Introduction to Baryogenesis and Leptogenesis

Transcription

1 Introducton to Baryogeness and Leptogeness 1 Introducton prepared by K.S.Lm MPIK Dvson for Partcle and Astropartcle Physcs PhD Student Semnar on Inflaton and Leptogeness 2011 Snce antpartcles were frst predcted and observed, t has been clear that there exst hgh degree of partcle-antpartcle symmetry. Ths observaton s a stark contradcton to the phenomena of everyday and cosmologcal evdence, partcularly our unverse conssts of almost entrely matter wth lttle prmordal antmatter. The evdence of unverse devod of antmatter comes n dfferent scale. conssts of matter up to Hubble-scale. Unverse Quanttatve estmate of relatve abundance of baryonc matter and antmatter may be obtaned from standard cosmology. Prmordal nucleosynthess allows accurate predctons of cosmologcal abundances of all lght elements wth only one nput parameter, the baryon to entropy rato: η = n b n b s (1) and η s measured to be < η < Whle the success of standard cosmology s encouragng, there remans the queston of ntal condtons. The values requred for many parameters are extremely unnatural n the sense that tnest devaton wll lead to a unverse dfferent from what we observe. The generaton of observed value of η n ths context s refered to as baryogeness. Three necessary condtons for generaton of baryonc asymmetry were dentfed and known as the Sakharov crtera: Volaton of baryon number symmetry Volaton of C and CP Departure from thermal equlbrum 2 Vacuum structure of SU(2) Consder the Yang-Mlls SU(2) gauge theory. L = 1 2g 2 tr (F µνf µν ) (2) We demand that the boundary condton for pure Yang-Mlls feld s F µν (x) 0, x S 3 1

2 For some open connected neghbourhood of x, A µ (x) s a pure gauge A µ (x) = ( µ U(x)) U 1 (x) (3) A vacuum state can be represented by the matrx U(x) SU(2), x S 3. Snce SU(2) = S 3, U(x) can be consdered as a map from 3-sphere to an element n SU(2) wth a sutable boundary condton. The vacuum state can be classfed as Π 3 (S 3 ) = Z. The degenerate vacuum state are physcally equvalent but topologcally dstnct. Topologcally the vacuum state can be dvded nto dfferent homotopy classes, classfed by nteger wndng number: n(u) = 1 24π 2 Note that ths defnton s group measure nvarant. dσ µ ɛ µνρσ tr [ ( ν U)U 1 ( ρ U)U 1 ( σ U)U 1] (4) Generally one can assgn the topologcal charge Q of confguraton to a four dmenson feld confguraton wth fnte Eucldean acton. Q = 1 16π 2 d 4 x tr(f µν F µν ) (5) = 1 16π 2 d 4 x µ K µ (6) wth F µν = 1 2 ɛ µνρσf ρσ and K µ defned as K µ = 2ɛ µνρσ tr(a ν ρ A σ Aν A ρ A σ ). The Chern-Smon number s defned by N CS = 1 16π 2 d 3 x K 0 (7) = 1 8π 2 d 3 x ɛ jk tr(a j A k A A j A k ) (8) By dong a large gauge transformaton, we see that N CS s changed by a unt of wndng number. Between 2 neghbourng vacuum state, there must be an energy barrer, snce they cannot be transformed nto one another wthout passng through non-vacuum state. Transton from one vacuum state to another can happen due to quantum tunnelng. In ths sense N CS can be regarded as a parameter for dfferent vacuum state. 1 t Q(t) = 16π 2 dt d 3 x µ K µ (9) 0 = N CS (t) N CS (0) (10) Notce that Chern-Smon number s not gauge nvarant, but the topologcal charge s. The soluton to ths feld confguraton s called the nstanton. 2

3 3 Chral anomaly and baryon/lepton number volaton Consder the lagrangan for massless Drac feld, coupled to U(1) gauge feld. L = Ψ /DΨ 1 4g 2 F µν F µν (11) Ths lagrangan s nvarant under local U(1) gauge transformaton. However, t s also nvarant under the axal U(1) transformaton. Ψ(x) e αγ 5 Ψ(x) (12) The assocated axal current should be conserved accordng to Noether theorem. But t has an anomalous dvergence: µ j µ A = 1 16π 2 tr(f µν F µν ) (13) Ths chral anomaly arses from the fact that the ntegral measure DΨDΨ s not nvarant under chral transformaton. If we couple the fermons to SU(2) gauge theory ala Standard Model, we would obtan the anomalous dvergence for baryonc and leptonc currents. B L s conserved n ths case. Snce L = B = N f (N CS (t f ) N CS (t )), we expect a contrbuton of 12 fermons lead by the SU(2) nstanton for N CS = 1. At zero temperature, B and L are volated respectvely by the tunnelng of nstanton. However, the tunnelng rate s heavly surpressed, Γ(T = 0) e 2S E By couplng the feld theory to thermal bath, one can obtan a hgher rate of anomalous process through sphaleron. Note that as sphaleron conserves B L, phase transton at equlbrum wll destroy any B + L should B L was zero before the phase transton, thus washng out all the baryon asymmetry. Sphaleron s a classcal statc soluton of feld confguraton, whch s a saddle pont n energy functonal. The energy of sphaleron E sp s gven as the nfmum of the maxmum energes for loops n a non-contractble homotopy class H, and t possesses topologcal charge of 1 2. For temperature greater than the temperature of electroweak phase transton, the rate of baryon number volatng events can be very profound, and t has been gven as Γ B+L αw 5 T 4 ln(1/α W ) (14) V 4 Relaton between Baryon and Lepton Asymmetres In a weakly coupled plasma, one can assgn a chemcal potental µ to each of the quark, lepton and Hggs felds. For a non-nteractng gas of massless partcles the asymmetry n partcle and antpartcle number denstes s gven by: { n n = gt 3 βµ + O((βµ ) 3 ), fermons 6 2βµ + O((βµ ) 3 (15) ), bosons 3

4 The effectve 12 fermons nteracton nduced by SU(2) nstanton mples: (3µ q + µ l ) = 0 (16) One has to take SU(3) QCD nstanton process nto account, whch generate an effectve nteracton between left-handed and rght-handed quark. (2µ q µ u µ d ) = 0 (17) The thrd condton, vald at all temperatures, arses from requrement that the total hypercharge of the plasma vanshes. (µ q + 2µ u µ d µ l µ e + 2 µ H ) = 0 (18) N f The Yukawa nteractons, supplemented by gauge nteractons, yeld relatons between chemcal potentals of left-handed and rght-handed fermons. We assume equlbrum between dfferent type of generatons, µ q = µ q, µ l = µ l. µ q µ H µ d = 0, µ q + µ H µ u = 0, µ l µ H µ e = 0 (19) And fnally usng the relaton of B and L we get: B = (2µ q + µ u + µ d ) (20) L = 2µ l + µ e, L = L (21) Solvng B and L under the constrans gven above, one obtans B = 8N ( ) f + 4 8Nf + 4 (B L), L = 22N f N f (B L) (22) The value of B L at tme t f, where Leptogeness process s completed, determnes the value of the baryon asymmetry today. 5 Recpe for Leptogeness B(t 0 ) = 8N f N f + 13 (B L)(t f ) (23) Frst one reles on a leptonc-volatng process, such as addng a rght-handed Majorana partcle and lettng t decay before the electroweak phase transton. In ths sense, we volate B L. Washout of lepton asymmetry occurs. Unfortunately ths unwanted process happens and one needs to solve the Boltzmann equaton to determne the decay and nverse decay of rght-handed neutrnos. Converson of remanng lepton asymmetry to baryonc asymmetry due to sphaleron. 4

5 6 Departure from Thermal Equlbrum I only dscuss the scenaro of out-of-equlbrum decay mechansm as the condton for departure from thermal equlbrum. Consder the scenaro when the decay rate Γ X of superheavy partcle X at tme they become nonrelatvstc s much smaller than the expanson rate of unverse, then the X partcles cannot decay on the tme scale of expanson and so they reman abundant as photon for T M X. Therefore they populate the unverse wth an abundance whch s larger than the equlbrum one. Ths overabundance s the departure from thermal equlbrum needed to produce a fnal nonvanshng baryon asymmetry when X undergoes B and CP volatng decays. References [1] W. Buchmuller, R.D. Pecce, and T. Yanagda. Leptogeness as the orgn of matter. Ann.Rev.Nucl.Part.Sc., 55: , [2] Ta-Pe Cheng and Lng-Fong L. Gauge Theory of elementary partcle physcs. Oxford Unversty Press, USA, [3] Frère. Introducton to Baryo- and Leptogeness. Surveys n Hgh Energy Physcs, 20(1):59 87, [4] M. Fukugta and T. Yanagda. Baryogeness Wthout Grand Unfcaton. Phys.Lett., B174:45, [5] Frans R. Klnkhamer and N.S. Manton. A Saddle Pont Soluton n the Wenberg- Salam Theory. Phys.Rev., D30:2212, [6] V.A. Kuzmn, V.A. Rubakov, and M.E. Shaposhnkov. On the Anomalous Electroweak Baryon Number Nonconservaton n the Early Unverse. Phys.Lett., B155:36, [7] N.S. Manton. Topology n the Wenberg-Salam Theory. Phys.Rev., D28:2019, [8] R. Rajaraman. Soltons and Instantons: An Introducton to Soltons and Instantons n Quantum Feld Theory. North Holland, 2nd repr edton, [9] S. Skadhauge. Sphalerons and electroweak baryogeness Hardcopy at DESY. [10] Mark Trodden. Electroweak baryogeness. Rev.Mod.Phys., 71: , [11] Mark Trodden. Baryogeness and leptogeness. page L018,