quantum field-theoretical description of neutrino oscillations...
TRANSCRIPT
Quantum field-theoretical
description of neutrino
oscillations in T2K experiment
V. Egorov, T. Rusalev
(SINP MSU, Faculty of Physics MSU)
28.09.2019
The XXIV International Workshop
High Energy Physics and Quantum Field Theory
Outline Introduction: difficulties of the standard
description, the idea of the novel approach
Oscillations with neutrino production in pion (kaon) decay and detection in charged-current weak interaction with nucleusTheory
T2K experiment
… in charged- and neutral-current weak interactions with electronTheory
T2K experiment
Conclusion
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IntroductionThe standard S-matrix formalism is not appropriate for describing finite space-time interval processes:
.Neutrino oscillations: the standard QM
description in terms of plane waves violation of energy-momentum conservation;the QM and QFT descriptions in terms of wave packets complicated calculations.
C. Giunti, C.W. Kim, J.A. Lee and U.W. Lee,«Treatment of neutrino oscillations without resort to weak eigenstates,» Phys. Rev. D 48 (1993) 4310.
Lt,
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We put forward a modification of the perturbative formalism, which allows one to describe the processes passing at finite distances during finite time intervals.
The approach is based on the Feynman diagram technique in the coordinate representation supplemented by modified rules of passing to the momentum representation. The latter reflect the geometry of neutrino oscillation experiments and lead to a modification of the Feynman propagators of the neutrino mass eigenstates in the momentum representation.
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The idea behind the approach comes from the paper
R.P. Feynman, «Space-Time Approach to Quantum Electrodynamics,» Phys. Rev. 76 (1949), 769.
It was developed in the papers
V.O. Egorov and I.P. Volobuev, «Neutrino oscillation processes in a quantum field-theoretical approach,» Phys. Rev. D 97 (2018) no.9, 093002,
V.O. Egorov and I.P. Volobuev, «Quantum field theory description of processes passing at finite space and time intervals,»Theor. Math. Phys. 199 (2019) no.1, 562,
V.O. Egorov and I.P. Volobuev, «Coherence length of neutrino oscillations in a quantum field-theoretical approach,»Phys. Rev. D 100 (2019) no.3, 033004.
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Oscillations with neutrino
production in pion (kaon) decay
and detection in charged-current
weak interaction with nucleusWe work in the framework of the minimal extension of
the SM by the right neutrino singlets. The charged-
current interaction Lagrangian of the leptons:
The field of the charged
lepton of the i-th generation
The field of the neutrino
mass eigenstate
The neutrino mixing
matrix (PMNS matrix)
.H.c.122
53
1,
cc
WUl
gL kik
ki
i
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The process is described in the lowest order by
the diagram:
The points of production x and detection y are
supposed to be separated by a fixed
macroscopic interval. The diagram must be
summed over all three neutrino mass
eigenstates k = 1, 2, 3.
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The amplitude in the coordinate representation in accordance with the usual Feynman rules. If we integrate with respect to x and y, we lose the information about the space-time interval between them. In order to save it we introduce the delta function into the integral. It is equivalent to the replacement:
We arrive at the time-dependent propagator of k-thneutrino mass eigenstate in the momentum representation:
Txy 00
.d, 0c4c
TzzSezTpS k
ipz
k
.00cc TxyxySxyS kk
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The integral can be evaluated exactly:
This propagator makes sense only for
macroscopic time interval T.
Particles which propagate over large macro-
scopic space-time intervals are close to mass shell
(GS theorem), i.e. . The time-
dependent propagator takes the simple form
.
2
ˆ 22200
2220
22200
0c
Tpmppi
k
kk
k
k
e
ipmp
mpmppp
ip,TS
9
12022 pmp k
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.2
ˆ 0
22
2
0
cT
p
pmi
kk
k
ep
mpip,TS
.
4sin41
4
1 3
1,0
n
222
2
2
220
n
2
2
2
1
2
ki
ki
ikki T
p
mUU
pMMM
Production
processDetection
process
The amplitude of the process in the momentum
representation when y0 – x0 = T , neglecting
neutrino masses everywhere except for the exp:
It contains three terms which interfere with each
other leading to oscillations.
The squared modulus of the amplitude
factorizes as follows:
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.1ˆ,cos2
5
n
3
1
22
2)(C0
n
2
F0n
2n
2
qpkuPPJeUmfp
GiM
k
Tp
pmi
k
k
11
Let us find the probability of the process. The ⟨|amplitude|2⟩is multiplied by the delta function of energy-momentum
conservation and integrated with
respect to the phase volume of the final particles.
The integration would result in variation of the virtual
neutrino momentum direction, which contradicts the
experimental setting since the neutrino propagates from
the production point to the detection point. Thus, one must
calculate the differential probability of the process where pn
is fixed.
It can be achieved by additional multiplying the
⟨|amplitude|2⟩ by , or, equivalently, substituting p
instead of pn there and multiplying the result by
.
PkqPp 4
2
pp n2
pqp 2
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.4
sin41d
d
d
d 3
1,
222
2
2
2
)(
23
)(
1
3
3
),(3
ki
ki
ikki L
p
mUUW
p
W
p
W
12
We find:
The additional delta function fixes not only the direction
of neutrino momentum, but also its length, thus we must
integrate the result above with respect to . The final
probability of detecting an electron in the considered
process:
p
pLpT0
Neutrino is
almost
on-shell
LpP ,
– the standard oscillating factor
SingularIs determined by energy-momentum
conservation in the production vertex
.,d
dd
d
d
d
d )(
2
)(
12
3
),(3),(
LpPWW
ppp
WW
pp
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The differential probability of neutrino production in the pion decay:
The differential probability of neutrino production in the
kaon decay is given by the same formula with
the replacements :KK mmff ,,sincos CC
.cos28
cos
d
d200
22
)(
22
)(
2
2
C
22
F
)(
1
ppp
mmmfGW
iK
.cos28
sin
d
d200
22
)(
22
)(
2
2
C
22
F
)(
1
KKK
KK
K
ppp
mmmfGW
The whole probability of finding an electron in the
detection process differs from the obtained one by the
replacements i.e.:,,)(
2
)(
2 e
e PPWW
.,d
dd
d
d
d
d )(
2
)(
12
3
),(3),(
LpPWW
ppp
WWe
pp
eee
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1,
2
2211
22
2211 2sinIm24sinRe4
kiki
ikkikiikkiki pLmUUUUpLmUUUU
Scheme of the T2K experiment:
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Main detection channels:
We should average the probability of the process with
detection of the lepton l (muon or electron) with respect to
the momenta of the initial mesons.
., pnpen ii
0
n
0n
Meson
To detector
Proton beam axis
.coscoscossinsin
coscos
,cos,sinsin,cossin
,cos,0,sin
00
00
000
nnnn
n
n
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The probability from both the π- and K-meson decays:
where
.d,d
d
d
d
d
d )(
2)(
),(),()(
pLpPpWp
Wa
Wa
Wl
llK
K
ll
.10Ka
a Total neutrino
momentum distribution:
For pion:
,,
,,,d
d,ddsind
d
d
)(
2)(
)(
2
)(
12),(
pWLpPppd
pWLpPpW
pppW
l
l
l
l
l
cos2 0
2
)(
2
pp
mmp
where the pion-produced neutrino momentum distribution
.,,
d
d,
,,
,,dsind
)(
12
)(
p
Wp
pD
pDpp
Redesignate
for short:Kaa )()()(
pp
Pion momentum distribution
Based on the work K. Abe et al., Nuclear Instruments and
Methods in Physics Research Section A 659 №1 (2011),
we fit the neutrino momentum distribution for θ0 = 2.5° by:
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.d8
cos2max
2min
22
)(4
22
)(2
2
)(2
C
22
F
2)(
2
Q
Q
lll
l QQCM
DQB
M
DQA
p
GMW
The probability of neutrino quasielastic scattering with
production of the lepton l has the form:
.GeV65.0,
GeV,65.0,
0
)(
2
pAey
peppap
zp
cbpnm
.14.0,151,03.0
,048.0,38.0,5.6
,006.0,16.0,87
0
zAy
cbn
ma
where A(l ), B(l ), C(l ) are extremely bulky and can be found
along with M and D in the book T. J. Leitner, "Neutrino
Interactions with Nucleons and Nuclei," Justus-Liebig
University, Germany (2005).
Let us consider the muon detection first. The product of the neutrino momentum distribution and the detection probability looks like:
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GeV,p
Norm
aliz
ed p
roduct
18
The results of numerical integration with the parameters
.753.3 ,1503.0 ,8657.0 ,5903.0
,eV 1053.2 ,eV 1039.7
CP132312
232
32
252
21
mmN
orm
aliz
ed p
robabili
ty
km ,L
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Norm
aliz
ed p
robabili
ty
km ,L
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Similarly one can consider the process where an electron is produced in the neutrino scattering by a neutron:
Norm
aliz
ed p
robabili
ty
km ,L
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Norm
aliz
ed p
robabili
ty
km ,L
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Neutrino detection in charged- and
neutral-current weak interactions
with electron
Let us consider the process which gives a contribution
in the T2K experiment, even though it is small:
k
)Amplitude(
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The same differential probability
of neutrino production
The amplitudes in the momentum representation,
corresponding to the diagrams, when y0 – x0 = T :
The final probability:
23
.d
dd
d
d
d
d2
)or (
12
3
)or (3)or (
pp
KKK
WW
ppp
WW
The probability of detection
contains the oscillating factor
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The detection probability reads
and can be written in the form
which is in accordance with the prediction of the
standard approach.
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The results of numerical integration with the same parameters:
Norm
aliz
ed p
robabili
ty
km ,L
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Norm
aliz
ed p
robabili
ty
km ,L
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Conclusion
A novel QFT approach to the description of processes passing at finite space-time intervals is discussed. It is based on the Feynman diagram technique in the coordinate representation supplemented by the modified rules of passing to the momentum representation, which reflect the experimental situation at hand.
It is explicitly shown that the approach allows to consistently describe neutrino oscillation processes. The main results of the standard approach are reproduced.
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The neutrino flavor states are redundant, we deal only with neutrino mass eigenstates.
The T2K experimental setting is considered and the oscillation fading out due to momentum distribution of the initial particles is taken into account. The obtained results confirm that the far detector position corresponds to the first maximum for e– production and the first minimum for μ– production.
The advantages of the approach are technical simplicity and physical transparency. Wave packets are not employed, we use only the description in terms of plane waves.
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Thank you for your attention!
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