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Schematic-depiction-of-catalyzed-baryogenesis-facilitated-by-an-electroweak-symmetric

https://www.researchgate.net/figure/Schematic-depiction-of-catalyzed-baryogenesis-facilitated-by-an-electroweak-symmetric_fig1_355399856

Схематическое изображение катализируемого бариогенеза, облегченного электрослабым симметричным шаром. По мере того, как шар движется через плазму, барионная асимметрия накапливается около его стенок из-за CP-нарушающего взаимодействия на стенке пузырька. Это зависит от относительной скорости плазмы и шара, так что больше антибарионов (синий) и барионов (оранжевый) приближается к стенкам с правой стороны, чем с левой. Затем асимметрия диффундирует из областей около стенки, но из-за относительной скорости между шаром и плазмой частицы около переднего края имеют тенденцию диффундировать внутрь, в то время как частицы около заднего края имеют тенденцию диффундировать наружу (обозначено зелеными пунктирными линиями). Это создает чистую асимметрию барионного числа внутри шара. Сфалероны, которые активны только внутри шара, затем преобразуют часть барионов в лептоны, сохраняя асимметрию даже при прохождении шара.

Катализированный бариогенез

https://youtu.be/5Kuhytb-NI8

Catalyzed_baryogenesis

ДомФизика элементарных частицСтандартная модель СтатьяДоступен PDF-файл Катализированный бариогенез Октябрь 2021 г.Журнал физики высоких энергий 2021(10) DOI: 10.1007/JHEP10(2021)147 ЛицензияCC BY 4.0 Авторы: Ян Бай Университет Луисвилля Джошуа Бергер Мрунал Корвар Индийский институт научного образования и исследований, Пуна Николас Орловский

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Абстракция и фигуры Аннотация Предлагается новый механизм, «катализированный бариогенез», для объяснения наблюдаемой барионной асимметрии в нашей Вселенной. В этом механизме движение шарообразного катализатора обеспечивает необходимое неравновесное состояние, его внешняя стенка имеет CP-нарушающие взаимодействия с частицами Стандартной модели, а его внутренняя часть имеет взаимодействия, нарушающие барионное число. Мы используем модель электрослабого симметричного шара в качестве примера такого катализатора. В этой модели электрослабые сфалероны внутри шара активны и превращают барионы в лептоны. Наблюдаемая асимметрия барионного числа может быть получена для легкой массы шара и большого радиуса шара. Из-за ограничений прямого обнаружения реликтовых шаров мы рассматриваем сценарий, в котором шары испаряются, что приводит к темному излучению на проверяемых уровнях. Схематическое изображение катализируемого бариогенеза, облегченного электрослабым симметричным шаром. По мере того, как шар движется через плазму, барионная асимметрия накапливается около его стенок из-за CP-нарушающего взаимодействия на стенке пузырька. Это зависит от относительной скорости плазмы и шара, так что больше антибарионов (синий) и барионов (оранжевый) приближается к стенкам с правой стороны, чем с левой. Затем асимметрия диффундирует из областей около стенки, но из-за относительной скорости между шаром и плазмой частицы около переднего края имеют тенденцию диффундировать внутрь, в то время как частицы около заднего края имеют тенденцию диффундировать наружу (обозначено зелеными пунктирными линиями). Это создает чистую асимметрию барионного числа внутри шара. Сфалероны, которые активны только внутри шара, затем преобразуют часть барионов в лептоны, сохраняя асимметрию даже при прохождении шара. Схематическое изображение катализируемого бариогенеза, облегченного электрослабым симметричным шаром. По мере того, как шар движется через плазму, барионная асимметрия накапливается около его стенок из-за CP-нарушающего взаимодействия на стенке пузырька. Это зависит от относительной скорости плазмы и шара, так что больше антибарионов (синий) и барионов (оранжевый) приближается к стенкам с правой стороны, чем с левой. Затем асимметрия диффундирует из областей около стенки, но из-за относительной скорости между шаром и плазмой частицы около переднего края имеют тенденцию диффундировать внутрь, в то время как частицы около заднего края имеют тенденцию диффундировать наружу (обозначено зелеными пунктирными линиями). Это создает чистую асимметрию барионного числа внутри шара. Сфалероны, которые активны только внутри шара, затем преобразуют часть барионов в лептоны, сохраняя асимметрию даже при прохождении шара. … Рисунки — доступны по лицензии: Creative Commons Attribution 4.0 International Контент может быть объектом авторского права. Логотип ResearchGate Откройте для себя мировые исследования

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Доступно по лицензии: CC BY 4.0 Контент может быть объектом авторского права. JHEP10(2021)147Опубликовано для SISSA СпрингерПолучено : 9 июля 2021 г.Изменено : 27 августа 2021 г.Принято : 30 сентября 2021 г.Опубликовано : 18 октября 2021 г.Катализированный барийогенезисИанг Бай, Джошуа Бергер, Мрунал Корвar a и Николас Орловский cКафедра физики, Университет ВисконсинаОнсин-Мэдисон,Мэдисон, Висконсин 53706, СШАб Кафедра физики, цветГосударственный университет Адо,Форт colЛинс, Колорадо 80523, СШАc Физический факультет, Карлтонский университет,Оттава, ON K1S 5B6, КанадаЭлектронная почта: [email protected] , [email protected] ,[email protected] , [email protected]Аннотация: НоябрьДля объяснения наблюдаемого явления предлагается использовать механизм «катализированного бариогенеза».служил барионной асимметрии в нашей универв другом месте. В этом механизме движение шарообразной кошки...alyst обеспечивает необходимое неравновесное состояние, его внешняя стенка имеет CP-нарушающуювзаимодействует с частицами Стандартной модели, а его внутренняя часть имеет барионное числоber vio-поздние взаимодействия. Втe использовать модель электрослабосимметричного шара в качестве примера такогокатализатор. В этой модели электрослабые сфалероны внутри шара активны и конвертбарионы в лептоны. Наблюдатьed барион числоасимметрия может бытье производится для светамасса шара и большой радиус шара. Из-за ограничений прямого обнаружения реликтовых шаров мырассмотрим сценарий, в котором шары evапорат, что приводит к темному излучению на контролируемых уровнях.Ключевые слова: Космология теорий за пределами СМ, За пределами Стандартной моделиArXiv ePrint: 2106.12589Открытый доступ , c Авторы.Статья финансируется SCOAP 3 . https://doi.org/10.1007/JHEP10(2021)147 JHEP10(2021)147Содержание1 Введение 12 Катализированный бариогенез с помощью шара EWS 42.1 Состояние неравновесия 42.2 Барионное числонарушение ber 62.3 нарушение КП 73 Распространенность шаров EWS и их эволюция 83.1 Шар EWS как темная материя 83.2 Уничтожение шаров EWS 93.3 Распад шара EWS 103.4 Асимметричные шары EWS 114 Испарение шариков EWS 115 Обсуждение и выводы 13А Электрослабосимметричные шаровые модели 14А.1 EWS Q-мяч 14А.2 EWS монополе 151 ВведениеОдной из самых больших проблем в физике элементарных частиц и космологии является пониманиеасимметрия материи-антиматерии в нашей Вселенной. Точнее, модели бариогенеза должныобъясните малое (безразмерное) отношение барионов к фотонам: η = ( N барина − N антибарион ) /N γв текущей Вселенной с N барионов , N антибарионов и N γ в качестве числа барийонс, анти-барионы и фотоны соответственно. Измерение содержания легких элементов вмежгалактическая среда вместе с теорией Большого взрыва нуклеозиндиссертация (BBN) имеетη = (5 . 931 ± 0 . 051) × 10 − 10 [ 1 ], тогда как спектр температурыколебания температурыкосмическая микроволнапять фонов (CMB) имеют η = ( 6,12 ± 0,04 ) × 10 - 10 [ 2 , 3 ]. Сахаровпредложил три общих условия, необходимых для любой модели бариогенеза: i) барион nумбранарушение (BNV), ii) нарушение C и CP и iii) отклонение от теплового равновесия [ 4 ].В литературе предложено много механизмов, некоторые из которыхве феномено-логически проверяемые предсказания (например, электрослабый бариогенез [ 5 ]). В этой статье мы предлагаемструктура бариогенеза, «катализируемого бариогенезаogenesis» (CAB), где взаимодействия BNVусиливаются в определенных областях пространства. Эти области подобны «катализаторам» в химии.история, в которой определенные молекулы способствуют взаимодействию других молекул. Если катализаторы естьдостаточно долгоживущие, они могли бы дажеen быть обнаружены как реликты, составляющие темную материюдень.– 1 – JHEP10(2021)147ТЧтобы проиллюстрировать эту структуру, мы исследуем в этой работе случай, когда электроек-Симметричные (EWS) шары действуют как катализатор. Такие состояния могут возникать во многих теориях, в том числевключая нетопологические солитоны [ 6 – 8 ], темные монополе [ 9 ], или магнитно заряженные черныеотверстия [ 10 – 14 ]. Схематически, катализируемый процесс бариогенеза для данного конкретного катализаторапоказано на рисунке 1 . Внутри области катализатора электроныклювые сфалероны активныпотому что электрослабая симметрия восстанавливается, аллокрылатые барионы будут преобразованы в лептоны, в то время каквнешние сфалерон-опосредованные процессы подавляются, обеспечивая условие (i). Сфалеронытакже нарушают C, частично удовлетворяя условию (ii). Ввзаимодействия между стенкой шара EWSи входящие частицы могут нарушать CP, завершая условие (ii). Окончательно, процесс ло-категорически выходит из равновесия (iii), когда существует относительная скорость между шаром EWS иокружающая плазма. Из-за относительной скорости, больше плазменных частиц падаетна правой стороне стенок шара, чем на левой на рисунке 1 . Затем избыток антибарияонспереданные через ведущую стенку, имеют тенденцию диффундировать в шар, где находятся сфалероныактивны; между тем, избыточные барионы отражалисьy задняя стенка имеет тенденцию к диффузии изшар, где скорость сфалерона подавляется после электрослабого фазового переходация (EWPT). Таким образом, избыток антибариОны внутри мяча истощаются, в то время как избытокбарионы за пределами шара не являются таковыми, что приводит к общей асимметрии после того, как шар пролетел мимо. Кудовлетворяют условию (iii), асимметрия CP, создаваемая на стенке, должна успеть рассеятьсядостаточно широко в мяче, чтобы иметьСфалероны действуют на асимметрию CP до того, как она распадетсячерез взаимодействия, нарушающие C, но сохраняющие CP, меняющие хиральность. Фдалее, мы будемпредположим, что система остается в динамическом равновесии, с плазмой внутришар быстро рассеивается, пока шар медленно сменяет направление, рассеиваясь от плазмы. ЭтотСитуация равновесия проиллюстрирована на рисунке 1 . Картинка рушится, если мяч меняется.направление достаточно быстро, чтобы плазма внутри шара не имелавремя вернутьсяк этому динамическому равновесию между порядками один изменяет направление.CAB имеет некоторое сходство с электрослабым баримодель огенеза, основанная на строгоEWPT первого порядка. Fили в обоих случаях движение стенки пузырька или шара обеспечивает выходсостояние равновесия. В отличие от CAB, электрослабый бариогенез имеет внешнюю областьактивен для процесса сфалерона. Фдалее пузырьки расширяются, сталкиваются, просачиваются иисчезают вскоре после возникновения барионной асимметрии. Напротив, шары EWS вCAB может существовать в течение космологически длительного времени. Если они доживут до настоящего времениу,их можно было искать напрямую. В противном случае, если они еапорат, полученные частицыможет оставить космологический отпечатоктс. Фдалее, CAB не полагается ни на какие предположения относительноприрода EWPT — за исключением, возможно, температуры, при которой он происходит. В CAB,бариогенез не обусловлен бда, фазовый переход.Поскольку рассматриваемые здесь катализаторы — это EWS, взаимодействия со Стандартной моделью(SM) частицы неизбежны, что приводит к большим взаимодействиям междуeen яйца и nu-Клеи [ 8 , 15 ]. Одним из важных выводов этой работы является то, что хотя шары EWS могли бы объяснить и то, и другоебариогенез и темная материя одновременно, они бы хауже наблюдалосьэксперименты по прямому обнаружению. Таким образом, шары EWS заданной массы и радиуса могут либоиграть роль темной материи или катализатора бариогенеза, но не то и другое одновременно. Чтобы стать катализатором,они должны распасться или исчезнутьапорировать в частицы SM или частицы темного сектора в ранней вселеннойстих. Этот opens узкая область пространства параметров, где симметричная популяция– 2 – JHEP10(2021)147        скорость плазмы в шаровой системе отсчета         диффузияРисунок 1 . Схематическое изображение катализируемого бариогенеза, осуществляемого электрослабо симметричныймяч. По мере того, как шар движется через плазму, барионная асимметрия нарастает вблизи его стенок из-заВзаимодействие, нарушающее CP, на стенке пузырька. Это зависит от относительной скорости плазменного шара, так чтобольше антибарионов (синих) и бари(оранжевые) подходят к стенам с правой стороны, чем слевый. Затем асимметрия рассеивается от областей вблизи стенки, но из-за относительногоскорость между шаром и плазмой, частицы вблизи передней кромки имеют тенденцию рассеиваться впалатыв то время как частицы вблизи задней кромки имеют тенденцию диффундировать наружу (обозначено by зеленые пунктирные линии).Это создает чистое барионное числоасимметрия внутри шара. Сфалероны, которыеh активны тольковнутри шара, затем преобразуйте часть барионов вк лептонам, сохраняя асимметрию даже какмяч проходит мимо.Шары EWS могут катализировать бариогенез. Если предположить, что шары EWS сами по себе несутасимметричный компонент, пространство параметров значительно расширяется. В последнюю секунду-ции, мы также размышляем о других возможных моделях катализаторов, которые могли бы подавить прямоепоперечные сечения обнаружения, отказываясь от необходимости их окончательного evапорация.Подобные сценарии ранее изучались в работах [ 16 , 17 ], при этом особое внимание уделялосьКосмические струны EWS вместо шаров. Некоторые из наших подробных анализов следуют за лечениемтиз этих работ. Для обоих сценариев, шарового и струнного, обнаружена окончательная барионная асимметрия.быть пропорциональна квадрату скорости катализатора. Однако строки, которые они считаютимеют радиус только порядка v − 1 при v = 246 ГэВ поле Хиггса vожидаемое значение акуума(VEV), тогда как мы рассматриваем шары EWS с мукомольнымh больший радиус. Фдалее, бариогенезограничено в их случае строкой netwэволюция орков: Струны, движущиеся под действием тренияслишком медленно, тогда как в режиме масштабирования строк слишком мало. Таким образом, в то время как ссылка [ 17 ] достиглаотрицательное заключение о том, что струны катализируют бариогенез, наша работа показывает, что шары могутобъяснить наблюдаемую барионную асимметрию из-за этих различий.Наша работа также отличается от моделей, которые включают как Q-шары, так и бариогенез. ПредварительноВ некоторых работах рассматриваются Q-шары, которые образуются в результате реакции Аﬄека-Дайна.огенез [ 18 ], илив качестве альтернативы, распады Q-шаров могли бы объяснить бариогенез [ 19 ]. В отличие от фор-mer, наш катализатор — не побочный продукт, а необходимый ингредиент для бариогенеза. В отличиев последнем случае наш катализатор не модифицируется и не разрушается в целях бариогенеза.– 3 – JHEP10(2021)147Данная статья организована следующим образом. В следующем разделе жe описать механизм CABподробно, включая то, как выполняются все условия Сахарова. В разделе 3 мы синтезируемусловия для детализации допустимого пространства параметров шаров EWS как катализаторов бариогенез.Механизмы и последствия распада шаров EWS описаны в разделе 4 . Заключительный разделв разделе 5 даны заключительные замечания.2 Катализированный бариогенез с помощью шара EWSВ этом разделе будет оценена величина создаваемой барионной асимметрии. Начнем с того,предполагая, что шары EWS существуют после электрослабого фазового перехода с массой M ирадиус R . Втe do not specify the detailed constituents or initial formation mechanism ofthe EWS ball, although we will occasionally reference model-speciﬁc considerations for thenon-topological soliton that is made of a complex scalar ﬁeld Φwith a Higgs-portal couplingto the SM (see appendix Aand ref. [8] for details). We express the EWS ball abundanceas a fraction of the dark matter (DM) abundance ρ=fDM ρDM. It is convenient to writethis in terms of the yield (with s=2π245 g∗sT3the entropy density and g∗sthe number ofeﬀective relativistic degrees of freedom):Y=ns≈5.1×10−17 fDM 107GeVM!.(2.1)Here, n=ρ /M is the EWS ball number density, and fDM can be a function of temper-ature Tif the ball has some non-trivial evolution history.2.1 Out-of-equilibrium conditionFirst, we address the out-of-equilibrium condition which creates a baryon asymmetry insideof the EWS ball. Here, we assume the existence of a CP-violating interaction at the EWSball wall, which we later justify.The basic mechanism at play is that there is a localized loss of equilibrium around thewall of the EWS ball as it passes through the plasma and disturbs it. In particular, aswe will show, the wall can generate a localized CP-asymmetry in its wake, speciﬁcally adiﬀerence in the abundance of left-handed top quarks compared to right-handed anti-topquarks. Chirality-ﬂipping interactions via the weak force tend to damp this asymmetry. Solong as the asymmetry persists locally within part of the EWS ball, the out-of-equilibriumcondition is satisﬁed. In the remainder of this subsection, we determine the chemicalpotential for this CP asymmetry using the diﬀusion equation.The diﬀusion equation for the CP asymmetric number density of fermions in the pres-ence of a CP-violating source is given by [20–22]∂nCP (~x, t)∂t −D∇2nCP (~x, t)+ΓnCP(~x, t) = S(~x, t).(2.2)Here, Dis the fermion diﬀusion constant and Γis the rate of chirality damping, includingspin-ﬂip interactions (for the top quark, D≈6/T and Γ≈T /100 [23]). As the EWSball moves in the plasma, before the ball changes directions, the incoming particle source– 4 – JHEP10(2021)147comes from one direction. In the limit of large ball radius with RpD/Γ, one canapproximately treat the problem as a one-dimensional problem. Provided that any changein the direction and velocity of the EWS ball rapidly leads to a quasi-static solution, wecan seek a steady state solution in which there is a ﬁxed moving source with velocity vw∼pT/M for the balls thermalized with plasma. The solution is nCP (x, t) = nCP (x−vwt),and the equation can be reduced to a single variable x:(−vw∂x−D ∂2x)nCP + Γ nCP =S(x).(2.3)This equation can be solved using a Green’s function [the solution when S(x) = δ(x−x0)]given byG(x−x0) = D−1k+−k−exp[−k+(x−x0)] x>x0exp[−k−(x−x0)] x<x0,(2.4)where,k±=vw2D 1±s1 + 4ΓDv2w!.(2.5)In the limit of v2wΓD,k±≈ ±pΓ/D.For large R, the ball can be approximated as two walls/sources propagating back-to-back but with opposite CP-violating sign. Thus, the source term is approximatelyS2(x) = S(x+R)−S(x−R), where the source function at each wall is given by [17,22]S(x)≈vwy2fD T 2θ0006.(2.6)Here yfis the fermion Yukawa coupling to the Higgs ﬁeld; θ(x)is the wall proﬁle of theCP-violating angle, which can be approximated by θ0(x) = ∆θ δ(x)with ∆θas the anglechanges from its outside to inside values [17]. The derivation of (2.6) uses the WKBapproximation, which is valid when the wall thickness ∼v−1is larger than the de Brogliewavelength of the particles ∼(3T)−1. Because of this, we must assumeR&v−1,(2.7)which is consistent with the previous condition RpD/Γto use a one-dimensionaldiﬀusion equation. With this source term, the CP asymmetric number density is given bynCP (x) = Z∞−∞G(x−x0)S2(x0)dx0=vwy2fT2∆θ6(k+−k−)k2+exp[−k+(x+R)] −k2−exp[−k−(x−R)].(2.8)For the EWS ball, the region of interest is x∈(−R , R ), where the sphalerons are active.In the limit of vw√ΓDand |k−R| ≈ |k+R|  1, the radius-averaged number densityinside the ball isnCP =RR−RnCP (x)dx2R≈v2wy2fT2∆θ24 R√ΓD.(2.9)– 5 – JHEP10(2021)147Notice that due to the cancellation of the contributions from the two walls in (2.8), theaverage number density is proportional to the square of the wall velocity. Also, from (2.8),the number density is localized around a distance 1/|k±| ≈ pD/Γwithin the ball wall.This is larger than the thickness of the wall, justifying the above approximation of θ(x)asa step function for the integral in (2.8).The CP-violating chemical potential can then be evaluated by µCP = 2 nCP /T 2[24].For the top quark, Γ≈T /100,D≈6/T , and yf≈1[23], givingµCP ≈v2w∆θR.(2.10)The EWS balls are not actually moving in one direction because of their inter-actions with SM particles in the plasma. The ball thermalization rate Γtherm 'Pi(niσivrel,i)T /M is related to the EWS ball’s cross section σi'πR2|Ri|2with SMparticles in the plasma (indexed by i) whose number density is ni. The reﬂection proba-bility |Ri|2∼1for semi-relativistic or non-relativistic species, while niis suppressed fornon-relativistic species. Thus, Γtherm ∼g T 4R2/M , with gthe number of semi-relativisticdegrees of freedom in the bath. To justify our previous estimation for nCP , the thermaliza-tion rate is required to be smaller than the diﬀusion rate, or Γtherm .D|k±|2≈Γ, whichgivesM&g T 4ΓR2≈103T3R2,(2.11)where g≈9, taken to be the Wand Zdegrees of freedom assuming CAB takes placeshortly after the EWPT. When the thermalization rate is very high in violation of (2.11),non-steady solutions should be obtained to estimate the nCP . In the ball rest-frame, thesource S(~x, t)in (2.2) should vary in time intervals of order 1/Γtherm. We do not calculatenCP for this case in this work and will restrict the model parameter space to satisfy (2.11).As we will later see in eq. (3.1), the region where (2.11) is not satisﬁed is anyways mostlyexcluded by other unrelated considerations.2.2 Baryon number violationWith the chemical potential inside the EWS ball derived in the previous section, a netbaryon asymmetry can be produced via baryon number violating sphalerons. When theball moves slowly, the EW sphaleron can convert the CP asymmetry into baryon asymmetryvia the “non-local” way [21].The baryon number generated per unit time by a single EWS ball is given by [17]dNBdt ≈ −Γsph R3µCPT,(2.12)where Γsph ≈25 α5wT4≈10−6T4is the sphaleron rate per unit volume [24]. Here, we haveused that Γsph/T 3<|k±|, implying particles undergo less than one sphaleron transitionon average before the asymmetric region near the wall diﬀuses. The density of producedbaryon asymmetry nBis related to the EWS ball density bydnBdt + 3HnB=−ΓsphR3µCPTn . (2.13)– 6 – JHEP10(2021)147Using dnBdt + 3HnB=s dYB/dt with sas the entropy density and dT/dt =−H T ,dYBdT =Γsph R3H TµCPTY . (2.14)Using H(T) = 16.6×(g∗/100)1/2T2/Mpl (assuming radiation domination), this equationcan be integrated from an initial temperature Tito obtain (assuming constant Y)YB= 1.9×10−10 fDM R1GeV−12 108GeVM!2∆θ−1 Ti100 GeV2.(2.15)In general, Tiwill be the smaller of either the EW phase transition temperature TEWPT(so that the Higgs has a nonzero VEV outside the EWS ball) or the EWS ball formationtemperature Tform. This quantity is related to the baryon-to-photon ratio by YB=η/7.04 ≈0.85 ×10−10 to explain the observed baryon asymmetry.2.3 CP violationCP violation can be introduced via a simple eﬀective operator coupling the EWS ball tothe SM. For example, the constituent scalar ﬁeld Φof the EWS ball (see appendix Aforexample models) can have a CP-violating interaction with the top quark [25] given byL ⊃ ytQLeH 1 + ηΦΦ†Λ2!tR+h.c. , (2.16)where eH=iσ2H∗and ηis a complex parameter taken to be eiπ/2. If |Φ| ≡ φ(r)variesinside and outside the ball, this induces a spatially varying complex mass for the top quark,mt(r) = yt√2h(r) 1 + iφ(r)2Λ2!=|mt(r)|eiθ(r),(2.17)where tan θ(r) = φ(r)2/Λ2. Outside the EWS ball, φ(r)is constant, and the phase can beabsorbed by a redeﬁnition of ﬁelds. Near the EWS ball, the change in phase is physical,inducing a CP-violating angle |∆θ| ≈ f2/Λ2for f.Λ, where fis the change in the φ(r)between the interior and exterior of the ball.For this model to provide an EWS ball, there must be a Higgs portal coupling tothe Φﬁeld [8,9] (see appendix A) with Lportal =λΦΦ†H H †with |λ|f2&v2. The signof λdepends on whether φ(r)2increases or decreases inside the ball, which will diﬀerdepending on the model. Then, this portal coupling in conjunction with the interactionterm in (2.16) leads via a Φloop diagram to a dimension-6 CP-violating operator involvingonly SM particles,L6∼iytλ16π2|H|2Λ2QLeHtR+h.c. (2.18)The coeﬃcient of this operator is constrained by the current upper limit on the elec-tron dipole moment |de|<1.1×10−29 ecm by the ACME collaboration [26]. Thus,Λ> v (λ/0.13)1/2[27]. If Λ&f&v, the EWS ball can simultaneously satisfy |λ|f2&v2,|∆θ|.1, and the ACME constraint. In other words, the scale of the new physics neednot be much larger than the electroweak scale while still allowing O(1) CP violation at theEWS ball wall.– 7 – JHEP10(2021)1473 Abundance of EWS balls and their evolutionWe now turn our attention to the parameter space where EWS balls can facilitate baryoge-nesis. We will require throughout that the mass-radius relation of the EWS balls satisﬁesM&4π3v4R3+ 4πv3R2.(3.1)This represents the minimum mass contribution from the Higgs ﬁeld, assuming the diﬀer-ence in the vacuum energy between the EWS region and the normal vacuum ∆V&v4forthe ﬁrst term. Depending on the EWS ball model, additional contributions to the masscould come from other ﬁelds or matter (e.g., from the Φﬁeld for a Q-ball, the dark Higgsand dark gauge ﬁelds for a dark monopole, or the black hole in a magnetically chargedblack hole). We saw in section 2.3 that the scale of the EWS ball physics could be near theweak scale or much larger. Thus, these additional mass contributions may be of the sameorder as (3.1) or much larger, making this a generic lower bound. As an aside, for somemodels, it is possible to have a ﬁne-tuned cancellation between terms so that ∆V≈0,suppressing the ﬁrst term of (3.1). In this case, the gradient energy of the ﬁelds near theball wall provides the irreducible contribution from the second term. We will not considerthis tuned scenario in what follows, and it would anyways not aﬀect our conclusions.3.1 The EWS ball as dark matterAs a simplest ﬁrst assumption, we can take the EWS ball yield Yto be constant beforethe EWPT, with its abundance determined at some higher temperature scale. Then, thebaryon asymmetry is ﬁxed by eq. (2.15). It is maximized when fDM = 1, in which case theEWS ball also plays the role of dark matter.The parameter space where EWS balls can catalyze baryogenesis is shown in ﬁgure 2,setting fDM = 1. The conditions of producing enough asymmetry in (2.15) (dashedblue line, excluding the region to its right by demanding |∆θ| ≤ 1and Ti< TEWPT ∼100 GeV [28,29]), having a long enough thermalization time in (2.11) (purple), satisfyingthe mass condition (3.1) (orange), and having a large enough radius in (2.7) (green) areimposed. It is clear that these conditions allow for successful baryogenesis.However, the relic abundance of EWS balls can be probed at direct detection ex-periments. Following [8,9,15], we approximate the EWS ball-nucleus scattering crosssection as the smaller of the expectation from the Born approximation and the geometriccross section,σA≈min 16π9m2NA4y2hN N v2R6,2πR2,(3.2)where mNis the nucleon mass, Ais the nucleus’s atomic mass number, and yhNN ≈1.1×10−3. Direct detection experiments like Xenon1T [30] can then place upper boundson surviving EWS ball radii as shown by the dotted red curve in ﬁgure 2. Due to thecombination of constraints, it is clear that it is not possible to explain both baryogenesisand dark matter simultaneously with EWS balls. Further, even if fDM is reduced, thedirect detection constraint on Rwill weaken only by f−1/6DM [see eq. (3.2)] while the baryonasymmetry is proportional to fDM , leaving no available parameter space that satisﬁes allconstraints.– 8 – JHEP10(2021)147Figure 2. Bounds on the parameter space for EWS balls to catalyze baryogenesis. Blue curvesshow the constraints from requiring suﬃcient baryogenesis to explain the observed asymmetry. Thesolid blue curve shows the constraint assuming the EWS ball abundance is set by annihilationsand includes an enhancement factor ξ(y) = 10 by assuming annihilations are taking place duringbaryogenesis and Tform =Ti; see (3.7). The dashed blue curve shows the constraint if fDM isﬁxed to unity, with no enhancement factor for annihilations; see (2.15). Both take Ti= 100 GeVand |∆θ| ≤ 1. The orange curve enforces the mass-radius relation in (3.1), while the green limitsR > v−1from (2.7). Purple enforces that the thermalization time must be long enough accordingto (2.11). Finally, the red dotted line shows the direct detection limits from Xenon1T [30]. Itassumes fDM = 1, but it does not impose any constraint if the EWS balls decay after catalyzingbaryogenesis.3.2 EWS ball annihilationsClearly, a ﬁxed value for fDM is constrained by the competition between direct detectionconstraints and creating a large enough baryon asymmetry. Instead, one could considerthat fDM evolves over time. If the EWS ball abundance is larger near T∼100 GeVthan it is today, then baryogenesis could still be suﬃciently eﬃcient while evading directdetection constraints. In this subsection we will show that fDM can evolve due to EWSball annihilations, but the impact on baryogenesis is relatively small.Here, consider a purely symmetric population of EWS balls. Further, if the EWS ballis a non-topological soliton, assume that most of the Φparticles are bound inside EWSballs. If an initial population of EWS balls form at some initial temperature Tform withabundance Yform, then some of them have a chance to annihilate. For simplicity, assumeall EWS balls have the same mass and radius. Further assume that EWS balls of oppositecharges will annihilate with geometric cross section σ∼πR2(i.e., there is no long range– 9 – JHEP10(2021)147force, just contact interactions). Then the evolution of EWS balls will go asdndt + 3Hn =−hσvrelin2⇒dYdT =hσvrel is Y 2H T .(3.3)Taking g∗=g∗s∼100 and vrel = (T /M )1/2,Y(T)−1−Y(Tform)−1=R2M−1/2g1/2∗Mpl (T3/2form −T3/2).(3.4)For this to make a diﬀerence in YB,Tform must be close to the EWPT temperatureTEWPT. The reason: if Tform is much larger, then Y(TTform)is constant, reducingto the previous case of constant fDM; on the other hand, if Tform ≥Tiis much smallerthan TEWPT, the baryon asymmetry is suppressed, going as T2iin (2.15). If we tuneTform ∼TEWPT, we can plug Y(T)into (2.14) to obtain the ﬁnal baryon asymmetry YB.This gives an enhancement factor to the right side of eq. (2.15):ξ(y) = 43log(y),for y≡Y(Tform)Y(TTform)1,(3.5)assuming Tform =Ti< TEWPT, where fDM in (2.15) would be calculated using Y(TTform). At most, we expect ξ(y)to give an O(10) enhancement due to the logarithmicdependence on y.Thus, including the time evolution of the EWS balls due to annihilations does notprovide a large enough baryogenesis enhancement to escape direct detection bounds. It isworth noting that EWS ball annihilations were assumed instantaneous and complete in thiscalculation. If the annihilations were to take a long enough time, e.g., because the EWSballs form metastable bound states before annihilating, then baryogenesis could proceedwith a larger EWS ball abundance than assumed here.3.3 EWS ball decayAnother possibility for time dependence is to allow the EWS balls to decay. Then, theycould exist shortly after the EWPT, but decay long before today, evading direct detec-tion constraints. They may even have an abundance larger than dark matter in theearly universe.The EWS ball abundance after annihilations in the limit Y(Tform)Y(T)andTform Tfrom (3.4) is,Y(TTform) = 10−18 M107GeV1/20.1GeVR2200 GeVTform 3/2.(3.6)Then, using (2.1), (2.15), (3.5), and (3.6),YB= 3.8×10−11107GeVM1/2∆θ−1200 GeVTform 3/2Ti100 GeV2ξ(y)10 .(3.7)Thus, the necessary baryon asymmetry could be generated for M.107GeV. This isshown in ﬁgure 2by the solid blue vertical line (the dotted red direct detection constraint– 10 – JHEP10(2021)147and dashed blue fDM = 1 constraint can be ignored here, since the abundance is assumedset by annihilations and decays well before today). Note that Tform cannot be much largerthan TEWPT, otherwise not enough baryon asymmetry is produced (in conjunction withlower bounds on Min ﬁgure 2). Also, since Ti≤Tform , the formation temperature cannotbe too small either. In all cases, Ti≤TEWPT is required.3.4 Asymmetric EWS ballsIf the dark sector responsible for EWS ball formation itself carries an asymmetry, then theEWS ball annihilations will halt once the symmetric component is depleted. This presentsadvantages compared to the prior symmetric case. First, it could allow Ywell in excessof (3.6), enhancing the amount of baryogenesis. Second, it removes any dependence for YBon Tform, as long as Tform ≥TEWPT . Third, a model-dependent advantage for the speciﬁccase of EWS Q-balls is that free Φantiparticles are also depleted, making it easier for Q-balls to survive without assuming every Φparticle must be bound in a Q-ball [31]. For theEWS ball abundance to be compatible with baryogenesis, it would be constrained by directdetection at low mass and overclosure at high mass if the EWS balls were stable. Thus,this possibility still depends on the EWS balls being unstable and evaporating between theEWPT and today.The asymmetric abundance is constrained by requiring the decay products of the EWSballs not contribute too much to the eﬀective radiation degrees of freedom during BBNor recombination. The most conservative bound comes from assuming EWS balls decayshortly after the EWPT. As shown in the next section, this amounts to the requirementfDM .2×1010 in eq. (2.15). Clearly, this opens a large swath of parameter space in R,M, and ∆θthat is not accessible when the EWS-ball-forming sector is symmetric, whichis depicted in ﬁgure 3.4 Evaporation of EWS ballsSupposing that the EWS ball can evaporate, we denote the rate of evaporation by Γevp .Such evaporation can happen, for example, by directly decaying the ﬁeld Φmaking upthe EWS ball into new dark sector particles. For instance, if there is a U(1)Φ-breakingcoupling of the form LΦχ=gΦχΦ¯χχ +h.c., then evaporation can happen via decaying intoχparticles, which we take to be massless for simpicity, each carrying energy of M /(2Q)with Qas the total U(1)Φcharge of an EWS ball. The massless products of this decaythen redshift as radiation. The energy contained in the χparticles will contribute to theeﬀective number of relativistic degrees of freedom, ∆Neﬀ, which is constrained by probesof BBN and CMB measurements. Alternatively, if the EWS balls decay into SM particlesbefore BBN (Tevp &1MeV), there is no constraint from ∆Neﬀ.We will take a model-independent approach to evaporation in which the temperatureat which EWS balls evaporate (Tevp ), i.e., the temperature at which H(Tevp)=Γevp ,is a phenomenological parameter. The total energy density contained in EWS balls isfDM ρDM(T)which redshifts as matter from T= 100 GeV to T=Tevp. Hence the total– 11 – JHEP10(2021)147Figure 3. Same as ﬁgure 2, but now allowing an asymmetric component of EWS balls thateventually decay into massless states in the dark sector. Constraints from ∆Neﬀ limit wherebaryogenesis can produce large enough η. Examples for present (projected) constraints areshown in solid (dashed) blue lines for two diﬀerent possible evaporation temperatures. BecauseTevp < TEWPT ∼100 GeV is required for baryogenesis by EWS balls, the shaded blue region on theright is completely excluded by current constraints. If the dark sector states are massive, additionalconstraints can apply as discussed in the text.energy density that is converted to χ-particles is given byρχ(T=Tevp)=0.14 fDM107g∗s(Tevp)60  Tevp1GeV3GeV4.(4.1)The contribution of ρχto ∆Neﬀ at late times that the CMB is sensitive to is given by∆Neﬀ ≈871144/3ρχ(TCMB)ργ(TCMB).(4.2)Since the radiation produced in the evaporation of the EWS balls is decoupled from theSM bath, their density scales as ρχ∝1/a4∝g4/3∗s(T)T4. From this, we determine thecontribution to the energy density at the temperature of recombination and ﬁnd that∆Neﬀ ≈0.025 fDM107 Tevp1GeV−1 60g∗s(Tevp)!1/3.(4.3)Large fDM, corresponding to more energy density in EWS balls, and small Tevp, correspond-ing to the late decay of EWS balls, are more constrained. The current strongest limit comesfrom the CMB epoch given by the Planck 2018 observations ∆Neﬀ <0.51 [2,32]. For suc-cessful baryogenesis, Tevp .TEWPT ∼100 GeV is required, setting a conservative bound– 12 – JHEP10(2021)147fDM .2×1010. Next-generation observations from CMB-S4 experiments are projected toimprove sensitivity by an order of magnitude to σ(Neﬀ)<0.03 [33].Alternatively, if the χhave mass mχ>0, they could explain dark matter providedmχ∼M /(2QfDM). They must not be too light, otherwise they will free stream fortoo long and suppress structure formation. They become non-relativistic at the time theuniverse has reached temperature TNR ∼Tevp/fDM. Their free-streaming length up tomatter-radiation equality (at temperature Teq), when perturbations become Jeans un-stable, is λFS ∼2MplT0TNR 1 + log[ TNRTeq ][34], where T0= 2.35 ×10−4eV is the temper-ature today. The bound on TNR is similar to the bound on the thermal warm darkmatter mass: TNR &1keV [35–38], corresponding to λFS .1Mpc. This sets a boundfDM .108×(Tevp/100 GeV), a bit stronger than the ∆Neﬀ constraint for this case. If mχis smaller than the quantity above, χonly makes up a subdominant component of darkmatter, and the free streaming constraint is correspondingly relaxed.5 Discussion and conclusionsWe have used the EWS ball as a representative model to implement the more generalcatalyzed baryogenesis mechanism. One could consider other possible models to introducebaryon number violating interactions. For instance, the constituents of the catalyst ballcan interact with SM particles via baryon number violating higher-dimensional operators.If those operators mainly contain the third or second-generation of quarks, the directdetection constraints could be dramatically relaxed and the catalyst balls can exist inthe current universe and contribute to a signiﬁcant fraction of dark matter. Similarly,lepton-number violating operators can also be adopted to catalyze leptogenesis before theelectroweak phase transition. The generated lepton asymmetry is then converted into abaryon asymmetry by the electroweak sphaleron process. Additionally, one could use thecatalyst mechanism to generate an asymmetry for a new particle that carries both baryonand dark matter number. The later decays of this new particle can provide a uniﬁed originfor baryon and dark matter asymmetry [39].In summary, we have proposed a novel mechanism to generate the baryon asymmetrythat is similar to the catalytic reaction in chemistry. To generate enough baryon asymmetry,the catalyst balls are preferred to have a smaller mass and a larger radius. We have used theelectroweak-symmetric ball to guide our discussion of this general catalyzed baryogenesismechanism. For the EWS ball abundance determined by their annihilations, the EWS ballmass and radius are required to be around 106GeV and 10−2GeV−1to explain the observedbaryon asymmetry. Interestingly, EWS ball relics within this region of parameter spaceare already excluded by direct detection constraints, so they are required to decay intoother states in the early universe to evade the constraints. We also discussed the case withasymmetric EWS balls with a much larger initial abundance and a wider parameter spacein Mand Rto accommodate the observed baryon asymmetry. The dark radiation fromEWS ball evaporations provide a large contribution to the eﬀective number of relativisticdegrees of freedom, which could be tested in future CMB experiments.– 13 – JHEP10(2021)147AcknowledgmentsWe thank Hooman Davoudiasl for insightful discussion. The work of YB and MK issupported by the U.S. Department of Energy under the contract DE-SC-0017647. The workof JB is supported by start up funds from Colorado State University. The work of NO issupported by the Arthur B. McDonald Canadian Astroparticle Physics Research Institute.A Electroweak-symmetric ball modelsIn this appendix, give details for possible EWS ball models to act as catalysts. While theresults of our work do not depend on the detailed model underlying the EWS ball, thissection demonstrates that such underlying models do exist. We give brief overviews of twosuch models here, and full details (including formation mechanisms) can be found in thecorresponding references. It is worth mentioning a third possibility for EWS objects is amagnetically charged black hole [12–14], but their masses are greater than the Planck scaleand thus too heavy to be of interest according to ﬁgures 2and 3.A.1 EWS Q-ballThe benchmark model that most closely replicates the EWS ball properties in the mainbody is the EWS Q-ball [8]. It is a nontopological soliton solution resulting from theinteraction of a complex scalar Φthat couples to the Higgs ﬁeld H. If the Φﬁeld ischarged under a global U(1) symmetry, the most general renormalizable potential for theseﬁelds isV=λh H†H−v22!2+λφhΦ†ΦH†H+m2φ,0Φ†Φ + λφ(Φ†Φ)2.(A.1)Here, all coupling coeﬃcients are taken to be positive, so the ﬁeld VEVs are h|H|2i=v2/2and hSi= 0.Consider the time-dependent, spherically symmetric ansatz Φ = e−iωtφ(r)/√2alongwith a spherically symmetric ansatz for the Higgs ﬁeld H>= (0, h(r)/√2). This ansatzhas a conserved global chargeQ=iZd3x(Φ†∂tΦ−Φ∂tΦ†)=4πω Z∞0dr r2φ2.(A.2)Then, the equation of motion for φis (omitting that of hfor brevity)φ00 +2rφ0+∂φUeﬀ(φ)=0,(A.3)Ueﬀ(φ) = 12ω2φ2−14λφhφ2h2−14λφφ4,(A.4)where ω2=ω2−m2φ,0. The ﬁelds are subject to the boundary conditions φ0(0) = h0(0) =φ(∞)=0and h(∞) = v. Further demanding an EWS solution with h(0) = 0 amounts torequiring φ(0) > vq2λh/λφh so that the Higgs mass is driven positive by the ﬁeld valueof Φ. The solution for φ(r)can be thought of as starting at some initial value φ(0), thenrolling down the potential Ueﬀ(φ)until coming to rest at the unstable ﬁxed point φ= 0– 14 – JHEP10(2021)147as r→ ∞. Such a solution exists if (λφλh/2)1/4≡ωc/v < ω/v < pλφh, with the lower(upper) limit corresponding to large (small) Qsolutions. In the large Qlimit, when ω≈ωc(and the corresponding value for ω≈ωc), it can be shown that [8]M≈Q ωc≈4π31λφω2cω2cR3.(A.5)For quartic couplings of order unity, ωc≈ωc≈v. Thus, this mass saturates the inequalityin (3.1), the latter of which is dominated by the ﬁrst R3term in this limit.A.2 EWS monopoleA second interesting case is that of the EWS monopole [9]. The renormalizable potentialfor the EWS monopole is very similar to (A.1), but changing Φto a triplet scalar of agauged SU(2) Φawith a= 1,2,3:V=λh(H†H)2+µ2hH†H−12λφh|Φ|2H†H−12m2φ,0|Φ|2+14λφ|Φ|4(A.6)with |Φ|2=Pa(Φa)2. Notice that the signs of the m2φ,0,µ2h, and λφh terms have ﬂippedwith respect to (A.1). The magnitudes of the terms should be arranged such that both Φand Hhave a VEV. I.e., when |Φ|=ftakes its VEV, the condition λφhf2> µ2hmeansthat Hwill also have a nonzero VEV as required in the SM.Because Φspontaneously breaks the gauged SU(2) to U(1), a topological monopoleconﬁguration exists with |Φ|= 0 at its center. Because of this, Hhas a positive mass-squared term inside of the monopole. Thus, the Higgs ﬁeld is driven to its symmetry-preserving value inside the monopole, leading to an EWS monopole.The mass-radius relation can also be determined. The monopole mass (not accountingfor the Higgs vacuum energy contribution) is known to beM=4πfgY(λφ/g2),(A.7)where gis the SU(2) gauge coupling of Φto the gauge ﬁeld and Y(x)is a monotonicfunction ranging from Y(0) = 1 to Y(∞)≈1.787. The EWS radius, i.e., the radius forwhich EW symmetry is restored, depends on the characteristic radius of the scalar ﬁeld Φ.Once Φis large enough, the EW symmetry is spontaneously broken. 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... Refs. [12][13][14][15]. ... ... [7]) and quadratic (see e.g. [14,49]) couplings between the singlet and the top quark ... Minimal Electroweak Baryogenesis via Domain Walls Preprint Dec 2024 Jacopo Azzola Oleksii MatsedonskyiAndreas Weiler View Show abstract ... An NTS can also have gauge charges [7][8][9][10][11] or even topological charges [12] in the presence of a gauge group. Other studies include [13][14][15][16][17][18][19][20][21][22]; see [23,24] for reviews. ... ... From the Q-ball domination we can also understand how the chemical potential µ evolves at late time. At T < T D , we expect almost all the global charge to be concentrated in (Q max ), i.e., n Qmax η n γ /Q max using (19), which gives an approximate analytic expression of µ: ... Origin of nontopological soliton dark matter: solitosynthesis or phase transition Preprint Aug 2022 Yang BaiSida LuNicholas Orlofsky View Show abstract ... An NTS can also have gauge charges [7][8][9][10][11] or even topological charges [12] in the presence of a gauge group. Other studies include [13][14][15][16][17][18][19][20][21][22]; see [23,24] for reviews. ... Origin of nontopological soliton dark matter: solitosynthesis or phase transition Article Full-text available Oct 2022J HIGH ENERGY PHYS Yang BaiSida LuNicholas Orlofsky View Show abstract Embedded domain walls and electroweak baryogenesis Article Full-text available Aug 2024 Tobias SchröderRobert Brandenberger View Show abstract

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HomeElementary Particle PhysicsStandard Model ArticlePDF Available Catalyzed baryogenesis October 2021Journal of High Energy Physics 2021(10) DOI:10.1007/JHEP10(2021)147 LicenseCC BY 4.0 Authors: Yang Bai University of Louisville Joshua Berger Mrunal Korwar Indian Institute of Science Education and Research, Pune Nicholas Orlofsky

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Abstract and Figures A bstract A novel mechanism, “catalyzed baryogenesis”, is proposed to explain the observed baryon asymmetry in our universe. In this mechanism, the motion of a ball-like catalyst provides the necessary out-of-equilibrium condition, its outer wall has CP-violating interactions with the Standard Model particles, and its interior has baryon number violating interactions. We use the electroweak-symmetric ball model as an example of such a catalyst. In this model, electroweak sphalerons inside the ball are active and convert baryons into leptons. The observed baryon number asymmetry can be produced for a light ball mass and a large ball radius. Due to direct detection constraints on relic balls, we consider a scenario in which the balls evaporate, leading to dark radiation at testable levels. Schematic depiction of catalyzed baryogenesis facilitated by an electroweak symmetric ball. As the ball moves through the plasma, a baryon asymmetry builds up near its walls due to a CP-violating interaction at the bubble wall. This relies on the relative plasma-ball velocity so that more antibaryons (blue) and baryons (orange) approach the walls from the right side than from the left. The asymmetry then diffuses away from the regions near the wall, but because of the relative velocity between the ball and the plasma, particles near the leading edge tend to diffuse inwards while particles near the trailing edge tend to diffuse outwards (indicated by the green dotted lines). This creates a net baryon number asymmetry inside the ball. Sphalerons, which are only active inside the ball, then convert some of the baryons into leptons, preserving the asymmetry even as the ball passes by. Schematic depiction of catalyzed baryogenesis facilitated by an electroweak symmetric ball. As the ball moves through the plasma, a baryon asymmetry builds up near its walls due to a CP-violating interaction at the bubble wall. This relies on the relative plasma-ball velocity so that more antibaryons (blue) and baryons (orange) approach the walls from the right side than from the left. The asymmetry then diffuses away from the regions near the wall, but because of the relative velocity between the ball and the plasma, particles near the leading edge tend to diffuse inwards while particles near the trailing edge tend to diffuse outwards (indicated by the green dotted lines). This creates a net baryon number asymmetry inside the ball. Sphalerons, which are only active inside the ball, then convert some of the baryons into leptons, preserving the asymmetry even as the ball passes by. … Figures - available via license: Creative Commons Attribution 4.0 International Content may be subject to copyright. ResearchGate Logo Discover the world's research

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Available via license: CC BY 4.0 Content may be subject to copyright. JHEP10(2021)147Published for SISSA by SpringerReceived:July 9, 2021Revised:August 27, 2021Accepted:September 30, 2021Published:October 18, 2021Catalyzed baryogenesisYang Bai,aJoshua Berger,bMrunal Korwaraand Nicholas OrlofskycaDepartment of Physics, University of Wisconsin-Madison,Madison, WI 53706, U.S.A.bDepartment of Physics, Colorado State University,Fort Collins, Colorado 80523, U.S.A.cDepartment of Physics, Carleton University,Ottawa, ON K1S 5B6, CanadaE-mail: [email protected],[email protected],[email protected],[email protected]: A novel mechanism, “catalyzed baryogenesis”, is proposed to explain the ob-served baryon asymmetry in our universe. In this mechanism, the motion of a ball-like cat-alyst provides the necessary out-of-equilibrium condition, its outer wall has CP-violatinginteractions with the Standard Model particles, and its interior has baryon number vio-lating interactions. We use the electroweak-symmetric ball model as an example of sucha catalyst. In this model, electroweak sphalerons inside the ball are active and convertbaryons into leptons. The observed baryon number asymmetry can be produced for a lightball mass and a large ball radius. Due to direct detection constraints on relic balls, weconsider a scenario in which the balls evaporate, leading to dark radiation at testable levels.Keywords: Cosmology of Theories beyond the SM, Beyond Standard ModelArXiv ePrint: 2106.12589Open Access,cThe Authors.Article funded by SCOAP3.https://doi.org/10.1007/JHEP10(2021)147 JHEP10(2021)147Contents1 Introduction 12 Catalyzed baryogenesis by an EWS ball 42.1 Out-of-equilibrium condition 42.2 Baryon number violation 62.3 CP violation 73 Abundance of EWS balls and their evolution 83.1 The EWS ball as dark matter 83.2 EWS ball annihilations 93.3 EWS ball decay 103.4 Asymmetric EWS balls 114 Evaporation of EWS balls 115 Discussion and conclusions 13A Electroweak-symmetric ball models 14A.1 EWS Q-ball 14A.2 EWS monopole 151 IntroductionOne of the biggest challenges in particle physics and cosmology is to understand thematter-antimatter asymmetry in our universe. More precisely, baryogenesis models mustexplain the small (dimensionless) baryon-to-photon ratio: η= (Nbaryon −Nantibaryon)/Nγin the current universe with Nbaryon,Nantibaryon, and Nγas the numbers of baryons, anti-baryons, and photons, respectively. The measurement of light element abundances in theintergalactic medium together with the theory of Big Bang nucleosynthesis (BBN) hasη= (5.931 ±0.051) ×10−10 [1], while the spectrum of the temperature ﬂuctuations ofthe cosmic microwave background (CMB) has η= (6.12 ±0.04) ×10−10 [2,3]. Sakharovproposed three general conditions necessary for any baryogenesis model: i) baryon numberviolation (BNV), ii) C and CP violation, and iii) departure from thermal equilibrium [4].There are many mechanisms proposed in the literature, some of which have phenomeno-logically testable predictions (e.g., electroweak baryogenesis [5]). In this paper, we proposea framework of baryogenesis, “catalyzed baryogenesis” (CAB), wherein BNV interactionsare enhanced in certain regions of space. These regions are similar to “catalysts” in chem-istry where certain molecules aid the interactions of other molecules. If the catalysts aresuﬃciently long-lived, they could even be detected as relics making up dark matter today.– 1 – JHEP10(2021)147To illustrate this framework, we explore in this work the case where electroweak-symmetric (EWS) balls act as the catalyst. Such states can arise in many theories, in-cluding non-topological solitons [6–8], dark monopoles [9], or magnetically charged blackholes [10–14]. Schematically, the catalyzed baryogenesis process for this particular catalystis illustrated in ﬁgure 1. Inside the catalyst region, electroweak sphalerons are active be-cause electroweak symmetry is restored, allowing baryons to be converted to leptons, whileoutside sphaleron-mediated processes are suppressed, providing condition (i). Sphaleronsalso violate C, partially satisfying condition (ii). Interactions between the EWS ball walland incoming particles can violate CP, completing condition (ii). Finally, the process lo-cally goes out of equilibrium (iii) when there is a relative velocity between the EWS ball andthe surrounding plasma. Due to the relative velocity, more plasma particles are incidenton the right side of the ball walls than the left in ﬁgure 1. Then, the excess antibaryonstransmitted through the leading wall tend to diﬀuse into the ball where sphalerons areactive; meanwhile, the excess baryons reﬂected by the trailing wall tend to diﬀuse out ofthe ball, where the sphaleron rate is suppressed following the electroweak phase transi-tion (EWPT). Thus, the excess antibaryons inside the ball are depleted, while the excessbaryons outside the ball are not, leading to a net asymmetry after the ball has passed. Tosatisfy condition (iii), the CP asymmetry produced at the wall should have time to diﬀusesuﬃciently widely in the ball to have sphalerons act on the CP asymmetry before it decaysvia C-violating, but CP-conserving, chirality-ﬂipping interactions. Furthermore, we willassume that the system remains in a dynamical equilibrium, with the plasma inside theball quickly diﬀusing as the ball slowly changes direction by scattering oﬀ the plasma. Thisequilibrium situation is illustrated in ﬁgure 1. The picture breaks down if the ball changesdirection suﬃciently quickly that the plasma inside the ball does not have time to returnto this dynamical equilibrium between order one changes in direction.CAB has some similarities to the electroweak baryogenesis model based on a stronglyﬁrst-order EWPT. For both cases, the motion of the bubble or ball wall provides the out-of-equilibrium condition. Diﬀerent from CAB, electroweak baryogenesis has the outside regionactive for the sphaleron process. Furthermore, the bubbles expand, collide, percolate, anddisappear shortly after generating baryon asymmetry. On the contrary, the EWS balls inCAB could survive for a cosmologically long time. If they survive until the present day,they could be searched for directly. Otherwise, if they evaporate, the resulting particlescan leave cosmological imprints. Further, CAB does not rely on any assumptions about thenature of the EWPT — excepting perhaps the temperature at which it occurs. In CAB,baryogenesis is not driven by a phase transition.Because the catalysts considered here are EWS, interactions with Standard Model(SM) particles are unavoidable, leading to large interactions between the balls and nu-clei [8,15]. One important ﬁnding of this work is that while EWS balls could explain bothbaryogenesis and dark matter simultaneously, they would have already been observed bydirect detection experiments. Therefore, EWS balls of a given mass and radius may eitherplay the role of dark matter or catalyst for baryogenesis, but not both. To be a catalyst,they should decay or evaporate into SM particles or dark-sector particles in the early uni-verse. This opens a narrow region of parameter space where a symmetric population of– 2 – JHEP10(2021)147        plasma velocity in ball frame    diffusionFigure 1. Schematic depiction of catalyzed baryogenesis facilitated by an electroweak symmetricball. As the ball moves through the plasma, a baryon asymmetry builds up near its walls due to aCP-violating interaction at the bubble wall. This relies on the relative plasma-ball velocity so thatmore antibaryons (blue) and baryons (orange) approach the walls from the right side than from theleft. The asymmetry then diﬀuses away from the regions near the wall, but because of the relativevelocity between the ball and the plasma, particles near the leading edge tend to diﬀuse inwardswhile particles near the trailing edge tend to diﬀuse outwards (indicated by the green dotted lines).This creates a net baryon number asymmetry inside the ball. Sphalerons, which are only activeinside the ball, then convert some of the baryons into leptons, preserving the asymmetry even asthe ball passes by.EWS balls could catalyze baryogenesis. If the EWS balls are assumed to themselves carryan asymmetric component, the parameter space signiﬁcantly expands. In the last sec-tion, we also speculate on other possible catalyst models that may have suppressed directdetection cross sections, forgoing the need for their eventual evaporation.Similar scenarios were previously studied in refs. [16,17], with a particular focus onEWS cosmic strings instead of balls. Some of our detailed analysis follows the treatmentfrom these works. For both ball and string scenarios, the ﬁnal baryon asymmetry is foundto be proportional to the catalyst velocity square. However, the strings they consideronly have radius of order v−1with v= 246 GeV the Higgs ﬁeld vacuum expectation value(VEV), whereas we consider EWS balls with much larger radius. Further, baryogenesisis limited in their case by the string network evolution: friction-dominated strings movetoo slowly, while in the scaling regime strings are too scarce. Thus, while ref. [17] reacheda negative conclusion for strings to catalyze baryogenesis, our work ﬁnds that balls canexplain the observed baryon asymmetry due to these diﬀerences.Our work also diﬀers from models which include both Q-balls and baryogenesis. Pre-vious works consider Q-balls that form as a result of Aﬄeck-Dine baryogenesis [18], oralternatively that decays of Q-balls could explain baryogenesis [19]. In contrast to the for-mer, our catalyst is not a byproduct but an essential ingredient for baryogenesis. In contrastto the latter, our catalyst is not modiﬁed or destroyed for purposes of baryogenesis.– 3 – JHEP10(2021)147This paper is organized as follows. In the next section, we describe the CAB mechanismin detail, including how all Sakharov conditions are satisﬁed. In section 3, we synthesize theconditions to detail the allowed parameter space of EWS balls as catalysts of baryogenesis.The mechanisms and eﬀects of EWS ball decays are described in section 4. The ﬁnal sectionin section 5gives concluding remarks.2 Catalyzed baryogenesis by an EWS ballThis section will estimate the amount of baryon asymmetry that is produced. We begin byassuming that EWS balls exist after the electroweak phase transition, with mass Mandradius R. We do not specify the detailed constituents or initial formation mechanism ofthe EWS ball, although we will occasionally reference model-speciﬁc considerations for thenon-topological soliton that is made of a complex scalar ﬁeld Φwith a Higgs-portal couplingto the SM (see appendix Aand ref. [8] for details). We express the EWS ball abundanceas a fraction of the dark matter (DM) abundance ρ=fDM ρDM. It is convenient to writethis in terms of the yield (with s=2π245 g∗sT3the entropy density and g∗sthe number ofeﬀective relativistic degrees of freedom):Y=ns≈5.1×10−17 fDM 107GeVM!.(2.1)Here, n=ρ /M is the EWS ball number density, and fDM can be a function of temper-ature Tif the ball has some non-trivial evolution history.2.1 Out-of-equilibrium conditionFirst, we address the out-of-equilibrium condition which creates a baryon asymmetry insideof the EWS ball. Here, we assume the existence of a CP-violating interaction at the EWSball wall, which we later justify.The basic mechanism at play is that there is a localized loss of equilibrium around thewall of the EWS ball as it passes through the plasma and disturbs it. In particular, aswe will show, the wall can generate a localized CP-asymmetry in its wake, speciﬁcally adiﬀerence in the abundance of left-handed top quarks compared to right-handed anti-topquarks. Chirality-ﬂipping interactions via the weak force tend to damp this asymmetry. Solong as the asymmetry persists locally within part of the EWS ball, the out-of-equilibriumcondition is satisﬁed. In the remainder of this subsection, we determine the chemicalpotential for this CP asymmetry using the diﬀusion equation.The diﬀusion equation for the CP asymmetric number density of fermions in the pres-ence of a CP-violating source is given by [20–22]∂nCP (~x, t)∂t −D∇2nCP (~x, t)+ΓnCP(~x, t) = S(~x, t).(2.2)Here, Dis the fermion diﬀusion constant and Γis the rate of chirality damping, includingspin-ﬂip interactions (for the top quark, D≈6/T and Γ≈T /100 [23]). As the EWSball moves in the plasma, before the ball changes directions, the incoming particle source– 4 – JHEP10(2021)147comes from one direction. In the limit of large ball radius with RpD/Γ, one canapproximately treat the problem as a one-dimensional problem. Provided that any changein the direction and velocity of the EWS ball rapidly leads to a quasi-static solution, wecan seek a steady state solution in which there is a ﬁxed moving source with velocity vw∼pT/M for the balls thermalized with plasma. The solution is nCP (x, t) = nCP (x−vwt),and the equation can be reduced to a single variable x:(−vw∂x−D ∂2x)nCP + Γ nCP =S(x).(2.3)This equation can be solved using a Green’s function [the solution when S(x) = δ(x−x0)]given byG(x−x0) = D−1k+−k−exp[−k+(x−x0)] x>x0exp[−k−(x−x0)] x<x0,(2.4)where,k±=vw2D 1±s1 + 4ΓDv2w!.(2.5)In the limit of v2wΓD,k±≈ ±pΓ/D.For large R, the ball can be approximated as two walls/sources propagating back-to-back but with opposite CP-violating sign. Thus, the source term is approximatelyS2(x) = S(x+R)−S(x−R), where the source function at each wall is given by [17,22]S(x)≈vwy2fD T 2θ0006.(2.6)Here yfis the fermion Yukawa coupling to the Higgs ﬁeld; θ(x)is the wall proﬁle of theCP-violating angle, which can be approximated by θ0(x) = ∆θ δ(x)with ∆θas the anglechanges from its outside to inside values [17]. The derivation of (2.6) uses the WKBapproximation, which is valid when the wall thickness ∼v−1is larger than the de Brogliewavelength of the particles ∼(3T)−1. Because of this, we must assumeR&v−1,(2.7)which is consistent with the previous condition RpD/Γto use a one-dimensionaldiﬀusion equation. With this source term, the CP asymmetric number density is given bynCP (x) = Z∞−∞G(x−x0)S2(x0)dx0=vwy2fT2∆θ6(k+−k−)k2+exp[−k+(x+R)] −k2−exp[−k−(x−R)].(2.8)For the EWS ball, the region of interest is x∈(−R , R ), where the sphalerons are active.In the limit of vw√ΓDand |k−R| ≈ |k+R|  1, the radius-averaged number densityinside the ball isnCP =RR−RnCP (x)dx2R≈v2wy2fT2∆θ24 R√ΓD.(2.9)– 5 – JHEP10(2021)147Notice that due to the cancellation of the contributions from the two walls in (2.8), theaverage number density is proportional to the square of the wall velocity. Also, from (2.8),the number density is localized around a distance 1/|k±| ≈ pD/Γwithin the ball wall.This is larger than the thickness of the wall, justifying the above approximation of θ(x)asa step function for the integral in (2.8).The CP-violating chemical potential can then be evaluated by µCP = 2 nCP /T 2[24].For the top quark, Γ≈T /100,D≈6/T , and yf≈1[23], givingµCP ≈v2w∆θR.(2.10)The EWS balls are not actually moving in one direction because of their inter-actions with SM particles in the plasma. The ball thermalization rate Γtherm 'Pi(niσivrel,i)T /M is related to the EWS ball’s cross section σi'πR2|Ri|2with SMparticles in the plasma (indexed by i) whose number density is ni. The reﬂection proba-bility |Ri|2∼1for semi-relativistic or non-relativistic species, while niis suppressed fornon-relativistic species. Thus, Γtherm ∼g T 4R2/M , with gthe number of semi-relativisticdegrees of freedom in the bath. To justify our previous estimation for nCP , the thermaliza-tion rate is required to be smaller than the diﬀusion rate, or Γtherm .D|k±|2≈Γ, whichgivesM&g T 4ΓR2≈103T3R2,(2.11)where g≈9, taken to be the Wand Zdegrees of freedom assuming CAB takes placeshortly after the EWPT. When the thermalization rate is very high in violation of (2.11),non-steady solutions should be obtained to estimate the nCP . In the ball rest-frame, thesource S(~x, t)in (2.2) should vary in time intervals of order 1/Γtherm. We do not calculatenCP for this case in this work and will restrict the model parameter space to satisfy (2.11).As we will later see in eq. (3.1), the region where (2.11) is not satisﬁed is anyways mostlyexcluded by other unrelated considerations.2.2 Baryon number violationWith the chemical potential inside the EWS ball derived in the previous section, a netbaryon asymmetry can be produced via baryon number violating sphalerons. When theball moves slowly, the EW sphaleron can convert the CP asymmetry into baryon asymmetryvia the “non-local” way [21].The baryon number generated per unit time by a single EWS ball is given by [17]dNBdt ≈ −Γsph R3µCPT,(2.12)where Γsph ≈25 α5wT4≈10−6T4is the sphaleron rate per unit volume [24]. Here, we haveused that Γsph/T 3<|k±|, implying particles undergo less than one sphaleron transitionon average before the asymmetric region near the wall diﬀuses. The density of producedbaryon asymmetry nBis related to the EWS ball density bydnBdt + 3HnB=−ΓsphR3µCPTn . (2.13)– 6 – JHEP10(2021)147Using dnBdt + 3HnB=s dYB/dt with sas the entropy density and dT/dt =−H T ,dYBdT =Γsph R3H TµCPTY . (2.14)Using H(T) = 16.6×(g∗/100)1/2T2/Mpl (assuming radiation domination), this equationcan be integrated from an initial temperature Tito obtain (assuming constant Y)YB= 1.9×10−10 fDM R1GeV−12 108GeVM!2∆θ−1 Ti100 GeV2.(2.15)In general, Tiwill be the smaller of either the EW phase transition temperature TEWPT(so that the Higgs has a nonzero VEV outside the EWS ball) or the EWS ball formationtemperature Tform. This quantity is related to the baryon-to-photon ratio by YB=η/7.04 ≈0.85 ×10−10 to explain the observed baryon asymmetry.2.3 CP violationCP violation can be introduced via a simple eﬀective operator coupling the EWS ball tothe SM. For example, the constituent scalar ﬁeld Φof the EWS ball (see appendix Aforexample models) can have a CP-violating interaction with the top quark [25] given byL ⊃ ytQLeH 1 + ηΦΦ†Λ2!tR+h.c. , (2.16)where eH=iσ2H∗and ηis a complex parameter taken to be eiπ/2. If |Φ| ≡ φ(r)variesinside and outside the ball, this induces a spatially varying complex mass for the top quark,mt(r) = yt√2h(r) 1 + iφ(r)2Λ2!=|mt(r)|eiθ(r),(2.17)where tan θ(r) = φ(r)2/Λ2. Outside the EWS ball, φ(r)is constant, and the phase can beabsorbed by a redeﬁnition of ﬁelds. Near the EWS ball, the change in phase is physical,inducing a CP-violating angle |∆θ| ≈ f2/Λ2for f.Λ, where fis the change in the φ(r)between the interior and exterior of the ball.For this model to provide an EWS ball, there must be a Higgs portal coupling tothe Φﬁeld [8,9] (see appendix A) with Lportal =λΦΦ†H H †with |λ|f2&v2. The signof λdepends on whether φ(r)2increases or decreases inside the ball, which will diﬀerdepending on the model. Then, this portal coupling in conjunction with the interactionterm in (2.16) leads via a Φloop diagram to a dimension-6 CP-violating operator involvingonly SM particles,L6∼iytλ16π2|H|2Λ2QLeHtR+h.c. (2.18)The coeﬃcient of this operator is constrained by the current upper limit on the elec-tron dipole moment |de|<1.1×10−29 ecm by the ACME collaboration [26]. Thus,Λ> v (λ/0.13)1/2[27]. If Λ&f&v, the EWS ball can simultaneously satisfy |λ|f2&v2,|∆θ|.1, and the ACME constraint. In other words, the scale of the new physics neednot be much larger than the electroweak scale while still allowing O(1) CP violation at theEWS ball wall.– 7 – JHEP10(2021)1473 Abundance of EWS balls and their evolutionWe now turn our attention to the parameter space where EWS balls can facilitate baryoge-nesis. We will require throughout that the mass-radius relation of the EWS balls satisﬁesM&4π3v4R3+ 4πv3R2.(3.1)This represents the minimum mass contribution from the Higgs ﬁeld, assuming the diﬀer-ence in the vacuum energy between the EWS region and the normal vacuum ∆V&v4forthe ﬁrst term. Depending on the EWS ball model, additional contributions to the masscould come from other ﬁelds or matter (e.g., from the Φﬁeld for a Q-ball, the dark Higgsand dark gauge ﬁelds for a dark monopole, or the black hole in a magnetically chargedblack hole). We saw in section 2.3 that the scale of the EWS ball physics could be near theweak scale or much larger. Thus, these additional mass contributions may be of the sameorder as (3.1) or much larger, making this a generic lower bound. As an aside, for somemodels, it is possible to have a ﬁne-tuned cancellation between terms so that ∆V≈0,suppressing the ﬁrst term of (3.1). In this case, the gradient energy of the ﬁelds near theball wall provides the irreducible contribution from the second term. We will not considerthis tuned scenario in what follows, and it would anyways not aﬀect our conclusions.3.1 The EWS ball as dark matterAs a simplest ﬁrst assumption, we can take the EWS ball yield Yto be constant beforethe EWPT, with its abundance determined at some higher temperature scale. Then, thebaryon asymmetry is ﬁxed by eq. (2.15). It is maximized when fDM = 1, in which case theEWS ball also plays the role of dark matter.The parameter space where EWS balls can catalyze baryogenesis is shown in ﬁgure 2,setting fDM = 1. The conditions of producing enough asymmetry in (2.15) (dashedblue line, excluding the region to its right by demanding |∆θ| ≤ 1and Ti< TEWPT ∼100 GeV [28,29]), having a long enough thermalization time in (2.11) (purple), satisfyingthe mass condition (3.1) (orange), and having a large enough radius in (2.7) (green) areimposed. It is clear that these conditions allow for successful baryogenesis.However, the relic abundance of EWS balls can be probed at direct detection ex-periments. Following [8,9,15], we approximate the EWS ball-nucleus scattering crosssection as the smaller of the expectation from the Born approximation and the geometriccross section,σA≈min 16π9m2NA4y2hN N v2R6,2πR2,(3.2)where mNis the nucleon mass, Ais the nucleus’s atomic mass number, and yhNN ≈1.1×10−3. Direct detection experiments like Xenon1T [30] can then place upper boundson surviving EWS ball radii as shown by the dotted red curve in ﬁgure 2. Due to thecombination of constraints, it is clear that it is not possible to explain both baryogenesisand dark matter simultaneously with EWS balls. Further, even if fDM is reduced, thedirect detection constraint on Rwill weaken only by f−1/6DM [see eq. (3.2)] while the baryonasymmetry is proportional to fDM , leaving no available parameter space that satisﬁes allconstraints.– 8 – JHEP10(2021)147Figure 2. Bounds on the parameter space for EWS balls to catalyze baryogenesis. Blue curvesshow the constraints from requiring suﬃcient baryogenesis to explain the observed asymmetry. Thesolid blue curve shows the constraint assuming the EWS ball abundance is set by annihilationsand includes an enhancement factor ξ(y) = 10 by assuming annihilations are taking place duringbaryogenesis and Tform =Ti; see (3.7). The dashed blue curve shows the constraint if fDM isﬁxed to unity, with no enhancement factor for annihilations; see (2.15). Both take Ti= 100 GeVand |∆θ| ≤ 1. The orange curve enforces the mass-radius relation in (3.1), while the green limitsR > v−1from (2.7). Purple enforces that the thermalization time must be long enough accordingto (2.11). Finally, the red dotted line shows the direct detection limits from Xenon1T [30]. Itassumes fDM = 1, but it does not impose any constraint if the EWS balls decay after catalyzingbaryogenesis.3.2 EWS ball annihilationsClearly, a ﬁxed value for fDM is constrained by the competition between direct detectionconstraints and creating a large enough baryon asymmetry. Instead, one could considerthat fDM evolves over time. If the EWS ball abundance is larger near T∼100 GeVthan it is today, then baryogenesis could still be suﬃciently eﬃcient while evading directdetection constraints. In this subsection we will show that fDM can evolve due to EWSball annihilations, but the impact on baryogenesis is relatively small.Here, consider a purely symmetric population of EWS balls. Further, if the EWS ballis a non-topological soliton, assume that most of the Φparticles are bound inside EWSballs. If an initial population of EWS balls form at some initial temperature Tform withabundance Yform, then some of them have a chance to annihilate. For simplicity, assumeall EWS balls have the same mass and radius. Further assume that EWS balls of oppositecharges will annihilate with geometric cross section σ∼πR2(i.e., there is no long range– 9 – JHEP10(2021)147force, just contact interactions). Then the evolution of EWS balls will go asdndt + 3Hn =−hσvrelin2⇒dYdT =hσvrel is Y 2H T .(3.3)Taking g∗=g∗s∼100 and vrel = (T /M )1/2,Y(T)−1−Y(Tform)−1=R2M−1/2g1/2∗Mpl (T3/2form −T3/2).(3.4)For this to make a diﬀerence in YB,Tform must be close to the EWPT temperatureTEWPT. The reason: if Tform is much larger, then Y(TTform)is constant, reducingto the previous case of constant fDM; on the other hand, if Tform ≥Tiis much smallerthan TEWPT, the baryon asymmetry is suppressed, going as T2iin (2.15). If we tuneTform ∼TEWPT, we can plug Y(T)into (2.14) to obtain the ﬁnal baryon asymmetry YB.This gives an enhancement factor to the right side of eq. (2.15):ξ(y) = 43log(y),for y≡Y(Tform)Y(TTform)1,(3.5)assuming Tform =Ti< TEWPT, where fDM in (2.15) would be calculated using Y(TTform). At most, we expect ξ(y)to give an O(10) enhancement due to the logarithmicdependence on y.Thus, including the time evolution of the EWS balls due to annihilations does notprovide a large enough baryogenesis enhancement to escape direct detection bounds. It isworth noting that EWS ball annihilations were assumed instantaneous and complete in thiscalculation. If the annihilations were to take a long enough time, e.g., because the EWSballs form metastable bound states before annihilating, then baryogenesis could proceedwith a larger EWS ball abundance than assumed here.3.3 EWS ball decayAnother possibility for time dependence is to allow the EWS balls to decay. Then, theycould exist shortly after the EWPT, but decay long before today, evading direct detec-tion constraints. They may even have an abundance larger than dark matter in theearly universe.The EWS ball abundance after annihilations in the limit Y(Tform)Y(T)andTform Tfrom (3.4) is,Y(TTform) = 10−18 M107GeV1/20.1GeVR2200 GeVTform 3/2.(3.6)Then, using (2.1), (2.15), (3.5), and (3.6),YB= 3.8×10−11107GeVM1/2∆θ−1200 GeVTform 3/2Ti100 GeV2ξ(y)10 .(3.7)Thus, the necessary baryon asymmetry could be generated for M.107GeV. This isshown in ﬁgure 2by the solid blue vertical line (the dotted red direct detection constraint– 10 – JHEP10(2021)147and dashed blue fDM = 1 constraint can be ignored here, since the abundance is assumedset by annihilations and decays well before today). Note that Tform cannot be much largerthan TEWPT, otherwise not enough baryon asymmetry is produced (in conjunction withlower bounds on Min ﬁgure 2). Also, since Ti≤Tform , the formation temperature cannotbe too small either. In all cases, Ti≤TEWPT is required.3.4 Asymmetric EWS ballsIf the dark sector responsible for EWS ball formation itself carries an asymmetry, then theEWS ball annihilations will halt once the symmetric component is depleted. This presentsadvantages compared to the prior symmetric case. First, it could allow Ywell in excessof (3.6), enhancing the amount of baryogenesis. Second, it removes any dependence for YBon Tform, as long as Tform ≥TEWPT . Third, a model-dependent advantage for the speciﬁccase of EWS Q-balls is that free Φantiparticles are also depleted, making it easier for Q-balls to survive without assuming every Φparticle must be bound in a Q-ball [31]. For theEWS ball abundance to be compatible with baryogenesis, it would be constrained by directdetection at low mass and overclosure at high mass if the EWS balls were stable. Thus,this possibility still depends on the EWS balls being unstable and evaporating between theEWPT and today.The asymmetric abundance is constrained by requiring the decay products of the EWSballs not contribute too much to the eﬀective radiation degrees of freedom during BBNor recombination. The most conservative bound comes from assuming EWS balls decayshortly after the EWPT. As shown in the next section, this amounts to the requirementfDM .2×1010 in eq. (2.15). Clearly, this opens a large swath of parameter space in R,M, and ∆θthat is not accessible when the EWS-ball-forming sector is symmetric, whichis depicted in ﬁgure 3.4 Evaporation of EWS ballsSupposing that the EWS ball can evaporate, we denote the rate of evaporation by Γevp .Such evaporation can happen, for example, by directly decaying the ﬁeld Φmaking upthe EWS ball into new dark sector particles. For instance, if there is a U(1)Φ-breakingcoupling of the form LΦχ=gΦχΦ¯χχ +h.c., then evaporation can happen via decaying intoχparticles, which we take to be massless for simpicity, each carrying energy of M /(2Q)with Qas the total U(1)Φcharge of an EWS ball. The massless products of this decaythen redshift as radiation. The energy contained in the χparticles will contribute to theeﬀective number of relativistic degrees of freedom, ∆Neﬀ, which is constrained by probesof BBN and CMB measurements. Alternatively, if the EWS balls decay into SM particlesbefore BBN (Tt from .We will take a model-independent approach to evaporation in which the temperatureat which EWS balls evaporate (Tevp ), i.e., the temperature at which H(Tevp)=Γevp ,is a phenomenological parameter. The total energy density contained in EWS balls isfDM ρDM(T)which redshifts as matter from T= 100 GeV to T=Tevp. Hence the total– 11 – JHEP10(2021)147Figure 3. Same as ﬁgure 2, but now allowing an asymmetric component of EWS balls thateventually decay into massless states in the dark sector. Constraints from ∆Neﬀ limit wherebaryogenesis can produce large enough η. Examples for present (projected) constraints areshown in solid (dashed) blue lines for two diﬀerent possible evaporation temperatures. BecauseTevp < TEWPT ∼V is required for baryogenesis by EWS balls, the shaded blue region on theright is completely excluded by current constraints. If the dark sector states are massive, additionalconstraints can apply as discussed in the text.energy density that is converted to χ-particles is given byρχ(T=Tevp)=0.14 fDM107g∗s(Tevp)60  Tevp1V3GeV4.(4.1)The contribution of ρχto ∆Neﬀ at late times that the CMB is sensitive to is given by∆Neﬀ ≈871144/3ρχ(TCMB)ργ(TCMB).(4.2)Since the radiation produced in the evaporation of the EWS balls is decoupled from theSM bath, their density scales as ρχ∝1/a4∝g4/3∗s(T)T4. From this, we determine thecontribution to the energy density at the temperature of recombination and ﬁnd that∆Neﬀ ≈0.025 fDM107 Tevp1GeV−1 60g∗s(Tevp)!1/3.(4.3)Large fDMonding to more energy density in EWS balls, and small Tevpond-ing to the late decay of EWS balls, are more constrained. The current strongest limit comesfrom the CMB epoch given by the Planck 2018 observations ∆Neﬀ <0.51 [2,32]. For suc-cessful baryogenesis, Tevp .TEWPT ∼V is required, setting a conservative bound– 12 – JHEP10(2021)147fDM .2×1010. Next-generation observations from CMB-S4 experiments are projected toimprove sensitivity by an order of magnitude to σ(Neﬀ)<0.03 [33].Alternatively, if the χhave mass mχ>0, they could explain dark matter providedmχ∼M /(2QfDM). They must not be too light, otherwise they will free stream fortoo long and suppress structure formation. They become non-relativistic at the time theuniverse has reached temperature TNR ∼Tevp/fDM. Their free-streaming length up tomatter-radiation equality (at temperature Teq), when perturbations become Jeans un-stable, is λFS ∼2MplT0TNR 1 + log[ TNRTeq ][34], where T0= 2.35 ×10−4eV is the temper-ature today. The bound on TNR is similar to the bound on the thermal warm darkmatter mass: TNR &1keV [35–38], corresponding to λFS .1Mpc. This sets a boundfDM .108×(Tevp/100 GeV), a bit stronger than the ∆Neﬀ constraint for this case. If mχis smaller than the quantity above, χonly makes up a subdominant component of darkmatter, and the free streaming constraint is correspondingly relaxed.5 Discussion and conclusionsWe have used the EWS ball as a representative model to implement the more generalcatalyzed baryogenesis mechanism. One could consider other possible models to introducebaryon number violating interactions. For instance, the constituents of the catalyst ballcan interact with SM particles via baryon number violating higher-dimensional operators.If those operators mainly contain the third or second-generation of quarks, the directdetection constraints could be dramatically relaxed and the catalyst balls can exist inthe current universe and contribute to a signiﬁcant fraction of dark matter. Similarly,lepton-number violating operators can also be adopted to catalyze leptogenesis before theelectroweak phase transition. The generated lepton asymmetry is then converted into abaryon asymmetry by the electroweak sphaleron process. Additionally, one could use thecatalyst mechanism to generate an asymmetry for a new particle that carries both baryonand dark matter number. The later decays of this new particle can provide a uniﬁed originfor baryon and dark matter asymmetry [39].In summary, we have proposed a novel mechanism to generate the baryon asymmetrythat is similar to the catalytic reaction in chemistry. To generate enough baryon asymmetry,the catalyst balls are preferred to have a smaller mass and a larger radius. We have used theelectroweak-symmetric ball to guide our discussion of this general catalyzed baryogenesismechanism. For the EWS ball abundance determined by their annihilations, the EWS ballmass and radius are required to be around 106GeV and 10−2GeV−1to explain the observedbaryon asymmetry. Interestingly, EWS ball relics within this region of parameter spaceare already excluded by direct detection constraints, so they are required to decay intoother states in the early universe to evade the constraints. We also discussed the case withasymmetric EWS balls with a much larger initial abundance and a wider parameter spacein Mand Rto accommodate the observed baryon asymmetry. The dark radiation fromEWS ball evaporations provide a large contribution to the eﬀective number of relativisticdegrees of freedom, which could be tested in future CMB experiments.– 13 – JHEP10(2021)147AcknowledgmentsWe thank Hooman Davoudiasl for insightful discussion. The work of YB and MK issupported by the U.S. Department of Energy under the contract DE-SC-0017647. The workof JB is supported by start up funds from Colorado State University. The work of NO issupported by the Arthur B. McDonald Canadian Astroparticle Physics Research Institute.A Electroweak-symmetric ball modelsIn this appendix, give details for possible EWS ball models to act as catalysts. While theresults of our work do not depend on the detailed model underlying the EWS ball, thissection demonstrates that such underlying models do exist. We give brief overviews of twosuch models here, and full details (including formation mechanisms) can be found in thecorresponding references. It is worth mentioning a third possibility for EWS objects is amagnetically charged black hole [12–14], but their masses are greater than the Planck scaleand thus too heavy to be of interest according to ﬁgures 2and 3.A.1 EWS Q-ballThe benchmark model that most closely replicates the EWS ball properties in the mainbody is the EWS Q-ball [8]. It is a nontopological soliton solution resulting from theinteraction of a complex scalar Φthat couples to the Higgs ﬁeld H. If the Φﬁeld ischarged under a global U(1) symmetry, the most general renormalizable potential for theseﬁelds isV=λh H†H−v22!2+λφhΦ†ΦH†H+m2φ,0Φ†Φ + λφ(Φ†Φ)2.(A.1)Here, all coupling coeﬃcients are taken to be positive, so the ﬁeld VEVs are h|H|2i=v2/2and hSi= 0.Consider the time-dependent, spherically symmetric ansatz Φ = e−iωtφ(r)/√2alongwith a spherically symmetric ansatz for the Higgs ﬁeld H>= (0, h(r)/√2). This ansatzhas a conserved global chargeQ=iZd3x(Φ†∂tΦ−Φ∂tΦ†)=4πω Z∞0dr r2φ2.(A.2)Then, the equation of motion for φis (omitting that of hfor brevity)φ00 +2rφ0+∂φUeﬀ(φ)=0,(A.3)Ueﬀ(φ) = 12ω2φ2−14λφhφ2h2−14λφφ4,(A.4)where ω2=ω2−m2φ,0. The ﬁelds are subject to the boundary conditions φ0(0) = h0(0) =φ(∞)=0and h(∞) = v. Further demanding an EWS solution with h(0) = 0 amounts torequiring φ(0) > vq2λh/λφh alueof Φ. The solution for φ(r)alue φ(0), thenrolling down the potential Ueﬀ(φ)until coming to rest at the unstable ﬁxed point φ= 0– 14 – JHEP10(2021)147as r→ ∞. Such a solution exists if (λφλh/2)1/4≡ωc/v < ω/v < pλφh, with the lower(upper) limit corresponding to large (small) Qsolutions. In the large Qlimit, when ω≈ωc(and the corresponding value for ω≈ωc), it can be shown that [8]M≈Q ωc≈4π31λφω2cω2cR3.(A.5)For quartic couplings of order unity, ωc≈ωc≈v. Thus, this mass saturates the inequalityin (3.1), the latter of which is dominated by the ﬁrst R3term in this limit.A.2 EWS monopoleA second interesting case is that of the EWS monopole [9]. The renormalizable potentialfor the EWS monopole is very similar to (A.1), but changing Φto a triplet scalar of agauged SU(2) Φawith a= 1,2,3:V=λh(H†H)2+µ2hH†H−12λφh|Φ|2H†H−12m2φ,0|Φ|2+14λφ|Φ|4(A.6)with |Φ|2=Pa(Φa)2. Notice that the signs of the m2φ,0,µ2h, and λφh terms have ﬂippedwith respect to (A.1). The magnitudes of the terms should be arranged such that both Φand Hhave a VEV. I.e., when |Φ|=ftakes its VEV, the condition λφhf2> µ2hmeansthat Hwill also have a nonzero VEV as required in the SM.Because Φspontaneously breaks the gauged SU(2) to U(1), a topological monopoleconﬁguration exists with |Φ|= 0 at its center. Because of this, Hhas a positive mass-squared term inside of the monopole. Thus, the Higgs ﬁeld is driven to its symmetry-preserving value inside the monopole, leading to an EWS monopole.The mass-radius relation can also be determined. The monopole mass (not accountingfor the Higgs vacuum energy contribution) is known to beM=4πfgY(λφ/g2),(A.7)where gis the SU(2) gauge coupling of Φto the gauge ﬁeld and Y(x)is a monotonicfunction ranging from Y(0) = 1 to Y(∞)≈1.787. The EWS radius, i.e., the radius forwhich EW symmetry is restored, depends on the characteristic radius of the scalar ﬁeld Φ.Once Φis large enough, the EW symmetry is spontaneously broken. 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