Standard Model Baryogenesis

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HEP-PH-9407403 y-94/71 on ation re- HD-THEP-94-24 ene hep-ph/yymmnnn hanism. A CP- CERN-TH. 7368/94. LPTHE Orsa t nite temp era- A The resulting bary ogenesis ela ts of quarks hitting the w what observ tre de Moriond, a 23, Switzerland v el. orks. harge transp ort mec yM.B.Ga e argue against a Standard Mo del ex- andez, J. Orlo and O.P y the damping rate of quasi-quarks in en at tree-lev Abstract ted b ts, w ed b ell on related w teractions and Uni ed Theories". y .Hern Presen tasw h induces loss of spatial coherence and suppresses ela, P eak In y orders of magnitude b elo v w ogenesis via the c tribution to the XXIXth Rencon e commen M.B. Ga \Electro eak phase b oundary created during a rst order phase transi- Standard Mo del Bary Con w CERN, TH Division, CH-1211, Genev Simply on CP argumen asymmetry is man quires. W a hot plasma, whic re ection on the b oundary ev planation of bary asymmetry is found inelectro the re ection co ecien tion. The problem iscase analyzed and b oth in inture, an the a academic realistic zero crucial nite temp erature role temp erature is one. pla

1 Intro duction

The baryon number to entropy ratio in the observed part of the universe

is estimated to b e n =s (4 6)10 [1] . In 1967, A.D. Sakharov [2]

established the three building blo cks required from any candidate theory

of baryogenesis: a) Baryon numb er violation, b) C and CP violation, c)

Departure from thermal equilibrium.

The Standard Mo del (SM) contains a)[3 ] and b)[4], while c) could also b e

large enough [5][6], if a rst order SU (2) U (1) phase transition to ok place

in the evolution of the universe [7]. An explanation within the SM would b e

avery economical solution to the baryon asymmetry puzzle. Unfortunately,

intuitive arguments lead to an asymmetry many orders of magnitude b elow

observation [8][9]. However, the study of quantum e ects in the presence of

a rst order phase transition is rather delicate, and traditional intuition may

fail. The authors of ref.[10]have recently studied this issue in more detail and

claim that, in the nite temp erature charge transp ort mechanism [11], the

SM is close to pro duce enough CP violation as to explain the observed n =s

ratio. In this talk, we summarize our study [12] of the Standard Mo del C

and CP e ects in an electroweak baryogenesis scenario. Even if one assumes

an optimal sphaleron rate and a strong enough rst order phase transition,

we discard this scenario as an explanation of the observed baryon number to

entropy ratio.

υ=0 υ=/ 0 ψ R inc

ψ L ref ψ

Figure 1: Artistic view of the charge transport mechanism, as describedin

the text. The hungry \pacman"represents rapid sphalerons processes. The

wiggly lines stand for col lisions with thermal gluons in the nite temperature

case; they are absent in the academic T =0 model. Only electroweak loops

are depicted, represented by dotted lines.

A rst order phase transition can b e describ ed in terms of bubbles of

\true" vaccuum (with an inner vaccuum exp ectation value of the Higgs eld

v 6= 0) app earing and expanding in the preexisting \false" vaccuum (with

v = 0 throughout). We can \zo om" into the vicinity of one of the bubbles.

There the curvature of its wall can b e neglected and the world is divided

in two zones: on the left hand side, say, v = 0; on the right v 6=0. The

actual bubble expands from the broken phase (v 6=0)towards the unbroken

one (v = 0). Wework in the wall rest frame in which the plasma ows in

the opp osite direction. Consider thus a baryonic ux hitting the wall from

the unbroken phase. The heart of the problem lies in the re ection and

transmission prop erties of quarks bumpimg on the bubble wall. CP violation

distinguishes particles from antiparticles and it is a pr ior i p ossible to obtain

a CP asymmetry on the re ected baryonic current, . The induced baryon

asymmetry is at most n =s 10 ,inavery optimistic estimation of

B CP

the non-CP ingredients [13][10].

The symmetries of the problem have b een analyzed in detail [14] for a

generic bubble. The analytical results corresp ond to the thin wall scenario.

The latter provides an adequate physical description for typical momentum

of the incoming particles j~pj smaller than the inverse wall thickness l , i.e.,

j~pj 1=l . For higher momenta, cuto e ects would show up, but it is

reasonable to b elieve that the thin wall approximation pro duces an upp er

b ound for the CP asymmetry.Wework in a simpli ed scenario with just

one spatial direction, p erp endicular to the wall surface: phase space e ects

in the 3 + 1 dimension case would further suppress the e ect.

The precise questions to answer in the ab ove framework are : 1) the nature

of the physical pro cess in terms of particles or quasi-particles resp onsible for

CP violation, 2) the order in the electroweak coupling constant, , at which

an e ect rst app ears, 3) the dep endence on the quark masses and the nature

of the GIM cancellations involved.

We shall consider the problem in two steps: zero temp erature scenario

(T = 0) in the presence of a wall with the non-equilibrium situation mim-

icked by assuming a ux of quarks hitting the b oundary from just one phase,

and nite temp erature case. Intuition indicates that an existing CP violating

e ect already present at zero temp erature will diminish when the system is

heated b ecause the e ective v.e.v. of the Higgs eld decreases and in con-

sequence the fermion masses do as well (only the Yukawa couplings already

presentat T = 0 remain unchanged). This intuition can b e misleading only if

a new physical e ect, absentat T = 0 and relevant for the problem, app ears

at nite temp erature. We discuss and compare the building blo cks of the

analysis in b oth cases. The T = 0 case provides a clean analysis of the novel

asp ects of the physics in a world with a two-phase vacuum.

At nite temp erature, a plasma is an incoherent mixture of states. CP

violation is a quantum phenomenon, and can only b e observed when quantum

coherence is preserved over time scales larger than or equal to the electroweak

time scales needed for CP violation. This is however not the case in the

plasma, where the scattering of quasi-quarks with thermal gluons induces a

large damping rate, .

We show that tree-level re ection is suppressed for any light avour bya

factor m=2 . The presently discussed CP-violation observable results from

the convolution of this re ection e ect with electroweak lo ops in which the

three generations must interfere coherently in order to pro duce an observable

CP-violation. It follows that further factors of this typ e app ear in the nal

result, which is many orders of magnitude b elow what observation requires

and has an \a la Jarlskog"[8] typ e of GIM cancellations.

The results of our analysis indicate that in the presence of a rst order

phase transition, a CP-asymmetry in the SM app ears at order , has a

conventional typ e of GIM cancellation and chiral limit, and it is well b elow

what observation requires in order to solve the baryon asymmetry puzzle.

2 Zero temp erature

The necessary CP-o dd couplings of the Cabibb o-Kobayashi-Maskawa (CKM)

matrix are at work. Kinematic CP-even phases are also present, equal for

particles and antiparticles, whichinterfere with the pure CP-o dd couplings

to make them observable. These are the re ection co ecients of a given

particle hitting the wall from the unbroken phase. They are complex when

the particle energy is smaller than its (broken phase) mass. Finally, as shown

in [12 ] [14] , the one lo op self-energy of a particle in the presence of the wall

cannot b e completely renormalized away and results in physical transitions.

Such an e ect is absent for on-shell particles in a world with just one phase.

The di erence is easy to understand: the wall acts as an external source of

momentum in the one-lo op pro cess. The transitions b etween anytwo avors

of the same charge pro duce a CP violating baryonic ow for any given initial

chirality.

The essential non-p erturbative e ect is the wall itself. The propagation

of any particle of the SM sp ectrum should b e exactly solved in its presence.

And this we do for a free fermion, leading to a new Feynman propagator

which replaces and generalizes the usual one. The propagator for quarks in

the presence of the wall contains massless and massive p oles:

! !

1 1 1 1 1

f i

+ + +

S (q ;q )=1=2

f f

f i f i

i i

=q =q =q m =q m

q q +i q q i

z z

1 1 1

z z

f i f i

=q m =q =q =q m

" #

m m m

1 (1) (1 )

z 0

f f 0 i i

=q (=q m) E + p =q (=q m)

where wehave assumed for simplicity zero momentum parallel to the wall

i i f f

(q = q = q = q = 0). Due to the wall the initial and nal z comp onents

x y x y

of the momentum need not b e equal. All denominators in the usual Feynman

propagators in eq. (1) should b e understo o d as containing a supplementary

+i factor. Besides this traditional source of phases, the propagator contains

2 2

E m , which b ecomes imaginary in the case

new CP-even ones in p =

of total re ection (E

With this exact, non-p erturbative to ol, p erturbation theory is then ap-

propiate in the gauge and Yukawa couplings of fermions to b osons, and the

one lo op computations can b e p erformed. Strictly sp eaking the gauge b oson

and Higgs propagators in the presence of the wall are needed, and it is p ossi-

ble to compute them with a similar pro cedure [15]. In particular this implies

to consider lo ops with unbroken, broken and mixed contributions. For the

time b eing, wework in a simpli ed case in which the wall do es not act inside

quantum lo ops. These are computed in the broken phase. We considered

one-lo op electroweak e ects which bring the CKM phase into the game. A

toy computation indicates that a negligible CP-asymmetry rst app ears at

order in amplitude, with two unitarity triangles describing the typ e of

GIM cancellations of the problem [14]. For a thin wall the non-lo cal charac-

ter of the internal lo op is imp ortant b ecause large particle momenta M

are present and l M . Our calculation suggests that an even smaller

result (although always at the same electroweak order) would follow for a

more realistic thickwall, l M , where a lo cal app oximation could b e

p ertinent.

3 Non zero temp erature

The three building blo cks are analogous to the T = 0 case: CKM CP viola-

tion, CP-even phases in the re ection co ecients [10 ] and the fact that the

fermion self-energy at nite T results in physical transitions.

A fundamental di erence with the T = 0 case is the damping rate, ,of

quasi-particles in a plasma. Due to incoherent thermal scattering with the

medium, their energy and momentum are not sharply de ned, but spread like

a resonance of width 2 [16]. The quasi-particle has thus a nite life-time,

turning eventually into a new state, out of phase with the initial one. Small

momenta are relevant for the problem under study, and it is known that at

zero momentum the QCD damping rate is of the order 0:15g T [16 ],

i.e. 19 GeV at T = 100 GeV. Although the imaginary part of the QCD

self-energy is smaller than its real part [17][20], which settles the overall

scale of the quasi-particle \masses", it is much larger than the real part of

the electroweak self-energy. It should weaken the e ect of electroweak level

splitting, essential to the asymmetry.

A rst step is the computation of the sp ectrum. The on-shell states

corresp ond to the zeros of the determinantof i (S ), and the corresp onding

eigenstates verify the e ective Dirac equation

i @ i + ! (z )

z z

3 2

(z; t)= 0 (2)

1 m

(z ) i@ + i @ i + !

t z z

2 3

0 0

where ! , ! are the zero momentum energies of the left/right quasiparticles

R L

in the unbroken phase. Notice that the group velo city of the solutions of eq.

(2) has b een approximated by 1/3. The sp ectrum far from the wall in b oth

the unbroken and broken phase is sketched in g. 2.

Let us start by considering the one avor case. The value of the damping

rate, , and the uncertainty principle imply to describ e the incoming quasi-

particles as wave packets whose size d cannot exceed the mean free path

1=6 . The e ective Dirac equation (2) determines then the time evolution

of the wave packet. Since eq. (2) has a non hermitean part prop ortional to ,

the total probability of our wave packets falls o exp onentially in time. This

re ects the fact that quantum coherence is lost after a time 1=2 . This

loss of probability is comp ensated by a continuous probability of creation of

new wave packets, tuned so as to keep constant the total particle density and

thus preserve unitarity. The new wave packets are assumed to b e pro duced

with a random phase, i.e. out of phase of the wave packets that have decayed.

This expresses the fact that thermal collisions with the plasma incoherently

turn the quasiparticle into a new one with a di erent energy and momentum

sp ectrum.

The details of the derivation are describ ed in [22]. We estimate the re-

ection probability of the incoming wave packet close to the wall, i.e. the

ux of particles re ected backby the wall into the unbroken phase p er unit

incoming ux. Forachirality (L or R)we de ne a re ection co ecient,

function of the energy:

m=2

; (3)

r (! )=

i 2 2

p(! )+e jp(! ) m =4j

with

0 0

! +!

arg (p(! ) m=2) + arg (p(! )+m=2)

; p(! )=! ; (4)

2 2

Notice that the re ection co ecient (3) b ecomes complex for an energy

range of width m: m=2

Using Gaussian wave packets and the analyticity of the functions r (! )

and via a formal extension of the t -integral to +1, it is p ossible to show

[22] that

Z Z

2 2 3d

9d (! E ) 2 2 3

e jr (! + i )j m (3d) 2

n (0; 0) = dE n (E ) d!

r 0 F 0

(5)

1=2

for m 1=3d. varies from 0 to ( 2)=8 dep ending on the

imp ortance of the would-b e t > 0 contribution. The last term in eq. (5)

can b e neglected.

In this limit the re ected densityisthus a gaussian smear-out of jr (! +

2 2 2 2

i )j , with a maximum value jr j = m =16 , instead of 1 when =0.

max

One way to understand the physical origin of this reduction is to notice

that, while the quasi-particles in the plasma are widely spread in energy

and momentum, d 6 m, re ection (i.e. CP-even phases) is only

imp ortantinavery narrow energy band, ! m. Hence quasi-particles can

hardly b e re ected, but for the top avor. In other words, it takes the wall

a long time ( 1=m) to emit the re ected comp onent of a small incoming

packet. If the packet decays rapidly in a time 1=2 , it is natural to see the

re ected wave strongly depleted by a factor m=2 .

Nowwe turn to several avors and compute the CP asymmetry. Using

the following values for the masses in GeV, M = 50, M = 57, m =0:006,

W Z d

m =0:09, m =3:1, m =0:003, m =1:0, m =93:7, the couplings [12]

s b u c t

4 3 2 5 2

=1:210 , =1:810 , =6:210 , =6:210 , =210 and

d s b u c

=1:88, and =0:1, =0:035 we obtain for the integrated asymmetry,

t s W

dbs

uct

21 24

=1:610 ; = 310 : (6)

T T

In b oth cases the asymmetry is dominated by the two heavier external quarks.

The induced baryon asymmetry n =s cannot exceed 10 times [10] these

b results.

Fig. 3 shows (! ) for up quarks.

In ref. [10] Farrar and Shap oshnikov (FS) obtain =T 10 , and

conclude n =s 10 (see eq. (10.3) in [10 ]). Their result is many orders

of magnitude ab ove ours, eq. (6). The main origin of the discrepancy is that

they have not considered the e ect of the damping rate on the quasi-particle

sp ectrum .

For the sake of comparison, we consider their approximation, i.e., with

just the unbroken phase inside the thermal lo ops, b oth with zero and non

zero damping rate, for a thin wall. In the energy region where the maximum

asymmetry was found for = 0 [10 ] and down quarks, the expansion

with non zero damping rate leads to:

2 2 2 2 2 2 2 2 2 2 2 2

T (m m )(m m )(m m ) (m m )(m m )(m m ) 3

t c t u c u b s s d b d

(! )= p

2 32 M (2 )

(7)

where J = c c c s s s s . This result shows the exp ected GIM cancellation

1 2 3 2 3

and regular chiral b ehaviour. Its magnitude, 410 ,islower than the

dominant one at order , shown in g. 4(b).

Furthermore, we con rm the validity of their numerical calculation with

zero damping rate, with no expansion involved and as can b e seen in g.

4(a). The same computation including the damping rate is also shown in g.

4(b).

A nal comment on the wall thickness l is p ertinent. The mean free

path for quasi-particles of lifetime 1=2 and group velo city1=3is1=6

(120GeV ) . The thin wall approximation is valid only for l 1=6 , while

p erturbative estimates[10] give l (10 GeV ) 1=6 . A realistic CP

asymmetry generated in such scenario will b e orders of magnitude b elow

the thin wall estimate in eq. (6), reinforcing thus our conclusions, b ecause

a quasi-particle would then collide and lo ose coherence long b efore feeling

awall e ect. This caveat should also b e considered in any non-standard

scenario of electroweak baryogenesis, where the wall thickness is larger than

the mean free path.

Wehave enjoyed several enlightening discussions with Tanguy Altherr

and we express our sorrow for his sudden death. Weacknowledge Luis

Alvarez-Gaume, Philipp e Boucaud, Gustavo Branco, Andy Cohen, Alvaro

De Rujula, Savas Dimop oulos, Jean Marie Frere, Jean Ginibre, Gian Giu-

dice, Patrick Huet, Jean-Pierre Leroy, Manolo Lozano, Jean-Yves Ollitreault,

Carlos Quimbay,Anton Rebhan, Eric Sather and Dominique Schi for many

fruitful discussions. Pilar Hernandez acknowledges partial nancial supp ort

from NSF-PHY92-18167 and the Milton Fund. This work was supp orted

in part by the Human Capital and Mobility Programme, contract CHRX-

CT93-0132.

More precisely, they takeinto account the nite mean free path of the quasi-particles

in the suppression factor, i.e. what fraction of the is transformed into a baryon

asymetry by the sphalerons, but not in the computation of . CP

4 Note added

After the Moriond Conference, Huet and Sather [21] have analyzed the nite

temp erature problem. These authors state that they con rm our conclu-

sions. As we had done in ref. [12 ], they stress that the damping rate is a

source for quantum decoherence, and use as well an e ective Dirac equation

which takes it into account. They discuss a nice physical analogy with the

microscopic theory of re ection of light. They do not use wave packets to

solve the scattering problem, but spatially damp ed waves. Subsequently,we

have submitted for publication two lengthy pap ers containing the details of

our computations [14 ] [22].

In a recent note [23] Farrar and Shap oshnikov (FS) have expressed doubts

on the technical reliability of b oth our work and that of Huet and Sather [21 ].

They claim that our schemes violate unitarity. This is incorrect, as particle

numb er is always conserved in our approaches. The e ective Dirac equation

for a given quasi-particle contains indeed an imaginary comp onent which

parametrizes the damping rate. An e ective description of the evolution

of a subsystem of a larger entity do es not have to b e hermitian. In fact,

consistency may imply an apparent lack of unitarity in a sub ensemble of a

whole unitary system. We had explicitely discussed this p oint in the detailed

version of our results [22 ]. We develop ed there a density matrix formal-

ism containing a creation term of quasi-particles due to collisions with the

medium, which exactly comp ensates disapp earance by the same pro cesses,

see eq. (4.30) in [22]. The density of quasi-particles is always normalized to

the equilibrium density. The quasi-particles created by the medium are out

of phase with resp ect to the ones destroyed, but total particle numb er, and

thus, unitarity, is preserved. FS also ob ject to our claim that the re ection

amplitude is suppressed when integrated over a wave packet by the inter-

ference b etween the contribution of di erent momenta. They state that, as

the re ection phase shift varies b etween 0 and , the di erent contributions

should still sum up to a signi cant result. This argument fails as the total

phase to consider is the combination of the ab ove mentioned one with the

ipz

optical path length, i.e. the phase of the e factors, as may b e seen in

section 4.1.2 in [22 ]. Consequently, the phase shifts range b etween 0 and 2

resulting in a very strong destructiveinterference. The rest of the note by

FS contains either comments which are unrelated to the main p oint under

discussion, or unproved sp eculations on alternative scenarios. In fact, these

authors have not demonstrated their implicit claim that the damping rate is

irrelevant to the problem, neither have they proven us or Huet and Sather

wrong in any concrete p oint of the calculations. And they have not included

the e ects of the damping rate in their explicit computation of the re ection

prop erties. There is no p oint in b eing rep etitive, and we refer the interested

reader to the published work [12] [21] [14] [22].

It may b e natural to wonder wether an analogous suppression of the re-

ection probabilityby damping phenomena has b een encountered elsewhere

in plasma physics. We did not nd in the litterature a completely analo-

gous scenario, and any new problem deserves a new analysis. However, some

general trends have b een found, for instance in the case of electromagnetic

waves propagating in non-homogeneous plasmas. We recall such an example

as an illustrative guideline to this typ e of physics, rather than as an argu-

ment in the present discussion. In ref. [24], several examples of the re ection

of electromagnetic waves on the b oundaries separating regions of di erent

dielectric prop erties are analyzed, b oth with and without absorption. Strong

re ection o ccurs whenever the wave has to cross a region of negative dielec-

tric constant, where it is spatially damp ed. With a prop er translation, it is

easy to see that our wave equation in the presence of a thin wall is equivalent

to the equation of the electromagnetic wave in the presence of the \transi-

tional" layer in [24]. There, the e ect of absorption is parametrized through

an imaginary part in the dielectric constant. They explicitly compare the

re ection co ecientforalayer of parab olic shap e, with and without absorp-

tion, see Fig.5 taken from [24]. We present this gure although the physical

situation di ers sensibly from ours in that their total photon densityisnot

constant. Nevertheless it illustrates the damping of the re ection co ecient

due to incoherentinteractions.

The horizontal axis measures the di erence b etween the frequency of the

electromagnetic wave and the critical frequency, for which the dielectric con-

stant b ecomes negative and re ection is strong. is the collision frequency

resp onsible for absorption. Notice the dramatic damping of the re ection

co ecient for non-zero : the curve corresp onding to =510 is scaled up

by a factor 2 10 .

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published by Gordon and Breach Science Publishers, Inc., New York, 1961. ω ω (a) (b)

0 0 ωL ωL

0 0 ωQCD ωQCD m 0 0 ωR ωR

k k

Figure 2: Dispersion relations for quasi-particles in the (a) unbroken and (b)

broken phases. The ful l (dashed) lines are normal (abnormal) branches. The

upper (lower) lines correspond to left (right) chirality. The vertical lines in

(b) represent the gaps of width ' m, in which total re ection occurs.

-20 2. 10 -20 1.5 10 -20 1. 10 -21 5. 10 0 -21 -5. 10

20 30 40 50 60 70 80

Figure 3: The dominant CP asymmetry when mass e ects are included inside

thermal lo ops, as a function of the energy. It corresp onds to charge 2/3 avors

and app ears at order (O ( )). W

-22 -5 5. 10 3. 10 -22 4. 10 -5 2. 10 -22 3. 10 -5 -22 1. 10 2. 10 -22 0 1. 10 0

47.9 48 48.1 48.2 48.3 48.4 20 30 40 50 60 70 80

(a) (b)

Figure 4: (a) shows the CP asymmetry pro duced bydown quarks in the

narrow energy range which dominates for zero damping rate, when masses

are neglected in the internal lo op. (b) shows the dramatic e ect of turning

on the damping rate e ects, in the same approximation. 2 |R| for ν=0 and 5x10 -4 3 |R| in units of 2x10 for ν=5x10

1.20 ν=0 .960

3 .720 ν=5x10

.480

2 ν=5x10 .240

0.00

-3.00 0.00 3.00 6.00 9.00 12.0 ∆ 2

f=(fcr-f), cps x 10

Figure 5: Coecient of re ection jRj for the parabolic layer z = 120km,

=30m for various values of the number of col lisions . =

f k