AFFLECK-DINE LEPTOGENESIS WITH VARYING PQ SCALE
Kyu Jung Bae,
Center for Theoretical Physics of the Universe,
based on JHEP 1702 (2017) 017 with H. Baer, K. Hamaguchi and K. Nakayama
"Testing CP-Violation for Baryogenesis" @UMass-Amherst
Mar. 29, 2018 INTRODUCTION • Baryogenesis via Leptogenesis - Due to (B-L)-conserving and (B+L)-violating process makes Lepton asymmetry Baryon asymmetry - Neutrino physics can show its footprints. • Affleck-Dine mechanism - scalar field dynamics in SUSY: CPV in SUSY breaking parameters - Along LHu direction: lepton number generation light neutrino mass required <10-9 eV; neutrinoless double beta decay
• Varying PQ scale
- PQ scale ~ Mp during leptogenesis but fa~109-12 GeV afterwards neutrino mass ~10-4 eV; suppress axion isocurvature INTRODUCTION • Dine-Fischler-Srednicki-Zhitnitsky model - SUSY DFSZ model provides strong CP solution, mu-term, also RHN mass - Dilution from saxion decay determines final lepton(baryon) asymmetry
- suppress unwanted lepton number violation during saxion oscillation 14
[ ] = = [ ] = / = 1012 1012 -7 -6 10-5 -4 10-8 10-5 -4 10 10 10 10 10-3 10-3 GeV -8 GeV 10 10 1 -2 < < 10 T T 10-2 1011 1011 10-7
] 10-1 ] 10-1
[ [ -6 10 GeV 10 < T m > 0.17 eV 1010 m > 0.17 eV 1010
m > 0.48 eV 8 m > 0.48 eV 8 fa N < 5 × 10 GeV fa NDW < 5 × 10 GeV DW
109 109 10. 102 103 104 105 102 103 104 105 [ ] [ ]
FIG. 1: Contours of m⌫1 [eV] on the (µ, fa) plane to reproduce the observed baryon asymmetry. In the left (right) panel we have taken m =5µ (m = µ/5) while msoft = m = 10m (msoft = m = 50m ). The light (dark) red shaded region corresponds to T < 10 GeV (1 GeV) and the grey shaded region is excluded by the KamLAND-Zen experiment. The light-gray shaded region is constrained by Planck+BAO [34]. The light-purple shaded region indicates bound from SN1987A [35]. The blue shaded region corresponds to µ =(0.01 1)f 2/M . a P
region shows parameter space where the lightest neutrino mass is larger than the KamLAND-Zen bound [33]. We also show a bound from Planck+BAO constraint on the sum of neutrino masses,
( m⌫) < 0.17 eV [34]. The light-purple shaded region shows the bound from SN1987A [35]. The (light-)redP shaded region shows parameter space where the saxion decay temperature is smaller than 1 GeV (10 GeV). The blue shade indicates the region for which µ =(0.01 1)f 2/M .We a P 6 consider fixed TR = 10 GeV since larger TR does not change or does suppress nB/s (see Eq. (28)).
In the case where nB/s is suppressed, it requires a smaller neutrino mass that is less attractive.
From the figure, it is clearly shown that neutrino mass is large for large µ and small fa while it
becomes smaller for small µ and large fa. This feature stems from the saxion decay temperature.
The saxion decay temperature is enhanced by the µ-term while suppressed by fa. It is also of great
importance that small fa is good for obtaining a flatter direction during lepton number generation 10 as shown in Eq. (6). For µ & 1 TeV and fa . 10 GeV, our model predicts a rather large neutrino mass so that it is constrained by recent neutrinoless double beta decay (0⌫ ) experiment. For
this constraint, we take a conservative bound from KamLAND-Zen, m⌫ < 0.48 eV [33]. From the lower-right corner (µ 105 GeV, f 1010 GeV) to the upper-left corner (µ 102 GeV, f 1012 ⇠ a ⇠ ⇠ a ⇠ GeV), the resulting neutrino mass scans over 10 1 10 8 eV. OUTLINE
1. Leptogenesis
2. AD mechanism along LHu direction
3. AD leptogenesis in DFSZ model with varying
PQ scale
4. Summary OUTLINE
1. Leptogenesis
2. AD mechanism along LHu direction
3. AD leptogenesis in DFSZ model with varying
PQ scale
4. Summary ber can be used to infer the ratio of the number density of baryons to photons in the Universe, a quantity that is measured independently from theprimordialnucleosynthesis of light elements. The WMAP results [4] are in agreement with the most recent nucle- osynthesis analysis of the primordial Deuterium abundance,buttherearediscrepancies with both the inferred 4He and 7Li values [5]. These latter values, however, may have an underestimated error [6]. Averaging the WMAP result only with that coming from the primordial abundance of Deuterium gives:
nB −10 ηB =6.1 0.3 10 . (1) nγ ≡ ± × Why does this ratio have this value? In this review, we will principally try to address this last question which, as we shall see, is intimately related to the existence of a primordial matter-antimatter asymmetry. Nevertheless, we shall try, when germane, to connect our discussion with the broader issues of what constitutes dark energy and dark matter. There is good evidence that the Universe is mostly made up of matter, although it is possible that small amounts of antimatter exist [7]. However, antimatter certainly does not constitute one of the dominant components of the Universe’s energy density. Indeed, as Cohen, de Rujula, and Glashow [8] have compellingly argued, if there were to exist large areas of antimatter in the Universe they could only be atacosmicdistancescale
from us. Thus, along with the question of why nB/nγ has the value given in Eq. (1), there is a parallel question of why the Universe is predominantly composed of baryons rather than antibaryons. In fact, these two questions are interrelated. If the Universe had been matter- antimatter symmetric at temperatures of O(1 GeV), as the Universe cools further and the inverse process 2γ B +B¯ becomes ineffective because of the Boltzmann factor, the → BARYONnumber density ASYMMETRY of baryons and antibaryons relative to photons would have been reduced ⌘B is determined both from light element productiondramatically in Big Bang as a result Nucleosynthesis of the annihilation (BBN) process B + B¯ 2γ.Astraightforward → and also from CMB measurements.Baryon Asymmetry Alternatively,calculation thisof isthe gives, sometimes Universe: in this case, expressed [9]: as the baryon-to-entropy ratio nB 10 cf) if universe were nB nB¯ −18 observed: 10 =(1.7) 10 . (2) s ' symmetric nγ nγ ≃ where s 7.04n in the present epoch. Thus, in a symmetric Universe the question is really why observationally n /n is so ' • Inflation dilutes all pre-existing particles. B γ Production of the baryon asymmetry of thelarge! Universe or BAU requires mechanisms which satisfy the three Sakharov criteria: 1. baryon number violation, 2. C and CP • We need a source ofIt B is asym. very diafterfficult inflation to imagine. processes at temperatures belowaGeVthatcould violation and 3. a departure from thermal equilibrium. Early proposals such as Planck enhance the ratio of the number density of baryons relative tothatofphotonsmuch scale or GUT scale baryogenesis seem no longer viable since the BAU would have been Sakharov’s conditions: 2 inflated away during the inflationary epoch of thebeyond Universe. the value Alternatively, this quantity most attains modern when baryon-antibaryon annihilation occurs. 2 proposals for developing the BAU take place afterAn the exception end of is the provided inflationary by some versions epoch, of at Affleck-Dine Baryogenesis [10] where a baryon excess • B violation or after the era of re-heating which occurs around the re-heat temperature T . In fact, R 4 the SM contains all the• ingredientsC & CP violation for successful electroweak baryogenesis since baryon (and lepton) number violating processes can take place at large rates at high temperature T>T 100 GeV via• departure sphaleron processes from thermal [43]. Unfortunately, equilibrium these first order phase weak ⇠ transition e↵ects require a Higgs mass . 50 GeV, and so has been excluded for many years. By invoking supersymmetry, then new possibilities emerge for electroweak baryogenesis. However, successful SUSY electroweak baryogenesis seems to require a Higgs mass mh . 113 GeV and a right-handed top-squark m 115 GeV [44]. These limits can be relaxed t˜R . to higher values so long as other sparticle/Higgs masses such as m 10 TeV. Such heavy A Higgs masses are not allowed if we stay true to our guidance from naturalness: after all, 2 2 2 Eq. 1.2 requires m / tan . m /2. For heavy Higgs masses, then mA mHd and then Hd Z ⇠ from naturalness we find m 4 8 TeV (depending on tan )[45]. A . In Sec. 2, we survey several leptogenesis mechanisms as the most promising baryogen- esis mechanisms: 1. thermal leptogenesis [46, 47], 2. non-thermal leptogenesis via inflaton decay [49] 3. leptogenesis from oscillating sneutrino decay [50, 52] and 4. leptogenesis via an A✏eck-Dine condensate [53, 54, 50].
Each of these processes requires some range of re-heat temperature TR and gravitino mass m3/2, and indeed some of them run into conflict with the so-called cosmological gravitino problem [55]. In this case, gravitinos can be thermally produced in the early universe at a rate proportional to TR [56]. If TR is too high then too much dark matter arises from thermal gravitino production followed by cascade decays to the LSP. Also, even if dark matter abundance constraints are respected, if the gravitino is too long-lived, then it may decay after the onset of BBN thus destroying the successful BBN predictions of the light element abundances [57, 58, 59]. In the case of natural SUSY with mixed axion-higgsino dark matter, then similar con- straints arise from axino and saxion production: WIMPs or axions can be overproduced, or light element abundances can be destroyed by late decaying axinos and saxions. After a brief review of the several leptogenesis mechanisms in Sec. 2,inSec. 3 we show con- straints on leptogenesis in the TR vs. m3/2 planes assuming a natural SUSY spectrum. In Sec. 4, we show corresponding results in the TR vs. fa planes. We vary the PQ scale fa from values favored by naturalness f 1010 1012 GeV to much higher values. While a ⇠
–5– 2.2 B + L Violation in the Standard Model
Due to the chiral nature of the electroweak interactions, baryon and lepton number are not conserved in theB Standard & L Model VIOLATION [17]. The divergence of the B and L currents, 1 J B = q γ q + u γ u + d γ d , (4) µ 3 L µ L R µ R R µ R • In the SM, baryon & leptongenerations! "number are (accidental)# symmetry L at the tree-level.Jµ = lLγµlL + eRγµeR , (5) generations! " # • Dueis given to bychiral the triangle nature anomaly, of leptons & quarks, B & L have anomalies
µ B µ L ∂ Jµ = ∂ Jµ N = f g2W I W Iµν + g′2B Bµν . (6) 32π2 − µν µν which describes processes" with ∆B = ∆L =3,suchas# I$ % Here Nf is the number of generations, and Wµ and Bµ are, respectively, the SU(2) and • At quantum level, (B-L) is conservedc c c ′ but (B+L) is violated. U(1) gauge fields with gauge couplingsu + gd and+ cg . d +2s +2b + t + νe + νµ + ντ . (11) E → As a consequence of the anomaly, the change in baryon and lepton number is related The transition rate is determined by the instanton action andonefinds[17] to the change inM thesph topological charge of the gauge field,
4π tf −S − Γ 3 µe Binst = e α B(tf ) B(ti)= dt d x∂ Jµ − ti ∼ & & = 10−165 . (12) = Nf [Ncs(tf ) NOcs(ti)] , (7) − ! "
B = b0 Nf B = b0 B = b0 +Nf where− Because[ Asph this, ϕsph ] rate is extremely[ A, ϕ ] small, (B + L)-violating interactions appear to be com- L = l0 Nf L = l0 L = l0 +Nf 3 − g 3 IJK Ii Jj Kk pletely negligibleNcs(t)= in the2 Standardd xϵijkϵ Model.W W W However,. this picture changes(8) dramatically Figure 1.1: A Schematic behavior of the energy dependence on theFigure96 configurationπ from of hep-ph/0212305 the gauge and Higgs fields [ A(x), ϕ(x)] [6]. The minima correspond to topologically& when one is in a thermalIi bath. distinct vacuaFor with(B+L) vacuumdifferent baryon violating to (B)andlepton( vacuum vacuumL)numbers.Theconfiguration transitions transitionW is a pure gauge configuration and the Chern- [ Asph(x), ϕsph(x)] represents the saddle point of the energy functional, the sphaleron solution. Simons numbers Ncs(ti)andNcs(tf )areintegers. to the next vacuum (B = b N and L = l N )occursattherate[6] In a0 ± non-abelianf 2.30 ± Sphaleronsf gauge theory there and are the infinitly KRS many Mechanism degenerate ground states, M (T ) Γ = C(T ) T exp sph , (1.5) which differ in their!− T value" of the Chern-Simons number, ∆Ncs = 1, 2,....Thecorre- 4 ± ± where dimensionless factor C(T )dependsontheratioAs emphasizedv(T )/T and the in coupling the constants. seminal paper of Kuzmin, Rubakov and Shaposhnikov [13], in the Msph(T )representsthefreeenergyofthesphaleronconfiguration(ponding points in field space areat temperature separatedT ), by a potential barrier whose height is given which is given by [31] by the so-calledthermal sphaleron bath energy providedEsph by[18]. the Because expanding of the Universe anomaly, one jumps can in make the Chern-transitions between the v(T ) Msph(T )=4πB(T ) , (1.6) Simons numbergauge are vacuag2 associated(T ) not by with tunneling, changes of but baryon through and lept thermalon number, fluctuations over the barrier. For where B(T )dependsonthegaugecouplingg2(T )andthe4-pointcouplingconstantof the Higgs potential λ(T )asB = B(λ/g2), varying from 1.5(λ/g2 0) to 2.7(λ/g2 temperatures2 larger2 → than2 → the height of the barrier, the exponential suppresion in the )[31].TherateinEq.(1.5)shouldbecomparedwiththeHubbleexpansionrate ∞ ∆B = ∆L = Nf ∆Ncs . (9) H =(π2g /90)1/2 T 2/M .(M =2.4 1018 GeV is the reduced Planck scale and g is ∗ × G G × ∗ defined in Appendix B.2.) Then, itrate is found provided that the sphaleron byrate in the Eq. (1.5) Boltzmann indeed factor disappears completely. Hence (B+L)-violating exceeds the HubbleObviously, expansion rate in for theT>T∗,where StandardT∗ is given Model by the smallest jump is ∆B = ∆L = 3. processes can−1 occur at a significant rate and these processes can be in equilibrium in the v(T∗) MG ± T∗ 4πB(T∗) ln . (1.7) In the≃ semiclassicalexpandingg2(T∗) × # $ T∗ %& approximation,Universe. the probability of tunneling between neighboring 4See commentsvacua below. is determined by instanton configurations. In the Standard Model, SU(2) instan- The finite-temperature transition rate in the electroweak theory is determined by tons lead to an e8ffective 12-fermion interaction the sphaleron configuration [18], a saddle point of the field energy of the gauge-Higgs
system. FluctuationsOB+L around= this(qLiq saddleLiqLilLi point) , have one negative eigenvalue,(10) which allows i=1'...3 one to extract the transition rate. The sphaleron energy is proportional to vF (T ), the 7 finite-temperature expectation value of the Higgs field, and one finds 8π E (T ) v (T ) . (13) sph ≃ g F Taking translational and rotational zero-modes into account, one obtains for the transi- tion rate per unit volume in the Higgs phase [19]
7 ΓB+L M = κ W e−βEsph(T ) , (14) V (αT )3
2 where β =1/T , MW = g vF (T )/2andκ is some constant. Extrapolating this semiclassical formula to the high-temperature symmetric phase, where v (T )=0,andusingforM the thermal mass, M g2T ,oneexpectsinthis F W W ∼ phase Γ /V (αT )4.However,detailedstudieshaveshownthatthisnaiveextrapo- B+L ∼ lation from the Higgs to the symmetric phase is not quite correct. The relevant spatial 8 scale for non-perturbative fluctuations is the magnetic screening length 1/(g2T ), but ∼ the corresponding time scale turns out to be 1/(g4T ln g−1), which is larger for small coupling [20, 21]. As a consequence one obtains for the sphaleron rate in the symmetric phase Γ /V α5 ln α−1T 4. (15) B+L ∼ It turns out that the dynamics of low-frequency gauge fields can be described by aremarkablysimpleeffective theory, derived by B¨odeker [21]. The color magnetic and electric fields satisfy the equation of motion
B & LD⃗ VIOLATIONB⃗ = σE⃗ ζ⃗ . (16) × − • HereAt ζ⃗highis Gaussian temperature, noise, a random vector field with variance
(B+L) violating′ transition4 ′ ζi(x)ζj(x ) =2σδijδ (x x ) . (17) scale forvia⟨ non-perturbative thermal⟩ fluctuation fluctuations− is the magnetic screening length 1/(g2T ), but E ∼ These equations definethe correspondinga stochastic three-dimensional time scale turns gaug outetheory.Theparameter to be 1/(g4T ln g−1), whichσ is is larger for small 2 “sphaleron” the ‘color conductivity’,couplingσ = [20,mD 21]./(3γ As), where a consequencemD gT oneis the obtains Debye for screening the sphal masseron and rate in the symmetric Msph ∼ γ g2T ln(1/g)isthehardgaugebosondampingrate.Toleading-logaccuraphase cy one has ∼ 1/σ ln g−1.Anext-to-leadingorderanalysisyieldsforthesphaleronΓ /V α5 ln α−1rateT 4. [22] (15) ∼ B+L ∼ Γ gT 2 m 1 B+L =(10.8It0 turns.7) out thatα5T the4 ln dynamicsD +3 of.041 low-frequency + gauge. fields(18) can be described by V ± mD # $ γ % $ln (1/g)%& B = b0 Nf B =aremarkablysimpleeb0 !B = b0 +"Nf ffective theory, derived by B¨odeker [21]. The color magnetic and − [ Asph, ϕsph ] [ A, ϕ ] L = l N L = l L = l +N The overall0 − f coefficientelectric0 has been fields determined satisfy0 f the byequation a numerical of motion lattice simulation [23]. From Figure from hep-ph/0212305 FigureEq. 1.1: A (18) Schematic one behavior easily of the energyobtains dependence the on temperaturethe configuration of range where sphaleron processes are in the gauge and Higgs fields [ A(x), ϕ(x)] [6]. The minima correspond to topologically D⃗ B⃗ = σE⃗ ζ⃗ . (16) distinctthermal vacua with di equilibrium:fferent baryon (B)andlepton(L)numbers.Theconfiguration × − [ A•sph(x(B+L)), ϕsph(x)] represents violating the saddle point ofinteraction the energy functional, the sphaleron is in thermal equilibrium for solution. Here ζ⃗ is Gaussian noise, a random vector field with variance 12 TEW 100 GeV 2.4 Electroweak Baryogenesis and its Experimental Con- straints An important ingredient in the theory of Baryogenesis is related to the nature of the elec- troweak transition from the high-temperature symmetric phase to the low-temperature 9 2.5 The Relation Between Baryon and Lepton Asymmetries In a weakly coupled plasma, one can assign a chemical potential µ to each of the quark, lepton and Higgs fields. In the Standard Model, with one Higgs doublet H and Nf 5 generations one then has 5Nf +1 chemicalpotentials. For a non-interacting gas of massless particles the asymmetry in the particle and antiparticle number densities is given by 3 3 gT βµi + (βµi) , fermions , 2.5 The Relation Betweenni ni = Baryon andO Lepton Asymmetries (21) − 6 ⎧ 2βµ + $(βµ )3% , bosons . ⎨ i O i In a weaklyThe coupled following plasma, analysis one is can based assign on a these chemical relations potenti$ foralβ%µµ to each1. However, of the quark, one should keep ⎩ i ≪ lepton andin mind Higgs that fields. the In plasma the Standard of the early Model, Universe with one is very Higgs diffdoubleterent fromH and a weaklyNf coupled 5 generationsrelativistic one then gas, has owing 5Nf to+1 the chemicalpotentials. presence of unscreenedFor non-a a non-interactingbelian gauge interactions, gas of where masslessnonperturbative particles the asymmetry effects are in important the particle in and some antipa cases.rticle number densities is given by Quarks, leptons and3 Higgs bosons interact3 via Yukawa and gauge couplings and, in gT βµi + (βµi) , fermions , n n = O (21) addition, viai thei nonperturbative⎧ sphaleron$ 3% processes. In thermal equilibrium all these − 6 2βµi + (βµi) , bosons . processes yield constraints⎨ betweenO the various chemical potentials [35]. The effective The following analysis is based on these relations$ for β%µ 1. However, one should keep interaction of Eq. (10) induced⎩ by the SU(2)i electroweak≪ instantons implies 2.5 Thein mind Relation that the Between plasma of theBaryon early Universe and Lepton is very Asymmetries different from a weakly coupled relativistic gas, owing to the presence of unscreened(3µ non-aqi + µbelianli)=0 gauge. interactions, where (22) In a weaklynonperturbative coupled plasma, eff oneects can are assign important a chemical in some& potentii cases.al µ to each of the quark, lepton and HiggsQuarks,One fields. leptons also In hasthe and toStandard Higgs take the bosons Model,SU(3) interact with Quantum one via Higgs Yukawa Chromodynamicsdoublet and gauH geand couplings (QCD)Nf instanton and, in processes generations one then has 5N +1 chemicalpotentials.5 For a non-interacting gas of 2.5 Theaddition, Relationinto via Between the account nonperturbativef [36], Baryon which generate sphaleron and Lepton an processes. effective Asymmetries In interactionthermal equilibrium between left-handed all these and right- massless particles the asymmetry in the particle and antiparticle number densities is processeshanded yield constraints quarks. The between corresponding the various relation chemical between potentials the chem [35].ical The potentials effective reads In agiven weakly by coupled2.5 plasma, The one Relation can assign a Between chemical potenti Baryonal µ to and each Lepton of the quark, Asymmetries interaction of Eq. (10)3 induced by the SU3(2) electroweak instantons implies lepton and Higgs fields. In thegT Standardβµ Model,i + (β withµi) one, fermions Higgs doublet, H and Nf n n = O (2µqi µui µdi)=0. (21) (23) In a weaklyi i coupled⎧ plasma, one$ can assign35% a chemical− − potential µ to each of the quark, generations one then has− 5Nf +16 chemicalpotentials.2βµi + (βµi) i For, bosons a non-interacting. gas of ⎨ O (3µqi&+ µli)=0. (22) massless particleslepton the asymmetry and Higgs in fields. the particle In the and Standard antipa Model,rticle number with one densities Higgs doublet is H and Nf $i % LEPTOGENESIS The following analysisAthirdcondition,validatalltemperatures,arisesfromth is based on⎩ these relations& for βµi 1. However,5 one shoulderequirementthatthetotal keep given by generations one then has 5Nf +1 chemicalpotentials.≪ For a non-interacting gas of in mind that the plasma of the early Universe is very different from a weakly coupled One alsomasslesshypercharge has to particles take3 the of theSU the asymmetry(3) plasma Quantum3 vanishes. in the Chromodynamics particle From and Eq. antipa (21) (QCD)rticle and instanton the number known densities processes hypercharges is one gT• Numberβµi + ( βforµi) asymmetry, fermions , relativistic gas,n owingi ni to= the presence of unscreenedO non-abelian gauge interactions,(21) where into accountgiven−obtains by [36],6 which⎧ 2β generateµ + $(β anµ e)3ff%ective, bosons interaction. between left-handed and right- ⎨ i i nonperturbative effects are important inO some3 cases.βµ + (βµ )3 , fermions2 , handed quarks.this is the The case corresponding for gauge$gT interactions.µ relationqi +2% µiui between Onµ thedi other theiµli chem hand,µeiical+ Yukawa potentialsµH interactions=0 reads. are in (24) The following analysis is based on⎩ theseni relationsni = for βµi 1.O However,3 one shouldN keep (21) Quarks, leptons and Higgs bosons− interacti '6 via⎧ 2 Yukawaβµ≪+− $ and(β−µ gau) %ge−, couplingsbosons .f and,( in in mind that the plasmaequilibrium of the early only in Universe a more& restricted is very⎨ di temperatureffi erentO fromi range a weakly thatdependsonthestrengthof coupled addition, via the nonperturbative sphaleron(2 processes.µqi µui In thermalµ$di)=0% equilibrium. all these (23) Thethe following YukawaThe Yukawa analysis couplings. is interactions, based In the on following⎩ these− supplemented relations we− shall ignorefor β byµi this gauge1. complication However, interactio one whichns, should has yield only keep relations be- relativistic gas, owing to the presence of unscreenedi non-abelian gauge interactions,≪ where processes yield constraintsasmalleff betweenect on our the discussion various& of chemical Leptogenesis. potentials [35]. The effective nonperturbative effinects mindtween are that important the• the chemicalchemical plasma in some potentials of the cases. potentials early of Universe left-handed in is very equilibrium and diff right-hanerent fromded a(SM) weakly fermions, coupled Athirdcondition,validatalltemperatures,arisesfromtherequirementthatthetotal interaction of Eq.relativistic (10)Using induced gas, Eq. owing by (21), the to theSU the baryon(2) presence electroweak number of unscreened density instantonsn non-a impliesgBTbelian2/6andtheleptonnumber gauge interactions, where Quarks, leptons and Higgs bosons interact via Yukawa and gaugeB couplings≡ and, in hypercharge of the plasma2 vanishes. From Eq. (21) and the known hypercharges one nonperturbativedensities nL µ eqiffLectsigTµH are/6canbeexpressedintermsofthechemicalpotentials: importantµdj =0,µ in someqi + cases.µH µuj =0,µli µH µej =0. (Yukawa)(25) i − − − − − addition, viaobtains the nonperturbative sphaleron≡ (3µqi processes.+ µli)=0 In. thermal equilibrium all these(22) Quarks, leptons andi Higgs bosons interact via Yukawa and gauge couplings and, in processes yield constraintsThese between relations the& hold various if theB chemical corresponding= (2 pµotentialsqi + µui interactions+ [35].µ2di) The, e areffectivein thermal equilibrium.(26) In the µqi +2µui µdi µli µei + µH =0. (24) (∑Y=0) addition, via the nonperturbative sphaleroni processes.12 In thermal equilibrium all these interactionOne also of has Eq. to (10) taketemperature induced the SU by(3) thei Quantum range'SU(2) 100 electroweak Chromodynamics GeV− 12 this is the case for gauge interactions. On the other hand, Yukawa interactions are in this is the case for gauge interactions. On the other hand, Yukawa interactions are in equilibrium only in a more restricted temperature range thatdependsonthestrengthof equilibrium only in a more restricted temperature range thatdependsonthestrengthof the Yukawa couplings. In the following we shall ignore this complication which has only the Yukawa couplings. In the following we shall ignore this complication which has only asmalleffect on our discussion of Leptogenesis. asmalleffect on our discussion of Leptogenesis. 2 Using Eq. (21), the baryon number density2 nB gBT /6andtheleptonnumber Using Eq. (21), the baryon number density nB gBT /6andtheleptonnumber≡ 2 ≡ densities2 nLi LigT /6canbeexpressedintermsofthechemicalpotentials: densities nLi LigT /6canbeexpressedintermsofthechemicalpotentials:≡ ≡ B = (2µqi + µui + µdi) , (26) B = (2µqi + µui + µdi) , (26) !i !i Li =2µli + µei ,L= Li . (27) Li =2µli + µei ,L= Li . (27) !i !i Consider now the case where all Yukawa interactions are in equilibrium. The asym- Consider now the case where all Yukawa interactions are in equilibrium. The asym- metries Li B/Nf are then conserved and we have equilibrium between the different metries Li B/Nf are then− conserved and we have equilibrium between the different − generations, µ µ , µ µ ,etc.Usingalsothesphaleronrelationandthehyper- generations, µ µ , µ µ li,etc.Usingalsothesphaleronrelationandthehyper-l qi q li ≡ l qi ≡ q ≡ ≡ charge constraint,charge one can constraint, express all one chemical can express potentials all chemical,andthereforeallasymmetries, potentials,andthereforeallasymmetries, in terms of a single chemical potential that may be chosen to be µl, in terms of a single chemical potential that may be chosen to be µl, 2Nf +3 6Nf +1 2Nf 1 2Nf +3 µe = 6Nf µ+1l ,µd = 2Nf µ1l ,µu = − µl , µe = µl ,µd =6Nf +3 µl ,µu−=6Nf +3− µl , 6Nf +3 6Nf +3 −6Nf +3 6Nf +3 1 4Nf 1 µq = 4Nfµl ,µH = µl . (28) µq = µl ,µH = −3 µl . 6Nf +3 (28) this is the case for gauge interactions. On the other hand, Yukawa interactions−3 are in6Nf +3 equilibrium only in a more restricted temperatureThe range corresponding thatdependsonthestrengthofThe baryon corresponding and lepton baryon asymmetries and lepton are asymmetries are the Yukawa couplings. In the following we shall ignore this complication which has only 2 4N2 14N +9Nf 4N 14N f+9Nf f asmalleffect on our discussion of Leptogenesis. B = f µ ,LB== f µl ,Lµ .= µl . (29) (29) LEPTOGENESISl − 3 l 6Nf +3 − 3 6Nf +3 Using Eq. (21), the baryon number density n gBT 2/6andtheleptonnumber B ≡ densities n L gT 2/6canbeexpressedintermsofthechemicalpotentials:• B & L relationsThis yields the importantThis yields connection the important between connection the B, B betweenL and theL asymmetriesB, B L and [37]L asymmetries [37] Li ≡ i − − B = c (B L); L =(c 1)(B L) , (30) B = (2µqi + µui + µdi) , B = c (B (26)L); L =(cs 1)(B L) , s (30) s − s − − − − − !i Li =2µli + µei ,L= Li . where c =(8N (27)+4)/(22N +13).TheaboverelationsholdfortemperaturesT v . where cs =(8Nf +4)/(22s Nf +13).Theaboverelationsholdfortemperaturesf f T vF . F i ≫ ≫ In general,! the ratioIn general,B/(B theL)isafunctionof ratio B/(B Lv)isafunctionof/T [38]. vF /T [38]. − − F Consider now the case where all Yukawa interactions are in equilibrium. The asym- The relations (30)The between relations B-, (30)(B-L)- between and L-number B-, (B-L)- suggest and L-number that (B-L)-violation suggest that is (B-L)-violation is metries L B/N are• thenNon-zero conserved and B we& haveL are equilibrium generated between if the (B-L) different≠0 in equilibrium6 i − f needed in order toneeded generate in order a B-asymmetry. to generate6 Because a B-asymmetry. the (B-L)-currentBecause the has (B-L)-current no anomaly, has no anomaly, generations, µli µl, µqi µq,etc.Usingalsothesphaleronrelationandthehyper- ≡ ≡ the value of B-L atthe time valuet ,wheretheLeptogenesisprocessiscompleted,determinest of B-L at time tf ,wheretheLeptogenesisprocessiscompleted,determinesthe he charge constraint, one can express all chemical potentialsB ,andthereforeallasymmetries,f value of the baryonvalue asymmetry of the baryon today, asymmetry today, in terms of a single chemical potential that may be chosenB L> to0 be µ , − l modified Sakharov ’s B L =0 − B(t )=c (B L)(t ) . (31) 2Nf +3 6Nf +1 2Nf 1 B(t0)=conditioncs (B L)(0tf ) . s − f (31) µe = µl ,µd = µl ,µu = −6 µl , − 6Nf +3 −6Nf +36 6Nf +3In the case of Dirac neutrinos, which have extremely small Yukawa couplings, one can construct In the case of DiracB L< neutrinos,0 which have extremely small Yukawa couplings, one can construct − 1 4Nf Leptogenesis modelsLeptogenesis where an asymmetry models where of lepton an asymmetry doublets isaccompaniedbyanasymmetryofright- of lepton doublets isaccompaniedbyanasymmetryofright- µq = µl ,µH = µl . (28)• −3 6Nf +3 handed neutrinos suchhanded that neutrinosthe total L-number such thatB is the conservedviolation total L-number and the is (B-L)-asymmetry conserved and the vanishes (B-L)-asymmetry [39]. vanishes [39]. (ii) L The corresponding baryon and lepton asymmetries are 12 12 2 4Nf 14Nf +9Nf • B = µl ,L(i) = µl . (29) (B-L) violation − 3 6Nf +3 B = C(B L) − (equilibrium) This yields the important connection between the B, B L and L asymmetries [37] − Figure from hep-ph/0212305 B = cs(B L); L =(cs 1)(B L) , (30) Figure 1.2: The relation− between baryon (B−)andlepton(−L)number.Thindottedlines correspond to constant B L,alongwhichB and L can move via the sphaleron process. At equilibrium, B and L reach− the thick dashed line, which represents B = C(B L). where c =(8N +4)/(22N +13).Theaboverelationsholdfortemperatures− T v . s f f ≫ F In general, the ratio B/(B L)isafunctionofvF /T [38]. lepton asymmetry (and hence B L asymmetry) is generated in an out-of-equilibrium − − The relations (30)way, between it is partially B-, converted (B-L)- into and baryon L-number asymmetry [9]. suggest (See the arrow that (ii) (B-L)-violation in Fig. 1.2.) is The amount of the baryon asymmetry at equilibrium is obtainedfromEq.(1.18)as 6 needed in order to generate a B-asymmetry.nB nBBecausenL thenL (B-L)-current has no anomaly, = C − = C , (1.20) s !eq s !eq − s !initial ! ! ! the value of B-L at time tf ,wheretheLeptogenesisprocessiscompleted,determinest! ! ! he where we have normalized! the number densities! by the! entropy density, so that they value of the baryon asymmetrybecome constant today, against the expansion of the universe. In the case of the MSSM, we have an additional Higgs doublet as well as supersym- metric partners.B( Lett )= us calculatec ( theB coeffiLcient)(t C) in. the presence of those particles. (31) First, the (gaugino)-(fermion)-(sfermion)0 s −∗ interactionsf ensure that the chemical poten- 6In the case of Dirac neutrinos,tial of each sfermion which7 is have the same extremely as that of small the corresponding Yukawa fermion. couplings, (Note one that tcanhe construct chemical potentials of the gauginos vanish since they are Majorana particles.) Next, the Leptogenesis models where an asymmetry of lepton doublets isaccompaniedbyanasymmetryofright- supersymmetric mass term W = µHuHd makes the chemical potentials of the up-type and handed neutrinos such that the total L-number is conserved and the (B-L)-asymmetry vanishes [39]. 7”sfermion” denotes a scalar partner of the fermion. 12 12 Q: How do we generate (B-L)≠0 in the early universe? thermal leptogensis mechanism is quite constrained depending on m3/2 and TR, the latter three mechanisms appear plausible over a wide range of TR, m3/2 and fa values which are consistent with naturalness. A summary and some conclusions are presented in Sec. 5. 2. Survey of some baryogenesis mechanisms 2.1 Thermal leptogenesis (THL) thermalThermal leptogensis leptogenesis mechanism [46, 47, is48 quite] is a constrained baryogenesis depending mechanism on m which3/2 and reliesTR, on the the latter intro- threeduction mechanisms of three appear intermediate plausible mass over scale a wide right range hand of singletTR, m3/ neutrinos2 and fa valuesN (i which=1 are3) so i consistentthat the with (type naturalness. I) see-saw mechanism A summary [31 and] elegantly some conclusions generates a are very presented light spectrum in Sec. of5. usual neutrino masses. The superpotential is given by 2. Survey of some baryogenesis mechanisms 1 W MiNiNi + hi↵NiL↵Hu (2.1) 2.1 Thermal leptogenesis (THL)3 2 Thermalwhere we leptogenesis assume a [46 basis, 47, for48] the is aN baryogenesisi masses which mechanism is diagonal which and relies real. on The the intro-index ↵ denotes the lepton doublet generations and hi↵ are the neutrino Yukawa couplings. From duction of three intermediate mass scale right hand singlet neutrinos Ni (i =1 3) so thermal leptogensis mechanism is quite constrained depending on m and TR, the latter thatthe the see-saw (type mechanism, I) see-saw mechanism one expects [31 a] spectrumelegantly generates of three sub-eV a very masslight spectrum Majorana of neutrinos usual 3/2 three mechanisms appear plausible over a wide range of T , m and f values which are neutrinom1, m2 masses.and m3 Theand superpotential three heavy neutrinos is given byM1 normally neglected by assuming the alignment ofeff final state lepton1 c and anti-lepton,1 i.e., ν = h1jNR1ℓLjH M1NR1NR1 + ηijℓLiHℓLjH +h.c., (56) L −102 M1 2 m⌫3 CP(L)=L¯. In general, however, one can consider✏1 the2 case10 in which final state lepton e↵ . (2.4) with 106 GeV 0.05 eV ⇠ ⇥ 3 and anti-lepton are–6– not aligned and thus the flavor e↵ect must✓ beT taken1 ◆ into⇣ account.⌘ ηij = hik hkj . (57) Mk Depending on the temperature atThe which ultimate dominant lepton lepton asymmetry asymmetry requiresk!=2 is evaluation generated, via flavor a coupled Boltzmann equation The asymmetry ε is then obtained from the interference of the Born graph and the one- calculation [61]. Once N1 is in thermal equilibrium, its number-density-to-entropy-density e↵ect can enhance the final asymmetry byloop up graph to involving an order1 the cubic of magnitude and the quartic [62 couplings.]. On Th theis includes other automatically ratio is simplyboth, vertex proportional and self-energy to corrections1/g where [59] and the yields degree an expression of freedom for ε directlyg = in 232.5 for MSSM. hand, one can also consider the case of nearly degenerate right⇤ handed neutrinos rather 1 ⇤ The lepton-number-densityterms of the light neutrino to mass entropy matrix: ratio is then given by than hierarchical spectrum. If the mass di↵erence is as small as its decay width, i.e., 3 M1 ∗ † ε1 Im h mνh . (58) 16nLπ (hh†) v2 nN1 11✏1 (M1 M2) N1 , the CP asymmetry factor is resonantly≃− enhanced= ✏11 F so that a successful (2.5) ⇠ s 1 s# ' $240 leptogenesis scenario is possible with (TeV)The CPright asymmetry handed then neutrino leads to a (B-L)-asymmetry mass [63]. [12], O n n ε In this work, we examine the viability of various leptogenesisY Y = scenariosL − L = κ for1 . natural (59) B−L ≃− L − s − g SUSY with mixed axion-higgsino dark matter. Nonetheless, we do not specify the∗ structure 20 of neutrino sector. For a clear discussion, we will consider only simple–6– scenario for the thermal leptogenesis. If one consider a specific neutrino sector in which flavor and/or resonant e↵ects are important, bound from thermal leptogenesis may be modified. 2.2 Non-thermal leptogenesis via inflaton decay (NTHL) As an alternative to thermal leptogenesis, non-thermal leptogenesis posits a large branching fraction of the inflaton field into N N : N N which is followed by asymmetric N 1 1 ! 1 1 1 decay to (anti-)leptons as before. In this case, the N1 number-density-to-entropy-density ratio is given by [49, 66] n ⇢ n n N1 rad N1 (2.7) s ' s ⇢ n 3 1 3 TR = TR 2Br = Br (2.8) 4 ⇥ m ⇥ 2 m where ⇢rad is the radiation density once reheating has completed and ⇢ is the energy density stored in the inflaton field just before inflaton decay. Thus, ⇢ ⇢ and ⇢ rad ' ' –7– 4. Cosmology with heavy axino/saxion and a gravitino as LSP 4.1 Two MSSM benchmark models: SUA and SOA In this section, we will discuss the cosmological implications of heavy axinos and saxions, concentrating on dark matter properties with the gravitino as the LSP. For definiteness, we will present most of our results with reference to two commonly used post-LHC8 benchmark scenarios. The first, labelled SUA for standard underabundance, is generated from IsaSugra using NUHM2 parameters m = 5000 GeV, m = 700 GeV, A = 8300 GeV and tan = 10 0 1/2 0 with µ = 200 GeV and mA = 1 TeV. With such a low µ value, the model has 3% electroweak fine-tuning with a Higgsino-like Z with mass m = 188 GeV where the standard thermal 1 Z1 abundance is given by ⌦stdh2 =0.013, a factor 10 below the measured value. Z1 ⇠e The second benchmark case,e labelled SOA for standard overabundance, is a mSUGRA/ Note, that the overall normalization differs from the expression given in [10] by the factor CMSSM model with parameterse chosen to be m = 3500 GeV, m = 500 GeV, A = 4(N 2 1). 0 1/2 0 7000 GeV,− tan = 10 with µ>0. It contains a bino-like Z1 with m = 224.2 GeV and The dependence of the soft part of the gravitino production rate onZ1 the cutoff k with a standard relic overabundance ⌦stdh2 =6.8, a factor 57 over the measured result.cut is again cancelled by the cutoff dependenceZ1 of the contributi⇠ on frome the hard 2 2 e → 4.2processes. Thermal There and are non-thermal 10 processes denoted gravitinoe by A production to J [5]: The axinoA: andga + saxiongb g˜ canc + G be˜ produced e ciently in the early universe by thermal scattering, decay• and inverse→ decays which can alter the standard dark matter property. The axino a a a a Note, that theand overall saxion normalization thermalg production differsG fromg has the expres been studiedsionG giveng extensively in [10] byG the for factor theg KSVZ caseG [12, 13, 14] gc 4(N 2 1). as well as for the DFSZ case+ [15, 16a , 17]. Depending+ b on the+ PQ breaking scale, reheat − g g The dependencetemperature, of the soft andgb part axino of mass, thegc gravitino itgb can be production eithergc hot, rgateb warm on the org cuto coldc ff darkgkbcut mattergc if the axino is is again cancelledsu ciently by the light cutoff [18dependence]. In such of circumstances, the contribution the from axion-axino the hard 2 mixed2 dark matter scenario → processes. Therecan are also 10 be processes realized denoted [24]. by Along A to with J [5]: the axino, the saxion can also play an important B: ga +˜gb gc + G˜ (crossing of A) role in• cosmology→ and astrophysics [25]. For conventional gravity mediation models with a b c ˜ A: g + g g˜ + G a • a typical→ C: massq ˜ + g spectrum,q˜ + G˜ ma˜ ms m3/2 msoft, the LSP is normally the lightest • i → j ⇠ ⇠ ⇠ gneutralino,a G andg thea decaysG of thega abundantG axinoga and saxinoG have to be taken into account c as theyg can a↵qecti the neutralinoG qi relic density.G Inqi such a case,G theqi axion-neutralinoG mixed Note, that the overall normalization+ differsg froma the expres+ siongb given in+ [10] by the factor qj q a 2 dark matter scenario can be+ realizedi either in+ the KSVZg model+ [26, 27, 28, 29] or in the 4(N 1). gb gc gb gc gb gc gb gc − DFSZ model [15a , 17, 30, 33, 34].a a a The dependence of the soft part ofg the gravitinoqj productiong rqatej ong the cutoff qkj g qj In this work, we address a di↵erent possibility: the heavycut axino/saxion with light is again cancelledB: g bya +˜ thegb cutogc +ff Gdependence˜ (crossing of of A) the contribution from the hard 2 2 • gravitino.→ Asa shown in Sec.˜ 2, the axino and saxion can be→ much heavier than not only the processes. There are 10 processesD: denotedg + qi byq˜j A+ toG J(crossing [5]: of C) a • → C:q ˜i + ggravitinoq˜j + G˜ but also the MSSM sparticles. In this case, we have two dark matter candidate: • → ¯ a ˜ a b c the˜ gravitinoE: q˜i + andqj theg axion.+ G (crossing The axion of C) dark matter is produced from coherent oscillations A: g + g g˜ + G • → • → qduringi theG QCDqi phase transition.G qi ConcerningG q thei gravitinoG production, there are three a b c ˜ a qj F:a g ˜ +˜g g˜ + Ga a g diG↵erent• g sources+ in→Gq ouri scenario:g + Gga g + G Note, that the overall normalizationc differs from the expression given in [10] by the factor GRAVITINOg a a aPROBLEMa a a b a a 2 g + thermalqj g productiong g+ qj g gG +g qj g G gqj G 4(N 1). c − b c • b c b c b c g g g g g g g g a g b The dependence of the soft part of the gravitinoThe gravitinos production are r producedate on the+ from cutoff thegkcut thermal+ bath viag interactionsproportional with to MSSMTR D: ga + q q˜ + G˜ (crossing of C) is again cancelled by the cuto•ff dependencei → particles.j of the contributi The gravitinobon from thermal thec hard densityb 2 2 is givenc by [b35, 36] c Gravitino problem: g g g → g g g B: ga +˜gb ¯gc + G˜ (crossinga ˜ of A) processes. There are 10 processesE: denotedq˜i + qj byg A+ toG J(crossing [5]: of C) m 2 1 GeV T • • → → ⌦TPh2 =0.21 g R (4.1) gravitinos are thermally produceda G 8 a b c a a b˜ c G: q +˜g q + G˜ 1 TeV m3/2 10 GeV C:˜q ˜i + g F:g ˜q˜j+˜+gG g˜ + G˜ i j ✓ ◆✓ ◆ A: g + g g˜• + G → • → ⇣ e ⌘ • → • → e Bolz, Brandenburg, Buchmüller; Strumia a a a a a a a g G qig GG g qig GGGqi g qig GGGG qigqi GGG qi G c c q g qj g aj q b a a b a + g + + ig + ++ g g +++ qgi + g –9– b c a b c b a b c c a b a b c c a b a c a g g g g qjg g g g g qjg g g g g qjgqjg ggg qgjqj g qj a b c a a˜ ˜ a B: g +˜g g +D:G˜g (crossing+ qiG: qq˜i ofj+˜+ A)gG (crossingqj +H:Gq ˜i of+˜ C)g q˜j + G˜ • → • • → → • → a ¯ a ˜ 12 C:q ˜i + gdecaysq˜j +E: G˜intoq˜i + q jLSPg with+ G (crossinglongqi life-time; of C)G eitherqi G qi G • → • → qj a producingF:g ˜a +˜toogb muchg˜c + G˜DM or spoiling+ BBN;q i + g qi • G qi → G qi G qi G upper bound for TR ga q ga q ga q qj q a j j j + iga + G g ga + G ga G c ga q ga qg ga q ga q j H:q ˜ +˜ga j q˜ + G˜ + aj + jb • i → j g g gb gc gb g12c gb gc D: ga + q q˜ + G˜ (crossing of C) • i → j ¯ a ˜ a E: q˜i + qj g +G:Gq(crossing+˜g q of+ C)G˜ • → • i → j F:g ˜a +˜gb g˜c + G˜ • → qi G qi G qi G a a qj a a Kawasaki, Kohri, Moroi, Yotsuyanagi g G g G+ g qi G+ g Figure 4: BBN constraints for the Case 3. c g a a a b a +g g qj +g g qj g qj gb gc gb gc gb gc H:q ˜ +˜ga q˜ + G˜ • i → j G: q +˜ga q + G˜ 12 • i → j qi G qi G qi G qj a + qi + g a a a g qj g qj g qj H:q ˜ +˜ga q˜ + G˜ • i → j 12 Figure 5: BBN constraints for the Case 4. 11 OUTLINE 1. Leptogenesis 2. AD mechanism along LHu direction 3. AD leptogenesis in DFSZ model with varying PQ scale 4. Summary 21 Many particle physics models lead to significant production of entropy at relatively late • times (Cohen, Kaplan and Nelson, 1993). This dilutes whatever baryon number existed previously. Co- herent production can be extremely efficient, and in many models, it is precisely this late dilution which yields the small baryon density observed today. In the rest of this section, we discuss Affleck-Dine baryogenesis in some detail. 21 B. Baryogenesis Through a Coherent Scalar Field: 21 In supersymmetricMany theories, particle the physics ordinary quarks models and lead lepton tosareaccompaniedbyscalarfields.Thesescalarfields significant production of entropy at relatively late • Many particle physics models lead to significant production of entropy at relatively late carry baryon andtimes lepton (Cohen, number. Kaplan• Atimes coherent and (Cohen, Nelson, field, Kaplani.e. and 1993).,alargeclassicalvalueofsuchafield,caninprinciplecarr Nelson, 1993).This dilutes This dilutes whatev whatever baryon baryon number number existed previously. existed previously.ya Co- Co- large amount ofherent baryon production number. As canherent we be will production extremely see, it can is be quite effi extremelycient, plau esible andfficient, in that and many in such many mode fields models,ls, were it it is is precisely excited precisely this in late the this dilution early late which universe. dilution yields which yields To understandthe the small basics baryon of the densitythe mechanism, small observed baryon density consider today. observed first today. a model with a single complex scalar field. Take the In the rest of this section, we discuss Affleck-Dine baryogenesis in some detail. lagrangian to beIn the rest of this section, we discuss Affleck-Dine baryogenesis in some detail. 2 2 2 B. Baryogenesis Through a= Coherent∂ φ Scalarm Field:φ (62) L | µ | − | | This lagrangianB. Baryogenesis has a symmetry, ThroughInφ supersymmetrica Coherenteiαφ,andacorrespondingconservedcurrent,whichwewillrefer Scalar theories, Field: the ordinary quarks and leptonsareaccompaniedbyscalarfields.Thesescalarfieldsto as baryon number: carry baryon→ and lepton number. A coherent field, i.e.,alargeclassicalvalueofsuchafield,caninprinciplecarrya large amount of baryon number. As we will see, it is quite plausible that such fields were excited in the early universe. In supersymmetric theories,To understand the the ordinaryµ basics of the quarksµ mechanism, andµ consider lepton firstsareaccompaniedbyscalarfields.Thesescalarfields a model with a single complex scalar field. Take the j = i(φ∗∂ φ φ∂ φ∗). (63) carry baryon and leptonlagrangian number. to be AB coherent field,− i.e.,alargeclassicalvalueofsuchafield,caninprinciplecarrya 21 = ∂ φ 2 m2 φ 2 (62) It also possesseslarge amount a “CP” of symmetry: baryon number. As we will see, it is quiteL plau| µ | sible− | that| such fields were excited in the early universe. To understand theThis basics lagrangian of the hasMany a mechanism, symmetry, particleφ e consideriαφ,andacorrespondingconservedcurrent,whichwewillrefer physics first models a model leadwith a to single significant complexto scalarproduction as baryon field.of Take entropy the at relatively late • φ → φ∗. (64) lagrangian to be number: times (Cohen,↔ Kaplan and Nelson, 1993). This dilutes whatever baryon number21 existed previously. Co- µ µ µ herent production canj = bei(φ∗ extremely∂ φ φ∂ φ∗) e. fficient, and in many models, it(63) is precisely this late dilution which yields With supersymmetry in mind, we will think of m as of order MWB. 2 −2 2 the small baryon= density∂µφ observedm φ today. (62) If we focusMany on the particle behavior physicsIt of also spatially possesses models a constant “CP” symmetry: lead fields, toLφ(⃗x, significant| t)=| φ−(t), this| production| system is equivalentof entropy to an at isotropic relatively late harmonic• oscillator in two dimensions. This remainsIn theiα rest the of cas thiseifweaddhigherordertermswhichrespectthephase section, we discuss Affleck-Dine baryogenesis in some detail. Thistimes lagrangian (Cohen, has Kaplan a symmetry, and Nelson,φ e 1993).φ,andacorrespondingconservedcurrent,whichwewillrefer This dilutesφ whatevφ∗. er baryon number existed previously.(64) to as baryonCo- symmetry. In supersymmetric models, however,→ we expect thathigherordertermswillbreakthesymmetry.Inthe↔ number:herent productionWith can supersymmetry be extremely in mind, efficient, we will and think inof m manyas of order modeMWls,. it is precisely this late dilution which yields isotropic oscillator analogy, thisIf corresponds we focus on the to behavior anharmo of spatiallynic terms constant which fields, φ break(⃗x, t)= theφ(t), rotational this system is invariance. equivalent to an With isotropic a the small baryon densityharmonic observed oscillator in two today. dimensions.µ This remainsµ the caseifweaddhigherordertermswhichrespectthephaseµ general initial condition, the system willB. develop Baryogenesis some non Throughj-zero= i angular( aφ∗ Coherent∂ φ momentum.φ∂ Scalarφ∗). Field: If the motion is damped, so that (63) In the rest of this section,symmetry. we In supersymmetric discuss Affl models,eck-DineB however, baryogene we expect− thasisthigherordertermswillbreakthesymmetry.Inthe in some detail. the amplitude of the oscillationsisotropic decreases, oscillator these analogy, rotatio this correspondsnally non-invariant to anharmonic terms terms which will break become the rotational less important invariance. With with a time. It also possesses a “CP”general symmetry: initial condition,In supersymmetric the system will develop theories, some thenon-zero ordinary angular momentum. quarks and If the lepton motionsareaccompaniedbyscalarfields.Thesescalarfields is damped, so that Let us add interactions in thethe following amplitudecarry of way, the baryon oscillations which and will decreases, lepton closely these number. parallel rotationally Awhat coherent non-invariant happens field, terms ini.e. the will,alargeclassicalvalueofsuchafield,caninprinciplecarr supersymmetric become less important case. with ya B. Baryogenesis Throughtime. a Coherent Scalar Field: φ φ∗. (64) Include a set of quartic couplings:Let us addlarge interactions amount in the of following baryon way, number. which↔ will As clos weely willparallel see, what it happens is quite in plau the supersymmetricsible that such case. fields were excited in the early universe. Include a set ofTo quartic understand couplings: the basics of the mechanism, consider first a model with a single complex scalar field. Take the With supersymmetry in mind, we will think4 of 3m as of4 order MW . I = λ φ + ϵφ φ∗ + δφ + c.c. (65) In supersymmetric theories, thelagrangian ordinary to quarks be and lepton4 sareaccompaniedbyscalarfields.Thesescalarfields3 4 If we focus on the behavior ofL spatially| | constantI fields,= λ φ +φϵφ(⃗x,φ∗ t+)=δφ +φc.c.(t), this system is equivalent(65) to an isotropic carry baryon and lepton number. A coherent field, i.e.L,alargeclassicalvalueofsuchafield,caninprinciplecarr| | ya These interactionsharmonic clearly oscillator violate inThese two “B”. interactions dimensions. For general clearly violate comple This “B”. remainsx ϵ Forand generalδ the,theyalsoviolateAFFLECK-DINE comple caseifweaddhigherordertermswhichrespectthephasex ϵ and δ,theyalsoviolateCP.Insupersymmetrictheories,= CP∂ .Insupersymmetrictheories,φ 2 m2 φ 2 (62) µ as welarge will shortly amount see, of the baryon couplings number.as we willλ, shortlyϵ, Asδ ... we see,will will the be couplings see, extremely itλ is, ϵ quite, δ ... small,will plau be extremelysible(M 2 that/M small,2 such)or(M fields2 /ML(M2)or2 were|/M(M2| excited2 −/M). 2 |). in| the early universe. symmetry. In supersymmetric models, however, we expect thathigherordertermswillbreakthesymmetry.IntheW p O W p W O GUTW ForGUT a review, Dine, Kusenko (03) In orderTo that understand these tiny the couplings basicsIn order of lead the thatThis to mechanism,these lagrangian an tiny appreciable couplings considerhas lead a bar tosymmetry,yon an first appreciable number,O a moφ bardel ityoneiα iswithφ number, necessary,andacorrespondingconservedcurrent,whichwewillrefer aO singleit is necessary that complex the that fields, the scalar fields, at at field. some some Take the to as baryon isotropic oscillator analogy,stage, were• this very large. corresponds To see how the to cosmic anharmo evolutionnic of thi termsssystemcanleadtoanon-zerobaryonnumber,firstnote→ which break the rotational invariance. With a stage,lagrangian were very large. to be To see howthat the at very cosmicnumber:Scalar early times, evolution whenfield the of Hubblewith thissystemcanleadtoanon-zerobaryonnumber,firstnote constant, B (orH L)m,themassofthefieldisirrelevant.Itisthusreasonable number, general initial condition, the system will develop some non-zero≫ angular momentum. If the motion is damped, so that that at very early times, when theto suppose Hubble that constant, at this early timeH φ =mφ,themassofthefieldisirrelevant.Itisthusreasonableo 0; later we will describe some specific suggestions as to how this might the amplitude of the oscillations decreases, these rotatio≫ 2 nally2 non-invariant2 µ terms willµ becomeµ less important with to suppose that at this early timecomeφ about.= φ How does0; later the field we then≫ will= evolve? describe∂µφ First ignore somem theφ specific quartic interactions. suggestionsjB = i In(φ a∗ gravitational∂ asφ to howφ∂ background,φ this∗). might the (62) (63) time. equation of motiono ≫ for the field is L | | − | | − come about. How does the field then evolve?It also First possessesiα ignore a the “CP” quartic symmetry: interactions. In a gravitational background, the22 This lagrangianLet us add has interactions a symmetry, in theφ followinge φ,andacorrespondingconservedcurrent,whichwewillrefer way, which will2 clos∂elyV parallel what happens in the supersymmetricto as baryon case. equation of motion for the field is → Dµφ + =0, (66) number:Include a set of quartic couplings:• small quartic couplings∂φ 22 φ φ∗. (64) In either case, the energy behaves, in termswhere D ofµ is the the covariant scale fact derivative.2 or,µ ∂R ForV(t a), spatially4 asµ homogeneous3 µ 4 field, φ(t), in a Robertson-Walker↔ background, this becomes DµφjI+== λi(φ=0∗∂+, φϵφ φφ∂∗ +φδφ∗). + c.c. B CP (for complex(66) couplings(63)(65)) In either case, the energy behaves, in termsWith supersymmetry of the scaleB ∂φ fact inor, mind,R(t−), weas will think of m as of order MW . L | | ∂V It alsoThese possesses interactions a “CP” clearly symmetry: violateIf “B”. we focus For2 2 general onR theo 3 comple behaviorφ¨ +3x Hϵ ofφand˙ + spatiallyδ,theyalsoviolate=0. constant fields,CPφ.Insupersymmetrictheories,(⃗x, t)=φ(t),(67) this system is equivalent to an isotropic where Dµ is the covariant derivative. For aE spatiallym φ homogeneous( ) R field, φ(t),∂φ in a Robertson-Walker background, this(69) as we will shortly see, the couplings• harmonicEq. of≈λ ,motionϵ oscillator, δE...o Rwillm2 inφ2 be( two extremelyo ) dimensions.3 small, This( remainsM 2 /M 2 the)or caseifweaddhigherordertermswhichrespectthephase(M 2 /M 2 ).(69) becomes At very early times, H m,andsothesystemishighlyoverdampedandessentiallyfrozo W p en at φoW.Atthispoint,GUT symmetry.≫ In supersymmetric≈ φR φ∗. models, however,O we expect o O thathigherordertermswillbreakthesymmetry.Inthe1 (64) In order that theseB tiny=0.However,oncetheuniversehasagedenoughthat couplings lead to an appreciable barHyonm number,, φ begins to it oscillate.= is necessary3/4 Substitutingsin(mt that)H (radiation)= the2t or fields, at some 3 2 ∂V ↔ ≪ (mt) i.e. it decreases like R ,aswouldtheenergyofpressurelessdust.Onecanthinkofth3 H = for theisotropic radiation¨ oscillator and matter˙ dominated analogy, eras, this respectively, correspondsoneis finds oscillating to that anharmo fieldnic terms as a coherentwhich break the rotational invariance. With a i.e. itWith decreasesstage, supersymmetry were like veryR ,aswouldtheenergyofpressurelessdust.Onecanthinkofth large. in mind, To3t see we how will the thinkφ cosmic+3 ofHmφ evolution+as of order=0 of. thiMssystemcanleadtoanon-zerobaryonnumber,firstnote. is oscillating field as a coherent(67) general initial condition,∂φ the systemW will develop some non o -zero angular momentum. If the motion is damped, so that state of φ particles with p⃗ =0. φo sin(mt) (matter) state ofthatφ particles at very with earlyp⃗ =0. times, when the Hubble constant, H 3/m2 sin(,themassofthefieldisirrelevant.Itisthusreasonablemt)(radiation) If we focus on the behavior ofthe spatially amplitude constant of theφ fields, oscillations= (mtφ) (⃗x, t decreases,)=φ(t), these this system rotatio(mt)nally is equivalent non-invariant to(68) an terms isotropic will become less important with φ≫o Now let’sAt consider very earlyto the suppose e times,ffects thatH of the atm quartic this,andsothesystemishighlyoverdampedandessentiallyfroz early couplings. time φ = φ Sincethefieldamplitudedampswithtime,theirsignificance0; later we willsin(mt describe)(matter) some. specificen at suggestionsφ .Atthispoint, as to how this might Nowharmonic let’s consider oscillator the e inffects two of dimensions. the quartictime. Thiscouplings.o remains Sinc theethefieldamplitudedampswithtime,theirsignificance! cas(mt) eifweaddhigherordertermswhichrespectthephaseo B =0.However,oncetheuniversehasagedenoughthat≫ ≫ H m, φ begins to oscillate. Substituting H = 1 or will decreasewill withsymmetry. decrease time.come about. with Suppose, In supersymmetric time. How initially, doesSuppose, the field that models,initially,IfLet thenφ us= however, that evolve? addφo φ interactionsis= First wereal.φo expectis ignore Then real. in tha the the Then thethigherordertermswillbreakthesymmetry.Inthe following quartic imaginary the imaginaryinteractions. way, which part part will of Inφ of a clos gravitationalsatisfies,φelysatisfies, parallel in2 int whatbackground, the the happens the in the supersymmetric case. 2 ≪ Im(✏ + ) 3 approximationHapproximation= that3t forϵequationand the radiationδ thatare of motionϵ small,and andδ formatterare the small, dominated field is eras, respectively, one finds that o isotropic oscillator analogy, thisInclude corresponds a set of to quartic anharmo couplings:nic terms which break thei =ar rotational2 3/4 sin( invariance.mt + r) (radiation) With a general initial condition, the system will develop some non-zero angular momentum. If them (mt motion) is damped, so that ¨ φo˙ 2 ∂V 3 4 3 3 4 φ +3H2 φ3/2+sin(mmtφ )(radiation)2Im(ϵ3+ δ)φ . = λ φ + ϵφIm(φ✏ ++ ) δφo + c.c. (70) (65) φ¨i +3Hφ˙ii+ m(mt)φi Im(iϵD+φδ+)φ . =0r , I ∗ (70) (66) the amplitude of the oscillationsφ decreases,= these rotatio≈µ nallyr non-invariantL terms| | willam become2 lesssin(mt important+(68) m) (matter) with φo ≈ ∂φ m (mt) time. 9/2! (mt) sin(mt)(matter). 3/2 For large times, the right hand falls asTheset− ,whereasthelefthandsidefallso interactions clearly violate “B”. Forff generalonly as complet− .Asaresult,justasinx ϵ and δ,theyalsoviolateCP.Insupersymmetrictheories, 9/2 4 3/2 2 2 2 2 For large times, theLetwhere right us handaddDµ is interactions falls the covariant as t− in the,whereasthelefthandsidefallso derivative.as following we will For way, shortly a spatially which see, will the homogeneous couplings closely parallelIm(ffλonly,✏ fieϵ+,ld,δ ...) aswhat φo(willtt−), happens in be.Asaresult,justasin a extremely Robertson-Walker in the small, supersymmetric(M background,/M )or case. this(M /M ). our mechanical analogy, baryon number• Baryon (angular number momentum) violation becomes negligible. The equation goes overW p W GUT nB =2ar 3/2 sin( r + ⇡/8) (radiation)O O our mechanicalto analogy, theInclude freebecomes equation, a baryon set of quartic with number a solution couplings: (angular of theIn momentum) orderform that these violation tiny couplings becomesm lead(mt negligible.) to an appreciable The equation baryon goes number, over it is necessary that the fields, at some to the free equation, with a solution of the formstage, were very large. To see how the cosmic evolution4 of thissystemcanleadtoanon-zerobaryonnumber,firstnote ¨4 3˙ ∂V Im(4 ✏ + ) o 3 I = λ φφ +3+ ϵφHφφ∗++2aδφm =0+ c.c.. 3 2 sin( m) (matter) (65)(67) Im(ϵ + δ)φo that at veryL early| times,| whenIm( theϵ Hubble+mδ)(φmto ) constant, H m,themassofthefieldisirrelevant.Itisthusreasonable φ = a 3 sin(mt + δ )(radiation), φ = a ∂φ 3 sin(mt + δ )(matter)≫ , (71) Im(ϵ +i δ)φr 2 3/4 to supposer that at thisiIm( earlyϵm+ timeδ)φφ 3= φo 0; later wem will describe some specific suggestions as to how this might φ = aThese interactionso sin(m (mt clearlymt)+ δ violate)(radiation) “B”. For, generalφ = complea x ϵ and δm,theyalsoviolateo tsin(mt≫ + δ )(matter)CP.Insupersymmetrictheories,, (71) i asr weAt2 will very shortly3 early/4 times, see, theHr couplingsm,andsothesystemishighlyoverdampedandessentiallyfrozcomeλ, about.ϵ, δ ... Howwilli be does extremelym the field3 then small, evolve?(M 2 First/Mm2 ignore)or the(M 2 qu/Marticen2 at interactions.).φo.Atthispoint, In a gravitational background, the m (mt) ≫ m t W p W GUT 1 in the radiationB =0.However,oncetheuniversehasagedenoughthat and matter dominatedequation cases, respectively. of motion forThe the constants fieldH is δ mO,,δφ, beginsa and toa oscillate.Ocan easily Substituting be obtained H = or In order that these tiny couplings lead to an appreciable baryon≪ number,m 4 m it is necessaryr that the fields, at some2t numerically,H = and2 for are the of radiation order unity: and matter dominated eras, respectively, one finds that in the radiation andstage, matter were3t very dominated large. To cases, see how respectively. the cosmic evolutionThe constants of thissystemcanleadtoanon-zerobaryonnumber,firstnoteδm, δ4, am and ar 2can easily∂V be obtained Dµφ + =0, (66) numerically, and arethat of at order very early unity: times, when the Hubble constant, Hφo m,themassofthefieldisirrelevant.Itisthusreasonable∂φ ar =0.85 am =0.85 δr3=≫/2 sin(0.91mt)(radiation)δm =1.54. (72) to suppose that at this early time φ = φo φ0;= later(mt we) will− describe some specific suggestions as to how this might(68) where Dµ ≫is the covariantφo sin( derivative.mt)(matter) For a. spatially homogeneous field, φ(t), in a Robertson-Walker background, this Butcome now about. we have How aa non-zeror does=0. the85 baryon fieldam then number;=0 evolve?.85 substitutingδ Firstr =! ignore(mt0.) in91 the theexpressionδ qumartic=1. interactions.54 for. the current, In a gravitational background,(72) the equation of motion for the field isbecomes − 2 2 ∂V But now we have a non-zero baryonφo number; substituting in the expression for the current,φoφ¨ +3Hφ˙ + =0. (67) nB =2arIm(ϵ + δ) sin(δr + π/8) (radiation) 2 nB∂=2V amIm(ϵ + δ) sin(δm)(matter). (73) m(mt)2 Dµφ + =0, m(mt)2 ∂φ (66) φ2 ∂φ φ2 o At very early times, H m,andsothesystemishighlyoverdampedandessentiallyfrozo en at φo.Atthispoint, nB =2arIm(Twoϵ + featuresδ) of thesesin( resultsδr + π/ should8) (radiation) be noted. First,nB if=2ϵ anda≫mIm(δ vanish,ϵ + δ)n vanishes.sin(δ Ifm)(matter) they are real,. and (73)φ is 1 where Dmµ(ismt the)2 covariant derivative.B =0.However,oncetheuniversehasagedenoughthat For a spatially homogeneous field,mB(φmt(t),)2 in a Robertson-WalkerH m, φ background,beginso to oscillate. this Substituting H = or real, n vanishes.It is remarkable that the lagrangian parameters can be real, and yet φ can be complex,≪ still giving 2t becomesB H = 2 for the radiation and matter dominated eras,o respectively, one finds that rise to a net baryon number. We will discuss3t plausible initialvaluesforthefieldslater,afterwehavediscussed Two features of these results should be noted. First, if ϵ and δ vanish,∂V nB vanishes. If they are real, and φo is supersymmetry breaking in the early universe. Finally,φ¨ we+3 shouldHφ˙ + point=0 out. that, as expected,φo sin(nBmtis)(radiation) conserved at late (67) real, nB vanishes.It is remarkable that the lagrangian parameters can be∂φ real, and yet φo can(mt)3 be/2 complex, still giving times. φ = φo (68) rise to a net baryon number. We will discuss plausible initialvaluesforthefieldslater,afterwehavediscussed! (mt) sin(mt)(matter). ThisAt mechanism very early times, for generatingH m baryon,andsothesystemishighlyoverdampedandessentiallyfroz number could be considered without supersymmetry. In thaten at case,φ .Atthispoint, it begs supersymmetry breaking in the early universe. Finally, we should point out that, as expected, nB is conservedo at late severalB =0.However,oncetheuniversehasagedenoughthat questions: ≫ H m, φ begins to oscillate. Substituting H = 1 or times. ≪ 2t HWhat= 2 arefor the the scalar radiation fields and carrying matter baryon dominated number? eras, respectively, one finds that This mechanism• for3 generatingt baryon number could be considered without supersymmetry. In that case, it begs several questions: Why are the φ4 terms so small? φo sin(mt)(radiation) • (mt)3/2 φ = φo (68) What are theHow scalar are fields the scalars carrying in the baryon condensate number? converted! (mt to) moresin(mt famil)(matter)iar particles?. • • Why are theIn theφ4 contextterms so of small? supersymmetry, there is a natural answer to each of these questions. First, as we have stressed, • there are scalar fields carrying baryon and lepton number. As we will see, in the limit that supersymmetry is unbroken, How arethere the scalars are typically in the directions condensate in the converted field space to in more which famil the quarticiar particles? terms in the potential vanish. Finally, the scalar • quarks and leptons will be able to decay (in a baryon and leptonnumberconservingfashion)toordinaryquarks. In the context of supersymmetry, there is a natural answer to each of these questions. First, as we have stressed, there are scalar fields carrying baryon and lepton number. As we will see, in the limit that supersymmetry is unbroken, there are typicallyC. Flat directions Directions in and the Baryogenesis field space in which the quartic terms in the potential vanish. Finally, the scalar quarks and leptons will be able to decay (in a baryon and leptonnumberconservingfashion)toordinaryquarks. To discuss the problem of baryon number generation, we first want to examine the theory in a limit in which we ignore the soft SUSY-breaking terms. After all, at very earlytimes,H MW ,andthesetermsareirrelevant.We want to ask whether in a model like the MSSM, some fields can havelargevev’s,i.e.whethertherearedirectionsin≫ C. Flat Directions and Baryogenesis the field space for which the potential vanishes. Before considering the full MSSM, it is again helpful to consider a + simpler model, in this case a theory with gauge group U(1), and two chiral fields, φ and φ− with opposite charge. To discuss theWe take problem the superpotential of baryon number simply to generation, vanish. In this we first case the wantpotential to examine is the theory in a limit in which we ignore the soft SUSY-breaking terms. After all, at very earlytimes,H MW ,andthesetermsareirrelevant.We 1 2 + +≫ want to ask whether in a model like the MSSM,V some= fieldsD canD = havg(φelargevev’s,i.e.whethertherearedirectionsin∗φ φ−∗φ−)(74) the field space for which the potential vanishes. Before2 considering the full− MSSM, it is again helpful to consider a + simpler model, in this case a theory with gauge group U(1), and two chiral fields, φ and φ− with opposite charge. We take the superpotential simply to vanish. In this case the potential is 1 2 + + V = D D = g(φ ∗φ φ−∗φ−)(74) 2 − POTENTIAL Complex quartic kicks scalar field to phase direction r i Q: How do we get large initial value? SUSY BREAKING BY INFLATION Dine, Randall, Thomas (95, 96) • Large vacuum energy during inflation breaks SUSY - SUSY breaking potential arises and ~H>>m V c H2 2 H H | | negative cH is possible in non-minimal Kähler potential - Together with 4 or 6 terms in F-term potential, 1 V = H2 2 + 6 pHM | | M 2 | | 0 ⇠ POTENTIAL When H>>m, scalar field stays at the potential minimum pHM 0 ⇠ r When H~m, scalar field starts oscillation i From these relations, one finds the lepton-number-to-entropy ratio: nL ⇢N1 1 3 TN1 = ✏1 = ✏1 s M1 s 4 M1 10 TN1 m⌫3 1.5 10 . (2.15) ' ⇥ 106 GeV 0.05 eV e↵ ✓ ◆ ⇣ ⌘ The baryon asymmetry is obtained via sphaleron process, and thus baryon number is given by n n B 0.35 L . (2.16) s ' s 6 Thus, enough baryon number can be generated for TN1 & 10 GeV. In this scenario, it is interesting that the e↵ective reheat temperature is (T ) for O N1 thermal relic particles, since sneutrino domination dilutes pre-existing particles when it decays [52].2 Therefore, in the numerical analyses of Sec’s. 3 and 4, we consider T a reheat AD LEPTOGENESISN1 temperature for production of gravitinos, axinos and saxions in the case of leptogenesis from Affleck, Dine; Dine, Randall, Thomas (95, 96) oscillating sneutrino decay.To realize AD mechanism, we need Murayama, Yanagida; Gherghetta, Kolda, Martin 2.4 A✏eck-Dine leptogenesis• light scalar (ADL) (flat direction) carrying B or L number 23 The last mechanism for baryogenesis is known as A✏eck-Dine (AD) [53,+54] leptogenesis. But D,andthepotential,vanishifφ = φ− = v.Itisnotdifficult to work out the spectrum in a vacuum of non-zero AD leptogenesis makes• usesmall of the BLH (oru flat L)v direction.Onefindsthatthereisonemasslesschiralfield,andamassiv and CP in theviolating scalar potential quartic [67 ,potential54, 68]. evectorfieldcontainingamassivegaugeboson,a massive Dirac field, and a massive scalar. This direction is lucrative in that it is not plagued by Q-balls which are problematic for flat directions carrying baryon number [69] andConsider, also because now, a somewhat the rate more for baryogenesiselaborate example. can Let us taketheMSSMandgiveexpectationvaluestothe In SUSY model, Higgs and the slepton fields of eqn. (46): be linked to the mass of the lightest neutrino, leading to a possible consistency check via 0 v observations of neutrinoless double beta decay (0⌫ )[70]. H = L = . (75) • LHu direction is flat (in SUSY limit) u v 1 0 In the case of the LHu flat direction, F -flatness is only broken by higher dimensional! " ! " operators which also give rise to neutrino massThe viaF term the vanishes see-saw in mechanism this direction, [ since31]: the potentially problematic HuL term in the superpotential is absent • quarticwhere can beby R generatedparity (the other possible contributions vanish because Q = HU =0).ItiseasytoseethattheD-term for hypercharge vanishes, 1 2 2 mSUSY 4 W (L H )(L VH )= m + (a + (2.17)h.c.2 )2 2 (2.20) i u i SBu Dm = g′ ( H LL)=0CP. (76) 3 2Mi | | 8M Y | u| − | | To see that the D terms for2 SU2(2) vanishes,H one4 can work directly with the Pauli matrices, or use, instead, the 3 VH = cH H + (aH + h.c.) (2.21) where Mi is the heavy neutrino mass scale. followingThe most device e whichcient works direction for| | a general is8M thatSU( forN)group.Justasonedefinesamatrix-valuedgaugefield, which 2 2 2 i = 1 corresponding to the lightest neutrino mass: mlinked⌫1 H tou /M(lightest1 in2 a2 basis) 2neutrino where9↵s(Ti) the4 massa a i V ⇠h= i c f T + (Aµ) j T= Alnµ(T |) j|, and (2.22) (77) neutrino mass matrix is diagonal. The A✏eck-Dine fieldTH then occursk k as | | 8 T 2 fk 4 In gravity mediation, it is natural to have mSUSY m . In our benchmark study in Sec. 3, however, ⇠ 3/2 mSUSY for SUSY particle spectrum is fixed while physical gravitino mass m3/2 varies from 1 TeV to 100 TeV. – 10 – From these relations, one finds the lepton-number-to-entropy ratio: nL ⇢N1 1 3 TN1 where = ✏1 = ✏1 s M1 s 4 M1 2 2 mSUSY T4 m VSB = m + 10 (am N1+ h.c.)⌫3 (2.20) 1.5 108 M e↵ . (2.15) '| | ⇥ 106 GeV 0.05 eV H✓ ◆ where V = c H2 2 + (a 4 + h.c.⇣)⌘ (2.21) H H H m The baryon asymmetry is obtained| | via8 sphaleronM process, and thus baryonV = numberm2 2 is+ givenSUSY (a 4 + h.c.) (2.20) 2 2 SB | | 8M m by 2 2 2 9↵s(T ) 4 VTH = ckfk T + T ln | |2 and2 (2.22)2 H 4 n| B| 8nL T VH = cH H + (aH + h.c.) (2.21) fk – 11 – – 11 – 30 14. Neutrino mixing 30 14. Neutrino mixingwhere ∗ ∗ JCP =Im Uµ3 Ue3 Ue2 Uµ2 , (14.41) where ! " is the “rephasing∗ invariant”∗ associated with the Dirac CP violation phase in U.Itis JCP =Im Uµ3 Ue3 Ue2 Uµ2 , (14.41) analogous to the rephasing invariant associated with the Dirac CP violating phase in ! " is the “rephasing invariant”the associated CKM quark with the mixing Dirac matrix CP vi [64].olation It phase is clear in fromU.Itis Eq. (14.40) that JCP controls the magnitude of CP violation effects in neutrino oscillations in the case of 3-neutrino mixing. analogous to the rephasing invariant associated with the Dirac CP violating phase in ′ the CKM quark mixing matrix [64]. It2 is clear from Eq. (14.40) that J controls the (l l) If sin(∆mij/(2p)L) ∼= 0for(ij)=(32), orCP (21), or (13), we get ACP ∼= 0. Thus, if as magnitude of CP violation effects in neutrino oscillations in the case of 3-neutrino mixing. aconsequenceoftheproduction,propagationand/ordetect′ ion of neutrinos, effectively (l l) If sin(∆m2 /(2p)L) = 0for(oscillationsij)=(32), dueor (21) only, toor one (13), non-zero we get A neutrino= 0. mass Thus, squared if as difference take place, the CP ij ∼ CP ∼ ′ aconsequenceoftheproduction,propagationand/ordetection of neutrinos, effectively (l l) violating effects will be strongly suppressed. In particular,wegetACP =0,unlessall oscillations due only to one non-zero neutrino2 mass squared difference take place, the CP three ∆mij =0,(ij)=(32), (21), (13). ′ violating effects will be strongly suppressed.̸ In particular,wegetA(l l) =0,unlessall If the number of massive neutrinos CPn is equal to the number of neutrino flavours, three ∆m2 =0,(ij)=(32), (21), (13). ij ̸ n =3,onehasasaconsequenceoftheunitarityoftheneutrinomixing matrix: If the number of massive neutrinos′ Pn(νis equalν ′ )=1, to the numberl = e, µ, ofτ, neutrino flavours,P (ν ν ′ )=1, l′ = e, µ, τ. l =e,µ,τ l → l l=e,µ,τ l → l n =3,onehasasaconsequenceoftheunitarityoftheneutrinomSimilar “probability conservation” equationsixing hold matrix: for P (¯νl ν¯ ′ ). If, however, the # ′# → l ′ P (νl ν ′ )=1,numberl = e, µ, ofτ, light massiveP neutrinos(νl ν ′ is)=1, biggerl than= e, the µ, τ. number of flavour neutrinos as l =e,µ,τ → l l=e,µ,τ → l Similar “probability conservation”aconsequence, equationse.g. hold, of for a flavourP (¯ν neutrinoν¯ ′ ). If, however, - sterile neutrino the mixing, we would have # # l → l number of light massive neutrinos′ is biggerP (ν thanν ′ )=1 the numberP (ν of flavourν¯ ), l neutrinos= e, µ, τ,wherewehaveassumedthe as l =e,µ,τ l → l − l → sL aconsequence,e.g., of a flavour neutrino - sterile neutrino mixing, we would have ′ existence of justAD one sterile LEPTOGENESIS neutrino. Obviously, in this case l′=e,µ,τ P (νl νl ) < 1if ′ P (νl ν ′ )=1 #P (νl ν¯sL), l = e, µ, τ,wherewehaveassumedthe → l =e,µ,τ → l −P (νl →ν¯sL) =0.Theformerinequalityisusedinthesearchesforoscillations between existence of just one sterile neutrino.→ Obviously,̸ in this case ′ P (ν ν ′ ) < 1if # # active and sterile neutrinos.1e+11 l =e,µ,τ l → l P (ν ν¯ ) =0.Theformerinequalityisusedinthesearchesforoscillations between l → sL ̸ Consider next neutrino oscillations# in the case of one neutrino mass squared difference active and sterile neutrinos. 1e+10 2 2 2 “dominance”: suppose that ∆mj1 ∆mn1 , j =2,...,(n 1), ∆mn1 L/(2p) ! 1and Consider next neutrino oscillations2 in the case of one neutr| ino| mass≪ |2 squared| difference − | | ∆m2j1 L/(2p) 2 1,1e+09 so that exp[i(∆mj12L/(2p)] = 1, j =2,...,(n 1). Under these “dominance”: suppose that |∆mj1 | ∆mn≪1 , j =2,...,(n 1), ∆mn1 L/(2p)∼! 1and − 2 |conditions| ≪ |2 we obtain| from Eq.− (14.35)| and Eq.| (14.36), keeping only the oscillating terms ∆mj1 L/(2p) 1, so that exp[i(∆mj1 L/2 (2Tp)] = 1, j =2,...,(n 1). Under these | | ≪ involving ∆mn1: R ∼1e+08 − conditions we obtain from Eq. (14.35) and Eq.[GeV] (14.36), keeping only the oscillating terms involving ∆m2 : n1 1e+07 P (ν ′ ν ′ ) = P (¯ν ′ ν¯ ′ ) = δ ′ 2 U 2 δ ′ U ′ 2 l(l ) → l (l) ∼ l(l ) → l (l) ∼ ll − | ln| ll − | l n| 1e+06 2 2 2 $ % P (ν ′ ν ′ ) = P (¯ν ′ ν¯ ′ ) = δ ′ 2 Uln δ ∆′ m U ′ l(l ) → l (l) ∼ l(l ) → l (l) ∼ ll − |(1 |cosll −n| 1 lLn)|. (14.42) ∆m2 − $ 2p % (1 cos n1 L) . 100000 (14.42) − 2p It follows from the10000 neutrino oscillation data discussed in Section 14.2 that in the case of 3-neutrino mixing,1e-12 one1e-11 of the1e-10 two independent1e-09 1e-08 1e-07 neutri1e-06no mass squared differences, It follows from the neutrino oscillation data discussed in Section 14.2 that in the Asaka, Fujii, Hamaguchi 2 mν1 [eV] 2 ∆m21 > 0, is much smaller than the absolute value of the second one, ∆m31 : case of 3-neutrino mixing, one of2 the two independent2 neutrino mass squared differences, | | 2 ∆m21 Figure∆m 3.2:31 .Thedata,aswehaveseen,imply: The contour plot of the baryon asymmetries nB2/s in the mν1–TR plane. The ∆m21 > 0, is much smaller than≪ the| absolute| value of the second one,−9 ∆−10m31 : −10 −11 −12 lines represent the contour plots for nB/s =10 ,10| ,0.4| 910 ,10 ,and10 ∆m2 ∆m2 .Thedata,aswehaveseen,imply:• Successful leptogenesis requires m⌫1 10 × eV 21 ≪ | 31| from the left to the right. ⇠ ∆m2 = 7.4 10−5 eV2 , 21 ∼ × ∆ism light,2 = like7. in4 gauge-mediated10−5 eV2 , SUSY∆ breakingm2 = models2.5 [45].10−113ThiseV is2 , the reason why we 21 ∼ × | 31| ∼ × have2 assumed gravity-mediated−3 2 2 SUSY-breaking2 and m3/2 1TeV. ∆m31 ∼= 2.5 10 eV∆m21, / ∆m31 ∼= 0.03 . ≃ (14.43) | Fig.| 3.2 shows× the contour plot| of the produced| baryon asymmetry in the m –T ∆m2 / ∆m2 = 0.03 . (14.43) ν1 R 21 plane,31 given∼ by the analytic formula Eq. (3.41). Here, we havetakenm = m =1TeV, | | October 6, 2016 11:02 φ 3/2 2 am =1andδph =1,andwehaveusedtherelationmν1 = Hu /Meff .(Althoughthe October| | 6, 2016 11:02 ⟨ ⟩ < 9 big-bang nucleosynthesis constraint on the gravitino abundance requires TR 10 GeV for m =1TeV,wehaveplottedthecontourinFig.3.2uptoT 1011 GeV,∼ keeping 3/2 R ≤ in mind that even T 1011 GeV can be allowed for a slightly heavier gravitino, say, R ≃ m 3TeV.SeeSec.1.5.) 3/2 ∼ Aremarkableobservationhereisthatthepresentbaryonasymmetry nB/s is deter- > 5 mined almost independently of the reheating temperature forawiderangeofTR 10 GeV. > 8 ∼ In particular, for a relatively high reheating temperature TR 10 GeV, the baryon asym- ∼ 11 If we introduce a gauged U(1)B−L,however,theLHu flat direction can produce enough lepton asymmetry even with a light gravitino. See Sec. 3.4. 90 OUTLINE 1. Leptogenesis 2. AD mechanism along LHu direction 3. AD leptogenesis in DFSZ model with varying PQ scale 4. Summary 3 saxion decay is dependent on the µ-term. Therefore the resulting baryon asymmetry after saxion decay is linked to electroweak fine-tuning, which is determined by the µ-term. In Sec. II, we analyze AD leptogenesis along the LHu flat direction with a dynamical PQ break- ing scale and then estimate the baryon asymmetry taking account of the dilution from saxion decay. In Sec. III, we investigate the dynamics of PQ breaking scalar fields and then examine conservation 3 of lepton asymmetry during the saxion oscillation. In Sec. IV we present the main results in the formsaxion of contours decay of is required dependentm on⌫1 values the µ-term. in the Thereforeµ vs. fa theplane, resulting which baryon show a asymmetry relation between after saxion the baryondecay asymmetry is linked and to electroweak the electroweak fine-tuning, fine-tuning. which In is determined Sec. V, we discuss by the µ some-term. cosmological impli- cations ofIn the Sec. PQ II, we sector: analyze the AD axion leptogenesis isocurvature along perturbation the LHu flat and direction axino with production. a dynamical We concludePQ break- in Sec.ing VI. scale and then estimate the baryon asymmetry taking account of the dilution from saxion decay. In Sec. III, we investigate the dynamics of PQ breaking scalar fields and then examine conservation ofII. lepton AFFLECK-DINE asymmetry during LEPTOGENESIS the saxion oscillation. WITH In DYNAMICAL Sec. IV we present PECCEI-QUINN the main results in the form of contours of required mSYMMETRY⌫1 values in the BREAKINGµ vs. fa plane, which show a relation between the baryon asymmetry and the electroweak fine-tuning. In Sec. V, we discuss some cosmological impli- Incations this section, of the wePQ show sector: how the dynamical axion isocurvature PQ symmetry perturbation breaking and accommodates axino production. enough We conclude baryon asymmetryin Sec. and VI. a relatively large neutrino mass. II. AFFLECK-DINE LEPTOGENESISA. The model WITH DYNAMICAL PECCEI-QUINN SYMMETRY BREAKING We consider the neutrino sector for AD leptogenesis and the PQ breaking sector, which are describedIn by this the section, following we show superpotential, how dynamical PQ symmetry breaking accommodates enough baryon asymmetry and a relatively large neutrino mass. W = WAD + WPQ, (1) 2 4 6 1 A. The model 2 gµY WAD = XNN + y⌫NLHu,WPQ = ⌘Z(XY f )+ HuHd, (2) 2 MP so is a massless mode correspondingX toY theZ axionN L H ofu theH spontaneously broken PQ symmetry. In We consider the neutrino sector for ADd leptogenesis and the PQ breaking sector, which are PQ 2 2 0 1 1 2 2 thewhere limitN of is0 theX RHN,0, the massiveL is the mode lepton is mostly doublet,a -like Hu and(Hd the) is massless the up-type mode is (down-type) mostly aX -like. Higgs doublet, described⌧ by the followingL superpotential,0 0 0 1 1 0 0 AD LEPTOGENESIS WITH PQ X and Y are the PQ fields, Z is a singlet scalar, , ⌘ and gµ represent numerical coe cients, f TABLE I: U(1) charges of the fields. 4 denotes the (present)• B. PQ Baryon breaking asymmetry scale and alongM LHtheu reduceddirection Planck scale. The PQ charges and W Scale= WAD of+ W MPQ ,(RHN mass)P can be generated by PQ breaking(1) lepton numbers of these fields1 are given in Table I. g Y 2 X Y Z N L Hu Hd The neutrinoNow let mass us evaluate is generated the bylepton the see-saw number mechanism generated at through low energy the ADX mechanism.f: 2 Theµ massive WAD = XNN + y⌫NLHu,WPQ = ⌘Z(XY f )+ HuHd, PQ 2 (2)2 0 1 1 2 2 2 h i⇠ MP phaseWhen modeX automaticallyobtains a largecancels field the value,imaginary the part RHN of becomes the Hubble massive induced andA-term can potential, be integratedL out0 to0 0 1 1 0 0 2 2 2 y⌫ Hu v soobtain it does the not e↵ significantlyective superpotential: contributem⌫1 = toh leptoni number, generation. Thus, as in ordinary(4) AD TABLE I: U(1) charges of the fields. 4 where N is the RHN,onceL isX the has lepton flarge doublet,' M veve↵ H(~f)u (Hd) is the up-type (down-type) Higgs doublet, leptogenesis, lepton number is determined by the ordinary A-term which depends on the hidden X and Y are the PQ fields, Z is a singlet scalar,2 , ⌘ and2 gµ represent numerical coe cients, f where v 174 GeV and 1 y (LH ) X Y Z N L Hu Hd ' ⌫ u The neutrino mass is generated by the see-saw mechanism at low energy X f: sector SUSY breaking (i.e., gravitino massW inAD gravity,e↵ = mediation). The. lepton number obeys the (3) h i⇠ denotes the (present) PQ breaking scale and 2MP the X reducedPQ Planck2 2 0 scale.1 1 The2 PQ2 charges and 4 2 4 equation, v 10 eV 2 2 2 M = =3.0 1017 GeV . L 0 0 0 1 (5)1 0 0 y⌫ Hu v lepton numbers ofe↵ thesem fields are⇥ given in Tablem I. m⌫1 = h i , (4) • ⌫1 ✓ ⌫1 ◆ f ' Me↵ AD leptogenesis works withTABLE M~ In what follows, we consider the dynamics of the AD field , which parameterizes the LHu D-flat direction as 1 1 0 L = ,Hu = . (8) p2 001 p2 0 1 @ A @ A 2 For simplicity, we assume that X = Y = f at the present universe. 4 X Y Z N L Hu Hd PQ 2 2 0 1 1 2 2 L 0 0 0 1 1 0 0 TABLE I: U(1) charges of the fields. The neutrino mass is generated by the see-saw mechanism at low energy X f: h i⇠ 2 2 2 6 y⌫ Hu v m⌫1 = h i , (4) f ' Me↵ so is a massless mode corresponding to the axion of the spontaneously broken PQ symmetry. In where v 174 GeV and the limit of X', the massive mode is mostly a -like and the massless mode is mostly a -like. 0 ⌧ 0 X v2 10 4 eV M = =3.0 1017 GeV . (5) e↵ m ⇥ m ⌫1 ✓ ⌫1 ◆ B. Baryon asymmetry along LHu direction Here and in the following, we assume that the lepton asymmetry is generated along the flattest Now letLH usu evaluatedirection, the which lepton corresponds number generated to the smallest through neutrino the ADmass mechanism.m⌫1 . As will The become massive clear, X phase modeis automatically di↵erent from cancels its low-energy theAD imaginary vacuum LEPTOGENESIS expectation part of the value Hubble (VEV) inducedf in theA-term early potential, universe. WITH2 In PQ so it does notparticular, significantlyX is shown contribute to be fixedto lepton at X number= X0( generation.MP ) until the Thus, saxion as begins in ordinary to oscillate AD (see • What if PQ scale ⇠is dynamical? leptogenesis,Sec. lepton III). number Thus one is can determined also define by an the e↵ective ordinary scaleA in-term the early which universe depends as on the hidden X AD X now f sector SUSY breaking (i.e., gravitino massh ini 2 gravity h mediation).i ⇠ The lepton number obeys the y⌫ 1 f 1 = . (6) 7 4 X0 Me↵ X0 ⌘ M equation, • LHu flat direction✓ is ◆“flatter”,⇤ AD works more efficiently with In a di↵erent way, Mlargeis expressed initial by 0 HM second and third⇤ terms inside themsoft bracket⇠ of4 Eq.⇤ (18). The detailed physical aspects are explained n˙ L +3HnL Im(am ). (16) ' 2M 23 in Ref. [20] and references therein.⇤10 If m4p eV= 1010 TeV12 GeV and M X=7.2 10 GeV, such early oscillation M =7.2 1023GeV ⇤ 0 .⇥ (7) ⇤ 7 ⇥ m⌫1 f MP (scale for AD mechanism) The baryon numberoccurs to entropy for TR & ratio10 isGeV. obtained as✓ [13] ◆✓ ◆✓ ◆ Thus M Foris much an illustration, larger than M wee↵ , show which a modifies formula the for ordinary a case where relation the between early oscillation low-energy does neu- not occur. In ⇤ nB M TR msoft am trino masssuch and a case, the eH ciency= m0. of029, AD so the leptogenesis⇤ baryon-number-to-entropy| [12].| ph, ratio becomes (17) oscs ' M 2 H P ✓ osc ◆ In what follows, we consider the dynamics of the AD field , which parameterizes the LHu 4 12 nB 8 TR 10 eV 10 GeV X0 where ph representsD-flat direction an e↵ asective CP violating phase, and Hosc is the Hubble parameter when the 3.6 10 ph 7 , (20) s ' ⇥ 10 GeV m⌫1 f MP field starts to oscillate. Taking account of thermal✓ e↵ects on◆✓ the AD potential◆✓ [18, 19], the◆✓ latter◆ 1 1 0 is determined by where we also• If assume PQ mscaleL == a mduringmsoft,H. In AD thisu = scenario, ~ MP however,. and becomes saxion will dominate f afterwards,(8) the universe, p|2 0|01 p2 0 1 and its decay produces entropy dilution. In order to obtain the final baryon asymmetry after saxion @ A 1/2@ A AD leptogenesis isag MpossibleP for sizable neutrino mass 2 decay, theH entropyosc max dilutionm ,H musti, ↵ beSTR taken into account., We will consider the(18) entropy production For simplicity, we assume' that X-4= Y = f at the present universe.M ~10 " eV ✓ ⇤ ◆ # in the following subsection. where Before closing this subsection, let us comment on the possible lepton number violation during the saxion oscillation. Since the2 field value of X can become small during its oscillation, the e↵ect 1 MP TR 2 4 2 1/3 Hi =min 4 2 , (ci fi MP TR) . (19) of the lepton number violation,fi M induced by the e↵ective superpotential (3) or the corresponding ⇤ A-term, may become large. Here, as shown in Sec. III, the µ-term interaction in Eq. (2), Here, the fi are coupling constants of with particles in thermal background, ci and ag(= 1.125) are real positive constants of order unity, and T is the reheating2 temperature after inflation. R gµY Wµ = HuHd , (21) From Eq. (18), Hosc is nearly m when the temperature is small.MP In the case of a large reheat temperature, the oscillation can commence earlier due to thermal e↵ects which result in the plays an important role. Assuming m m , the lepton number violation during the saxion X ⌧ 4 oscillation is small enough to2 maintain the generated lepton asymmetry by the AD mechanism. As While the lepton number is violated by the (LHu) term in the superpotential after integrating out N,thePQ number is exactlywe conserved shall see (except in the for the next small subsection, instanton e the↵ect).µ-term interaction (21) also plays a key role to determine the saxion decay, and hence the final baryon asymmetry. C. Saxion decay in DFSZ model We have discussed how the dynamical PQ breaking scale can enhance the baryon asymmetry. For the final result, one crucial point to consider is the entropy production from saxion decay. In the DFSZ model, saxion interactions with the standard model particles and their superpartners are realized in the µ-term interaction (21). Once X and Y settle down to the current value of the PQ symmetry breaking scale, X Y f, this superpotential generates the µ-term, ⇠ ⇠ g f 2 µ µ , (22) ⇠ MP 9 For the case where saxion dominantly decays into Higgsinos, 4 12 2 1 1/4 1/2 nB 12 10 eV 10 GeV X0 90 µ m =1.1 10 ph , (29) s ⇥ m⌫1 fa MP g TeV 10 TeV ✓ ◆✓ ◆ ✓ ◆ ✓ ⇤ ◆ ⇣ ⌘⇣ ⌘ or for the case where saxion dominantly decays into light Higgs and gauge bosons, 4 12 2 1 1/4 2 1/2 nB 12 10 eV 10 GeV X0 90 µ 100 GeV =3.0 10 ph . 9 s ⇥ m⌫1 fa MP g TeV m ✓ ◆✓ ◆ ✓ ◆ ✓ ⇤ ◆ ✓ ◆ ⇣ ⌘ (30) For the case where saxion dominantly decays into Higgsinos, From this it is easily seen that the observed baryon-number-to-entropy-ratio can be obtained for 10 4 2 1 1/4 relatively large neutrino mass m⌫1 = 10 eV if the PQ breaking scale is4 near the Planck12 scale in 1/2 nB 12 10 eV 10 GeV X0 90 µ m =1.1 10 ph , (29) the beginning and settles to 1011 GeV at the presents universe.⇥ m⌫1 fa MP g TeV 10 TeV VEV. Let us✓ see the scalar◆✓ potential◆ in✓ detail.◆ ✓ ⇤ ◆ ⇣ ⌘⇣ ⌘ We will see numerical results for some example parameter regions in Sec. IV. or for the caseIn where this discussion, saxion dominantlyZ obtains decays a mass into of lightX Higgsand andY gauge(>f), bosons, so its VEV is zero up to of the order | | | | 2 of the gravitino mass:4 (m 12). Thus we can safely1 neglect1/4 the2 dynamics of1/Z2 .IfI MP during III. DYNAMICS OFn PQB BREAKING12 FIELDS10 eV 103/2 GeV X0 90 µ 100 GeV =3.0 10 ph O . ⌧ s ⇥ m⌫1 fa 10MP g TeV m inflation, the✓ inflaton energy◆✓ is dominantly◆ ✓ ◆ determined✓ ⇤ ◆ by the F✓-term potential,◆ i.e. KI MP , ⇣ ⌘ (30) ⌧ In this section, we discuss the dynamics of the PQW breaking/M fieldsW and in orderD W to investigateW F the. One can simplify the inflaton potential: V (I) F 2 VEV. Let us see the scalar potentialFrom in detail. this| it| is easilyP ⌧ seenI that theI observed' I baryon-number-to-entropy-ratio' I can be obtained for ⇠ | I | ⇠ 2 2 realization of theIn this Planck discussion, scale PQZ obtains breaking a mass in of theX earlyandH M stageY (>f. In and), these so lepton its VEV circumstances, number is zero up conservation4 to of the the scalar order at potential of X and Y is given by relatively| | large| |P neutrino mass m⌫1 = 10 eV if the PQ breaking scale is near the Planck scale in of the gravitino mass: (m3/2). Thus we can safely neglect the dynamics of Z.IfI MP during the late stage. O 11 ⌧ the beginning and settles to 10 GeV at the present universe. 2 inflation, the inflaton energy is dominantly determined by the F -term potential,(i.e.X 2+KIY 2)/MMP2, 2 2 2 FI V = e | | | |⌧ P ⌘ XY f + | | , (34) We will see numerical results for some example2 parameter regions in Sec. IV. 2 2 W /MP WI and DI W WI FI . One can simplify the inflaton potential: V (I) FI | | 1+b X /MP | | ⌧ A. PQ' breaking' at the Planck scale ⇠ | | ⇠ ✓ | | ◆ 2 2 H MP . In these circumstances, the scalar potential of X and Y is given by Let us define X = x and Y = y. The extremum condition is obtained as III.h i2 DYNAMICSh i OF PQ BREAKING FIELDS Let us first examine the scalar potential( X 2+ Y 2)/M of2 X and2 Y for2 2 large HFIto check if the PQ scale is V = e | | | | P ⌘ XY f + | | , (34) | | 1+b X 2/M 2 ✓ P ◆ 2 (MP ). We have to consider the supergravity potential which@V is given| | by2 2 2x 2 FI 2x 2bx 1 O In this section,0= we= discuss⌘ ( thexy dynamicsf ) of2 ( thexy PQf breaking)+2y fields in| order2| 2 to investigate2 2 the 2 2 Let us define X = x and Y = y. The extremum condition@x is obtained as M 1+bx /M M M 1+bx /M h i h i ✓ P ◆ P ✓ P P P ◆ realization of the Planck scale2 PQ2 breaking2 in the early stage and lepton number conservation at K/M2 i¯j 3 2 (x +y )/MP V = e P DiWK D¯W ⇤ W2 , e , (31) (35) @V 2 2 2x 2 j 2 FI ⇥ 2x 2bx 1 10 0= = ⌘ (xy f ) (xythef late)+2 stage.y MP|| | | @x M✓ 2 1+@bxV 2/M◆2 2M 2 M 221+bxy2/M 2 2 (x2+y2)/M 2 ✓ P ◆0= =P ⌘✓ (xyP fP ) (xyP ◆ f )+2x e P . (36) (x2+y2)/M 2 2 e 2 P , @y MP (35) whereVEV.DiW Let= usWi see+ K theiW/M scalar⇥ P . potential We assume in detail. that the e↵POTENTIALect of the AD field is negligible.✓ The ◆ @V 2 2 y 2 (x2+y2)/M 2 0= = ⌘ (xy f ) (xy f )+2x e P . A. PQ breaking at(36) the Planck scale K¨ahler potential and superpotential are2 given by In this discussion,@y Z obtains aM massP of X andFromY the(>f second), so its equation, VEV is zero we up find to that of the order • Hubble✓ induced| |potential◆ | | realizes such a scenario. of the gravitinoFrom the mass: second equation,(m3/2). we Thus find that we canLet safely us first neglect examine the the dynamics scalar potential of Z.IfI of XMandP duringY for large H to check if the PQ scale is O 2 2 2 2 b 2 2 ⌧ K = X + Y + Z + I + 2 I X , 2 (32) 2 2 2 inflation, the inflaton energy| is| dominantly| | |(M|P determined).| We| haveMP by to| considerthe| | F| -term the potential, supergravityxy =i.e.f potentialKI or M whichP ,xy is givenf y by+ MP x =0. (37) 2 2 2 2 xy = f 2O or xy f y + M x =0. (37)⌧ W /M W and WD W= ⌘ZW(XY Ff.) One, can simplify theP inflaton potential: V (I)(33)F 2 | | P ⌧ I I ' I ' I ⇠ | I | ⇠ I: inflaton K/M2 i¯j 3 2 2 2 Since the secondV solution= e P leadsDiWK to aD trivial¯W ⇤ solutionW x =, y = 0 we select the(31) first solution. From H MP .Since In these the second circumstances, solution leads the to scalar a trivial potential solution ofx =Xyand= 0Y weis select given the by first solution. Fromj M 2 | | where I is the inflaton field. Note that only X has non-minimal coupling with the✓ inflaton in K. P ◆ Eq. (35), we obtain the solution for x, Eq. (35), we obtain the solution for x, 2 2 2 2 FI 2 If b>1, one can obtain a negative( X Hubble-induced+ Y )/MwhereP 2 D massiW = term2W2i for+ KXiW/MandP thus. WeX assumedevelops that a large the e↵ect of the AD field is negligible. The V = e | | | | ⌘ XY 1f/2 + | |2 2 , (34) | 1 | 1+b X /MP 1/2 K¨ahlerx✓= 1 potentialM andP . superpotential| | ◆ are given by (38) 1 b x = 1 M . (38) ✓ ◆ b P Let us define X = x and Y = y. The extremum condition is obtained as 1/2 2 2 2 2 2 ✓ ◆ For b>h 1,i x developsh ani (MP ) VEV as long as FI /M m . It(H is alsom evident) that X b X =(1O 1/b) MP for | | P 3/2 2 3/22 2 2 2 2 h i K = X + Y + Z + I + 2 I X , (32) obtains a (negative) mass squared of the order of H2 during the inflaton| domination.| | | | | | | M | | | |2 2 2 For b>1, x2 develops an (MP ) VEV as longP as FI /MP m3/2. It is also evident that X @V 2 2 2x 2 FI 2x 2Obx 1 | | 0= = ⌘ (xy f ) (xy f )+2y | | W = ⌘Z(XY f 2), (33) @x M 2 obtains 1+ abx (negative)2/M 2 M mass2 M squared2 1+bx of2/M the2 order of H2 during the inflaton domination. ✓ P ◆ P ✓ P P P ◆ (x2+y2)/MB.2 Saxion oscillation and lepton number conservation e• P , (35) ⇥ When H~mwhere3/2, PQI is thefield inflaton (saxion) field. Note starts that only oscillationX has non-minimal with coupling with the inflaton in K. @V 2 2 y 2 (x2+y2)/M 2 0= As= argued⌘ (xy in thef ) previous( subsection,xy f )+2X staysx e at X MPP. until theB. Hubble Saxion parameter oscillation drops (36) and lepton number conservation @y amplitude M 2 M IfP.b>1, one can obtain⇠ a negative Hubble-induced mass term for X and thus X develops a large down to m . After that,✓ theP saxion begins a coherent◆ oscillation around the minimum x y f 3/2 ⇠ ⇠ with an initial amplitude of MP . Since the scalar field orthogonal to the F -flat direction (saxion) From the second equation, we findsaxion-dominated⇠ that As argued universe in the previous after subsection,reheatingX stays at X MP until the Hubble parameter drops has a mass of f, which is much higher than the soft mass scale, we can safely set XY = f 2 to ⇠ ⇠ down to m3/2. After that, the saxion2 begins a coherent oscillation around the minimum x y f integrate out either X or Y . Then2 the scalar potential2 2 along the2 F -flat direction XY = f and ⇠ ⇠ xy = f or xy f y + MP x =0. (37) with an initial amplitude of MP . Since the scalar field orthogonal to the F -flat direction (saxion) ⇠ has a mass of f, which is much higher than the soft mass scale, we can safely set XY = f 2 to Since the second solution leads to a trivial solution x = y = 0⇠ we select the first solution. From integrate out either X or Y . Then the scalar potential along the F -flat direction XY = f 2 and Eq. (35), we obtain the solution for x, 1 1/2 x = 1 M . (38) b P ✓ ◆ For b>1, x develops an (M ) VEV as long as F 2/M 2 m2 . It is also evident that X O P | I | P 3/2 obtains a (negative) mass squared of the order of H2 during the inflaton domination. B. Saxion oscillation and lepton number conservation As argued in the previous subsection, X stays at X M until the Hubble parameter drops ⇠ P down to m . After that, the saxion begins a coherent oscillation around the minimum x y f 3/2 ⇠ ⇠ with an initial amplitude of M . Since the scalar field orthogonal to the F -flat direction (saxion) ⇠ P has a mass of f, which is much higher than the soft mass scale, we can safely set XY = f 2 to ⇠ integrate out either X or Y . Then the scalar potential along the F -flat direction XY = f 2 and POTENTIAL • Along flat direction XY = f 2 H m H m 3/2 ⇠ 3/2 X M X M 0 ⇠ P 0 ⇠ P "saxion oscillation" 3 saxion decay is dependent on the µ-term. Therefore the resulting baryon asymmetry after saxion decay is linked to electroweak fine-tuning, which is determined by the µ-term. In Sec. II, we analyze AD leptogenesis along the LHu flat direction with a dynamical PQ break- ing scale and then estimate the baryon asymmetry taking account of the dilution from saxion decay. In Sec. III, we investigate the dynamics of PQ breaking scalar fields and then examine conservation of lepton asymmetry during the saxion oscillation. In Sec. IV we present the main results in the form of contours of required m⌫1 values in the µ vs. fa plane, which show a relation between the baryon asymmetry and the electroweak fine-tuning. In Sec. V, we discuss some cosmological impli- cations of the PQ sector: the axion isocurvature perturbation and axino production. We conclude in Sec. VI. II. AFFLECK-DINE LEPTOGENESIS WITH DYNAMICAL PECCEI-QUINN SYMMETRY BREAKING In this section, we show how dynamical PQ symmetry breaking accommodates enough baryon asymmetry and a relatively large neutrino mass. A. The model We consider the neutrino sector for AD leptogenesis and the PQ breaking sector, which are described by the following superpotential, W = WAD + WPQ, (1) 2 1 2 gµY WAD = XNN + y⌫NLHu,WPQ = ⌘Z(XY f )+ HuHd, (2) SAXION2 OSCILLATION MP • Saxionwhere oscillatesN is the along RHN, L isXY the lepton= f 2 doublet, Hu (Hd) is the up-type (down-type) Higgs doublet, X and Y are the PQ fields, Z is a singlet scalar, , ⌘ and gµ represent numerical coe cients, f Y denotes the (present) PQ breaking scale and MP the reduced Planck scale. The PQ charges and lepton numbers of these fields are given in Table I. When X obtains a large field value, the RHN becomes massive and can be integrated out to obtain the e↵ective superpotential:When X is very small, 1 y2(LH )2 W = ⌫ u . valid? (3) AD,e↵ 2 X f X M 0 ⇠ P X 3 saxion decay is dependent on the µ-term. Therefore the resulting baryon asymmetry after saxion decay is linked to electroweak fine-tuning, which is determined by the µ-term. In Sec. II, we analyze AD leptogenesis along the LHu flat direction with a dynamical PQ break- ing scale and then estimate the baryon asymmetry taking account of the dilution from7 saxion decay. In Sec. III, we investigate the dynamics of PQ breaking scalar fields and then examine conservation second and third termsof lepton inside asymmetry the bracket during of Eq. the (18).saxion The oscillation. detailed In physical Sec. IV aspects we present are explainedthe main results in the 23 in Ref. [20] and referencesform of contours therein. of If requiredm = 10m TeV⌫1 values and M in the=7µ.2vs.10fa plane,GeV, such which early show oscillation a relation between the ⇤ ⇥ baryon7 asymmetry and the electroweak fine-tuning. In Sec. V, we discuss some cosmological impli- occurs for TR & 10 GeV. For an illustration,cations we of show the PQ a formula sector: thefor a axion case isocurvature where the early perturbation oscillation and does axino not production. occur. In We conclude in Sec. VI. such a case, Hosc = m , so the baryon-number-to-entropy ratio becomes 4 12 nB II. AFFLECK-DINE8 TR LEPTOGENESIS10 eV 10 WITHGeV DYNAMICALX0 PECCEI-QUINN 3.6 10 , (20) s ' ⇥ ph 107 GeV m f M ✓ ◆✓ SYMMETRY⌫1 ◆✓ BREAKING◆✓ P ◆ where we also assume m = am msoft. In this scenario, however, saxion will dominate the universe, In this section,| | we show how dynamical PQ symmetry breaking accommodates enough baryon and its decay produces entropy dilution. In order to obtain the final baryon asymmetry after saxion asymmetry and a relatively large neutrino mass. decay, the entropy dilution must be taken into account. We will consider the entropy production in the following subsection. A. The model Before closing this subsection, let us comment on the possible lepton number violation during 12 the saxion oscillation.We Since consider the field the neutrino value of X sectorcan for become AD leptogenesis small during and its the oscillation, PQ breaking the e↵ sector,ect which are SAXION2 OSCILLATION Using nL msoft , we obtain of the lepton numberdescribed violation, by the induced following by superpotential, the e↵ective⇠ superpotential (3) or the corresponding n y2 2 A-term, may become large. Here,• asDFSZ shown inplays Sec. III,a role the µ-term interaction inL Eq.⌫ (2), . (44) nL ⇠ mX Xmax W = WAD + WPQ, (1) 2 2 1 g Y2 2 g8 Y 2 Sinceµ in the numerator decreases faster than2 Xmaxµ in the denominator, this takes a maximum WAD = W XNNµ = + y⌫HNLHuHdu,,WPQ = ⌘Z(XY f )+guf HuH(21)d, (2) 2 M Veff M value justP after the X begins to oscillate H Mm 2. PX2 XP ⇠ plays an importantwhere role.N is Assuming the RHN,mLXis them lepton, the doublet, lepton numberHu (Hd) violation is the up-type during (down-type) the2 saxion2 Higgs doublet, 2 n1L/2 y⌫ H =mX ⌧ f gµ . (45) n ⇠ m M X and Y are the PQ fields, Z is a singletXmin scalar, , ⌘ and| |gµL representH mX numericalX P coe cients, f oscillation is small enough to maintain the generated lepton asymmetry⇠ M m by✓ the AD◆ ⇠ mechanism. As P ✓ X ◆ we shall see in thedenotes next subsection, the (present) the PQµ-term breakingThis interaction must scale be smaller and (21)M thanP alsothe 1 plays to reduced ensure a key Planck the role conservation scale. to determine The of lepton PQ charges number. and If thermal e↵ects are lepton numbers of these fieldsneglected are given and m in Tablem I.,wehave 2 m M (m /m )2 and itm becomes the saxion decay, and hence the finalDuring baryon saxion asymmetry. oscillation,X AD field H =mX X X ⌧ ⇠ ⇤ When X obtains a large field value, the RHN becomes massive and can be integrated out to nL mX obtain the e↵ective superpotential:RHN mass is much larger than soft mass scale 1 . (46) nL H m ⇠ m ⌧ C. Saxion decay in DFSZ model ✓ ◆ ⇠ X 2 2 Thus, the lepton number1 violationy⌫(LHu during) the saxion oscillation can be neglected as far as mX m WAD,e↵ = . is valid (3) ⌧ We have discussed how the dynamical PQis satisfied. breaking scale can 2 enhance X the baryon asymmetry. For the final result, one crucial point to consider is the entropy production from saxion decay. In the DFSZ model, saxion interactions with the standard modelIV. particles BARYON and NUMBER their superpartners AND SUSY SCALE are realized in the µ-term interaction (21). Once X and Y settle down to the current value of the We have discussed how PQ symmetry breaking accommodates the baryon asymmetry with a PQ symmetry breaking scale, X Y f, this superpotential generates the µ-term, ⇠ ⇠ sizable neutrino mass when the PQ scale varies during and after inflation. In this scenario, the entropy dilution2 from saxion decay indeed plays a substantial role for determining the final value gµf ofµ the baryon, asymmetry. The saxion decay rate depends(22) on its mass and the µ-term as shown ⇠ MP in Eqs. (23) and (25). In many cases, the saxion mass and µ-term are related to the soft SUSY breaking scale. In particular, µ-term is a measure of fine-tuning of the electroweak symmetry breaking. Therefore, it leads us to discuss the soft SUSY scale and fine-tuning from the measured baryon asymmetry. Since the saxion is linked to the axion which is the Nambu-Goldstone boson of broken PQ symmetry, it is massless in the supersymmetric limit. When SUSY is broken, however, the saxion (and also the axino) acquires a mass. The saxion mass is typically of the gravitino mass order although it can be either larger or smaller than the gravitino mass in some models [20–23]. On the other hand, as shown in the Sec. III B, the saxion mass (i.e. mX ) is required to be smaller than the AD field mass in order to not spoil lepton number generation. In this regard, we consider a rather small saxion mass compared to the AD field mass, i.e. m m m /10. ⇠ X . 11 the AD field reads f 4 g2f 8 2 V m2 X 2 + m2 2 + m2 + µ , (39) ' X | | | | Y X 2 M 2 X2 P | | where mX and mY are the soft SUSY breaking mass of X and Y , respectively. The last term comes from Wµ (21). The third and fourth terms act as the e↵ective potential for X and they prevent X from being very small during the oscillation. Let us denote by Xmax the maximum value of X during each X oscillation, which adiabatically becomes smaller due to the Hubble expansion (X a 3/2). Then we can define X , the minimum value of X during each oscillation. For a max / min large AD field value , the last term is important to determine Xmin. Thus we can evaluate Xmin as f 2 g X 1/2 m f 2 X max µ max| | , Y . (40) min ⇠ X M m m X " max ✓ P X ◆ X max # Since X M and m just after the saxion oscillation, X is generically much larger max ⇠ P | | X min than the soft mass scale, meaning that the RHN masses cannot be as small as the soft mass 12 during saxion oscillation and hence the procedure to integrate out the RHN to obtain the e↵ective 2 SAXION OSCILLATION Using nL msoft , we obtain potential of the AD field⇠ is justified. Now let us consider the lepton number violation n aftery the2 X 2begins to oscillate. The lepton • Lepton number violationL ⌫ during. saxion oscillation (44) n ⇠ m X number follows: L X max (after AD works) 12 12 Since 2 in the numerator decreases faster than X in the denominator, this takes a maximum y2m max ⌫ soft 2 4 2 n˙ L +3HnUsingL =nL msoftIm( ,a wem obtain). (41) Using nL msoftvalue , we just obtain after the X begins to oscillate⇠ X H mX . ⇠ ⇠ n y2 2 When X is small,2 it 2could make large2 2 Lepton1 Lnumber⌫ change As discussed above, X oscillates between nL Xymax⌫ and nLXmin in a timey⌫ scaleH=mXm where Xmin is given. (44) . .XnL ⇠ mX Xmax(44) (45) nL ⇠ mXnXLmaxH m ⇠ mX MP 2 1/2 by Eq. (40). The most dangerous L violation✓ may◆ happen⇠ X around X Xmin atf whichgµ the L- DFSZ prevents2 X from being too small⇠ Xmin | | Since in the numerator decreases faster than⇠ MXmaxP inm theX denominator, this takes a maximum violating2 operator becomes large. The time interval t during which X X is estimated✓ from◆ Since in theThis numerator must be decreases smaller faster than than1 to ensureXmax in the the conservation denominator, of leptonthis takesmin number. a maximum If thermal e↵ects are value just after the X begins to oscillate⇠ H mX . the equation of motion 2 2 ⇠ value just afterneglected the X begins and tomX oscillatem ,wehaveH mX . H=m m M (mX /m ) and it becomes • Total⌧ Lepton numberX ⇠ change⇤ is ⇠ 2 2 nL y⌫ H=mX 2 4 2 2 3 . (45) µ f 2 2 XminnL ⇠ mX MP ¨ nL y ⌫¨nLH=mX mX H mX X 5 Xmin X( t) . t ✓1 .2 .◆ ⇠ (45)(42) (46) ⇠ X ! ⇠ n !⇠ m ⇠⌧µf nL H m ⇠ LmX MH PmX ✓ ◆ This⇠ X must✓ be smaller◆ ⇠ than 1 to ensure the conservation of lepton number. If thermal e↵ects are During this time interval, the L number changes as 2 2 This must be smallerThus, the than lepton 1 to number ensure the violationneglected conservation duringfor and them ofX lepton saxionm ,wehave number.oscillation H If can= thermalm be neglectedm eM↵ects(mX as are/m far ) asandmX it becomesm ⌧ neccessaryX ⇠ for⇤ AD before saxion⌧ oscillation 2 2 neglected and mis satisfied.m ,wehave m M (m /m ) and4 it2 becomes X H=m2X 4 X 2 ⌧ y⌫m⇠soft ⇤ y⌫ gµmsoft f nL mX nL t . (43)1 . (46) ⇠ X ⇠ µmX MP Xmax nL ⇠ m ⌧ ✓ ◆H mX nL mX ⇠ IV. BARYON NUMBER1 . AND SUSY SCALE (46) nL HThus,m the⇠ leptonm ⌧ number violation during the saxion oscillation can be neglected as far as mX m ✓ ◆ ⇠ X ⌧ is satisfied. Thus, the lepton numberWe have violation discussed during how the PQ saxion symmetry oscillation breaking can be accommodates neglected as far the as baryonm m asymmetry with a X ⌧ is satisfied. sizable neutrino mass when the PQ scale varies during and after inflation. In this scenario, the IV. BARYON NUMBER AND SUSY SCALE entropy dilution from saxion decay indeed plays a substantial role for determining the final value of the baryonIV. BARYON asymmetry. NUMBER TheWe saxion have AND discusseddecay SUSY rate how SCALE depends PQ symmetry on its mass breaking and accommodates the µ-term as the shown baryon asymmetry with a in Eqs. (23) and (25). In manysizable cases, neutrino the masssaxion when mass the and PQµ scale-term varies are related during and to the after soft inflation. SUSY In this scenario, the We have discussedbreaking how scale. PQ symmetry In particular, breakingentropyµ-term dilution accommodates is a from measure saxion the of decay baryon fine-tuning indeed asymmetry plays of the a substantial electroweak with a role symmetry for determining the final value sizable neutrinobreaking. mass when Therefore, the PQ it scale leads variesof us the to during baryon discuss andasymmetry. the after soft SUSY inflation. The saxion scale In and decay this fine-tuning scenario, rate depends from the on the its measured mass and the µ-term as shown entropy dilutionbaryon from saxion asymmetry. decay indeed playsin Eqs. a (23) substantial and (25). role In for many determining cases, the saxion the final mass value and µ-term are related to the soft SUSY of the baryon asymmetry.Since the The saxion saxion is linked decaybreaking rateto the depends scale. axion In whichon particular, its massis theµ and Nambu-Goldstone-term the isµ a-term measure as shown of boson fine-tuning of broken of the PQ electroweak symmetry breaking. Therefore, it leads us to discuss the soft SUSY scale and fine-tuning from the measured in Eqs. (23) andsymmetry, (25). In many it is massless cases, the in the saxion supersymmetric mass and µ-term limit. are When related SUSY to isthe broken, soft SUSY however, the saxion baryon asymmetry. breaking scale.(and In particular, also the axino)µ-term acquires is a measure a mass. of The fine-tuning saxion mass of the is electroweak typically of symmetry the gravitino mass order Since the saxion is linked to the axion which is the Nambu-Goldstone boson of broken PQ breaking. Therefore,although it leads it can us be to either discuss larger the soft or smaller SUSY scale than the and gravitino fine-tuning mass from in thesome measured models [20–23]. On the symmetry, it is massless in the supersymmetric limit. When SUSY is broken, however, the saxion baryon asymmetry.other hand, as shown in the Sec. III B, the saxion mass (i.e. mX ) is required to be smaller than (and also the axino) acquires a mass. The saxion mass is typically of the gravitino mass order Since the saxionthe AD is linked field mass to the in order axion to which not spoil is the lepton Nambu-Goldstone number generation. boson of In broken this regard, PQ we consider a although it can be either larger or smaller than the gravitino mass in some models [20–23]. On the symmetry, it israther massless small in the saxion supersymmetric mass compared limit. to Whenthe AD SUSY field mass, is broken,i.e. m however, mX the. m saxion /10. other hand, as shown in the Sec. III B, the⇠ saxion mass (i.e. mX ) is required to be smaller than (and also the axino) acquires a mass. Thethe saxion AD field mass mass is typically in order to of not the spoil gravitino lepton mass number order generation. In this regard, we consider a although it can be either larger or smaller thanrather the small gravitino saxion mass mass in compared some models to the [20–23].AD field mass, On thei.e. m m m /10. ⇠ X . other hand, as shown in the Sec. III B, the saxion mass (i.e. mX ) is required to be smaller than the AD field mass in order to not spoil lepton number generation. In this regard, we consider a rather small saxion mass compared to the AD field mass, i.e. m m m /10. ⇠ X . 7 second and third terms inside the bracket of Eq. (18). The detailed physical aspects are explained 23 7 in Ref. [20] and references therein. If m = 10 TeV and M =7.2 10 GeV, such early oscillation ⇤ ⇥ occurs for T 107 GeV. second and third terms insideR & the bracket of Eq. (18). The detailed physical aspects are explained For an illustration, we show a formula for a case23 where the early oscillation does not occur. In in Ref. [20] and references therein. If m = 10 TeV and M =7.2 10 GeV, such early oscillation ⇤ ⇥ such7 a case, Hosc = m , so the baryon-number-to-entropy ratio becomes occurs for TR & 10 GeV. For an illustration, we show a formula for a case where the early oscillation4 does12 not occur. In nB 8 TR 10 eV 10 GeV X0 3.6 10 ph , (20) such a case, H = m , so thes baryon-number-to-entropy' ⇥ 107 GeV ratio becomesm f M osc ✓ ◆✓ ⌫1 ◆✓ ◆✓ P ◆ 4 12 nwhereB we also assume8 mT R= am m10soft .eV In this10 scenario,GeV however,X0 saxion will dominate the universe, 3.6 10 | | , (20) s ' ⇥ ph 107 GeV m f M and its decay produces✓ entropy◆✓ dilution.⌫1 In◆✓ order to obtain◆✓ theP ◆ final baryon asymmetry after saxion where we also assumedecay, them = entropya m dilution. In this must scenario, be taken however, into account.saxion will We dominate will consider the universe, the entropy production | m| soft and its decay producesin the following entropy dilution. subsection. In order to obtain the final baryon asymmetry after saxion decay, the entropyBefore dilution closing must this be taken subsection, into account. let us comment We will consider on the thepossible entropy lepton7 production number violation during in the followingthe subsection. saxion oscillation. Since the field value of X can become small during its oscillation, the e↵ect second and third terms inside the bracket of Eq. (18). The detailed physical aspects are explained Before closingof the this lepton subsection, number let us violation, comment induced on the bypossible the e lepton↵ective number superpotential violation (3) during or the corresponding 23 in Ref. [20] and references therein. If m = 10 TeV and M =7.2 10 GeV, such early oscillation the saxion oscillation.A-term, Since may thebecome field large. value of Here,X⇤ can as become shown⇥ in small Sec. during III, the itsµ oscillation,-term interaction the e↵ect in Eq. (2), 7 occurs for TR & 10 GeV. of the lepton number violation, induced by the e↵ective superpotential2 (3) or the corresponding For an illustration, we show a formula for a case where the early oscillationgµY does not occur. In Wµ = HuHd , (21) A-term, may become large. Here, as shown in Sec. III, the µ-termMP interaction in Eq. (2), such a case, Hosc = m , so the baryon-number-to-entropy ratio becomes g Y 2 8 plays an important role. Assuming4 µ 12mX m , the lepton number violation during the saxion nB 8 TR 10W µ =eV 10HuHGeVd ,⌧ X0 (21) 3.6 10 ph 7 MP , (20) s ' ⇥oscillation is10 smallGeV enoughm to⌫1 maintainf the generatedMP lepton asymmetry by the AD mechanism. As and also interactions between✓ the axion superfield◆✓ and the◆✓ Higgs supermultiplets.◆✓ ◆ playsThrough an important thiswe interaction, shall role. see the Assuming in saxion the dominantly nextmX subsection, decaysm , into the the Higgsino leptonµ-term statesnumber interaction if they violation are (21) kine- during also plays the saxion a key role to determine where we also assume m = am msoft. In this scenario,⌧ however, saxion will dominate the universe, matically allowed. Its| decay| rate is approximately given by [36]5 and its decayoscillation produces is small entropythe saxion enough dilution. decay, to maintain In and order hence the to obtain generated the the final final lepton baryon baryon asymmetry asymmetry. asymmetry by the after AD saxion mechanism. As 2 decay, thewe entropy shall see dilution in the must next be subsection, taken into the account.µ1-termµ We interaction will consider (21) the also entropy plays a production key role to determine ( 2H) m . (23) ! ' 4⇡ fa in the followingthe saxion subsection. decay, and hence the final baryon✓ asymmetry.◆C. Saxion decay in DFSZ model e Note that we have used here f =2f under assumptionSAXION of X = Y = f in the present universe,DECAY so Before closing this subsection, leta us comment on the possible lepton number violation during quantities related to axionWe have dark matter discussed is determined how the by f dynamicala/NDW (NDW PQ: domain breaking wall number) scale canas enhance the baryon asymmetry. the saxion oscillation. Since the field valueC. of X Saxioncan become decay insmall DFSZ during model its oscillation, the e↵ect the usual normalization.• TheSaxion decay temperature osc. with is ~MP dominates the Universe. of the lepton number violation,For the final induced result, by the one e↵ crucialective superpotential point to consider (3) or is the the entropycorresponding production from saxion decay. In 90 1/4 µ 1012 GeV m 1/2 We have discussedthe(m > DFSZ2µ) how model, the dynamical saxion interactions PQ breaking with scale the can standard enhance modelthe baryon particles asymmetry. and their superpartners A-term, may become large.T • Here,25 as GeV shown in Sec. III, the µ-term interaction. in Eq. (2),(24) Saxion' g decayTeV is determinedfa 10 TeV by ✓ ⇤ ◆ ✓ ◆ 8 For the final result,are realized one crucial in the pointµ-term⇣ to consider interaction⌘ is the (21).⇣ entropy Once⌘ productionX and Y fromsettle saxion down decay. to the In current value of the 2 If saxion decays into Higgsino states are disallowed,gµY it dominantly decays into the light Higgs and the DFSZ model,PQ saxion symmetry interactionsW breakingµ = with scale,Hu theHXd , standardY f model, this superpotential particles and their generates(21) superpartners the µ-term, gaugeand bosons. also The interactions decay rate in between such a case theM isP axion given by superfield⇠ ⇠ and the Higgs supermultiplets. are realized in the µ-term interaction (21). Once X and Y settle down to the current value of the Through this interaction, the saxion dominantly4 decays into2 Higgsino states if they are kine- plays an important role. Assuming mX m+ , the lepton2 µ number1 violationgµf during the saxion PQ symmetry breaking scale, ( Xhh,⌧ WY W f,ZZ, this) superpotential2 , µ generates, the µ(25)-term, (22) ! ⇠ ⇠ ' ⇡ fa m ⇠ M 5 oscillation is smallmatically enough allowed. to maintain Its decay the generated rate is approximately lepton asymmetry given by by the [36] ADP mechanism. As and the decay temperature becomes g f 2 we shall see in the next subsection, the µ-term interaction (21)µ also plays2 a key role to determine µ , 1 µ (22) ( ⇠2HM) P m . (23) the saxion decay, and hence the final baryon1/ asymmetry.4 2 12 1/2 (m <2µ) 90 µ !10 GeV' 4⇡100f GeVa T 70 GeV ✓ ◆ . (26) ' g TeV fa m ✓ ⇤ ◆ ⇣ ⌘ ✓ e ◆✓ ◆ Note that we have used here fa =2f under assumption of X = Y = f in the present universe, so From the above decay• temperatureSaxionC. Saxion for eachdecay decay case, in one dilutes DFSZ finds the model entropy generated dilution factor lepton number quantities related to axion dark matter is determined by fa/NDW (NDW: domain wall number) as 9 2 1 X0 4 = max TR , 1 . (dilution factor)(27) We have discussedthe usual how normalization. the dynamical The PQ decay8 breaking temperatureM scale3T can is enhance the baryon asymmetry. " For the✓ caseP ◆ where saxion# dominantly decays into Higgsinos, For the final result, one crucial point to consider is the entropy production from saxion decay. In 1/4 12 2 Here we have included the case where TR is small so90 that the saxionµ decays410 beforeGeV12 the reheatingm 1/12 1/4 1/2 (m >2µ) nB 12 10 eV 10 GeV X0 90 µ m the DFSZ model, saxion interactionsT with25 the GeV standard=1.1 10 model ph particles and their superpartners. (24) , (29) process is over. The final baryon asymmetry' iss determinedg ⇥ by theTeV amountm⌫1 of asymmetryfa fa when10 TeV theMP g TeV 10 TeV ✓ ⇤ ◆ ⇣ ✓⌘ ✓ ◆✓ ◆ ⇣ ◆ ✓ ⌘◆ ✓ ⇤ ◆ ⇣ ⌘⇣ ⌘ are realizedAD in mechanism the µ-term completes, interaction Eq. (17) (21). and byOnce theX dilutionand factor,Y settle Eq. down (27): to the current value of the or for the case where saxion dominantly decays into light Higgs and gauge bosons, If saxion decays into Higgsino states are disallowed, it dominantly decays into the lightµ(m Higgssax)-dependent! and PQ symmetry breaking scale, X Y f, this superpotential generates the µ-term, n⇠B ⇠ M TR msoft am 1 2 1 1/4 1/2 ⇤ 4 12 2 gauge bosons. The decay=0 rate.029 inn such2B a case| | is12 givenph 10 by . eV 10 GeV (28)X0 90 µ 100 GeV s final MP =3.0Hosc10 ph⇥ . ⇣ ⌘ s ✓2 ⇥ ◆ m⌫1 fa MP g TeV m gµf ✓ ◆✓ ◆ ✓ ◆ ✓ ⇤ ◆ ⇣ ⌘ ✓ ◆ µ , 4 (22) (30) 5 In the numerical calculation in Sec.IV, we use the⇠ saxionMP decay rates+ including phase space2 µ and mixings1 in [36]. From( thishh, it W is easilyW seen,ZZ that) the observed2 , baryon-number-to-entropy-ratio(25) can be obtained for ! ' ⇡ fa m 4 relatively large neutrino mass m⌫1 = 10 eV if the PQ breaking scale is near the Planck scale in and the decay temperature becomesthe beginning and settles to 1011 GeV at the present universe. We will see numerical results for some example parameter regions in Sec. IV. 1/4 2 12 1/2 (m <2µ) 90 µ 10 GeV 100 GeV T 70 GeV . (26) ' g TeV III. DYNAMICSfa OF PQm BREAKING FIELDS ✓ ⇤ ◆ ⇣ ⌘ ✓ ◆✓ ◆ From the above decay temperatureIn for this each section, case, we discuss one finds the dynamics the entropy of the dilution PQ breaking factor fields in order to investigate the realization of the Planck scale PQ breaking in the early stage and lepton number conservation at 1 X 2 4 the late= max stage. T 0 , 1 . (27) 8 R M 3T " ✓ P ◆ # A. PQ breaking at the Planck scale Here we have included the case where TR is small so that the saxion decays before the reheating process is over. The final baryon asymmetryLet us first examine is determined the scalar by potential the amount of X and ofY asymmetryfor large H to when check the if the PQ scale is (M ). We have to consider the supergravity potential which is given by AD mechanism completes, Eq.O (17)P and by the dilution factor, Eq. (27): K/M2 i¯j 3 2 V = e P DiWK D¯jW ⇤ W , (31) nB M TR msoft am 1 M 2 | | ⇤ ✓ P ◆ =0.029 2 | | ph . (28) s final M Hosc ⇥ P ✓ 2 ◆ ⇣ ⌘where DiW = Wi + KiW/MP . We assume that the e↵ect of the AD field is negligible. The K¨ahler potential and superpotential are given by 5 In the numerical calculation in Sec.IV, we use the saxion decay rates including phase space and mixings in [36]. 2 2 2 2 b 2 2 K = X + Y + Z + I + 2 I X , (32) | | | | | | | | MP | | | | W = ⌘Z(XY f 2), (33) where I is the inflaton field. Note that only X has non-minimal coupling with the inflaton in K. If b>1, one can obtain a negative Hubble-induced mass term for X and thus X develops a large 16 16 It significantlyIt significantly suppresses suppresses the axion isocurvature the axion isocurvature perturbation perturbation [39–42]. Since [39–42]. the massless Since axion the massless axion mode almostmode consists almost of consists the phase of component the phase of componentX for X of Xf,theefor X↵ectivef,thee PQ scale↵ective during PQ16 scale during16 | | | | 8 8 inflation isinflation simply given is simply by X given= X by XM=. X MP . inf | | P inf ⇠ It significantly suppressesIt significantly the| axion| suppresses isocurvature⇠ the axion perturbation isocurvature [39–42]. perturbation Since [39–42]. the massless Since the axion massless axion The magnitudeThe magnitude of CDM isocurvature of CDM isocurvature perturbation perturbation is then given is by then given by mode almost consistsAXIONmode of the almost phase consists component ISOCURVATURE of the phase of X componentfor X of fX,theefor X↵ectivef,thee PQ↵ scaleective during PQ scale during | | | | H 2 8 2 inflation is simply giveninflation by X is simply= X givenM by .8X2 = Xinfinf M2P . Hinf infSCDM P |r | ⇠ r . . (49) (49) | | P ⇠ ' S⇡CDMX ✓ PQ is broken Theduring magnitude inflation of CDM isocurvatureand✓P neverinf 'a perturbation◆ restored⇡Xinf ✓a is then given by The magnitude of CDM isocurvature perturbation is then✓ given by◆ where• IsocurvatureHinfwheredenotesH thedenotes Hubble pert. the scale Hubble during scale inflation, during✓a inflation,denotes the✓H initialdenotes2 misalignment the initial anglemisalignment of angle of inf 2 r2 ainf . (49) 2 HinfSCDM the axion and r denotes the fraction of presentr axionP energy' density. ⇡X ininf ✓ thea matter energy density:(49) the axion and r denotes theSCDM fraction of present axion✓ energy◆ density in the matter energy density: P ' ⇡Xinf ✓a 2 2 ✓ ◆ r (⌦ah )/(⌦mh ).2 The final2 axion density is given by [45] ⌘ r (⌦ahwhere)/(⌦mHhinf ).denotes The final the Hubble axion scale density during is given inflation, by [45]✓a denotes the initial misalignment angle of where H denotes⌘ the Hubble scale during inflation, ✓ denotes the initial misalignment angle of inf the axion and r denotes the fraction ofa present1.19 axion energy density in the matter energy density: 2 2 fa/NDW 1.19 the axion and r denotes the fraction2 ⌦ah2 of present0.18 ✓a axion12 energy densityfa./NDW in the matter energy density:(50) r (⌦ah )/(⌦mh ).' The final axion210 densityGeV2 is given by [45] ⌘ ⌦ah✓ 0.18 ✓a◆ 12 . (50) 2 2 ' 10 GeV r (⌦ah )/(⌦mh ). The final axion density is given by [45]✓ ◆ ⌘ 1.19 Here we have assumed that there is no dilution of the2 axion density2 fa/N dueDW to the saxion decay. The ⌦ah 0.18 ✓a . (50) Here we have assumed that there is no dilution' of1.19 the10 axion12 GeV density due to the saxion decay. The Planck constraint on the uncorrelated isocurvaturef perturbation/N ✓ [46] reads◆ • Planck constraint ⌦ h2 0.18 ✓2 a DW . (50) Planck constraint on thea uncorrelated' a 10 isocurvature12 GeV perturbation [46] reads Here we have assumed that there✓ is no dilution◆ of the axion density due to the saxion decay. The 12 1.19 13 1 10 GeV Xinf H 7 10 GeV ✓ . (51) Here we have assumedPlanck thatinf constraint there. ⇥ is no on dilution the uncorrelateda off the/N axion isocurvature density12M perturbation due to1.19 the [46] saxion reads decay. The ✓13 a DW 1◆ 10✓ GeVP ◆ Xinf Hinf . 7 10 GeV ✓a . (51) Planck constraint on the uncorrelated isocurvature⇥ perturbationfa/N [46]12DW reads 1.19 MP 13 ✓ 1 10 GeV◆ ✓ Xinf◆ This constraint is easily satisfied for mostH inflation7 models10 GeV since✓ Xinf = X0 MP in our. scenario. (51) inf . ⇥ a f /N ⇠ M ✓ a DW ◆ ✓ P ◆ X ThisM constraintf is easily satisfiedaccommodates for most12 inflation most1.19 modelsof inflation since modelsXinf = X0 MP in our scenario. inf P a 13 1 10 GeV Xinf ⇠ ⇠ Hinf . 7 10 GeV ✓a . (51) This constraint⇥ is easilyB. Axino satisfied production forf /N most inflation modelsM since Xinf = X0 MP in our scenario. ✓ a DW ◆ ✓ P ◆ ⇠ B. Axino production This constraint is easily satisfied for most inflation models since X = X M in our scenario. The axino is the fermionic superpartner of the axion,B. consisting Axino productioninf of the0 fermionic⇠ P components of X and Y with a small mixture of higgsino. It obtains a mass of m = ⌘ Z m . It has The axino is the fermionic superpartner of the axion, consistinga˜ h i' of the3/2 fermionic components The axino is the fermionic superpartner of the axion, consisting of the fermionic components a relatively long lifetime if it is not theB. lightest Axino SUSY production particle (LSP) [47–49]. Its decay width is of X and Y with a small mixture of higgsino. It obtains a mass of ma˜ = ⌘ Z m3/2. It has of X and Y with a small mixture of higgsino. It obtains a mass of m = ⌘ hZ i'm . It has approximately given by a˜ h i' 3/2 a relatively long lifetime if it is not the lightest SUSY particle (LSP) [47–49]. Its decay width is The axino is thea fermionic relatively superpartner long lifetime if of it is the not axion, the lightest consisting SUSY of particle the fermionic (LSP) [47–49]. components Its decay width is 2 approximatelyapproximately given by given by 2 µ ma˜ of X and Y with a small mixture of higgsino. It obtains a mass of ma˜ = ⌘ Z m . It has a˜ 2 , 3/2 (52) ⇠ ⇡ fa h i' a relatively long lifetime if it is not the lightest SUSY particle2 µ22m (LSP)µ2m [47–49]. Its decay width is a˜ a˜ , (52) a˜ a˜ 2 2, (52) approximatelywhich is comparable given by to the saxion. Thus the axino can have⇠ ⇡ significant⇠ ⇡fa fa impacts on cosmology. The dominant axino production process is the thermal one.9 The axino thermal production which is comparable to the saxion. Thus the axino can have significant impacts on cosmology. which is comparable to the saxion. Thus2 µ2m the axino can have significant impacts on cosmology. in the DFSZ model comes from the combination of higgsinoa˜ decay/inverse decay, scatterings of The dominant axino productiona˜ 2 process, is the thermal one.9 The axino thermal(52) production The dominant axino production⇠ ⇡ processfa is the thermal one.9 The axino thermal production 8 in the DFSZ model comes from the combination of higgsino decay/inverse decay, scatterings of If Hu or Hd has a larger field value than X,thee↵ective PQ scale is given by Hu or Hd [43, 44]. This is not in the DFSZ model comes from the combination of higgsino| | decay/inverse| | decay, scatterings of whichthe is case comparable in our model. to the saxion. Thus the axino can have significant impacts on cosmology. 9 The direct saxion decay8 into the axino pair can be kinematically forbidden for m < 2ma˜. If Hu or Hd has a larger field value than X,thee↵ective9 PQ scale is given by Hu or Hd [43, 44]. This is not The dominant8 axino production process is the thermal one. The axino thermal| | production| | If Hu or Hthed has case a in larger our model. field value than X,thee↵ective PQ scale is given by Hu or Hd [43, 44]. This is not 9 | | | | in the DFSZthe model case comes inThe our direct model. from saxion the decay combination into the axino of pair higgsino can be kinematically decay/inverse forbidden decay, for m scatterings< 2ma˜. of 9 The direct saxion decay into the axino pair can be kinematically forbidden for m < 2ma˜. 8 If Hu or Hd has a larger field value than X,thee↵ective PQ scale is given by Hu or Hd [43, 44]. This is not | | | | the case in our model. 9 The direct saxion decay into the axino pair can be kinematically forbidden for m < 2ma˜. RESULTS 14 [ ] = = [ ] = / = 1012 1012 -7 -6 10-5 -4 10-8 10-5 -4 10 10 10 10 10-3 10-3 GeV -8 GeV 10 10 1 -2 < < 10 T T 10-2 1011 1011 10-7 ] 10-1 ] 10-1 [ [ -6 10 GeV 10 < T m > 0.17 eV 1010 m > 0.17 eV 1010 m > 0.48 eV 8 m > 0.48 eV 8 fa N < 5 × 10 GeV fa NDW < 5 × 10 GeV DW 109 109 10. 102 103 104 105 102 103 104 105 [ ] [ ] FIG. 1: Contours of m [eV] on the (µ, f ) plane to reproduce the observed baryon asymmetry. In the left ⌫1 -10 a • (right)Contours panel we for have n takenB/s=10m =5µ (.m = µ/5) while msoft = m = 10m (msoft = m = 50m ). The light (dark) red shaded region corresponds to T < 10 GeV (1 GeV) and the grey shaded region is excluded by the KamLAND-Zen experiment. The light-gray shaded region is constrained by Planck+BAO [34]. The light-purple shaded region indicates bound from SN1987A [35]. The blue shaded region corresponds to µ =(0.01 1)f 2/M . a P region shows parameter space where the lightest neutrino mass is larger than the KamLAND-Zen bound [33]. We also show a bound from Planck+BAO constraint on the sum of neutrino masses, ( m⌫) < 0.17 eV [34]. The light-purple shaded region shows the bound from SN1987A [35]. The (light-)redP shaded region shows parameter space where the saxion decay temperature is smaller than 1 GeV (10 GeV). The blue shade indicates the region for which µ =(0.01 1)f 2/M .We a P 6 consider fixed TR = 10 GeV since larger TR does not change or does suppress nB/s (see Eq. (28)). In the case where nB/s is suppressed, it requires a smaller neutrino mass that is less attractive. From the figure, it is clearly shown that neutrino mass is large for large µ and small fa while it becomes smaller for small µ and large fa. This feature stems from the saxion decay temperature. The saxion decay temperature is enhanced by the µ-term while suppressed by fa. It is also of great importance that small fa is good for obtaining a flatter direction during lepton number generation 10 as shown in Eq. (6). For µ & 1 TeV and fa . 10 GeV, our model predicts a rather large neutrino mass so that it is constrained by recent neutrinoless double beta decay (0⌫ ) experiment. For this constraint, we take a conservative bound from KamLAND-Zen, m⌫ < 0.48 eV [33]. From the lower-right corner (µ 105 GeV, f 1010 GeV) to the upper-left corner (µ 102 GeV, f 1012 ⇠ a ⇠ ⇠ a ⇠ GeV), the resulting neutrino mass scans over 10 1 10 8 eV. SUMMARY • Simple AD leptogenesis along LHu requires very light neutrino. • If AD leptogenesis works with varying PQ scale, successful leptogenesis is possible with (relatively) large neutrino mass. • Non-minimal Kähler for a PQ field realizes varying PQ scale. • DFSZ is good to suppress unwanted L violation during saxion oscillation; Saxion decay determines the final BAU. • Axion isocurvature is suppressed. 5 Ca Zr Nd tainties from neutrino oscillation parameters [29, 30]. We ob- Se Te Cd tain a 90% C.L. upper limit of mlightest < (180 − 480) meV. Te 1 In conclusion, we have demonstrated effective background Ge Mo reduction in the Xe-loaded liquid scintillator by purifica- tion, and enhanced the 0νββ decay search sensitivity in Xe KamLAND-Zen. Our search constrains the mass scale to lie −1 136 10 KamLAND-Zen ( Xe) below ∼100 meV, and the most advantageous nuclear matrix (eV) element calculations indicate an effective Majorana neutrino IH mass limit near the bottom of the quasi-degenerate neutrino m mass region. The current KamLAND-Zen search is limited by −2 10 backgrounds from 214Bi, 110mAg, muon spallation and par- tially by the tail of 2νββ decays. In order to improve the NH search sensitivity, we plan to upgrade the KamLAND-Zen ex- periment with a larger Xe-LS volume loaded with 800 kg of −3 10 enriched Xe, corresponding to a twofold increase in 136Xe, contained in a larger balloon with lower radioactive back- −4 −3 −2 −1 10 10 10 10 50 100 150 ground contaminants. If further radioactive background re- mlightest (eV) A duction is achieved, the background will be dominated by muon spallation, which can be further reduced by optimiza- Figure from Phys. Rev. Lett. 117, 082503 (2016) FIG. 3: Effective Majorana neutrino mass ⟨mββ⟩ as a function of tion of the spallation cut criteria. Such an improved search m the lightest neutrino mass lightest.Thedarkshadedregionsare will allow ⟨mββ⟩ to be probed below 50 meV, starting to con- the predictions based on best-fit values of neutrino oscillation pa- strain the inverted mass hierarchy region under the assump- rameters for the normal hierarchy (NH) and the inverted hierarchy tion that neutrinos are Majorana particles. The sensitivityof (IH), and the light shaded regions indicate the 3σ ranges calculated the experiment can be pushed further by improving the en- from the oscillation parameter uncertainties [29, 30]. The horizon- 136 ergy resolution to minimize the leakage of the 2νββ tail into tal bands indicate 90% C.L. upper limits on ⟨mββ⟩ with Xe from KamLAND-Zen (this work), and with other nuclei from Ref. [2, 26– the 0νββ analysis window. Such improvement is the target of 28], considering an improved phase space factor calculation[17,18] afuturedetectorupgrade. and commonly used NME calculations [19–25]. The side-panel shows the corresponding limits for each nucleus as a functionofthe The authors wish to acknowledge Prof. A. Piepke mass number. for providing radioactive sources for KamLAND. The KamLAND-Zen experiment is supported by JSPS KAKENHI nism is dominated by exchange of a pure-Majorana Standard Grant Numbers 21000001 and 26104002; the World Pre- Model neutrino. 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