A quantum field theoretical model of neutrino oscillation without external wave packets

0.58 MB·

by  Z. Y. Law

A quantum field theoretical model of neutrino oscillation without external
wave packets
Z. Y. Law,∗ A. H. Chan,† and C. H. Oh‡
Department of Physics, National University of Singapore,
2 Science Drive, Singapore 115742, Singapore
(Dated: March 4, 2014)
We develop a general and consistent model of neutrino oscillation based on the quantum
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1 field theoretical (QFT) description of the neutrino production and detection processes. Em-
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phasisisplacedonthelocalityoftheinteractionsoftheseprocesses,whereontopoftheusual
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r application of the 4-Fermion local Hamiltonian, we assume that weak interactions switched
a
M on only when the wave functions of the particles involved are overlapping and switched off
upon their separation.
3

Akeyassumptioninourtreatmentisthatthewavepacketsizesoftheparticles,inpartic-
]
h
ular, the neutrino-producing source particles and the neutrino absorbing detector particles,
p
-
p are taken to be negligible compared with their mean free path in their respective medium.
e
With this assumption, and taking into considerations of the finite time of neutrino produc-
h
[
tion, neutrino wave packets with well defined edges are generated. This fact, together with

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thelocalityofweakinteractions,enableustorelatethepropagationtimetothepropagation
v
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distance, thus doing away with the ad hoc time averaging procedure normally employing in
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7 derivations of neutrino oscillation formula. No ansatzs on the particular forms of particle
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. wave functions (for example, gaussians) need to be presupposed.
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0 As a result, for the case of ultra-relativistic neutrinos, we have derived a neutrino count
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1 rate formula, which will be useful for making direct connection to experiments. A neutrino
:
v flavor oscillation probability arises naturally from this formula, which, when compared to
i
X
the standard oscillation formula, shows modifications dependent on the relative velocities
r
a of the source/detector particles and the decoherence time taken for the unstable source to
collapse into its undecayed or decayed states. This correction could be significant for short
baselineneutrinooscillationexperiments. Agoodfeatureofourapproachisthattheneutrino
oscillation formula is automatically normalized if the in-going states of the production and
detection processes are normalized.
We also show that causality and unitarity cannot both be satisfied in virtual neutrino
models.
PACS numbers: 14.60.Pq, 13.15.+g, 12.15.Ff
∗ Email:[email protected]
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† Email:[email protected]
‡ Email:[email protected]
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I. INTRODUCTION
With the discovery of the Higgs particle, all the elements of the Standard Model are now
complete,andneutrinomassesremainasthesoletangibleindicationofphysicsbeyondtheStandard
Model. This window into new physics is built on the foundation of the phenomenon of neutrino
flavoroscillations,whichremaintobethechiefavenuethroughwhichexperimentaldataonneutrino
masses and mixing parameters are gathered.
Given the inherent challenges involved in conducting neutrino oscillation experiments, and the
equally stringent standards required in seeking experimental evidences for new physics; it is im-
portant that the steps from any theoretical mass models of the neutrino leading to the neutrino
oscillation probability formula [1], which is to make direct connections with experiments, be free
of ambiguities and errors.
The main aim of this paper is to strengthen this link and provides a more rigorous derivation
of the neutrino oscillation formula, while identifying the conditions under which deviations from
the standard oscillation formula may be expected.
This is not the first attempt towards such a goal. From the original derivation presented
by Pontecorvo [2], based on the planewave description of the neutrinos, modifications have been
suggested to give a more physical description of the neutrino itself, by describing them with wave
packets [3–6] in the quantum mechanical formalism, to mirror the way neutrinos are produced and
detected locally. These wave packets are put in by hand without considerations as to how they are
formed. This gives a set of conditions under which neutrino oscillation can occur, namely, there
exist a coherence length and that the neutrino wave packet must have a size small compared to
the oscillation length. This last condition rules out the plane wave description.
Further investigations [7–10, 12, 15, 16, 20], taking into considerations the production processes
of the neutrinos, reveal that neutrinos are produced in weak decays in an entangled state with
the associated charged lepton, and that on tracing out the lepton state (or the rest of the decay
products), the remaining neutrino state can only be plane waves. This reveals the problem of
disentangling the neutrino while maintaining its wave packet form.
The most common way around this problem in the quantum field theoretical treatment of
neutrinooscillationis,insteadoftracingouttheleptonstatefromthejointneutrino-leptondecayed
(fromanunstableparent)stateinsomebasis(forexampleeigen-states),theleptonisassumedtobe
measured as a wave packet. Thus, a corresponding wave packet is induced, through entanglement,
on the neutrino state.
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This is done for the external wave packet model (see [16] and the references therein), in which
the neutrino is described as a virtual particle (therefore, not an external particle; see for example
[17–19]), and all external particles are each assigned a wave packet. For the intermediate wave
packet model [16] (for example [12]), in which a neutrino is treated as a real particle, the neutrino
is simply assumed to exist as a wave packet, as in the quantum mechanical wave packet treatment,
and thus does not address the issue of neutrino localization.
We feel that since in most of the treatments just mentioned, the neutrino oscillation formula is
dependent on the sizes of the particle wave packets, including the measured lepton wave packet;
this means that the ways in which the leptons are measured (for example, the spatial resolution
of the detector measuring it) must be discernable by the neutrino observer. In a hypothetical
neutrino-lepton EPR (Einstein-Podolsky-Rosen) scenario, causality might be violated (effects of
themeasurementoftherecoilparticleonneutrinooscillationthroughentanglementisalsodiscussed
in [7] and [8], with the conclusion that neutrino oscillation is not observable if phases due to the
recoil particle is significant; but this effect would be discernable to the neutrino detector).
Thus, to avoid potential complications, we build a full quantum field theoretical model based
on local interactions (for example the 4 fermion weak interactions), assuming wave packets only
for the source particle (producing neutrinos through decay) and the detector particle (absorbs the
neutrino to give a lepton). The neutrino wave packet in our model results from the decay process
itself, as the lepton state is traced out in a convenient basis (momentum basis).
Weconsideradecoherencetime(T )betweenthedecayedandtheundecayedstateofthesource,
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after which the neutrino is emitted (source collapsed to the decayed state), and propagates for the
time, T , when the neutrino wave function overlap with that of the detector particle and the
prop
detection interaction begins.
During this interaction, the neutrino-detector state evolves into a superposition of the detection
(neutrino absorbed) and non-detection (neutrino not absorbed) state, which collapsed into the
former at the end of the time interval T (decoherence time associated with detection). Thus, we
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are concerned with 3 time scales, T , T and T which are present in all neutrino oscillation
1 prop 2
scenario, and show how they enter into the neutrino oscillation formula.
We assume no particular forms for the wave packets describing the source and the detector
particles, except that their sizes are much smaller than the decoherence time scales (T and T ).
1 2
Our model is considered as an intermediate wave packet model.
More examples of earlier works done on the field theoretical treatment of neutrino can be found
in [7, 10–20], which include considerations of neutrino propagators both in vacuum and matter
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[11, 14], as well as the possibility of constructing the Hilbert space for neutrino flavor states [11].
In this paper, we begin by highlighting a potential problem which occurs in modeling neutrino
oscillation as a process involving the exchange of virtual neutrinos between a source and a detector
(which is common in such field theoretical models). It is shown that there is a conflict between
unitaryandcausalityintheprobabilitiesobtaininsuchtreatments. Thiswillserveasamotivation
for the development of a model that is free from such problems, which will be the subject of the
rest of the paper.
We conclude with the derivation of a neutrino count rate formula, which will be useful for
making direct connection with experiments, followed by a discussion of results.
II. THE PROBLEM WITH VIRTUAL NEUTRINO MODELS
As mentioned in the last section, virtual neutrino models consider neutrino oscillation as a
single process including both the neutrino production and detection interactions. To calculate
the probability for such a process, one computes the corresponding Feynman as shown in Fig.
1. S /D (source/detector) is the in-going states of the scattering, that include the state of
in in
the source which will emit the neutrino, and the state of the detector before neutrino detection.
S /D represents the products of this interaction. The neutrino is the propagator connecting
out out
the production/detection vertices located at the space-time points, y/x, while V is the space-
S/D
time uncertainties of the production/detection. We shall show that under such a context, this
ampltiude cannot be both causal and unitary.
By causal, we mean the energy resolution of the process allows us to distinguish the source
(emitting energy/neutrino) from the detector (receiving energy/neutrino); this naturally implies a
temporalordering(emissionbeforedetection),sincetheenergyoftheneutrinoispositive. Causality
is ensured in such models by choosing the appropriate external wave packets (wave functions of
the in/out-going particles) such that y > x and V do not overlap.
0 0 S/D
The unitarity condition is given by
(cid:88)
(cid:104)f|iT |i(cid:105)+(cid:104)f|(iT)†|i(cid:105) = − (cid:104)f|(iT)†|n(cid:105)(cid:104)n|(iT)|i(cid:105) (1)
n
where T is the transfer matrix, |i/f(cid:105) are the initial/final states, |n(cid:105) are the intermediate states. By
the cutting equation [22], the cut diagram corresponding to that in Fig.1 is shown in Fig.2 (Note
that momentum flows from the shaded to the unshaded side of the cut), and this diagram must
corresponds to terms on the R.H.S. of (1) in order for unitarity to be satisfied. Since causality
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is imposed via the external wave packets, the cut diagram does not vanished. But because the
neutrino is exchange through the t-channel, this diagram could not represent any of the terms on
the R.H.S. of (1), hence violating unitarity.
This implies that field theoretical approaches to neutrino oscillation employing the exchange of
virtual neutrinos is inherently inconsistent, and the probabilties thus derived may appear unitary
only because they are normalized by hand. In general, the only meaningful temporal ordering in
S-matices are between the in/out states and not the interaction points, which exist in a cloud of
space-time uncertainty.
Since, as shown in [16], such a virtual neutrino model, with appropriately chosen external wave
packets could reproduce the results of most neutrino oscillation models (even those that do not
involve virtual neutrinos), it is therefore prudent to re-examine the neutrino oscillation probabili-
ties, putting special emphasis on the issues of unitarity and causality. With these considerations
in mind we will proceed to construct a field theoretical model of neutrino oscillation that will be
free from the defects highlighted here.
FIG. 1. Neutrino exchange amplitude. ν is a neutrino of mass m . y/x are the space-time coordinates of
j j
the production/detection vertices, V are the corresponding space-time uncertainties
S/D
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FIG. 2. Neutrino line is cut. Momentum flows from the shaded to unshaded side.
III. AN ALTERNATIVE MODEL
For the sake of generality, we assume the following neutrino producing decay
A −→ ν +B (2)
α
with A as the unstable source particle, α as the initial flavor of the neutrino, ν , and B is the
α
decay products inclusive of the charged lepton, l .
α
The neutrino detection process is given by
ν +D −→ l +C (3)
β β
where D is the detector particle, β is the flavor of the charge lepton produced from neutrino
detection, and C is the remaining products of this process.
The interaction governing these 2 processes is taken to be local and can be describe by some
(S/D)
Hamiltonian density, H (x) (which is assumed to be first order in the Fermi constant, G ),
int F
where the superscript indicates source/detection, which may involve different interactions. Work-
ing in the Schrodinger picture, the time evolution operator describing neutrino production and
detection is given by
U(t) = e−iH0tS(t) (4)
where t is the interaction time of the processes, H is the Hamiltionian describing the free propa-
0
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gation of particles in vacuum, and S(t) is the scattering matrix
(cid:32) t (cid:33)
(cid:90) (cid:90)
S(t) = Texp −i dt(cid:48) d3xH(S/D)((cid:126)x,t(cid:48)) (5)
int
0
(the effects on neutrino oscillation due to the finite time of its production and detection is also
given in [20], using a planewave treatment. See also [19] for virtual neutrino oscillation with an
unstable source).
As previously stated, wave packets will be assigned only to the source particle and the detector
particle (in-going states of interactions) and these are (We follow the convention of [7, 16])
(cid:90)
|A(cid:105) = [dp(cid:126) ]Ψ (p(cid:126) )|p(cid:126) (cid:105) (6)
A A A A
(cid:113)
for the source, with [dp(cid:126)A] ≡ d3pA(2π)−3(2EA)−12, EA = p(cid:126)2A+m2A. ΨA(p(cid:126)A) is the wave packet of
A, localizing it within the source. The following set of normalizing conditions holds
(cid:104)p(cid:126) |p(cid:126)(cid:48) (cid:105) = (2π)32E δ3(p(cid:126) −p(cid:126)(cid:48) ) (7)
A A A A A
(cid:90)
d3p (2π)−3|Ψ |2 = 1 (8)
A A
(cid:104)A|A(cid:105) = 1 (9)
Similarly, for the detector particle, D
(cid:90)
|D(cid:105) = [dp(cid:126) ]Ψ (p(cid:126) )e−ip(cid:126)D·L(cid:126) |p(cid:126) (cid:105) (10)
D D D D
where ΨD(p(cid:126)D) localized D to within the source, and the translation e−ip(cid:126)D·L(cid:126) shifts it across the
macroscopic displacement L(cid:126), into the detector. Thus |D(cid:105) is located at L(cid:126) from |A(cid:105).
During the weak decay of A, its state evolves such that at T
1
|A(cid:105) −→ Φ (T )|A(cid:105)+Φ (T )|ν ,l (cid:105) (11)
sur 1 decay 1 α α
Φ (T ) and Φ (T ) are the survival and decay amplitude at T . T is the time scale associated
sur 1 decay 1 1 1
with the decoherence of state (11) within the source, through interactions with the environment.
This could occur via collisions between A particles within the source or similar electromagnetic
collisions, experienced by l (ν is assumed to escape the source and only interacts with the detec-
α α
tor). These predominantly electromagnetic collisional monitoring by the environment is assumed
to occur much faster than the weak process of the decay; hence it makes sense to speak of the start
and end of the interval T .
1
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Thestate(11)collapsedeitherintothefirstterm, atwhichtheprocessisreset, andevolvesuntil
sufficient decoherence effects build up again at the end of the next interval T , or collapsed into the
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second term, whereby the decay occurs and the neutrino is emitted. T can thus be understood
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as the time interval between incidents of environment monitoring beginning with an undecayed A
and ending with its decay. It is apparent that this second term is sufficient for one to construct
the state of emitted neutrinos. The neutrinos are assumed to be external states and on mass-shell
to avoid the problems discussed in Section II.
The transfer matrix element corresponding to (2) is given by
(cid:90)T1
(cid:104)(p(cid:126) ,s),{p(cid:126) }|iT(S)|p(cid:126) (cid:105) ≡ U∗ iT(S)(p(cid:126) ,s;p(cid:126) )(2π)3δ3(p(cid:126) −p(cid:126) ) dte−i(EA−Ef)t (12)
k B A αk B k A A f
0
where the convention, S = 1+iT is adopted for the definition of the transfer matrix. (p(cid:126) ,s) refers
k
to the momentum of the neutrino of mass m , with helicity s, and {p(cid:126) } labels the momentum
k B
states of the set of particles B (possible discrete quantum numbers are suppressed for simplicity).
(cid:113)
p(cid:126) is the total momentum of B, p(cid:126) = p(cid:126) +p(cid:126) and E = E +E (E = p(cid:126)2+m2), E is the
B f k B f k B k k B
total energy of B). The dependence on the unitary neutrino mixing matrix U , is factored out
αk
explicitly. And because we are working to the first order of G (at the amplitude level), for a finite
F
time evolution, the usual energy conserving delta function is replaced by a time integral.
(12) can be rewritten as
(cid:104)(p(cid:126) ,s),{p(cid:126) }|iT(S)|p(cid:126) (cid:105) ≡ U∗ (cid:90) d∆ e−i∆1T1/2sin(cid:16)∆1T1(cid:17)
k B A αk 1 π∆ 2
1
·iT(S)(p(cid:126) ,s;p(cid:126) )(2π)4δ3(p(cid:126) −p(cid:126) )δ(E −E −∆ ) (13)
B k A A f A f 1
where we have used
(cid:90)T1dte−i(EA−Ef)t = (cid:90) d∆ e−i∆1T1/2sin(cid:16)∆1T1(cid:17)2πδ(E −E −∆ ) (14)
1 A f 1
π∆ 2
1
0
Allowing us to reintroduced the usual energy conserving delta function, with the uncertainty con-
trolled by the function ∆−1sin(∆1T1).
1 2
Assuming that only neutrinos traveling along L(cid:126) (L(cid:126) = Lxˆ) are detected, the energy-momentum
constraints imposed by the delta functions are
p(cid:126) = p xˆ+p(cid:126) ,E = E +E +∆ (15)
A k B A k B 1
(15) could then be solved for p(cid:126) ({p(cid:126) },∆ ) and p(cid:126) ({p(cid:126) },∆ ). With (4) and (6), the joint ν −B
k B 1 A B 1
state after decay is given by
(cid:90) (cid:90)
|Ψ (cid:105) ∝ [dp(cid:126) ]Ψ (p(cid:126) ) D |(p(cid:126) ,s),{p(cid:126) }(cid:105)(cid:104)(p(cid:126) ,s),{p(cid:126) }|e−iH0T1iT(S)|p(cid:126) (cid:105) (16)
ν⊗B A A A ν⊗B k B k B A
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(cid:82)
D is the sum over the relevant final ν-B state (over xˆ propagating neutrino). Performing the
ν⊗B
momentum integral (cid:82) d3p d3p , using (13) and (15), (16) becomes
A k
|Ψ (cid:105) ∝(cid:90) D e−iEBT1(cid:88)U∗ (cid:90) d∆ e−i(∆1T1/2+Ek(p(cid:126)k({p(cid:126)B},∆1))T1) 1 sin(cid:16)∆1T1(cid:17)
ν⊗B B αk 1 ∆ 2
1
k,s
1 1
· ·
(2π)3(cid:112)2EA(p(cid:126)A({p(cid:126)B},∆1)) (2π)32Ek(p(cid:126)k({p(cid:126)B},∆1))
(S)
·iT (p(cid:126) ({p(cid:126) },∆ ),s;p(cid:126) ({p(cid:126) },∆ ))·Ψ(p(cid:126) ({p(cid:126) },∆ ))|(p(cid:126) ({p(cid:126) },∆ ),s),{p(cid:126) }(cid:105)(17)
B k B 1 A B 1 A B 1 k B 1 B
The factor (2π)−3(2E (p(cid:126) ({p(cid:126) },∆ )))−1 come from the neutrino phase space factor of the
k k B 1
(cid:82)
integral, D . In the next section, we shall derive the neutrino state from |Ψ (cid:105).
ν ν⊗B
A. The neutrino state
To obtain the neutrino state right after emission, we need to trace out the B − state from
|Ψ (cid:105). This is easily done using (17)
ν⊗B
(cid:90)
ρ = tr{|Ψ (cid:105)(cid:104)Ψ |} = D g({p(cid:126) })|ν (T ,{p(cid:126) })(cid:105)(cid:104)ν (T ,{p(cid:126) })| (18)
ν ν⊗B ν⊗B B B α 1 B α 1 B
whereg({p(cid:126) })isaprobabilitydistributionoftheclassicalsumoftheneutrinostate|ν (T ,{p(cid:126) })(cid:105),
B α 1 B
whichisthestatethatoccursinaproductwiththemomentumB−state|{p(cid:126) }(cid:105)in(17)thatsatisfies
B
the orthonormality condition
(cid:104){p(cid:126) }|{p(cid:126)(cid:48) }(cid:105) ∝ δ3(p(cid:126) −p(cid:126)(cid:48) ) (19)
B B B B
where the proportionality sign indicates the we are comparing entire configurations of B −states
(not just their total momentum), and that a host of other quantum numbers associated with B
are not expressed explicitly. From (17) this neutrino state is
(cid:88)
|ν (T ,{p(cid:126) })(cid:105) = U∗ |ν (T ,{p(cid:126) })(cid:105) (20)
α 1 B αk k 1 B
k
with the neutrino mass eigenstate
|ν (T ,{p(cid:126) })(cid:105) ∝(cid:88)(cid:90) d∆ e−i(∆1T1/2+Ek(p(cid:126)k({p(cid:126)B},∆1))T1) 1 sin(cid:16)∆1T1(cid:17)
k 1 B 1
∆ 2
1
s
1 1
· ·
(2π)3(cid:112)2EA(p(cid:126)A({p(cid:126)B},∆1)) (2π)32Ek(p(cid:126)k({p(cid:126)B},∆1))
(S)
·iT (p(cid:126) ({p(cid:126) },∆ ),s;p(cid:126) ({p(cid:126) },∆ ))
B k B 1 A B 1
·Ψ(p(cid:126) ({p(cid:126) },∆ ))|p(cid:126) ({p(cid:126) },∆ ),s(cid:105) (21)
A B 1 k B 1