Theoretical Particle Limiting Velocity From The Bicubic Equation: Neutrino Example

0.11 MB·

by  Josip Soln

Theoretical Particle Limiting Velocity From The
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Bicubic Equation: Neutrino Example
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n Josip Sˇoln
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J JZS Phys-Tech, Vienna, Virginia 22182

8 E mail: [email protected]
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January 2014
]
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p
- Abstract
n
e
Therehasbeenalot of interestin measuringthevelocities of massive
g
elementary particles, particularly the neutrinos. Some neutrino experi-
.
s mentsatfirstobservedsuperluminalneutrinos,thusviolatingthevelocity
c
oflightcasalimitingvelocity. But,aftereliminatingsomemistakes,such
i
s as,fortheOPERAexperimentspluggingthecablecorrectlyandcalibrat-
y
ing the clock correctly, the measured neutrino velocity complied with c.
h
Pursuing the theoretical side of particle limiting velocities, here directly
p
from the special relativistic kinematics, in which all physical quantities
[
are in the overall mathematical consistency with each other, one treats

1 formally the velocity of light c as yet to be deduced particle limiting ve-
v
locity,andderivesthebicubicequationfortheparticlelimitingvelocityin
3
thearbitrary reference frame. The Lorentzinvariance(LI)of theenergy-
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momentum dispersion relation assumes the velocity of light c to be uni-
6
versallimitingvelocityofanyparticle. Thisexpectsthephysicalsolutions
2
ofthebicubicequationtobeconstrainedinasensethatanyphysicallim-
.
3 iting velocity solution should equal numerically to c, preferably exactly,
0
or at least, in aextremely good approximation. Arather large numerical
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departure from c means solution which, if it is physical, would indicate
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the significant degree of Lorentz violation (LV). However, this LV could
:
v be false if experimentally particle parameters were read wrongly yielding
i different from c solution for the physical limiting velocity. Still, one may
X
allow possible anticipation of finding some LV in neutrino physics. The
ar threesolutionsforthesquaresoflimitingvelocities, denotedasc21,c22 and
c23,dependontheparticlem,Eandv(mass,energyandordinaryvelocity)
throughinversesinusoidalfunctions. Asc21,c23 >0andc22<0,onlyc21and
c23 have chances to be physical while c22 is unphysical. Furthermore, with
the inverse sinusoidal functions principal values dependences, c21 and c23
beingpositivearecomplementarylimitingvelocitysquares,atleastoneof
themphysicalandpresumedluminal,whilec22 beingnegativeisdefinitely
unphysical. However, c22 can become c21 and c23 when transformed from
the principal values region into the multiple values region of the inverse
sinusoidal functions. With thesesolutions one can treat physicallimiting
1
velocities, for any particle, electron, neutrino, photon, etc. The OPERA
17GeV muonneutrinovelocityexperimentsarediscussedthroughthelim-
itingvelocityc3 becausethecalculatedneutrinomνc2 of0.076eV ,being
negligible, makesc1 unphysical. Furthermore,because in OPERA exper-
iments, mνc2 <<Eν, one findsout that c3 =c(1+∆)≃c because ∆ is
negligible (it varies from O(−10−6) to O(10−6) ). This implies basically
the LI of the neutrinoenergy-momentum dispersion relation.
1. Introduction
There have been the whole series of neutrino velocity experiments such as
theOPERAcollaborationswiththedetectorintheCNGSbeam[1](versions1,
2 and 3), the OPERA detector the CNGS beam using the 2012 dedicated data
[2] (versions 1 and 2) as well as the ICARUS detector in the CNGS beam [3]
(versions 1 and 2).
These neutrino velocity experiments are not simple to carry out and a lot
of them had a variety mistakes . For instance in [1] (version 1) a cable was
incorrectlypluggedandtherewasamiscallibratedatomicclock. Bothmistakes
were found and the OPERA collaborators made proper corrections and after a
precise measurment of the neutrino velocity in agreement with c, the velocity
of light, published the result in [1] (version 3). OPERA collaborators also
published the results from a measurement of a special bunched neutrino beam
[2](version2)givingtheprecisionmeasurementofthemuonneutrinoandmuon
anti-neutrino velocities, in good agreement with c, the velocity of light. Other,
socalledGranSassolaboratoryrepeatedthemeasurementsandobtainedc,the
velocity of light, for the neutrino velocity [3] (version 2). Now, the masses of
flavor neutrinos, whose velocities one can measure, are not yet known precisely
but are calculated as exactly as possible from the provided masses of the mass
state neutrinos. Nevertheless, the accepted notion from the special relativity,
also in cases like these, expects the neutrino velocity not to exceed the velocity
of light c, considered in the special relativity as the universal limiting velocity.
InSection2,fromrelativistickinematicsoneformulatesthesixthorderbicu-
bicequationforthesquareofthelimitingvelocityand,asexposedin[4],canbe
solvedasacubicequationforc2. Thebicubicequationyieldsthreesolutionsc2,
i
i=1,2,3, depending on m,v, and E (particle mass, velocity and energy). One
expects that at least one solution is physical and luminal and as such supports
the LI; that is when evaluated to be numerically, either exactly or practically
exactlyequaltoc,sothatitssubstitutioninplaceofc,willnotchangeatallthe
energy-momentum dispersion relation or it will change it insignificantly. The
three limiting velocity solutions depend on the inverse sinusoidalfunction prin-
cipal values in such a way that the complementary c and c are real and, at
1 3
least one of them, physical while c is imaginary and as such unphysical. How-
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ever, for the specific multiple values of the inverse sinusoidal function, c can
2
become c and c . The important thing is the fact that c and c are comple-
1 3 1 3
mentarylimitingvelocitiessincetheytogethercancoveralltheallowedparticle
parameters, m,v, and E , while each of them is limited to particular values.
2
Besides the exact solutions, Section 2 contains also the perturbative solutions
for c2, i=1,2,3, basically in terms of (mv2/E) . These perturbative solutions
i
are often very convenient for determining as to which of the limiting velocities
is physical, c or c , either luminal (=, c) or not (=c) that is, applicable for
1 3
≃ 6
the particle in question.
In Section 3, after deducing the three flavor neutrino masses, one finds out
that the physical parameter structure of the OPERA [2] muon neutrino veloc-
ity experiment is such that the Taylor series expansion, in terms of (mv2/E),
strongly suggests c as the luminal solution. Along the same lines, one notices
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thatc is unphysical inthe OPERA[2]experiments.The sameis true for other
1
neutrino velocity experiments [1] and [3]. Furthermore, because the muon neu-
trinomassbeing negligible,onefinds outthatinfactc c. Toverifythis per-
3
≃
turbative result, one performs the calculation also with exact non-perturbative
expression for c . The result is the same as from the pertubative calculation.
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Conclusion and final remarks are given in Section 4. Here also the com-
parisons with other approaches from the literature for discussing the Lorentz
invariance and Lorentz violation, either through changes in the Dirac equation
or by explicitly changing the relativistic kinematics, are given.
2. Particle limiting velocities from the velocity bicubic equation
The velocity of light, c, in the special relativity particle kinematics is con-
sidered the universal relativistically invariant limiting velocity. Here, with the
desireofhavingconanequalbasiswithotherphysicalparameters,onetreatsit
as yetto be analythicallyformulatedlimiting velocity andstarts with the same
kinematics.
−→p = Ec2−→v ,E2 = 1m2cv42 (1,2)
− c2
which defines the particle momentum −→p, and energy E, through its mass m
and velocity −→v. Momentum and energy from (1) and (2) are related through
the mass shell condition,
−→p2c2 E2 = m2c4 (3.1)
− −
whosechange,ifany,causedbyreplacingcwithlimitingvelocitysolutionsfrom
thebicubicequation(tobediscussed),couldindicateeitherLIorLV,providing
that particle parameters, m,v, and E are known.
As in the neutrino velocity experiments [1,2,3], one has the known energy
andthevelocityoffixeddirection,itisconvenienttocontinuewithjustrelation
(2)bytransformingitintothebicubicequationfortheparticlelimitingvelocity
c:
m2c6 =E2c2 E2v2 (3.2)
−
Next, one rewrites it in the mathematically more familiar forms with solutions
characterized by the discriminant satisfying D <0 ,
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c 2 3 E 2 c 2 E 2 E 2
+ = 0, q = p= ,
v − mv2 v mv2 − mv2
(cid:20)(cid:16) (cid:17) (cid:21) (cid:18) (cid:19) (cid:16) (cid:17) (cid:18) (cid:19) (cid:18) (cid:19)
q 2 p 3
D = +
2 3
(cid:16) (cid:17) (cid:16)4(cid:17) 2
1 E 4 E
= 1 <0,
4(cid:18)mv2(cid:19) " − 27(cid:18)mv2(cid:19) #
(3.3)
3√3mv2
z = ; D <0: 1<z <1 (4)
2E −
According to [4], the solutions for (3.2,3) plus (4) can be written as
p θ π
c2 = 2v2 | |cos + ,
1 3 3 6
r (cid:18) (cid:19)
p θ π
c2 = 2v2 | |cos ,
2 − 3 3 − 6
r (cid:18) (cid:19)
p θ π
c2 = 2v2 | |cos +
3 − 3 3 2
r (cid:18) (cid:19)
π q 3√3mv2
cos θ+ = = − ,
(cid:16) 2(cid:17) −2 |p| 32 2E
3
(cid:16) (cid:17)
π 3√3mv2 3√3mv2
θ = +cos−1 − =sin−1 (5)
−2 2E ! 2E !
However,in order to see more ofthe physics,the exactsolutions from (5), with
the help from relations (3.3) and (4), are rewritten in such a way as to exhibit
more explicitly the m, v, and E parameters in them:
2E π 1 3√3mv2
c2 = sin sin−1 >0, (6.1)
1 √3m "3 − 3 2E !#
2E 1 3√3mv2 π
c2 = cos sin−1 <0, (6.2)
2 −√3m "3 2E !− 6#
2E 1 3√3mv2
c2 = sin sin−1 >0 (6.3)
3 √3m "3 2E !#
Noting thatwith the variablez fromrelation(4) the inversesinusfunction,
sin−1(z),in(6.1,2,3)refertotheprincipalvaluesthatlieinthe( π/2 to π/2)
−
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range where either of the positive c2 and c2 can be physical while the negative
1 3
c2 is definitely unphysical. However, c2 can become physical in the multiple
2 2
valuesranges. Denote c2,i=1,2,3dependence onz asc2 sin−1(z) ,i=1,2,3
i i
. Then assume that alternately in c2 the range of sin−1(z) is changed from
2 (cid:2) (cid:3)
( π/2 to π/2) to (3π/2 to 5π/2) and to (( 5π/2)to ( 3π/2)).This is simply
a−chieved by replacing c2 sin−1(z) in (6.2) a−lternately w−ith c2 sin−1(z)+2π
2 2
andc2 sin−1(z) 2π . Thenthesimpleevaluations,withthehelpfrom(6.1,2,3),
2 − (cid:2) (cid:3) (cid:2) (cid:3)
shows that
(cid:2) (cid:3)
c2 sin−1(z)+2π =c2 sin−1(z) ; c2 sin−1(z) 2π =c2 sin−1(z)
2 3 2 − 1
(6.4,5))
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
In(6.4,5)therespectivemultivaluerangesinc2are(3π/2 to 5π/2)and( 5π/2 to 3π/2)
2 − −
while in c2and c2 the range is ( π/2 to π/2). As one sees , here the comple-
3 1 −
mentary limiting velocities c and c are the only ones of the physical signifi-
1 3
cances. Importance of (6.4,5) is in the fact that sometimes one has to change
the”coordinates”inordertofindoutthesameorperhapseventhenewphysics.
These different ranges, principle values and multiple values do not change the
fact that according to (4) z < 1. The imaginary c moves the ”imaginary”
2
physics to the ”real” physi|cs| with sin−1(z), being changed to sin−1(z) 2π
±
now with the multiple value ranges.
Next,itisillustrativetoperformtheTaylorseriesexpansionsofc2,i=1,2,3
i
(6.1,2,3) . Except for the few first terms, they are done basically in terms of
(mv2/E)withinequalitiesbetweenv,E andmasindicatedineachoftheseries.
Extra v2 factor in each term makes the whole expression to have dimension of
v2 .
E v2 3mv4 m2v6 mv2 3 2E
c2 = +O v2 >0,v2 < ,(7.1)
1 m − 2 − 8E − 2E2 " (cid:18) E (cid:19) # 3√3m
E v2 3mv4 m2v6 mv2 3 2E
c2 = + +O v2 <0,v2 < (7,.2)
2 −m − 2 8E − 2E2 " (cid:18) E (cid:19) # 3√3m
m2v6 69m4v10 mv2 6 2E
c2 = v2+ + +O v2 >0,m< (7.3)
3 E2 32 E4 " (cid:18) E (cid:19) # 3√3v2
Already fromexactsolutions(6.1,2,3)aswellas now fromthe Taylorseries,
one sees that all three limiting velocities are different from each other in the
principle values region. For instance, c2 and c2 diverge for m = 0 but are
1 2
finite for v = 0. So c2 and the unphysical c2 need to have m = 0. The
1 2 6
different behaviors of c2 and c2 for either m 0 or v 0 emphasizes their
1 3 → →
complementarity. Thissmallexcursion,suggestsdefining the physicalc2 andc2
1 3
in the principle values region satisfying
Phyhsical:c2, c2 =0, (8.1)
1 3 6 ∞
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Here, consistent with (8.1), is the summary of important situations that can
occur for c2 and c2
1 3
E
v = 0,m=0: 0 =c2 =c2; c2 =0 (unphysical), (8.2)
6 m 1 3
m = 0,E finite;c =v =c (photon); c = (unphysical), (8.3)
3 1
∞
m = 0,E ;c v; c (unphysical). (8.4)
3 1
6 →∞ → →∞
What examples (8.2,3,4) show clearly is that in these particular situations the
physically acceptable limiting velocity is either given by c or c which further
1 3
indicates to their complementarity. The relation (8.2) is to be understood as a
definitionofE(v)atv =0wherec2 =c2. As c andc arethe limiting velocity
1 1 3
solutionsofthebicubicequation(3.3),itisappropriatetoseewhateffectwillbe
caused if one sets either c or c in place of c in the energy momentum relation
1 3
(3.1). These substitutions leave the energy momentum relation (3.1) either
LI or, to a degree, LV under the Lorentz transformations with the following
respective general possible values for c or c :
1 3
LI :c =c or c =c; LV :c =c or c =c (8.5)
1 3 1 3
6 6
Here it is assumed that either c or c is LI but not both of them at the same
1 3
time. Also,thedegreeoftheLVwoulddependonhowstronglyc =corc =c.
1 3
6 6
Despite their complementarity, It is necessary to investigate whether it can
happen that for a given particle one can have c =c ? Imposing this equality,
1 3
from (6.1) and (6.2), with z as defined in (4), one arrives at the following
sequence of equations,
π 1 1
c2 = c2 :sin sin−1(z) =sin sin−1(z) , (9.1)
1 3 3 − 3 3
(cid:20) (cid:21) (cid:20) (cid:21)
π
sin−1(z) = : z =1 (9.2)
2
Since relation (9.2) is in contradiction to the relation (4), which excludes
z = 1, one concludes that c and c , while complementary, cannot be equal to
1 3
each other for the same particle,c = c . Hence, if for instance c = c then
1 3 3
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c =c and so on.
1
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3. Limiting velocity of the neutrino
Here one is specifically interested in applying the formalism of obtaining
the limiting velocity for the muon neutrino, ν , with the physical parameters
µ
from the OPERA experiment [2]. From the perturbative expressions (7) , the
indicationisthatc andc arerespectively,theunphysicalandphysicallimiting
1 3
velocities, in this case. To see whether the physical c is also luminal, that is,
3
leading to c and LI, one first expresses c perturbatively from (7.3) and then,
3
for verification purposes, also exactly from (6.3).
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The formalism in relations (6) and (7) demand, in addition to E and v also
the value of the mass, here denoted for the muon neutrino as m . As in the
ν
reference [2] the value of m is not given, one has to first find which value is
ν
presentlyfavored,althoughinOPERAexperiment[2]withthe neutrinoenergy
of E (µ) = 17GeV, the calculated neutrino mass even if exact, will be very
ν
likelynegligibleascomparedtotheenergy. Now,therearethreeflavorneutrinos,
denoted as ν , ν and ν , the electron, muon and tau neutrino.Their masses
e µ τ
m (e), m (µ)andm (τ) arederivedfromthe massesofthe independent mass-
ν ν ν
statethreeneutrinoswithmassesm ,m andm . Inthediscussionoftheµ τ
1 2, 3
−
symmetry, these masses have been given by S. Gupta et al. [5] as,
m c2 =0.067 eV, m c2 =0.068 eV, m c2 =0.084 eV, (10.1)
1 2 3
Theflavorneutrinomassesaredefinedin[5]withthehelpoftheHarrisonetal.
neutrino mixing matrix [6], U , α = e,µ.τ ;i = 1,2,3, connecting the flavor
α,i
neutrino states to the mass-state neutrino states (see,also [7] ]). Hence, using
U asinreferences[6]and[7],accordingtoGuptaetal. [5],theflavorneutrino
α,i
masses are defined as
(1)
α = e,υ,τ; i=1,2,3: m (α)= U 2m2 2 ;
ν i| α,i| i
h i
2 1 0 P
3 3
(U ) =  q 1 q1 1  (10.2)
α,i − 6 3 − 2
 q1 q1 q1 
 − 6 3 2 
 q q q 
yielding
m (e)c2 =0.067 eV, m (µ)c2 =0.076 eV, m (τ)c2 =0.076 eV (11)
ν ν ν
With these values and from [2] the collection of data for the OPERA muon
neutrino velocity experiment is
E (µ) = 17GeV, m (µ)c2 =0.076 eV,
ν ν
v (µ) = c(1+∆)
ν
1.8 10−6 ∆ 2.3 10−6 (12)
− × ≤ ≤ ×
Because one has that m (µ) < 2E (µ)/3√3v2(µ) for any v (µ) from
ν ν ν ν
(12), one easily deduces, according to (7.1), that approximate numerical value
(cid:0) (cid:1)
of c is c 4.73 105c .Such a large value makes c unphysical. Expecting
1 1 1
≃ ×
that c is physical, one calculates it with more precision first perturbatively
3
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from (7.3),
m (µ)c2 2 v (µ) 4 m (µ)c2 4 v (µ) 8
c2 =v2(µ) 1+ ν ν +O ν ν
3 ν E (µ) c E (µ) c
" (cid:18) ν (cid:19) (cid:18) (cid:19) (cid:18) ν (cid:19) (cid:18) (cid:19) !#
(13)
Furthermorewith m (µ)c2/E (µ) 4.5 10−12 and ∆ <<1, one obtains
ν ν
≃ × | |
for c the perturbative solution in the form,
3
(cid:0) (cid:1)
c 1 m (µ)c2 2
3 (1+∆) 1+ ν (1+∆)4 =(1+∆) (14.1)
c ≃ 2(cid:18) Eν(µ) (cid:19) !
Exact expression for c from (6.3) with the OPERA physical parameters is
3
as follows
c 2 2E (µ) 1 3√3m (µ)c2(1+∆)2
3 = ν sin sin−1 ν
c √3mν(µ)c2 "3 2Eν(µ) !#
(cid:16) (cid:17)
= (1+∆)2 (14.2)
In deriving (14.2), one simply takes into account that m (µ)c2/E (µ) <<
ν ν
1 and that in (1 + ∆) , ∆ << 1 so that only first terms in Taylor ex-
| | (cid:0) (cid:1)
pansions of sin and sin−1 functions need to be retained. Hence , both solu-
tions, being equal and basically luminal, yield LI with c as the solution for
c :
3
(1 1.8 10−6)c c (1+2.3 10−6)c : c c (14.3)
3 3
− × ≤ ≤ × ≃
The result in (14.3) is what Einstein envisioned long time ago.
What one notices here is the fact that for OPERA experiments [2],
through a particular collection of neutrino physical parameters, such as mass,
ordinary velocity and energy, the bicubic equation yields the luminal limiting
velocity solution, that is, with the velocity of light c and with the LI.
Now,ononeexampleonecanshowhowtheluminallimitingvelocitysolution
withtheLI,canbecomethesuperluminalsolutionwiththeLV.Simply,in(14.1)
and (14.2) replace the negligible ∆ with a small but finite and positive ∆. In
doing so, one basically obtains he LV from reference [8],implied by the change
1
in the particle special relativistic velocity, written as c/ p2+m2c2 2 +∆c.It
is easily seen that in the situation where the mass is negligible, as is in the
(cid:0) (cid:1)
OPERAexperiments,this ∆ shouldbe the same as∆ inrelations(14.1,2),and
if negligible, as in relations (14), should allow the LI rather than the LV under
the Lorentz transformations. Although so far no verifiable LV showed up in
neutrino physics, on should, nevertheless, keep an open mind also for such a
possibilitywithsubluminalorsuperluminalanticipations. The LVformulations
with superluminal particles through the Dirac equation have been done, for
example, in [9] and [10].
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4. Conclusion and final remarks
Identifying the velocity of light c in the relativistic kinematics as a limit-
ing velocity yet to be determined, one is lead naturally to the bicubic equation
for the limiting velocity. Of the three resulting solutions one, c , is imaginary
2
while two other solutions, c and c , are real and complementary with different
1 3
emphasis on particle parameter dependences. Of course, if the particle param-
eters choose, say c = c ,then as argued in (8.1) to (8.4) c will be unphysical.
1 3
The remarkable point in determining the limiting velocity of any particle from
the bicubic equation is that the particle physical parameters will yield for it
most likely c, no matter where one measures its mass, energy and the ordinary
velocity.
Itappearsthatwhatoneneedsarethevelocityexperimentsdonewithrather
a masssive particle which allow full participation of the particle mass in deter-
mining ofits limiting velocity . A naturalcandidate for sucha limiting velocity
determinationistheelectronwhosemassisverywellknownandtheenergycan
be chosen so as not to render the mass negligible.
Acknowledgment
TheauthorisgratefultoDr. M.Dracosforinforminghimthattheneutrino
beam has 17 GeV average energy in the OPERA experiments. Descriptions
and explanations of neutrino velocity experiments by an anonymous physicist
knowledgeablewithOPERAandother experimentsis gratefullyacknowledged.
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