Possibility of bottom-catalyzed matter genesis near to primordial QGP hadronization
PPossibility of bottom-catalyzed matter genesis near to primordial QGP hadronization
Cheng Tao Yang a , and Johann Rafelski a a Department of Physics, The University of Arizona, Tucson, Arizona 85721, USA (Dated: April 14, 2020)We study bottom flavor abundance in the early Universe near to a temperature T H (cid:39)
150 MeV,the condition for hadronization of deconfined quark-gluon plasma (QGP). We show bottom flavorabundance nonequilibrium lasting microseconds. In our study we use that in both QGP, and thehadronic gas phase (HG) b and ¯ b quarks near T H are bound in B-mesons and antimesons subject to CP violating weak decays. A coincident non-equilibrium abundance of bottom flavor can lead tomatter genesis at required strength: a) The specific thermal yield per entropy is n thb /σ = 10 − ∼ − . b) Considering time scales, millions of cycles of B-meson decays, and b ¯ b -pair recreationprocesses occur. I. INTRODUCTIONA. Overview
An epoch in the primordial Universe evolution al-lowing matter genesis (baryogenesis, leptogenesis) atthe level observed has not been established. Usuallythe temperature range between GUT phase transition T G (cid:39) GeV and the electroweak phase transition near T W (cid:39)
130 GeV is explored [1–9]. Here we present argu-ments that the Sakharov conditions [10] for matter asym-metry to form also appear during quark-gluon plasma(QGP) hadronization era near to T H (cid:39)
150 MeV. Weshow substantial departure from equilibrium and C and CP violation.The third condition, a violation of baryon B , lepton L number conservation coincident with the above situationneeds future theoretical and experimental consideration,constrained by the experimental limit on proton life span O (10 y). The relevant thermal environment can be ex-plored in the laboratory since T H is readily available inrelativistic heavy ion (RHI) collision experiments [11].However, T H may not suffice for catalysis of baryogene-sis [1–3]. On the other hand, if leptoquarks exist [12], itis possible that the decay of heavy bottomnium flavoredmesons could generate, via B − L = 0 processes, thematter excess required. We defer this question reachingbeyond the scope of this work to future studies.The observed baryon and lepton number cannot begenerated in a full thermal (chemical and kinetic) equi-librium, because even if the required processes are occur-ring, the net effect is cancelled out by the equal numberof back-reactions. We believe that the presence of abun-dance ( i.e. chemical) non-equilibrium is more relevant –kinetic (equipartition of energy) equilibrium is usually es-tablished much quicker and has less impact on the actualparticle abundances [13, 14]. In this work we show thata relatively large bottom flavor chemical non-equilibriumarises near to the QGP hadronization condition.The Sakharov condition requiring C and CP assuresthat we can recognize a universal difference between mat-ter and antimatter, thus one abundance can be enhancedcompared to the other. We are seeking CP violation of relevance to QGP, considering a mechanism grosslydifferent and presumably entirely independent from thechiral magnetic effect [25]. Specifically, given that thenon-equilibrium of bottom flavor arises at relatively lowQGP temperature, the bottom quark decay occurs frompreformed [15–20] B x meson states, x = u, d, s, c . Thesedecays violate aside of C also the CP symmetry, seefor example [21, 22]. The exploration of the here in-teresting CP symmetry breaking in B c ( b ¯ c ) decay is inprogress [23, 24]. While in the following we focus our at-tention on the QGP deconfined phase, in qualitative andnearly quantitative manner our results apply to bottom-nium non-equilibrium and CP violation in the hadrongas (HG) phase.The CP violation is well established in all bottommesons including B c ( b ¯ c ) decays [26]. In general, violationof CP asymmetry can occur in the amplitudes of hadrondecay. The weak interaction CP violation arises from thecomponents of Cabibbo-Kobayashi-Maskawa (CKM) ma-trix associated with quark-level transition amplitude and CP -violating phase. In this case, the charged B c mesondecay can be the source of required CP violation [21]. B. Is there enough bottom flavor to matter?
Considering that the expanding Universe evolves con-serving entropy, and that baryon and lepton number fol-lowing on the era of matter genesis is conserved, thecurrent day baryon B to entropy S , B/S -ratio must beachieved during matter genesis. The PDG [26] estimatesthe present day baryon-to-photon ratio 5 . × − (cid:54) η (cid:54) . × − . This small value quantifies the matter-antimatter asymmetry in the present day Universe. Theparameter η allows the determination of the present valueof B/S ≈ . × − [11, 27–29] in the Universe dom-inated by photons and free-streaming low mass neutri-nos [30].In chemical equilibrium the ratio of bottom quark(pair, b , ¯ b ) density n thb to entropy density σ = S/V just above quark-gluon hadronization temperature T H (cid:39) ∼
160 MeV is n thb /σ = 10 − ∼ − , see Fig. 1 Wediscuss how these results arise in appendix.Considering the n thb /σ value, there is sufficient abun- a r X i v : . [ h e p - ph ] A p r FIG. 1: The bottom b ,¯ b and charm c ,¯ c (pair) number densitynormalized by entropy density, as a function of temperaturein the primordial Universe: b -quark mass parameters shownare m b = 5 . m b = 4 . m b = 4 . c -quarkmass: m c = 0 . m b = 1 . dance of b , ¯ b quarks for the proposed matter genesismechanism to be relevant. This is true even if b , ¯ b quarksdisappear from particle inventory below T H . In Fig. 1we see that the charm (quark pair c ,¯ c ) abundance at T H (cid:39) ∼ ,
000 times greater: b ,¯ b quarksare embedded in a background comprising all lighter u, d, s, c quarks and antiquarks, as well as gluons g . II. BOTTOM QUARK FREEZE-OUT PROCESSA. Bottom production and annihilation
Bottom quark freeze-out process occurs near to theQGP phase transition. This is so since in the expandingUniverse at ever decreasing temperature the strong inter-action gluon g + g → b + ¯ b , and quark pair q + ¯ q → b + ¯ b fusion processes become slower compared to the relativelyslow WI decay process of bottom flavor: the lifespanof the preformed B x meson states in empty-space hasa 0 . ∼ .
64 picosecond lifespan [26]: τ B ± c = 0 . × − s , τ B s = 1 . × − s , (1) τ B d = 1 . × − s , τ B ± u = 1 . × − s . Considering the energy balance in quark binding, andquark exchange reactions, we recognize that ultimatelyB c ( b ¯ c ) mesons are always formed. The large binding ofheavy B ± c and its slow thermal motion protects this state,see also Ref. [31, 32]. In the following we assume thatdue to the enhanced binding effect of B ± c all bottom b , ¯ b quarks are found in B ± c .As noted, by means of quark exchange through moreabundant light quark states B x , x = u, d, s , ultimatelythe most bound B c state arises with picosecond lifespan. FIG. 2: The ratio between reaction rates involving B ± c mesonand Hubble expansion rate H as a function of temperature T in the primordial Universe: b + c → B c + g (purple and greendotted line at the top with σ ∈ . ,
10 mb), g + g → b + ¯ b (black solid line, mid figure), g + g → Υ+ g (blue dashed line),B c → anythings (yellow solid line, horizontal mid-figure), andthe density dilution term of B ± c mesons (see text, brown dot-dashed line at the very bottom) The rapid formation rate of B c ( b ¯ c ) states in primordialplasma is shown by dotted lines in Fig. 2. We believethat this process is fast enough to allow consideration ofbottom decay from the B c ( b ¯ c ), B c (¯ bc ) states.While B c is strongly bound and thus protected fromfollow-up chemical reactions, we need further to be surethat only a small fraction of produced b, ¯ b pairs formsthe rapidly annihilating b ¯ b -onium state such as Υ( b ¯ b ). Anexample could be the processB c + B c ⇔ b ¯ b + c ¯ c , Q (cid:39) . (2)In a study of chemical process in QGP by Yao andM¨uller [33], their relative yield Υ /b (cid:39) − withoutcharm. Using Eq. (A.1a) and Eq. (A.1b) we find therelative charm abundance c/ ( u + d + s ) ≈ . T H ( m c = 1 .
24) GeV. This means that charm catalyzedΥ( b ¯ b ) formation is sufficiently small not to alter our re-sults in qualitative manner.The B c ( b ¯ c ) lifespan is controlled by weak interactionprocesses and we will use the free space τ B c = 10 − s(nearly horizontal solid line in Fig. 2). In comparison,the speed of Universe expansion with characteristic timeat the scale of 10 µ s is nearly 10 times slower, as seen inFig. 2 where the ratio of rates to the Hubble expansionrate H is shown. In Fig. 2 we further see that the linesdepicting the rate of heavy flavor production and decaycross near to T H (cid:39)
150 MeV. This result is of pivotalimportance in this work as it establishes the temperatureera for the abundance non-equilibrium of bottom quarks.
FIG. 3: The fugacity of B ± c meson as a function of temper-ature in early Universe with different mass of bottom quark.We have m b = 5 . m b = 4 . m b = 4 . B. Abundance nonequilibrium
We obtain the bottom quark abundance fugacity byevaluating B c fugacity as shown in Fig. 3 for several bot-tom quark masses entering the QGP gluon-fusion bottompair production process. For bottom mass below 4.6 GeVour results are also representative of the competition be-tween strong interaction B c ( b ¯ c ) forming processes in pre-dominantly HG phase competing with WI decay process.The results seen in Fig. 3 are obtained by neglectingthe time dependence of the fugacity, that is called adi-abatic approximation – we show in appendix that thisapproach is valid. The curves are thus analytical func-tionsΥ B c = Γ DecayB c SourceB c (cid:34)(cid:114) (cid:16) SourceB c / Γ DecayB c (cid:17) − (cid:35) , (3)with characteristic rates of approach to chemical equilib-rium – the so called relaxation ratesΓ SourceB c ≡ R SourceB c n th B c = R gg → b ¯ b n th B c , Γ DecayB c ≡ τ c . (4)Here R gg → b ¯ b , Eq. (A.3), is the thermal reaction rate pervolume of g + g −→ b + ¯ b , and n th B c is the thermal equilib-rium number density of B ± c mesons, obtained accordingto Eq. (A.1a).The key result seen in Fig. 3 is that the large massof bottom quark slows the strong interaction formationrate to the value of weak interaction B x decays just nearthe phase transformation of QGP to HG phase, were amixed phase governs the transformation of phases lastingas long as 10 µ s [27, 29, 36]. III. POSSIBILITY OF BOTTOM-CATALYZEDMATTER GENESISA. Repetitive formation and annihilation ofbottom flavor
We can consider the duration of the phase transfor-mation from the QGP to a hot hadronic gas to last inprimordial Universe τ H (cid:39) µs . In this scenario, thenumber of bottom formation and annihilation cycling canbe written as C ycle = τ H τ c ≈ × . (5)Inspecting Fig. 1 at the phase transition temperature T = 150 MeV, we see the particle number per entropyfor bottom quark n thb /σ = n b /σ ∼ . × − . To ob-tain the observed value of baryon asymmetry today weneed to derive from each cycle an asymmetric excess ofbaryons over antibaryons (cid:15) = (cid:18) BS (cid:19) (cid:18) σn b (cid:19) ycle ≈ . × − . (6)To conclude, a small violation of matter-antimatter sym-metry at the level of (cid:15) ∼ − associated with bottomquarks cycling during the hadronization can result in theobserved baryon number today. B. How unique is the role of bottom flavor?
It is fortuitous that the here explored bottom freeze-out process happens near to the QGP cross-over to HG, T H . This coincidence is, however, is not a required con-dition for our non-equilibrium mechanism to operate,as already noted, our results can be refined by explor-ing bottomnium in HG environment. Since charm fla-vor is much lighter, a similar computation reveals charmflavor disappearance at considerably lower temperature T ≈
10 MeV, thus well in the hadron phase and af-ter most baryons and antibaryons emerging in QGPhadronization have annihilated [28]. At this temperaturethe residual charm would be bound in D-mesons.Nothing protects the D-mesons from fusing into fastdecaying charmonia. Qualitative studies we carried outlead us to believe that charm disappears sufficientlyrapidly from particle inventory once HG phase is formedand cannot contribute decisively to matter genesis. Wehave performed a similar study of strangeness flavor ingreat detail. Given the slower expansion of the Universe,we know with certainty that strangeness never occursoutside of full thermal equilibrium. We will return tothis matter under separate cover.We next look at the top quark flavor: If the top quarksfreeze-out at higher temperature, the non-equilibriumconditions required for matter genesis will also operate.What stops a repetition of the here reported mechanismof matter genesis is the fast decay t → W + b , withΓ t = 1 . ± . t we realize that top in hot QGP is produced in the W + b → t fusion process – given the strength of thisprocess there is no freeze-out of the top quark till W it-self freezes out. However W has an even greater widthΓ W = 2 . Z = 2 . W, Z never freeze out, just like π [34]. We conclude that t , W , Z disappear gradually, retaining their thermal equi-librium abundance during the cooling of the primordialUniverse. IV. CONCLUSIONS
We have demonstrated the special role of the bottomflavor for matter genesis in primordial-QGP hadroniza-tion. The bottom quark flavor is just at the ‘sweet-spot’allowing the two Sakharov conditions in the same rangeof temperature, near to T H , the era of QGP hadroniza-tion. For those believing in the Anthropic principle thehere proposed bottom flavor role in matter genesis pro-vides beyond CKM-phase another reason for the exis-tence of the third quark family.Bottom flavor disappearance from particle inventorydefines the epoch of required departure from equilibrium.Formation of a matter excess over antimatter at a rela-tively late Universe evolution period has the additionalmerit of depending on experimentally accessible Universeevolution stage: Further insight can be derived from ex-perimental study of bottom flavor in RHI collisions, andmore generally, exploration of the the properties of bot-tom flavored particles, including but not limited to thesearch for a specific baryon non-conservation mechanism.Our results provide as we believe a strong motivationto explore physics of baryon nonconservation involvingthe bottomnium mesons or/and bottom quarks in ther-mal environment. We have shown that millions of de-cays and reformation processes of bottom flavored quarksduring the hadronization era can occur. This circularUrca-like situation can amplify even a tiny value of mat-ter conservation violation producing the today observedmatter-antimatter asymmetry. Acknowledgments
We thank Berndt Mueller for reading the manuscriptand kind comments and suggestions.
Appendix: Technical details1. Bottom flavor equilibrium abundance
In the primordial-QGP Universe evolution era, at T (cid:39) T H (cid:39)
150 MeV freely propagating light quarks u, d, s and gluons dominate the strongly interacting plasma inthe Universe. The dominant mechanism for producingquarks is the gluon-fusion reaction gg → q ¯ q , hence theabundance of quarks is strongly coupled to gluons, bothfollow thermal equilibrium. The thermal equilibriumnumber density of heavy Q = b, c quarks near to T H with m b,c /T H (cid:29) n thQ = g q π T n = ∞ (cid:88) n =1 ( − n +1 Υ n n (cid:16) n m q T (cid:17) K ( n m q /T ) . (A.1a)For s quarks several terms are needed. For light quarkswe have to evaluate the massless limit n thq = g q π T F (Υ) , F = (cid:90) ∞ x dx − e x , (A.1b)where F (Υ = 1) = 3 ζ (3) / ζ (3) ≈ . σ of the early Universe is given by σ = 2 π g s ∗ T , (A.2)where g s ∗ counts the total number of effective degrees offreedom from entropy [28, 29, 35]. Near T H only lightparticles matter in establishing the value of g s ∗ ; thus theresult we consider is independent of actual abundance of c, b and other heavy particles.In Fig. 1 the ratio of Eq. (A.1a) with Eq. (A.2) is seen,the equilibrium number density per entropy density ofheavy quarks. We evaluated this ratio for the bottomquark as a function of temperature T , allowing for dif-ferent mass m b = 4 . , . , . T (cid:39) T H , the lower values applyat higher energy scale [26]. The vale m b (cid:39) . m b = 4 . , . c mesons of mass m B c = 6 .
275 GeV should be smaller.However, such an abundance has to be computed allow-ing for a free supply of charmed quarks and antiquarks.This introduces a chemical potential that effectively re-duces m B c by charm mass. This enhances the B c mesonyield to be in the realm of where the here presented re-sults for unbound bottom quarks are shown. The sameapplies to other bottom quark preformed states in QGP.
2. Reaction rates involving bottom quarks
The thermal reaction rate per volume for bottom quarkproduction can be written as [27] R gg → b ¯ b = (cid:90) ∞ s th ds dR gg → b ¯ b ds = (cid:90) ∞ s th ds σ gg → b ¯ b P g , (A.3)where σ gg → b ¯ b ( s ) is the cross section of the reaction chan-nel gg → b ¯ b and P g ( s ) is the number of collisions perunit time and volume. The cross section of gluon fusionis given by σ gg → b ¯ b = πα s s (cid:20) (cid:18) m b s + m b s (cid:19) ln (cid:18) w ( s )1 − w ( s ) (cid:19) − (cid:18)
74 + 31 m b s (cid:19) w ( s ) (cid:21) , (A.4)where the function w ( s ) ≡ (cid:112) − m b /s , and m b is themass of bottom quark, α s is the QCD coupling constant.The number of collisions per unit time and volume formassless gluons fusions into bottom quarks is given by P g ( s ) = 4 Tπ ( √ s ) ∞ (cid:88) l,n =1 K ( √ lns/T ) √ ln (A.5)Hence from Eq. (A.3) we can calculate the thermal pro-duction rate per volume for the gluon fusion as a functionof temperature with given parameters.The α s value we consider is based on required gluoncollisions above b + ¯ b energy threshold; we adopt α s =0 . H = 8 πG ρ R + ρ SI ) , (A.6)where G is the Newtonian constant of gravitation, ρ R is the energy density of relativistic species, and ρ SI isthe energy density from strong interaction in the earlyUniverse [27].After formation, the heavy b, ¯ b quark can bind withany of the available lighter quarks, with the most likelyoutcome being a chain of reactions b + q → B + g , B + s → B s + q , B s + c → B c + s , (A.7)with each step providing a gain in binding energy and re-duced speed due to the diminishing abundance of heavierquarks s, c . To capture the lower limit of the rate of B c production we show in Fig. 2 the expected formation rateby considering the direct process b + c → B c + g , con-sidering the range of cross section σ = 0 . ∼
10 mb. WeobtainΓ( b + c → B c + g ) ≈ H × (10 ∼ ) . (A.8)Despite the low abundance of charm, the rate of B c formation is relatively fast, and that of lighter flavoredB-mesons is substantially higher. Note that as long as wehave bottom quarks made in gluon fusion bound practi-cally immediately with any quarks u, d, s into B-mesons,we can use the production rate of b, ¯ b pairs as the rateof B-meson formation in the primordial-QGP, which alldecay with lifespan of pico-seconds, see Eq. (1). In Fig. 2 the mid-figure horizontal line shows B c decay rate nor-malized with the Hubble parameter.In Fig. 2 we also show the rate of direct productionof Upsilonium (dashed blue line), which we obtain usingUpsilonium three gluon lifetime τ and unitarity of thereaction matrix element. We have R gg → Υ g = 2 g g (2 π ) g Υ Tm Υ τ (cid:90) ∞ m ds s − m √ s K ( √ s/T ) , (A.9)where m Υ = 9 .
460 GeV is the mass of Upsilonium. Thisrate is 7 orders of magnitude smaller compared to thedirect production of open flavor. This is so since neartemperature T H it is hard to find two gluons above re-quired energy threshold.
3. Bottom quarks abundance in dynamic model
We consider the production and decay reaction pro-cesses based on the hypothesis that all bottom flavor isbound rapidly into B ± c mesons. The master equation hastwo components, as we can jump into B ± c meson statesfrom b -pairs produced in gluon fusion reactions g + g ←→ b + ¯ b [ b (¯ b ) + ¯ c ( c )] ←→ B ± c + g, (A.10)B ± c −→ anything . (A.11)For reactions Eq. (A.10) and Eq. (A.11), the master equa-tion can be written as:1 V dN B c dt = (cid:0) − Υ c (cid:1) R SourceB c − Υ B c R DecayB c , (A.12)where R SourceB c and R DecayB c are the thermal reaction rateper volume of production and decay of B ± c meson respec-tively. The bottom source rate is the gluon fusion rateEq. (A.3), while the decay rate is the natural lifespan,Eq. (1) weighted with density of particles, see Eq. (A.1a) R DecayB c = n th B c τ c = n th B c Γ DecayB c . (A.13)We wish to replace the variation of particle abundanceseen on LHS in Eq. (A.12) by the time variation of abun-dance fugacity Υ. Considering the expansion of Universewe have1 V dN B c dt = dn B c d Υ B c d Υ B c dT dTdt + dn B c dT dTdt + 3 Hn B c . (A.14)The expanding Universe conserves entropy and hence therelation between temperature and cosmic time can bewritten as dTdt = − HT T g s ∗ dg s ∗ dT ≡ − H F , (A.15) F ≡ T (cid:18) T g s ∗ dg s ∗ dT (cid:19) . (A.16)where g s ∗ is the degree of freedom describing the entropyin Universe [35].Substituting these relations into Eq. (A.12) the fugac-ity equation becomes d Υ B c dT =(Υ c − F (cid:32) Γ SourceB c H (cid:33) + Υ B c F (cid:32) Γ DecayB c H + 3 − d ln ( n th B c ) F dT (cid:33) . (A.17)In Fig. 2 we show that when temperature is near to T H ,we have Γ SourceD s /H ≈ Γ DecayD s /H ∼ , which is muchlarger than the term d ln n thD s / F dT ∼ O (10). In this case,the last two terms in Eq. (A.17) can be neglected.
4. Bottom quark disappearance – quantitativeresults
We verify that the last two terms in Eq. (A.17) arenegligible; thus we solve d Υ B c dT = (Υ c − F (cid:32) Γ SourceB c H (cid:33) + Υ B c F (cid:32) Γ DecayB c H (cid:33) . (A.18)Furthermore we can use the adiabatic solution for thefugacity equation setting d Υ B c /dT = 0 and solve alge-braically the equation for Υ B c . We so obtain Eq. (3).In Fig. 3 the fugacity of B ± c meson Υ B c as a functionof temperature, Eq. (3) is shown around the tempera-ture T = 200 ∼
100 MeV for different masses of bottomquarks. In all cases we see prolonged non-equilibrium.This happens since the decay and reformation rates of bottom quarks are comparable to each other as we havenoted in Fig. 2, where both lines cross.
FIG. 4: The quantity d Υ /dT as a function of temperaturein the Universe. We found that the values of d Υ / F dT (cid:28) Γ Source /H, Γ Decay /H .
5. Validity of approximations
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