Standard Model CP-Violation and Baryon Asymmetry Part I: Zero

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o phases. 1 AMNSE-12-94 NSF-ITP-94-64 F w hep-ph/9406288 HD-THEP-94-19 and CERN-TH.7262/94 d LPTHE-Orsa ene y triangles. tum and the on-shell e compute the exact t 2 , O.P c e discuss the p ossible quark- en phases in the propagator. o unitarit all separates the t w a 23, Switzerland o co existing symmetry phases, rance, yp e of GIM cancellations of the w all. W ,F . The asymmetries stem from the y y a , J. Orlo w ysik, Univ. Heidelb erg b emp erature e and massless p oles, and new CP-ev T CP-violatio n orld with t he Ph asymmetry Abstract all acts as a source of momen en, as a result of a rst order phase transition. The on , M. Lozano Zero ts indicate that an e ect rst app ears at order a omica, Molecular y Nuclear, Sevilla, Spain, t ys b oth massiv Mo del ur Theoretisc I: ela, LPTHE, F 91405 Orsa v d isica A Bary tum e ects in a w viour can b e expressed in terms of t taneously brok art y computation that indicates the t Institut f c P CERN, TH Division, CH-1211, Genev M.B. Ga a Dpto. de F b en and sp on e consider quan Standard t Green function for a free fermion, when a thin w W = 0 case. General argumen brok tiquark CP asymmetries pro duced in the Standard Mo del(SM) for the academic e p erform a to terference of the SM CP-o dd couplings and the CP-ev W one-lo op self-energy cannotin b e renormalized a problem. The b eha phases resulting from the fermion re ection on the w the re ection amplitude, as the w an T un discrete symmetries of the problem arep oin discussed in general. W The Dirac propagator displa

1 Intro duction

In ref. [1] we presented recently a summary of our ideas and results on Standard Mo del (SM)

baryogenesis, in the presence of a rst order phase transition. The aim of the presentwork

is to describ e in detail our zero temp erature analysis. The nite temp erature T scenario is

treated in the acompanying pap er, ref. [14 ].

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

be n =s (4 6)10 [3]. In 1967 A.D. Sakharov[4 ] established the three building blo cks

required from any candidate theory of baryogenesis:

(a) Baryon numb er violation,

(b) C and CP violation,

The Standard Mo del (SM) contains (a)[5] and (b)[6], while (c) could also b e large enough

[7][8], if a rst order SU (2) U (1) phase transition to ok place in the evolution of the universe

[9]. We will not enter the discussion on the latter: it will b e assumed that a rst order phase

transition did take place, as strong as wished by the reader. An optimal sphaleron rate can

b e assumed as well. Our aim is to argue, on a quantitative estimation of the electroweak

C and CP e ects exclusively, that the current SM scenario is unable to explain the ab ove

mentioned baryon number to entropy ratio.

Intuitive CP arguments lead to an asymmetry many orders of magnitude b elow observa-

tion [10][11]. Assume a total ux of baryonic current, where all quark avours are equally

weighted. An hyp othetical CP-violating contribution in the SM with three generations[6]

should b e prop ortional to

2 2 2 2 2 2 2

s s s c c c s (m m )(m m )(m m )

1 2 3 1 2 3 t c t u c u

2 2 2 2 2 2

(m m )(m m )(m m ); ( 1.1)

b s b d s d

times some p ower of the electroweak coupling constant, to b e determined. In eq. ( 1.1) the

mixing angles and phase are the original Kobayashi-Maskawa ones[6]. In order to obtain a

dimensionless result, the expression in ( 1.1) has to b e divided by the natural mass scale

of the problem M or, at nite temp erature, T , at the 12 power. It follows[11] that the

resulting asymmetry is negligible, n =s 10 or smaller if other reduction factors are

considered. When the physical b oundary conditions are such that some sp eci c avour is

singled out by nature (or exp eriment as, for instance, in the K K system, electric dip ole

0 0

moment of the neutron, etc.), some of the fermionic mass di erences in ( 1.1) are no more

comp elling, and a stronger e ect is p ossible. Nevertheless it seems dicult to comp ensate the

10 orders of magnitude di erence in the ab ove reasoning. Such could b e the case if the desired

observable came ab out as a ratio of ( 1.1) with resp ect to another pro cess which contained

by itself some fermionic suppression factors: an example is the CP-violating parameter in

K decays, where the CP-violating amplitude is compared to a CP-conserving denominator,

Re (K K ), which is a GIM suppressed ob ject. This is not what happ ens with the

0 0

baryon asymmetry of the universe where, as shown b elow, although nature may single out

some sp eci c avours, there is nothing to divide with (other than the overall normalization 2

to the total incoming ux, which b ears no suppression factor). In this resp ect, the pro cess

has some analogy with the electric dip ole moment of the neutron where the exp eriment

preselects the up and down avours, and of course the results are normalized to the total

incoming ux of neutrons.

On the light of the ab ove considerations, the reader may b e astonished that we give

the problem further thought. Our motivation is that the study of quantum e ects in the

presence of a rst order phase transition is rather new and delicate, and traditional intuition

may fail. A SM explanation would b e a very economical solution to the baryon asymmetry

puzzle. To discard this p ossibility just on CP basis suggests that new physics is resp onsible

for it, without any need to settle whether a rst order phase transition is p ossible at all.

Furthermore, the detailed solution in the SM gives the \knowhow" for addressing the issue

in any theory b eyond the standard one where CP violation is rst present at the one-lo op

level. On top of the ab ove, the authors of ref.[12]have recently studied the issue in more

detail and claim that, at nite temp erature, the SM is close to pro duce enough CP violation

as to explain the observed n =s ratio. B

υ=0 υ=/ 0 x,y ψinc pinc tr 0 z ψ out out p ψ tr

Figure 1: Quarks scattering o a true vacuum bubble. Some notations in the text are sum-

marised in the \zoomed-in" view.

A rst order phase transition can b e describ ed in terms of bubbles of \true" vacuum

(with an inner vacuum exp ectation value of the Higgs eld v 6= 0) app earing and expanding

in the preexisting \false" vacuum (with v = 0 throughout). We can \zo om" into the vicinity

of one of the bubbles, see Fig. 1. There, the curvature of its wall can b e neglected and

the world is divided into two zones: on the left hand side, say, v = 0; on the right v 6=0

and masses app ear. 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. Far enough to the left no signi cant CP-violating e ect is p ossible as all fermions are

massless. In consequence, the heart of the problem lies in the re ection and transmission

prop erties of quarks bumping 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 3

current, . The induced baryon asymmetry is at most n =s 10 ,inavery

CP B CP

optimistic estimation of the non-CP ingredients [13][12].

The symmetries of the problem are analyzed in detail for a generic bubble. The analyt-

ical 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~p j 1=l .For higher momenta the cuto e ects would show up[17], but it

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

b ound for the CP asymmetry.

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: T = 0 in the present pap er, and nite

temp erature case in the subsequent one [14]. The cosmological rst order phase transition

is a temp erature e ect. In order to disentangle the physical implications of the presence

ofawall, with the consequent breaking of translation invariance, from the pure thermal

e ects, we consider here an hyp othetical world at zero temp erature but with two phases of

sp ontaneous symmetry breaking. We will see that this academic mo del is illuminating.

Intuition indicates that an existing CP violating e ect already present at zero temp era-

ture will diminish when the system is heated, b ecause the e ective v.e.v. of the Higgs eld

decreases and in consequence the fermion masses do as well (only the Yukawa couplings

already presentat T = 0 remain unchanged). The exp ected decrease of CP asymmetries

for increasing T follows, then, from the well known fact that the Kobayashi MaskawaCP

violating e ects of the standard mo del disapp ear when at least two fermion masses of the

same charge vanish or b ecome degenerate. 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.

Treating rst the T = 0 case allows a clean analysis of the novel asp ects of physics in a world

with two phases of sp ontaneous symmetry breaking. The quantum mechanical problem is

well de ned and in particular the de nition of particles, elds, in and out states, etc. is

transparent.

At T = 0 the building blo cks of the CP violating e ect are threefold. First, the necessary

CP-o dd couplings of the Cabibb o-Kobayashi-Maskawa (CKM) matrix are at work. Second,

there exist CP-even phases, equal for particles and antiparticles, whichinterfere with the

pure CP-o dd ones 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 mass. Third, as argued b elow, 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. The e ect, thus, is present at order in amplitude. Such an e ect is absent

in the usual world with just one phase of sp ontaneous symmetry breaking, where the self-

energy is renormalized away for an on-shell particle. The di erence is easy to understand:

the wall acts as an external source of momentum in the one-lo op pro cess.

The truly and 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. With this exact, non-p erturbative to ol, p erturbation theory is then appropriate in the 4

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 particles propagators in the presence of

the wall are required, and it is p ossible to compute them with a similar pro cedure[16]. Fur-

thermore, at T = 0 the eld theoretical problem is ill-de ned in the unbroken phase where

the fermions are massless, due to infrared divergences. Some cuto is needed, although

no dep endence on it should remain when the cancellations b etween di erent diagrams con-

tributing to a CP-violation e ect are considered. By the time b eing, wework in a simpli ed

case in which the one-lo op e ect itself is computed in one of the two phases. The nal

result describ es a transition b etween anytwo avours of the same charge, resulting in a CP

violating baryonic ow for a given initial chirality, of order in the total rate. The GIM

times logarithmic corrections, for the internal cancellations app ear as p owers of m =M

f W

fermionic masses in lo ops. The dep endence in the external masses is not trivial either, and

its GIM implications are discussed in detail. It follows numerically that the CP factor is

orders of magnitude smaller than the CP factor needed for baryon numb er generation. This

T = 0 mo del is academic, and we develop e it to sharp en our to ols for the physical nite

temp erature case[14].

The scop e of the present pap er go es b eyond the particular issue of baryon numb er genera-

tion in the SM. For instance, the ab ove mentioned exact fermion propagator in the presence

ofawall, whichwe derive, should b e useful in other scenarios when a rst order phase

transition is present. The understanding at the quantum level of physical pro cesses in the

presence of phase transitions still is in its infancy, and we give it a mo dest try. CP violation

in the SM is just an example of an e ect whichwould disapp ear in a classical or semiclassical

statistical physics approach.

2 Notations and symmetries

In this section we settle the formalism and express some general results. Consider several

quark avours in a gauge theory with a vacuum presenting a \wall" structure, parallel to the

x y plane. More precisely, assume that the Higgs eld (or elds if there are several Higgses)

has a vanishing v.e.v. for z !1 and some constant v.e.v. for z !1.Inbetween there

is a \wall" with arbitrary width and pro le. The interactions of quarks with the gauge

and scalar b osons are given by some gauge invariant Lagrangian. As previously stated, the

quark-b oson interactions will b e treated in p erturbation, while the non-p erturbative e ect

of the wall on the fermion propagation is solved exactly.

As a rst step let us consider the non-interacting case and treat in rst quantization

formalism the free Dirac Hamiltonian, in the presence of the wall. The second quantized

elds in the \interaction representation" are built subsequently.

The quarks/antiquarks hitting the wall from the unbroken phase are re ected or trans-

mitted. The question is whether there exists a CP asymmetry that pro duces a di erent

re ection probability for quarks and antiquarks .

Note that current conservation relates transmission to re ection, so that an asymmetry in the re ection

probability automatically implies an asymmetry in the transmission probability. 5

In the unp erturb ed case, we will diagonalize the free Hamiltonian. The eigenstates pro-

vide then the re ection and transmission co ecients. The elds and the quark propagators

are made out of these eigenstates. Furthermore, they will b e the building blo cks of the in-

coming (from the unbroken phase) and outgoing (back to the unbroken phase) wave packets

when higher orders are considered later on.

The symmetries of the problem and the conserved or partially conserved quantum num-

b ers are relevant to classify the eigenstates. It should b e obvious that discrete symmetries

are of particular relevance. In the second subsection we shall derive in the exact theory the

consequences of the only unbroken discrete symmetry, CPT, and study in that context the

e ect of CP violation. Note that the consideration of a ux of particles approaching the wall

with a non-zero average velo city (which will corresp ond later on to the physical pro cess)

breaks the CPT symmetry in a global sense, but the microscopic relations for two-particle

transition amplitudes derived b elow will still hold.

2.1 Free quarks in the presence of a wall.

Let us consider a static \wall" parallel to the x y plane. The Higgs vacuum exp ectation

value dep ends only on the z co ordinate, resulting in hermitian mass matrices m(z ) and m (z )

for the quarks. The time indep endent Dirac Hamiltonian is then

H = ~ ~p + m(z)+ i m (z ) ; (2:1)

5 5

where = and = .

i 0 i 0

For the SM, in the basis where the mass matrices are diagonal, the study simpli es to the

one avour case and m(z ), m (z ) b ecome real. Furthermore, it is always p ossible to rotate

m (z )away through a chiral rotation. In other mo dels, a diagonalization of b oth matrices

is not always p ossible and the matrix m (z )may remain, for instance, in two-Higgs mo dels

when the relative phase b etween the corresp onding v.e.v.'s changes with z (see for example

ref [15]).

Let E; p ;p ;p denote the four-momentum of the quark. For a static wall the particle

x y z

energy, E , is conserved, while p is not when the fermion crosses the wall or b ounces back, due

to lack of translational invariance. The system is symmetric under re ections with resp ect

to the x y plane and with resp ect to rotations around the z axis, the latter implying

conservation of total angular momentum in the z direction, J . It is also invariant under

Lorentz b o osts parallel to the x y plane (K and K commute with H ). From the two

x y

preceding symmetries follows a conserved quantum numb er for anywave function whichis

an eigenvector of E; p and p . Bo ost the reference frame to obtain p = p = 0. In that

x y x y

frame, the helicity states of the incoming plane waves in the unbroken phase corresp ond

to eigenstates of J = S . This quantum numb er is more convenient to use than helicity

z z

b ecause it is conserved. Denote generically by j (j = 1=2) the corresp onding eigenvalue

z z

of J : it will b e used when lab eling the eigenstates of the Hamiltonian.

Charge conjugation, C, is conserved when there exists a basis where b oth m(z )and m (z )

are real matrices.

De ne P' as the pro duct of parity times a rotation of angle around the y axis, i.e.,

parity with resp ect to the x z plane, which will maintain the unbroken phase on the initial

side of the z axis. P' is a symmetry when m (z ) = 0 in some basis. CP' is conserved when a

5 6

basis exists such that m(z )+i m (z ) is a real matrix. The unp erturb ed SM Hamiltonian

5 5

is invariant under C, P' and CP'. In general, the following quantum numb ers are conserved:

p ;p ;J ;E and CP'T.

x y z

To b e sp eci c wehave assumed that the unbroken phase is on the negative side of the z

axis. We de ne the eigenstates by

E = H (2:2)

n n n

where the \energy" E may b e p ositive or negative, and n is a short hand notation for the

quantum numb ers that lab el the states. For the latter wecho ose n = p ;p ;j ;E ;f; .

x y z n

p ;p and E are conserved real quantum numb ers. They can takeany p ositive or negative

x y n

2 2 2

value provided E >p +p. j is a discrete conserved quantum numb er de ned ab ove.

n x y

f is the avour of the state in the broken region. will b e relevant when the dimension

of the eigenspace is bigger than one, for xed values of the other quantum numb ers. The

eigenspace has dimension one in the case of total re ection, i.e. when the energy is smaller

than the height of the wall, jE j

m (1) = 0. In the opp osite case, when the energy is larger than the height of the wall, it

has dimension two. serves then to lab el the state in any convenient basis. Examples will

b e given in the case of the thin wall. With the ab ove tagging, two states with di erent lab els

are linearly indep endent.

Given a complete set of orthonormal eigenstates of the Hamiltonian, elds are de ned in

the usual way,

X X

iE t iE t

n n

+ +

(~x; t)= b

(~x)e + d (~x)e ; (2:3)

n n n

n n

+ +

where n (n ) lab el the p ositive (negative) energy states. b and d and their hermi-

n n

tian conjugates are annihilation and creation op erators with the usual anticommutation

relations. The elds (~x; t)verify the Dirac equation in the presence of the wall. The

completeness relation for the eigenstates (~x) forces up on them the standard Fermi-Dirac

anticommutation relations. They are the Heisenb erg quark elds in the presence of the wall,

when the coupling of quarks to the gauge and scalar b osons are switched o : they constitute

the starting p oint of usual p erturbation theory.

Strictly sp eaking, although we study transitions b etween two given fermions, the prop-

agation of b osons in the presence of the wall is needed. The reason is that a CP-violating

e ect in the SM is at least a one-lo op pro cess. The b osonic propagators in the presence of

the wall are thus p ertinent and viable in analogy to the fermion treatmentwe are developing

here[16]. For technical simplicity,we leave such a task for a forthcoming publication, as the

subsequent mo di cations should not change neither the order in the electroweak coupling

constant at which the e ect rst shows up, nor the typ e of GIM cancellations.

Table 1 shows the transformation laws for the discrete symmetries of the system.

2.2 Symmetries of the exact theories.

Consider now the exact theory with the quark-b oson interactions incorp orated. The states

in eq. ( 2.2) are no more eigenstates of the total Hamiltonian. The elds de ned in eq. (

The annihilation op erators verify b j0 >= d j0 >= 0, the state j0 > b eing the "vacuum" that contains

n n

the wall, i.e. with a Higgs exp ectation value that dep ends on z . 7

symmetry wave function eld momentum helicity

Identity (~x) (~x;t) p h

P i (~x) i (x ~;t) p~ -h

y 0 n y 0

C i (~x) i (~x; t) p h

2 2

CP (~x) (~x; t) p~ -h

5 5

T (~x) (~x; t) p h

y y

TCP (~x) (~x; t) p~ -h

y 5 n y 5

Table 1: Transformation laws for the wave functions and the elds under the discrete sym-

metries of the wal l, with p~ =(p ;p ;p ). In the case of the symmetries C, CP' and CP'T

x y z

the transformed wave function corresponds to a negative energy state. However the trans-

formed momenta and helicities we quote are those of the anti-particles, i.e. opposite to those

for negative energy states.

2.3) are not Heisenb erg elds of the full theory, but they can b e used as the quark elds

in the \interaction representation". The discrete symmetries apply to the elds exactly as

stated in Table 1. In general all these symmetries are broken except TCP'.

Consider, for de niteness, the unbroken phase. From Table 1 the TCP'-transformed of a

state with avour f , momentum ~p and helicity h is given by

TCP

jf; ~p; h > = jf; p;~ h>; (2:4)

where f corresp onds to the antiquark with the same avour. TCP' is an antiunitary op era-

tion, and invariance under it implies that

0 0 0 0

TCP iH (t t ) TCP iH (t t )

1 2

= ()

0 0

iH (t t )

1 2

= ; ( 2.5)

0 0

where a; b are two given quark states and t = t and t = t . The relation

1 2

0 0

iH (t t ) iH (t t ) y

1 2

e =(e ) has b een used. Equations ( 2.4) and ( 2.5) imply, for instance

for a top to charm transition, that

iH (t t )

1 2

x y z

iH (t t )

1 2

< t; (p ;p ;p );R je jc; (p ;p ;p );L >; ( 2.6)

x y z x y z

and the analogous formula for all avours and helicities.

Letting t !1 and t !1,two imp ortant consequences follow from this equation:

1 2

i) All CP' asymmetries obtained by summing over all avours, helicities and incoming

momenta will necessarily vanish[15 ]:

2 inc 0 out 2 inc 0 out 2

jA (f ; h; p ! f ; h; p )j jA(f ; h; p ! f ; h; p )j =0; (2:7)

0 inc

f;f ;h;p

inc 0 out

where A(f ; h; p ! f ; h; p ) represents the amplitude for an incoming particle f

inc 0

with helicity h and mo emntum p to b e re ected into an outgoing particle f with 8

out inc inc inc

helicity h and momentum p =(p ;p ; p ;E). This strong result do es not kill,

x y z

however, the scenario for baryogenesis during the electroweak phase transition, since

the latter is based on the role of sphalerons, which is totally di erent for left-handed

and right-handed quarks. In the following we will consider CP' asymmetries summed

over avours and momenta but only for a given initial helicity.

ii) A straightforward consequence of equation 2.7 is

inc 0 out 2 inc 0 out 2

jA(f ; L; p ! f ;R;p )j jA(f ; R; p ! f ;L;p )j =

0 inc

f;f ;p

inc 0 out 2 inc 0 out 2

jA(f ; L; p ! f ;R;p )j jA(f ; R; p ! f ;L;p )j ; ( 2.8)

0 inc

f;f ;p

which is a useful simpli cation, as it allows to express a CP' asymmetry without any

use of anti-quarks.

As we are interested in the breaking of CP' symmetry, it is useful to express the conse-

quences of an hyp othetical theory invariant under this transformation:

iH ((t t )

1 2

x y z

iH ((t t )

1 2

< c; (p ; p ; p ;Lj e jt; (p ; p ;p );R >; ( 2.9)

x y z x y z

and the analogous formulae for all avours and helicities.

3 The thin wall.

From nowonwe consider the SM in the presence of a thin wall, which allows to p erform

easier analytic calculations. The unp erturb ed Hamiltonian is, for each avour,

H = ~ ~p + m(z): (3:1)

3.1 Eigenstates

Consider a p ositive energy fermion with given p ;p ;E.

x y

2 2 2

E p p > On the left of the wall (z<0), an incoming particle has a z -momentum p =

x y

0, and helicity h. De ne

inc

p =(p ;p ;p ;E): (3:2)

x y z

Up on hitting the wall, a re ected particle has z -momentum p and

out

p =(p ;p ;p ;E); (3:3)

x y z

while a transmitted one has z -momentum p given by

2 2

p = p m if p >m

z z z

2 2

p =i m p if p : ( 3.4)

z z z

2 2

E> p +p .

x y 9

ip z 2

With these de nitions, e is a falling exp onential in the case of total re ection (p <

m ). Wethus de ne the transmitted 4-momentum as

tr 0

p =(p ;p ;p ;E): (3:5)

x y

The ab ovephysical situation is describ ed by the following eigenstate of the hamiltonian,

which will b e denoted \incoming" eigenstate,

inc out

inc

inc i~p ~x inc i~p ~x

(z ) (~x)= u (~p ) e + Ru (~p ) e

h h

inc i~p ~x

+(1 + R) u (~p ) e (z ); h = j : ( 3.6)

h z

inc

It is directly related to an incoming wave packet. u (~p ) is a solution of the Dirac equation

in the unbroken phase,

inc inc

=p u (~p )=0; (3:7)

and the re ection matrix R is given by

R = : (3:8)

p + p

The Dirac structure of R re ects the opp osite chirality of the incoming and re ected

state. Note as well that R is complex in the case of total re ection. Its imaginary phase

do es not change from particles to antiparticles, and constitutes an example of the CP-even

phases which will later on interfere with the CP-o dd ones to pro duce observable e ects.

inc inc

It follows from the de nition of R that (1 + R)u (~p ) is a solution of the Dirac equation

in the broken phase,

tr inc

(=p m)(1+ R)u (~p )=0 ; (3:9)

as well as

out inc

=p Ru (~p )=0 : (3:10 )

2 2

In fact, eq. ( 3.9) was used to compute R. In the case of total re ection, p m < 0,

the wave function ( 3.6) is the only p ositive energy eigenstate.

When transmission o ccurs, one more eigenstate is needed to span the eigenspace. A

second linearly indep endent solution, to b e called \outgoing" state (b ecause directly related

to the outgoing wave packets), is given by

out inc

out

out i~p ~x y out i~p ~x

(~x)= u (~p ) e + R u (~p ) e (z )

h h

y out i~p ~x

+(1 + R ) u (~p ) e (z ); h = j ; ( 3.11)

h z

where

br 0

p =(p ;p ;(p ) ;E): (3:12 )

x y

out inc

This wave function is however not orthogonal to . An orthogonal eigenstate is

provided by the solution coming from the broken phase:

br tr

br i~p ~x br i~p ~x

(~x)= p =p u (~p )e + Ju (~p )e (z )

z s s

n z

out

br i~p ~x

)e (z ) ; ( 3.13) +(1 + J )u (~p

s 10

where J is the re ection matrix for a particle b ouncing on the wall from the broken phase,

given by

(1 + R); (3:14 ) 1+J =

br out

and s is a spin index dep endenton j , such that (1 + J )u (~p )=u (~p ) is a massless

z s h

spinor with helicity h = j , and u (~p ) satis es the Dirac equation in the broken phase

z s

br br

(=p m) u (~p )=0 : (3:15 )

For later reference, we will name the solution in eq. ( 3.13) the \broken incoming" wave

function.

The negative energy solutions, , describ e the propagation of antiparticles. In terms

of the quantum numb ers for p ositive energy antifermions, the corresp onding wave functions

are

inc out

inc

inc i~p ~x inc i~p ~x

v (~p ) e + (~x)=

Rv (~p ) e (z )

h h

inc i~p ~x

+(1 + R ) v (~p ) e (z ); h = j ; ( 3.16)

h z

out inc

out

out i~p ~x out i~p ~x

(~x)= v (~p ) e + R v (~p ) e (z )

h h

y br

out i~p ~x

+(1 + R ) v (~p ) e (z ); h = j ; ( 3.17)

h z

and

br tr

br i~p ~x br i~p ~x

(~x)= p =p v (~p )e + Jv (~p )e (z )

z s s

n z

out

br i~p ~x

+(1 + J )v (~p )e (z ) ; ( 3.18)

0 inc

where R = R and (1 + J )=p=p (1 + R ). v (~p ) is a negative energy Dirac spinor

z h

verifying the equation

inc inc

=p v (~p )=0: (3:19 )

with the helicity h de ned as ~ pv^ (~p)=hv (~p). The spin s in eq. ( 3.18) is de ned

h h

br out

as for eq. ( 3.13), in sucha way that (1 + J )v (~p )=v (~p ) with helicity h = j .It

s h z

inc

follows from the de nition of R that (1 + R)v (~p ) is a solution of the Dirac equation in

the broken phase:

tr inc

(=p + m)(1+ R)v (~p )=0; (3:20 )

as well as

out inc

=p Rv (~p )=0; (3:21 )

and

br br

(=p + m) v (~p )=0: (3:22 )

It is convenient to consider the Fourier transformed wave functions

iq z

e (z ): (3:23 ) (q )=

2 11

From now on it will b e assumed that the b o ost leading to p = p = 0 has b een p erformed.

x y

We will only write here the z dep endence, since the x y part is trivial ( (q ) (q )). The

x y

results are

1 1 R 1+R

inc inc

(q )= u (~p ); (3:24 )

z h

2i p (q + i) p + q + i p (q i)

z z z z z

y y

1 R 1+R 1

out out

u (~p ); (3:25 ) (q )=

h z

2i p (q + i) p + q + i (p ) (q i)

z z z z z

1 1 J 1+J

br br

(q )= p =p u (~p ); (3:26 )

z z s

n z

0 0

2i p (q i) p (q i) p (q +i)

z z z z

z z

R R 1 1 1+

inc inc

(q )= v (~p ); (3:27 )

z h

2i p (q + i) p + q + i p (q i)

z z z z z

y y

1+ 1 R R 1

out out

(q )= v (~p ); (3:28 )

z h

2i p (q + i) p + q + i (p ) (q i)

z z z z z

1 1+ 1 J J

br br

(q )= p =p v (~p ); (3:29 )

z z s

n z

0 0

2i p (q i) p (q i) p (q +i)

z z z z

z z

inc out

e e

with h =+j in and h = j in .

z z

n n

3.2 Wave packets

The physical pro cess under study is describ ed bya wave packet coming from the unbroken

phase, b ouncing on the wall, and generating re ected and transmitted wave packets. The

transmission probability follows from the re ection one. It is thus sucient to concentrate

on re ection prop erties.

Eachwave packet has xed avour and helicity h (equivalent in the unbroken phase to

j , j , resp ectively for the incoming, outgoing wave packets). Its energy and momenta

z z

are clustered around average values. Somewhere in the pro cess the electroweak interactions

will act as a p erturbation, and induce avour changes through lo ops, as will b e discussed in

Section 5.

The x and y comp onents of the wave packet are not mo di ed by the wall, and will b e

ignored in what follows. Consider the following incoming wave packet, approaching the wall

from the unbroken phase,

inc 2

inc (k p ) ik (z +Z tT )

z z

z ~

P (~p ;z;t) = N dk e e u (k )

z h

(z +Z tT )

inc

ip (z +Z tT ) inc

e e u (~p ); ( 3.30)

where N is a normalization constant, d denotes the spatial extension of the wave packet,

which is lo cated at time T (T > 0) around p osition Z (Z>0), assuming Z=d >> 1,

inc inc 2

T=d >> 1. d is chosen so that p d>>1. Terms exp onentially suppressed by exp [(p d) ]

z z

have b een neglected in the second line of eq. ( 3.30).

At t T, the wave packet is an almost mono chromatic wave lo cated far from the wall

in the unbroken phase, with group velo city p ointing towards the wall. To study its time 12

evolution when approaching the wall, an expansion on the eigenstates de ned in subsection

3.1 is convenient. Up to exp onentially suppressed terms, the result is

inc 2

inc (k p ) inc ik (t+T )

z z

P (~p ;z;t)= N dk e (z ) e : (3:31 )

In other words, the incoming wave packet totally expands on the eigenstates denoted by

inc

,thus justifying their name. Analogous conclusions hold for the outgoing wave packet

in the unbroken phase. It follows that the physical amplitude can b e expressed at order

out inc

A(i ! f )=< jH j >; (3:32 )

int

n n

where H is the interaction Hamiltonian induced by electroweak lo ops.

int

4 The quark propagator.

The free quark propagator is de ned as usual:

S (x; y )=< 0jT( (x) (y ))j0 >: (4:1)

It is not translational invariant due to the presence of the wall, and veri es the equation

(@ + iH (x))S (x; y )= (xy); (4:2)

x 0 4

where H (x) is the Hamiltonian in eq. ( 3.1). Furthermore, the time ordered pro duct in

equation ( 4.1) insures the Feynman b oundary conditions. From eq. ( 2.3), we get

iE (x y )

+ 0 0

S (x; y ) =((x y ) (~x) (~y)e

0 0 0

iE (x y )

0 0

(y x ) (~x) (~y )e ); ( 4.3)

0 0

where E is the p ositive/negative energy of the eigenstate, cf. eq. ( 2.2), and where the

de nition

Z Z

X X

(4:4) dp dp dE

x y

j ;f ;

has b een used.

i f

The Fourier transformed propagator dep ends on two momenta, q and q , unlike in the

translationally invariant case,

f f

q =(E; q ;q ;q );

x y

i i

q =(E; q ;q ;q ): ( 4.5)

x y

We can write

f i 0 0 0 0 0

S (q ;q ) = d d d( )d( )d( ) S ( ;)

z x y 0

0 i 0 0 0

iq +iq iq ( )iq ( )+iE ( )

z x x y y

z z x y

e ; ( 4.6) 13

where translation invariance in x; y and time directions has b een used. From nowonwe use

the lib erty of b o osting in the x; y plane to bring q and q to 0. At the end of the section

x y

we will return to the general case.

In terms of the momentum-space wave functions, the ab ove translates into

6 f i

inc f inc i

e e

(2 ) (q S (q ;q ) = ) (q )

+ +

z z

n n

E E +i

f i inc inc

e e

(q ) (q ) +

z z

n n

E E i

f i br br

e e

(q ) (q ) +

+ +

z z

n n

E E +i

f i br br

e e

(q ) (q ) + : ( 4.7)

z z

n n

E E i

Notice in the ab ove expression that an orthonormal basis for the wave functions was used,

inc br

e e

i.e., the one spanned by and , given in eqs. ( 3.24), ( 3.27),( 3.26) and ( 3.29). In

particular, orthogonality requires to restrict the energy integration on the broken phase to

E>m.Wehave explicitly veri ed the completeness relation for our system of eigenstates ,

y y

inc f inc i br f br i

e e e e

(q ) (q ) + (q ) (q ) ( 4.8)

+ + + +

n z n z n z n z

y y

i f br i br f inc inc f i

e e e e

(q ) = (q ) (q ) + (q ) + (q q ):

z z n z n z n n z z

(2 )

The obvious consequence of eqs. ( 2.2) and ( 4.8) is the Green's function equation for the

propagator:

f i 3 f i

i(E H ) S (q ;q ) =(2) (q q ): (4:9)

z z

The Fourier transform of eq. ( 4.9) is:

0 4 0

@ + ~ r + i m ( ) S (; )= ( ): (4:10 )

~ z 0

Details on the computation of the propagator can b e found in App endix A. Let us

summarise the results in 1+1 dimensions (time and z -directions). An interesting interme-

diate step is given by the expressions which follow. The contribution of states propagating

exclusively on the left-hand side of the wall (v = 0, \unbroken" region) is

1 1

f i f

S (q ;q ) = (E + q )

lef t 0 z

f f

i 2 2

q q + i (sE + i) (q + i)

z z

1 1 s(1 + s )

sE q + i 2

(sE + i) (q + i)

1 1 (1 s ) sm

z 0

+ : (4:11 )

f 0

sE q + i 2 sE + p

sE + q + i

The negative energy eigenfunctions describ e the propagation of antiparticles: in the notation presently

f i

used an initial antiparticle will have momentum q , while a nal antiparticle has momentum q . 14

The contribution from states propagating exclusively on the right-hand side of the wall (v.e.v.

6= 0), that is, in the \broken" region is

1 1

f i f

S (q ;q ) = (E + q + m )

right 0 z 0

f f

i 0 2 2

q q i (p + i) (q + i)

z z

1 1 E 1 p m

1 +

z 0

0 i 0

p + q + i p 2 E E

(p + i)+ (q i)

z z z

E 1 sm 1 1 p m

: (4:12 ) 1+ +

z 0

0 f

0 i 0

p + q + i p 2 E E sE + p

p q + i

z z z

Finally, the contribution from states propagating across the wall is

1 1 (1 + s ) sm

z 0

f i

S (q ;q ) = s 1+

acr oss 0

sE q + i sE + p 2

p q + i

sm 1 1 (1 s )

0 z

1+ + ; (4:13 )

0 i

p + q + i 2 sE + p

sE + q + i

z z

where s is a parameter which takes values +1 or 1 for p ositive or negative energy particles,

resp ectively. p is de ned as follows

2 2

p =+ E m +i; (4:14 )

and the factor sm =(sE + p ) originates from the re ection matrix R, eq. ( 3.8).

Eqs. ( 4.11)-( 4.13) show sometimes two i terms with opp osite signs present in the same

denominator, for instance 1=((sE + i) (q + i)) in the second line of eq. ( 4.11). The

reason for this phenomenon is clear. The rst i in our example is there to implement the

Feynman b oundary conditions for the propagator in time direction. The second i sp eci es

in which spatial phase (z<0or z>0) this terms acts. This may seem ambiguous: is

the p ole lo cated in the upp er or lower half plane? Both prescription are not equivalent,

unless the residue at the p ole vanishes. In other words, since 1=(x i)= P(1=x) (x),

the ambiguity disapp ears only if the factor multiplying (x)vanishes at x =0. A close

scrutiny of eqs. ( 4.11)-( 4.13) shows that this is indeed the case. For instance, the residue

in the second line of eq. ( 4.11) is exactly comp ensated by the residue of the at sE = q

equally \ambiguous" term in the rst line of eq. ( 4.11). It results that the sum of b oth

terms is not ambiguous. It is p ossible to chose inaconsistent way any sign for the i in the

rst and second lines of eq. ( 4.11). Flipping from one convention to the other exchanges a

term prop ortional to (sE q )between the rst and second line of eq. ( 4.11), and the sum

do es not change. For later use denote \time i" the usual Feynman convention and \space

i" convention the other one. In the following the \time i" convention will b e assumed

unless sp eci ed otherwise.

The propagator is given by the sum of eqs. ( 4.11), ( 4.12) and ( 4.13) ab ove,

f i f i f i f i

S (q ;q ) = S(q ;q ) + S (q ;q ) + S (q ;q ) : (4:15 )

0 lef t right acr oss 0

In App endix A it is explicitly shown that eq. ( 4.15) veri es the Dirac equation. Using the

relations ( 5.38- 5.39), as well as the ab ove mentioned freedom to shift certain p oles, it is 15

p ossible to combine the various terms in a more compact and familiar way. One can either

show that

1 1 1 1

f i

S (q ;q )= +

f f

i i

=q =q m

q q +i q q i

z z

" #

1+s m 1 ms

z 0

1 s

f 0 i i

=q 2 E + p =q (=q m)

" #

1 m ms 1 s

0 z

1 s ; (4:16 )

f 0 i i

=q m E + p 2 =q (=q m)

which simpli es the action of the q -Dirac op erator, or that

1 1 1 1

f i

S (q ;q )= +

f f

i i

=q =q m

q q +i q q i

z z

" #

ms m 1 s 1

0 z

1 s

f f 0 i

=q (=q m) E + p 2 =q

" #

ms m 1+s 1

0 z

1 s ; (4:17 )

f f 0 i

=q (=q m) 2 E + p =q m

which prepares for the action of the q -Dirac op erator. In fact, eq. ( 4.16) is the CP'

i f

conjugate of ( 4.17). An elegant expression, b ecause more symmetric in the q and q

dep endences, can b e obtained combining ( 4.16) and ( 4.17) and is given by

1 1 1 1

f i

S (q ;q )=1=2 +

f f

f i

i i

=q =q

q q +i q q i

z z

1 1 1 1 1 1

+ +

z z

f i f i f i

=q m =q m =q m =q =q =q m

" #

m ms m

1 (1 s ) s (4:18)

z 0

f f 0 i i

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

In ( 4.16), ( 4.17) and ( 4.18): all =q 's in the denominators are understo o d as regularized \a

la Feynman", i.e. as =q + i, the \time i" convention. This convention allows to check easily

that our propagator ob eys Feynman b oundary conditions, i.e. that the p ositive frequencies

(q > 0) are \retarded", and the negative ones (q < 0) \advanced".

0 0

If the \space i" convention was used instead, then the regularisation implies to consider

f i f f i i

all massless p oles (1==q ; 1==q ) with q ! q + i and q ! q i.UponFourier transform it

z z z z

corresp onds to (z ) and (z ) resp ectively, i.e. it lo cates the massless p oles in the unbro-

i f

f i

ken region. Analogously, all massive p oles (1=(=q m ); 1=(=q m )) are to b e understo o d

f i

f f i i

as q ! q i and q ! q + i, lo cating them in the broken region. In this convention

z z z z

it is easy to check that the propagator in eqs. ( 4.16), ( 4.17) and ( 4.18) veri es the Dirac

equation ( 4.10). 16

For some terms in eqs. ( 4.16), ( 4.17) and ( 4.18) the \time i" and the \space i"

conventions are apparently in con ict. For instance,

f f

=q =q

f f 2 f 2

= +2i =q (q ) (q (q ) ): (4:19 )

z 0 z

f f

2 2

q (q + i) q (q ) + i

z z

0 0

The two conventions di er by the last term on this equation. However, this term, as all

matching terms of this typ e, do cancel among the di erent factors in any of the eqs. ( 4.16),

( 4.17) and ( 4.18). This cancellation is in fact the same phenomenon as the cancellation of

the residues corresp onding to \ambiguous i's" that we discussed after eqs. ( 4.11), ( 4.12)

and ( 4.13). We are thus lead to the striking conclusion that the \time" and the \space" i

conventions are strictly equivalent in eqs. ( 4.16), ( 4.17) and ( 4.18).

This equivalence is physically understandable. It means that all the p oles for which the

two conventions give con icting \i" signs havea vanishing residue: they corresp ond to

physically non acceptable asymptotic states. For instance, consider a massless p ole with a

p ositive q in eq. ( A.7). Massless asymptotic states can only exist in the unbroken phase,

i.e. on the left side of the wall. A nal state in the unbroken phase must necessarily move

away from the wall, i.e. with a negative q . It is then physically welcome that the residue

vanishes when q is p ositive. All other apparent con icts are resolved in a similar way.

We shall denote the rst line of expressions ( 4.16), ( 4.17) and ( 4.18) as the \non-

homogeneous" part. The rest of the propagator in any of its versions will b e denoted \ho-

mogeneous part", as the latter is a solution of the Dirac equation without a source, contrary

to the former. By the same token, it is p ossible to verify that the advanced and retarded

Green functions in the presence of a wall di er from those for the usual Dirac propagator by

terms whichvanish up on the action of the Dirac op erator.

The presence of b oth massless and massive p oles re ects the fermion sailing b etween the

unbroken and broken phases. It is trivial to verify in eqs. ( 4.16), ( 4.17) and ( 4.18) that

the usual i==p Feynman propagator for a massless fermion is recovered. Notice as well that,

b esides the usual i dep endence which corresp onds to absorptive contributions for on-shell

fermions, new imaginary phases app ear which are generated by the re ection co ecients of

2 2

the fermion when E

are induced by the re ection matrix, eq.( 3.8).

For completeness we giveaswell the fermion propagator in the presence of the wall for

the 4-dimensional case, i.e., when q and q have not b een b o osted to zero values:

x y

! !

1 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 1

z z

f i f i

=q m =q =q =q m

" #

E q q + (E q q + s )

0 x x y y 0 0 x x y y z

m m

p +p

s (4:20 )

f f i i

=q (=q m) p =q (=q m)

q q

2 2 2 2 2 2 2

where p = E q E q q q + i and p m + i. =

x x y y z 17

During the (long) preparation of this manuscript, we b ecame aware of ref.[17] which

derives the Dirac propagator in presence of a thickwall and uses it to compute the CP-

asymmetry in the re ection pro cess. The source of CP-violation they use is not a SM lo op

but rather the tree-level axial quark mass in a 2 scalar mo del. Although this di erence makes

the comparison b etween b oth works delicate, their results seem to supp ort our statements

that increasing the wall thickness would further reduce the asymmetry.

5 One lo op computation.

The propagators in the presence of the wall will b e depicted by a line screened by crosses (see

Fig. 2), while the usual ones are drawn as solid lines. Consider a one-lo op transition b etween

two external avours, i and and f , such as the ones in Fig. 2. The analogous diagrams with

full lines, i.e., with the usual Feynman propagators, are completely renormalized away for

on-shell external states. This will b e changed by the presence of the wall, which acts on the

diagram as a external source of momentum.

qW H+,H0

(a) (b) (c)

(d) (e)

Figure 2: The kinky propagators for quarks, W 's and scalars can be assembled into loop

diagrams. Vertices, being local, are unchanged in this formalism.

Let us review the prop erties exp ected in general, indep endently of the details of a given

calculation. The pro cess under consideration describ es an asymptotically massless quark i,

approaching the wall from the unbroken phase, which is re ected into the same phase as an

asymptotic massless f quark. A ip of chirality has thus necessarily o ccurred. The avour

change happ ens through one electroweak lo op, and we will denote by M the internal fermion

mass.

First of all, the amplitude should not vanish at order . There is no symmetry reason

known to us implying that the co ecient of the CP-violating op erator should b e zero at this

order.

Second, the amplitude should vanish at least as m m when the internal lo op is computed

i f

in just one of the two phases, and Lorentz covariance is preserved. A priori this b ehaviour is

not mandatory, as just one chirality ip is needed, and an amplitude vanishing with masses 18

as either m or would seem adequate. But there are more constraints: in the unbroken phase,

the computation of the internal lo op should b ehaveas

I (q )/=q; (5:1)

due to Lorentz covariance. This lo op is to b e inserted on the propagator in the presence

of the wall. Imagine that re ection has happ ened b efore the lo op insertion (providing an

mi factor), but not afterwards. The outgoing state is then a plane waveverifying the Dirac

equation for a massless particle. The action of eq.( 5.1) on it gives zero. In the opp osite

case, when re ection happ ens only after the lo op insertion, the analogous argument for the

incoming massless state holds as well. Consequently, the action of the wall must b e present

b efore and after the internal lo op insertion, and a non trivial dep endence of the amplitude

on b oth m and m results. The total e ect should go to zero as some p ositivepower of

i f

b oth of them, with an o dd overall dep endence on external masses, since chirality ips up on

re ection. The argument can b e generalized to the situation where the broken phase is

considered inside the lo op, the di erence with eq. ( 5.1) b eing terms indep endentof =q, but

prop ortional to the external masses, as it will b e exempli ed b elow.

Third, it is p ossible to argue the typ e of GIM cancellations for the quark masses inside the

lo op. The relevant terms in the re ection amplitude A(i ! f ) corresp ond to the interference

of diagrams with two di erentinternal quark masses, M and M , to b e summed up on. Each

individual diagram can b e written as

2 4

M M M M

A(i ! f ) / F (M )= a+ (b+cl og )+ (d+el og )+::::: ; (5:2)

2 4

M M M M

W W

with a; b; c; d; e... dep ending only on external masses and momenta. Due to CKM unitarity

(see eq. ( 5.25)), the CP observable is prop ortional to Im[F (M )F (M ) ]. As each

M;M

term in this sum is an antisymmetric function of M and M , the squared terms in the devel-

opment ( 5.2) of the pro duct cancel, while the cross-terms involving the constant piece a fall

2 02

M M

out once the sum is p erformed. The lowest order contribution is thus log (M ).

2 04

M M

In the practical example b elow, we nd log M , i.e. a further internal mass suppres-

sion. It is imp ortant to notice that, whenever only the unbroken phase is considered inside

the electroweak lo op, c = d = e = 0, b ecause the fermionic mass dep endence stems from

pure Yukawa couplings. No antisymmetric function in M; M is viable, and the e ect should

vanish at order in total rate. There is no reason, though, to neglect the broken phase,

and we exp ect an O ( ) contribution.

Finally, the observable should b e prop ortional to the interference of the imaginary parts

of the CKM couplings with the CP-even phases pro duced in the re ection on the wall.

The ab ove considerations apply as well for the thickwall scenario. Some remarks sp ecif-

ically related to the wall thickness are appropriate, b efore entering into the details of our

computation. The internal electroweak lo op is in general a complicate function of q , and

thus non-lo cal. Its typical \inverse-size" is of the order of the dimensional parameters in-

volved which means M ,aswe will see. To neglect its non-lo cal character is only adequate

This dep endence is in general a complicate function. For transparency, only the b ehaviour for M

is describ ed here. As the internal lo op is IR convergent when M ! 0, no pure logM terms can app ear. 19

for wall thicknesses l larger than the lo op \size", l>>M .For a thin wall, l ! 0, a lo-

cal approximation for the internal lo op is incorrect. Arbitrary large particle momenta are

relevant, and it will b e shown that indeed the non-lo cal character of the internal lo op is

imp ortant. The role played by the latter e ects in our calculation suggests that an even

smaller result would follow in a more realistic thickwall computation, l>>M , where a

lo cal internal lo op could b e a go o d approximation.

We will see that the ab ove four general characteristics do app ear in the toy computation

given b elow.

5.1 One particle irreducible lo op

The complete computation of the diagrams in Fig. 2 requires not only the knowledge of

the fermion propagator in the presence of the wall, but the corresp onding one for gauge and

Higgs b osons as well. They can b e obtained in complete analogy with the computations of

the previous section. However, they need rather lengthy calculations. Our goal here is rather

to get a feeling of how the di erent building blo cks work together than to solve exactly a

problem whichanyhow is academic to a large extent. Wehavechosen to simplify our task,

leaving the full calculation for a forthcoming publication. What we compute in fact are the

diagrams in Fig. 3, with the internal lo op computed in the broken phase. Notice that a

similar choice has b een made in recent publications which try to solve the problem at nite

temp erature[12], where only the unbroken phase was considered inside the electroweak lo ops.

q-k q-k

i f i f q k , Ml q q k , Ml q

(a) (b)

Figure 3: The subset of Feynmann diagrams used in this paper.

Let us then consider the broken phase inside the lo op. In the 't Ho oft Feynman gauge

the integral to p erform can b e written in general as

ka= + b

; (5:3) I (q )=c d k

2 2

) )(k M ((q k ) M

l W

where q is the external 4-momentum and a; b and c are k -indep endent. indicates that the

integral has b een regularized someway, for instance by dimensional regularization.

Wecho ose to substract at q = 0. Because Lorentz covariance is preserved at T = 0, the

result of the substraction will have the general form

subst: subst: 2 subst: 2

I (q )==qA (q )+B (q ) ; (5:4)

subst: 2 2

where A and B are matrices with 1 and/or comp onents, A (q )= A(q )A(0) and

subst: 2 2

B (q )=B(q )B(0). Finally,

Z Z

1 1

q x(1 x)

subst: 2

I (q )= c dx dy (=qax + b) : ( 5.5)

2 2

q xy (1 x) M x M (1 x)

0 0

W l

Notice that with, the ab ovechoice of substraction, the internal lo op is completely renor-

malized away for an on-shell massless fermion which su ers no interactions with the wall.

Concentrate for the moment in the case i; f = d; s or b. The scalar-exchange contribution is

then

Z Z

2 2 2

1 1

g q x(1 x)

subst:

I (q )= dx dy K K

scalar lf

2 2 2

4 2

2M (2 ) q xy (1 x) M x M (1 x)

0 0

W W l

2 2

fx=q (m m R + M L) M (m L + m R)g;

i f f i

l l

( 5.6)

where L and R denote the chiral pro jectors, L =(1 )=2, R =(1+ )=2. K describ es the

5 5

CKM couplings of the quarks and M is the mass of the internal quark of l -th generation.

The W-exchange contribution to the substracted internal lo op is

Z Z

2 2

1 1

q x(1 x)

subst: 2

I (q )=g K K dx dy x=qL:

W lf

2 2

4 2

(2 ) q xy (1 x) M x M (1 x)

0 0

W l

( 5.7)

The ab ove represent only one contribution among many that should b e considered in

the full calculation. When this internal lo op is inserted b elow on our propagator, gauge

invariance is preserved, as the fermion interaction with the wall is of Yukawatyp e and thus

SU (3) SU (2) U (1) gauge invariant. In the full theory, the substraction pro cedure will

b e more complicated, though . Our aim in a toy computation is to get an idea of how the

ingredients of CP vilolation, GIM mechanism, and wall re ection combine together, and to

settle useful calculational to ols for this typ e of problem.

5.2 Lo op insertion in the propagator

The diagrams in Fig. 3 are computed inserting the internal lo ops given ab ove on the fermion

propagator in the presence of the wall, derived in the preceding section. The desired am-

plitude should describ e the transition of an incoming massless avour i into an outgoing

massless avour f . The selection of these asymptotic states is obtained pro jecting onto the

massless p oles of the propagator, for the 4-momenta app earing at b oth extremes of Fig. 3.

What has b een done ab ove is close to the complete result in a limited extension of the standard mo del,

with two Higgs's, one ( ) coupled to down quarks and the other ( ) to up quarks. We assume the following

d u

exp ectation values: < (z ) >= v (z ) and < (z ) >= v , with v v . In this mo del the equation (

d d u u d u

5.7) turns out to b ecome the exact contribution from W exchange, while eq. ( 5.6), or rather its Fourier

transform, is mo di ed by the following substitution: m ! m (z );m ! m (z ), where z (z ) is the

i i i f f f i f

co ordinate of the incoming (outgoing) vertex of the lo op. An analysis of the changes in the computations

runs parallel to the one p erformed ab ove, but it do es not seem p ertinent to pursue this mo del here. 21

The relevant contributions of Fig.(3) are then those terms which contain b oth p oles in

i f

1==q and 1==q . It can b e shown that the non-homogeneous part of the fermion propagator

in the presence of the wall, gives no contribution of this typ e. That is, among the four

p ossible combinations of homogeneous and non homogeneous on b oth sides of the internal

lo op, only the homogeneous-internal lo op-homogeneous contributes, yielding for a p ositive

energy particle:

" #

dq 1 1+ m m

z z f f 0

A(i ! f )= 1

2 =q 2 =q(=q m )

E +p

" #

h i

m 1 m 1

i i 0 z

subst: subst:

I (q )+I (q ) 1 ; ( 5.8)

W scalar

=q (=q m ) E + p 2 =q

where

p =+ E m +i (5:9)

and

p =+ E m +i: (5:10 )

The q integration re ects the arbitrary o -shellness of the quarks up on re ection on the

wall. The relevantintegrals are the following ones:

dq 1 1 1

2 2

2 q m q m

q M

2 2

M m

1 1 i

l f

( 5.11) (p + )

i f i f

0 0 0 0

2 2 2 2 e

f f

p p p + p

2(m M )(m M )

z z z z

l f l

and

dq 1 1 1

q =

2 2

2 q m q m

q M

2 2

M m

M i 1

l f

2 2 0

M + m ; ( 5.12) (p )

l i z

i f i f

0 0 0 0

2 2 2 2

f f

p p p + p

2(m M )(m M )

z z z z

l f l

where

xM +(1 x)M

2 l

M = ; (5:13 )

xy (1 x)

e f

+ i : (5:14 ) Q = E M

After some lenghty algebraic manipulations, the resulting amplitude for an incoming left

asymptotic quark can b e written as

Z Z

2 2

1 1

dy g i

dx A(i ! f ) = K K

R L li

2 2 2 2

f f

2 (2 ) y

0 0

(m M )(m M )

l f l

2 2 2

f f

1 M 1 M m

l i l

2 0 2 2

m m [xa~ + b]+ (p ) M + m

f z l f

i f i f

0 0 0 0

p p p + p

z z z z

2 2

M 1 m 1 1+ 1

l i

+ (p ) [xc~ + d] R ; ( 5.15)

i f i f

0 0 0 0

4 =q

p p p + p

z z z z 22

where

i f

2 0 2 0

M p m p

l z i z

~a = (2 + )(1 )+ (1 ); ( 5.16)

2 2

M M

E + p

W W

2 0

M E p

l z

b = (1 + ); ( 5.17)

E + p

# "

E M E

(2 + )+ ; ( 5.18) c~ = m

2 2 f

M M

E + p

E +p

W W

2 0

M E p

i f

l z

2 0 0

d = (E p p )(1 + ): ( 5.19)

z z

E + p

Notice that Q , obtained from the convolution of the electroweak lo op with the free

fermion propagator in the presence of the wall, is in general complex, and will b e one of

the relevant CP-even phases which will interfere with the CKM complex couplings. This

new phase stems from the non-lo cal character of the internal electroweak lo op, and would

have disapp eared in a linear approximation. Other contributions of the same typ e should

b e present in a full calculation with the wall accounted for inside the lo op, and although

cancellations are p ossible, they are not mandatory byany symmetry argument.

For an incident left-handed particle, the result is obtained from eqs. ( 5.15) to ( 5.19)

i f

0 0

replacing everywhere m by m and viceversa (and consequently p by p ). The CKM

i f

z z

couplings remain identical.

Consider transitions b etween charge 1=3 quarks. Concentrate as well on quark energies

2 2 2 2

b elow or equal to M . The approximation m ;m ;E < M is then justi ed, and eq. (

i f l

5.15) simpli es to

Z Z

2 2

1 1

g dy i

A(i ! f ) = K K dx

R L li

2 (2 ) y

0 0

1 1 1

2 2 0

m m M [xa + b]+ (p + )

f l z

i f i f

0 0 0 0

p p p + p

z z z z

1 1 M 1 1+

[xc + d] R ; ( 5.20) (p + )

i f i f

0 0 0 0

4 =q

p p p + p

z z l z z

2 2 2 2

where a; b; c; d; are given by eq. ( 5.19), neglecting terms in m =M ,m =M .

i W f W

The amplitude can b e rewritten as

2 2

g 1 m m 1+ 1

i f z

A(i ! f ) = i K K : ( 5.21) F (M ) L

R L li

4 f i

2 (2 ) =q M 4 =q

The function F (M ) is de ned as follows:

1 E p

F (M ) = m (1 + )[I I ] ( 5.22)

f 1 2

i f f

0 0 0

p + p E + p

z z z

" #

i i

0 0

m E p p

z z

+i (1 + )I (1 )I ; ( 5.23)

3 4

f f

0 0

E + p E + p

z z 23

and the integrals I are

1;2;3;4

3 2

x(1 x)

5 4

x I (M )= dx ; (5:24 )

n n

x + (1 x)

2 2 2 2

with =3, =M=M , =3=2=2+M=M , =(0;1;0;1) and =

1 3 2 4 n n

W W

(1; 1; 3=2; 3=2).

Consider i 6= f , i.e., a avour changing transition. The CP-violating observable is pro-

p ortional to

2 2

g m m

i f

2 2 2 2

~ ~

[jA(i ! f )j jA(i ! f )j ]= [ ] ( )

R L L R

4 (2 ) M

l;l

0 0

Im K K (K K ) Im F (M )F (M ) ; ( 5.25) (2)

li l i l l

lf l f

l;l

i f

where A(i ! f ) is obtained from A(i ! f ) taking the residues of 1==q and 1==q at q =

i f

q = q . A(i ! f ) corresp onds to the matrix element of the e ective Hamiltonian b etween

z z

states ji> and

Eq. ( 5.25) shows explicitly the interference b etween the CP-o dd phases of the CKM

couplings K and the CP-even phases of F . The latter can havetwo di erent origins: either

p when i or f are totally re ected, or Q , which leaves a trace of the non-lo cality of the

internal lo op in the i factor b efore I and I .

3 4

It is well known that

= J ; (5:26 ) Im K K (K K )

li l i

lf l f

0 0

where =1 iff i=l l (mo d 3), = 1 when f i 6= l l (mo d 3) and where J

is twice the area of the CKM unitarity triangle, J = c c c s s s s . Let us now consider in

1 2 3 2 3

the complex plane the triangle spanned by the three numb ers F (M );l = u; c; t. Let B be

the area of this triangle with a sign +1 (-1) for a cyclic clo ckwise (anti-clo ckwise) ordering

of u; c; t around the triangle. Then

0 0

Im K K (K K ) Im F (M )F (M ) =8JB: (5:27 )

li l i l l

lf l f

l;l

This expression exhibits clearly the GIM mechanism for the internal avours, since the

area vanishes obviously whenever two p oints coincide. Additional vanishing happ ens when

three p oints are aligned. The function F dep ends b oth on the internal and external masses

as can b e seen from eq. ( 5.23). In order to exhibit the GIM mechanism for the external

masses as well, Im F (M )F (M ) can b e expanded, leading to:

l l

2 2

2 2 2 2

~ ~

[jA(i ! f )j jA(i ! f )j ]= [ ] (c c c s s s s )

R L L R 1 2 3 2 3

4 (2 )

l;l

X X

(2) S (M ;M )b (E; m ;m ): ( 5.28)

jk l l jk i f

l;l =l+1 (mo d 3) 24

In this expression, the function S derives from the integrals I in eq. ( 5.24), and contains

1;2;3;4

the dep endence on the masses for the quarks inside the lo op,

0 0 0

S (M; M )=I (M)I (M )I (M )I (M); (5:29 )

jk j k j k

and the b are antisymmetric functions in j; k . They contain the dep endence on the masses

of the incoming and outgoing quarks. In the approximation used from eq. ( 5.20) on, they

are equal to

b (E; m ;m ) = 0; ( 5.30)

12 i f

f i

0 0 2

m m j2E + p p j 1

i f

z z

b (E; m ;m ) = ( ) Re ; ( 5.31)

13 i f

f i f

0 2 0 0

M M

jE + p j p + p

z z z

b (E; m ;m ) = b (E; m ;m ); ( 5.32)

23 i f 13 i f

m m 1

i f

b (E; m ;m ) = +( )

14 i f

2 f

0 2

M M

jE + p j

f i f i

0 0 0 0

(2E + p p )(E + p p

z z z z

; ( 5.33) Re

i f

0 0

p + p

z z

b (E; m ;m ) = b (E; m ;m ); ( 5.34)

24 i f 14 i f

m m E

i f

f i

0 0 2

b (E; m ;m ) = ( Im p p : ( 5.35) )

34 i f

z z

f 2 2

0 2

M M

jE + p j

W W

For each j; k pair, fI (M );I (M)g can b e seen as the co ordinates of the p oint M in

j k

the (j; k )-plane. Then the p oints (m ;m ;m ) span a triangle in this plane whose area

u c t

1=2 S (M ;M ) exhibits again the GIM mechanism for internal masses. Fur-

jk l l

l;l =l+1 (mo d 3)

0 0

thermore, b ecause Im (K K )(K K ) is antisymmetric in the exchange of i and f ,

i;l l;f i;l l ;f

the sum over external avours is prop ortional to

/ (b (E; m ;m ) b (E; m ;m )); (5:36 )

jk i f jk f i

i;f =i+1 (mo d 3)

which shows explicitly the GIM mechanism op erating on external quarks. This pattern

should b e present in a complete computation, with p ertinent mo di cations in the functions

S ;b .

jk jk

All b but b require the existence of the CP-even phase in Q found ab ove, and dep end

jk 12 0

as well on the complex re ection b ehaviour of the initial and nal quarks, contained in p

and p . b dep ends only on the latter, and could b e 6= 0 only when m ,m and E are not

12 i f

neglected in frontof M .

Note that all b vanish when either m or m go to zero, resp ecting the second p oint

jk i f

of the general b ehaviour announced at the b eginning of this section. In each sector of

quark charges (1=3or+2=3) the heavier masses will dominate the e ect, as exp ected from

intuition. Note that a two-threshold structure is present, corresp onding to m , m .

i f

). It is In our numerical results we use the exact values of the fonctions S (M; M

instructive, though, to show a t for the particular case M; M << M , an appropriate

expansion for all quarks but the top, 25

" #

2 02 2 2 02 02

M M M M M M

S (M; M ) ! log log ; (5:37 )

1;2

4 2 2 2 2

M M M M M

W W W W W

and the ratio of the remaining S (M; M ) to the result in eq. ( 5.37) is given by S =S !

j;k 1;3 1;2

7 1 1 1

, S =S ! +1:5, S =S ! , S =S ! , and S =S ! + , in the same

1;4 1;2 2;3 1;2 2;4 1;2 3;4 1;2

15 30 4 6

limit. This b ehaviour is consistent with the third p oint of the general exp ectations develop ed

at the b eginning of this section.

5.3 Transition amplitude in terms of wave functions

The computation of the amplitude describ ed in the previous subsection was obtained by

pro jecting over the massless p oles of the propagator in the presence of the wall, once a

avour-changing electroweak lo op was inserted.

The result should b e equivalent to the more direct one in terms of incoming and outgoing

wave functions (wave packets), as given by eq. ( 3.32), with no need to consider the full

propagator in the presence of the wall derived in Section 4, as only the usual propagators

were used inside lo ops. To prove that such is the case, consider the incoming wave function

inc 9

in eq. ( 3.24). The following identities

1 1 1 1+ 1

z z

= + ; ( 5.38)

=q p q + i 2 p +q +i 2

z z z z

0 0

1 1 1 1 p + p + m p p + m

z z 0 z z 0

z z

= + ; ( 5.39)

0 0 0

=q m p 2 p q + i 2 p + q + i

z z

z z z

lead to the equivalent expression

1 m m 1+

0 z

inc inc

(q )=

(1 u (~p ): ( 5.40) ) + (p q )

z h 0 z z

2i =q(=q m) p + p 2

In the neighb ourho o d of the massless p oles of the propagator of a given quark,

inc f inc

(q ) u (~p )y

n+ z

f i

2i S (q ;q ) ; ( 5.41)

!E q E q + i

out out i

u (~p ) (q )y

n+ z

f i

S (q ;q ) 2i : ( 5.42)

q !E

z E + q + i

In eq. ( 5.40) the rst term corresp onds to the residue at the massless p ole of the

\homogeneous" part of the propagator, while the term in (p q ) corresp onds to the residue

at the same p ole of the \non homogeneous" part of the propagator. For the particular choice

of substraction (at q = 0) in subsection 5.1, the latter obviously gives no contribution.

The conclusion found in the previous subsection, namely that only the homogeneous parts

of the propagator participated in the amplitude under consideration, gets thus a simple

out inc

interpretation in terms of the wave function contributions to < jH j >. And the

int

n n

equivalence of b oth metho ds to compute the amplitude is proved.

recall that p = E = q in this case.

z 0 26

6 CP Asymmetry

6.1 Re ection probability

The task is to estimate the probability of re ecting a quark of avour i into another avour f ,

for a given ux of incoming particles (antiparticles), i.e. the re ected ux p er unit incoming

ux. Let us consider a nite b ox of length L and unit section in the plane x y , which

incorp orates the wall. For simplicity,we will ignore in the following the x and y directions.

inc out

We normalize the wave functions and in that b ox. Denote the matrix elementof

the interaction Hamiltonian, generated by the lo op b etween the latter states, by

A(i ! f )

out inc

; (6:1) < jH j >=

int

n n

where A(i ! f ) is dimensionless, see eqs.( 5.25), ( 5.28). For any time interval T such that

A(i ! f )T =L << 1,

T =2

A(i ! f )

out iH T inc iE T i (E E )t

< je j > e dte (6:2)

n n

T =2

whose mo dulus squared gives the re ection probability p er particle (E =(E +E)=2 in this

f i

equation). Taking now the usual limit:

" #

T =2

i (E E )t

dte T2(E E ): (6:3)

f i

T!1

T =2

For an incoming ux p er unit time of particles with velo city v , the numb er of particles in

the volume will b e vL . The re ection probability p er unit time and unit ux is then:

jA(i ! f )j

v 2(E E ): (6:4) P (i ! f )

f i

R R

f f

In the in nite volume limit, f =L ! dp =(2 ), and with dp (E E )=1=v ,it

z f i

z z

follows that

P (i ! f )=jA(i!f)j : (6:5)

The numb er of particles re ected p er unit time (or re ected ux) is = jA(i ! f )j ,

r i

and thus related to the incoming ux through the previously computed jAj .

Assume that the probability of the incoming state i in the unbroken phase is given by

a distribution (i), which dep ends in general on the avour, chirality and energy of i.For

instance, at nite temp erature close to equilibrium, this distribution will b e close to the

Fermi-Dirac one, with prop er mo di cation of the disp ersion relation b etween momentum

and energy.We de ne useful CP asymmetriesby:

(i)(P (i ! f ) P (i ! f ))

i;f

= ; (6:6)

2(i )

i 27

where the sum over i; f in the numerator implies the sum over the energies and avours for

10 0

the isosinglet quarks (or the iso doublet quarks) , while the sum i in the denominator acts

over avour and helicities and the factor 2 stems from summing over quarks and antiquarks.

In the case of zero temp erature, if we assume an equal incoming probability for all the

avours and helicities, and considering the asymmetry for one given energy (for instance the

most favorable one) the formula ( 6.6) simpli es to:

T =0 T =0

(!) = (!) ; (6:7)

CP CP

i;f

where

P (i ! f ) P (i ! f )

T =0

(6:8) (!) =

i;f

is the contribution to the asymmetry of a given pair of avours. N denotes the numb ers of

avours, 6. The colour factor has b een obviated as it will nally cancel b etween nominator

and denominator.

6.2 Numerical results

Consider rst the re ection asymmetries in the \down" quark sector. The numerical con-

tribution of the functions S (M ;M ), which contain the dep endence on the internal quark

jk l l

masses (l; l = u; c and t), is the following one

P P

5 5

0 0

S (M ;M )= 3:82 10 S (M ;M )= 1:84 10

1;2 l l 1;3 l l

P P

5 7

0 0

S (M ;M )= 3:68 10 S (M ;M )= 9:96 10

(6:9)

1;4 l l 2;3 l l

P P

7 7

0 0

S (M ;M )= 5:74 10 S (M ;M )= 6:81 10 ;

2;4 l l 3;4 l l

0 0

where the sums on l; l are constrained by the relation l = l +1 (mo d 3).

The following values have b een used for the particle masses: M =80:22GeV, m =

W u

5MeV, m = 10MeV, m = 200MeV, m =1:3GeV, m =4:7GeV, m = 120GeV. Our

d s c b t

numerical results are presented in Fig. 4. As exp ected the heavier quark masses dominate.

This is well understo o d from the analytical b ehaviour eq. ( 5.35). The dominant contribution

for charge 1=3 quarks is thus given by the s b pair. The two spikes in the gures corresp ond

to the thresholds for the external quark masses involved. Indeed, the analytical formulae in

eq. ( 5.35) display such a structure, with for instance in the case s ! b :

R L

m m

b =0; b = ( ; at E = m )

12 14 s

M M

2 2

(6:10 )

m m m m m m

s s

b b b b s

b b + ; at E = m

2 2 2

13 14 b

M M 2

M M m

W W

W W b

while for the complementary transition b ! s ,

R L

m m

b s

b =0; b =+ ; at E = m

12 14 s

(6:11 )

2 2

m m

m m m m

9 3

s s

b b s s

; b + ; at E = m : b

2 2

14 b 13

4m M 2 m M

M M

b W b W

W W

The reason not to add in the asymmetry the isosinglet and iso doublet quarks is the TCP' symmetry,

eq. ( 2.7), which implies that their incoming probability (i) are equal. The eventual e ect of sphalerons to

transform the CP asymmetry generated by the wall into a baryon numb er asymmetry,isanyhow di erent

for doublets and singlets. Wethus need to distinguish the asymmetry for doublets and singlets. 28

The total asymmetry, eq. ( 6.7), is clearly dominated by the s ! b contribution, with

R L

T =0 26

a maximum value (! m )=510 around the m threshold.

b b

Consider nowcharge 2=3 quarks. Strictly sp eaking, our computation is not a go o d

2 2 2 2

approximation for this sector as in eq.( 5.20) wehave neglected factors of m =M , m =M ,

i W f W

etc., which is not legitimate for the top quark. Nevertheless the typ e of GIM cancellations,

whichwe justi ed in terms of general arguments in Sect. 5, should still hold. In particular,

the analytical b ehaviour for the internal mass dep endence (d; s; b in this case), given in eq.

( 5.37), are sp ecially accurate b ecause all internal masses are really small in frontofM .

The exact numerical result for these quantities is

P P

10 10

0 0

S (M ;M )= 1:12 10 S (M ;M )= 2:37 10

1;3 l l 1;2 l l

P P

12 10

0 0

S (M ;M )= 7:37 10 S (M ;M )= 3:33 10

(6:12 )

2;3 l l 1;4 l l

P P

11 11

0 0

S (M ;M )= 3:45 10 ; S (M ;M )= 4:910

3;4 l l 2;4 l l

0 0

where the sums now run on l; l = d; s; b under the usual cyclicity constraint l = l +1

(mo d 3).

In this toy computation, the \up" typ e quarks contribution is exp ected to b e b e ab out

the same as the \down" typ e one: while the external mass dep endence gives a bigger factor,

the internal GIM cancellation pro duces the opp osite e ect. Applying for the sakeofthe

argument our analytical formulae to this sector, gives an external mass dep endence at the

5 5 6 6

dominant threshold m =M , while the internal GIM cancellation go es as m =M . The

q W q W

contributions of b oth typ es of quarks should not di er much. Using our formulae at face

T =0 25

value results in a dominant c ! t contribution of order (! m ) 2:510 .In

R L t

the present case b is not zero, which reinforces the e ect.

As previously discussed, in a complete computation, an external suppression factor

m m

( ) could disapp ear, leading to a large enhancement for certain avours (up to 13 orders

of magnitude for the d s pair, still insucient for the observed asymmetry), although this

maybevery optimistic as the pattern of unitarity triangles must p ersist.

7 Conclusions

Realistic mo dels of baryogenesis imply the inclusion of nite temp erature e ects. We con-

sider scenarios with a rst order phase transition. It is interesting to disentangle whether

temp erature e ects, other than the existence of the transition itself, are resp onsible for p os-

sible mo di cations of standard CP arguments. Wethus revert in a rst step to the analysis

of particle propagation at T =0.

The essential non p erturbative element is the wall separating the two phases of sp onta-

neous symmetry breaking. The propagation of any particle of the SM sp ectrum should b e

exactly solved in its presence. And this wehave done for a free fermion and a thin wall,

leading to a new Feynman propagator which replaces and generalizes the usual one. It con-

tains b oth massless and massive p oles, and new CP-even phases, present b oth for on-shell

and o -shell fermions. They stem from the tree-level re ection of fermions on the bubble

wall. This propagator can b e useful in physical pro cess other than baryogenesis, where a

rst order phase transition is relevant.

When electroweak lo ops are considered, a CP-asymmetry is found in the one-lo op re ec-

tion prop erties of quarks and antiquarks. The one-lo op self-energies cannot b e completely 29

renormalized away for on-shell fermions in the presence of a wall, as the latter acts as a

source of momenta. The asymmetry is then found at order in amplitude and requires

the interference of the mentioned CP-even phases with the CP-o dd CKM ones. The de-

sired e ect is thus present without any consideration of temp erature e ects in the quark

propagation.

A complete one-lo op SM calculation is b eyond the scop e of the present pap er. Wehave

instead p erformed a toy computation in which only the broken phase is considered inside

the lo ops, in a Lorentz covariantway. The result is many orders of magnitude b elow what

observation requires, and it can b e expressed in terms of two unitarity triangles. They show

the internal and external GIM cancellations and the interplaybetween CP-even and CP-

o dd phases. This pattern of triangles should survive in a complete computation, whichwe

leave for later publication. We argue that suppression factors in external masses could b e

absent then, due to the breaking of Lorentz invariance in the lo ops analysis, although the

quantitative enhancement is insucient.

With the exp ertise acquired in this pap er, we face in ref.[14] the physical nite temp er-

ature case, where we discuss the common building blo cks and new features of the scenario.

Acknowledgments

Weacknowledge Luis Alvarez-Gaume, Alvaro De Rujula, Jean-Marie Frere, Pilar Hernandez

and Jean-Pierre Leroy for many inspiring discussions. We are sp ecially indebted to Andy

Cohen, who to ok part in the early discussions. M.B. Gavela is indebted to ITP (Santa

Barbara) for hospitality in the nal p erio d of this work, where her researchwas supp orted

in part by the National Science Foundation under Grant No. PHY89-04035.

A Feynman b oundary conditions for the fermion prop-

agator

We prove here that the propagator derived in eqs. ( 4.16), ( 4.17) and ( 4.18) is a unique

solution of the inhomogeneous Dirac equation ( 4.10). In a rst step, the equivalence b etween

eqs. ( 4.16) and ( 4.17) will b e established. We then prove that the two`i" conventions are

indeed equivalent, as stated in section (4). Next we show, in the \space i" convention, that

the propagator satis es the inhomogeneous Dirac equation. Finally, in the (Feynman) \time

i" convention, we demonstrate that the Feynman b oundary conditions are indeed satis ed.

A.1 Equivalence b etween eqs. (4.16) and (4.17).

Let us consider the di erence b etween the right hand sides of eq. ( 4.17) and eq. ( 4.16).

The di erence b etween the rst lines of these equations is

! !

1 1 1 1 1 1

; (A:1)

f f

i f i f

i i

=q =q =q m =q m

q q q + i q i

z z

where the Feynman convention is understo o d in all all 1==q ', 1=(=q m)'s. 30

The di erence b etween the second and third lines of the same equations is

1 1 1 1

s s + s s : (A:2)

z 0 z 0

f i f i

=q =q =q m =q m

f i f i f i

q and q di er only in their z comp onent, implying that =q =q = (q q ). Using

i f f f i i

this equality, together with 1==q 1==q =1==q (=q =q )1==q , it is straightforw to show that

the sum of eqs. ( A.1) and ( A.2) vanishes, whence the equality of r.h.s. of ( 4.16) and of (

4.17).

A.2 Equivalence b etween the two\i" conventions.

2 2

The usual\time i" simply amounts to q ! q + i, while the \space i" convention has

0 0

b een de ned in Sec. 4. The di erence b etween the two\i" conventions has b een exhibited

in eq. ( A.7).

Consider the equality

1 1 m

= + :

f f f f

=q (=q m) =q m =q

Aplying it to eq. ( 4.17), we can separate the terms containing 1==q and those containing

1==q m. Let us dismiss momentarily the latter, which will b e discussed later.

f f f 2

It is p ossible to prove that the substitution of all 1==q factors in eq. ( 4.17 by2i =q (q ) (q

z 0

f 2

(q ) ) gives a vanishing result. Indeed, from

f f 2 f 2 0 f 2 f 2

=q (q ) (q (q ) )=sjq j (1 s ) (q ) (q (q ) ); (A:3)

0 z

z 0 z z 0 z

the third line in eq. ( 4.17) gives then a vanishing co ecient, since (1 s )(1 + s )= 0.

z z

Applying now in the second line the equality

" #

ms 1 s

0 z

(1 s ) 1 =1s ; (A:4)

z z

E + p 2

it follows that

f i

1 q + q 1

z z

(1 s )s = s(1 s ) = s(1 s ) ; (A:5)

z 0 z z

f f

2 i 2 i

(q ) (q ) + i q q + i

z z

f f f

where wehave used sq = q (since sq > 0by de nition of s, and q > 0 due to the (q )

0 0

z z z

factor). Finally, using once more eq. ( A.3), the second line gives

f f 2 f 2

=q (q ) (q (q ) ) ; (A:6)

z 0 z

q q + i

. whichobviously cancels the contribution from the rst line.

Let us now turn to the 1=(=q m) term. The equivalent of eq. ( A.7) is

f f

+ m =q =q + m

f f 2 f 2 2

= +2i (=q + m) (q ) (q (q ) m ): (A:7)

z 0 z

f f

2 2

2 2 2 2

q (q q (q + i) m ) + i m

z z

0 0 31

The pro of pro ceeds in a way similar to the preceding one but slightly more involved. It

can b e shown that

f 0

p +m

f f 2 f 2 2 0

(=q + m) (q ) (q (q ) m )=sjq j 1+ ; (A:8)

z 0 z

sjq j

where p q m . The last bracket in ( A.8) is twice a pro jector that plays the same =

role as 1 s in the preceding derivation.

Using

!" #

f 0

p +m ms 1 s

z 0 z

1+ 1 =0; (A:9)

sjq j E + p 2

it follows that the contribution from the second line of ( 4.17) vanishes. From

! " # !

f 0 f 0

p +m 1+s ms p +m

z z 0 z

z z

1+ 1+ = 1+ (A:10)

sjq j 2 E + p sjq j

0 0

it is p ossible to check that the third line cancels the contribution from the rst one, which

ends the pro of for the for 1=(=q m) term.

i i

The 1=(=q m) and 1==q may b e treated in a similar way leading to the anounced

conclusion: the quark propagator in eq. ( 4.17) has the same value whichever \i" convention

is used. From the equality derived in the preceding subsection, this applies also to eqs. (

4.16) and ( 4.18).

A.3 The inhomogeneous Dirac equation.

We prove here that the propagator veri es the inhomogeneous Dirac equation. Let us con-

sider the equation ( 4.10). Multiplying b oth sides by i , the Dirac op erator b ecomes

0 0

( )i=@ + ( )(i=@ m): (A:11)

z z

f f

We use the \space i" convention, so that 1==q has a p ole in the q complex plane b elow the

real axis. Its Fourier transform has then a p ole ab ove the real axis,

iq f 1 0

e dq )[i=@] : (A:12) / (

z z

Similarly,1=(=q m) has a p ole ab ove the real axis, leading to

iq f 1 0

e dq )[i=@ m] : (A:13) / (

z z

=q m

f f

As a consequence, the op erator ( A.11) applied to m==q (=q m) gives zero. It follows

that the second and third lines in eq. ( 4.17) vanish under the action of the Dirac op erator.

Its action on the rst line gives

1 1

f i

+ =2i (q q ): (A:14)

z z

f f

i i

q q + i q q i

z z

z z 32

This ends the pro of. The same argument holds when the Dirac equation is applied on

the right hand side of the propagator. In this case, it is more convenient to p erform the

analysis on eq. ( 4.16), sticking again to the \space i" convention.

A.4 Feynman b oundary conditions.

In the \time i" convention, it is trivial to pro of that the propagator veri es the Feynman

b oundary conditions. Indeed, the latter state that the p ositive frequencies are \retarded"

while the negative ones are \advanced". By de nition, the \time i" prepscription achieves

2 2

it, as it amounts to the replacement q ! q + i, which leads to p oles in the q complex

0 0

plane b elow (ab ove) the real axis when its real part is p ositive (negative).

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hep-ph/9402204. 34 A d(R)->s(L) A s(R)->d(L) E(GeV) 0.1 0.2 0.3 0.4 -33 8. 10 -34 -2. 10 -33 6. 10 -34 -33 -4. 10 4. 10 -34 -33 -6. 10 2. 10 E(GeV) -34 -8. 10 0.1 0.2 0.3 0.4

-26 A s(R)->b(L) A b(R)->s(L) E(GeV) 5. 10 -28 2 4 6 8 10 -26 -5. 10 4. 10 -27 -1. 10 -26 -27 3. 10 -1.5 10 -27 -26 -2. 10 2. 10 -27 -2.5 10 -26 -27 1. 10 -3. 10 E(GeV) -27 -3.5 10 2 4 6 8 10

A d(R)->b(L) E(GeV) -31 A b(R)->d(L) 2 4 6 8 10 5. 10 -29 -2. 10 -31 -29 4. 10 -4. 10 -31 -29 3. 10 -6. 10 -29 -31 -8. 10 2. 10 -28 -31 -1. 10 1. 10 -28 -1.2 10 E(GeV)

0.05 0.1 0.15 0.2

Figure 4: Asymmetries for incoming down quarks, as a function of energy. 35