Dark Matter and Dark Energy via Non-Perturbative (Flavour) Vacua

0.61 MB·English

by  Walter Tarantino

Dark Matter and Dark Energy via Non-Perturbative (Flavour) Vacua
Walter Tarantino∗
King’s College London, Department of Physics, Strand, London WC2R 2LS, UK.
A non-perturbative field theoretical approach to flavour physics (Blasone-Vitiello formalism) has
been shown to imply a highly non-trivial vacuum state. Although still far from representing a
satisfactory framework for a coherent and complete characterization of flavour states, in recent
years the formalism has received attention for its possible implications at cosmological scales. In a
previouswork,weimplementedtheapproachonasimplesupersymmetricmodel(freeWess-Zumino),
withflavourmixing,whichwasregardedasamodelforfreeneutrinosandsneutrinos. Theresulting
effectivevacuum(calledflavourvacuum)wasfoundtobecharacterizedbyastrongSUSYbreaking.
Inthispaperweexplorethephenomenologyofthemodelandwearguethattheflavourvacuumisa
consistentsourceforbothDarkEnergy(thankstothebosonicsectorofthemodel)andDarkMatter
(viathefermionicone). Quiteremarkably,besidestheparametersconnectedwithneutrinophysics,
in this model no other parameters have been introduced, possibly leading to a predictive theory
2
of Dark Energy/Matter. Despite its oversimplification, such a toy model already seems capable
1
to shed some light on the observed energy hierarchy between neutrino physics, Dark Energy and
0
Dark Matter. Furthermore, we move a step forth in the construction of a more realistic theory, by
2
presentinganovelapproachforcalculatingrelevantquantitiesandhenceextendingsomeresultsto
n interactive theories, in a completely non-perturbative way.
a
J
PACSnumbers: 14.60.Pq,95.36.+x,95.35.+d
1
1
INTRODUCTION by the theory. Such a flavour vacuum (which can be
] regarded as a vacuum condensate) represents the physi-
h
p NeutrinoFlavourOscillationisnowadaysafairlywell- cal state with no (flavour) particles in it. Despite being
- established fact, thanks to a wide range of experimen- merely an “empty” state, the flavour vacuum is charac-
p
tal evidences [59]. A simple quantum mechanical model terized by a rich structure revealed by the non-vanishing
e
h (based on the work of Pontecorvo, Maki, Nagawa, and expectation value of the stress-energy tensor and the re-
[ Sakata [36, 48, 56, 57]) is commonly considered as suf- latedequationofstate. WithinBVformalismoneisable
ficient for accounting for experimental data. How- to fully describe it at a non-perturbative level, and its
2
v ever, this hides non-trivial difficulties in the formula- features depend on the specific model considered (spin,
1 tion of flavour oscillations in a Quantum Field Theoret- interactions,numberofparticlescharacterizedbyflavour
6 ical (QFT) framework [3]. Flavour states indeed do not mixing, etc.).
4 represent correct asymptotic states (by definition, since In a series of papers it was suggested that the flavour
3 their oscillating behaviour), which are required in the vacuum might behave as a source of Dark Energy [6–
2. usual perturbative approach to QFT (in the Lehmann, 10, 13, 23–25, 27]. Recently, it has been shown that
1 Symanzik and Zimmermann scheme). in a simple supersymmetric (Wess-Zumino) model with
1 Morethanadecadeago,anon-perturbativeapproachfor flavourmixing,inwhichtwoMajoranafields,twoscalars
1 buildingflavourstateswassuggestedbyBlasone,Vitiello and two pseudo-scalars were present (a simple model for
v: andcoworkers(BVformalism forflavourphysics)[12]. A neutrinos and sneutrinos), the flavour vacuum was actu-
i first version was proposed in 1995 [12], but some incon- ally characterized by a strong Supersymmetry breaking
X
sistencies in the derivation of oscillation formulae were [28, 54]. In the present work we argue that this breaking
ar noticed [17, 20, 31, 32] shortly after; a revisited version, istheoriginofaninterestingphenomenology,thatmight
in which these discrepancies were clarified and removed, shed some lights both on the Dark Energy and the Dark
was suggested and developed [15, 18, 19, 41, 42] later on Matter problem. More precisely, in the supersymmetric
[3]. However, for some aspects, the formalism remains context of [28, 54], the flavour vacuum can be thought
controversial and its physical relevance is still matter of as made of two different fluids that fill in all universe: a
debate [16, 34, 35]. first one related to the bosonic sector of the model, and
Theapproachcorrectlyreducestothecommonquantum a second induced by the fermionic one. The former is
mechanicalapproachinthesmall(neutrinos)masslimit, characterized by negative pressure, equal in modulus to
but leads to corrections to those formulas that are cur- its energy density (acting as a source of Dark Energy).
rently beyond experimental sensitivity [3, 26]. However, The latter is characterized by zero pressure, giving rise
perhaps the most interesting feature of BV formalism is to a source of Cold Dark Matter.
the non-trivial vacuum (called flavour vacuum) implied
Thefirstpartofthepaper(SectionI)willbededicated
to review BV formalism, complementing the original lit-
erature with a discussion on non-perturbative theories
∗ [email protected] and Fock spaces.
2
In the second part of the paper (Section II) we shall that connects flavour fields ν , ν with massive ones ν ,
A B 1
explore in details the phenomenology of the model stud- ν . Such a relation is connected with the linearization of
2
ied in [28, 54]. We shall clarify why the flavour vacuum the following Lagrangian for free spin-1/2 fields
is a good Dark Energy and Dark Matter candidate, with
emphasis on this latter. More importantly, we will ex- L=iν¯ (x)∂/ν (x)+iν¯ (x)∂/ν (x)+
A A B B
plain how to relate all parameters of the model to ob-
−m ν¯ (x)ν (x)−m ν¯ (x)ν (x)+
servational data. This is a quite important aspect of A A A B B B
−m (ν¯ (x)ν (x)+ν¯ (x)ν (x)) (4)
the approach: the model introduces very few parameters AB A B B A
whichareallrelatedtoneutrinophysics. Hopefully,more
which becomes
realisticmodelswillnotrelyonuncontrolledfreeparam-
eters, leading to a truly falsifiable theory for both Dark
Energy and Dark Matter. A first encouraging result in L=iν¯1(x)∂/ν1(x)+iν¯2(x)∂/ν2(x)+
this direction comes already from the simple model here −m ν¯ (x)ν (x)−m ν¯ (x)ν (x) (5)
1 1 1 2 2 2
studied: this is indeed fairly consistent with a choice of
the parameters modeled on real world data, as we shall when
see in the relevant section.
A first step towards more realistic models will be mA =m1cos2θ+m2sin2θ
moved in the last part of this work (Section III). We will m =m sin2θ+m cos2θ
B 1 2
present a novel method for calculating relevant quanti-
m =(m −m )sinθcosθ. (6)
AB 2 1
ties,specificallythoughtforanalyzingthefeaturesofthe
flavour vacuum, which might enable us to study inter-
However,inthefield-theoreticalframeworkthedecom-
active theories completely at a non-perturbative level.
position of the fields (3) into ladder operators associated
In particular, we shall show how, under reasonable as-
withflavourparticlestatesishighlynon-trivial[3]. Ithas
sumptions, the method can discriminate which interac-
been shown, indeed, that states defined as the relativis-
tions preserve the behaviour of the condensate as Dark
ticequivalentof(1),forwhich|ν (cid:105)belongstothemass-
1,2
Energy/Matter source.
m irreduciblerepresentationofthePoincar´egroup,are
1,2
not eigenstates of the flavour charge operators [11, 14],
which for the theory (4) read [11]
I. BV FORMALISM
(cid:90)
Q (t)= d(cid:126)xν†(x)ν (x),
Neutrino oscillations can be described in a non- A A A
relativisticquantummechanicalframeworkbyconstruct- (cid:90)
Q (t)= d(cid:126)xν†(x)ν (x). (7)
ingparticlestates,labelledbyaflavour number,thatare B B B
not eigenstates of the Hamiltonian. In its simplest for-
mulationfortwodistinctflavours,thePontecorvomodel, BVformalismcompensateforthis[12,39,42],bydefin-
flavour states are constructed as follow [4]: ing appropriate flavour eigenstates via the action of a
certain operator G on massive states:
θ
|ν (cid:105)=cosθ|ν (cid:105)+sinθ|ν (cid:105)
A 1 2
|νB(cid:105)=−sinθ|ν1(cid:105)+cosθ|ν2(cid:105) (1) |(cid:126)k ,f ;(cid:126)k ,f ;(cid:126)k ,f ;...(cid:105)≡
1 1 2 2 3 3
where |ν1(cid:105) and |ν2(cid:105) are massive eigenstate of the free G†θ|(cid:126)k1,m(1);(cid:126)k2,m(2);(cid:126)k3,m(3);...(cid:105) (8)
Hamiltonian (particles with well defined mass m , with
i
i=1,2), and from which where |(cid:126)k ,f ;(cid:126)k ,f ;(cid:126)k ,f ;...(cid:105) denotes a state with
1 1 2 2 3 3
different flavour particles described by their mo-
℘A→B =|(cid:104)νB|νA(t)(cid:105)|2 =|(cid:104)νB|e−iHt|νA(cid:105)|2 menta (cid:126)k and their flavours f = A,B, whereas
(cid:18) (cid:19) i i
=sin2θsin2 ω1(k)t−ω2(k)t (2) |(cid:126)k1,m(1);(cid:126)k2,m(2);(cid:126)k3,m(3);...(cid:105) denotes a state with dif-
2 ferent massive particles described by their momenta (cid:126)k
i
(cid:113) and their masses m(i) = m1,m2, defined from the lin-
with ωi(k) ≡ (cid:126)k2+m2i, describing the non-vanishing earized theory (5). The operator Gθ is defined by the
probability of a flavoured particle with momentum (cid:126)k to equations:
be created with a certain flavour (A) and be detected
ν (x)=G−1ν (x)G
later on with a different flavour (B). A θ 1 θ
Theformofthetransformationbetweenflavouredand νB(x)=G−θ1ν2(x)Gθ (9)
massive particles (1) is reflected in the relativistic field
formalism by the relation and its explicit form depends on the specific theory con-
sidered. For the theory (4), G is written as [12]
θ
ν (x)=ν (x)cosθ+ν (x)sinθ
A 1 2
νB(x)=−ν1(x)sinθ+ν2(x)cosθ (3) Gθ(t)=eθ2(cid:82)d(cid:126)x(ν1†(x)ν1(x)−ν1†(x)ν2(x)). (10)
3
AmongallflavourstatesdefinedintheBVapproach,the framework,ascatteringprocessisrepresentedbyagraph
one called flavour vacuum and defined by with a certain number of external legs (incoming and
outgoing particles). The total number of internal lines
|0(cid:105)f ≡G†θ|0(cid:105) (11) is connected to the precision of the approximation used
in the perturbative expansion: the higher the order of
plays a special role, since it represents the physical vac-
the perturbation, the higher the number of vertices, and
uum. In this context, by physical vacuum we mean the
thereforethehigherthenumberofinternallinesinvolved.
state that represents the physical empty state, i.e. with
Eachlinecanbena¨ıvelyinterpretedasasinglefreeparti-
no particle in it. Since only particles with well defined
clestate,whichisemittedinthestartingvertexandthen
flavour, rather than mass (i.e. Hamiltonian eigenstates),
absorbed in the ending one. At each order in perturba-
can be created/detected, one expects the physical vac-
tion theory, a finite number of free particle states enters
uumtoberepresentedbythestatethatcountsnoflavour
in the description of the scattering process. However,
particles in it. It can be shown that this is state is |0(cid:105) ,
f under the assumption that the perturbative series con-
rather than |0(cid:105) [12].1
verges, its limit would be described by an infinite num-
Furthermore, it has been proven that all flavour
beroflines/one-free-particle-states. Inbra-ketformalism
states|(cid:126)k1,f1;(cid:126)k2,f2;(cid:126)k3,f3;...(cid:105)areorthogonal toeachmas- suchalimitstatewouldthereforeberepresentedbyavec-
sive state |(cid:126)k ,m ;(cid:126)k ,m ;(cid:126)k ,m ;...(cid:105), and therefore tor of H, the space of all physical states, but not of F ,
1 (1) 2 (2) 3 (3) 0
(cid:104)0|0(cid:105) = 0 follows as a particular case. This result en- the space of states with finite number of free particles.
f
ables us to talk of a Fock space for flavour states, in With this example we want to remark the non-trivial
opposition of the usual Fock space, whose basis is given difference existing from a free theory and an interactive
by {|(cid:126)k ,m ;(cid:126)k ,m ;(cid:126)k ,m ;...(cid:126)k ,m (cid:105)|∀n∈N}. one: wecanexpressinteractiveprocessesintermsoffree
1 (1) 2 (2) 3 (3) n (n)
Theorthogonalityofthetwospacesisnotverysurpris- states (which have no direct physical meaning or inter-
ing if one regards BV formalism as a non-perturbative pretation,beingjustabasisinwhichwechoosetoexpress
approach to the interactive theory defined by (4). our process) but only in a weakly-interactive/perturbed
In the Second-quantization framework, the Hilbert framework. A full non-perturbative treatment for inter-
space representing physical states is defined by vectors active particle states requires subspaces of H, that are
in the number occupation representation: assuming that orthogonal to the Fock space of free states F [38].
0
a single-particle state can be classified by a discrete set Coming back to our case, a Lagrangian with flavour
of states labeled by the index i=1,2,3,..., a vector rep- mixing, such as (4), can be regarded as an interactive
resenting a many-particle state can be identified by the theory, thanks to its non-diagonal terms. Flavour par-
number ni of particles occupying the i-th state and it is ticle states defined `a la BV form a Fock space Ff that
denotedwith|n1,n2,n3,...(cid:105);theHilbertspaceofphysical is therefore orthogonal to F0. In other words, we could
states H is therefore defined as the vector space gener- express flavour states in a perturbative way by means
ated by the basis {|n1,n2,n3,...(cid:105)}. For bosons ni ∈ N0, of F0; however, BV formalism enables us to construct
whereas for fermions ni = 0,1. It can be shown that flavour states in a completely non-perturbative manner,
in both cases the set {|n1,n2,n3,...(cid:105)} is uncountable and andthereforeitrequiresstatesthatarepartofHbutnot
therefore H is non-separable [60]. In particle physics a of F .
0
separable subset of H, a Fock space that we will denote
Different Fock spaces are ordinarily used in QFT on
with F , is usually considered [58]. F carries an irre-
0 0 curved backgrounds and other contexts [5, 22]. In the
ducible representation of the Poincar´e group and par-
former, for instance, one identifies Fock spaces for physi-
ticle states belonging to it have well-defined mass and
cal states in flat regions. However, these Fock spaces do
spin. Such a subset is spanned by the countable basis of
not coincide (they are different/orthogonal subset of H)
all states with an arbitrary, yet finite in total, number of
if the regions are not connected and curved regions exist
freeparticles. Althoughthisbasisdoesnotfullydescribe
inbetween. Asaconsequence,thevacuumdefinedbyan
H (for instance the vector |1,1,1,...(cid:105) which counts an in-
observer in a certain region is not necessarily described
finite number of particles is not included), it is sufficient
by the state with no particles by an observer in another
for accounting for scattering processes at a perturbative
region. The particle creation phenomenon may occur:
level. In the usual perturbation theory all interactive
a state that is empty for an observer can actually con-
processes are indeed approximated by means of a super-
tain particles according with a different observer. This
position of a finite number of free particle states. This is
mechanism characterizes both of the two main results of
quite clear in the functional formalism, when Feynman
QFT in curved spacetime: the Unruh effect [61] and the
diagrams are considered. In a simplified picture of this
Hawking radiation [40].
Inaformalanalogy,BVformalismintroducesaground
state,calledflavourvacuum,whichisnotastrivialasthe
groundstateforthefreetheory. Asalreadysaid, sinceit
1 In short: flavour states can be built by means of specific cre-
ation/annihilationoperatorsforflavourparticles;itcanbeshown is the state in which no flavour particles are present, it
thattheonlystatethatisannihilatedbyallannihilationopera- correctly represents the physical vacuum. Even though
torsistheflavourvacuumdefinedvia(11). it is empty, it is characterized by a non-zero expectation
4
value of the stress-energy tensor (cid:104)0|T |0(cid:105) , whose ef- condensate, might provide a suitable description of the
f µν f
fects must be gravitationally testable. This is true as low energy limit of the model.
long as we fix as zero-point of our theory the usual vac- In [54] we presented the behaviour of the flavour vac-
uum |0(cid:105) for the free theory and belonging to F , or, uum,inasimplesupersymmetrictheory. Themodelthat
0
in other words, we consider the usual normal ordering wasconsideredinvolvestwofreerealscalarsS (x),S (x)
A B
(cid:104)0|:T :|0(cid:105) ≡ (cid:104)0|T |0(cid:105) −(cid:104)0|T |0(cid:105), which is valid with mixing, two free real pseudo-scalars P (x), P (x)
f µν f f µν f µν A B
inperturbationtheoryaswellasinthisnon-perturbative withmixingandtwofreeMajoranaspinorsψ (x),ψ (x)
A B
approach. Onecommonlyreferstotheflavourvacuumas with mixing:
a condensate for the following reason: once expressed in
(cid:88) (cid:104)
terms of particles with well defined mass (eigenstates of L= ∂ S (x)∂µS (x)+∂ P (x)∂µP (x)+
µ ι ι µ ι ι
the Hamiltonian), the flavour vacuum contains an non-
ι=A,B
vanishing number of those particles, per unit of volume. +iψ¯(x)∂/ψ (x)(cid:105)− (cid:88) (cid:104)m2 S2(x)+m2 S2(x)+
In our example, they are characterized by the following ι ικ ι ικ ι
distribution over the momentum space [26] ι,κ=A,B
(cid:105)
+m ψ¯(x) ψ (x) (13)
sin2θω (k)ω (k)−m m −k2 ικ ι κ
(cid:104)0|n((cid:126)k)|0(cid:105) = 1 2 1 2 (12)
f f 4π3 ω (k)ω (k) with m =m . Terms involving products of fields of
1 2 AB BA
differentflavoursdisappearwhenoneexpressesthemodel
with n((cid:126)k) ≡ (cid:80) (ar†((cid:126)k)ar((cid:126)k) + ar†((cid:126)k)ar((cid:126)k)), k ≡ |(cid:126)k|, in terms of new fields, obtained by appropriate rotations
r 1 1 2 2
andar(†) representingladderoperatorsforparticleswith of the previous ones:
i
well-defined mass. However, since the physical degrees
φ (x)=cosθφ (x)+sinθφ (x)
A 1 2
of freedom of the theory are flavour particles (the only
φ (x)=−sinθφ (x)+cosθφ (x) (14)
kind of particle that can be produced and detected), the B 1 2
interpretation as a gas or collection of particles remains with φ=S,P,ψ, leading to
at a mere mathematical level, the flavour vacuum be-
ing absolutely “empty” from a physical point of view (in L= (cid:88) (cid:104)∂ S (x)∂µS (x)−m2S2(x)+
µ i i i i
the sense that no flavoured particles are present in it),
i=1,2
and only characterized by a non vanishing stress-energy
+∂ P (x)∂µP (x)−m2P2(x)+
tensor expectation value which is detectable via gravita- µ i i i i
(cid:105)
tional effects. +ψ¯(x)(i∂/−m )ψ (x) . (15)
i i i
The features of the flavour vacuum depend on the
model considered and a preliminary investigation on a From this latter it is possible to build the usual Fock
simple supersymmetric model [54] showed that it might space for massive particles, previously denoted as F ,
0
behave very differently, according with the spin of the which has as ground state the “massive” vacuum |0(cid:105).
particles involved. The flavour vacuum is hence defined as
|0(cid:105) ≡eθ(cid:82)d(cid:126)x(X12(x)−X21(x))|0(cid:105) (16)
f
II. PHENOMENOLOGY OF A SUSY FLAVOUR
with
VACUUM
1
X (x)≡ ψ†(x)ψ (x)+iS˙ (x)S (x)+iP˙ (x)P (x).
A. Free WZ `a la BV 12 2 1 2 2 1 2 1
(17)
Its features have been explored via its stress-energy ten-
Our interest in BV formalism was firstly motivated
sor expectation value, being
by physics beyond the Standard Model. The Wess-
Zumino model here discussed has been considered in (cid:88)(cid:104)
T (x)= 2∂ S (x)∂ S (x)+2∂ P (x)∂ P (x)+
[54] after two works in which the flavour vacuum has µν (µ i ν) i (µ i ν) i
i=1,2
been regarded as an effective vacuum arising in a string-
(cid:105)
theoretical framework [51, 53]. Indeed, a specific model +iψ¯(x)γ ∂ ψ (x) −η L. (18)
i (µ ν) i µν
from the braneworld scenario, called D-particle foam
model [29, 30, 49, 50, 52], seems to explain neutrino It has been shown that the flavour vacuum behaves as a
flavour oscillations in terms of flavour oscillation of fun- perfect relativistic fluid, i.e.
damental strings, in presence of a “cloud” (or foam) of
(cid:104)0|T |0(cid:105) =diag{ρ,P,P,P} (19)
point-like topological defects in the bulk space. In the f µν f
spirit of weak coupling string theory, the interaction be- with
tween the foam and strings/branes in the theory can be
ρ≡ (cid:104)0|T (x)|0(cid:105) =
regardedas“vacuumdefects”fromthepointofviewofa f 00 f
macroscopical observer. Therefore it has been suggested (m −m )2 (cid:90) K (cid:18) 1 1 (cid:19) (20)
=sin2θ 1 2 dk k2 +
that BV formalism, together with its “flavour vacuum” 2π2 ω (k) ω (k)
0 1 2
5
P≡ (cid:104)0|T (x)|0(cid:105) =
f jj f cannotexplaintheobservedstructuresatgalacticscales,
m2−m2 (cid:90) K (cid:18) 1 1 (cid:19) (21) thereforeDarkMatterisexpectedtobemadeoutoffairly
=−sin2θ 1 2 dk k2 −
2π2 ω (k) ω (k) massiveand“cold”(non-relativistic)particles. Big-Bang
0 2 1
nucleosynthesislimitsontheaveragebaryoniccontentof
ρ representing its energy density, P its pressure and K a the Universe exclude that (the majority of) Dark Mat-
momentum cutoff (cf [28, 54]).2 terismadeoutofordinarybaryonicmatter(i.e. atoms).
In particular, disentangling the contribution of the Furthermore,although“dark”,inthesensethatdoesnot
bosonic sector from the fermionic one, one finds emit nor absorb light (i.e. electromagnetically neutral),
Dark Matter might couple to ordinary matter in other
(cid:90) K k2 ways (besides gravity); however, arguments on its den-
ρb =sin2θ dkπ2 (ω1(k)+ω2(k))× sity and thermal production at early times imply that
0 such a coupling must be weak.
(cid:32) (cid:33)
(ω (k)−ω (k))2
× 1 2 (22) Both astrophysics and particle physics have been
2ω (k)ω (k)
1 2 proposing suitable candidates for Dark Matter through
thelastthreedecades, givingrisetoanenormouswealth
of choice. However, because of the absence of direct de-
(cid:90) K k2 tectionsandthelackofpredictionsbytheoreticalmodels,
ρ =sin2θ dk (ω (k)+ω (k))×
f π2 1 2 plaguedbyanundesirableabundanceoffreeparameters,
0
(cid:18)ω (k)ω (k)−m m −k2(cid:19) the nature of Dark Matter remains elusive.
× 1 2 1 2 (23)
The fermionic sector of the flavour vacuum in the
ω (k)ω (k)
1 2
model here presented clearly fulfills basic requests for
a Dark Matter candidate: it contributes to the energy
P =−ρ , P =0 (24) content of the universe; it is “dark” (i.e. it is an electro-
b b f
magnetically neutral object, since (s)neutrino fields do
in which the standard normal order has being adopted. not couple with the electromagnetic field); furthermore,
it does not interact with any other of the SM particles
(excluding gravitational effects), being the empty state
B. Flavour vacuum as a source of Dark Matter forthe(s)neutrinosector; unlikeitsbosoniccounterpart,
it is purely pressureless.3
An important result emerges from the above outlined
A possible concern about its uniform distribution in
analysis: the equation of states w ≡P/ρ for the bosonic
space, incontrastwiththeobserveddistributionofDark
and the fermionic sectors are different, w = −1 and
b Matter which is usually gathered in clusters around and
w = 0 holding. The emphasis on the novelty of this
f inside galaxies, can be easily dispelled by recalling that
SUSY breaking mechanism has already being remarked
we are actually modeling a simple “empty” universe. If
[54]. We are now aimed to explore the interesting phe-
a non-uniform matter distribution is considered in our
nomenology connected with such a result. Our simple
toy universe in addition to the flavour vacuum, it would
model implies a physical vacuum that is a combination
start to interact gravitationally with our vacuum con-
oftwofluidswhichbehavequitedifferently: bothperme-
densate. Thanks to initial irregularities in the matter
atetheemptyspaceuniformlyandstatically,butonehas
distribution, we expect them to form clusters via gravi-
a cosmological-constant-like behaviour (w = −1), while
tational instability (gravity tends to enhance irregulari-
theotherbehavesasdust(w =0). Theroleoftheflavour
ties, pulling matter towards denser regions [47]), as the
vacuum as source of Dark Energy (which now is played
system evolves with time. It is known that such an ef-
onlybythebosonicsectorofthetheory)hasbeenexten-
fect, on the other hand, does not necessarily occurs for
sively discussed in literature [6–10, 13, 23–25, 27]. Here
Dark Energy-like fluids [1], as the bosonic component of
we present a new feature of the flavour vacuum: its con-
the flavour vacuum, which can persist in their state of
tribution to Dark Matter.
spatial uniformity even in presence of clustered matter.
Dark Matter is the name given to unknown sources
Theevolutionofourflavourvacuum,consideringbothits
of gravitational effects, whose presence, primarily within
bosonic and fermionic components, in presence of other
andaroundgalaxies,hasbeenestablishedbymanyastro-
matterandgravitationalinteraction,representsnecessar-
physical data [2, 43]. Numerical simulations of structure
ily an object of future studies.
formation have shown that “hot” (relativistic) particles
2 FromtheperspectiveofconsideringBVformalismasaneffective
formalismforphysicsbeyondtheStandardModel[51,54],such 3 This is certainly true for the free above mentioned model; the
acutoffmustbeinterpretedastheenergyscaleuptowhichthe possibility of extending this result to interactive models will be
formalismprovidestheframeworkforagoodeffectivetheory. discussedlateron.
6
C. Testability gles, and we can constrain parameters induced by SUSY
breaking, the cutoff is the only parameter left to deter-
mine.
An interesting aspect of the model concerns the in-
As our toy universe expands, we assume that the two
terplay between the two fluids. Supersymmetry imposes
fluidsobeyEinsteinequationsandthereforetheyscaleas
that the energy density of the bosonic component is tied
up to the energy density of the fermionic component;
ρ (t) a(t)3 =ρ a3
in a more realistic theory, therefore, one should be able f f 0 (25)
to reproduce the current experimental value of the ratio ρb(t)=ρb.
between the Dark Energy density and the Dark Mat-
This means that today their value is
ter energy density (∼ 2.8), in the optimist belief that
the flavour vacuum is the only responsible for both of
ρ (t )=ρ a3
them. The role of a curved background in the formu- f now f 0 (26)
lation of the theory might be crucial, since the energy ρb(tnow)=ρb.
density of a dust-like fluid gets diluted by the expansion
of the universe, whereas such an effect does not occur respectively, being a(tnow) = 1 by convention. Those
for a cosmological-constant type, and therefore the ratio two quantities depends on the following parameters:
between those two quantities changes dramatically with (s)neutrino masses, mixing angles, cutoff energy, scale
time. factor at t0. Provided with these expressions, we can
then test our model in two ways.
Within a momentum cutoff regularization framework,
as the one here presented, the two energy densities de- 1.If observational data enable us to constrain
pend on such a cutoff, which is the same for both quan- (s)neutrino masses, mixing angles, Dark Matter and
tities. The ratio between them can in general be cutoff Dark Energy densities, from (26) we can derive the
dependent, as it actually is in the case here presented. otherparametersleft: thecutoffenergyandthescale
On one side, one might hope that in a more realistic factor a0. Well equipped with all the parameters of
theory (on a curved background, for instance) the ratio thetheory,wewillthenbeabletocheckifthemodel
might be cutoff independent. On the other hand, one isinreasonableagreementwithotherstandardmod-
could consider the opposite situation, in which the ratio els. Forexample,ifthescalefactora0 fittingalldata
varies with the cutoff, as highly desirable: if the ratio corresponds to a time in the future (for a0 >1), the
is fixed from the cutoff, the same value for the cutoff model has to be rejected, or corrected at least.
would also fix the value of the energy. This implies that 2.On the other hand, theoretical reasons might sug-
once the cutoff is decided on the basis of experimental gest specific values for the cutoff (if for instance the
data on the ratio, the model gives a precise prediction flavourvacuumrisesinthelowenergylimitofanun-
fortheabsolutevaluesoftheenergydensities, whichcan derlyingtheory,and/orthescalefactora0,beingthe
be compared with their observational estimates. temperature of the universe inversely proportional
In order to illustrate these ideas we will present a con- to its scale factor. In this case, we might be able
crete example. Let us assume that our supersymmetric to make a prediction on the value of the Dark Mat-
modeliseffectiveuptotheenergyscaleK (whichcomes terandDarkEnergydensity,viaformulae(26),that
from deeper theories, as, for instance, in [54] ). In the mightbecomparedwithobservationalestimates. On
standard Big-Bang picture, this means that when the thisbasisourmodelisthereforeacceptedorrefused.
universe cools down to that energy, the flavour vacuum
starts to be the effective description of the vacuum state
of the (unknown) underlying theory. We call t the time D. A Preliminary Test
0
correspondingwiththistransitionanda thecorrespond-
0
ing scale factor. The simple toy model discussed in Section IIA is not
Inourtoyuniverse,weassumethatatt ,inabsenceof realisticenoughtoholdthecomparisonwithdataalready
0
anyothersourcesofenergyormatter,theenergy/matter available: only two generations of neutrinos have been
contentofourtoyuniverseisonlyduetotheflavourvac- considered, neither matter or interactions are present,
uum. Moreover, we assume that it can be describe, at SUSY is unbroken, there is no prescription for the cutoff
a classic level (i.e. on sufficiently large scales), in terms K.4 However, some preliminary tests can be performed.
of two fluids: a first one, due to the bosonic component As just explained, in a realistic theory with three gen-
of the flavour vacuum and described by ρ and w =−1, erations instead of two, it is possible to constrain the
b
and a second one, due to its fermionic component and parameter space of mixing angles and masses thanks to
described by ρ and w = 0. We will regard the bosonic observational data. In absence of such a theory, we will
f
component as the only source of Dark Energy and the
fermionicastheonlysourceofDarkMatter. Bothρ and
b
ρ are function of the following parameters: (s)neutrino
f
masses, mixing angles, and the cutoff K. If we know 4 The assumption of treating neutrinos as Majorana particles
(from observations) the neutrino masses and mixing an- mightalsobequestioned.
7
limit our selves to check if our simpler model admits a (26) hold). We will further assume that some radiation
choiceofparametersthatgivesrisetophysically“plausi- and matter are present in our toy universe, whose den-
ble” estimations for Dark Energy and Dark Matter den- sity is at least one order lower than the flavour vacuum
sities. Moreprecisely,weshallcheckthecompatibilityof density. Their presence justifies the notion of “tempera-
our model with the relation ture” and defines the profile of the time-evolution of our
toy universe. Since we assume the neutrino sector not
ρΛ ∼ρDM ∼(∆m2ij)2, (27) being coupled with any other fields, the flavour vacuum
and the matter/radiation content of the universe inter-
with ρΛ/DM the Dark Energy/Matter density today and act only gravitationally. As mentioned, we expect this
∆mij the difference of the squared masses of neutrinos5, interaction to lead the fermionic flavour vacuum to clus-
which for our model becomes ter together with ordinary matter, leaving the bosonic
flavourvacuumhomogeneouslydistributed. Theseeffects
ρb ∼ρf(tnow)∼(∆m2)2, (28) are reasonably expected as long as gravitational effects
are relevant only on cosmological scales, at which the
intheassumptionthatallDarkEnergyandDarkMatter
flavour vacuum is well approximated by a classical fluid.
ofourtoyuniverseisduetotheflavourvacuum. Inother
As a consequence of a possible non-uniform distribution
words, is there a sensible region of the parameter space
inspaceofthefermionicfluid,weshallconsiderthevalue
that gives rise to (28)?
of (23) not as a local attribute but as a global, or “aver-
Once provided with a realistic theory, the reasoning aged” over sufficiently large scales, property.
goestheotherwayaround: giventhespaceofparameters
In order to check the compatibility of our model with
constrainedbyobservationaldata, does(28)hold? How- (28),westartbydefiningthequantityξ ≡(1−m2/m2)2,
2 1
ever, the analysis on which we are embarking is neither
withm thesmallerofthetwomasses(m <m ). Since
2 2 1
irrelevantnornegligible: previousanalyseshardlyconcil-
0<ξ <1, we can expand (22) and (23) in series around
iate the very different scales of energies entering into the
ξ =0 and get to
problem, such as the momentum cutoff, which presum-
ably is greater then the TeV scale, and neutrino mass sin2θ
ρ = m4f(K/m )ξ+O(ξ3/2)
differences (cf for instance [24]). The aim of this section b π2 1 1
is therefore to show that even in our simple toy model, sin2θ
non-perturbative formulae describing the features of the ≈ π2 (∆m2)2f(K/m1)
flavourvacuumcanaccommodateverydifferentscalesin
sin2θ
anaturalway,givingrisetophysicallysensiblevaluesfor ρ = m4g(K/m )ξ+O(ξ3/2)
f π2 1 1
Dark Energy and Dark Matter densities.
sin2θ
Recapitulating, in the following we shall assume that ≈ (∆m2)2g(K/m ) (29)
thephysicalvacuumiseffectivelydescribedbytheflavour π2 1
vacuum defined by (16) for energies lower than K; over
with
largedistancescales,suchaflavourvacuumbehavesasa
classicalfluids, obeyingEinsteinequations(i.e. (25)and (cid:90) K/m1 x2
f(K/m )= dx (30)
1 4(1+x2)3/2
0
(cid:90) K/m1 x4
g(K/m )= dx (31)
5 The energy scale of Dark Energy is far away from all natural 1 0 4(1+x2)3/2
scalesprovidedbytheSMviaparticlemasses. Onlyonefunda-
mental scale is known to be comparable with the Dark Energy and m4ξ = (∆m2)2 ≡ (m2 −m2)2. Relations (29) are
1 1 2
one: thescaleofneutrinophysics. Boundaryontotalmassesof good approximations of the exact values, as long as the
neutrinos show that they are much lighter then all other parti-
two masses are very similar m ∼ m . All divergen-
cles: astrophysicaldataindicatethatΣmν <058eV,with95% ciesconnectedwithourproblem1areinc2ludedinfunction
ofconfidence[44](thesumrunsoverallpossiblespecies-possi-
blymorethenthree-thatwherepresentintheearlyUniverse). f(K/m1) and g(K/m1), for their argument running to
Moreover, direct observations on solar and atmospheric neutri- infinity. Inthefollowinganalysisaphysicalcutoffofmo-
(nboesinshgomw2th−atm∆2m≡212∆≈m28·,1c0f−[54]eVan2danredfe∆remnc22e3s≈th2e.r5e·in1)0.−T3heVes2e menta (K, rescaled by the neutrino mass m1) will be
i j ij considered, in the belief that flavour physics `a la BV
massscales10−1÷10−2eV havetobecomparedwiththescale
must be regarded as an effective description at low en-
10−3eV, that one obtains from ρΛ = 3×10−11eV4. This “co-
ergy scales of a deeper theory [51]. Clearly other renor-
incidence” gave rise to many works, besides the ones connected
with BV formalism, aimed to provide a theoretical explanation malization tools might be required if this assumption is
forit(see[37]andreferencestherein). dropped, for instance, in a pure self consistent quantum
The more famous coincidence problem regarding Dark Energy field theoretical approach. Despite this choice, it is re-
concerns the similar density of Dark Energy and Dark Matter
markable, however, that the relation ρ ∝(∆m2)2 has
(ρΛ ≈ 2.8ρDM) as measured today, which requires a notable b/f
fine-tuning of initial conditions considering their very different been derived entirely analytically.
evolutionintime. Inthefollowing,wewouldliketoshowthatinacutoff-
Thesetwo“coincidences”arecombinedtogetherinformula(27). regularization scheme the two functions in (30) can give
8
rise to physically sensible values, i. e. the function sin2 Θ
f(K/m1) remains relatively small even when the cutoff 1.0
isveryhigh(givingrisetothehierarchy: highcutoff/low
DarkEnergydensity),whereasg(K/m )canbeconsider-
1 0.8 (cid:72) (cid:76)
ablygreater then f(K/m )for thesamechoice ofcutoff,
1
motivating the observed discrepancy between the Dark
0.6
Energy and Dark Matter densities at early times. We
should stress once more that a strict comparison with
available experimental data would be possible once a 0.4
more realistic model will be given.
We will now focus on the former of (29). Being SUSY 0.2
LHC Planck
unbrokeninourtoymodel,neutrinosandsneutrinoshave K
thesamemasses. Sothem andm appearingin(29)are
the masses of two neutrino1s, even t2hough ρ encodes the 105 1010 1015 1020 1025 1030 m1
b
contribution of the bosonic sector of the theory, which
in a realistic case would be affected by the breaking of FIG. 1. The parameter space (K/m1,sin2θ) is plotted. In
SUSY via effective masses greater than m . Recalling blue,theacceptanceregionforthecondition(32)(boundaries
i
oftheregioncorrespondtolhs=0.9andlhs=1.1). Physical
theobservedrelation(27), wewondernowifisthereany
mixinganglesforthreegenerationofneutrinosare: sin2θ =
region of the parameter space (θ,K) that might generate 13
0.000+0.028, sin2θ = 0.30+0.04, sin2θ = 0.50+1.4 (within
a similar situation (i.e. ρ ∼ (∆m2)2) in our model. 12 −0.05 23 −1.2
b 2σ) [33]. These values are also plotted in the graph (red
Quite interestingly, the condition
regions). DottedlinesrepresentsLHCenergyscale(∼1TeV)
andthePlanckenergyscale(∼1028eV)fixingm =10−2eV.
sin2θ 1
f(K/m )∼1 (32)
π2 1
is satisfied by a region of the plane (θ,K), which is not We already know that a must be extremely small, but
0
that far from the expected value for a realistic theory. what is the ratio between g(K/m ) and f(K/m ) for
1 1
As shown in Figure 1, if the cutoff of the theory lies large K/m (K/m > 1014)? As shown in Figure 2,
1 1
somewhere between the TeV (1012 eV) scale and the equation (35) is indeed satisfied for large values of K
Plankscale(1028 eV),thevalueofsin2θ mustbearound and very small values of a , as one would expect from a
0
the 0.5÷1 region. Moreover, a complete overlap with realistic theory. If, for instance, the cutoff is set equal
one real mixing angle is obtained in a region with a to the Planck scale, relation (35) is satisfied for a =
0
very high cutoff, close to the Planck scale. It should be 10−20, which sets the transition phase (when the flavour
emphasized that the existence itself of such a region is vacuum became effective) well far in the past, when our
highlynon-trivial, sinceitappearsfromthecombination toy universe was 1020 times smaller than “now”. Once
of parameters spanning a wide range of energies, being more, parameters characterized by very different values
∆m2 ∼10−4eV2 and K >1012 eV. (a ∼ 10−20, K/m ∼ 1030) combine together to give
0 1
Focusing now on the latter of (29), we shall proceed rise to a third scale, which has a physical significance
withasimilaranalysisinordertotestthehypothesisthat (nowadays Dark Matter density).
thefermionicsectorofthemodelprovidesasensibleDark Despite the unrealistic nature of our toy model, these
Matter candidate. As explained in the previous section, first quantitative tests go in the right direction and cer-
under the assumption that the flavour vacuum behaves tainlymotivatethestudyofmorerealisticmodels,which
as a perfect classical fluid on large scales, the fermionic hopefully will share with our toy model all the good fea-
contribution would get diluted with time as the universe tures here discussed.
expands. Its value today would then be
ρ (t )=ρ a3 (33)
f now f 0
III. TOWARDS INTERACTIVE FLAVOUR
with a the scale factor corresponding to the time at VACUA
0
which the model became effective. In order to reproduce
the relation (∆m2)∼ρ (t ) we expect
f now A. A new method of calculation: Free WZ Revised
sin2θ
a30 π2 g(K/m1)∼1 (34) Following the standard literature, the results in [54]
discussed so far have been derived by the following ap-
that can be obtained by combining (33), (28) and (29). proach:
BecauseoftheconstrainontheDarkEnergydensity(32)
1.stress-energy tensor has been written in terms of
the above condition becomes
massive fields;
g(K/m ) 2.massive fields have been decomposed in terms of
a3 1 ∼1. (35)
0f(K/m ) massive ladder operators;
1
9
a Recalling the discussion in Section IIA, in the study of
0
1020 the free WZ model we can consider the bosonic and the
fermionic component separately, by evaluating relevant
quantities in two separated contexts (a bosonic theory
1010 and a fermionic one) and eventually combining together
theresults. Furthermore,thepseudoscalarandthescalar
EXPANDINGUNIVERSE field are indistinguishable for our purposes, therefore we
1
CONTRACTINGUNIVERSE are allowed to consider just the scalar field, keeping in
mind to sum its contribution to the relevant quantities
10(cid:45)10 Planck twice.
In the real scalar case, we have
LHC K
10(cid:45)21005 1010 1015 1020 1025 1030 m1 T0b0(x)=(cid:88)(cid:18)πi2(x)+(cid:16)∇(cid:126)φi(x)(cid:17)2+m2iφ2i(x)(cid:19) (36)
i
FIG.2. Theparameterspace(K/m1,a0)isplotted. Inblue, with π ≡φ˙ , the conjugate momentum of φ . Since
the acceptance region for the condition (32) (boundaries of i i i
the region correspond to lhs = 10 and lhs = 0.1). The red G (t)=eiθ(cid:82)d(cid:126)x(π2(x)φ1(x)−π1(x)φ2(x)) (37)
line depicts the scale factor of the real universe as a function θ
of its average temperature, fixing m = 10−2eV. In a more
1 from which
realistictheory,theacceptanceregion(whichmaydifferfrom
the one here presented) is further reduced by comparison of G (t)φ (x)G†(t)=φ (x)cosθ−φ (x)sinθ
the first of (29) with Dark Energy data, which gives a con- θ 1 θ 1 2 (38)
straintonpossiblemomentumcutoffs;themodelisruledout G (t)φ (x)G†(t)=φ (x)sinθ+φ (x)cosθ
θ 2 θ 1 2
if the resulting region lies below the red line, since it would
predict an amount of Dark Matter greater than what is ob- and
served; the model is consistent with data on Dark Matter
if the region overlaps or lies above the red line; in this latter Gθ(t)π1(x)G†θ(t)=π1(x)cosθ−π2(x)sinθ (39)
casetheflavourvacuumcontributesonlytoafraction oftotal G (t)π (x)G†(t)=π (x)sinθ+π (x)cosθ
Dark Matter density. θ 2 θ 1 2
via the Baker-Campbell-Hausdorff formula
3.massiveladderoperatorshavebeenwritteninterms 1 1
eYXe−Y =X+[Y,X]+ [Y,[Y,X]]+ [Y,[Y,[Y,X]]]+...
of flavour ladder operators; 2 3!
4.hence,stress-energytensorhasbeenwritteninterms (40)
of flavour ladder operators; we can write
5.flavour-vev of the stress-energy tensor has been re-    
ducedbyactingwiththeflavourladderoperatorson (cid:88) (cid:88)
the flavour vacuum. Gθ(t) πi2(x)G†θ(t)= πi2(x), (41)
i=1,2 i=1,2
However, such a long procedure can be avoided and the
sameexactresultmaybeobtainedinamuchshorterway,
following the steps:  
1.flavour vacuum is written as |0(cid:105)f = G†θ|0(cid:105), the Gθ(t)(cid:88) (cid:16)∇(cid:126)φi(x)(cid:17)2G†θ(t)=
flavour-vev of T becoming a vev of O ≡G T G†; i=1,2
µν θ µν θ
2.the stress-energy operator is transformed under the  
action of the Gθ(cid:4)G†θ, using (9); the transformed op- (cid:88) (cid:16)∇(cid:126)φi(x)(cid:17)2 (42)
erator O will be expressed in terms of the flavour
i=1,2
fields, rather then the massive ones;
3.(3) is used to write the flavour fields in terms of the and
massive fields; O is again expressed in terms of the
massive fields, but this time the operator G , and (cid:104)0|G (t)(m2φ2(x)+m2φ2(x))G†(t)|0(cid:105)=
θ θ 1 1 2 2 θ
its complicated exponential structure, is not present (cid:104)0|(m2φ2(x)+m2φ2(x))|0(cid:105)+
any more; 1 1 2 2
4.a vev of O, expressed as a simple combination of sin2θ(m21−m22)(cid:104)0|(φ22(x)−φ21(x))|0(cid:105). (43)
massive fields, is left, which can be reduced by de-
It follows that
composingthemassivefieldsintomassiveladderop-
erators.
(cid:104)0|T (x)|0(cid:105) =(cid:104)0|T (x)|0(cid:105)+
f 00 f 00
As a neat example of the above procedure, we will
+sin2θ(m2−m2)(cid:104)0|(φ2(x)−φ2(x))|0(cid:105) (44)
derive the results of Section IIA via this new method. 1 2 2 1
10
and therefore its (anti-)commutation rules, regardless of the tensorial
or spinorial structure of the field itself. Furthermore, ex-
ρ = (cid:104)0|:T (x):|0(cid:105) = tending these results to more then two flavours is rather
b f 00 f
straightforward.
=sin2θ(m2−m2)(cid:104)0|(φ2(x)−φ2(x))|0(cid:105) (45)
1 2 2 1 Finally, the method enables us to distinguish among
all the terms of the stress-energy tensors the ones that
Equivalently for T (x) we have
jj
reallycontributetothefinalresult. Thismightbehelpful
in understanding the behaviour of the flavour vacuum in
P = (cid:104)0|:T (x):|0(cid:105) =−sin2θ(m2−m2)×
b f jj f 1 2 morerealistictheories,asweshallseeintheforthcoming
×(cid:104)0|(φ22(x)−φ21(x))|0(cid:105)=−ρb. (46) sections.6
Oncethefieldsin(45)and(46)aredecomposedinterms
of the ladder operators and the quantum algebra is sim- B. Self-interactive Bosons
plified, expressions (22) and (23) are correctly repro-
duced.
Anexampleoftheapplicationsjustdiscussedisoffered
In the fermionic case, a similar procedure leads to
by a λφ4 model. The theory
ρf =f (cid:104)0|:T0f0(x):|0(cid:105)f =sin2θ(m1−m2)× L= (cid:88) (cid:0)∂µφi∂µφi−m2iφ2i −λφ4i(cid:1) (53)
×(cid:104)0|(cid:0)ψ¯2(x)ψ2(x)−ψ¯1(x)ψ1(x)(cid:1)|0(cid:105) (47) i=1,2
and can be regarded as derived from a model with flavour
mixing:
P = (cid:104)0|:Tf(x):|0(cid:105) =0. (48)
f f jj f
L=∂ φ ∂µφ +∂ φ ∂µφ −m2φ2 −m2φ2+
µ A A µ B B A A B B
By comparing (47) and (45), the analogy between the
(cid:88)
fermionic and the bosonic condensate that earlier was −m2ABφAφB − gικλρφιφκφλφρ (54)
hidden in formulae (22) and (23) is now more evident. ι,κ,λ,ρ=A,B
Again,formula(23)iscorrectlyreproduced,oncetheop-
with the usual rotation
eratorial structure of the fields is simplified with respect
to (cid:104)0|(cid:4)|0(cid:105). The expression (47) dispels any doubts con- φ =cosθφ −sinθφ
A 1 2
cerning formula (23) and its possible dependency on the (55)
φ =sinθφ +cosθφ
specific form of the gamma matrices and spinors used to B 1 2
achievetheresultsof[54],being(47)independentofsuch
and a specific choice of the coupling constants g(cid:4). Since
a choice [55].
theexpressionofG intermsofthefieldscanbededuced
θ
Furthermore, Supersymmetry enables us to rewrite
from
this result in terms of the bosonic fields only. For the
massive vacuum |0(cid:105) we know that G†φ G =cosθφ −sinθφ
θ 1 θ 1 2 (56)
(cid:104)0|Tµν(x)|0(cid:105)=0 (49) G†θφ2Gθ =sinθφ1+cosθφ2
which leads to justusingcommutationrelationsbetweenfieldsandcon-
jugate momenta, which are not modified by the form of
(cid:104)0|ψ¯(x)ψ (x)|0(cid:105)=−4m (cid:104)0|φ2(x)|0(cid:105) (50) the Lagrangian [55], expression
i i i i
and hence G =eiθ(cid:82)d(cid:126)x(φ˙2φ1−φ˙1φ2) (57)
θ
ρ =2sin2θ(m −m )2(cid:104)0|(cid:0)φ2(x)+φ2(x)(cid:1)|0(cid:105) (51)
WZ 1 2 1 2
P =−2sin2θ(m2−m2)(cid:104)0|(φ2(x)−φ2(x))|0(cid:105) (52) 6 In addition, we would like to stress that the method does not
WZ 1 2 2 1 requireanexplicitdecompositionoftheflavourfieldsintermsof
flavourladderoperators. Suchadecompositionhasbeenobject
in accordance with (20) and (21).
ofadebateinliterature,raisedbytheauthorsof[31]. Although
The procedure exemplified in the previous section can
the problem was exhaustively discussed in [20] and [32], not all
beeasilyimplementedforotherfields: onemightwantto the community was convinced by the arguments presented [34].
considerDiracortwocomponentWeylspinorsaswellas Without entering into the details of the dispute, here we would
complex scalar fields, getting to analogous results. For like to suggest that a different point of view on the formalism,
suchastheoneofferedbyformulae(47)and(45),whereanob-
mere speculative reasons, applications to vector fields
servablequantityconcerningaflavourstatehasbeencalculated
or even more complex objects might be thought: the withouttheexplicituseofthecontroversialdecomposition,might
methodinvolvesamanipulationofthestress-energyten- helpinadeeperunderstandingoftheproblemandtheformalism
sor, with the use of equation of motion of the field and itself.