Thomas Durt
email: thomas.durt@centrale-marseille.fr

Antonio Di Domenico
email: antonio.didomenico@roma1.infn.it

Beatrix C Hiesmayr
email: beatrix.hiesmayr@univie.ac.at

Falsification of a Time Operator Model based on Charge-Conjugation-Parity violation of neutral K-mesons

Keywords: decay time, CP violation, kaon systems, Time Operator

The founding fathers of the quantum theory already struggled with the different roles of space and time in quantum theory. Position is by default represented by an operator, whereas time is usually treated as a parameter. Time operator models exist, in particular a certain Time Operator model denoted the Temporal Wave Function (T.W.F.). This one extends the Born's rule to the time domain. The statistics of pure exponential decay does not allow to exclude this Born rule for time. However the tiny violation of the combined discrete symmetries P (parity) and C (charge conjugation) observed in the Kmeson system allows us, as we prove in this paper, to discriminate between the standard predictions, made in the framework of the so-called mass-decay matrix formalism, and the T.W.F. predictions. We also present experimental data of a particular CP violating decay channel, that is in contrast with the prediction of T.W.F. approach model, resulting in its falsification.

Introduction.

In the present paper, we investigate fundamental aspects of time, which is still a controversial issue in the quantum theory (see for example Ref. [START_REF]Time in Quantum Mechanics[END_REF]).

Our main goal is to discriminate the Temporal Wave Function (T.W.F.) approach [START_REF] Courbage | A wave-function model for the CP-violation in mesons[END_REF][START_REF] Durt | Crucial tests of the existence of a Time Operator[END_REF][START_REF] Durt | Correlations of decay times of unstable quantum systems[END_REF] from the standard approach applied to decaying meson-systems. The T.W.F. approach is in the last resort motivated by the idea that, similar to the complementary observables assigned to the position and momentum of a particle in the non-relativistic formulation of the quantum theory, there would exist an operator representing time which would be complementary to the Hamiltonian. A Time Operator would be the temporal component of a relativistic quadrivector containing as spatial components the observables associated with the position of the particle. As is well known in non-relativistic quantum mechanics, each spatial component is complementary relatively to the corresponding component of the momentum. We expect thus that the Time Operator would be complementary to the Hamiltonian, so that ultimately the quadrivector time-position would be complementary to the quadrivector energy-momentum. The temporal degree of freedom would no longer be an external, classical, parameter in this approach, contrarily to the standard approach. In this view the time of occurence of an event, for instance of the decay of an unstable quantum particle, would become a stochastic variable, distributed according to a generalized Born rule, in analogy with the position of a quantum particle in non-relativistic quantum mechanics [START_REF] Courbage | A wave-function model for the CP-violation in mesons[END_REF][START_REF] Durt | Crucial tests of the existence of a Time Operator[END_REF][START_REF] Durt | Correlations of decay times of unstable quantum systems[END_REF][START_REF] Courbage | Time decay probability distribution of the neutral meson system and CP-violation[END_REF][START_REF] Champenois | Quest for the Time Operator with a single trapped ion[END_REF].

Our main result is the following: a precise measurement of the asymmetry of the decay rates of the two possible strangeness eigenstates K 0 and K 0 into pion pairs, A 2π (t), makes it possible to falsify a Time Operator approach, the T.W.F. approach.

The paper is structured as follows.

In the first section, we motivate the Time Operator approach, summarizing (section 1.1) debates around the status of time that can be traced back to the founding fathers of the quantum theory (Bohr, von Neumann, Dirac, Heisenberg Schrödinger and Pauli). We also explain briefly why exponential decay does not make it possible to end the debate (section 1.2) [START_REF] Champenois | Quest for the Time Operator with a single trapped ion[END_REF]. We also generalize the reasoning to the most general situation of the exponential decay treated as a multi-channel process (section 1.3).

In section 2, we briefly recall the phenomenology of unstable neutral kaon particles and their description in terms of the mass-decay matrix in the standard Weisskopf-Wigner approximation (WWA) [START_REF] Weisskopf | Uber die naturliche linienbreite in der strahlung des harmonische oszillators[END_REF][START_REF] Lee | Remarks on possible noninvariance under time reversal and charge conjugation[END_REF][START_REF] Chiu | Decay and evolution of the neutral kaon[END_REF].

In section 3 we confront the standard WWA formulation of decay processes to the Time Operator approach. We show how a certain model of Time Operator, denoted the T.W.F. approach, can agree with the standard WWA predictions in the neutral kaon system -including the description of kaon oscillations in semi-leptonic decay-provided we neglect the violation of the combined parity (P ) and charge conjugation (C) symmetries (from now on CP violations). Moreover, we show that the precise measurement of the A 2π (t) asymmetry at short times performed by the CPLEAR experiment [START_REF]Physics at CPLEAR[END_REF][START_REF] Gabathuler | The CPLEAR experiment[END_REF] -in perfect agreement with the WWA description -does not confirm the prediction of our T.W.F. model, which results therefore falsified.

The last section is devoted to discussions and conclusions of the falsification.

Extra-material can be found in appendix, which contains a detailed treatment of the Fermi golden rule when the decay is a multi-channel process.

1 Standard and non-standard conceptions of time in quantum mechanics.

1.1 Time in quantum mechanics: is there a problem?

As intensively discussed in the literature, see e.g. Ref. [START_REF] Hilgevoord | Time in Quantum Mechanics, a story of confusion[END_REF], the status of time in the quantum theory has been in the past, and is still to some extent today, the source of much confusion. Let us summarize the discussions of [START_REF] Hilgevoord | Time in Quantum Mechanics, a story of confusion[END_REF], inviting the reader to consult this very detailed reference for more precise explanations and bibliography.

The main source of confusion could be formulated through the question:

...Is time a classical variable (c-number)?..., by which we mean that time can be treated as an external parameter (similar to time as it appears in classical, Newtonian, and/or relativistic, Minkowski space-time).

The other alternative can be formulated as follows:

...Is time a quantum quantity (q-number) represented by an operator?..., by which we mean that it is represented by or associated with an internal parameter (example: phase) of some quantum system, represented by an operator (similar to position components x, ŷ and ẑ in the non-relativistic quantum theory).

As noted in [START_REF] Hilgevoord | Time in Quantum Mechanics, a story of confusion[END_REF], various answers were given by the founding fathers of the quantum theory to this deep foundational issue. Bohr, for instance, treated time as a c-number and concluded that there is no problem in doing so, which is close to the mainstream view of contemporary physicists on the subject. Von Neumann acknowledged that time is treated as a c-number in the first quantized, non-relativistic, quantum theory but at the same time he considered that this is a source of problems, mainly for the following reasons:

-The coordinates x, y, z are described by operators (in the non-relativistic quantum theory), and Lorentz transformation treats space and time on the same footing (see [START_REF] Vaccaro | Quantum asymmetry between time and space[END_REF] for a fresh view on this subject) so that time ought to be treated as a q-number, and, consequently, represented by an operator. Let us e.g. reconsider the usual quantization rule that associates the energy E with i ∂ ∂t , p x to ∂ i∂x , p y to ∂ i∂y , and p z to ∂ i∂z . It obviously possesses a strong relativistic flavour, which suggests that time ought to be represented by an operator conjugate to the Hamiltonian.

Dirac also felt strongly concerned by this question and, as noted in [START_REF] Hilgevoord | Time in Quantum Mechanics, a story of confusion[END_REF], he wrote his famous equation in order to formulate a Lorentz covariant quantum theory (of the electron), where space and time would be treated on the same footing. Finally, Pauli remarked that when an Hamiltonian possesses a continuous bounded spectrum, it is not possible to construct an operator T such that [ Ĥ, T ] = i Î. In other words, it is not always possible to treat a quantum system as a clock.

As it was noted by J. Hilgevoord [START_REF] Hilgevoord | Time in Quantum Mechanics, a story of confusion[END_REF], many discussions about the status of time focused on Pauli's former objection that was since then interpreted as a no-go theorem about the existence of a Time Operator. In the meanwhile, the question of the existence of a Time Operator somewhat lost interest [START_REF]Time in Quantum Mechanics[END_REF], and we can claim without risk that the main opinion today is that the time that appears in Schrödinger equation IS a classical variable (c-number), and can thus be considered as an external parameter (universal or general time). Several theoretical arguments invite us indeed to orient the debate to this conclusion, and it is commonly accepted nowadays that in quantum field theory time and even space [START_REF] Newton | Localized States for Elementary Systems[END_REF] correspond to classical parameters (the spacetime "arena"). Despite of these theoretical arguments, some confusion remains concerning these questions [START_REF]Time in Quantum Mechanics[END_REF] and there were several attempts to associate with time an operator [START_REF]Time in Quantum Mechanics[END_REF][START_REF] Bauer | A Time Operator in quantum mechanics[END_REF] or a positive-operator valued measure (POVM) [START_REF] Kijowski | On the Time Operator in quantum mechanics and the heisenberg uncertainty relation for energy and time[END_REF][START_REF] Moyer | Timelines and Quantum Time Operators[END_REF] (or Superoperator [START_REF] Courbage | On necessary and sufficient conditions for the existence of time and entropy operators[END_REF][START_REF] Misra | Lyapounov variable: Entropy and measurement in quantum mechanics[END_REF]) in order to deal with arrival times and/or decaying processes 4 (photon in a cavity, particle tunneling from a trap, decaying radio-active particle and so on).

Two approaches to exponential decay.

In the non-standard approach (Time Operator and/or Superoperator approaches), the time of decay of an individual quantum unstable system is considered to be a quantum quantity (q-number), an internal parameter, represented by an operator or a Superoperator. The Time Super Operator approach of Misra, Prigogine and Courbage et al. [START_REF] Courbage | On necessary and sufficient conditions for the existence of time and entropy operators[END_REF][START_REF] Misra | Lyapounov variable: Entropy and measurement in quantum mechanics[END_REF]), makes it for instance possible to associate a generalized Time Operator (Super Operator) with any Hamiltonian provided its spectrum is not bounded by above. It has been applied to kaon phenomenology in the past [START_REF] Courbage | Time decay probability distribution of the neutral meson system and CP-violation[END_REF]. In this paper we shall consider a simpler model, which fits into the Time Operator approach, through which we associate with the statistical distribution of decay processes a temporal wave function (T.W.F. [START_REF] Courbage | A wave-function model for the CP-violation in mesons[END_REF][START_REF] Durt | Crucial tests of the existence of a Time Operator[END_REF][START_REF] Durt | Correlations of decay times of unstable quantum systems[END_REF][START_REF] Courbage | Time decay probability distribution of the neutral meson system and CP-violation[END_REF][START_REF] Champenois | Quest for the Time Operator with a single trapped ion[END_REF]). This model is for instance very natural and straightforward in the case of exponential decay [START_REF] Durt | Crucial tests of the existence of a Time Operator[END_REF], in which case it is common, following Gamow, to associate with the unstable system a wave function of the form Ψ S (t) = Ψ S (0)exp -i( mc 2

-i Γ 2 )•t
which is interpreted as the amplitude of a complex energy state.

According to the standard interpretation, |Ψ S (t)| 2 |Ψ S (0)| 2 is interpreted to be equal to the survival probability P s (t) up to time t (0 ≤ t).

Alternatively, let us define the Temporal Wave Function ΨT.W.F. (t) through5

ΨT.W.F. (t) = ΨT.W.F. (0)exp -i( mc 2 -i Γ 2 )•t , (1)
with ΨT.W.F.

(0) = √ Γ. It is straightforward to check that -dP s (t) dt = | ΨT.W.F. (t)| 2 . (2)
The choice ΨT.W.F. (0) = √ Γ also ensures that the survival probability decreases from its initial value (1) at time t = 0 to its final value (0), when t goes to +∞.

As we see, by a formal renormalisation, we can define a temporal wave function ΨT.W.F. (t) (T.W.F.), normalised to unity over time in the interval [0, ∞], such that the probability of decay by unit of time (probability density function) is equal to its modulus squared, in full analogy with the usual Schrödinger wave function in position representation. Considered so, the T.W.F. provides a time representation assigned to the distribution of decay times of an unstable quantum system.

It is worth noting that this is not only a formal trick, but that this procedure resists to experimental falsification in the case of purely exponential decay. Indeed, lifetimes of particles are often measured indirectly in particle physics, by fitting the energy distribution of decay products with a Breit-Wigner (Lorentzian) distribution. Also in this case the standard and Time Operator approaches cannot be distinguished [START_REF] Champenois | Quest for the Time Operator with a single trapped ion[END_REF]. Roughly summarized, the argument goes as follows:

-Suppose that the Hamiltonian is time-independent and that at time t = 0 the amplitude of probability that the particle energy is E equals ψ E (E). Then, developing the wave function in the energy eigenbasis it is straightforward to show that

P s (t) = | < ψ(0), ψ(t) > | 2 = | Emax. E min. dEe -i Et |ψ E (E)| 2 | 2
, where the spectrum of the Hamiltonian runs from E min. to E max. . When the spectrum of the Hamiltonian is large enough

| Emax. E min. dEe -i Et |ψ E (E)| 2 | is close to | +∞ -∞ dEe -i Et |ψ E (E)| 2 |
and the survival probability is equal to the squared modulus of the Fourier transform of the energy distribution. Now, the characteristic function (Fourier transform) of the Breit-Wigner distribution is equal to C • e -i(mt-i 2 Γ|t|) . Thus fitting with a Breit-Wigner distribution is consistent with the "standard" recipe:

P s (t) = | < ψ(0), ψ(t) > | 2 (disregarding negative times).
-Assume that the decay probability density function is associated with a "temporal wave function": -dPs(t)

dt = Γθ(t)e -Γt =| ΨT.W.F. (t)| 2 with | ΨT.W.F. (t)| 2 =| √ Γθ(t)e -i(m-i 2 Γ)t) | 2 (where θ(t) is the Heaviside function). Then, | ΨT.W.F. (t)| 2 =| +∞ -∞ dEe +i Et ΨT.W.F. (E)| 2 , with ψT.W.F. (E)= 1 2π +∞ -∞ dte -i Et ΨT.W.F. (t)=-i Γ 2π 1 (E-(m-i 2 Γ))
. The modulus squared of ψT.W.F. (E) is precisely equal to the Breit-Wigner distribution:

| ψT.W.F. (E)| 2 =N. 1 (E-m) 2 +(Γ/2) 2
with N a normalisation factor. To fit with a Breit-Wigner distribution is thus also consistent with the "non-standard" recipe: -dPs(t)

dt = Γθ(t)e -Γt =| ΨT.W.F. (t)| 2 .
It is worth noting that in the previous paragraph we performed two different Fourier transforms: in one case we transformed amplitudes, in the other case we transformed the modulus square of these amplitudes. By chance, in the case of exponential decay, both approaches lead to the same prediction. In order to study more in depth the two approaches, it is thus necessary to investigate more complex decay processes which exhibit a non-exponential statistics. This motivates our choice to consider neutral kaons which show also subtle non trivial effects due to CP violation.

In order to tackle these more sophisticated decay processes, we shall have to generalize the T.W.F. model described here in the case of exponential decay. This model will be outlined in section 3.1, on the basis of kaon phenomenology that we outline in section 2.

Before doing that we will show in the next paragraph how the T.W.F. approach can be applied in the case where several decay channels are present, a situation that has not been investigated in previous literature. In particular we show how the summation rule of the partial decay widths Γ i assigned to various decay channels can be accomodated in the framework of the T.W.F. approach.

1.3 Multi-channel exponential decay processes.

1.3.1 Multi-mode exponential decay processes in the standard approach.

In the standard approach, a Γ factor is associated with each outgoing mode (decay product) through the Fermi golden rule. As it is shown in appendix, in the case of multi-channel exponential decay processes, each individual decay mode contributes to the decay of the survival probability by a factor which can be shown, at the first order of perturbation theory, to increase proportionally to

|G i,λ | 2 t 2 sinc 2 (ω i,λ -ω in ) t
2 , where G i,λ represents the coupling constant with the mode labelled by λ of the ith decay species. Summing over internal degrees of freedom λ of different decay products i and also over the energies ω, which is taken into account through the introduction of the D.O.S. (density of states) factor aimed at converting the discrete sum into an integral, via the prescription D.O.S.(ω in , i, λ)δω=number of modes labelled by i and λ with an energy comprised between (ω in -δω/2) and (ω in + δω/2), and making use of the property [START_REF] Englert | Lectures on Quantum Mechanics[END_REF] lim

t→+∞ (t sinc 2 δω t 2 ) = 2πδ Dirac (δω). (3)
We find, after integrating over a continuum of frequencies centered around ω in , i,λ,ω

|c i,λ (t) | 2 = lim t→+∞ i,λ +∞ -∞ dωD.O.S.(ω, i, λ)|G i,λ (ω)| 2 t 2 sinc 2 (ω in -ω) t 2 = ... = i Γ i t = Γt. (4)
We observe that i,λ,ω |c i,λ (t) | 2 increases linearily with time. In other words the amplitude that a decay occurs increases with √ t which is the essence of the Fermi Golden Rule.

The important lesson that must be retained from ( 4) is that the resulting Γ factor is the sum of the contributions of all individual modes (integrated over species and internal degrees of freedom). In particular, Γ = i Γ i where i labels the different species of decay products (for instance for the decay process of a kaon distinct values of i are assigned e.g. to π + π -, π 0 π 0 , π ν, 3π 0 , π + π -π 0 , ... decay products. Γ i in turn obeys

Γ i = λ D.O.S.(ω in , i, λ)2π|G i,λ (ω in )| 2 ,
where λ does label the internal degrees of freedom of the species under concern, in agreement with the Fermi Golden Rule.

1.3.2 Multi-channel exponential decay processes in the T.W.F. approach.

As it is shown in appendix, the multi-mode exponential decay process does not fundamentally differ from a single mode process, being given that only one outgoing mode (|ef f. >) is effectively coupled to the ingoing mode. Of course, this effective mode is in general a coherent superposition of modes of different species, which will never interfere in practice, so that in practice we can as well consider that the sum over the different possible outgoing channels is incoherent.

Then, when the Fermi golden rule applies, it is straightforward to interpret a generation of decay products labelled by the index i at a rate Γ i dt either in the standard approach where this rate is equal to | √ Γ i √ dt| 2 in agreement with Fermi Golden Rule, or in the T.W.F. approach where this rate is seen to be equal to | √ Γ i | 2 dt. The so-called sum rule for the Γ i factors is seen here to derive from (4).

2 Single kaons and CP-violation in the standard approach.

2.1 Short overview of kaon phenomenology.

Neutral Kaons are bosons that were discovered in the forties during the study of cosmic rays.

They are constituted by a quark-antiquark pair, K 0 (= ds) with strangeness S = +1, and K 0 (= ds) with strangeness S = -1. They are usually produced in processes dominated by strong interactions, while weak interactions -which do not conserve strangeness -are responsible for their decay or induce oscillations through transitions with an intermediate virtual state of the type K 0 ↔ ππ ↔ K 0 .

In absence of CP -violation, the following combinations

|K 1 = 1 √ 2 |K 0 -|K 0 , |K 2 = 1 √ 2 |K 0 + |K 0 , (5)
which are eigenstates of the CP operator, are also eigenstates of the total Hamiltonian of the system.

In 1964 Christenson et al. discovered [START_REF] Christenson | Evidence for the 2π decay of the K 2 meson[END_REF] that the CP symmetry is slightly violated (by a factor of O(10 -3 ) ) in the neutral kaon system, so that the CP eigenstates K 1 and K 2 do not exactly coincide with the eigenstates of the Hamiltonian. In fact these contain a small CP impurity and can be expressed as coherent superpositions of the K 1 and K 2 states6 :

|K S = 1 1 + | | 2 |K 1 + |K 2 , (6)
|K L = 1 1 + | | 2 |K 1 + |K 2 , (7)
where is a complex CP -violation parameter [23],

| | = (2.228 ± 0.011) × 10 -3 , arg( ) = (43.5 ± 0.5) • . (8)
K S state and K L are the short-and long-lived states, with quite different lifetimes, τ S ≈ 8.92 × 10 -11 s and τ L ≈ 5.17 × 10 -8 s, respectively.

Direct CP violation in the decay is negligible [23,[START_REF] Abouzaid | Precise measurements of direct CP violation, CPT symmetry, and other parameters in the neutral kaon system[END_REF], therefore one can distinguish the K 1 state, which cannot decay into 3π 0 , a pure CP = -1 final state, from the K 2 one, which cannot decay into ππ (π + π -or π 0 π 0 ), a pure CP = +1 final state. Moreover, the sign of the lepton (or of the pion) in semileptonic decays, π -+ ν or π + -ν, exclusively identifies a K 0 or a K 0 state, respectively, assuming the validity of the ∆S = ∆Q rule.

Inverting equations [START_REF] Champenois | Quest for the Time Operator with a single trapped ion[END_REF][START_REF] Weisskopf | Uber die naturliche linienbreite in der strahlung des harmonische oszillators[END_REF] in the basis |K 0 and |K 0 , we get

|K 0 = 1 + | | 2 √ 2(1 + ) |K S + |K L , (9)
|K 0 = 1 + | | 2 √ 2(1 -) |K S -|K L . ( 10
)
2.2 Mass-decay matrix.

In the WWA mass-decay formalism, it is assumed that the evolution law describing the surviving kaon can be cast in Schrödinger form [START_REF] Weisskopf | Uber die naturliche linienbreite in der strahlung des harmonische oszillators[END_REF] i

∂ ∂t |ψ(t) = H|ψ(t) , (11)
where H, the effective, not hermitian, Hamiltonian, can be written as a mass-decay matrix [START_REF] Lee | Remarks on possible noninvariance under time reversal and charge conjugation[END_REF][START_REF] Chiu | Decay and evolution of the neutral kaon[END_REF] which has the following form (for instance in the basis |K 0 and |K 0 , taking here = c = 1):

H = M - i 2 Γ ≡ M 11 -i 2 Γ 11 M 12 -i 2 Γ 12 M 21 -i 2 Γ 21 M 22 -i 2 Γ 22 (12)
where M and Γ are individually Hermitian. Γ incorporates contributions from all possible decay channels. For instance, the decay of a |K 1 (|K 2 ) state into ππ (3π 0 ) contributes to the mass-decay matrix by a factor Γ K

1 →ππ |K 1 K 1 | (Γ K 2 →π 0 π 0 π 0 |K 2 K 2 |).
In principle all Γ factors associated with the various decay channels can be computed thanks to the Fermi Golden Rule.

The Γ matrix can then be expressed by summing the contributions from all possible decay channels:

Γ = i Γ |K i →i |K i K i | (13)
For instance, we find, writing explicitly some of the decay products:

Γ = i Γ |K i →i |K i K i | ≈ Γ |K 1 →2π |K 1 K 1 | + Γ |K 2 →3π |K 2 K 2 | + Γ |K 0 →π -+ ν |K 0 K 0 | + Γ | K0 →π + -ν | K0 K0 | + ...
If for instance we have to compute the decay rate that a prepared K 2 state decays into a + through a semi-leptonic decay process, we find:

Γ |K 2 →π -+ ν = K 2 |Γ|K 2 π -+ ν = Γ |K 0 →π -+ ν • K 2 |K 0 K 0 |K 2
K L and K S are the eigenstates of H. The eigenvalues of the mass-decay matrix corresponding to the long and short states are equal to

E L = m L - i 2 Γ L , and E S = m S - i 2 Γ S . (14)
Reintroducing dimensional factors we get

E L = m L c 2 - i 2 Γ L , and E S = m S c 2 - i 2 Γ S . (15)
These features are taken into account by noting that when a |K 0 state is prepared at time t = 0, the state of the kaon at time t is, in virtue of equations [START_REF] Gabathuler | The CPLEAR experiment[END_REF][START_REF] Champenois | Quest for the Time Operator with a single trapped ion[END_REF][START_REF] Weisskopf | Uber die naturliche linienbreite in der strahlung des harmonische oszillators[END_REF][START_REF] Chiu | Decay and evolution of the neutral kaon[END_REF], equal to

|K 0 (t) = 1 + | | 2 √ 2(1 + ) (exp -iE S t/ |K S + exp -iE L t/ |K L ) (16)
Accordingly, in the WWA standard approach, the probability per unit of time that at time t the kaon particle decays say in the CP=+1 sector by emission of a pair of pions can be computed by estimating the corresponding contribution to the decay matrix which is, in virtue of ( 14),

equal to Γ K 1 →ππ • | K 1 |K 0 (t) | 2
where Γ K 1 →ππ is the probability of transition per unit of time (or production rate) between a K 1 state and a pair of pions.

Thus we find:

P K 0 (0)→2π(t) = Γ K 1 →ππ 2|1 + | 2 |exp -iE S t/ + exp -iE L t/ | 2 , (17)
where 2π(t) denotes the production rate at time t of a pair of pions (for instance 2π = π 0 π 0 , π + π -).

Anagously for the CP = -1 decay into 3π 0 we get:

P K 0 (0)→3π(t) = Γ K 2 →π 0 π 0 π 0 2|1 + | 2 |exp -iE L t/ + exp -iE S t/ | 2 . ( 18
)
The production rates of all possible species of decay products can be in principle computed in a similar fashion.

3 Experimental proposal: falsifying the Time Operator approach in presence of CP violation related effects.

3.1 Time Operator Approach: a binary model.

Let us now stick to the Time Operator approach, in order to investigate whether or not the statistical distribution of decay times exhibited by a neutral kaon might be derived from a "temporal wave function". This T.W.F. would constitute a generalisation in the time domain of Born's interpretation of the quantum wave function as representing an amplitude of probability of "spatial position". According to the Born rule, the probability that at a given time a particle is located around a given position is supposed to be proportional to the modulus squared of a complex field, the so-called quantum wave function. In the temporal wave function approach the probability that a "click" occurs at a given time in a given detector is also supposed to be linked to a complex field, the Temporal Wave Function, and we shall propose in the next section an experiment aimed at challenging/falsifying this hypothesis. Before we do so, let us develop a Time Operator formalism aimed at modeling neutral kaon phenomenology.

Our model which has been developed elsewhere [START_REF] Courbage | A wave-function model for the CP-violation in mesons[END_REF][START_REF] Durt | Crucial tests of the existence of a Time Operator[END_REF][START_REF] Durt | Correlations of decay times of unstable quantum systems[END_REF] is directly inspired from the spin 1/2 formalism, in particular from the non-relativistic spinorial wave function approach developed by Pauli. In this approach, the wave function of a spin 1/2 particle splits into two components, the spin "up" component Ψ + (x, y, z, t) and the spin "down" component Ψ -(x, y, z, t), relatively to an a priori defined axis of reference. Both components depend on space and time, and according to the standard interpretation, the probability that at time t and location (x, y, z) a measurement of the spin projection along the axis of reference (traditionally denoted the z axis) provides the result spin up (down) is equal to

|Ψ + (x, y, z, t)| 2 /(|Ψ + (x, y, z, t)| 2 +|Ψ -(x, y, z, t)| 2 ) (|Ψ -(x, y, z, t)| 2 /(|Ψ + (x, y, z, t)| 2 + |Ψ -(x, y, z, t)| 2 )).
Besides, the probability that the particle is present in a neighbourhood dxdydz around (x, y, z) is equal to

(|Ψ + (x, y, z, t)| 2 + |Ψ -(x, y, z, t)| 2 )dxdydz.
The Temporal Wave Function differs from the standard spin 1/2 wave function considered above in the following sense: the binary value taken by the spin projection along an axis of reference (± /2) is in our case replaced by the CP eigenvalue (±1), while space is replaced by time. Accordingly, instead of dealing with probabilities to observe spin up or spin down we are now considering the probabilities that a single kaon decays in the CP=+1 or -1 sectors. We assume thus in the T.W.F. approach that there exists a two-components temporal wave function ( ΨT.W.F.

+ (t), ΨT.W.F. - (t)
) such that the probability that the kaon decays by emitting a CP=+1 (-1) labelled decay product in the time interval [t, t + dt] (where t ≥ 0 and dt > 0) is equal to | ΨT.W.F.

+ (t)| 2 dt (| ΨT.W.F. - (t)| 2 dt
). This binary model is somewhat reductionist because it implicitly assumes that only CP labelled decay products are emitted during the decay process of a kaon. A priori, this is a very crude assumption, because it is known for intance that a K L decays 67 % of the times into semileptonic decays with undefined CP value, and only 20 % of the times in 3π 0 , and 12 % of the times into π + π -π 0 which both have a well-defined CP-value.

As we shall discuss now, even semi-leptonic decay can be accomodated in the framework of the T.W.F. model, so that our reductionist hypothesis is fully justified as far as we may consistently neglect CP violation.

Without CP violation. If there were no CP violation ( = 0), then, in the standard WWA approach, the mass matrix and the decay matrix as well would both be diagonal in the CP eigen-basis (|K 1 , |K 2 ): |K S = |K 1 , |K L = |K 2 and only CP-labelled decay products would be emitted during the decay process, for instance CP=+1 π 0 π 0 , π + π -states and CP=-1 π 0 π 0 π 0 states, but also 1

√ 2 • (|π -+ ν + |π + -ν ) and 1 √ 2 • (|π -+ ν -|π + -ν ), which are CP=±1 decay products.
In the case of the semileptonic decay this approach is not conventional, but in practice it is equivalent to the usual approach, for several reasons.

The first reason is that, resulting from the fact that semileptonic decay is a CP respecting process,

Γ |K 0 →π -+ ν = Γ | K0 →π + -ν so that Γ |K 0 →π -+ ν |K 0 K 0 | + Γ | K0 →π + -ν | K0 K0 | = Γ |K 0 →π -+ ν (|K 1 K 1 | + |K 2 K 2 |).
The second reason is that, making use of the linearity of Schrödinger equation, it is equivalent to describe a decay process of the type |K 0 → |π -+ ν (| K0 → |π + -ν ) as a symmetric (antisymmetric) fifty fifty coherent superposition of the process

|K 1 → (1/ √ 2)(|π -+ ν + |π + -ν ) with the process |K 2 → (1/ √ 2)(|π -+ ν -|π + -ν ).
Finally, it is worth noting that detectors are not sensitive to the phase between π -+ ν and π + -ν decay products, which are experimentally detected in different outgoing channels, as already emphasised in the section 1.3.2, where we showed that in last resort decay products are actually coherent superpositions of distinct decay species. Taken all together, these features make it possible to properly reproduce strangeness oscillations during the semileptonic decay in the framework of the T.W.F. approach which is in a sense remarkable for such a crude model.

From now on, we shall only be concerned with the 2π production rate, due to the fact that this quantity has been measured with very high accuracy. This does not mean that we fully neglect other decay channels. Actually they are still implicitly present through the normalisation which imposes the following conditions:

i=allpossibleCP+1decayproducts Γ K 1 →i = Γ S = Γ 1 j=allpossibleCP-1decayproducts Γ K 2 →j = Γ L = Γ 2
In absence of CP violation, the expression of the CP=±1 components of the temporal wave function in terms of complex energies E S(L) would in turn be extremely simple:

( ΨT.W.F. =0 + (t), ΨT.W.F. =0 - (t)) = (α Γ S exp -iE S t/ , β Γ L exp -iE L t/ ), (19)
where α and β are two complex amplitudes of which the value is fixed by the preparation of the kaon at time t = 0.

For instance, whenever a source produces |K 0 states at time t = 0, which are fifty-fifty superpositions of CP=+1 and -1 states, (see e.g. equations [START_REF] Courbage | Time decay probability distribution of the neutral meson system and CP-violation[END_REF][START_REF] Champenois | Quest for the Time Operator with a single trapped ion[END_REF][START_REF] Kijowski | On the Time Operator in quantum mechanics and the heisenberg uncertainty relation for energy and time[END_REF][START_REF] Moyer | Timelines and Quantum Time Operators[END_REF][START_REF] Courbage | On necessary and sufficient conditions for the existence of time and entropy operators[END_REF])), we would have

α = β = 1/2.
The tilded expressions7 differ from the non-tilded ones by the normalisation factors

√ Γ S and √ Γ L : (α, β) = (Ψ + (t = 0), Ψ -(t = 0)) = ( ΨT.W.F. =0 + (t = 0)/ √ Γ S , ΨT.W.F. =0 - (t = 0)/ √ Γ L ).
It is easy to check that the moduli squared of the components of ( ΨT.W.F. =0 + (t), ΨT.W.F. =0 -(t)) correctly reproduce [START_REF] Durt | Crucial tests of the existence of a Time Operator[END_REF] the standard predictions concerning decays in the CP=±1 sectors [START_REF] Moyer | Timelines and Quantum Time Operators[END_REF][START_REF] Courbage | On necessary and sufficient conditions for the existence of time and entropy operators[END_REF] in the case = 0, for instance we get

P =0 K 0 (0)→ππ(t) = Γ K 1 →ππ 2 |exp -iE S t/ | 2 = | Γ K 1 →ππ Γ S ΨT.W.F. =0 + (t)| 2 = P =0 K 0 (0)→ππ(t) , (20)
where we dropped the T.W.F. subscripts but kept the "tilded" notations for the production rates in the CP=+1 and -1 sectors derived in the T.W.F. approach, in order to differentiate them from the "non-tilded" predictions of the standard WWA approach [START_REF] Moyer | Timelines and Quantum Time Operators[END_REF][START_REF] Courbage | On necessary and sufficient conditions for the existence of time and entropy operators[END_REF]). We made use here of the results of section 1.3.2 according to which the T.W.F. amplitude of a particular species of decay products (i) is equal to √ Γ i . In the next section we shall show that in the case of departure from exponential decay due to CP violation, the standard and "tilded" predictions made in the framework of the binary T.W.F. model presented in the previous section differ so that a discrimination is possible.

With CP violation.

If now we wish to incorporate CP related effects into our T.W.F. model, we are forced to introduce a short-lived and a long-lived T.W.F.:

| KS = 1 1 + |˜ S | 2 |K 1 + ˜ S |K 2 ; | KL = 1 1 + |˜ L | 2 ˜ L |K 1 + |K 2 , (21)
It is worth noting that we can express the short-lived and long-lived states in this way without losing generality, because the four amplitudes KS |K 1 , KS |K 2 , KL |K 1 , and KL |K 2 obviously differ from zero. Expressed in matricial form, the corresponding pseudo-spinorial T.W.F. explicitly depend on time through ( ΨT.W.F.

+S

(t), ΨT.W.F.

-S

(t)) = 1 1 + |˜ S | 2 Γ S exp -iE S t/ (1, ˜ S ), (22)
( ΨT.W.F.

+L (t), ΨT.W.F. -L (t)) = 1 1 + |˜ L | 2 Γ L exp -iE L t/ (˜ L , 1), (23)
where the indices + and -refer to the CP=+1 and -1 sectors respectively. ˜ S and ˜ L are complex CP -violation parameters. According to the well-established kaon phenomenology, CP violation is small so that they are supposedly small parameters. Otherwise, we do not assume anything about their (complex) value 8 .

Consequently, we shall consider in the following temporal wave functions of the form ( ΨT.W.F.

+ (t), ΨT.W.F. - (t)) = (α Γ S exp -iE S t/ + β˜ L Γ L exp -iE L t/ , β Γ L exp -iE L t/ + α˜ S Γ S exp -iE S t/ ), (24)
where α and β are fixed by the initial preparation process.

3.2 Experimental discrimination between the WWA and T.W.F. predictions.

3.2.1 Binary T.W.F. predictions in presence of CP-violation.

Having in mind that CP-violation is small, it is temptating to generalize our binary T.W.F. model in presence of CP violation to a binary model where all decay products would have a clear CP signature.

In the binary T.W.F. approach, we assume that the probability that a CP=+1 (-1) decay product is produced between time t and time t + dt is equal (in very good approximation) to the decrease of the CP=+1 (-1) component during that period. In particular, when a short (long) state is prepared at time t = 0, the decay is purely exponential and the production rates of CP=+1 ππ decay products are predicted to be equal to :

PK S (0)→2π(t) = | Γ K 1 →ππ Γ S | 2 (1/(1 + |˜ S | 2 ))Γ S exp -Γ S t dt ( PK L (0)→2π(t) = | Γ K 1 →ππ Γ S | 2 (|˜ L | 2 /(1 + |˜ L | 2 ))Γ L exp -Γ L t dt).
This is an assumption motivated by the fact that most often the decay process is CP-respecting, which justifies to "import" here an equation ( 20) valid in the = 0 limit. From this point of view the binary T.W.F. model is a minimal extension of the = 0 case9 .

It is instructive to compare these prediction with the standard WWA ones. For instance, when a short (long) state is prepared at time t = 0, the production rate of a ππ pair between time t and time t + dt is: [START_REF] Gabathuler | The CPLEAR experiment[END_REF][START_REF] Champenois | Quest for the Time Operator with a single trapped ion[END_REF][START_REF] Weisskopf | Uber die naturliche linienbreite in der strahlung des harmonische oszillators[END_REF],

P K S (0)→2π(t) = | Γ K 1 →ππ Γ S | 2 (1/(1 + | | 2 ))Γ S exp -Γ S dt (P K L (0)→2π(t) = | Γ K 1 →ππ Γ S | 2 (| | 2 /(1 + | | 2 ))Γ S exp -Γ L dt).
It is already obvious at this level that the standard predictions a priori differ from those made in the binary T.W.F. model. Considering e.g. that , ˜ L and ˜ S are small parameters, we notice that discrepancies occur between the expression of P K L (0)→2π(t) and of its tilded counterpart which is of the order of ( √ Γ S / √ Γ L ) 2 , which is not a small parameter. Discarding all contributions quartic in , S and L which are supposedly small parameters (see also appendix for a more detailed discussion), the contradiction is avoided at first sight provided we assume that

˜ L = Γ S Γ L (25)
However, it is possible in principle to discriminate experimentally the standard predictions from those derived in the binary T.W.F. approach modulo equation ( 25), as we shall show now.

Incompatibility between T.W.F. and WWA approaches.

In agreement with previous notations, let us denote P X(0)→Y (t) the production rate at time t of a specific decay product denoted Y (in what follows, Y most often represents a pair of pions (ππ)), obtained after preparing a kaon in the X (X=K 0 or K 0 ) state at time t = 0, evaluated in the standard WWA approach. Let us denote PX(0)→Y (t) the same rate derived in the framework of the T.W.F. approach.

The initial preparation fixes the coefficients α and β in expression [START_REF] Abouzaid | Precise measurements of direct CP violation, CPT symmetry, and other parameters in the neutral kaon system[END_REF], making use of the identities

(1, 1) = 1 1-˜ S ˜ L [(1 -˜ L )(1, ˜ S ) + (1 -˜ S )(˜ L , 1)] and (1, -1) = 1 1-˜ S ˜ L [(1 + ˜ L )(1, ˜ S ) -(1 + ˜ S )(˜ L , 1
)], which is the non-standard counterpart of the standard expressions (9,10) which can be rewritten in a similar fashion as:

(1, 1) = 1 1+ [(1, ) + ( , 1)] and (1, -1) = 1 1-[(1, ) -( , 1
)]. Straightforward computations similar to those of the previous section show that, in the case that a K 0 state is prepared at time t = 0, the tilded quantities are:

PK 0 (0)→2π(t) = Γ K 1 →ππ 2|1 -˜ S ˜ L | 2 |(1 -˜ L )exp -iE S t/ + (1 -˜ S )˜ L Γ L Γ S exp -iE L t/ | 2 , (26)
to compare with the standard WWA relation ( 17) that we reproduce below

P K 0 (0)→2π(t) = Γ K 1 →ππ 2|1 + | 2 |exp -iE S t/ + exp -iE L t/ | 2 ,
In the case that a K 0 state is prepared at time t = 0 we get:

PK 0 (0)→2π(t) = Γ K 1 →ππ 2|1 -˜ S ˜ L | 2 |(1 + ˜ L )exp -iE S t/ -(1 + ˜ S )˜ L Γ L Γ S exp -iE L t/ | 2 , (27)
to compare with its standard counterpart

P K 0 (0)→2π(t) = Γ K 1 →ππ 2|1 -| 2 |exp -iE S t/ -exp -iE L t/ | 2 , (28)
In order to discriminate the T.W.F. approach from the WWA approach on the basis of experimental data, let us now consider the asymmetry in the production rates of pion pairs, denoted A ππ (t) and defined as follows:

A ππ (t) = P K 0 (0)→2π(t) -P K 0 (0)→2π(t)

P K 0 (0)→2π(t) + P K 0 (0)→2π(t) (29)
In the standard WWA approach, making use10 of equations [START_REF] Moyer | Timelines and Quantum Time Operators[END_REF][START_REF] Bertlmann | Strangeness measurements of kaon pairs, CP violation and Bell inequalities[END_REF], it is easy to show that the asymmetry vanishes when t goes to 0:

lim t→0 A ππ (t) = 0 (30)
On the contrary, if we use equations [START_REF] Bernabeu | Direct test of time reversal symmetry in the neutral kaon system at a φ-factory[END_REF][START_REF] Bertlmann | Bell inequality and CP violation in the neutral kaon system[END_REF] in order to evaluate the same limit in the T.W.F. approach, adopting the value ˜ L = Γ S Γ L in accordance with equation ( 25), we find instead that

lim t→0 Ãππ (t) = ≈ 2Re.((1 - Γ L Γ S ) • ˜ L ) ≈ 2Re(˜ L ) ≈ 8% = 0 (31)
The discrepancy between the predictions ( 30) and [START_REF] Massar | Investigating the origin of time with trapped ions[END_REF] shows the possibility to experimentally discriminate between the standard and T.W.F. approaches.

Experimental falsification of the T.W.F. model.

The transition probabilities assigned to various CP-labelling processes (for instance |K 0 (t = 0) → |ππ(t) , |K 0 (t = 0) → |ππ ) are well-documented.

In particular, the asymmetry A ππ (t) has been measured by the CPLEAR experiment [START_REF]Physics at CPLEAR[END_REF][START_REF] Gabathuler | The CPLEAR experiment[END_REF] in the range 1 < t < 20 τ S . Figure 1 shows two plots of the asymmetry as a function of t computed in the standard WWA approach (solid curve) and in the T.W.F. approach (dotted curve). The left plot corresponds to the values ˜ L = Γ S Γ L and ˜ S = ; the right was obtained after imposing ˜ L = Γ S Γ L and ˜ S = Γ L Γs (see appendix for more explanation). In each figure, the discrepancy between the two curves is very evident in the range 1 < t < 10 τ S . Even a simple visual inspection of the CPLEAR measurement in this region (Fig. 14b of Ref. [START_REF]Physics at CPLEAR[END_REF]) shows that it is in perfect agreement with the WWA prediction and some tens of standard deviations away from the T.W.A. prediction. Therefore we can conclude that the T.W.F. model prediction is falsified at a very high degree of confidence level.

Conclusions and discussions

Neutral kaons constitute a unique physical system where a variety of spectacular phenomena at the root of Quantum Mechanics and Particle Physics show-up [START_REF]Physics at CPLEAR[END_REF][START_REF] Gabathuler | The CPLEAR experiment[END_REF][START_REF] Ambrosino | First observation of quantum interference in the process φ → K S K L → π + ππ + π --: A test of quantum mechanics and CPT symmetry[END_REF][START_REF] Bernabeu | Direct test of time reversal symmetry in the neutral kaon system at a φ-factory[END_REF][START_REF] Bertlmann | Bell inequality and CP violation in the neutral kaon system[END_REF][START_REF] Bertlmann | Strangeness measurements of kaon pairs, CP violation and Bell inequalities[END_REF][START_REF]Handbook of neutral kaon interferometry at a φ-factory[END_REF][START_REF] Hiesmayr | Revealing Bells nonlocality for unstable systems in high energy physics[END_REF]. If there were no CP-violation ( = 0), it would be impossible to discriminate the T.W.F. model from the standard WWA model.

In the present work, we have shown that CP violation (the discovery was awarded in 1980 with the Nobel Prize) makes instead possible to discriminate the standard approach in which Time is treated as an external, classical, parameter from the Time Operator (in particular T.W.F.) approach in which, in analogy with the spin 1 Ultimately, our analysis allows us to bring to the realm of experiments an old debate that can be traced back to the original developments of the quantum theory concerning the role and status of Time. Here CP violation turned out to be an essential ingredient of our derivation. Despite CP violation is tiny, the measurements of its manifestations are so accurate [START_REF]Physics at CPLEAR[END_REF] that neutral kaons provide a successful candidate for implementing our ideas. Trapped ions also offer promising perspectives [START_REF] Champenois | Quest for the Time Operator with a single trapped ion[END_REF] to test other fundamental ideas concerning the status of time in the quantum theory [START_REF] Massar | Investigating the origin of time with trapped ions[END_REF].

Actually our analysis is also valid when one considers other models as for instance the Time Super-Operator proposed by Misra Prigogine and Courbage [START_REF] Misra | Lyapounov variable: Entropy and measurement in quantum mechanics[END_REF]. Here Time is a quantized variable (a q-number) [START_REF] Hilgevoord | Time in Quantum Mechanics, a story of confusion[END_REF]. This Time Super-Operator has been recently also applied to kaon phenomenology [START_REF] Courbage | Time decay probability distribution of the neutral meson system and CP-violation[END_REF]. Even in that case structural discrepancies can be found and, in principle, it should be possible to discriminate these Super-Operator models from the standard theory. concerning the Fermi Golden Rule. Let us now define [START_REF] Debierre | Fermi golden rule beyond the Zeno regime[END_REF] the state |ef f.(ω) > through |ef f.(ω) >= 1 i,λ,ω i,λ =ω |G i,λ | 2 i,λ,ω i,λ =ω G i,λ |i, λ, ω >. It is easy to check that, in each energy sector characterized by an energy ω, all superpositions of modes orthogonal to |ef f.(ω) > are decoupled from the decay process, so that in practice everything happens as if there was only one decay product, |ef f.(ω) >, which is a coherent superposition of different decay products, and is coupled to the ingoing mode with the effective coupling constant G ef f. (ω) = i,λ,ω i,λ =ω |G i,λ | 2 . The Fermi Golden Rule treatment can also be applied in this case, and we find, following the computation sketched in section 1.3.2, that the decay into this effective single mode is characterized by an effective Gamma factor Γ ef f. which obeys Γ ef f. = i λ D.O.S.(ω in , i, λ)2π|G i,λ (ω in )| 2 = i Γ i , where Γ i is the Gamma factor associated to a particular species of decay product, in agreement with section 1.3.2.

Figure 1 :

 1 Figure 1: Asymmetry A 2π (t) as a function of t in units of τ S : standard WWA (solid curve) and T.W.F model predictions (dotted curve). The left top plot corresponds to ˜ S =

Decay processes are appealing candidates for a Time Operator approach because to the contrary of stable systems, the probability of decay by unit of time (probability density function) of such systems is normalised to unity.

From now on the upperly tilded quantities will always refer to quantities derived in the framework of the T.W.F. approach.

At this level, CPT invariance will be assumed according to the CPT theorem[START_REF] Lueders | Proof of the TCP theorem[END_REF].

We remind the reader that in the following, tilded quantities are evaluated within the T.W.F. approach, while non-tilded ones denote standard WWA expressions.

In previous works[START_REF] Courbage | A wave-function model for the CP-violation in mesons[END_REF][START_REF] Durt | Correlations of decay times of unstable quantum systems[END_REF][START_REF] Courbage | Time decay probability distribution of the neutral meson system and CP-violation[END_REF], we sticked to the particular choice ˜ S = ˜ L = ˜ but here we do no want to diminish the level of generality of the T.W.F. model that we consider.

It is exactly at this level that the binary T.W.F. model departs from the WWA mass-decay formalism.

We are neglecting here the tiny contribution O(10 -6 ) to the A ππ (t) given by direct CP violation[23,[START_REF] Abouzaid | Precise measurements of direct CP violation, CPT symmetry, and other parameters in the neutral kaon system[END_REF].

theory, the time of occurrence of decay processes in the CP=±1 sectors would obey a generalized Born rule (as explained in section

3.1). Indeed, as we have shown (in section 3.2), accurate experimental measurements of neutral kaon decay rates into two pions and of the asymmetry A 2π (t) showing evidence of CP violation[START_REF]Physics at CPLEAR[END_REF][START_REF] Gabathuler | The CPLEAR experiment[END_REF], allow us to falsify a model based on this Time Operator approach, which constitutes the main result of our paper.

Acknowledgement:

B.C.H. gratefully acknowledges the Austrian Science Fund (FWF-P26783). All of us acknowledge support from the COST MP 1006 action (Fundamental Problems in Quantum Physics) and in particular support from the Working Group 4 From theory to experiment.

Let us describe the decay process by the system of coupled equations

where we denoted ω in and d in the energy and amplitude of the ingoing mode (e.g. a kaon produced in a particle accelerator) while the indices i and λ are respectively assigned to different species of decay products (e.g. pion pairs and pion triplets) and to their internal degrees of freedom (energy, direction and so on).

It is convenient to introduce c in = d in e iω in t and c i,λ = d i,λ e iω i,λ t , so that we can cast [START_REF] Debierre | Fermi golden rule beyond the Zeno regime[END_REF] in the form

In a perturbative approach, it is standard to impose that, at short time, c in (t) = c in (t = 0) = 1 in first approximation so that we get after a straightforward integration:

), from which we derive all results of section 1.3

Appendix 2: constraints on ˜ S .

Due to the fact that CP-violation is small, we expect that ˜ S is a small parameter. It can be shown that if we impose that the theoretical prediction of P K S (0)→3π(t) , made in the standard WWA approach, is the same as the corresponding prediction, PK S (0)→3π(t) , made in the T.W.F. approach, we find, after computations similar to those that led to the prediction (25), the constraint

according to which ˜ S is a very small parameter (of the order of 10 -6 ). This constraint however is not based on experimental data because no precise measure of the 3π production rate has been realized up to now [START_REF] Babusci | KLOE-2 collaboration: "A new limit on the CP violating decay K S → 3π 0 with the KLOE experiment[END_REF].

The measure of the asymmetry parameter provides instead another constraint on ˜ S . Indeed, it can be shown by straightforward computation that in the limit t

If now we impose to get identical results in both approaches, we find

Obviously, the above result would not be true in the case ˜ S = .

The two predictions made above (˜ S = Γ L Γ S and ˜ S = ) are plotted in figure 1. Without zooming (bottom plots), the difference between the two top plots is nearly imperceptible with the naked eye, but this does not really matter: what is important is the constraints derived from observations on ˜ L alone, and those constraints suffice to discriminate the T.W.F. approach as we have shown in the paper.


Jacob Szeftel

Nicolas Sandeau

Michel Abou Ghantous

An observable prerequisite for the existence of persistent currents

Keywords: 74, 20, Mn Nonconventional mechanisms -74, 25, Bt Thermodynamic properties -74, 25, Fy Transport properties

A classical model is presented for persistent currents in superconductors. Their existence is argued to be warranted because their decay would violate the second law of thermodynamics. This conclusion is achieved by analyzing comparatively Ohm's law and the Joule effect in normal metals and superconducting materials. Whereas Ohm's law applies in identical terms in both cases, the Joule effect is shown to cause the temperature of a superconducting sample to decrease. An experiment is proposed to check the validity of this work in superconductors of both types I and II.

Introduction

The prominent signature of superconductivity, i.e. the property to sustain persistent currents [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Parks | [END_REF][START_REF] Tinkham | Introduction to Superconductivity[END_REF] in vanishing electric field, has remained unexplained, since its discovery [START_REF] Onnes | The disappearance of the resistivity of mercury[END_REF], as stressed by Ashcroft and Mermin [START_REF] Ashcroft | Solid State Physics[END_REF] (see [START_REF] Ashcroft | Solid State Physics[END_REF] p.750, 1 st paragraph, line 1) : the property for which the superconductors are named is the most difficult to extract from the microscopic (i.e. BCS [START_REF] Bardeen | Theory of superconductivity[END_REF]) theory. This is all the more dis-turbing, since the mainstream narrative on superconductivity relies heavily on the phenomenological equations, proposed by London [6] and Ginzburg and Landau [START_REF] Ginzburg | On superconductivity theory[END_REF], for which the existence of persistent currents is merely assumed.

In order to understand why this long-standing riddle has withstood every attempt [START_REF] Matricon | The Cold Wars, a History of Superconductivity[END_REF] at elucidating it so far, it is helpful to recall the basic tenets of electric conductivity in normal conductors [START_REF] Ashcroft | Solid State Physics[END_REF]. The applied electric field E accelerates the electrons in the conduction band, which gives rise to a current j. Eventually, the driving force ∝ E is counterbalanced by a friction one ∝ j, exerted by the lattice, as conveyed by Ohm's law :

j = σE , σ = c 0 e 2 τ m , (1)
where σ, c 0 , e, m, τ stand for the conductivity, the electron concentration, the electron charge, its effective mass, and the decay time of j due to friction, respectively. Simultaneously, the work performed by the electric force is entirely transformed into heat, to be released in the lattice, through the Joule effect. As a consequence of Eq.( 1), the observation [START_REF] Onnes | The disappearance of the resistivity of mercury[END_REF] of j = 0 despite E = 0 seemed indeed to suggest τ → ∞ ⇒ σ → ∞. However, it is wellknown nowadays that both τ, σ are finite, provided the measurement is carried out with an ac current, as emphasized by Schrieffer [START_REF] Schrieffer | Theory of Superconductivity[END_REF] (see [START_REF] Schrieffer | Theory of Superconductivity[END_REF] p.4, 2 nd paragraph, lines 9, 10): at finite temperature, there is a finite ac resistivity for all frequencies > 0. For instance, the conductivity, measured in Y Ba 2 Cu 3 0 7 below the critical temperature T c , has been reported [START_REF] Geshkenbein | AC absorption in the high-T c superconductors: Reinterpretation of the irreversibility line[END_REF][START_REF] Samarappuli | Comparative study of AC susceptibility and resistivity of a superconducting YBa2Cu3O7 single crystal in a magnetic field[END_REF] to be such that σ ≈ 10 5 σ n , where σ n stands for the normal conductivity, measured just above T c . Additional evidence is provided by commercial microwave cavity resonators, made up of superconducting materials, displaying a very high, albeit finite conductivity (see [START_REF] Tinkham | Introduction to Superconductivity[END_REF] lowest line in p.38). Finally, the observable consequences of finite σ, regarding the skin [START_REF] Jackson | Classical Electrodynamics[END_REF][START_REF] Born | Principles of Optics[END_REF] and Meissner [6,[START_REF] Meissner | Ein neuer effekt bei eintritt der supraleitfähigkeit[END_REF] effects, have been discussed recently [START_REF] Szeftel | Study of the skin effect in superconducting materials[END_REF][START_REF] Szeftel | Comparative Study of the Meissner and Skin Effects in Superconductors[END_REF] and the finite resistivity of superconductors has been ascribed solely to superconducting electrons [START_REF] Szeftel | Revisiting low-frequency susceptibility data in superconducting materials[END_REF] on the basis of susceptibility data.

Therefore the issue of persistent currents will be tackled here from quite different a starting point. Likewise we shall show how the very properties of the BCS state [START_REF] Bardeen | Theory of superconductivity[END_REF] cause the Joule dissipation to be thwarted in a superconductor, undergoing no electric field. This goal will be achieved by making a comparative study of Ohm's law and the Joule effect in normal and superconducting metals, based on Newton's law and the two laws of thermodynamics.

The outline is as follows : the conditions for a superconductor to be in thermal equilibrium are discussed in sections 2, while stressing the different properties of the BCS state [START_REF] Bardeen | Theory of superconductivity[END_REF] versus those of the Fermi gas [START_REF] Ashcroft | Solid State Physics[END_REF]; Ohm's law and the Joule effect are studied in sections 3 and 4, respectively; a necessary condition for the existence of persistent currents is worked out in section 5, while an experiment, enabling one to check the validity of this analysis in superconducting materials of both kinds, is described in section 6. Our observable predictions will turn out to concur very well with a remark by De Gennes [START_REF] De Gennes | Superconductivity of Metals and Alloys[END_REF]. The results of this work are summarized in the conclusion.

The two-fluid model

The conduction properties of a superconducting material will be analyzed within the two-fluid model [START_REF] Parks | [END_REF][START_REF] Tinkham | Introduction to Superconductivity[END_REF][START_REF] Schrieffer | Theory of Superconductivity[END_REF][START_REF] De Gennes | Superconductivity of Metals and Alloys[END_REF]. In this framework, the conduction electrons make up a homogeneous mixture, in thermal equilibrium, of normal and superconducting electrons, in concentration c n , c s , respectively.

All of the electronic properties of the normal state are governed by the Fermi-Dirac statistics, and thence accounted for within the Fermi gas [START_REF] Ashcroft | Solid State Physics[END_REF] model. In particular, its Helmholz free energy per unit volume F n depends on two parameters, the temperature T and the Fermi energy E F , defined [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Landau | Statistical Physics[END_REF] as the chemical potential of independent electrons, i.e. E F = ∂Fn ∂cn .

By contrast, the BCS wave-function [START_REF] Bardeen | Theory of superconductivity[END_REF] describes the motion of superconducting electrons, as a many-body bound state, which entails that the BCS energy per unit volume E s depends only on the concentration of superconducting electrons c s . Because E s is T -independent, the BCS state [START_REF] Bardeen | Theory of superconductivity[END_REF], unlike the Fermi gas, is inferred to carry no entropy [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Parks | [END_REF][START_REF] Tinkham | Introduction to Superconductivity[END_REF], so that its free energy is equal to E s (this property is confirmed experimentally by the weak thermal conductivity [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Parks | [END_REF][START_REF] Tinkham | Introduction to Superconductivity[END_REF], measured in superconductors, in marked contrast with the high one, typical of normal metals). Thus the chemical potential µ of the BCS state reads µ = ∂Es ∂cs .

The equilibrium, achieved in the two-fluid model, stems from Gibbs and Duhem's law [START_REF] Landau | Statistical Physics[END_REF], which requires the free energy of the whole electron system F e = F n (T, c n ) + E s (c s ) to be minimum with respect to c n , c s , under the constraints c n + c s = c 0 (c 0 refers to the total concentration) and T kept constant, and thence leads to

E F (T, c n ) = µ(c s ) . (2)
The peculiar properties of the Joule effect, taking place in a BCS state, will appear below to be solely determined by the sign of ∂µ ∂cs = ∂ 2 Es An early, phenomenological attempt [START_REF] Gorter | The thermodynamics of the superconducting state[END_REF], aimed at explaining the specific heat data, measured in superconducting materials, made use of Eq.( 2) too. However our approach differs from that one, inasmuch as it refrains from assuming specific, but arbitrary expressions for F n (T, c n ), E s (c s ), so that our conclusions do not suffer from any loss of generality.

∂c 2 s . ε (k) k k F 0 E F j n = 0 -k F

Ohm's law

Owing to Fermi-Dirac statistics and T << T F = EF kB ≈ 3 × 10 4 K (k B stands for the Boltzmann constant), the electrons in a normal metal make up a degenerate Fermi gas [START_REF] Ashcroft | Solid State Physics[END_REF], for which each one-electron state, with energy ranging from the bottom of the conduction band up to E F , is doubly occupied (due to the two spin directions), whereas those states with energy > E F remain empty. The corresponding one electron dispersion curve ǫ(k) has been projected onto the direction of the applied electric field E, as pictured in Fig. 1. Since the electron velocity [START_REF] Ashcroft | Solid State Physics[END_REF] is 2. schematic projected one electron dispersion ǫ(k) for jn = 0 in the laboratory frame; the solid line represents most of electrons, which contribute nothing to jn, whereas the dashed line corresponds to the few electrons responsible for jn = 0; actually jn is proportional to the length of the dashed line but the tiny difference ǫ1 -ǫ2 has been hugely magnified for the reader's convenience, that is ǫ1 ≈ ǫ2 ≈ EF ; the dotted line represents empty one electron states; the arrows illustrate electron transitions, from occupied (dashed line) back to empty (dotted line) states, driven by the friction force equal to ∂ǫ(k) ∂k and thanks to ǫ(k) = ǫ(-k), the resulting current j n vanishes.

ε (k) k k F 0 ε 2 ε 1 j n ≠0 -k F Fig.
The applied field E arouses a finite current j n = 0 by accelerating δc n of electrons (δc n << c n ) from their initial wave-vector -k F up to their final one k F , with k F being such that ǫ(k F ) = E F , due to Pauli's principle. Therefore all electrons, contributing to j n , have about the same velocity v F = ∂ǫ(kF ) ∂k , so that the resulting current reads j n = 2δc n ev F (see the dashed line in Fig. 2). Inversely, the friction force, exerted by the lattice on those electrons making up j n , tends to bring 2 δcn τn of electrons per unit time from k F back to -k F , where τ n , showing up in Eq.( 1) as τ , represents the average time between two successive scattering events [START_REF] Ashcroft | Solid State Physics[END_REF] (see the arrows pointing to the dotted line in Fig. 2). As the momentum change rate, involved in this process, is equal to

δp τ n = -2 mv F δc n τ n = - m eτ n j n ,
Newton's law reads [START_REF] Szeftel | Study of the skin effect in superconducting materials[END_REF][START_REF] Szeftel | Comparative Study of the Meissner and Skin Effects in Superconductors[END_REF] finally

m e dj n dt = c n eE - m eτ n j n . (3)
Because the inertial force m e djn dt has been shown to be negligible [START_REF] Szeftel | Study of the skin effect in superconducting materials[END_REF][START_REF] Szeftel | Comparative Study of the Meissner and Skin Effects in Superconductors[END_REF], the electric force c n eE and the friction one -m eτn j n cancel each other, so that Eq.( 3) boils down to Ohm's law, as expressed in Eq.( 1).

Ohm's law will be worked out now for a superconductor by proceeding similarly as hereabove. The j s = 0 superconducting state (j s refers to the superconducting current) is assumed to consist in two subsets, each of them comprising the same number of electrons. It ensues, from the very properties of the BCS state [START_REF] Bardeen | Theory of superconductivity[END_REF], flux quantization and Josephson's effect [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Parks | [END_REF][START_REF] Josephson | Possible new effects in superconductive tunnelling[END_REF], that the electrons in each subset are organized in pairs, moving in opposite directions with respective velocity v s , -v s , which ensures j s = 0. The driving field E causes δc s of electrons (δc s << c s ) to be transferred from one subset to the other, which results into a finite current j s = 2δc s ev s .

The friction force is responsible for the reverse mechanism, whereby an electron pair is carried from the majority subset of concentration c s + δc s back to the minority one of concentration c sδc s . Hence if τ -1 s is defined as the transfer probability per time unit of one electron pair, the electron transfer rate is equal to

c s + δc s -(c s -δc s ) τ s = 2 δc s τ s .
Then Newton's law reads similarly as Eq.( 3), valid for independent electrons

m e dj s dt = c s eE - m eτ s j s . (4)
As for independent electrons, the electric force c s eE and the friction force -2 mvsδcs τs = -mjs eτs cancel each other, which yields the searched result, identical to Eq.( 1)

c s eE = m eτ s j s ⇒ j s = σ s E , σ s = c s e 2 τ s m . (5)
Although Ohm's law displays the same expression for normal and superconducting metals as well, it should be noted that τ s >> τ n [START_REF] Geshkenbein | AC absorption in the high-T c superconductors: Reinterpretation of the irreversibility line[END_REF][START_REF] Samarappuli | Comparative study of AC susceptibility and resistivity of a superconducting YBa2Cu3O7 single crystal in a magnetic field[END_REF].

Finally note that the inter-electron forces, responsible for the binding energy of the BCS state with respect to the corresponding Fermi gas of same electron concentration and also for the two-electron scattering within the Fermi gas, do not show up in Eqs. [START_REF] Tinkham | Introduction to Superconductivity[END_REF][START_REF] Onnes | The disappearance of the resistivity of mercury[END_REF]. In order to understand this feature, let us consider two electrons labelled i, j. They exert the forces f i→j , f j→i on each other, respectively. Due to f i→j +f j→i = 0, the net force, resulting from all i, j pairs, vanishes and thence does not contribute to Ohm's law, although the inter-electron coupling will turn out to play a paramount role in the Joule effect.

The Joule effect

Because no electron contributes to j n , but the few ones in concentration 2δc n with ǫ(k) ≈ E F , showing up as the dashed line in Fig. 2, they are also the only ones to be instrumental in the Joule effect. Besides all of them have the same velocity v F . Thus, the well-known formula of the power released by the Joule effect, ẆJ = dWJ dt (t refers to time), ensues from Ohm's law j n = σ n E, which implies that the friction force equals 2δc n eE, as

ẆJ = 2δc n eE.v F = E.j n = j 2 n σ n , (6)
for which we have made use of j n = 2δc n ev F .

The Joule effect takes place via two different processes in a superconductor. The calculation of the Joule power Ẇ1 , released through process I, is identical to that one leading to Eq.( 6)

Ẇ1 = m eτ s j s .v = j 2 s σ s , ( 7
)
where v is the mass center velocity of superconducting electrons (⇒ v = j s cse ) and advantage has been taken of Ohm's law in Eq.( 5) to express the resulting friction force (= m eτs j s ), exerted on the mass center of superconducting electrons. The physical significance of Eq.( 7) is such that the work W 1 > 0 performed by the driving force ∝ E is turned into heat by the friction force.

However, the calculation of the Joule power Ẇ2 , released through process II, proceeds otherwise, because the work W 2 , to be turned into heat by the friction force, is performed by the inter-electron forces, rather than the driving one, as seen for W 1 in Eq.( 7). Accordingly, while any electron in a normal metal may lose, due to Pauli's principle, an energy randomly distributed from 0 up to ǫ 1ǫ 2 (see Fig. 2), conversely the corresponding internal energy change, experienced by the BCS electrons, due to the scattering of one electron pair, is uniquely defined, as will be shown hereafter.

In case of j s = 0, the chemical potential of majority (minority) electrons, characterized by the average velocity v s (-v s ) reads µ(c s + δc s ) (µ(c sδc s )). During each elementary scattering process, a single pair is brought back from the majority subset to the minority one, which results into δE s , the energy lost by the BCS electrons to the lattice, reading

δE s = µ(c s + δc s ) -µ(c s -δc s ) = 2 ∂µ ∂c s δc s .
Since the transfer rate is equal to 2 δcs τs , the Joule power Ẇ2 = 2 δcs τs δE s reads finally, due to j s = 2δc s ev s

Ẇ2 = 4 ∂µ ∂c s δc 2 s τ s = j 2 s σ J , σ J = (ev s ) 2 τ s ∂µ ∂cs . ( 8
)
The result in Eq.( 8) is noteworthy in two respects :

even though Ẇ2 is still proportional to j 2 s as Ẇ1 in Eq.( 7), the conductivity σ s , deduced from Ohm's law in Eq.( 5), differs from σ J σ s = c s e 2 τ s m = (ev s ) 2 τ s ∂µ ∂cs = σ J ;

unlike σ s > 0, the sign of σ J , which sets whether the Joule heat W 2 will flow from the conduction electrons towards the lattice (⇔ W 2 > 0), as is always the case in a normal conductor, or conversely will flow into the reverse direction (⇔ W 2 < 0), is to be determined by the sign of ∂µ ∂cs . As a matter of fact, the searched criterion for the existence of persistent currents will be worked out by taking advantage of this peculiarity.

It is worth elaborating upon the significance of σ s = σ J . The Joule power ẆJ reads in general

ẆJ = i f i .v i , (9)
where the sum is carried out on every electron in the conduction band, labeled by the index i, moving with velocity v i and undergoing the friction force f i . Owing to Ohm's law, which implies that the resulting friction force equals 2δc n eE, and v i = v F for all electrons contributing to j n = 2δc n ev F , Eq.( 9) can be recast, for a normal metal, as

ẆJ = i f i .v i = i f i .v F = 2δc n eE. j n 2δc n e = j 2 n σ n ,
which is seen to be identical to Eq.( 6). Hence the fact, that the same conductivity σ n shows up in both expressions of Ohm's law j n = σ n E and the Joule effect ẆJ = j 2 n σn , is realized to result from the typical property of a degenerate Fermi gas, that all electrons, contributing to j n , have the same and one velocity v F . Besides, Eq.( 6) expresses also the fact that the Joule heat W J is equal to the work performed by the driving force ∝ E. However this is no longer true for the BCS state, because of the additional contribution W 2 , expressed in Eq.( 8). Accordingly, in case of a BCS state, the whole Joule power, reads as

ẆJ = Ẇ1 + Ẇ2 = j 2 s σ -1 s + σ -1 J ⇒ W J = W 1 .
Finally it remains to be shown that W 2 = 0 in a normal metal. The time-derivative of the work done by the inter-

electron forces f i→j , f j→i reads Ẇij = f i→j .v j + f j→i .v i . Meanwhile f i→j + f j→i = 0 and v i = v j = v F imply that Ẇij = 0. Q.E.D.

Prerequisite for the existence of persistent currents

The applied field E gives rise to the total current j = j n + j s , where j n = σ n E and j s = σ s E, as required by Ohm's law. After E has vanished, j n is quickly destroyed by the Joule effect. However whether j s will decay down to 0 or conversely will turn to a persistent current, will be shown hereafter to depend solely upon the sign of the whole Joule power ẆJ , generated via processes I and II.

In case of E = 0, the kinetic energy, associated with

j s = 0, E K = csm 2 v 2 = m 2cse 2 j 2
s , due to j s = c s ev, is turned into heat by the friction force. The expression of ĖK is obtained, thanks to j s = 2δnev s and v = -2δnvs csτs , as

ĖK = -c s mv v = - m c s e 2 τ s j 2 s = - j 2 s σ s ,
so that the expression of Ẇ1 = -ĖK remains unaltered with respect to that one in Eq.( 7) and finally we get the same expression as in the E = 0 case, i.e.

ẆJ = j 2 s σ -1 s + σ -1 J .
If ẆJ > 0, the Joule effect will cause eventually j s = 0 and the associated kinetic energy will be converted into heat, to be dissipated in the lattice, as occurs in a normal metal. Inversely in case ẆJ < 0, which requires both σ J < 0 ⇔ ∂µ ∂cs < 0 (see Eq.( 8)) and σ J + σ s > 0, the Joule heat is seen to be bound to flow from the lattice towards the superconducting electrons, which will cause the lattice temperature to decrease. However, since such a spontaneous cooling of the system, comprising all of electron and lattice degrees of freedom, which can furthermore exchange neither heat, nor work with the outer world due to E = 0, would cause its whole entropy to decrease, and would thence be tantamount to violating the second law of thermodynamics, the searched criterion is deduced to say that persistent currents can be observed, only if both following conditions are fulfilled

∂ 2 E s ∂c 2 s = ∂µ ∂c s < 0 ⇒ σ J < 0 , σ J + σ s > 0 . ( 10
)
That those conditions in Eqs.( 10) are necessary, but by no means sufficient ones, can be understood by looking back at Eq.( 2). The equilibrium of the mixture of normal and superconducting electrons will be stable provided

∂E F ∂c n + ∂µ ∂c s > 0 .
Both stable and instable cases are illustrated in Figs. 3,4,

where E F (T, c n ), µ(c s ) have been plotted versus c n , c s , respectively. Note that ∂EF ∂cn ≈ ρ(E F ) -1 > 0 where ρ(ǫ)

is the density of one electron states in the conduction band [START_REF] Ashcroft | Solid State Physics[END_REF]. The infinite slope ∂EF ∂cn (c n → 0) → ∞ is then typical of a 3 dimensional van Hove singularity [START_REF] Ashcroft | Solid State Physics[END_REF], associated with the bottom of the conduction band, where

ρ(ǫ → 0) ∝ √ ǫ.
The inequality in Fig. 3,E

F (T i , c n ) < E F (T f , c n ), ∀c n with T i > T f , ensues from ∂ρ ∂EF (E F ) > 0
via the Sommerfeld integral [START_REF] Ashcroft | Solid State Physics[END_REF], which will be shown elsewhere to be another prerequisite for the occurrence of superconductivity. At last in case c s → 0, there is E s ≈ ǫc 2 c s where ǫ c refers to the Cooper pair energy [START_REF] Cooper | Bound electron pairs in a degenerate Fermi gas[END_REF], which en-

tails that µ(0) = ∂Es ∂cs (0) = ǫc 2 .
The experiment, to be discussed below, is aimed primarily at bringing evidence of the anomalous (σ J < 0 ⇒ ẆJ < 0) Joule effect, associated with a BCS state. Since every superconducting material is claimed here to be characterized by ẆJ < 0, the experimental procedure will look for evidence of the sample temperature being lowered by the Joule effect.

There are in general two ways to have a current flowing through any conductor, i.e. either directly by feeding an externally controlled, time-dependent current I(t) into the sample, or indirectly by inducing the current j s (t) via a time-dependent magnetic field H(t) according to Faraday's law [START_REF] Jackson | Classical Electrodynamics[END_REF]. Though the latter has been overwhelmingly favored [START_REF] Parks | [END_REF][START_REF] Tinkham | Introduction to Superconductivity[END_REF][START_REF] Matricon | The Cold Wars, a History of Superconductivity[END_REF][START_REF] Meissner | Ein neuer effekt bei eintritt der supraleitfähigkeit[END_REF][START_REF] De Gennes | Superconductivity of Metals and Alloys[END_REF] so far in experiments involving superconductors, the former procedure should be given preference for two reasons :

as the Meissner effect [START_REF] Szeftel | Comparative Study of the Meissner and Skin Effects in Superconductors[END_REF] gives rise to a spatially inhomogeneous current j s (t, r) with r referring to the local coordinate inside the sample, the Joule power ẆJ (t, r)

will thereby vary with r, whereas both j s (t), ẆJ (t) will remain r-independent within the former procedure;

because of an irreversible consequence [START_REF] Szeftel | Comparative Study of the Meissner and Skin Effects in Superconductors[END_REF] of the finite conductivity σ s , there can be no one-to-one correspondence between the applied magnetic field H(t) and j s (t, r), so that the current distribution remains unknown, by contrast with j s (t) = I(t) S , ∀t (S refers to the area of the sample cross-section) within the former procedure.

Experimental outlook

Consider a thermally isolated, superconducting sample, taken in its initial state 0, c n (t = 0) = c n (T i ), c s (t = 0) = c s (T i ) (see A in Fig. 3).

Then let a direct current I(t) flow through this sample.

I(t) grows from I(0) = 0 up to its maximum value I(t M ), reached at t = t M , such that I(t M ) = I c (T (t M )), with I c (T (t M )) standing for the maximum persistent current [START_REF] Parks | [END_REF] at T = T (t M ), which causes the sample to go normal at 3).

t = t M , i.e. c n (t M ) = c 0 ⇒ c s (t M ) = 0 (see B in
t = t f and characterized by T f = T (t f ), I(t f ) = 0, c n (t f ) = c n (T f ), c s (t f ) = c s (T f ) (see C in Fig.
The work W (t f ), performed by the external electric field during the thermodynamical process, described hereabove and pictured as a dashed-dotted line in Fig. 3, is then given by

W (t f ) = t f 0 U (t)I(t)dt , (11)
with U (t) designating the measured voltage drop across the sample. Since the sample is thermally isolated, applying the first law of thermodynamics to the system, comprising the independent and superconducting electrons and the lattice, driven from A to C via B through an adiabatic process, yields then

Q 2 = T f Ti (C φ (T ) + C s (T )) dT -W (t f ) , (12)
with W (t f ) being defined in Eq. [START_REF] Samarappuli | Comparative study of AC susceptibility and resistivity of a superconducting YBa2Cu3O7 single crystal in a magnetic field[END_REF]. C φ (T ), C s (T ) stand for the respective contributions [START_REF] Ashcroft | Solid State Physics[END_REF] to the specific heat of the phonons (Debye), which is I independent, and of the conduction electrons, the latter being measured at T ≤ T c , I = 0. Then the integral over T represents the difference in internal energy of the thermodynamical system, defined above, between T i and T f . Besides,

Q 2 = V t f 0 j 2 s (t) σJ dt = V S 2 t f 0 I 2 (t)
σJ dt and V are the Joule heat released via process II, and the sample volume, respectively (j s (t), being r-independent, warrants j s (t) = I(t) S , ∀t and

Q 2 ∝ V ).
As, due to σ J < 0 and σ J + σ s > 0 (see Eqs.( 10)), the Joule effect is expected to cool down the sample, we predict that Eq.( 12) will be fulfilled with Q 2 < 0 and

dT dt (t ∈ [0, t f ]) < 0 ⇒ T f < T M < T i , in full agreement
with a remark by De Gennes [START_REF] De Gennes | Superconductivity of Metals and Alloys[END_REF] (see [START_REF] De Gennes | Superconductivity of Metals and Alloys[END_REF] footnote in p.18):

if one passes from the superconducting state to the normal one in a thermally isolated specimen, the temperature of the sample decreases. Although this experiment could be done as well with I(t M ) < I c (T (t M )), the condition which includes all high-T c compounds, C s (T ) is negligible [START_REF] Ashcroft | Solid State Physics[END_REF] with respect to C φ (T ), so that Eq.( 12) gets simpler

I(t M ) = I c (T (t M )) secures the largest T i -T f ,
Q 2 ≈ T f Ti C φ (T )dT -W (t f ) . (13)
Due to C φ (T ) being I independent, unlike C s (T ), taking the time derivative of Eq.( 13) yields in addition

V S 2 I 2 (t) σ J (t) = C φ (T ) dT dt -U (t)I(t) ,
which enables one to assess σ J (t) < 0 for t ∈ [0, t f ] and thence to check σ J (t)+σ s (t) > 0, the necessary conditions for the existence of persistent currents (see Eqs.( 10)), provided σ s has been measured independently [START_REF] Geshkenbein | AC absorption in the high-T c superconductors: Reinterpretation of the irreversibility line[END_REF][START_REF] Samarappuli | Comparative study of AC susceptibility and resistivity of a superconducting YBa2Cu3O7 single crystal in a magnetic field[END_REF] (the t dependences of σ s = cse 2 τs m and σ J = (evs) 2 τs ∂µ ∂cs are both mediated by the j s dependence of c s (t), as demonstrated hereafter in the concluding section).

Although T f < T i entails that the entropy of the twofluid system decreases, the second law of thermodynamics is thereby not violated, because the electrons remain coupled with the outer world via I(t) during the experiment. At last, note that the state, illustrated by B in Fig. 3, refers to a metastable equilibrium, because the stable position at T f is rather inferred to be at C in Fig. 3, as required by Eq.( 2). However, were the electron system to go spontaneously from B to C, e.g. along the dasheddotted line, this process would result [START_REF] Szeftel | Comparative Study of the Meissner and Skin Effects in Superconductors[END_REF] into djs dt = 0, due to j s = 0 at B versus j s = 0 at C, while the accompanying Joule effect would give rise to a negative entropy variation ∆S B→C < 0, at odds with the second law of thermodynamics, as noted hereabove.

Conclusion

The anomalous Joule effect is characterized by σ s = σ J , i.e. the conductivity σ s , deduced from Ohm's law, should differ from σ J , the conductivity pertaining to the Joule power released through process II. It ensues solely from the inter-electron coupling, which causes the BCS electrons to gain the internal energy δE s = W 2 < 0 through process II at the expense of the lattice, while losing si-multaneously the kinetic energy W 1 > 0 through process I to the lattice, so that W 1 + W 2 < 0 gives rise eventually to the cooling effect, embodied by Eqs. [START_REF] Jackson | Classical Electrodynamics[END_REF][START_REF] Born | Principles of Optics[END_REF]. Due to W 2 = 0 in a normal metal as shown above, the anomalous Joule effect can be observed solely for a many-body bound state, such as the BCS one. Likewise, the existence of persistent currents is warranted as a consequence of σ J < 0 and σ J + σ s > 0 (see Eqs.( 10)), because the resulting Joule dissipation ẆJ < 0 would run afoul at the second law of thermodynamics, which lends itself to an experimental check, as discussed above.

Besides, the property σ s = σ J implies that Eq.( 2)

can never be fulfilled in presence of a persistent current j s = 0. Here is a proof : consider the electron system in the equilibrium state, defined by T = T f , j s = 0 and represented by C in Fig. 3, for which Eq.( 2) is fulfilled. As j s grows from 0 up to its maximum value, the electron system shifts away from C : the Fermi gas, represented by P n in Fig. 3

Fig. 1 .

 1 Fig. 1. schematic projected one electron dispersion ǫ(k) of occupied states (ǫ(k) ∈ [0, EF ]) for jn = 0 as a solid line

Fig. 3 .

 3 Fig. 3. schematic plots of EF (Ti, cn), EF (T f , cn) and µ(cs) as dotted, solid and dashed lines, respectively, in case ∂µ ∂cs < 0, ∂E F ∂cn + ∂µ ∂cs > 0; ∂µ ∂cs has been taken to be constant for simplicity; the origin EF = µ = 0 is set at the bottom of the conduction band; the crossing points A, C of EF (Ti, cn), EF (T f , cn), respectively, with µ(cs), exemplify stable solutions of Eq.(2); the tiny differences EF (T f , cn)-EF (Ti, cn), EF (Ti, cn)-µ(c0 -cn) have been hugely magnified for the reader's convenience; the dashed-dotted line, linking A, B, C together represents the adiabatic process, discussed in section VI; the points Pn, Ps and the arrow linking them illustrate a superconducting-normal transition in progress (i.e. cn(T f ) < cn(Pn) < c0, 0 < cs(Ps) < cs(T f )), taking place at T f , under the constraint cn + cs = c0

Fig. 4 .

 4 Fig. 4. schematic plots of EF (T, cn) (solid line) and µ(cs) (dotted line) in case ∂µ ∂cs < 0 and ∂E F ∂cn + ∂µ ∂cs < 0; the crossing point E of EF (T, cn) with µ(cs) represents an instable solution of Eq.(2)

  , moves, along the solid line, towards D, corresponding to the normal state c n = c 0 ⇒ c s = 0, while the BCS state, represented by P s , goes, along the dashed line, towards the single Cooper pair state, characterized by µ (c s = 0) = ǫc 2 , provided the sample re-mains connected to a heat bath at T f . Meanwhile, whenever the thermodynamical state of the two-fluid system is represented by the pair {P n , P s } in Fig.3, Eq.(2) is nolonger fulfilled because of E F (T f , c n (P n )) > µ(c s (P s ) = c 0c n (P n )), which demonstrates the first order nature of the j s -driven superconducting-normal transition[START_REF] Parks | [END_REF][START_REF] Tinkham | Introduction to Superconductivity[END_REF][START_REF] Schrieffer | Theory of Superconductivity[END_REF][START_REF] De Gennes | Superconductivity of Metals and Alloys[END_REF], by contrast with the second order transition, observed

at T c with j s = 0, for which Eq.( 2) is indeed fulfilled, i.e.