Fermion Damping in a Fermion-Scalar Plasma

19 Pages·0.26 MB·English

by  D. Boyanovsky

Fermion Damping in a Fermion-Scalar Plasma
D. Boyanovsky(a), H.J. de Vega(b), D.-S. Lee(c), Y.J. Ng(d), S.-Y. Wang(a)
(a) Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh PA. 15260, U.S.A
(b) LPTHE, Universit´e Pierre et Marie Curie (Paris VI) et Denis Diderot (Paris VII), Tour 16, 1er. ´etage, 4, Place Jussieu,
75252 Paris, Cedex 05, France
(c) Department of Physics, National Dong Hwa University, Shoufeng, Hualien 974, Taiwan, R.O.C.
(d)Department of Physics and Astronomy, University of North Carolina, Chapel Hill, N.C. 27599, U.S.A.
(October 16, 1998)
In this article we study the dynamics of fermions in a fermion-scalar plasma. We begin by
obtainingtheeffectivein-mediumDiracequationinrealtimewhichisfullyrenormalizedandcausal
and leads to the initial value problem. For a heavy scalar we find the novel result that the decay
of the scalar into fermion pairs in the medium leads to damping of the fermionic excitations and
9 their in-medium propagation as quasiparticles. That is, the fermions acquire a width due to the
9 decayoftheheavierscalarinthemedium. Wefindthedampingratetolowest orderintheYukawa
9 coupling for arbitrary values of scalar and fermion masses, temperature and fermion momentum.
1 An all-order expression for the damping rate in terms of the exact quasiparticle wave functions is
n established. A kinetic Boltzmann approach to the relaxation of the fermionic distribution function
a confirms the damping of fermionic excitations as a consequence of the induced decay of heavy
J scalars in themedium. Alinearization of theBoltzmann equation near equilibrium clearly displays

3 therelationshipbetweenthedampingrateoffermionicmeanfieldsandthefermion interactionrate
1 to lowest order in the Yukawacoupling directly in real time.

2 12.15Ji,12.38.Mh
v
3
9
3 I. INTRODUCTION
0
1
8 The propagation of quarks and leptons in a medium of high temperature and/or density is of fundamental im-
9 portance in a wide variety of physically relevant situations. In stellar astrophysics, electrons and neutrinos play a
/ major role in the evolution of dense stars such as white dwarfs, neutron stars and supernovae [1]. In ultrarelativistic
h
heavy ion collisions and the possibility of formation of a quark-gluon plasma, electrons (and muons) play a very
p
- important role as clean probes of the early, hot stage of the plasma [2]. Furthermore, medium effects can enhance
p neutrino oscillations as envisaged in the Mikheyev-Smirnov-Wolfenstein (MSW) effect [3] and dramatically modify
e
the neutrino electromagnetic couplings [4]. The propagation of quarks during the non-equilibrium stages of the elec-
h
troweakphase transition is conjectured to be an essentialingredientfor baryogenesisatthe electroweakscale both in
:
v non-supersymmetric and supersymmetric extensions of the standard model [5].
i
X In-mediumpropagationisdramaticallydifferentfromthatinvacuum. Themediummodifiesthedispersionrelation
ofthe excitationsandintroducesawidthtothepropagatingexcitation[6–9]thatresultsindampingofthe amplitude
r
a of the propagating mode. In this article we focus on several aspects of propagation of fermionic excitations in a
fermion-scalar plasma:
• We begin by deriving the effective and fully renormalized Dirac equation in real time. This is achieved by
relating the expectation value of a fermionic field induced by an external fermionic source via linear response
to an initial value problem for the expectation value. This initial value problem is in terms of the effective real
time Dirac equation in the medium that is i) renormalized, ii) retarded and causal.
The necessity for a consistent Dirac equation in a medium has been recognized in the literature within the
contextof neutrino oscillations in the medium [10–13]. In particular,in references [10,11]a proposalfor a field-
theoretical treatment of neutrino oscillations in the medium starting from the Dirac equation was presented.
Inthis articlewe introduce the fully renormalized,in-medium effective Diracequationinreal timethatallowsa
more transparent study of damping and oscillations in time. There are definite advantages in such formulation
since the real time evolution is obtained at once from an initial value problem and allows a straightforward
identification of the damping rate.
• We apply the effective Dirac equation in a medium to study the real-time evolution of fermionic excitations
in a fermion-scalar plasma. Whereas the propagation of quarks and leptons in a QED or QCD plasma has
been studied thoroughly (see ref. [8,9] for details) a similar study for a scalar plasma has not been carried
1
out to the same level of detail. Recently some attention has been given to understanding the thermalization
time scales of bosonic and fermionic excitations in a plasma of gauge [14] and scalar bosons [15], furthermore
fermion thermalization is an important ingredient in models of baryogenesis mediated by scalars [16]. Most
of the studies of fermion thermalization focus on the mechanism of fermion scattering off the gauge quanta in
the heat bath and/or Landau damping in the hard thermal loop (HTL) resummation program [17]. Although
the scalar contribution to the fermionic self-energy to one loop has been obtained a long time ago [7], scant
attentionhasbeenpaidtoamoredetailedunderstandingofthe contributionfromthe scalardegreesoffreedom
to the fermionrelaxationandthermalization. As mentioned abovethis issue becomes of pressingimportance in
models of baryogenesisand more so in models in which the scalars carry baryon number [16].
Whereasthecontributionfromscalarstothefermionicthermalizationtimescale(dampingrate)hasbeenstudied
for massless chiral fermions [10,18], in this article we offer a detailed and general study of fermion relaxation
andthermalizationthrough the interactions with the scalarsin the plasma in realtime and for arbitraryvalues
of the scalar and fermion masses, temperature and fermion momentum. More importantly, we focus on a novel
mechanism of damping of fermionic excitations that occurs whenever the effective mass of the scalar particle
allows its kinematic decay into fermion pairs. This phenomenon only occurs in a medium and is interpreted as
aninduced decay of the scalarin the medium. It is a process different from collisions with particles in the bath
and Landau damping which are the most common processes that lead to relaxation and thermalization.
This process results in new thermal cuts in the fermionic self-energy and for heavy scalars,this cut results in a
quasiparticlepolestructureforthefermionandprovidesawidthforthefermionicquasiparticle. Theremarkable
andperhapsnon-intuitiveaspectofthisprocessisthatthedecayofthescalarresultsindampingofthefermionic
excitations and their propagation as quasiparticle resonances. Our real time analysis reveals that amplitude of
the ~k-mode of the expectation value of the fermion field decreases as e−Γkt while it oscillates with frequency
ω (k) which is determined by the position of the resonance.
p
Theeffective real-timeDiracequationinthe mediumallowsadirectinterpretationofthe dampingofthe quasi-
particle fermionic excitation and leads to a clear definition of the damping rate. By analyzing the quasiparticle
wave functions we obtain an all-order expression for the damping rate Γ and confirm and generalize recent
k
results for the massless chiral case [10,18].
• In order to provide a complementary understanding of the process of induced decay of the heavy scalars in the
medium and the resulting fermion damping, we study the kinetics of relaxation of the fermionic distribution
function via a Boltzmann equation to lowest order. Linearizing the Boltzmann equation near the equilibrium
distribution,weobtaintherelationbetweenthethermalizationrateforthedistributionfunctionintherelaxation
timeapproximation(linearizednearequilibrium)andthedampingratefortheamplitude ofthe fermionicmean
fields to lowest order in the Yukawa coupling. This analysis provides a real-time confirmation of the oft quoted
relation between the interaction rate (obtained from the Boltzmann kinetic equation in the relaxation time
approximation) and the damping rate for the mean field [7,9]. More importantly, this analysis reveals directly,
via a kinetic approachin real time how the process of induced decay of a heavy scalar in the medium results in
damping and thermalization of the fermionic excitations. A study of the relation between the interaction rate
and the damping rate has been presented recently for gauge theories within the context of the imaginary time
formulation [19]. Our results provide a real-time confirmation of those of reference [19] for the scalar case.
The article is organized as follows: in section II we obtain the effective in-medium Dirac equation in real time
starting from the linear response to an external Grassmann-valued source that induces a mean field. We obtain the
fully renormalized Dirac equation with the real time self-energy to one loop order by turning the linear response
problem into an initial value problem for the mean field. The renormalization aspects are addressed in detail in this
section. InsectionIIIwestudyindetailthestructureoftherenormalizedself-energyandestablishthepresenceofnew
cuts of thermal origin. We then note that for heavy scalars such that their decay into fermion pairs is kinematically
allowed, the fermionic pole becomes embedded in this thermal cut resulting in a quasiparticle (resonance) structure,
whichisanalyzedindetail. Thedecayrateisevaluatedinthenarrowwidthapproximation(justifiedforsmallYukawa
couplings) for arbitrary values of the scalar and fermion masses, temperature and fermion momentum.
In section IV we present a real time analysis of the evolution of the mean-fields. In this section we clarify the
differencebetweencomplexpolesandresonances(oftenmisunderstood). Thisanalysisrevealsclearlythattheinduced
decay of the scalar results in an exponential damping of the amplitude of the mean field and yields to a clear
identification of the damping rate bypassing the conflicting definitions of the damping rate offered in the literature.
An analysis of the structure of the self-energy and an interpretation of the exact quasiparticle spinor wave functions
allowsustoprovideanall-order expressionforthe dampingrateofthefermionicmeanfields. InsectionVwepresent
an analysis of the evolution of the distribution functions in real time by obtaining a Boltzmann kinetic equation
2
for the spin-averaged fermionic distribution function. In the linearized approximation near equilibrium (relaxation
time approximation), we clarify to lowest order the relation between the damping rate of the quasiparticle fermionic
excitationsandthethermalizationrate(linearized)ofthedistributionfunction,directlyinrealtime. Intheconclusions
we summarize our results and suggest new avenues including the contribution from gauge fields. In this section we
also assess the importance of the scalar contribution in theories with gauge and scalar fields.
II. EFFECTIVE DIRAC EQUATION IN THE MEDIUM
Asmentionedintheintroduction,whereasthedampingofcollectiveandquasiparticleexcitationsviatheinteractions
with gauge bosons in the medium has been the focus of most attention, understanding of the influence of scalars has
not been pursued so vigorously.
Although we are ultimately interested in studying the damping of fermionic excitations in a plasma with scalars
andgaugefieldswithinthe realmofelectroweakbaryogenesisineitherthe StandardModelorgeneralizationsthereof,
wewillbeginby consideringonlythe couplingofamassiveDiracfermiontoascalarviaasimpleYukawainteraction.
The model dependent generalizations of the Yukawa couplings to particular cases will differ quantitatively in the
details ofthe groupstructure but the qualitative featuresofthe effective Dirac equationin the medium aswellas the
kinematics of the thermal cuts that lead to damping of the fermionic excitations will be rather general.
We consider a Dirac fermion with the bare mass M coupled to a scalar with the bare mass m via a Yukawa
0 0
coupling. The bare fermion mass could be the result of spontaneous symmetry breaking in the scalar sector, but for
the purposes of our studies we need not specify its origin.
The Lagrangiandensity is given by
1 1
L=Ψ¯(i6∂−M )Ψ+ ∂ φ∂µφ− m2φ2−L [φ]−y Ψ¯φΨ+η¯Ψ+Ψ¯η+jφ , (2.1)
0 2 µ 2 0 I 0
wherey isthebareYukawacoupling. Theself-interactionofthescalarfieldaccountedforbythetermL [φ]neednot
0 I
be specified to lowestorder. The η and j are the respective external fermionic and scalarsources that are introduced
inorderto provideaninitial value problemforthe effectiveDirac equation. We nowwrite the barefields andsources
Ψ, φ, η and j in terms of the renormalized quantities (hereafter referred to with a subscript r) by introducing the
renormalizationconstants and counterterms:
1/2 1/2 −1/2 −1/2
Ψ=Z Ψ , φ=Z φ , η =Z η , j =Z j ,
ψ r φ r ψ r φ r
y =y Z1/2Z /Z , m2 = δ +m2 /Z , M =(δ +M)/Z . (2.2)
0 φ ψ y 0 m φ 0 M ψ
With the above definitions, the L can be express(cid:0)ed as: (cid:1)
1 1
L=Ψ¯ (i6∂−M)Ψ + ∂ φ ∂µφ − m2φ2−Lr[φ ]−yΨ¯ φ Ψ +η¯ Ψ +Ψ¯ η +j φ
r r 2 µ r r 2 r I r r r r r r r r r r
1 1
+ δ ∂ φ ∂µφ − δ φ2+Ψ¯ (iδ 6∂−δ )Ψ −yδ Ψ¯ φ Ψ +δLr[φ ] , (2.3)
2 φ µ r r 2 m r r ψ M r y r r r I r
where m and M are the renormalized masses, and y is the renormalized Yukawa coupling. The terms with the
coefficients
δ =Z −1 , δ =Z −1 ,
ψ ψ φ φ
δ =M Z −M , δ =m2Z −m2 , (2.4)
M 0 ψ m 0 φ
δ =Z −1
y y
and δLr are the counterterms to be determined consistently in the perturbative expansion by choosing a renormal-
I
ization prescription. As it will become clear below this is the most natural manner for obtaining a fully renormalized
Dirac equation in a perturbative expansion.
The dynamics of expectation values and correlation functions of the quantum field is obtained by implementing
the Schwinger-Keldysh closed-time-path formulation of non-equilibrium quantum field theory [20–23]. The main
ingredient in this formulation is the real time evolution of an initially prepared density matrix and its path integral
representation. It requires a path integral defined along a closed time path contour. This formulation has been
described elsewhere within many different contexts and we refer the reader to the literature for details [20–23].
3
Our goal is to understand the non-equilibrium relaxationaldynamics of the inhomogeneous fermionic mean fields
ψ(~x,t)≡hΨ (~x,t)i
r
from their initial states in the presence of the fermion-scalar medium.
This statement requires clarification. In states of definite fermion number (either zero or finite temperature) the
expectation value of the fermion field must necessarily vanish. However, in order to understand the non-equilibrium
dynamics, we prepare the system by coupling an externalGrassmansource to the fermionic field in the Hamiltonian.
This source creates a coherent state of fermions which is a superposition of states with different fermion number. As
will be discussed in detail below, this source term is switched-on adiabatically from t = −∞ in the full interacting
theory up to a time t and switched-off at t = t . The resulting fermionic coherent state at t = t does not have a
0 0 0
definitefermionnumberandtheexpectationvalueofthefieldoperatorinthisstateisnon-vanishing. Afterthesource
is switched-off at time t the expectation value evolves in time with the full interacting Hamiltonian in absence of
0
sources. This is the standard approach to linear response, in this approach the source term couples to a given mode
of wavevector k of the fermionic fields, thus displacing this degree of freedom off-equilibrium. The other modes are
assumed to be remain in thermal equilibrium.
Since we are interested in real time correlation functions and the initial density matrix is assumed to be that of
thermal equilibrium at initial temperature T = 1/β with respect to the free (quadratic) Lagrangian, only the real
time branches,forwardandbackwardsarerequired. The contributionfromthe imaginarytime branchcorresponding
to the thermal component of the density matrix cancels in the connected non-equilibriumexpectation values [21–23].
The effective non-equilibrium Lagrangiandensity that enters in the contour path integral is therefore given by
L =L Ψ+,Ψ¯+,φ+ −L Ψ−,Ψ¯−,φ− . (2.5)
non-eq r r r r r r
Fields with (+) and (−) superscripts are defi(cid:2)ned respectiv(cid:3)ely o(cid:2)n the forwar(cid:3)d (+) and backwards (−) time contours
and are to be treated independently. The external sources are the same for both branches.
The essential ingredients for perturbative calculations are the following real time Green’s functions [23]:
• Scalar Propagators
G++(t,t′)=G>(t,t′)Θ(t−t′)+G<(t,t′)Θ(t′−t),
~k ~k ~k
G−−(t,t′)=G>(t,t′)Θ(t′−t)+G<(t,t′)Θ(t−t′),
~k ~k ~k
G+−(t,t′)=−G<(t,t′),
~k ~k
G−+(t,t′)=−G>(t,t′),
~k ~k
G>(t,t′)=i d3xe−i~k·~x hφ (~x,t)φ (~0,t′)i
~k r r
Z
= i (1+n )e−iωk(t−t′)+n eiωk(t−t′) ,
k k
2ω
k
h i
G<(t,t′)=i d3xe−i~k·~x hφ (~0,t′)φ (~x,t)i
~k r r
Z
= i n e−iωk(t−t′)+(1+n )eiωk(t−t′) ,
k k
2ω
k
h i
1
ω = ~k2+m2 , n = . (2.6)
k k eβωk −1
q
• Fermionic Propagators(Zero chemical potential)
S++(t,t′)=S>(t,t′)Θ(t−t′)+S<(t,t′)Θ(t′−t),
~k ~k ~k
S−−(t,t′)=S>(t,t′)Θ(t′−t)+S<(t,t′)Θ(t−t′),
~k ~k ~k
S+−(t,t′)=−S<(t,t′),
~k ~k
S−+(t,t′)=−S>(t,t′),
~k ~k
S>(t,t′)=−i d3xe−i~k·~x hΨ (~x,t)Ψ¯ (~0,t′)i
~k r r
Z
4
=− i (6k+M)(1−n¯ )e−iω¯k(t−t′)+γ (6k−M)γ n¯ eiω¯k(t−t′) ,
k 0 0 k
2ω¯
k
h i
S<(t,t′)=i d3xe−i~k·~x hΨ¯ (~0,t′)Ψ (~x,t)i
~k r r
Z
= i (6k+M)n¯ e−iω¯k(t−t′)+γ (6k−M)γ (1−n¯ )eiω¯k(t−t′) ,
k 0 0 k
2ω¯
k
h i
1
ω¯ = ~k2+M2 , n¯ = . (2.7)
k k eβω¯k +1
q
The perturbative evaluation of correlation functions proceeds as usual, but now the Feynman rules involves two
types of vertices with opposite signs and the four different non-equilibrium propagators. The symmetry factors are
the usual ones.
These free propagators(2.6) and (2.7) are thermal since the initial state is assumed to be in thermal equilibrium.
A. In-medium Dirac equation from linear response
Consider the fermionic mean field obtained as the linear response to the externally applied (Grassmann-valued)
source η :
r
∞
hΨ+(~x,t)i=hΨ−(~x,t)i=ψ(~x,t)=− dt′d3x′S (~x−~x′,t−t′)η (~x′,t′) , (2.8)
r r ret r
Z−∞
with the exact retarded Green’s function
S (~x−~x′,t−t′)= S>(~x−~x′,t−t′)−S<(~x−~x′,t−t′) Θ(t−t′)
ret
=−ih{Ψ (~x,t),Ψ¯ (~x′,t′)}iΘ(t−t′) , (2.9)
(cid:2) r r (cid:3)
where the expectation values are in the full interacting theory but with vanishing sources. An initial value problem
is obtained by considering that the external fermionic sources are adiabatically switched on in time from t → −∞
therebyinducinganexpectationvalueofthefermionicfields,andswitching-offthesourcetermatsometimet . Then
0
for t > t this expectation value or mean field will evolve in the absence of a source and will relax because of the
0
interactions. The evolution for t>t is an initial value problem, since the source term was used to prepare an initial
0
state and switched off to let this state evolve in time. This initial value problem can therefore be formulated by
choosing the source term to be of the form
η (~x,t)=η (~x)eǫt Θ(t −t) r, ǫ→0+ . (2.10)
r r 0
In what follows we choose t =0 for convenience. The adiabatic switching-onof the source induces an expectation
0
value that is dressed adiabatically by the interaction. The retarded and the equilibrium nature of S (~x−~x′,t−t′)
ret
(which depends on the time difference) and the form of the source (2.10) guarantee that
ψ(~x,t=0)=ψ (~x) ,
0
ψ˙(~x,t<0)=0 ,
where ψ (~x) is determined by η (~x) (or vice versa, the initial conditions for the condensates ψ (~x) can be used to
0 r 0
find η (~x)). This can be seen by taking the time derivative of ψ(~x,t) in eq. (2.8), using the fact that the retarded
r
propagator depends on the time difference, integrating by parts, and using the form of the external source and the
retarded nature of the propagator.
In order to relate this linear response problem to the initial value problem for the dynamical equation of the mean
field, let us consider the (integro-) differential operator O which is the inverse of S (~x−~x′,t−t′) so that
(~x,t) ret
O ψ(~x,t)=−η (~x,t) , ψ(~x,t=0)=ψ (~x) , ψ˙(~x,t<0)=0 , (2.11)
(~x,t) r 0
where the source is given by eq. (2.10).
It is at this stage where the non-equilibrium formulation provides the most powerful framework. The real-time
equations of motion for the mean fields can be obtained via the tadpole method [21–23], which automatically leads
5
to a retarded and causal initial value problem for the expectation value of the field. The implementation of this
methodis asfollows. Letus introducethe inhomogeneousmeanfields ψ(~x,t)=hΨ(~x,t)iandψ¯(~x,t)=hΨ¯(~x,t)i. The
dynamics of these fermionic mean fields in the plasma can be analyzed by treating ψ(~x,t) and ψ¯(~x,t) as background
fields, i.e., the expectation values of the corresponding fields in the non-equilibrium density matrix, by expanding
the non-equilibrium Lagrangian density about these mean fields. Therefore, we write the full quantum fields as the
c-number expectation values (mean fields) and quantum fluctuations about them:
Ψ±(~x,t)=ψ(~x,t)+χ±(~x,t) , Ψ¯±(~x,t)=ψ¯(~x,t)+χ¯±(~x,t) ,
r r
ψ(~x,t)=hΨ±(~x,t)i , ψ¯(~x,t)=hΨ¯±(~x,t)i , (2.12)
r r
where the expectation values of the field operators are taken in the time-evolved density matrix.
Theequationsofmotionforthemeanfieldscanbeobtainedtoanyorderintheperturbativeexpansionbyimposing
the requirement that the expectation value of the quantum fluctuations in the time evolved density matrix vanishes
identically. This is referred to as the tadpole equation [21–23] which follows from eqs. (2.12):
hχ±i=0 , hχ¯±i=0 . (2.13)
The procedure consists in treating the linear terms in χ± as well as the non-linearities in perturbation theory and
keeping only the connected irreducible diagrams, as in usual perturbation theory. The equations obtained via this
procedure are the equations of motion obtained by variations of the non-equilibrium effective action [20–23].
ByapplyingthetadpolemethodandtakingspatialFouriertransform,wefindtheeffectivereal timeDiracequation
for the~k-mode of the expectation value of the fermion:
∂ ∂ t
iγ −~γ·~k−M +δ iγ −~γ·~k −δ ψ (t)+ dt′ Σ (t−t′)ψ (t′)=−η (t) ,
0∂t ψ 0∂t M ~k ~k ~k r,~k
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21) Z−∞
where Σ (t−t′) is the fermion self-energy and
~k
ψ (t)≡ d3xe−i~k·~x ψ(~x,t) .
~k
Z
Using the non-equilibrium Green’s functions (2.6) and (2.7), we find to one loop order, that Σ (t−t′) is given by
~k
Σ (t−t′)=iγ Σ(0)(t−t′)+~γ·~k Σ(1)(t−t′)+Σ(2)(t−t′) , (2.14)
~k 0 ~k ~k ~k
where
d3q ω¯
Σ(0)(t−t′)=y2 q ×
~k (2π)32ω ω¯
Z k+q q
cos[(ω +ω¯ )(t−t′)](1+n −n¯ )+cos[(ω −ω¯ )(t−t′)](n +n¯ ) ,
k+q q k+q q k+q q k+q q
h d3q 1 ~k·~q i
Σ(1)(t−t′)=y2 ×
~k (2π)32ω ω¯ k2
Z k+q q
sin[(ω +ω¯ )(t−t′)](1+n −n¯ )−sin[(ω −ω¯ )(t−t′)](n +n¯ ) ,
k+q q k+q q k+q q k+q q
h d3q M i
Σ(2)(t−t′)=y2 ×
~k (2π)32ω ω¯
Z k+q q
sin[(ω +ω¯ )(t−t′)](1+n −n¯ )−sin[(ω −ω¯ )(t−t′)](n +n¯ ) ,
k+q q k+q q k+q q k+q q
h i
with ωk+q = (~k+~q)2+m2 andnk+q =(eβωk+q−1)−1 being,respectively,the energyandthe distribution function
for scalars ofqmomentum~k+~q.
As mentioned before, the source is takento be switchedon adiabatically fromt=−∞ andswitched off at t=0 to
provide initial conditions
ψ (t=0)=ψ (0) , ψ˙ (t<0)=0 . (2.15)
~k ~k ~k
Defining σ (t−t′) as
~k
6
d
σ (t−t′)=Σ (t−t′) (2.16)
dt′ ~k ~k
and imposing that η (t>0)=0, we obtain the equation of motion for t>0
r,~k
∂ ∂ t
iγ −~γ·~k−M +δ iγ −~γ·~k +σ (0)−δ ψ (t)− dt′ σ (t−t′)ψ˙ (t′)=0 . (2.17)
0∂t ψ 0∂t ~k M ~k ~k ~k
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21) Z0
The equation of motion (2.17) can be solved by Laplace transform as befits an initial value problem. The Laplace
transformed equation of motion is given by
iγ s−~γ·~k−M +δ iγ s−~γ·~k −δ +σ (0)−sσ˜ (s) ψ˜ (s)
0 ψ 0 M ~k ~k ~k
h (cid:16) (cid:17) i
= iγ +iδ γ −σ˜ (s) ψ (0) , (2.18)
0 ψ 0 ~k ~k
where ψ˜ (s) and σ˜ (s) are the Lap(cid:2)lace transforms of ψ(cid:3) (t) and σ (t) respectively:
~k ~k ~k ~k
∞ ∞
ψ˜ (s)≡ dte−st ψ (t) , σ˜ (s)≡ dte−st σ (t).
~k ~k ~k ~k
Z0 Z0
B. Renormalization
Beforeproceedingwiththesolutionoftheaboveequation,weaddressthe issueofthe renormalizationbyanalyzing
the ultraviolet divergences of the kernels. As usual the ultraviolet divergences are those of zero temperature field
theory, since the finite temperature distribution functions are exponentially suppressed at large momenta. Therefore
the ultraviolet divergences are obtained by setting to zero the bosonic and fermionic occupation numbers.
With eq. (2.16), σ (t−t′) can be written as
~k
σ (t−t′)=iγ σ(0)(t−t′)+~γ·~k σ(1)(t−t′)+σ(2)(t−t′) . (2.19)
~k 0 ~k ~k ~k
A straightforwardcalculation leads to
σ(0)(0)=0 , σ(1)(0)=− y2 ln Λ +finite , σ(2)(0)= y2M ln Λ +finite ,
~k ~k 16π2 K ~k 8π2 K
σ˜(0)(s)=− y2 ln Λ +fin(cid:0)ite(cid:1), σ˜(1)(s)=finite , σ˜(2)(s)=(cid:0) fi(cid:1)nite ,
~k 16π2 K ~k ~k
(cid:0) (cid:1)
where σ˜(0)(s), σ˜(1)(s) and σ˜(2)(s) are the Laplace transform of σ(0)(t), σ(1)(t) and σ(2)(t) respectively, Λ is an
~k ~k ~k ~k ~k ~k
ultraviolet momentum cutoff and K is an arbitrary renormalizationscale. Therefore the counterterms δ and δ are
ψ M
chosen to be given by
y2 Λ y2M Λ
δ =− ln +finite , δ = ln +finite , (2.20)
ψ 16π2 K M 8π2 K
(cid:18) (cid:19) (cid:18) (cid:19)
and the respective kernels are rendered finite, i.e.,
σ˜ (s)−iδ γ =σ˜ (s)=finite , σ (0)−δ ~γ·~k−δ =σ (0)=finite . (2.21)
~k ψ 0 r,~k ~k ψ M r,~k
The finite parts of the counterterms in eq. (2.20) are fixed by prescribing a renormalization scheme. There are
two important choices of counterterms: i) determining the counterterms from an on-shell condition, including finite
temperature effects, and ii) determining the counterterms from a zero temperature on-shell condition. Obviously
thesechoicesonlydifferbyfinitequantities,howeverthesecondchoiceallowsustoseparatethe dressingeffectsofthe
medium from those in the vacuum. For example by choosing to renormalize with the zero temperature counterterms
on-shell, the pole in the particle propagator will have unit residue at zero temperature; however in the medium, the
residue atthe finite temperature poles (or the positionofthe resonances)are finite,smaller than one anddetermined
solely by the properties of the medium. Thus the formulationof the initial value problem as presentedhere yields an
unambiguous separationof the vacuum and in-medium renormalization effects.
Hence we obtain the renormalized effective Dirac equation in the medium and its initial value problem
7
iγ s−~γ·~k−M +Σ˜ (s) ψ˜ (s)= iγ −σ˜ (s) ψ (0) , (2.22)
0 r,~k ~k 0 r,~k ~k
hΣ˜ (s)=σ (0)−sσ˜ (s)i, h i
r,~k r,~k r,~k
with σ (0) and σ˜ (s) the fully renormalized kernels (see eq. (2.21)), and Σ˜ (s) the Laplace transform of the
r,~k r,~k r,~k
renormalized fermion self-energy which can be written in its most general form as follows
Σ˜ (s)=iγ sε˜(0)(s)+~γ·~k ε˜(1)(s)+M ε˜(2)(s) . (2.23)
r,~k 0 ~k ~k ~k
The solution to eq. (2.22) is
1
ψ˜ (s)= 1+S(s,~k) ~γ·~k+M −Σ˜ (0) ψ (0) , (2.24)
~k s r,~k ~k
h (cid:16) (cid:17)i
where
−1
S(s,~k)= iγ s−~γ·~k−M +Σ˜ (s)
0 r,~k
hiγ s(1+ε˜(0)(s))−~γ·~k(1−i ε˜(1)(s))+M(1−ε˜(2)(s))
= 0 ~k ~k ~k (2.25)
−s2(1+ε˜(0)(s))2−k2(1−ε˜(1)(s))2−M2(1−ε˜(2)(s))2
~k ~k ~k
is the fermionpropagatorintermsofthe Laplacevariables. The squareofthe denominatorof(2.25)is recognizedas
det iγ s−~γ·~k−M +Σ˜ (s)
0 r,~k
h i
Thereal-timeevolutionofψ (t)isobtainedbyperformingtheinverseLaplacetransformalongtheBromwichcontour
~k
inthe complexs-planeparallelto the imaginaryaxisandto the rightofallsingularitiesofψ˜ (s). Thereforeto obtain
~k
the real time evolution we must first understand the singularities of the Laplace transform in the complex s-plane.
III. STRUCTURE OF THE SELF-ENERGY AND DAMPING PROCESSES
To onelooporder,the Laplacetransformofthe components ε˜(i)(s)ofthe fermionself-energyΣ˜ (s) (see eq.(2.23))
~k ~k
can be written as dispersion integrals in terms of spectral densities ρ(i)(k )
~k 0
ε˜(0)(s) ρ(0)(k )
~k 1 ~k 0 δψ
 εε˜˜~~(k(k21))((ss))=Z dk0s2+k02  kk00 ρρ~~(k(k21))((kk00))+ −−δδMMψ  , (3.1)
with the one-loop spectraldensitiesgiven by the expressions   
d3q ω¯
ρ(0)(k )=y2 q ×
~k 0 (2π)32ω ω¯
Z k+q q
[δ(k −ω −ω¯ )(1+n −n¯ )+δ(k −ω +ω¯ )(n +n¯ )] ,
0 k+q q k+q q 0 k+q q k+q q
d3q 1 ~k·~q
ρ(1)(k )=y2 ×
~k 0 (2π)32ω ω¯ k2
k+q q
Z
[δ(k −ω −ω¯ )(1+n −n¯ )−δ(k −ω +ω¯ )(n +n¯ )] ,
0 k+q q k+q q 0 k+q q k+q q
d3q 1
ρ(2)(k )=y2 ×
~k 0 (2π)32ω ω¯
Z k+q q
[δ(k −ω −ω¯ )(1+n −n¯ )−δ(k −ω +ω¯ )(n +n¯ )] . (3.2)
0 k+q q k+q q 0 k+q q k+q q
The analytic continuation of the self-energy and its components ε˜(i)(s) in the complex s-plane are given by
Σ˜ (s=−iω±0+)=Σ (ω)±iΣ (ω) ,
r,~k R,k I,k
ε˜(i)(s=−iω±0+)=ε(i) (ω)±iε(i)(ω) , (3.3)
~k R,~k I,~k
8
where the real parts are even functions of ω and given by
ε(0) (ω) ρ(0)(k )
R,~k 1 ~k 0 δψ
 εε(R(R12,,))~~kk((ωω))=Z dk0P(cid:18)k02−ω2(cid:19) kk00ρρ~~(k(k12))((kk00)) +−−δδMMψ  , (3.4)
and the imaginary parts are odd functions of ω and given by   
π
ε(0)(ω)= sgn(ω) ρ(0)(|ω|)+ρ(0)(−|ω|) ,
I,~k 2|ω| ~k ~k
π h i
ε(1)(ω)= sgn(ω) ρ(1)(|ω|)−ρ(1)(−|ω|) ,
I,~k 2 ~k ~k
π h i
ε(2)(ω)= sgn(ω) ρ(2)(|ω|)−ρ(2)(−|ω|) . (3.5)
I,~k 2 ~k ~k
h i
The denominator of the analytically continued fermion propagatoreq. (2.25) can be written in the compact form
ω2−ω¯2+Π(ω,~k)
k
with ω¯ = ~k2+M2 and
k
p
Π(ω,~k)=2 ω2ε(0)(ω)+k2ε(1)(ω)+M2ε(2)(ω)
~k ~k ~k
+ω2h[ε(0)(ω)]2−k2 [ε(1)(ω)]2−M2 [ε(2i)(ω)]2 . (3.6)
~k ~k ~k
We recognize that the lowest order term of this effective self-energy can be written in the familiar form [7–9]
1
Π(ω,~k)=2 ω2ε(0)(ω)+k2ε(1)(ω)+M2ε(2)(ω) = Tr[(6k+M)Σ˜ (s=−iω+0+)] , (3.7)
~k ~k ~k 2 ~k
h i
but certainly not the higher order terms.
The imaginary part of Π on-shell will be identified with the damping rate (see below). The expression given by
eq. (3.7) leads to the familiar form of the damping rate, but we point out that eq. (3.7) is a lowest order result. The
full imaginary part must be obtained from the full function Π(ω,~k) and the generalizationto all orders will be given
in a later section below.
(i)
ForfixedM andm,theδ-functionconstraintsinthespectraldensitiesρ (ω)canonlybesatisfiedforcertainranges
~k
ofω. Since the imaginarypartsareoddfunctions ofω we onlyconsiderthe caseofpositive ω. The δ(|ω|−ω −ω¯ )
k+q q
has supportonly for |ω|> k2+(m+M)2 andcorrespondsto the normaltwo-particlecuts that arepresentatzero
temperature corresponding to the process ψ → φ+ψ. These cuts (both for positive and negative ω) do not give a
p
contribution to the imaginary part on-shell for the fermion.
The terms proportional to n +n¯ give the following contribution to the lowest order effective self-energy:
k+q q
d3q n +n¯
Π (ω,~k)=πy2sgn(ω) k+q q |ω|ω¯ −~k·~q−M2 δ(|ω|−ω +ω¯ )
I (2π)3 2ω¯ ω q k+q q
Z q k+q h(cid:16) (cid:17)
+ |ω|ω¯ +~k·~q+M2 δ(|ω|+ω −ω¯ ) . (3.8)
q k+q q
(cid:16) (cid:17) i
The first delta function determines a cut in the region 0 < |ω| < k2+(m−M)2 and originates in the physical
process φ → ψ+ψ¯ whereas the second delta function determines a cut in the region 0 < |ω| < k and originates in
p
the process φ+ψ → ψ. Whereas the first cut originates in the process of decay of the scalar into fermion pairs,
the second cut for ( ω2 < k2) is associated with Landau damping. Both delta functions restrict the range of the
integrationvariableq (see below). For m>2M the scalarcandecay on-shellinto a fermionpair, andin this case the
fermion pole is embedded in the cut 0<|ω|< k2+(m−M)2 becoming a quasiparticle pole and only the first cut
contributes to the quasiparticle width. This is a remarkable result, the fermions acquire a width through the induced
p
decay of the scalar in the medium. This process only occurs in the medium (obviously vanishing at T = 0) and its
originis very different from either collisionalbroadeningor Landaudamping. A complementaryinterpretationof the
origin of this process as a medium induced decay of the scalars into fermions and the resulting quasiparticle width
for the fermion excitation will be highlighted in section V within the kinetic approachto relaxation.
9
The width is obtained to lowest order from Π (ω ,~k), and the on-shell delta function is recognized as the energy
I k
conservation condition for the decay φ → ψ+ψ¯ of the heavy scalar on-shell. To lowest order we find the following
expressionfor Π (ω¯ ,~k)for arbitraryscalarand fermionmasseswith m>2M andarbitraryfermionmomentumand
I k
temperature:
d3q ω¯ ω¯ −~k·~q−M2
Π (ω¯ ,~k)=πy2 k q (n +n¯ )δ(ω¯ +ω¯ −ω ) ,
I k (2π)3 2ω¯ ω k+q q k q k+q
Z q k+q
y2m2T 4M2 1−e−β(ω¯q+ω¯k) ω¯q2∗
= 1− ln , (3.9)
16πk m2 1+e−βω¯q
(cid:18) (cid:19) (cid:20) (cid:21)(cid:12)(cid:12)ω¯q1∗
(cid:12)
where q∗ and q∗ are given by (cid:12)
1 2
m2 2M2 4M2
q∗ = k 1− − (k2+M2) 1− ,
1 2M2 (cid:12) m2 s m2 (cid:12)
(cid:12) (cid:18) (cid:19) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
m2 (cid:12) 2M2 4M2 (cid:12)
q∗ = (cid:12)k 1− + (k2+M2) 1− (cid:12) , (3.10)
2 2M2 " (cid:18) m2 (cid:19) s (cid:18) m2 (cid:19)#
with |~q|∈(q∗,q∗) being the support of δ(ω¯ −ω +ω¯ ).
1 2 k k+q q
IV. REAL TIME EVOLUTION
The real time evolution is obtained by performing the inverse Laplace transform as explained in section II. This
requires analyzing the singularities of ψ˜ (s) given by eq. (2.25) in the complex s-plane. It is straightforward to see
~k
that the putative pole at s = 0 has vanishing residue, therefore the singularities are those arising from the inverse
fermion propagator S(s,~k). If the fermionic pole is away from the multiparticle cuts, the singularities are: i) the
isolated fermion poles at s = −iω with ω the position of the isolated (complex) poles (corresponding to stable
p p
fermionicexcitations),andii)themultiparticlecutsalongtheimaginaryaxiss=−iω for0<|ω|< k2+(m−M)2
and |ω|> k2+(m+M)2.
p
The Laplacetransformis performedby deforming the contour,circlingthe isolatedpoles and wrappingaroundthe
p
cuts. When the scalarparticle can decay into fermion pairs,i.e., m>2M,the fermion pole is embedded in the lower
cut and we must find out if it becomes a complex pole in the physical sheet (the domain of integration) or moves off
the physical sheet.
For m < 2M the fermion pole at ω = ω is real and the fermion mean field oscillates for late times with constant
p
amplitude and frequency ω .
p
A. m>2M: complex poles or resonances?
When m > 2M the fermion pole is embedded in the cut 0 < |ω| < k2+(m−M)2 and the pole becomes
complex. The position of the complex poles are obtained from the zeros of ω2 −ω¯2 +Π(ω,~k) in the analytically
p k
continued fermion propagator for ω = ω (k)−iΓ with ω being the real part of the complex pole. For Γ ≪ ω
p k p k p
(narrowwidthapproximation)andwiththeexpressionsforthediscontinuitiesinthephysicalsheetgivenbyeq.(3.3),
the equation that determines the position of the complex pole is given by the solution of the following equation
(ω −iΓ )2−ω¯2+Π (ω ,~k)−isgn(Γ )Π (ω ,~k)=0 , (4.1)
p k k R p k I p
where we have used the narrow width approximation. To lowest order, the real and imaginary parts of this equation
become
ω2−ω¯2+Π (ω ,~k)=0 , (4.2)
p k R p
Π (ω ,~k)
I p
Γ =−sgn(Γ ) , (4.3)
k k
2ω
p
and the lowest order solution of eq. (4.2) is given by
10