Superluminal self-interacting neutrino

0.14 MB·English

by  Ernst Trojan

Superluminal self-interacting neutrino
2
Ernst Trojan
1
0
Moscow Institute of Physics and Technology
2
PO Box 3, Moscow, 125080, Russia
n
a
J
January 11, 2012
9

]
h
p Abstract
-
p
e The effect of nonlinear self-interaction can be associated with su-
h perluminal velocity of neutrino. The power energy spectrum E =
[
p+Cpa is derived from the nonlinear Dirac equation when interac-

1 tion term V = λ(ψ¯γ ψψ¯γµψ)a is added to the Lagrangian of a free
µ
v
spin-1/2 particle. The superluminal velocity recorded by the OPERA
5
8 and MINOS collaborations is achieved when the coupling constants
0 are taken in the range a = 0.4÷1.18 and λ = −(0.5÷1.6)×10−4.
2
The self-interaction Lagrangian V = λψ¯γ ψψ¯γµψ with the coupling
. µ
1 constant λ = −(0.7÷0.9)×10−4 yields the same result. Scalar inter-
0
2 action V = λ(ψ¯ψ)b and scalar-vector interaction λ ψ†ψ b+1/ ψ¯ψ b
1
cannot be responsible for the observed superluminal neutrino.
: (cid:0) (cid:1) (cid:0) (cid:1)
v
i
X
1 Introduction
r
a
Neutrino was believed to be a massless spin-1/2 fermion with energy
E = pc (1)
and group velocity v = c equal to the speed of light c = 1 (in relativistic
units). The modern theory expects, however, that neutrino has finite mass
[1]
m = m < 0.28eV (2)
ν
1
that implies deviation from the energy spectrum (1) and velocity
dE
v = 6= 1 (3)
dp
Recent experiments of the OPERA Collaboration [2] revealed superluminal
motion of neutrino with energy E = 17GeV at the average velocity
v = 1+2.37×10−5 (4)
while the the MINOS Collaboration detected velocity
v = 1+5.1×10−5 (5)
for the low energy neutrino with energy spectrum peaked at approximately
E = 3GeV. Superluminal neutrino was also observed in supernova explosion
SN1987a [4].
This fact is a serious puzzle to the researchers. Superluminal velocity (4)
cannotbelongtoafreemassiveparticlewithenergyspectrumE = p2 +m2
whose velocity is always subluminal because
p
1m2 1m2
v −1 = − ≃ − < 0 (6)
2 p2 2 E2
The tachyonic energy spectrum E = p2 −m2 results in velocity above the
speed of light
1mp2 1m2
v −1 = ≃ > 0 (7)
2 p2 2E2
that does not exceed 3 × 10−22 even at maximum possible neutrino mass
m (2) and E = 17GeV. Nevertheless, in the frames of the accuracy of
measurements, the energy spectrum of superluminal neutrino of OPERA [2]
and MINOS [3] can be fitted to a power law [5, 6, 7]
E = p+Cpa (8)
where the coefficients must be taken in the range [8]
a = 0.40÷1.18 C = 4.15×10−4 ÷1.5×10−5 (9)
and C = (2.2 ÷3.03)×10−5 if we choose a ≡ 1. Indeed, neutrino is not
a free particle, but there are several interesting hypotheses to explain its
superluminal motion [9].
2
In the present paper we consider massive neutrino whose Lagrangian
L = ψ¯(iγµ∂ −m)ψ +V ψ¯,ψ (10)
µ
includes nonlinear self-interaction term V ψ¯,ψ(cid:0) . T(cid:1)here is no additional
interaction with external fields and the medium, while the superluminal ve-
(cid:0) (cid:1)
locity is hidden in the very nature of neutrino. We need to find the energy
spectrum of nonlinear Dirac equation
∂V
(iγµ∂ −m)ψ + = 0 (11)
µ ¯
∂ψ
and check whether the Lagrangian (10) is adjustable to reproduce the super-
luminal velocity (4)-(5) detected by the OPERA [2] and MINOS [3] collab-
orations.
2 Effect of self-interaction
Consider the Lagrangian (10) with a simple self-interaction term
V ψ¯,ψ = λ ψ†ψ b+1 ψ† = ψ¯γ0 (12)
b+1
(cid:0) (cid:1) (cid:0) (cid:1)
The relevant Dirac equation (11) is written
(iγµ∂ −m+F)ψ = 0 (13)
µ
where
0
F = γ ω (14)
and
ω = λ ψ†ψ b (15)
The Dirac equation (13) at ω = 0 ha(cid:0)s we(cid:1)ll-known solution in the form of
plane wave
φ ~σ ·~p
ψ = 0 exp(−iE t) χ = φ E = m2 +p2 (16)
0 0 0 0 0
χ0 E0 +m
(cid:18) (cid:19)
p
Substituting stationary wave function
ψ = ϕ(~r)exp(−iEt) (17)
3
in (13), we have equation
(−iα~ ·∇+βm)ϕ = (E +ω)ϕ (18)
where
0
α~ = β~γ β = γ = −γ (19)
0
Substituting a plane-wave bispinor
φ
ϕ = exp(ip~·~r) (20)
χ
(cid:18) (cid:19)
in (18), we obtain a linear system of equations
~σ ·~pχ = (E +ω −m)φ
(21)
~σ ·p~φ = (E +ω +m)χ
that has solution if and only if
E = m2 +p2 −ω (22)
p
where p = |p~|.
We can estimate the energy spectrum (22) in the frames of mean-field
approximation if we neglect correlations between the field operators in (15)
and apply effective interaction
ω ≃ ω = λnb (23)
∗
where
n = ψ†ψ (24)
is the particle number density. The (cid:10)latter(cid:11)can be adjusted as n = 1/V for
1 particle in volume V. For a many-particle system the quantity (24) is
determined according to formula
∞
2
n = f dqk (25)
q k
(2π)
Z
0
where f is the distribution function in q-dimensional momentum space. The
p
latter is the Fermi-Dirac distribution function if it is an ideal gas in equilib-
rium, however, the OPERA [2] and MINOS [3] collaborations considered a
4
neutrino beam with the energy peaked at k = p, and its distribution function
corresponds to a delta-function
k
f = δ 1− (26)
k
p
(cid:18) (cid:19)
in 1-dimensional momentum space, so that the particle number density (25)
is estimated so
p
n = (27)
π
and, according to (22) and (23), the energy spectrum is
λ
E = p2 +m2 −λnb = p2 +m2 − pb (28)
πb
p p
3 Vector self-interaction
Consider more general form of vector self-interaction [10]
V = λ ψ¯γ ψψ¯γνψ b+1 (29)
ν
b+1
(cid:0) (cid:1)
corresponding to the Dirac equation
(iγµ∂ −m+F)ψ = 0 (30)
µ
where
F = λ ψ¯γ ψψ¯γνψ bγ ψ¯γµψ (31)
ν µ
At b = 1 it is no more than th(cid:0)e Heisenber(cid:1)g mo(cid:0)del of(cid:1)self-interaction [11]
F = λγ ψ¯γµψ (32)
µ
When we consider 1-dimensional n(cid:0)eutrino(cid:1) beam, we take into account
that gamma-matrices in (1+1)-dimensional representation [10]
1 0 0 i
γ0 = γx = (33)
0 −1 i 0
(cid:18) (cid:19) (cid:18) (cid:19)
satisfy standard commutation relations
1 0
{γ ,γ } = 2η η = (34)
µ ν µν µν 0 −1
(cid:18) (cid:19)
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Substituting stationary plane wave solution
u
ψ = exp(ip x−iEt) p~ = (p ,0,0) (35)
v x x
(cid:18) (cid:19)
in the Dirac equation (45), we have [10]
ip v = (E +ω −m)u
x (36)
−ip u = (E +ω +m)v
x
instead of (21), while
ω = λ |u|2 +|v|2 b = λ ψ†ψ b (37)
formally coincides with (15).(cid:0)The linear(cid:1)system(cid:0)(36)(cid:1)has solution if and only
if condition (22) is satisfied. Again, applying the mean-field approximation
ω ≃ ω = λ ψ†ψ b = λnb (38)
∗
we obtain the same formula for the en(cid:10)ergy(cid:11)spectrum (28) of a 1-dimensional
neutrino beam.
The group velocity is immediately calculated
p λb
v = − pb−1 (39)
p2 +m2 πb
that tends to p
λb m2
v ≃ 1− pb−1 − (40)
πb 2p2
intheultra-relativistic limit E ≃ p ≫ m. Thelattertermin(40) isnegligible
even at the upper bound of neutrino mass (2), and superluminal velocity (4)-
(5) can be explained by the second term if λb < 0. Indeed, the power energy
spectrum (8) is compatible with (40) if
b = a = 0.4÷1.18 (41)
λ = −(0.5÷1.6)×10−4 (42)
Particularly, the coupling constant may vary in the range
λ = −(0.7÷0.9)×10−4 (43)
when a = b = 1 that corresponds to the Heisenberg model (32). Therefore,
the origin of superluminal velocity of neutrino [2, 3] can be associated with
vector self-interaction (29) if the coupling constants are properly identified
(41)-(42).
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4 Scalar self-interaction
Consider the Lagrangian (10) with scalar self-interaction
λ ¯ b+1
V = ψψ (44)
b+1
(cid:0) (cid:1)
The Dirac equation (11) has the form
(iγµ∂ −m+W)ψ = 0 (45)
µ
where
¯ b
W = λ ψψ (46)
Substituting stationary wave funct(cid:0)ion ψ(cid:1) = ϕ(~r)exp(−iEt) and the ef-
fective mass
m = m−W (47)
∗
in (45), we obtain equation
(−iα~ ·∇+βm )ϕ = Eϕ (48)
∗
that has plane-wave solution
φ ~σ ·p~
ϕ = exp(ip~·~r) χ = φ (49)
χ E +m
(cid:18) (cid:19) ∗
for a free particle with the energy spectrum
E = p~2 +m2 (50)
∗
p
and relevant group velocity
1 dm
∗
v = p+m (51)
∗
p2 +m2 dp
∗ (cid:18) (cid:19)
p
Solution (49) also implies
m
ψ¯ψ = kφk2 −kχk2 = ∗ψ†ψ (52)
E
where
ψ†ψ = kφk2 +kχk2 (53)
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Applying the mean-field approximation, we neglect correlations in (46)
and use the effective interaction
W ≃ W = λnb (54)
∗ s
where
m
¯ ∗
n = ψψ = n (55)
s
E
is the scalar density and n is the p(cid:10)arti(cid:11)cle number density (24). For a many-
particle system it is determined by formula
∞
2 m (k)
n = ∗ f dqk (56)
s q k
(2π) E(k)
Z
0
The distribution function f of a neutrino beam with the energy peaked at
k
k = p is presented by a delta-function (26) in 1-dimensional momentum
space, so that the scalar density (55) is estimated
p m (p)
∗
n = (57)
s
π E(p)
Substituting (54) and (57) in (47), we obtain self-consistent equation for the
effective mass
b
p m (p)
m (p) = m−λnb = m−λ ∗ (58)
∗ s "π (p2 +m2)#
∗
Consider the ultra–relativistic limit E ≃ pp≫ m . The energy (50) is
∗
reduced to
m2 m2 mW W2
E = p+ ∗ = p+ − ∗ + ∗ (59)
2p 2p p 2p
while the effective interaction (54) is reduced to
λ
W ≃ mb (60)
∗ πb ∗
and velocity (51) is estimated
m2
v ≃ 1− ∗ (61)
2p2
Although the deviation from the speed of light may be visible at large W ,
∗
the velocity (61) is subluminal at any choice of constants λ and b. Therefore,
thescalarself-interactioncannotberesponsibleforthesuperluminal neutrino
velocity.
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5 Scalar-vector self-interaction
Now consider a mixed variant of scalar-vector self-interaction in the form
[12, 13]
ψ†ψ b+1
V = λ (62)
¯ b
(cid:0) ψψ(cid:1)
that corresponds to the Dirac equation(cid:0) (cid:1)
iγµ∂ −m+W +γ0ω ψ = 0 (63)
µ
where (cid:0) (cid:1)
ψ†ψ b ψ†ψ b+1
ω = λ(b+1) W = −λb (64)
¯ b ¯ b+1
(cid:0)ψψ(cid:1) (cid:0)ψψ(cid:1)
Substituting a stationary p(cid:0)lane(cid:1)wave solution (cid:0) (cid:1)
φ
ϕ = exp(ip~·~r −iEt) (65)
χ
(cid:18) (cid:19)
and the effective mass (47) in (63), we have a linear system of equations for
the spinors
~σ ·~pχ = (E +ω −m )φ
∗ (66)
~σ ·~pφ = (E +ω +m )χ
∗
that has solution
~σ ·p~
χ = φ (67)
E +ω +m
∗
if and only if the particles has the energy spectrum
E = m2 +p2 −ω (68)
∗
p
where the effective mass is
m = m−W (69)
∗
According to (65) and (67) we have
ψ†ψ kφk2 +kχk2 m2 +p2
= = ∗ (70)
ψ¯ψ kφk2 −kχk2 m
p ∗
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and the effective mass (69) is determined by self-consistent equation
(m2 +p2)b+1
∗
m = m−λb (71)
∗ q mb+1
∗
that is reduced to
pb+1
m = m−λb (72)
∗ mb+1
∗
in the ultra-relativistic limit p ≫ m , so that
∗
b+2 m
m −m m′ = (m −m) ∗ (73)
b+1 ∗ ∗ ∗ p
(cid:20) (cid:21)
The energy spectrum (68) is, then, reduced to
m2 pb
E ≃ p+ ∗ −λ(b+1) (74)
2p mb
∗
At large coupling constant λ, when
m ≫ m (75)
∗
equation (73) implies
b+1m
m′ = ∗ (76)
∗ b+2 p
and, according to equation (74), the neutrino velocity is subluminal
m2
v ≃ 1− ∗ (77)
2p2
When
m → m (78)
∗
equation (73) implies m′ → 0 and, according to equation (74), the neutrino
∗
velocity is also subluminal
m2
v ≃ 1− (79)
2p2
When
m ≪ m (80)
∗
10