Mrejesho wa annihilation, ni hali ambapo michakato isiyo elastiki ya chembe za mada nyeusi, kama annihilation, haiwaondoi tu chembe kutoka kwenye njia elastiki, bali pia hubadilisha kernel ya self-energy ya hali ya chembe mbili zinazoingia na hivyo kutoa mrejesho kwenye amplitudo ya mtawanyiko elastiki. Utafiti huu unachunguza ni lini mrejesho huu huwa muhimu katika mada nyeusi yenye mwingiliano binafsi chini ya nguvu ya Yukawa ya masafa marefu inayotokana na chembe mpatanishi nyepesi. Hitimisho kuu ni kwamba athari si kubwa katika nafasi yote ya vigezo; huwa ya kuamua hasa karibu na resonance za Sommerfeld na sifuri za baseline za mtawanyiko elastiki.

Msingi mkuu wa kifizikia wa waandishi ni unitarity. Unitarity ya S-matrix na optical theorem zinaonyesha kwamba uwezekano wa mtawanyiko elastiki na usio elastiki ndani ya partial wave ileile hauwezi kuchaguliwa kwa kujitegemea. Katika normalization inayotumika, \(x_\ell=\sigma_\ell^{\rm elas}/\sigma_\ell^{\rm U}\) na \(y_\ell=\sigma_\ell^{\rm inel}/\sigma_\ell^{\rm U}\) zinapofafanuliwa, ikiwa cross section isiyo elastiki si sifuri, cross section elastiki hupata si kikomo cha juu tu bali pia kikomo cha chini kilicho juu ya sifuri. Kwa hiyo, wakati annihilation ni kubwa, kuhesabu mtawanyiko elastiki kwa kutumia potential ya Hermitian ya masafa marefu pekee kunaweza kutoa matokeo yasiyolingana na unitarity katika baadhi ya maeneo.

Utafiti unashughulikia tatizo hili kwa kuongeza kwenye mlinganyo wa Schrödinger wa hali ya chembe mbili zinazoingia michango absorptive, yaani anti-Hermitian, inayotokana na michakato isiyo elastiki. Wakati mwingiliano purely elastic wa Yukawa wa masafa marefu unaunda tatizo la baseline Hermitian, annihilation huleta mchango non-Hermitian kwenye kernel ya self-energy ya chembe mbili. Resummation ya mara kwa mara ya mchango huu hubadilisha amplitudo zote mbili, za elastiki na zisizo elastiki.

Nguvu ya masafa marefu inaweza kuongeza annihilation kupitia athari ya Sommerfeld. Hasa karibu na \(s\)-wave bound-state threshold, yaani resonance ya Sommerfeld, baseline annihilation cross section inaweza kukaribia au hata kuvuka kikomo cha partial-wave unitarity kadiri kasi inavyopungua. Katika chanzo resonance ya kwanza ya \(s\)-wave iko karibu na \(\xi_{1,0}\simeq0.84\). Feedback inapojumuishwa, ukuaji huu usiodhibitiwa wa kasi ndogo hukatwa; cross sections za regulated elastic na annihilation huelekea kwenye tabia yenye kikomo.

Katika antiresonance ambapo baseline elastic amplitude ni sifuri, hali tofauti hutokea. Amplitudo ya annihilation kwa kawaida si sifuri katika hatua hii. Kwa sababu ya sharti la coupled unitarity, inelasticity isiyo sifuri hairuhusu mtawanyiko elastiki kuwa sifuri kabisa. Hivyo annihilation feedback hubadilisha baseline elastic zero kuwa kiwango cha chini kisicho sifuri. Katika chanzo, kwa exact \(s\)-wave antiresonance, baseline physical elastic cross section huenda hadi sifuri kama \(k^4\) katika momentum ndogo, ilhali regulated cross section yenye feedback hukaribia plateau thabiti ya kasi ndogo.

Kwa upande wa fenomenolojia ya halo, utafiti unatumia kama benchmark zinazowakilisha: \(\sigma^V/m_X\lesssim1\) barn/GeV kwa \(v_c\sim10^3\) km/s katika kiwango cha cluster, \(\sigma^V/m_X\gtrsim1\) barn/GeV kwa \(v_c\sim10\) km/s katika kiwango cha dwarf-galaxy, na takriban \(10^2\) barn/GeV kwa \(v_c\sim1\) km/s katika kiwango cha subhalo. Hivi si vikomo vya jumla na thabiti, bali ni vipimo vya kifonomenolojia vinavyotumiwa kulinganisha regimes tofauti za SIDM.

Kikomo cha tafsiri ya utafiti ni muhimu: makala hii haionyeshi kuwepo kwa mada nyeusi, kwamba modeli fulani ya mediator inatokea katika maumbile, wala kwamba galaksi fulani ziliundwa kwa sababu ya annihilation feedback. Utafiti unachunguza hesabu self-consistent ya scattering inayotakiwa na unitarity katika modeli mahususi ya kinadharia ya dark sector na unaonyesha maeneo ya vigezo ambapo hesabu za baseline zinaweza kupoteza uaminifu.

Kwa Nini Mtawanyiko Elastiki na Usio Elastiki Hauwezi Kuhesabiwa kwa Kujitegemea?

Unitarity ya S-matrix inahitaji uhifadhi wa uwezekano wote; optical theorem hubadilisha sharti hili kuwa uhusiano kati ya scattering amplitudes na cross sections. Kwa hiyo, njia isiyo elastiki inapokuwa na nguvu katika partial wave fulani, eneo linaloruhusiwa la elastic cross section pia hubadilika, na inelasticity yenye nguvu ya kutosha haiwezi kuongezwa kwa kujitegemea bila kutoa mrejesho kwenye hesabu ya elastic amplitude.

Sharti la partial-wave unitarity

Coupled constraint kuu ya chanzo:

\[ \left( \frac{ \sigma_\ell^{\rm elas} + \sigma_\ell^{\rm inel} }{ \sigma_\ell^{\rm U} } \right)^2 \leq \frac{ \sigma_\ell^{\rm elas} }{ \sigma_\ell^{\rm U} } \tag{2.1} \]

iko katika umbo hili.

Hapa:

• \(\ell\) ni partial wave iliyoainishwa kwa orbital angular momentum.
• \(\sigma_\ell^{\rm elas}\) ni elastic scattering cross section ya partial wave husika.
• \(\sigma_\ell^{\rm inel}\) ni jumla ya cross section za michakato yote isiyo elastiki katika partial wave hiyo.
• \(\sigma_\ell^{\rm U}\) ni kipimo cha partial-wave unitarity.

Chanzo hufafanua unitarity cross section kama:

\[ \sigma_\ell^{\rm U}(k) = \mathscr C_\ell \frac{4\pi}{k^2} (2\ell+1) \tag{2.2} \]

kwa umbo hili.

\(k\) ni ukubwa wa momentum ya mojawapo ya chembe zinazoingia katika mfumo wa center-of-momentum. \(\mathscr C_\ell\) ni symmetry factor inayotegemea ikiwa chembe zinaweza kutofautishwa au ni sawa:

\[ \mathscr C_\ell = \begin{cases} 1, & \text{ayırt edilebilir parçacıklar},\\ 2, & \text{özdeş parçacıklarda izin verilen modlar}. \end{cases} \tag{2.3} \]

Matokeo yake ya moja kwa moja ya jumla ni:

\[ \sigma_\ell^{\rm elas} \leq \sigma_\ell^{\rm U}, \qquad \sigma_\ell^{\rm inel} \leq \frac{\sigma_\ell^{\rm U}}{4} \tag{2.4} \]

hivyo.

Hata hivyo, jambo kuu kwa utafiti si vikomo hivi viwili vya juu pekee. Chanzo:

\[ x_\ell \equiv \frac{\sigma_\ell^{\rm elas}}{\sigma_\ell^{\rm U}}, \qquad y_\ell \equiv \frac{\sigma_\ell^{\rm inel}}{\sigma_\ell^{\rm U}} \tag{2.5} \]

baada ya kutoa ufafanuzi huu, huandika coupled constraint kama:

\[ \frac{1}{2} \left( 1-2y_\ell-\sqrt{1-4y_\ell} \right) \leq x_\ell \leq \frac{1}{2} \left( 1-2y_\ell+\sqrt{1-4y_\ell} \right) \tag{2.6a} \]

kwa umbo hili.

Pia:

\[ x_\ell\leq1, \qquad y_\ell\leq\frac14. \tag{2.6b} \]

Mlinganyo huu unabeba mantiki kuu ya kifizikia ya utafiti. Kadiri \(y_\ell\) inavyoongezeka, si kikomo cha juu cha elastic cross section pekee kinachoshuka; kikomo chake cha chini pia hupanda. Yaani mtawanyiko usio elastiki usio sifuri unahitaji mtawanyiko elastiki usio sifuri. Katika thamani ya juu zaidi ya inelastic:

\[ y_\ell=\frac14 \]

thamani elastiki pia lazima iwe:

\[ x_\ell=\frac14 \]

hivyo.

Kwa nini unitarity hapa haimaanishi tu “cross section isiwe kubwa sana”?

Mara nyingi unitarity huelezwa kupitia kiwango cha juu cha scattering cross section pekee. Katika makala hii, coupled constraint yenye nguvu zaidi ndiyo muhimu. Uwezekano usio elastiki katika partial wave ileile unapoongezeka, eneo fulani pia hulazimishwa kwa uwezekano elastiki. Kwa hiyo, ikiwa baseline elastic cross section inaenda hadi sifuri lakini annihilation inabaki isiyo sifuri, matokeo hayo mawili hayawezi kwa pamoja kuwa nadharia kamili.

Potential Non-Hermitian Inawakilishaje Annihilation Feedback?

Mwingiliano purely elastic huchangia kwenye kernel ya Hermitian ya chembe mbili, huku njia wazi zisizo elastiki zikiunda mchango anti-Hermitian, yaani absorptive, kupitia hali za kati za on-shell; mchango huu huondoa probability flux kutoka kwenye njia elastiki, na unapofanyiwa resummation mara kwa mara ndani ya mlinganyo wa Schrödinger, amplitudo za scattering elastiki na zisizo elastiki hubadilika kwa namna self-consistent.

Mlinganyo wa jumla wa Schrödinger

Chanzo kinatumia katika regime nonrelativistic:

\[ -\frac{\nabla^2}{2\mu} \psi_{\mathbf k}(\mathbf r) + \int d^3r' \, \mathcal V(\mathbf r',\mathbf r) \psi_{\mathbf k}(\mathbf r') = \mathcal E_{\mathbf k} \psi_{\mathbf k}(\mathbf r) \tag{2.7} \]

mlinganyo huu.

Hapa \(\mu\) ni reduced mass, \(\psi_{\mathbf k}\) ni scattering wavefunction ya chembe mbili zinazoingia, na \(\mathcal V\) kwa ujumla ni potential effective iliyo nonlocal na non-Hermitian.

Nishati katika kikomo nonrelativistic:

\[ \mathcal E_{\mathbf k} \equiv \sqrt{s}-m_T \simeq \frac{k^2}{2\mu}. \tag{2.8} \]

\(m_T\) ni jumla ya mass ya jozi inayoingia; \(s\) ni kigezo cha kwanza cha Mandelstam.

Kernel ya 2PI ni nini?

Kernel ya two-particle-irreducible, yaani 2PI, ni jumla ya michoro ya Feynman yenye nukta nne ambayo haiwezi kugawanywa katika sehemu mbili kwa kukata propagator mbili za chembe mbili zinazoingiliana. Kernel hii inaporudiwa katika mlinganyo wa Schrödinger, michango inayolingana ya two-particle-reducible hufanyiwa resummation kiotomatiki.

Potential hufafanuliwa kutoka Fourier transform ya kernel nonrelativistic ya 2PI:

\[ \mathcal V(\mathbf r',\mathbf r) = -\frac{1}{4m_T\mu} \int \frac{d^3p'}{(2\pi)^3} \frac{d^3p}{(2\pi)^3} e^{i\mathbf p\cdot\mathbf r} \, \mathcal K^{\rm 2PI}(\mathbf p',\mathbf p) \, e^{-i\mathbf p'\cdot\mathbf r'} \tag{2.9} \]

kwa umbo hili.

Mgawanyo wa Hermitian na anti-Hermitian

\[ \mathcal K^{\rm 2PI} = \mathcal K_H^{\rm 2PI} + \mathcal K_A^{\rm 2PI} \tag{2.10a} \]

na:

\[ (\mathcal K_H^{\rm 2PI})^\dagger = \mathcal K_H^{\rm 2PI}, \qquad (\mathcal K_A^{\rm 2PI})^\dagger = -\mathcal K_A^{\rm 2PI}. \tag{2.10b} \]

Mwingiliano purely elastic wa one-boson-exchange huchangia \(\mathcal K_H\). Open inelastic channels huunda sehemu absorptive ya \(\mathcal K_A\) kwa sababu ya hali za kati za on-shell; aidha off-shell intermediate states zinaweza pia kutoa michango dispersive Hermitian.

Sehemu anti-Hermitian kimwili inawakilisha probability flux inayotoka kwenye njia elastiki inayoingia kwenda kwenye njia nyingine kama annihilation. Kwa hiyo “absorption” hapa si ufyonzaji wa kawaida ndani ya mada, bali uhamishaji wa quantum scattering probability kwenda kwenye njia nyingine za final-state.

Unregulated na regulated maana yake nini?

General regulated inelastic cross section

Baada ya kufafanua quantity mbili halisi zinazoitwa \(w_\ell\) na \(\widetilde w_\ell\), chanzo hupata normalized inelastic cross section kama:

\[ y_{\ell,\rm reg} = \frac{ w_\ell }{ (1+w_\ell)^2+\widetilde w_\ell^2 } \tag{2.27b} \]

kwa umbo hili.

\(w_\ell\) hubeba hasa athari absorptive, huku \(\widetilde w_\ell\) ikibeba phase modification inayohusiana na michango dispersive. Matokeo muhimu ya formula hii ni kwamba regulated cross section haikui bila kikomo kadiri baseline inelastic quantity inavyoongezeka.

Modeli ya Dark Sector Katika Utafiti Imejengwaje?

Chanzo huunda modeli kama dark sector rahisi yenye field ya complex scalar dark matter \(X\), complex scalar mediator \(\varrho\) yenye mara mbili ya chaji ya \(U(1)_D\), na mwanzoni dark gauge field \(V_\mu\); katika uchanganuzi wa kifonomenolojia gauge interaction hupuuzwa na dynamics ya masafa marefu ya \(X X^\dagger\) hupunguzwa hadi potential ya Yukawa inayotokana na kubadilishana \(\varrho\) nyepesi.

Fields na chaji

Kuwa na chaji mara mbili kwa \(\varrho\) kuliko \(X\):

\[ X^2\varrho^\dagger \]

huruhusu trilinear interaction.

Lagrangian husika

Chanzo:

\[ \begin{aligned} \mathcal L ={}& -\frac14F_{\mu\nu}F^{\mu\nu} + \frac12m_V^2V_\mu V^\mu + |D_\mu X|^2 - m_X^2|X|^2 \\ &+ |D_\mu\varrho|^2 - m_\varrho^2|\varrho|^2 - \frac{y\,m_X}{2} \left[ X^2\varrho^\dagger + (X^\dagger)^2\varrho \right] + \mathcal L_{\rm quartic} \end{aligned} \tag{3.1} \]

hutumia usemi huu.

Utafiti:

\[ m_\varrho\ll m_X \tag{3.2} \]

unakubali hierarchy hii. Kwa hiyo mediator nyepesi inaweza kuunda nguvu effective ya masafa marefu kwenye mizani ya halo.

Coupling zisizo na vipimo

\[ \alpha_V = \frac{g^2}{4\pi}, \qquad \alpha_\varrho = \frac{y^2}{16\pi}. \tag{3.3} \]

Mwingiliano quartic:

\[ \mathcal L_{\rm quartic} = -\frac{\lambda_X}{4}|X|^4 -\frac{\lambda_\varrho}{4}|\varrho|^4 -\lambda_{X\varrho}|X|^2|\varrho|^2. \tag{3.4} \]

Chanzo kinakubali masharti yafuatayo ili potential iwe bounded from below:

\[ \lambda_X>0, \qquad \lambda_\varrho>0, \qquad \lambda_{X\varrho}> -\frac12\sqrt{\lambda_X\lambda_\varrho} \tag{3.5a} \]

na kwamba coupling zibaki katika perturbative regime.

Potential ya Yukawa ya masafa marefu

Katika kikomo global-\(U(1)_D\), gauge interaction inapopuuzwa:

\[ V_{XX^\dagger,\ell}(r) = - (-1)^\ell \alpha_\varrho \frac{e^{-m_\varrho r}}{r}, \qquad V_{XX,\ell} = V_{X^\dagger X^\dagger,\ell} = 0. \tag{3.7} \]

Hivyo tatizo la msingi kwa scattering ya \(X X^\dagger\) hupunguzwa kwa potential ya jumla:

\[ V(r) = -\alpha \frac{e^{-m_{\rm med}r}}{r} \tag{3.8} \]

ya Yukawa.

Kwa \(\ell\) even, potential ya \(X X^\dagger\) huwa attractive; kwa \(\ell\) odd huwa repulsive. Mgawanyo huu ni muhimu kwa tabia ya resonance; attractive partial waves zinaweza kusaidia bound state, ilhali repulsive partial waves haziwezi.

Vigezo \(\zeta\) na \(\xi\) Vinapima Nini Kimwili?

\(\zeta=\alpha/v_{\rm rel}\) ni kigezo kisicho na vipimo kinachopima nguvu isiyo ya perturbative kwa kulinganisha coupling ya mwingiliano na kasi jamaa; \(\xi=\mu|\alpha|/m_{\rm med}\) hupima athari ya pamoja ya masafa ya potensheli na nguvu ya mwingiliano. Rejimu za mtawanyiko wa Yukawa, resonansi na Sommerfeld enhancement huainishwa kwa kiasi kikubwa na thamani za vigezo hivi viwili.

Chanzo:

\[ \zeta \equiv \frac{\mu\alpha}{k} = \frac{\alpha}{v_{\rm rel}}, \tag{3.10a} \]

\[ \xi \equiv \frac{\mu|\alpha|}{m_{\rm med}}, \tag{3.10b} \]

na:

\[ \tau \equiv \frac{m_{\rm med}}{\mu} \tag{3.10c} \]

hutumia ufafanuzi huu.

\(\zeta\) inaweza kuwa chanya kwa mwingiliano wa kuvutana na hasi kwa mwingiliano wa kusukumana; \(\xi\geq0\).

Rejimu kuu za mtawanyiko wa Yukawa

Idadi ya takriban ya angular momentum muhimu:

\[ \ell_{\rm clas} \sim \frac{\mu v_{\rm rel}}{m_{\rm med}} = \frac{\xi}{|\zeta|}. \tag{3.11} \]

Usemi huu hukadiria ni channel ngapi za angular momentum zinaweza kutoshea kwa ufanisi ndani ya masafa ya mwingiliano.

Kwa Nini Elastic Cross Section na Viscosity Cross Section ni Tofauti?

Total elastic cross section huhesabu pembe zote za scattering kwa uzito sawa. Hata hivyo, kwa usafirishaji wa momentum na joto ndani ya halo, forward scattering ndogo sana haina ufanisi sawa na mkengeuko wa pembe kubwa. Kwa hiyo chanzo hutumia pia transfer na viscosity cross sections.

\[ \sigma^{\rm elas} = \int d\Omega \, \frac{d\sigma}{d\Omega}, \tag{2.29a} \]

\[ \sigma^T = \int d\Omega \, (1-\cos\vartheta) \frac{d\sigma}{d\Omega}, \tag{2.29b} \]

\[ \sigma^V = \int d\Omega \, \sin^2\vartheta \frac{d\sigma}{d\Omega}. \tag{2.29c} \]

Transfer weighting hupunguza uzito wa forward scattering, huku viscosity weighting ikipunguza forward na backward scattering na kuweka uzito mkubwa zaidi kwenye transverse deflections.

Katika Appendix A, chanzo kutoka kwenye partial-wave decomposition:

\[ \sigma^T(k) = 4\pi \sum_{\ell=0}^{\infty} (\ell+1) |f_{k,\ell+1}-f_{k,\ell}|^2 \tag{A.8} \]

na:

\[ \sigma^V(k) = 4\pi \sum_{\ell=0}^{\infty} \frac{(\ell+1)(\ell+2)}{2\ell+3} |f_{k,\ell+2}-f_{k,\ell}|^2 \tag{A.12} \]

hutoa misemo hii. Kwa hiyo transfer cross section ina interference kati ya partial waves jirani za even/odd, huku viscosity cross section ikiwa na interference kati ya partial waves za \(\ell\) na \(\ell+2\) zenye parity sawa.

Kwa Nini Sommerfeld Enhancement Huimarisha Annihilation?

Ingawa annihilation ni mchakato wa masafa mafupi, kabla ya annihilation kutokea wavefunction ya jozi inayoingia ya \(X X^\dagger\) inaweza kuundwa upya na potensheli ya Yukawa ya masafa marefu ya kuvutana; amplitude ya wavefunction katika umbali mfupi inapoongezeka, probability ya annihilation pia huongezeka, na kizidishi hiki huelezwa na Sommerfeld factor \(S_\ell\).

Njia ya annihilation

Mchakato katika utafiti:

\[ X+X^\dagger \rightarrow \varrho+\varrho^\dagger. \tag{3.17} \]

Chanzo kinafafanua leading low-velocity partial-wave irreducible amplitude kama:

\[ \widehat{\mathcal A}_\ell(p) \simeq \left(\frac{p}{\mu}\right)^\ell \]

na absorptive coupling katika Mlinganyo (4.1).

Baseline annihilation cross section:

\[ \left(\frac{k}{\mu}\right) \sigma_{\ell,\rm unreg}^{\rm Ann} \simeq \eta_{A,\ell}^{\,2} \frac{4\pi(2\ell+1)}{\mu^2} \frac{\sqrt{1-\tau^2/4}}{2} \left(\frac{k}{\mu}\right)^{2\ell} S_\ell(k), \qquad \tau\leq2. \tag{4.2} \]

Hapa \(S_\ell(k)\) ni Sommerfeld factor inayotokana na potensheli ya Yukawa.

Sommerfeld factor katika kikomo cha Coulomb

\[ S_\ell^{C} = \frac{2\pi\zeta}{1-e^{-2\pi\zeta}} \prod_{j=1}^{\ell} \left( 1+\frac{\zeta^2}{j^2} \right). \tag{4.3} \]

Kwa mwingiliano wa kuvutana, \(\zeta>0\) ikiwa, kasi ndogo yaani \(\zeta\) kubwa inaweza kuipa annihilation enhancement kubwa. Kwa mwingiliano wa kusukumana, \(\zeta<0\) ikiwa, exponential suppression hutokea katika kasi ndogo.

Sommerfeld resonance ni nini?

Potensheli ya Yukawa ya kuvutana inapoleta bound state moja kwa moja kwenye threshold, scattering wavefunction inaweza kuongezeka sana katika umbali mfupi. Pointi hizi za parameter huitwa Sommerfeld resonance au threshold resonance. Katika chanzo, resonance ya kwanza ya \(s\)-wave:

\[ \xi_{1,0} \simeq 0.84 \]

iko karibu na thamani hii.

Katika exact resonance, \(s\)-wave Sommerfeld factor inaweza kukua zaidi ya Coulomb enhancement ya kawaida katika kasi ndogo. Kwa hiyo baseline annihilation calculation inaweza kukaribia kikomo cha unitarity katika kasi ndogo vya kutosha hata ikiwa coupling ni ndogo.

Unitarity Inakiukwa Vipi Feedback Ikipuuzwa?

Katika Kielelezo 9, chanzo kinaonyesha kwamba kwa \(\alpha_\varrho=0.1\) na exact resonance ya kwanza ya \(s\)-wave, baseline normalized annihilation cross section:

\[ y_{0,\rm unreg}^{\rm Ann} > \frac14 \]

inaweza kuingia takriban katika eneo ambalo:

\[ v_{\rm rel}\lesssim50~{\rm km/s} \]

chini ya thamani hiyo.

Hii haimaanishi kwamba theory ya kimwili kweli inakiuka unitarity. Maana yake ni kwamba baseline approximation iliyohesabiwa kwa Hermitian incoming-state dynamics pekee na inayopuuza annihilation feedback si self-consistent katika eneo hili.

Kwa Nini Contact-Like Annihilation Inahitaji Renormalization?

Kwa kuwa leading nonrelativistic annihilation vertex hutenda kama contact-like, ikiwa huru na momentum au ikiwa katika nguvu chache za momentum, kurudiwa kwake ndani ya self-energy kernel hadi arbitrarily high off-shell momenta huzalisha integrali zenye ultraviolet divergence; kwa hiyo utafiti hufafanua absorptive na dispersive couplings za masafa mafupi kupitia renormalization conditions.

Tatizo la UV

Leading contact approximation:

\[ \widehat{\mathcal A}_\ell(p) \propto p^\ell \]

ikiwa itaendelezwa hadi \(p\) kubwa bila kikomo, continuum integral ndani ya projected Green function huwa UV divergent.

Kwa \(\ell=0\), sehemu divergent haitegemei external momentum na inaweza kufyonzwa na complex contact coupling moja, yaani renormalization conditions mbili za real.

Kwa \(\ell\geq1\), ikiwa leading derivative expansion itaendelezwa hadi momentum kubwa bila kikomo, derivative counterterms za juu zaidi zinahitajika. Hata hivyo, chanzo kinaonyesha kwamba tabia halisi ya UV ya full irreducible amplitude ni:

\[ \widehat{\mathcal A}_\ell(p) \propto p^{-\ell-2}, \qquad p\rightarrow\infty \tag{5.1} \]

na kwamba hii hurejesha convergence ya UV ya integral.

Tofauti hii ni muhimu: divergence si infinity ya lazima ya theory ya kimwili, bali ni matokeo ya kuendeleza low-energy contact approximation nje ya eneo lake la uhalali.

Renormalized quantities za \(s\)-wave

Kwa \(s\)-wave, chanzo kinatumia:

\[ w_0(k;k_\diamond) = \frac{ y_{0,\rm unreg}^{\rm ren}(k;k_\diamond) }{ [1+\beta_0(k_\diamond)z_0(k;k_\diamond)]^2 + z_0^2(k;k_\diamond) } \tag{5.2a} \]

na:

\[ \widetilde w_0(k;k_\diamond) = y_{0,\rm unreg}^{\rm ren} \frac{ \beta_0+ (1+\beta_0^2)z_0 }{ (1+\beta_0z_0)^2+z_0^2 } \tag{5.2b} \]

misemo hii.

Hapa:

\[ y_{0,\rm unreg}^{\rm ren} = [\eta_{A,0}^{\rm ren}]^2 |\widehat{\mathcal M}_{0,\rm unreg}|^2, \tag{5.3a} \]

\[ z_0 = [\eta_{A,0}^{\rm ren}]^2 W_0^{\rm fin}, \tag{5.3b} \]

\[ \beta_0 = \frac{ \eta_{H,0}^{\rm ren} }{ [\eta_{A,0}^{\rm ren}]^2 }. \tag{5.3c} \]

\(\eta_A\) huwakilisha absorptive short-distance coupling, \(\eta_H\) dispersive Hermitian contact interaction, na \(W_0^{\rm fin}\) sehemu finite iliyorenormalishwa ya projected Green function.

Subtraction scale

Chanzo huchagua kama benchmark:

\[ k_\diamond = \mu\alpha_\varrho, \qquad \zeta_\diamond=1 \tag{5.12} \]

.

Scale hii inalingana na momentum scale ambapo long-range dynamics huanza kuwa nonperturbative.

Benchmark ya chanzo

Kwa absorptive coupling, contribution huru ya contact \(\lambda_{X\varrho}\) huzimwa katika benchmark:

\[ \eta_{A,0}^{\rm ren}(k_\diamond) \rightarrow \frac{ \alpha_\varrho }{ 2(1-\tau^2/8) }. \tag{5.13a} \]

Kwa dispersive coefficient, badala yake:

\[ \beta_0(k_\diamond)=0. \tag{5.13b} \]

huchaguliwa.

Uchaguzi huu haumaanishi dispersive correction ni sifuri katika momenta zote. Wakati \(k\neq k_\diamond\), athari nonzero ya dispersive hujitokeza tena kupitia \(z_0\), na subtraction scale ikibadilishwa \(\beta_0\) kwa ujumla pia huwa nonzero kwa sababu ya running coupling.

Benchmark regulated cross sections

Chini ya benchmark hii, chanzo kinatumia:

\[ x_{0,\rm reg}^{\rm elas} = \frac{ x_{0,\rm unreg}^{\rm elas} + \left[ z_0 \sqrt{x_{0,\rm unreg}^{\rm elas}} \pm y_{0,\rm unreg}^{\rm ren} \sqrt{1-x_{0,\rm unreg}^{\rm elas}} \right]^2 }{ (1+y_{0,\rm unreg}^{\rm ren})^2+z_0^2 } \tag{5.15a} \]

na:

\[ y_{0,\rm reg} = \frac{ y_{0,\rm unreg}^{\rm ren} }{ (1+y_{0,\rm unreg}^{\rm ren})^2+z_0^2 } \tag{5.15c} \]

formula hizi.

Katika denominator:

\[ (1+y_{0,\rm unreg}^{\rm ren})^2 \]

Athari ya absorptive flux loss; \(z_0^2\) hubeba athari ya dispersive correction kwenye phase dynamics. Katika elastic numerator kuna interference kati ya baseline amplitude na absorptive/dispersive corrections.

Feedback Hubadilishaje Cross Sections Katika Sommerfeld Resonance?

Katika exact \(s\)-wave resonance, baseline elastic cross section hukaribia unitarity limit katika kasi ndogo huku baseline annihilation ikikua hata kwa kasi zaidi; annihilation feedback inapofanyiwa resummation, probability-flux conservation hukata ukuaji huu, hufanya regulated elastic na annihilation rates kuwa finite, na hupunguza sana unyeti uliopitiliza kwa resonance detuning ndogo.

Exact resonance

Kwa mujibu wa chanzo, katika kasi ndogo:

\[ x_{0,\rm unreg}^{\rm elas} \rightarrow1, \]

lakini:

\[ y_{0,\rm unreg}^{\rm ren} \propto k^{-1} \propto \zeta. \]

Baseline annihilation quantity hivyo huendelea kukua katika kasi ndogo.

Feedback inapokuwa muhimu:

\[ y_{0,\rm unreg}^{\rm ren} \simeq1 \]

karibu na thamani hiyo regulated inelastic quantity hufikia global maximum:

\[ y_{0,\rm reg} \simeq\frac14, \]

na coupled unitarity wakati huohuo:

\[ x_{0,\rm reg}^{\rm elas} \simeq\frac14 \]

hulazimisha thamani hii.

Katika kasi za chini zaidi:

\[ y_{0,\rm unreg}^{\rm ren}\gg1 \]

ikiwa:

\[ x_{0,\rm reg}^{\rm elas} \propto \zeta^{-2}, \qquad y_{0,\rm reg} \propto \zeta^{-1}. \]

Unitarity cross section:

\[ \sigma_0^{\rm U} \propto k^{-2} \]

kwa kuwa hivyo, regulated physical elastic cross section na \(\sigma^{\rm Ann}v_{\rm rel}\) hukaribia thamani thabiti katika kasi ndogo. Yaani normalized cross section inapopungua, scale ya unitarity ya \(k^{-2}\) husawazisha upungufu huo na inaweza kuunda plateau katika physical cross section.

Mkengeuko mdogo sana kutoka resonance

Katika baseline calculation, mabadiliko madogo sana ya \(\xi\) yanaweza kubadilisha low-velocity cross section kwa orders of magnitude nyingi. Feedback inapojumuishwa, interference ya absorptive na dispersive corrections pamoja na denominator regularization hupunguza unyeti huu uliopitiliza wa fine-tuning.

Katika Kielelezo 11, chanzo:

• \(\alpha_\varrho=0.1\),
• resonance ya kwanza ya \(s\)-wave,
• \(v_{\rm rel}=1\) km/s na 10 km/s

hulinganisha masharti haya. Katika exact resonance, regulated elastic cross section inaweza kuwa orders of magnitude ndogo kuliko baseline; resonance ikiachwa nyuma, hesabu hizi mbili hukaribiana kwa haraka.

Benchmark ya viscosity cross section

Kwa Kielelezo 13, chanzo hutumia:

• \(\alpha_\varrho=0.1\),
• \(m_X=100\) GeV,
• \(m_\varrho\simeq5.95\) GeV,
• \(\xi_{1,0}\simeq0.84\),
• exact resonance na detuning ya \(\Delta\xi=10^{-4}\)

.

Katika exact resonance, baseline viscosity cross section huendelea kukua katika kasi ndogo, huku regulated result ikiunda finite plateau. Kwa detuning ndogo sana, baseline tayari huelekea finite plateau; lakini regularization hupunguza sana urefu wa plateau. Regulated results za tabia hizi mbili tofauti za baseline hukaribiana sana katika kasi ndogo.

Kwa Nini Feedback Inaweza Kuongeza Cross Section Kwenye Elastic Scattering Zero?

Baseline elastic amplitude inapokwenda sifuri, annihilation amplitude kwa ujumla si sifuri; kwa hiyo coupled unitarity hairuhusu elastic cross section ibaki sifuri kabisa. Incoming-state self-energy feedback ya njia ya annihilation hubadilisha baseline zero kuwa minimum isiyo sifuri na, katika exact \(s\)-wave antiresonance, hufanya physical elastic cross section ya kasi ndogo ikaribie finite plateau.

Elastic zeros na antiresonance

Baseline:

\[ x_{0,\rm unreg}^{\rm elas} = \sin^2\theta_0 \]

kwa kuwa:

\[ \theta_0=n\pi \]

elastic cross section huwa sifuri katika pointi hii.

Katika attractive Yukawa potential, zeros hizi zinaweza kukaribia thamani fulani finite za \(\xi\) kadiri kasi inavyopungua. Chanzo huita mipaka hii ya zero-momentum antiresonance na kuihusisha na Ramsauer–Townsend suppression.

Elastic floor inayolazimishwa na unitarity

Kwa \(y_{\ell,\rm reg}\) ndogo, coupled inequality ni:

\[ y_{\ell,\rm reg}^{\,2} \lesssim x_{\ell,\rm reg} \lesssim 1-2y_{\ell,\rm reg}. \tag{6.1} \]

Chanzo kinapoweka \(x_{0,\rm unreg}^{\rm elas}=0\) katika baseline elastic zero:

\[ x_{0,\rm reg}^{\rm elas} = y_{0,\rm unreg}^{\rm ren} y_{0,\rm reg} \geq y_{0,\rm reg}^2 \tag{6.2} \]

hupata matokeo haya.

Katika kikomo cha correction dhaifu:

\[ x_{0,\rm reg}^{\rm elas} \simeq \left( y_{0,\rm unreg}^{\rm ren} \right)^2 \simeq y_{0,\rm reg}^2. \tag{6.3} \]

Huu ndio msingi wa kihisabati wa hitimisho la makala kwamba ‘inelasticity inaweza wakati mwingine kuongeza elastic scattering’.

Scaling ya kasi ndogo katika exact antiresonance

Baseline:

\[ x_{0,\rm unreg}^{\rm elas} \propto k^6 \propto \zeta^{-6}. \tag{6.4} \]

Annihilation quantity:

\[ y_{0,\rm unreg}^{\rm ren} \propto k \propto \zeta^{-1}. \tag{6.5} \]

Baada ya feedback:

\[ x_{0,\rm reg}^{\rm elas} \simeq \left( y_{0,\rm unreg}^{\rm ren} \right)^2 \propto k^2 \propto \zeta^{-2}. \tag{6.6} \]

Kwa sababu:

\[ \sigma_0^{\rm U}\propto k^{-2}, \]

baseline physical elastic cross section:

\[ \sigma_{0,\rm unreg}^{\rm elas} \propto k^4 \rightarrow0 \]

ilhali regulated physical cross section:

\[ \sigma_{0,\rm reg}^{\rm elas} \sim {\rm constant} \]

hivyo ndivyo inavyokuwa.

Kwa hiyo feedback haibadilishi tu zero kuwa namba ndogo; pia hubadilisha low-momentum asymptotic scaling.

Je, Antiresonance ni ya Manufaa Kiotomatiki kwa SIDM?

Hapana. Chanzo kinasisitiza kwamba velocity dependence karibu na antiresonance inaweza kuwa kinyume na lengo la kawaida la SIDM. Ingawa feedback huunda finite plateau katika kasi ndogo, antiresonant suppression hudhoofika kasi inapoongezeka na cross section inaweza kukua. Baadhi ya parameter zinazozalisha cross section kubwa vya kutosha katika dwarf scale zinaweza kutoa scattering kubwa zaidi katika cluster scale na kugongana na cluster benchmark.

Kwa Nini Higher Partial Waves Ni Muhimu Kidogo Katika Utafiti Huu?

Partial waves za \(\ell\geq1\) hukandamiza wavefunction ya umbali mfupi kwa sababu ya centrifugal barrier; kwa hiyo annihilation enhancement katika threshold resonance na feedback yake havikui kuelekea kasi ndogo kwa kiwango sawa na \(s\)-wave, na narrow finite-velocity quasi-bound-state peaks zinaweza kufutwa kwa kiasi kikubwa na halo velocity averaging.

Katika Appendix D, chanzo kinatoa kwa threshold resonance:

\[ x_{\ell,\rm unreg}^{\rm elas} \propto k^{4\ell-2}, \qquad \ell\geq1 \]

scaling hii.

Katika modeli hii higher resonance ya kwanza ya kuvutana ni \(d\)-wave, yaani \(\ell=2\):

\[ x_{2,\rm unreg}^{\rm elas} \propto k^6. \]

Katika pointi hiyo hiyo generic off-resonant \(s\)-wave:

\[ x_{0,\rm unreg}^{\rm elas} \propto k^2 \]

hutenda hivyo na huwa dominant zaidi kuliko \(d\)-wave katika kasi ndogo.

Kwa annihilation, chanzo katika threshold resonance hupata:

\[ y_{\ell,\rm unreg} \propto k^{2\ell-3}, \]

. Kwa hiyo:

\[ y_{2,\rm unreg} \propto k, \]

hupungua katika kasi ndogo, huku katika \(s\)-wave resonance:

\[ y_{0,\rm unreg} \propto k^{-1} \]

hukua.

Chanzo hakifanyi full numerical unitarization kwa higher-partial-wave feedback. Appendix D inajadili jinsi hili lingeweza kufanywa na kueleza kwamba narrow finite-velocity peaks zinatarajiwa kukatwa na feedback; lakini hili si final numerical result iliyohesabiwa moja kwa moja katika makala hii.

Cross Section Hupewa Wastani Vipi Ndani ya Halo?

Kwa kuwa halo ya astrofizikia haiundwi na chembe zenye relative velocity moja, microscopic cross section lazima iwekwe wastani juu ya velocity distribution. Utafiti hutumia isotropic Maxwellian distribution iliyokatwa kwenye escape speed ili kuhesabu velocity-weighted cross sections zinazohusiana na elastic, momentum-transfer na transverse-energy-transfer.

Katika Appendix B, chanzo kinafafanua single-particle velocity distribution kama:

\[ F(v,v_c,v_{\rm esc}) = \mathcal N(v_c,v_{\rm esc}) \Theta(v_{\rm esc}-v) e^{-v^2/v_c^2} \tag{B.1} \]

.

\(v_c/\sqrt2\) ni one-dimensional velocity-dispersion parameter ya untruncated Maxwellian inayolingana nayo.

Katika uchanganuzi wa phenomenology, chanzo hukadiria:

\[ v_{\rm esc}\simeq\sqrt2\,v_c \]

.

Velocity-weighted cross sections zinazotumika kwa halo transport

\[ \overline{\sigma}^{\rm elas} = \frac{ \langle \sigma^{\rm elas}v_{\rm rel} \rangle }{ \langle v_{\rm rel}\rangle }, \]

na kwa namna sawa \(\overline{\sigma}^{T}\) na \(\overline{\sigma}^{V}\), zenye velocity moments za juu kwa momentum na transverse-energy transport, hufafanuliwa.

Katika sehemu ya halo phenomenology ya utafiti, viscosity cross section hutumika zaidi.

Ukaguzi exact wa velocity average

Katika Appendix B, kwa kutumia isotropy na velocity distributions huru za chembe mbili, chanzo hupata:

\[ \langle v_{\rm rel}^2\rangle = 2 \langle v_{\rm DM}^2\rangle \tag{B.11} \]

uhusiano exact huu.

Feedback ni Muhimu Kwa Kiasi Gani Katika SIDM Benchmarks za Kiwango cha Halo?

Uchunguzi wa parameter space wa utafiti unaonyesha kwamba annihilation feedback haibadilishi upya parameter space pana kwa ujumla; mabadiliko hujikita hasa katika bendi za \(s\)-wave resonance na antiresonance, na katika maeneo ya resonant SIDM ambapo annihilation hubaki coupling-suppressed kwa thamani ndogo za \(\alpha_\varrho\), athari zinaweza kuwa ndogo zaidi.

Phenomenological benchmarks zilizotumika

Chanzo kinasisitiza kwamba vigezo hivi si universal bounds. Uhusiano kati ya microscopic scattering na halo evolution hutegemea mfumo, observable, na transport process gani ni muhimu.

Symmetric dark matter

Kwa symmetric DM, chanzo:

\[ n_X=n_{X^\dagger} = \frac{n_{\rm DM}}{2} \]

hukubali na:

\[ \sigma_{\rm SIDM}^{V} = \frac12 \left( \sigma_{XX^\dagger}^{V} + \sigma_{XX}^{V} \right). \tag{6.8} \]

Katika utafiti huu, contribution ya Yukawa ya \(XX^\dagger\) inachunguzwa. Katika kikomo cha global-\(U(1)_D\), hakuna baseline long-range elastic amplitude ya \(XX\). Mtawanyiko wa \(XX\) unaweza kuzalishwa na radiative bound-state formation feedback; lakini contribution hii imeachwa kwa companion paper na inapuuzwa hapa.

Uchambuzi wa parameter space wa Kielelezo 15

Chanzo kinalinganisha parameter planes mbili:

• plane ya \(m_X-\alpha_\varrho\) kwa \(m_\varrho=1\) MeV iliyowekwa thabiti,
• plane ya \(m_X-m_\varrho\) kwa \(\alpha_\varrho=0.1\) iliyowekwa thabiti.

Katika kila plane, baseline Yukawa result inalinganishwa na regulated result inayojumuisha \(s\)-wave annihilation feedback.

Karibu na antiresonance, feedback hujaza baseline \(s\)-wave suppression na kukata baadhi ya narrow parameter-space excursions. Kwa \(m_X\) kubwa, higher partial waves huvunja muundo wa antiresonance, hivyo umuhimu wa jamaa wa feedback hupungua.

Katika maeneo ya resonance, feedback hukata enhancement baseline \(s\)-wave contribution inapokaribia unitarity limit. Hata hivyo, kwa kuwa baadhi ya resonance bands zinazohusiana na SIDM benchmarks ziko kwenye \(\alpha_\varrho\) ndogo, annihilation inaweza kubaki coupling-suppressed huko na uhamisho wa eneo la parameter space unaweza kuwa mdogo kuliko katika eneo la antiresonance.

Mbinu na Matokeo ya Utafiti

Mtiririko wa uchanganuzi

1. Partial-wave elastic–inelastic coupled bounds ziliandikwa kutoka S-matrix unitarity.
2. Michango ya anti-Hermitian na dispersive ya michakato isiyo elastiki kwenye incoming two-particle self-energy kernel ilifafanuliwa.
3. General unitarization relations zilitumika kwa full non-Hermitian effective potential.
4. Elastic, transfer na viscosity cross sections zilitenganishwa kwa partial waves.
5. Truncated isotropic Maxwellian distribution ilifafanuliwa kwa halo velocity averaging.
6. Modeli ilijengwa ikiwa na complex scalar \(X\), scalar mediator \(\varrho\), na awali dark gauge field.
7. Uchanganuzi wa phenomenology ulipunguzwa hadi kikomo cha global-\(U(1)_D\) na potensheli ya Yukawa ya \(X X^\dagger\) ikatumika.
8. Baseline Yukawa phase shifts na Sommerfeld factors zilipatikana kwa numerical variable-phase method.
9. Annihilation amplitudes za \(XX^\dagger\rightarrow\varrho\varrho^\dagger\) zilitolewa.
10. Ilionyeshwa kwamba baseline annihilation inayopuuza feedback inaweza kuzidi unitarity bound karibu na resonance.
11. Contact-like \(s\)-wave annihilation contribution ilifanyiwa renormalization.
12. Benchmark subtraction scale na short-distance couplings zilichaguliwa.
13. Regulated elastic, viscosity na annihilation cross sections zilihesabiwa.
14. Detuning na velocity dependence karibu na \(s\)-wave resonance ya kwanza zilichunguzwa.
15. Asymptotic scaling karibu na elastic scattering zero/antiresonance ilichunguzwa.
16. Higher \(s\)-wave resonances na short-distance coupling dependence zilijadiliwa.
17. Halo-averaged viscosity cross section ililinganishwa kwenye parameter space dhidi ya cluster/dwarf/subhalo benchmarks.

Benchmarks na namba muhimu

Matokeo makuu 1 — feedback kwa kawaida ni ndogo nje ya resonance

Ikiwa baseline annihilation cross section iko mbali na partial-wave unitarity limit, absorptive na dispersive corrections hubaki ndogo. Katika hali hii:

\[ x_{\rm reg}\simeq x_{\rm unreg}, \qquad y_{\rm reg}\simeq y_{\rm unreg}^{\rm ren}. \]

Kwa hiyo standard Hermitian Yukawa calculation hubaki approximation nzuri katika maeneo mapana ya parameter space.

Matokeo makuu 2 — low-velocity growth hukatwa katika resonance

Katika Sommerfeld resonance, baseline annihilation inapokua na kuvunja self-consistency, feedback hurudisha elastic na inelastic cross sections kwenye eneo la coupled unitarity. Physical elastic cross section na annihilation rate huonyesha finite behavior badala ya uncontrolled growth katika kasi ndogo.

Matokeo makuu 3 — unyeti wa parameter fine-tuning hupungua

Katika baseline calculation, mabadiliko madogo sana ya \(\xi\) karibu na resonance au antiresonance yanaweza kubadilisha cross sections kwa orders of magnitude. Tofauti hii hupungua sana katika regulated results. Kwa kuwa self-energy feedback hubadilisha finely tuned pole/zero structure, parameter points zilizo karibu sana hutoa physical predictions zinazofanana zaidi.

Matokeo makuu 4 — elastic zero haibaki sifuri kabisa

Kwa muda annihilation inabaki nonzero, coupled unitarity inakataza exact elastic zero. Feedback hubadilisha baseline zero kuwa minimum isiyo sifuri. Matokeo haya yanatokana moja kwa moja na probability conservation.

Matokeo makuu 5 — mabadiliko katika SIDM parameter space yamejikita kwenye maeneo maalum

Halo-averaged results zinaonyesha kwamba annihilation feedback haibadilishi parameter space yote kwa kiwango kikubwa. Athari zinazoonekana zaidi ziko karibu na mistari ya \(s\)-wave resonance na antiresonance. Higher partial waves zinapokuwa muhimu zaidi, relative contribution ya \(s\)-wave feedback pekee hupungua.

Matokeo yanayoungwa mkono na utafiti

• Elastic na inelastic partial-wave scattering si huru kwa sababu ya coupled unitarity.
• Open annihilation channels huzalisha absorptive contribution kwenye incoming two-particle self-energy kernel.
• Long-range attractive Yukawa force inaweza ku-enhance annihilation kwa nguvu kupitia Sommerfeld effect.
• Katika exact \(s\)-wave resonance, baseline annihilation inayopuuza feedback inaweza kuzidi unitarity bound katika velocity ndogo vya kutosha.
• Resummation ya annihilation kernel hurejesha unitarity.
• Karibu na Sommerfeld resonance, feedback inaweza kukandamiza low-velocity elastic enhancement.
• Karibu na elastic scattering zero, feedback inaweza kuzalisha nonzero elastic contribution.
• Feedback hupunguza extreme parameter sensitivity karibu na resonance na antiresonance.
• Katika benchmarks zilizochunguzwa, baseline Hermitian result kwa kawaida ni approximation nzuri mbali na maeneo ya resonance/zero.
• Katika halo parameter-space analysis, athari kubwa za feedback hujikita hasa karibu na miundo ya \(s\)-wave resonance na antiresonance.

Matokeo ambayo utafiti hauungi mkono

• Makala hairipoti ugunduzi wa moja kwa moja au usio wa moja kwa moja wa mada nyeusi kwa majaribio.
• Haionyeshi kwamba scalar dark-sector model iliyochunguzwa ndiyo inayotokea kiasili.
• Haithibitishi kwamba galaxy au subhalo fulani iliundwa kwa sababu ya annihilation feedback.
• Thamani za cross section zinazotumika kwa cluster, dwarf na subhalo si mipaka sahihi ya ulimwengu mzima kwa mifumo yote.
• Utafiti hauchunguzi modeli zote zinazowezekana za SIDM.
• Radiative bound-state formation feedback haijajumuishwa katika numerical halo result ya makala hii.
• Hali ya jumla ya gauge interaction \(\alpha_V\neq0\) haichunguzwi katika main phenomenological scan.
• Higher-partial-wave annihilation feedback haifanyiwi full numerical unitarization.
• Benchmark \(\beta_0(k_\diamond)=0\) si thamani ya kimwili inayolazimika katika microscopic models zote.
• Thamani zote zinazowezekana za short-distance couplings \(\lambda_{X\varrho}\) na \(\lambda_X\) hazijachunguzwa.

Vikwazo vya kimbinu na wigo

• Utegemezi wa modeli: matokeo ya quantitative hutegemea modeli iliyochaguliwa ya complex-scalar \(X\) na scalar mediator \(\varrho\).
• Global-\(U(1)_D\) simplification: uchanganuzi mkuu unapuuza gauge-mediated interaction.
• Mkabala wa \(s\)-wave feedback: katika halo scan, annihilation feedback inatumika tu kwa resonant \(s\)-wave huku higher partial waves zikiachwa kwenye baseline values.
• Short-distance benchmark: benchmark prescription moja hutumika kwa absorptive na dispersive renormalized coefficients; comprehensive coupling scan haifanywi.
• Mgawanyo wa BSF: radiative bound-state formation inashughulikiwa katika companion paper tofauti.
• Halo distribution: isotropic truncated Maxwellian velocity distribution hutumika; real halo velocity distributions zinaweza kutofautiana nayo.
• Phenomenological benchmark: thamani za cluster/dwarf/subhalo hutumika kama representative SIDM scales, si universal constraints.
• Higher-wave calculation: Appendix D inajadili scaling na tabia inayotarajiwa badala ya full higher-partial-wave feedback calculation.

Matokeo ya short-distance annihilation ya Appendix C

Chanzo huandika full short-distance amplitude kwa kuchukua incoming \(X X^\dagger\) particles kuwa off-shell na outgoing \(\varrho\varrho^\dagger\) pair kuwa on-shell kama:

\[ \mathcal A(p,k_\varrho) = \frac{ 16\pi\alpha_\varrho m_X^2 }{ s/4+m_X^2-m_\varrho^2+p^2-2pk_\varrho\cos\vartheta } - \lambda_{X\varrho}, \tag{C.1} \]

.

Partial-wave expansion ya expression hii inaeleza leading contact approximation inatoka wapi na kwa nini leading polynomial behavior pekee haipaswi kutumika katika high momentum.

Nonrelativistic leading absorptive coupling:

\[ \eta_{A,\ell} \simeq \frac{\alpha_\varrho}{2} \frac{\ell!^2}{(2\ell+1)!} \left( 1-\frac{\tau^2}{4} \right)^{\ell/2} \left( 1-\frac{\tau^2}{8} \right)^{-\ell-1} - \frac{\lambda_{X\varrho}}{16\pi} \delta_{\ell0}. \tag{C.8} \]

Mlinganyo huu unaonyesha kwamba \(s\)-wave inaweza kupata additional annihilation contribution kutoka independent biquadratic contact interaction \(\lambda_{X\varrho}\).

Ni upanuzi gani wa baadaye ulio wa asili katika modeli hiyo hiyo?

Katika sehemu ya hitimisho, chanzo kinasisitiza kwamba annihilation na radiative BSF zinaweza kuwa muhimu kwa wakati mmoja kwa incoming state ileile. Gauge interaction inapowashwa tena, jozi za \(X X^\dagger\) zinaweza annihilate na pia kufanya radiative bound-state formation kupitia vector emission. Ku-unitarize pamoja inelastic contributions hizi mbili ndani ya self-energy ya pamoja kunatajwa kama mwendelezo wa asili wa utafiti.

Maelezo ya Chanzo na Mbinu

Kichwa asili: Annihilation feedback on elastic scattering in self-interacting dark matter

Waandishi: Kalliopi Petraki, Thomas Pignard.

Mpangilio wa waandishi: Kalliopi Petraki → Thomas Pignard.

Mchango sawa: Hakuna tamko la mchango sawa/uandishi wa kwanza wa pamoja lililotambuliwa katika preprint iliyopakiwa.

Mwandishi wa mawasiliano: Chanzo kinatoa anwani za barua pepe za waandishi wote wawili kwa footnotes za nyota na dagger; pia hakitumii lebo moja ya ‘corresponding author’. Kwa hiyo Verianla hamtangazi kiholela mwandishi mmoja tu kuwa mwandishi wa mawasiliano.

Barua pepe: kalliopi.petraki@phys.ens.psl.eu; thomas.pignard@phys.ens.psl.eu.

Taasisi: Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité, Paris, F-75005, France.

Jukwaa: arXiv.

Kitambulisho cha arXiv: arXiv:2610.27056v1 [hep-ph].

Makundi mengine: Cosmology and Nongalactic Astrophysics (astro-ph.CO); Astrophysics of Galaxies (astro-ph.GA).

Tarehe ya kuwasilishwa arXiv: 19 Oktoba 2026.

Tarehe inayoonekana kwenye PDF: 21 Oktoba 2026.

DOI: 10.48550/arXiv.2610.27056; ni kitambulisho cha arXiv/DataCite.

Aina ya chanzo: Preprint ya nadharia na numerics ya high-energy physics / dark-matter phenomenology.

Hali ya peer review: Katika ukaguzi wa bibliografia uliofanywa kufikia 4 Novemba 2026, hakuna toleo la jarida lililopitiwa na wahakiki lililothibitishwa lililotambuliwa. Chanzo kinapaswa kutathminiwa kama arXiv preprint.

Kurasa / vielelezo: Kurasa 60, vielelezo 17.

Leseni: Hakuna tamko wazi la Creative Commons au leseni nyingine mahususi ya makala lililotambuliwa katika PDF iliyopakiwa; Verianla haipaswi kudhani aina ya leseni.

Mbinu kuu: Partial-wave unitarity na optical theorem; generally non-Hermitian two-particle self-energy kernel; Schrödinger-equation resummation; Yukawa scattering; numerical phase shifts na Sommerfeld factors; short-distance annihilation amplitude; contact-interaction renormalization; regulated/unregulated cross-section comparison; truncated-Maxwellian halo velocity averaging.

Modeli: Dark sector yenye complex scalar dark matter \(X\), complex scalar mediator \(\varrho\) na awali dark Abelian gauge field \(V_\mu\); uchanganuzi mkuu wa phenomenology umepunguzwa hadi scalar-mediated Yukawa interaction katika kikomo cha global-\(U(1)_D\).

Mchakato mkuu usio elastiki: \(X+X^\dagger\rightarrow\varrho+\varrho^\dagger\).

Msingi mkuu wa numerical Yukawa: Phase shifts, Sommerfeld factors na baseline numerical functions zinazohusiana zinategemea mbinu ya ‘Variable-phase methods and data for long-range scattering’ iliyotajwa kama Ref. [37] katika chanzo.

Ufadhili: European Union Horizon 2020 research and innovation program, grant agreement No. 101002846, ERC CoG CosmoChart.

Shukrani: Waandishi wanamshukuru Marcos M. Flores kwa majadiliano na ushirikiano kuhusu numerical framework ya baseline Yukawa calculations.

Upatikanaji wa data/kodi: Preprint hii haina ‘Data Availability Statement’ tofauti. Sehemu ya utangulizi inaeleza kwamba numerical method, code na data zinazotumika katika baseline Yukawa problem zimewasilishwa katika Ref. [37].

Mgongano wa maslahi: Hakuna tamko tofauti la competing-interest/conflict-of-interest lililotambuliwa katika preprint iliyopakiwa.

Michango ya waandishi: Preprint iliyopakiwa haina sehemu tofauti ya CRediT/michango ya waandishi.

Hitimisho kuu: Annihilation feedback huregulate baseline low-velocity elastic na annihilation growth karibu na Sommerfeld \(s\)-wave resonance; karibu na baseline elastic-scattering zero, kutokana na coupled unitarity, huzalisha nonzero elastic contribution.

Kikomo kikuu cha tafsiri: Matokeo yanahusu theoretical/numerical scattering calculation ya dark-sector model iliyofafanuliwa. Astrophysical halo benchmarks si ugunduzi wa moja kwa moja wa mada nyeusi kwa majaribio wala uthibitisho wa microscopic model ya galaxy fulani.

Kikomo cha higher-partial-wave: Main phenomenological regularization inajumuisha tu \(s\)-wave annihilation feedback. Higher partial waves zimeachwa na baseline phase shifts; Appendix D inajadili tabia inayotarajiwa ya full higher-wave unitarization.

Kikomo cha BSF: Radiative bound-state formation feedback haijajumuishwa katika halo results za utafiti huu; ni mada ya companion paper.