-range interactions in thespin-singlet channels can be expressed as L zero − range = − λ ( w c † w d † ) ( δ ac δ bd − δ ad δ bc )( w a w b ) − λ ( w c † + w d †− ) ( δ ac δ bd − δ ad δ bc )( w a w b ) − λ ( w c † w d † ) ( δ ac δ bd − δ ad δ bc )( w a + w b − ) − λ ( w c † + w d †− ) ( δ ac δ bd − δ ad δ bc )( w a + w b − ) , (2.6)where λ , λ , and λ are bare coupling constants. The factor ( δ ac δ bd − δ ad δ bc ) is theprojector onto the spin-singlet channel.In the Zero-Range Model, the zero-range interactions must be treated nonpertur-batively by summing bubble diagrams involving the vertices from the interaction termin eq. (2.6) to all orders. For wino-wino scattering with α = 0, the first few terms inthe sum are shown in figure 2. In the Zero-Range Model with electromagnetism, theelectromagnetic interactions must also be treated nonperturbatively by summing to allorders ladder diagrams in which photons are exchanged between charged winos. In theabsence of electromagnetic interactions, the Zero-Range Model is nonperturbativelyrenormalizable, at least in the two-wino sector. With electromagnetic interactions in-cluded, the Zero-Range Model in Coulomb gauge is probably renormalizable as aneffective field theory.The Zero-Range Model has two coupled scattering channels with different energythresholds. This model is analogous to the leading order (LO) approximation to the pion-less effective field theory ( π/ EFT) that has been widely used in nuclear physics todescribe low-energy nucleons [19, 20]. In π/ EFT at LO, nucleon pairs have two decoupledS-wave scattering channels (the spin-singlet isospin-triplet channel and the spin-tripletisospin-singlet channel) with the same energy threshold. Zero-range models that havetwo coupled scattering channels with different energy thresholds were first considered– 8 –n Ref. [21]. They have been applied previously to ultracold atoms [22], to charm mesonpairs [23], and to nucleon-nucleus interactions [24].Coulomb resummation in a zero-range model was first carried out by Kong andRavndal for the proton-proton system in pion-less effective field theory [25, 26]. TheCoulomb resummation for the two-nucleon system was recently revisited in Ref. [27],where it was also applied to the three-nucleon system. Coulomb resummation hasalso been carried out in a zero-range model with two coupled scattering channels withdifferent energy thresholds by Lensky and Birse [24].
Range corrections can be incorporated into the Zero-Range Model by adding termsto the Lagrangian with more and more gradients acting on the fields. Alternatively,for S-wave interactions, range corrections can be incorporated by adding terms to theLagrangian with more and more time derivatives acting on the fields [28]. If all pos-sible range corrections are included, the theory has infinitely many parameters. An effective field theory can be defined as a sequence of models with an increasing finitenumber of parameters that take into account corrections with systematically improvingaccuracy. A nonrelativistic effective field theory called
ZREFT for winos that have anS-wave resonance near the neutral-wino-pair threshold was introduced in Ref. [13]. Thewinos interact through zero-range self-interactions and through their couplings to theelectromagnetic field.An effective field theory can be defined most rigorously through deformations of arenormalization-group (RG) fixed point. Systematically improving accuracy is ensuredby adding to the Lagrangian operators with increasingly higher scaling dimensions. InRef. [24], Lensky and Birse carried out a careful RG analysis of the two-particle sectorfor a nonrelativistic field theory for distinguishable particles with two coupled scatteringchannels and with zero-range S-wave interactions. They identified three distinct RGfixed points. The first RG fixed point is the noninteracting fixed point at which the 2 × E : T ∗ ( E ) = 0. The second RG fixed point is the two-channel-unitarity fixed point , in which the cross sections saturate the S-wave unitaritybounds in both scattering channels. At this fixed point, the two scattering channelshave the same threshold at E = 0 and the T-matrix with the standard normalizationof states in a nonrelativistic field theory is T ∗ ( E ) = 4 πiM √ M E (cid:18) (cid:19) , (2.7)where M is the mass of the particle. The cross sections have the scaling behavior 1 /E .The power-law dependence on E reflects the scale invariance of the interactions. In– 9 –ef. [24], Lensky and Birse pointed out that there is a third RG fixed point: the single-channel-unitarity fixed point . At this fixed point, the two scattering channels have thesame threshold at E = 0 and the T-matrix is T ∗ ( E ) = 4 πiM √ M E (cid:18) cos φ cos φ sin φ cos φ sin φ sin φ (cid:19) . (2.8)There is nontrivial scattering in a single channel that is a linear combination of the twoscattering channels with mixing angle φ . In that channel, the cross section saturatesthe S-wave unitarity bound. There is no scattering in the orthogonal channel. Thesingle-channel-unitarity fixed point is the most natural one for describing a systemwith a single fine tuning, such as the tuning of the wino mass M to a unitarity valuewhere there is an S-wave resonance at the threshold.In ref. [24], Lensky and Birse diagonalized the RG flow near the single-channel-unitarity fixed point whose T-matrix T ∗ ( E ) is given in eq. (2.8), identifying all thescaling perturbations and their scaling dimensions. The scaling perturbations providea basis for the vector space of perturbations near the fixed point. Their coefficientsprovide a complete parametrization of the T-matrix. There is one relevant scalingperturbation that corresponds to changing √ M E in the denominator in eq. (2.8) to √ M E + iγ , where γ is a real parameter that can be interpreted as an inverse scatteringlength. There are two marginal scaling perturbations. One of them corresponds toturning on the splitting 2 δ between the thresholds in the two channels, and the othercorresponds to changing the mixing angle φ . All the other scaling perturbations areirrelevant. The inclusion of scaling perturbations with increasingly higher scaling di-mensions defines the successive improvements of ZREFT. The parameters in ZREFTat leading order (LO) are M , δ , the mixing angle φ , and the parameter γ . There aretwo additional parameters in ZREFT at next-to-leading order (NLO), and there is oneadditional parameter at next-to-next-to-leading order (NNLO).The systematic improvement provided by the effective field theory can be formu-lated in terms of an expansion in powers of the ratio of the generic momentum scale Q described by the effective field theory and the smallest momentum scale Λ beyondits domain of applicability. In the case of winos, Λ can be identified with m W . Wetake the energy E and the mass splitting δ to be order Q /M . The natural scale for γ is Λ, but we assume it is reduced to order Q by the fine tuning responsible for theS-wave resonance near the threshold. The mixing angle scales as ( Q/ Λ) . All otherparameters scale as negative powers of Λ. Instead of taking the interaction parametersto be coefficients in the Lagrangian, it is convenient to take them to be parametersin the inverse of the 2 × Q/ Λ.ZREFT can be extended to an effective field theory for winos and photons. InZREFT at LO, the only electromagnetic coupling is that of the charged winos throughthe covariant derivatives acting on the charged wino fields in eq. (2.4). Thus includingelectromagnetism does not introduce any additional adjustable parameters at LO. InZREFT beyond LO, gauge invariance requires some of the terms proportional to pow-ers of E in the inverse of the T-matrix to be accompanied by additional interactionterms involving the time component A of the photon field. They do not introduceany additional parameters. There may also be additional interaction terms involvingthe gauge-invariant electromagnetic field strengths E and B , which would introduceadditional parameters. In this section, we use NREFT to calculate cross sections for nonrelativistic wino-winoscattering. We keep the wino mass splitting fixed at δ = 170 MeV, and we study thedependence of the two-wino observables on the wino mass M . We also consider theeffect of turning off the electromagnetic coupling constant α . Ladder diagrams from the exchange of electroweak gauge bosons between a pair ofwimps can be summed to all orders in NREFT by solving a Schr¨odinger equation. Thecoupled-channel radial Schr¨odinger equation for S-wave scattering in the spin-singletchannel is (cid:34) − M (cid:18) (cid:19) (cid:18) ddr (cid:19) + 2 δ (cid:18) (cid:19) + V ( r ) (cid:35) r (cid:18) R ( r ) R ( r ) (cid:19) = E r (cid:18) R ( r ) R ( r ) (cid:19) , (3.1)where R ( r ) and R ( r ) are the radial wavefunctions for a pair of neutral winos and apair of charged winos, respectively. The 2 × V ( r ) = − α (cid:18) √ e − m W r /r √ e − m W r /r c w e − m Z r /r (cid:19) − α (cid:18) /r (cid:19) , (3.2)where c w = cos θ w . There is a continuum of positive energy eigenvalues E that cor-respond to S-wave scattering states. There may also be discrete negative eigenvaluesthat correspond to S-wave bound states.The coupled-channel radial Schr¨odinger equation in eq. (3.1) can be solved forthe radial wavefunctions R ( r ) and R ( r ). For energy E above the charged-wino-pair– 11 –hreshold 2 δ , the asymptotic solutions for R ( r ) and R ( r ) as r → ∞ determine adimensionless, unitary, and symmetric 2 × S ( E ). The dimensionless 2 × T ( E ) is defined by S ( E ) = + i T ( E ) , (3.3)where is the 2 × T ( E ) = T † ( E ) T ( E ) . (3.4)For energy in the range 0 < E < δ , the asymptotic solutions for R ( r ) determine a1 × S ( E ) = exp (cid:0) iδ ( E ) (cid:1) , where δ ( E ) is the real-valued S-wave phase shift.We denote the contribution to the cross section for elastic scattering from channel i to channel j at energy E from scattering in the S-wave spin-singlet channel by σ i → j ( E ).The expressions for these cross sections in terms of the T-matrix elements T ji are σ → j ( E ) = 2 πM v ( E ) (cid:12)(cid:12) T j ( E ) (cid:12)(cid:12) , (3.5a) σ → j ( E ) = πM v ( E ) (cid:12)(cid:12) T j ( E ) (cid:12)(cid:12) , (3.5b)where v ( E ) and v ( E ) are the wino velocities in the center-of-mass frame for a neutral-wino pair and a charged-wino pair with total energy E : v ( E ) = (cid:112) E/M , (3.6a) v ( E ) = (cid:112) ( E − δ ) /M . (3.6b)For the neutral-wino elastic cross section σ → , the energy threshold is E = 0. For theother three cross sections σ → , σ → , and σ → , the energy threshold is E = 2 δ . Thecross sections in eqs. (3.5) have been averaged over initial spins and summed over finalspins. The S-wave unitarity bounds for the scattering of w w , which are identical spin- particles, and for the scattering of w + w − , which are distinguishable spin- particles,are σ → ( E ) ≤ πM E , (3.7a) σ → ( E ) ≤ πM ( E − δ ) . (3.7b)– 12 – EM α
02 4 6 8 10 1210 M [ TeV ] σ → m W Figure 3 . Neutral-wino elastic cross section σ → at zero energy as a function of the winomass M . The cross section is shown for α = 1 /
137 (solid curve) and for α = 0 (dashed curve).The darker shaded region is σ → < π/m W and the lighter shaded region is σ → < π/M δ .If σ → is above the darker shaded region, the ZREFT for neutral and charged winos isapplicable. If σ → is above the lighter shaded region, a ZREFT for neutral winos only isapplicable. The neutral-wino elastic cross section σ → ( E = 0) at zero energy for δ = 170 MeV isshown as a function of the wino mass M in figure 3. The cross section diverges at criticalvalues of M . The first critical mass is M ∗ = 2 .
39 TeV and the second is 9.23 TeV.The divergence indicates that there is a zero-energy resonance at the neutral-wino-pairthreshold. At a critical mass where there is an S-wave resonance at the neutral-wino-pair threshold, the neutral-wino elastic cross section saturates the unitarity bound ineq. (3.7a) in the limit E →
0. We therefore refer to such a critical mass as a unitaritymass , and we refer to a system with such a mass as being at unitarity .The neutral-wino elastic cross section at zero energy depends sensitively on thestrength α of the Coulomb potential. The Coulomb potential can be turned off bysetting α = 0 in the potential matrix in eq. (3.2). The resulting cross section forneutral winos with zero energy is compared to the cross section at the physical value α = 1 /
137 in figure 3. If the Coulomb potential is turned off by setting α = 0, the cross– 13 – nitarity α EM α = δ δ δ δ E σ → m W Figure 4 . Neutral-wino elastic cross section σ → as a function of the energy E . The crosssection for M ∗ = 2 .
39 TeV is shown for α = 1 /
137 (solid curve) and for α = 0 (dashed curve).The S-wave unitarity bound is shown as a dotted curve. section at M = 2 .
39 TeV is reduced to 123 /m W . The shape of the curve is almost thesame, but the first two unitarity masses are shifted upward by about 20% to 2.88 TeVand 11.18 TeV.The neutral-wino elastic cross section σ → ( E ) has the most dramatic energy depen-dence at a unitarity mass, such as M ∗ = 2 .
39 TeV. The cross section for M ∗ = 2 .
39 TeVis shown in figure 4. As E approaches 0, the cross section approaches the unitaritybound in eq. (3.7a) from below, saturating the bound in the limit. Just below thecharged-wino-pair threshold 2 δ , the cross section with the Coulomb potential has a se-quence of narrow resonances whose peaks saturate the unitarity bound. The resonancescan be interpreted as bound states in the Coulomb potential for the charged-wino pair w + w − . Just above the threshold at 2 δ , the cross section is 34 . /m W , and it decreasesslowly as E increases. The cross section with α = 0 and M = 2 .
39 TeV is also shownin figure 4, and it has a qualitatively different behavior. As E approaches 0, the crosssection has a finite limit. The resonances just below the charged-wino-pair thresholddisappear. As E increases from 0, the cross section increases monotonically until thethreshold 2 δ , where it has a kink, and it then decreases as E increases further.Neutral winos with energies well below the charged-wino-pair threshold 2 δ have– 14 – REFTPadé - - - M [ TeV ] a m W Figure 5 . Neutral-wino scattering length a as a function of the wino mass M (solid curve).The dashed curve is the Pad´e approximant given in eq. (3.9). The vertical dotted lines indicatethe first and second unitarity masses M ∗ = 2 .
39 TeV and 9.23 TeV. The darker shaded regionis | a | < /m W and the lighter shaded region is | a | < / √ M δ . If a is outside the darkershaded region, the ZREFT for neutral and charged winos is applicable. If a is outside thelighter shaded region, a ZREFT for neutral winos only is applicable. short-range interactions, because the Coulomb interaction enters only through virtualcharged winos. The short-range interactions guarantee that v ( E ) /T ( E ) can be ex-panded in powers of the relative momentum p = √ M E :2 M v ( E ) T ( E ) = − a − ip + 12 r p + 18 s p + O ( p ) . (3.8)The only odd power of p in the expansion is the pure imaginary term − ip . Thecoefficients of the even powers of p are real valued. The leading term in the expansiondefines the neutral-wino S-wave scattering length a . It diverges at a unitarity mass.The coefficients of p and p define the effective range r and a shape parameter s .The coefficients in the range expansion in eq. (3.8) can be determined numericallyby solving the Schr¨odinger equation. The scattering length a for δ = 170 MeV isshown as a function of the wino mass M in figure 5. The dependence of a on M canbe fit surprisingly well by a Pad´e approximant in M of order [2,2] whose poles matchthe first and second resonances at M ∗ = 2 .
39 TeV and M (cid:48)∗ = 9 .
23 TeV and whose zeros– 15 –atch the first and second zero crossings at M = 0 . M (cid:48) = 7 .
39 TeV.The only adjustable parameter is an overall prefactor. We can improve the fit near theresonance at M ∗ significantly by fitting M as well as the prefactor. The resulting fit is a ( M ) = 0 . m W ( M − M )( M − M (cid:48) )( M − M ∗ )( M − M (cid:48)∗ ) , (3.9)where M = 0 .
845 TeV. Near a unitarity mass where a diverges, the scattering lengthis necessarily very sensitive to α . If α is set to 0, the scattering length at M = 2 .
39 TeVis reduced to − . /m W .The winos can be described by a zero-range effective field theory (ZREFT) forneutral and charged winos if the neutral-wino scattering length is large compared tothe range of the weak interactions: | a | > /m W . Figure 5 shows that the region of M near M ∗ = 2 .
39 TeV in which the 2-channel ZREFT is applicable is roughly from1.8 TeV to 4.6 TeV. The energy region in which it is applicable is total wino-pairenergy E below about m W /M , which at M ∗ is about 2700 MeV. There is a narrowerrange of M is which neutral winos can be described by a ZREFT for neutral winosonly. The neutral-wino scattering length must be large not only compared to 1 /m W but also compared to the range associated with the transition between a neutral-winopair and a virtual charged-wino pair: | a | > / ∆, where ∆ = (2 M δ ) / . Figure 5shows that the region of M in which the ZREFT for neutral winos only is applicable isroughly from 2.1 TeV to 2.9 TeV. The energy region in which it is applicable is totalneutral-wino-pair energy E below about δ = 170 MeV.The coefficients of terms with higher powers of p in the range expansion in eq. (3.8)can also be determined numerically by solving the Schr¨odinger equation. For δ =170 MeV and α = 1 / M ∗ = 2 .
39 TeV are r ( M ∗ ) = − . / ∆ ∗ , (3.10a) s ( M ∗ ) = − . / ∆ ∗ , (3.10b)where ∆ ∗ = √ M ∗ δ = 28 . ∗ is an appropriate momentum scale. If the Coulomb potential isturned off by setting α = 0, the coefficients on the right sides of eqs. (3.10a) and (3.10b)are changed to − .
224 and − . α . The effective range r for δ = 170 MeV is shown as a function of the mass M in figure 6. The dependence of r on M can be fit surprisingly well by a [4,4] Pad´eapproximant in M . If an offset equal to the local maximum near M (cid:48) = 5 .
13 TeV is– 16 –
REFTPadé - - - M [ TeV ] r m W Figure 6 . Neutral-wino effective range r as a function of the wino mass M (solid curve).The dashed curve is the Pad´e approximant given in eq. (3.11). The vertical dotted lineindicates the first unitarity mass M ∗ = 2 .
39 TeV. The grey region is the range of M in which | a | < /m W , so a ZREFT for neutral and charged winos is not applicable. subtracted, the remainder can be fit by a [3,4] Pad´e with double poles at the zerocrossings M and M (cid:48) of a ( M ), a double zero at M (cid:48) = 5 .
13 TeV, and a single zero at M (cid:48)(cid:48) = 9 .
11 TeV. The only adjustable parameter in the Pad´e approximant is an overallprefactor. We choose to improve the fit near the resonance at M ∗ by fitting M as wellas the prefactor. The resulting Pad´e approximant is r ( M ) = (5 . /m W ) (cid:18) M ∗ ( M − M (cid:48) ) ( M − M (cid:48)(cid:48) )( M − M ) ( M − M (cid:48) ) − . (cid:19) , (3.11)where M = 0 .
124 TeV.Particles with short-range interactions that produce an S-wave resonance suffi-ciently close to their scattering threshold have universal low-energy behavior that iscompletely determined by their S-wave scattering length a [18]. The universal pre-dictions are just those of the single-channel ZREFT at leading order. The universalapproximation to the cross section is σ → ( E ) = 8 π /a + M E . (3.12)– 17 – EM α = δ δ δ E σ → m W unitarity α EM α = δ δ δ E σ → m W Figure 7 . Neutral-to-charged transition cross section σ → (left panel) and the charged-winoelastic cross section σ → (right panel) as functions of the energy E . The cross sections for M ∗ = 2 .
39 TeV are shown for α = 1 /
137 (solid curves) and for α = 0 (dashed curves). In theleft panel, the dotted curve is the cross section σ → for α = 0 multiplied by the Sommerfeldfactor C ( E ) and normalized to the cross section for α = 1 /
137 at the threshold. In the rightpanel, the dotted curve is the S-wave unitarity bound in eq. (3.7b).
The universal region is where | a | is large compared to the range set by the interactions.For neutral winos, the appropriate range is the maximum of 1 /m W and 1 / ∆. Theuniversal region of M is inside the region from 2.1 TeV to 2.9 TeV. The universalapproximation becomes increasingly accurate as M approaches the unitarity mass M ∗ =2 .
39 TeV. The universal region of the energy is E (cid:28) δ . The energy dependence of the neutral-to-charged transition cross section σ → ( E ) atthe unitarity mass M ∗ = 2 .
39 TeV is illustrated in the left panel of figure 7. As E decreases to the threshold 2 δ , the cross section increases monotonically to 36 . /m W .The cross section with α = 0 is also shown in the left panel of figure 7, and it hasa qualitatively different behavior. As E decreases towards 2 δ , the cross section with α = 0 increases to a maximum near E = 2 . δ , and it then decreases to zero. Thezero comes from a factor of v ( E ) from the phase space of the final-state w + w − . Thenonzero cross section at the threshold for α = 1 /
137 is due to a Sommerfeld factorfor Coulomb rescattering of the final-state w + and w − . The nonperturbative effect of– 18 –he Coulomb rescattering of charged particles was derived by Sommerfeld around 1920[29]. As the energy E approaches the threshold 2 δ , the cross section differs from thecross section for α = 0 by a multiplicative factor that is the product of a constant anda Sommerfeld factor that depends on the velocity v ( E ) of the charged particles. Forparticles with charges ±
1, the Sommerfeld factor is C ( E ) = πα/v − exp( − πα/v ) . (3.13)The Sommerfeld factor approaches 1 as v increases, and it approaches πα/v as v → /v cancels the factor of v from the phase space of the final-state w + w − , so the cross section has a nonzero limit as v →
0. If the momentum scale αM in the Sommerfeld factor was much smaller than the other relevant momentumscales, the cross section for α = 1 /
137 at relative momentum of order αM could beapproximated by the cross section for α = 0 multiplied by the Sommerfeld factor C ( E )and normalized to the cross section for α = 1 /
137 at the threshold. This approximationis shown as a dotted line in the left panel of figure 7. It is not a good approximation,because the momentum scale √ M δ = 28 . αM = 17 . σ → ( E ) at theunitarity mass M ∗ = 2 .
39 TeV is illustrated in the right panel of figure 7. As E decreases to the threshold 2 δ , the cross section appears to increase monotonically toinfinity. However at energies E extremely close to the threshold at 2 δ , there are rapidoscillations in the cross section that are too large to be visible in figure 7. The crosssection with α = 0 is also shown in the right panel of figure 7, and it has a qualitativelydifferent behavior. As E decreases towards 2 δ , it increases monotonically to a finitemaximum. In this Section, we calculate analytically the transition amplitudes for w w and w + w − in the Zero-Range Model with Coulomb resummation. Observables in the sector consisting of two neutral winos w w or two charged winos w + w − are conveniently encoded in the amplitudes for transitions among the two coupledchannels. We denote the neutral channel w w by the index 0 and the charged channel w + w − by the index 1. In the Zero-Range Model, the S-wave spin-singlet transition– 19 –mplitudes A ij ( E ) are functions of the total energy E of the wino pair only. TheT-matrix elements T ij ( E ) for S-wave wino-wino scattering are obtained by evaluatingthe transition amplitudes A ij ( E ) on the energy shell. The constraints on the T-matrixelements from S-wave unitarity can be derived from the unitarity condition for theamplitude matrix A ( E ) at real E , which can be expressed as A ( E ) − A ( E ) ∗ = − π A ( E ) M / (cid:104) κ ( E ) − κ ( E ) ∗ (cid:105) M / A ( E ) ∗ , (4.1)where M is the 2 × M = (cid:18) M
00 2 M (cid:19) (4.2)and κ is a diagonal matrix whose entries are functions of E : κ ( E ) = (cid:18) κ ( E ) 00 κ ( E ) (cid:19) . (4.3)The functions κ and κ of the complex energy E have branch points at 0 and 2 δ ,respectively: κ ( E ) = √− M E − iε, (4.4a) κ ( E ) = (cid:112) − M ( E − δ ) − iε. (4.4b)The different diagonal entries of the matrix M in eq. (4.2) are a convenient way totake into account that the neutral channel w w consists of a pair of identical fermionswhile the charged channel w + w − consists of two distinguishable fermions.The amplitudes A ij ( E ) have a diagrammatic representation. We represent thepropagator for the neutral wino w by a solid line without an arrow. We represent thepropagator for the charged winos w + and w − by solid lines with a forward arrow anda backward arrow, respectively. We represent the photon propagator by a wavy line.The interaction term in the Lagrangian for the Zero-Range Model in eq. (2.6) providesvertices for w w → w w , w w → w + w − , w + w − → w w , and w + w − → w + w − . Thecovariant derivatives in the kinetic term in eq. (2.4) provide vertices in which one ortwo photons attach to a w + line or to a w − line. The transition amplitude A ij ( E ) canbe expressed as the sum of all diagrams with the appropriate pair of incoming solidlines and the appropriate pair of outgoing solid lines. If the only interactions between winos were the Coulomb interactions between chargedwinos, the only nonzero transition amplitude in the two-wino sector with zero total– 20 – igure 8 . Amplitude for w + w − → w + w − with Coulomb interactions only. The ladderdiagrams from the exchange of a photon between w + and w − must be summed to all orders. charge would be the amplitude for w + w − → w + w − . The amplitude would be given bythe sum of ladder diagrams in figure 8. The projection A of that amplitude onto theS-wave channel is the S-wave Coulomb transition amplitude: A C ( E ) = (cid:18) − Γ(1 + iη )Γ(1 − iη ) (cid:19) πM κ ( E ) , (4.5)where κ is given in eq. (4.4b) and η is an energy variable defined by η ( E ) ≡ i α M κ ( E ) = i αM (cid:2) − M ( E − δ ) − i(cid:15) (cid:3) − / . (4.6)For a real energy E = 2 δ + p /M above the charged-wino-pair threshold, η is real andnegative: η = − αM/ p . For a real energy E below the charged-wino-pair threshold, η is pure imaginary. The amplitude in eq. (4.5) has poles in E at real energies E n thatcorrespond to Coulomb bound states of w + w − : E n = 2 δ − α M n , (4.7)where n is a positive integer. The value of η at the energy E n of a Coulomb boundstate is η n = in . For real energy E , the Coulomb transition amplitude satisfies theunitarity condition A C ( E ) − A C ( E ) ∗ = − M π A C ( E ) (cid:2) κ ( E ) − κ ( E ) ∗ (cid:3) A C ( E ) ∗ . (4.8)For E < δ , κ is real and η is pure imaginary, so the unitarity condition in eq. (4.8)is satisfied because both sides vanish. For E > δ , κ is pure imaginary and η is real.The unitarity condition in eq. (4.8) then follows from the explicit expression for theCoulomb transition amplitude in eq. (4.5). In the absence of electromagnetic interactions, the transition amplitude A ( E ) for w w → w w is given by the sum of all bubble diagrams, as illustrated in figure 9.– 21 – igure 9 . Feynman diagrams for the transition amplitude A ( E ) for w w → w w in theZero-Range Model without electromagnetism. The bubble diagrams must be summed to allorders. Each bubble can be either a neutral-wino pair without arrows or a charged wino pairwith arrows. The Feynman diagrams for w w → w + w − , w + w − → w w , and w + w − → w + w − areobtained by putting arrows on the outgoing pair of lines, on the incoming pair of lines,and on both, respectively. The transition amplitudes can be determined analyticallyby solving Lippmann-Schwinger equations. The Lippmann-Schwinger equations forthe Zero-Range Model without Coulomb interactions were solved nonperturbatively inAppendix A of Ref. [13]. The solution is expressed most simply by giving the inverseof the 2 × A ( E ): A − ( E ) = 18 π M / (cid:104) − γ + κ ( E ) (cid:105) M / , (4.9)where γ is a symmetric matrix of renormalized parameters: γ = (cid:18) γ γ γ γ (cid:19) . (4.10)The unitarity equation in eq. (4.1) is automatically satisfied if the parameters γ , γ ,and γ are real.Since charged winos also have electromagnetic interactions, there are additionaldiagrams for wino-wino scattering beyond those in figure 9. The additional diagramshave photons exchanged between charged wino lines. Most of the diagrams have effectsthat are suppressed by one or more factors of the electromagnetic coupling constant α = 1 / αM or smaller, there are photon-exchange diagrams that are not suppressed. InCoulomb gauge, the diagrams that are not suppressed are ladder diagrams in whichstatic Coulomb photons are exchanged between a pair of charged winos. The summa-tion of these diagrams is called Coulomb resummation .For the amplitude for w w → w w , Coulomb resummation involves addingall ladder diagrams in which photons are exchanged between the w + and w − inside– 22 – igure 10 . Feynman diagrams for the transition amplitude A ( E ) for w w → w w inthe Zero-Range Model with Coulomb resummation. The bubble diagrams must be summedto all orders. Each bubble can be either a neutral-wino-pair bubble, which is a one-loopsubdiagram, or a charged-wino-pair bubble, which is the sum of the diagrams in figure 11. Figure 11 . Feynman diagrams for the bubble amplitude for w + w − in the Zero-Range Modelwith Coulomb resummation. The ladder diagrams from the exchange of a photon between w + and w − must be summed to all orders. Figure 12 . Feynman diagrams for the amplitude for creation of w + w − at a point in theZero-Range Model with Coulomb resummation. The ladder diagrams from the exchange ofa photon between w + and w − must be summed to all orders. The sum is equal to the treediagram multiplied by the amplitude W ( E ) in eq. (4.12). the charged-wino bubbles, as illustrated in figure 10. The charged-wino bubble withCoulomb resummation is the sum of the diagrams in figure 11. For amplitudes with w + w − in the initial state and/or in the final state, Coulomb resummation also involvesadding all ladder diagrams in which photons are exchanged between the incoming w + and w − lines and/or the outgoing w + and w − lines. For the outgoing w + and w − lines,Coulomb resummation involves replacing the final vertex by the sum of diagrams infigure 12. Finally, Coulomb resummation for w + w − → w + w − also requires adding thediagrams in figure 8 in which photons are exchanged between w + and w − .If winos have short-range interactions as well as the Coulomb interactions between– 23 –harged winos, the matrix of S-wave transition amplitudes can be expressed in the form A ( E ) = (cid:18) A C ( E ) (cid:19) + (cid:18) W ( E ) (cid:19) A s ( E ) (cid:18) W ( E ) (cid:19) , (4.11)where W ( E ) is the dimensionless amplitude for creating or annihilating w + and w − with total energy E at a point in the presence of Coulomb interactions, and the matrix A s ( E ) is the contribution to A ( E ) from diagrams in which the first interaction andthe last interaction are both short-range interactions. We refer to its entries as the short-distance transition amplitudes . The amplitude W ( E ) for creating w + w − at apoint can be obtained diagrammatically by expressing the sum of diagrams in figure 12as the tree diagram multiplied by W ( E ). It is determined in Appendix A: W ( E ) = C ( E ) (cid:18) Γ(1 + iη )Γ(1 − iη ) (cid:19) / , (4.12)where η is the function of E in eq. (4.6) and C is the square root of the Sommerfeldfactor in eq. (3.13): C ( E ) = 2 πη exp(2 πη ) − . (4.13)In the Zero-Range Model, the short-distance transition amplitudes A s,ij ( E ) can bedetermined analytically by solving the Lippmann-Schwinger equations in Appendix A.The solution is expressed most simply by giving the inverse of the 2 × A s ( E ): A − s ( E ) = 18 π M / (cid:104) − γ + K ( E ) (cid:105) M / , (4.14)where γ is the symmetric matrix of renormalized parameters in eq. (4.10) and K is thediagonal matrix K ( E ) = (cid:18) κ ( E ) 00 K ( E ) (cid:19) . (4.15)Its first diagonal entry is the function κ in eq. (4.4a), and its second diagonal entry is K ( E ) = αM (cid:20) ψ ( iη ) + 12 iη − log( − iη ) (cid:21) , (4.16)where ψ ( z ) = ( d/dz ) log Γ( z ) and η ( E ) is defined in eq. (4.6). This function has alogarithmic branch point at E = 2 δ and poles at the Coulomb bound-state energies ineq. (4.7). Since η is negative for E > δ , the argument of the logarithm in eq. (4.16)should be interpreted as e + iπ iη . As z → ∞ in any direction of the complex plane exceptalong the negative real axis, the asymptotic behavior of ψ ( z ) is ψ ( z ) −→ log( z ) − z − z + . . . . (4.17)– 24 –his implies that K ( E ) approaches a constant as E approaches the threshold 2 δ fromabove: K (2 δ + ) = − iπαM. (4.18)The function K ( E ) does not have a limit as E approaches 2 δ from below, because ψ ( z ) has poles at the negative integers. In this section, we present the transition amplitudes for w w and w + w − in ZREFT atLO with Coulomb resummation. We determine the adjustable parameters of ZREFTat LO by matching low-energy w w scattering amplitudes from NREFT. We comparepredictions of ZREFT at LO for wino-wino cross sections and for the binding energyof a wino-pair bound state with results from NREFT. In order to give an explicit parametrization of the transition amplitudes A ij ( E ) forZREFT, we introduce two 2-component unit vectors that depend on the mixing angle φ : u ( φ ) = (cid:18) cos φ sin φ (cid:19) , v ( φ ) = (cid:18) − sin φ cos φ (cid:19) . (5.1)We use these vectors to define two projection matrices and another symmetric matrix: P u ( φ ) = u ( φ ) u ( φ ) T = (cid:18) cos φ cos φ sin φ cos φ sin φ sin φ (cid:19) , (5.2a) P v ( φ ) = v ( φ ) v ( φ ) T = (cid:18) sin φ − cos φ sin φ − cos φ sin φ cos φ (cid:19) , (5.2b) P m ( φ ) = u ( φ ) v ( φ ) T + v ( φ ) u ( φ ) T = (cid:18) − sin(2 φ ) cos(2 φ )cos(2 φ ) sin(2 φ ) (cid:19) . (5.2c)The superscript T on u or v indicates the transpose of the column vector. The threematrices defined in eqs. (5.2) form a basis for 2 × P (cid:48) u ( φ ) = P m ( φ ) , (5.3a) P (cid:48) v ( φ ) = − P m ( φ ) , (5.3b) P (cid:48) m ( φ ) = − P u ( φ ) + 2 P v ( φ ) . (5.3c)– 25 –he T-matrix at the RG fixed point for ZREFT with α = 0 is T ∗ ( E ) = 8 πi √ M E M − / P u ( φ ) M − / , (5.4)where M is the diagonal matrix in eq. (4.2).A possible choice for the interaction parameters of ZREFT with α = 0 are thecoefficients of the scaling perturbations to the Lagrangian near the RG fixed point thatcorresponds to the T-matrix in eq. (5.4). A more convenient choice are coefficients inthe expansion in powers of E of the inverse T − ( E ) of the T-matrix. The correspondingparameterization for the inverse of the matrix of transition amplitudes is A − ( E ) = 18 π M / (cid:104)(cid:0) − γ u + r u p + . . . (cid:1) P u ( φ ) + (cid:0) − /a v + . . . (cid:1) P v ( φ )+ (cid:0) r m p + . . . (cid:1) P m ( φ ) + κ ( E ) (cid:105) M / , (5.5)where p = M E and κ ( E ) is the diagonal matrix in eq. (4.3). The coefficients of P u , P v , and P m have been expanded in powers of p . The mixing angle φ has beenchosen so that the p term in the expansion of the coefficient of P m is 0. The interactionparameters of ZREFT with α = 0 are the mixing angle φ , the parameters γ u and a v , andthe coefficients of the positive powers of p , such as r u and r m . These parameters shouldall be regarded as functions of M and δ with expansions in powers of ∆ = 2 M δ . Thesuccessive improvements of ZREFT can be obtained by successive truncations of theexpansions in p . At leading order (LO), the only nonzero term in the three expansionsis the coefficient − γ u of P u ( φ ). By setting the coefficients of positive powers of p tozero in A − ( E ) in eq. (5.5), inverting the matrix, and then taking the limit a v →
0, weobtain the matrix of transition amplitudes for ZREFT at LO with α = 0: A ( E ) = 8 π − γ u + cos φ κ ( E ) + sin φ κ ( E ) M − / P u ( φ ) M − / . (5.6)At next-to-leading order (NLO), there are two additional interaction parameters: a v and r u . At NNLO, there is one additional interaction parameter: r m .If α is not zero, it is necessary to resum the effects of the exchange of Coulomb pho-tons between charged winos to all orders. ZREFT at LO with Coulomb resummationis just a limiting case of the Zero-Range Model. The matrix of transition amplitudes A ( E ) has the form in eq. (4.11), where A C ( E ) is the Coulomb amplitude in eq. (4.5), W ( E ) is the amplitude for creating w + w − at a point in eq. (4.12), and A s ( E ) is thematrix of short-distance transition amplitudes, which can be expressed as: A s ( E ) = lim a v → π M − / (cid:2) − γ u P u ( φ ) − (1 /a v ) P v ( φ ) + K ( E ) (cid:3) − M − / , (5.7)– 26 –here K ( E ) is the diagonal matrix in eq. (4.15). The limit must be taken after evalu-ating the inverse of the matrix between the factors of M − / in eq. (5.7). The matrix A ( E ) in eq. (4.11) reduces to A ( E ) = (cid:18) A C ( E ) (cid:19) + 8 πL u ( E ) (cid:18) W ( E ) (cid:19) M − / P u ( φ ) M − / (cid:18) W ( E ) (cid:19) . (5.8)The denominator in the second term is L u ( E ) = − γ u + cos φ κ ( E ) + sin φ K ( E ) , (5.9)where κ ( E ) is given in eq. (4.4a) and K ( E ) is given in eq. (4.16).The neutral-wino scattering length a can be obtained by evaluating the transitionamplitude A ( E ) at the neutral-wino-pair threshold: A ( E = 0) = − πa /M. (5.10)The inverse neutral-wino scattering length γ ≡ /a is γ = (1 + t φ ) γ u − t φ K (0) , (5.11)where t φ ≡ tan φ . This equation can be solved for γ u as a function of γ : γ u = t φ K (0) + γ t φ . (5.12)If γ = 0, the elastic neutral-wino cross section σ → ( E ) saturates the unitarity boundin eq. (3.7a) in the limit E →
0. For this reason, we refer to the critical value γ = 0as unitarity . If | γ | (cid:28) √ M δ , there are large cancellations in the denominator L u ( E )in eq. (5.9). These cancellations can be avoided by eliminating γ u in favor of γ . Theresulting expression for the matrix of transition amplitudes is A ( E ) = (cid:18) A C ( E ) (cid:19) + 8 πL ( E ) (cid:18) W ( E ) (cid:19) M − / (cid:18) t φ t φ t φ (cid:19) M − / (cid:18) W ( E ) (cid:19) , (5.13)where A C ( E ) is the Coulomb amplitude in eq. (4.5) and W ( E ) is the amplitude ineq. (4.12) for w + w − created at a point to become w + w − with energy E . The denomi-nator in the second term is L ( E ) = − γ + t φ (cid:2) K ( E ) − K (0) (cid:3) + κ ( E ) , (5.14)where κ ( E ) is given in eq. (4.4a) and the function K ( E ) is given in eq. (4.16).– 27 – .2 Wino-wino scattering The cross section for elastic scattering from channel i to channel j at energy E , averagedover initial spins and summed over final spins, is denoted by σ i → j ( E ). The expressionsfor these cross sections in terms of the T-matrix elements T ij ( E ) for states with thestandard normalizations of a nonrelativistic field theory are σ i → ( E ) = M π (cid:12)(cid:12) T i ( E ) (cid:12)(cid:12) v ( E ) v i ( E ) , (5.15a) σ i → ( E ) = M π (cid:12)(cid:12) T i ( E ) (cid:12)(cid:12) v ( E ) v i ( E ) , (5.15b)where v i ( E ) and v j ( E ) are the velocities of the incoming and outgoing winos, whichare given in eqs. (3.6). The extra factor of 1 / σ i → in eq. (5.15a)for producing a neutral-wino pair compensates for overcounting by integrating overthe entire phase space of the two identical particles. The T-matrix elements T ij ( E )are obtained by evaluating the transition amplitudes A ij ( E ) on the appropriate energyshell. For a neutral-wino pair w w with relative momentum p , the energy shell is E = p /M . For a charged-wino pair w + w − with relative momentum p , the energy shellis E = 2 δ + p /M .For center-of-mass energy in the range 0 ≤ E < δ below the charged-wino-pair threshold, only the neutral-wino-pair channel is open. The T-matrix elementfor w w → w w in ZREFT at LO is given by the 00 entry of the matrix in eq. (5.13): T ( E ) = 8 π/ML ( E ) , (5.16)where L ( E ) is given in eq. (5.14). The reciprocal of the T-matrix element T ( E ) forneutral-wino elastic scattering can be expanded in powers of the relative momentum p = √ M E : 8 π/M T ( E ) = − γ − ip + r p + s p + O ( p ) . (5.17)The only odd power of p in the expansion is the pure imaginary term − ip . Thecoefficients of the even powers of p are real valued. The leading term − γ vanishesat unitarity. The effective range r and the shape parameter s can be determined byexpanding the real part of 1 / T ( E ) from eq. (5.16) in powers of p and comparing tothe expansion in eq. (5.17): r = 2 t φ K (cid:48) (0) /M, (5.18a) s = 4 t φ K (cid:48)(cid:48) (0) /M . (5.18b)– 28 –he predictions for these coefficients are independent of γ .For energy in the range E > δ above the charged-wino-pair threshold, the w w and w + w − channels are both open. The T-matrix elements in ZREFT at LO for w w → w w is given in eq. (5.16). The T-matrix elements in ZREFT at LO for w w → w + w − and w + w − → w + w − are given by the 01 and 11 entries of the matrixin eq. (5.13): T ( E ) = (4 √ π/M ) t φ W ( E ) L ( E ) , (5.19a) T ( E ) = A C ( E ) + (4 π/M ) t φ W ( E ) L ( E ) , (5.19b)where L ( E ) is given in eq. (5.14), W ( E ) is given in eq. (4.12), and A C ( E ) is theon-shell Coulomb amplitude in eq. (4.5). The interaction parameters of ZREFT can be determined by matching T-matrix el-ements in ZREFT with low-energy T-matrix elements in NREFT. The dimensionlessT-matrix elements T ij ( E ) for wino-wino scattering in NREFT can be calculated numer-ically by solving the coupled-channel Schr¨odinger equation in eq. (3.1). The T-matrixelements T ij ( E ) for wino-wino scattering in ZREFT at LO are given analytically ineqs. (5.16) and (5.19). For E > δ , the relation between the T-matrix in NREFT andthe T-matrix in ZREFT is [13]12 M v ( E ) − / T ( E ) v ( E ) − / = 18 π M / T ( E ) M / , (5.20)where M is the diagonal matrix of masses in eq. (4.2) and v ( E ) is the diagonal matrixof the velocities defined in eq. (3.6): v ( E ) = (cid:18) v ( E ) 00 v ( E ) (cid:19) . (5.21)For 0 < E < δ , the relation between the T-matrix elements for neutral-wino scatteringis 12 M v ( E ) T ( E ) = M π T ( E ) . (5.22)The interaction parameters of ZREFT at LO are α , φ , and γ = 1 /a . An accurateparametrization of neutral-wino scattering length a ( M ) for NREFT with δ = 170 MeVand M near the critical mass M ∗ is provided by the Pad´e approximant in eq. (3.9).The angle φ can be determined by matching some other physical quantity in ZREFT– 29 –nd in NREFT. If δ is fixed, it is better to use a value of M close to the unitarity value M ∗ ( δ ) and to match a T-matrix element at an energy E close to 0. The expansionof the reciprocal of the T-matrix element T ( E ) for neutral-wino elastic scattering inpowers of the relative momentum p = √ M E is given in eq. (5.17). The correspondingexpansion in powers of p in NREFT is given in eq. (3.8). At unitarity, the lowest-energyquantity that can be used for matching is the effective range r .If we choose the effective range at some mass M as the matching quantity, thematching condition for ZREFT at LO is t φ ( M ) = − ∆ / z ψ (cid:48) ( z ) − − z r ( M ) , (5.23)where ∆ = √ M δ and z = − αM/ (2∆). In the limit α →
0, the matching conditionreduces to r = − tan φ/ ∆. The derivative of ψ ( z ) can be expanded as a power seriesthat converges for | z | < ψ (cid:48) ( z ) = 1 z + ∞ (cid:88) n =0 ( − n ( n + 1) ζ ( n + 2) z n , (5.24)where ζ ( z ) is the Riemann zeta function. This can be used to expand the right sideof eq. (5.23) as a power series in z = − αM/ (2∆). The convergence rate of theexpansion is determined not by the size of α = 1 / αM/ (2∆). For δ = 170 MeV, the value of this ratio at unitarity is 0.306. Thusalthough the effective range provides a matching condition that is perturbative in α ,matching at α = 0 is not quantitatively useful.A specific choice for the matching point in eq. (5.23) is α = 1 / δ = 170 MeV,and the unitarity mass M ∗ = 2 .
39 TeV. Using the numerical result for the effectiverange in NREFT in eq. (3.10a), our matching condition in eq. (5.23) gives tan φ = 0 . φ is about 40 ◦ . A differentchoice for the matching mass M near M ∗ would give a different value for tan φ . For M near M ∗ , the effective range r ( M ) can be accurately approximated by the Pad´eapproximant in eq. (3.11). The value of tan φ ( M ) determined by inserting this Pad´eapproximant into the matching condition in eq. (5.23) is shown as a function of M infigure 13. It varies significantly with M within the range of validity of ZREFT. In thepredictions of ZREFT at LO at the mass M , it is therefore better to use the valueof tan φ ( M ) from matching at the mass M than the value tan φ ( M ∗ ) = 0 .
877 frommatching at unitarity.If α was small enough, we could determine tan φ by matching predictions fromZREFT with α = 0 to results from NREFT with α = 0. The effective range for– 30 – M [ TeV ] t an ϕ Figure 13 . Interaction parameter tan φ for ZREFT at LO as a function of the wino mass M for δ = 170 MeV and α = 1 / φ ( M ) (solid red curve) is determinedfrom the matching condition for the effective range r in NREFT in eq. (5.23). The constantvalue tan φ ( M ∗ ) = 0 .
877 (dashed red line) is determined by matching r at unitarity. Thevertical dotted line marks the unitarity mass M ∗ = 2 .
39 TeV. α M (TeV) δ (MeV) γ / (2 M δ ) / r (2 M δ ) / tan φ − .
653 0.8770 2.39 170 − . − .
224 1.1060 2.88 170 0 − .
693 0.8320 2.22 0 0 − .
552 1.246
Table 1 . Interaction parameter tan φ for ZREFT at LO from matching to NREFT at variousmatching points. The inverse scattering length γ and the effective range r are calculated us-ing NREFT. The parameter tan φ is determined by the matching condition for r in eq. (5.23). α = 0, δ = 170 MeV, and M ∗ = 2 .
39 TeV is given in the text after eq. (3.10). Bymatching it to the prediction r = − tan φ/ ∆ from ZREFT at LO with α = 0, weobtain tan φ = 1 . φ is listed in table 1, along with the valuesobtained in ref. [13] at two other matching points with α = 0. Significant differencesin the value of tan φ imply significant differences in the predictions of ZREFT at LO.Matching at α = 0, δ = 170 MeV, and the corresponding unitarity mass M ∗ = 2 .
88 TeV– 31 – - - M [ TeV ] r Δ - - - M [ TeV ] s Δ Figure 14 . Neutral-wino effective range r (left panel) and shape parameter s (right panel)as functions of the wino mass M : NREFT (thicker grey curve), ZREFT at LO with tan φ ( M )(solid red curve), and ZREFT at LO with tan φ ( M ∗ ) = 0 .
877 (dashed red curve). The verticaldotted lines indicate the unitarity mass M ∗ = 2 .
39 TeV. gives a value of tan φ that is only 5% lower than that from matching at α = 1 / δ = 170 MeV, and the unitarity mass M ∗ = 2 .
39 TeV.
The predictions of ZREFT at LO as a function of the wino mass M can be obtained byusing the Pad´e approximant for the inverse scattering length γ ( M ) = 1 /a ( M ) givenby eq. (3.9) and the M -dependent mixing angle φ ( M ) determined by inserting the Pad´eapproximant for the effective range r ( M ) in eq. (3.11) into the matching condition ineq. (5.23). As shown in the left panel of figure 14, the Pad´e approximant for r ( M ) isvery accurate over the entire range of validity of ZREFT. In the right panel of figure 14,the prediction of ZREFT at LO for the shape parameter s ( M ) as a function of M iscompared to the result from NREFT. At unitarity, the prediction for s differs fromthe result from NREFT in eq. (3.10b) by a multiplicative factor of 0.81. The accuracyof the prediction remains comparable at other values of M within the range of validityof ZREFT. If M is very close to the unitarity mass M ∗ = 2 .
39 TeV, we can use theconstant mixing angle given by tan φ ( M ∗ ) = 0 . r at unitarity. However, as shown in figure 14, the resulting predictions for r ( M )– 32 – REFTZREFT LOunitarity δ δ δ δ E σ → m W Figure 15 . Neutral-wino elastic cross section σ → as a function of the energy E . The crosssection at the unitarity mass M ∗ = 2 .
39 TeV is shown for NREFT (thicker grey curve) andfor ZREFT at LO with tan φ = 0 .
877 (red curve). The S-wave unitarity bound is shown as adotted curve. and s ( M ) as functions of M have the wrong slopes. The accuracy of the predictionstherefore deteriorates quickly as | M − M ∗ | increases.The energy dependence of the wino-wino cross sections is most dramatic at aunitarity mass. ZREFT at LO can be applied at the unitarity mass M ∗ = 2 .
39 TeVby setting γ = 0 and by setting tan φ = 0 . φ = 0 .
877 with the resultsfrom NREFT. In the limit E →
0, both cross sections saturate the unitarity bound.The mixing angle φ ( M ∗ ) was tuned so that the next-to-leading term in the low-energyexpansions also agrees. The prediction of ZREFT at LO also agrees well with theresult from NREFT above the charged-wino-pair threshold at 2 δ . Just above 2 δ , theprediction is smaller by a factor of 0.857. The prediction for σ → at E > δ can beimproved by decreasing tan φ at the cost of decreasing the accuracy of the predictionfor E close to 0. The mixing angle determined by matching σ → in the limit E → δ + is given by tan φ = 0 . .96 1.98 200.20.40.60.81 E / δ σ → / ( π / M * E ) Figure 16 . Neutral-wino elastic cross section σ → divided by the S-wave unitarity boundas a function of the energy E near the charged-wino-pair threshold. The cross section at theunitarity mass M ∗ = 2 .
39 TeV is shown for NREFT (thicker grey curve) and for ZREFT atLO with tan φ = 0 .
877 (red curve).
ZREFT at LO and the results from NREFT in the resonance region just below thethreshold at 2 δ . A blow-up of the threshold region, with the cross section divided bythe S-wave unitarity bound, is shown in figure 16. Just below the threshold, there isa sequence of increasingly narrow resonances associated with Coulomb w + w − boundstates that saturate the unitarity bound. ZREFT at LO reproduces the qualitativebehavior of the dramatic energy dependence. It predicts that the cross section haszeros and resonant peaks at the energies where the real part of the function L ( E ) ineq. (5.14) has poles and zeros, respectively. At unitarity where γ = 0, the predictionsfor the zeros and resonant peaks are independent of the mixing angle φ . The zeros inthe cross section are predicted to be at the energies E n of the Coulomb bound states ineq. (4.7). More accurate predictions for σ → in the resonance region could be obtainedby using ZREFT at NLO, which has two additional relevant parameters.In the left panel of figure 17, we compare the energy dependence of the cross sectionfor the neutral-to-charged transition predicted by ZREFT at LO with tan φ = 0 . E = 4 δ . The prediction for σ → ( E ) can be improved by increasing tan φ at the cost– 34 – REFTZREFT LO δ δ δ E σ → m W NREFTZREFT LO δ δ δ E σ → m W Figure 17 . Neutral-to-charged transition cross section σ → (left panel) and the charged-wino elastic cross section σ → (right panel) as functions of the energy E . The cross sectionsat the unitarity mass M ∗ = 2 .
39 TeV are shown for NREFT (thicker grey curve) and forZREFT at LO with tan φ = 0 .
877 (red curve). of decreasing the accuracy of the prediction for σ → . The mixing angle determined bymatching the cross section σ → at E = 2 δ is given by tan φ = 1 . E = 4 δ then differs from the result of NREFT by a factor of 0.99.In the right panel of figure 17, we compare the cross section for charged-winoelastic scattering predicted by ZREFT at LO with tan φ = 0 .
877 with the results fromNREFT. The prediction seems to have the correct qualitative behavior. It is smallerthan the NREFT cross section, differing by a factor that decreases to about 0.6 at E = 4 δ . The prediction for σ → can be improved by increasing tan φ at the cost ofdecreasing the accuracy of the prediction for σ → . Very near the charged-wino-pairthreshold, both cross sections have dramatic oscillations that are too large to be visiblein the right panel of figure 17. A blow-up of the threshold region, with the cross sectionsdivided by the S-wave unitarity bound, is shown in figure 18. As E approaches 2 δ , theoscillations become increasingly narrow. They are not resonances, because they donot saturate the unitarity bound. The prediction of ZREFT at LO has the correctqualitative behavior. The predicted oscillations have amplitude smaller by about afactor of 0.66 and average value smaller by about a factor of 0.87. More accuratepredictions for σ → in the oscillation region could be obtained by using ZREFT atNLO, which has two additional relevant parameters.– 35 – E / δ σ → / ( π / M * ( E - δ )) Figure 18 . Charged-wino elastic cross section σ → divided by the S-wave unitarity boundas a function of the energy E near the charged-wino-pair threshold. The cross section at theunitarity mass M ∗ = 2 .
39 TeV is shown for NREFT (thicker grey curve) and for ZREFT atLO with tan φ = 0 .
877 (red curve).
If the wino mass M is larger than the unitarity mass where the neutral-wino scatteringlength a ( M ) diverges, the S-wave resonance is a bound state below the neutral-wino-pair threshold. The bound state is a superposition of a neutral-wino pair and a charged-wino pair, and we denote it by ( ww ). The coupled-channel radial Schr¨odinger equationfor NREFT in eq. (3.1) has a negative eigenvalue − E ( ww ) , where E ( ww ) is the bindingenergy. In figure 19, the binding energy for δ = 170 MeV is shown as a function of M .The binding energy goes to zero as M approaches the unitarity mass M ∗ = 2 .
39 TeVfrom above. The binding energy depends sensitively on the electromagnetic couplingconstant α . If the Coulomb potential between the charged winos is turned off by setting α = 0, the unitarity mass where E ( ww ) vanishes is shifted to 2.88 TeV.In ZREFT, the binding energy E ( ww ) of the wino pair bound state can be obtainedby solving an analytic equation numerically. If γ >
0, each of the transition amplitudes A ij ( E ) given by the matrix in eq. (5.13) has a pole at a real energy − E ( ww ) belowthe neutral-wino-pair threshold. The pole in E is at a zero of the function L ( E ) ineq. (5.14). The binding energy can be expressed as E ( ww ) = γ /M , where the binding– 36 – REFTZREFT LO: ϕ ( M ) ZREFT LO: ϕ ( M * ) universal M [ TeV ] E ( ww ) / δ Figure 19 . Binding energy E ( ww ) of the wino-pair bound state as a function of the wino mass M : NREFT (thicker grey curve), ZREFT at LO with tan φ ( M ) (red solid curve), ZREFT atLO with tan φ ( M ∗ ) = 0 .
877 (red dashed curve), and the universal approximation in eq. (5.26)(dotted curve). momentum γ is a positive solution to the equation0 = γ − γ + t φ (cid:2) K ( − γ /M ) − K (0) (cid:3) . (5.25)The correct root of this equation is the one that approaches 0 as γ decreases to 0from above. In figure 19, the predictions for the binding energy in ZREFT at LOare compared to the result from NREFT. Using the M -dependent parameter tan φ ( M )obtained by matching r ( M ) gives a prediction for E ( ww ) that tracks the result fairlywell as a function of M . As M → M ∗ , the prediction approaches the result fromabove. Its error decreases to less than 5% for M − M ∗ < . φ ( M ∗ ) = 0 .
877 obtained by matching r at unitarity gives a predictionfor E ( ww ) whose error deteriorates quickly as M increases. As M → M ∗ , the predictionapproaches the result from below. Its error decreases to less than 5% for M − M ∗ < . a [18]. If a is positive, the– 37 –-wave bound state closest to the threshold is universal. The universal approximationfor its binding energy is E ( ww ) = 1 / ( M a ) . (5.26)For neutral winos with mass near the unitarity mass M ∗ = 2 .
39 TeV, the universalapproximation in eq. (5.26) is applicable for M inside the region between M ∗ and2.9 TeV. The universal approximation becomes increasingly accurate as M approaches M ∗ . In figure 19, the universal approximation in eq. (5.26) with the Pad´e approximantfor a ( M ) in eq. (3.9) is compared to the result from NREFT. As M → M ∗ , theuniversal approximation approaches the result from above. Its error decreases to lessthan 5% for M − M ∗ < .
004 TeV.The bound state ( ww ) is a superposition of a neutral-wino pair w w and a charged-wino pair w + w − . The probabilities of the w w and w + w − components of the boundstate ( ww ) can be deduced from the transition amplitudes A ( E ) and A ( E ) ineq. (5.13). Both of these amplitudes have a pole in the energy at E = − γ /M , where γ satisfies eq. (5.25). We denote the residues of the poles in A ( E ) and A ( E ) by −Z and −Z , respectively. The absolute values of the residues Z and Z are proportionalto the probabilities for the w w and w + w − components of the bound state, respectively.The residue factor for the w w channel at LO is Z = 16 πγ/M − t φ K (cid:48) ( − γ /M ) γ/M . (5.27)The ratio of the residue factors at LO is Z / Z = 12 t φ W ( − γ /M ) , (5.28)where the function W ( E ) is given in eq. (4.12). The ratio of the probabilities for w + w − and w w is |Z ||Z | / φ Γ (1 − | η | ) , (5.29)where | η | = αM/ (2∆). In the limit α →
0, the probability for w w reduces to cos φ .Given the numerical value tan φ = 0 .
877 from the LO fit, the probability for w w isapproximately 43%. One of the options for a wimp is the neutral wino w , which belongs to an SU (2)multiplet that also includes the charged winos w + and w − . The splitting δ between acharged wino and a neutral wino is small compared to the mass M of the wino. Thephysics of nonrelativistic winos involves many momentum scales, including– 38 – the weak gauge boson mass scale m W , • the scale α M of nonperturbative effects from exchange of weak gauge bosons, • the Bohr momentum αM , which is the scale of nonperturbative effects from theCoulomb interaction, • the scale √ M δ associated with the transition between a neutral-wino pair anda charged-wino pair, • the inverse scattering length γ = 1 /a of the neutral wino.A fundamental description of winos is provided by a relativistic quantum field theory.Nonrelativistic effective field theories provide simpler descriptions for low-energy winosin which some of the momentum scales are not described explicitly.If the winos are nonrelativistic, the momentum scale M does not need to be treatedexplicitly. The winos can be described by the nonrelativistic effective field theory calledNREFT. In NREFT, low-energy winos interact instantaneously at a distance througha potential generated by the exchange of weak gauge bosons, and charged winos alsohave local couplings to the electromagnetic field. If M is large enough that α M iscomparable to m W , interactions between nonrelativistic winos from the exchange ofthe W ± and Z are nonperturbative. The effects of Coulomb interactions betweencharged winos are also nonperturbative. Calculations in NREFT then require thenumerical solution of a coupled-channel Schr¨odinger equation. The power of NREFThas recently been demonstrated by a calculation of the capture rates of two neutralwinos into wino-pair bound states through the radiation of a photon [9].There are critical values of the wino mass at which there is an S-wave resonance atthe neutral-wino-pair threshold. We refer to such a critical value as a unitarity mass,because the cross section saturates the S-wave unitarity bound in the low-energy limit.If M is near a unitarity mass, the inverse scattering length γ is much smaller thanthe momentum scales m W and α M . If the relative momentum of winos is smallerthan m W and α M , those momentum scales do not need to be described explicitly. InRef. [13], we developed a zero-range effective field theory called ZREFT to describewinos with mass M near a unitarity mass. The effects of the exchange of weak gaugebosons between winos is reproduced by zero-range interactions between the winos thatmust be treated nonperturbatively. Charged winos also have local couplings to theelectromagnetic field. The effects of Coulomb interactions between charged winos mustalso be treated nonperturbatively. The power of ZREFT was illustrated in Ref. [13] bycalculating the rate for the formation of the wino-pair bound state in the collision of– 39 –wo neutral winos through a double radiative transition in which two soft photons areemitted.NREFT is more broadly applicable than ZREFT. NREFT can describe nonrela-tivistic winos with any mass M , while ZREFT is only applicable if the wino mass isin a window around a unitarity mass. If the wino mass splitting is δ = 170 MeV, thefirst such unitarity mass is M ∗ = 2 .
39 TeV, and the window for the applicability ofZREFT is M from about 1.8 TeV to about 4.6 TeV. NREFT describes nonrelativisticwinos, while ZREFT can only describe winos with relative momentum less than m W .NREFT can describe the interactions of a pair of winos in any angular-momentumchannel, while ZREFT can only describe S-wave interactions. Despite its more limitedapplicability, ZREFT has distinct advantages over NREFT. In particular, two-bodyobservables can be calculated analytically in ZREFT. This makes it easier to explorethe impact of an S-wave near-threshold resonance on dark matter.In the absence of electromagnetism, ZREFT is a systematically improvable ef-fective field theory. The improvability is guaranteed by identifying a point in theparameter space in which the S-wave interactions of winos are scale invariant in thelow-energy limit, and can therefore be described by an effective field theory that isa renormalization-group fixed point. At the RG fixed point, the mass splitting δ be-tween the charged wino and the neutral wino is 0, and the corresponding unitaritymass is M ∗ = 2 .
22 TeV. In Ref. [13], it was verified explicitly that, in the absence ofelectromagnetic interactions, ZREFT at NLO provides systematic improvements in thepredictions of ZREFT at LO at δ = 170 MeV and the corresponding unitarity mass M ∗ = 2 .
88 TeV.In this companion paper to Ref. [13], we carried out the Coulomb resummationthat is needed to calculate the quantitative predictions of ZREFT at LO. The T-matrix elements for wino-wino scattering are given analytically in eqs. (5.16) and (5.19).An analytic equation for the binding energy of the wino-pair bound state is given ineq. (5.25). The parameters of ZREFT at LO are the kinematic parameters M and δ andthe interaction parameters α = 1 / φ , and γ = 1 /a . The interaction parameters φ and γ can be determined by matching predictions of ZREFT at LO for scatteringamplitudes with results calculated by solving the Schr¨odinger equation for NREFTnumerically. An accurate Pad´e approximant of a ( M ) for M near the first unitaritymass M ∗ = 2 .
39 TeV is given in eq. (3.9). The mixing angle φ can be determinedfrom NREFT calculations of the effective range r by using the matching conditionin eq. (5.23). An accurate Pad´e approximant of r ( M ) for M near M ∗ is given ineq. (3.11). The M -dependent mixing angle φ ( M ) obtained by matching r ( M ) as afunction of M is shown in figure 13. The mixing angle φ ( M ∗ ) determined by matching r at unitarity is given by tan φ ( M ∗ ) = 0 . M was illustrated by the neutral-wino shape parameter s and by the binding energy E ( ww ) of the wino-pair bound state. Accurate predictions away from unitarity requireusing an M -dependent mixing angle φ ( M ), such as that shown in figure 13. The errorin the prediction of s remains small thoughout the region of validity of ZREFT, asshown in the right panel of figure 14. The error in the prediction of E ( ww ) increaseswith M , but it also remains small in the region of validity of ZREFT, as shown infigure 19.The accuracy of the predictions of ZREFT at LO as functions of the energy E wasillustrated by using the wino-wino cross sections at the unitarity mass M ∗ = 2 .
39 TeV.ZREFT at LO gives accurate predictions for the neutral-wino elastic cross section σ → for E < δ and for E > δ , as shown in figure 15. Its predictions in the resonance regionjust below 2 δ have the correct qualitative behavior, as shown in figure 16. ZREFTat LO gives reasonably good predictions for the charged-to-neutral transition crosssection σ → and the charged-wino elastic cross section σ → , as shown in figure 17. Itspredictions for σ → in the oscillation region just above 2 δ have the correct qualitativebehavior, as shown in figure 18. More accurate predictions could be obtained by usingZREFT at NLO, which has two additional relevant parameters.One of the primary motivations for the development of ZREFT for winos was thecalculation of the “Sommerfeld enhancement” of the annihilation of a pair of winosinto electroweak gauge bosons when the wino mass is near a resonance at the neutral-wino pair threshold. Wino-pair annihilation also affects other aspects of the few-bodyphysics for low-energy winos. For example, the neutral-wino elastic cross section doesnot actually diverge at a unitarity mass, but it instead has a very narrow peak [30].The effects of wino-pair annihilation on low-energy winos can be taken into account inZREFT by analytically continuing real parameters to complex values. Since two-bodyobservables for winos can be calculated analytically in ZREFT, the effects of wino-pairannihilation can also be taken into account analytically. The results are presented inanother companion paper [31]. Acknowledgments
This research project was stimulated by a discussion with M. Baumgart. We thankS K¨onig for useful discussions of Coulomb effects in zero-range effective field theory.We thank T. Slatyer for helpful discussions on the partial wave expansion for theCoulomb interaction. This work was supported in part by the Department of Energyunder grant DE-SC0011726. – 41 – igure 20 . Diagrammatic representation of the coupled-channel Lippmann-Schwinger inte-gral equations for the short-distance transition amplitudes A s, ( E ), A s, ( E ), A s, ( E ), and A s, ( E ). Each of the charged-wino bubbles is the sum of diagrams in figure 11. A Lippmann-Schwinger equation with Coulomb resummation
In the Zero-Range Model with Coulomb resummation, the 2 × A ( E ) of transi-tion amplitudes can be expressed in the form in eq. (4.11), where A C ( E ) is the Coulombamplitude in eq. (4.5), W ( E ) is the amplitude for the creation of w + w − at a pointin eq. (4.12), and A s ( E ) is the 2 × A s ( E ). Wealso use unitarity to determine the amplitude W ( E ). A.1 Short-distance transition amplitudes
In the Zero-Range Model discussed in section 2.3, there are two wino-wino channelsfor which there are zero-range interactions: a pair of neutral winos in the S-wavespin-singlet channel, which we label by 0, and a pair of charged winos in the S-wavespin-singlet channel, which we label by 1. The S-wave spin-singlet transition amplitudeshave the same Pauli spinor structure as the zero-range interaction vertices. They canbe expressed as A ij ( E ) multiplied by the spin-singlet projector ( δ ac δ bd − δ ad δ bc ), where i and j are the incoming and outgoing channels, a and b are Pauli spinor indices forthe incoming lines, and c and d are Pauli spinor indices for the outgoing lines. Thetransition amplitudes A ij ( E ) are functions of the total energy E in the center-of-mass– 42 –rame. They do not depend separately on the energies and momenta of the incomingand outgoing lines.The Lippmann-Schwinger integral equations for the short-distance transition am-plitudes can be expressed as the diagrammatic equations in figure 20. In the momentumrepresentation, the Lippmann-Schwinger equation for the 2 × A s ( E )can be expressed as a matrix equation: A s ( E ) = − λ + λ I ( E ) A s ( E ) , (A.1)where λ is a symmetric matrix of bare coupling constants, λ = (cid:18) λ λ λ λ (cid:19) , (A.2)and I ( E ) is a diagonal matrix of bubble amplitudes: I ( E ) = (cid:18) I ( E ) 00 J ( E ) (cid:19) . (A.3)The factor of in the upper diagonal entry is a symmetry factor. The loop integralsare ultraviolet divergent. They can be regularized using dimensional regularization in d = 3 − (cid:15) spatial dimensions. After integrating over the loop energy by contours, theneutral-wino bubble amplitude I ( E ) is I ( E ) = − M (cid:18) Λ2 (cid:19) − d (cid:90) d d k (2 π ) d k − M E − i(cid:15) , (A.4)where Λ is an arbitrary renormalization scale. The integral can be evaluated analyti-cally. The linear ultraviolet divergence in d = 3 spatial dimensions appears as a polein d − M Λ / π . The integral can be renormalized by power divergencesubtraction [19], in which the limit d → d − I ( E ) = − M π (cid:2) Λ − κ ( E ) (cid:3) , (A.5)where κ ( E ) is the function of the complex energy E defined in eq. (4.4a). The charged-wino bubble amplitude J ( E ) was evaluated analytically using dimensional regulariza-tion by Kong and Ravndal [26]. It can be expressed as the sum of discrete contributions The function J was denoted by ¯ J in Ref. [26]. – 43 –rom Coulomb bound states and a dimensionally regularized integral over the relativemomentum of scattering states: J ( E ) = α M π ∞ (cid:88) n =1 n ( E − E n ) − M (cid:18) Λ2 (cid:19) − d (cid:90) d d k (2 π ) d πη ( k )exp (cid:0) πη ( k ) (cid:1) − k − M ( E − δ ) − i(cid:15) , (A.6)where E n is the energy of the Coulomb bound state in eq. (4.7) and η ( k ) = − αM/ (2 k ).The integral has a linear ultraviolet divergence in d = 3 spatial dimensions that appearsas a pole in d − M Λ / π . In the power divergence subtraction regular-ization scheme [19], the linear divergence is canceled by subtracting the pole in d − d = 3 spatial dimensionsthat appears as a pole in d −
3. After subtracting from the integrand the terms thatgive the poles in d − d −
3, the remaining integral can be evaluated analyticallyin d = 3 dimensions. The final result for the bubble amplitude in the limit d → J ( E ) = − M π (cid:20) Λ + αM (cid:18) − d + log √ π Λ αM + 1 − γ (cid:19) − K ( E ) (cid:21) , (A.7)where K ( E ) is the function of the complex energy E defined in eq. (4.16) and γ isEuler’s constant. This result was first calculated by Kong and Ravndal in ref. [26], andit was verified in ref. [27].To solve the integral equation in eq. (A.1), we multiply by A − s on the right and λ − on the left and then rearrange: A − s ( E ) = − λ − + I ( E ) . (A.8)The dependence of the amplitudes A s,ij ( E ) on the renormalization scale can be elimi-nated by choosing the bare parameters λ ij to depend on Λ in such a way that λ − = M π (cid:18) γ √ γ √ γ γ (cid:19) − M Λ8 π (cid:18) (cid:19) − αM π (cid:18) − d + log √ π Λ αM + 1 − γ (cid:19) (cid:18) (cid:19) . (A.9)This defines physical scattering parameters γ , γ , and γ with dimensions of mo-mentum. Substituting these relations into eq. (A.8), we have A − s ( E ) = M π (cid:18) − γ + κ ( E ) −√ γ −√ γ (cid:2) − γ + K ( E ) (cid:3)(cid:19) . (A.10)The inverse A s ( E ) of this 2 × − R d Π M
22 Im
Figure 21 . The discontinuity in the bubble amplitude J ( E ) at a real energy E > δ isproportional to the absolute square of the amplitude W ( E ) for creating w + w − at a point. A.2 Amplitude for creating a charged-wino pair at a point
To complete the calculation of the transition amplitudes for the Zero-Range Model withCoulomb resummation, we must determine the amplitude W ( E ) for creating w + w − ata point with total energy E . We use the unitarity condition for A ( E ) and the opticaltheorem for the charged-wino bubble amplitude J ( E ).The unitarity condition for A ( E ) in eq. (4.1) can be reduced to a similar equationfor the short-distance amplitudes: A s ( E ) − A s ( E ) ∗ = − π A s ( E ) (cid:18) W ( E ) (cid:19) × M / (cid:2) κ ( E ) − κ ( E ) ∗ (cid:3) M / (cid:18) W ( E ) ∗ (cid:19) A s ( E ) ∗ , (A.11)provided W ( E ) at a real energy E satisfies the identity W ( E ) − W ( E ) ∗ = − M π A C ( E ) (cid:2) κ ( E ) − κ ( E ) ∗ (cid:3) W ( E ) ∗ . (A.12)Using the explicit expression for the Coulomb amplitude in eq. (4.5), this implies thatfor real energies E > δ , W must satisfy W ( E ) /W ( E ) ∗ = Γ(1 + iη ) / Γ(1 − iη ) , (A.13)where η is the function of E defined in eq. (4.6).The optical theorem for the charged-wino bubble amplitude is represented dia-grammatically in figure 21. It determines the imaginary part of the function J ( E ) atreal energies E > δ : J ( E ) − J ( E ) ∗ = − i (cid:12)(cid:12) W ( E ) (cid:12)(cid:12) M (cid:112) M ( E − δ )2 π . (A.14)The last factor is the phase space integral for w + w − with total energy E . The function J ( E ) is given explicitly in eq. (A.7). Its imaginary part comes from the imaginary– 45 –art of the function K ( E ) defined in eq. (4.16). For real values of E , that functionsatisfies the identity K ( E ) − K ( E ) ∗ = C ( E ) (cid:2) κ ( E ) − κ ( E ) ∗ (cid:3) , (A.15)where C is the Sommerfeld factor in eq. (4.13). For E < δ , both sides are 0. For E > δ , the identity follows from a property of the function ψ ( z ): ψ ( z ) − ψ ( − z ) = − z − π tan( πz ) . (A.16)Comparing eqs. (A.14) and (A.15), we find that for real energies E > δ , W ( E ) mustsatisfy W ( E ) W ( E ) ∗ = C ( E ) . (A.17)Combining eqs. (A.13) and (A.17), we obtain the amplitude W ( E ) for creating w + w − at a point in eq. (4.12). This expression was derived for real E > δ , but it can beextended to complex E by analytic continuation.On the right side of eq. (A.11), all the diagonal matrices between A s and A ∗ s com-mute. Since the product of diag(1 , W ) and diag(1 , W ∗ ) is diag(1 , C ), they can bothbe replaced by diag(1 , C ). By multiplying eq. (A.11) by a prefactor of diag(1 , C ) andby a postfactor of diag(1 , C ), we find that the matrix diag(1 , C ) A s diag(1 , C ) satisfiesthe unitarity condition in eq. (4.1). References [1] E. W. Kolb and M. S. Turner, The early universe, Front. Phys. , 1 (1990).[2] G. Steigman, B. Dasgupta and J. F. Beacom, Precise relic WIMP abundance and itsimpact on searches for dark matter annihilation, Phys. Rev. D , 023506 (2012)[arXiv:1204.3622].[3] J. Hisano, S. Matsumoto and M. M. Nojiri, Unitarity and higher order corrections inneutralino dark matter annihilation into two photons, Phys. Rev. D , 075014 (2003)[hep-ph/0212022].[4] J. Hisano, S. Matsumoto and M. M. Nojiri, Explosive dark matter annihilation, Phys.Rev. Lett. , 031303 (2004) [hep-ph/0307216].[5] J. Hisano, S. Matsumoto, M. M. Nojiri and O. Saito, Non-perturbative effect on darkmatter annihilation and gamma ray signature from galactic center, Phys. Rev. D ,063528 (2005) [hep-ph/0412403]. – 46 –
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