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

2

QFTHEP’2019

3

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,

QFTHEP’2019

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.

4

QFTHEP’2019

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.

5

QFTHEP’2019

6

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

QFTHEP’2019

7

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.

QFTHEP’2019

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

8

QFTHEP’2019

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

QFTHEP’2019

.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:

10

QFTHEP’2019

.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

QFTHEP’2019

.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

QFTHEP’2019

13

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

QFTHEP’2019

3

1,

2

2211

22

2211 2sinIm24sinRe4

kiki

ikkikiikkiki pLmUUUUpLmUUUU

Scheme of the T2K experiment:

14

QFTHEP’2019

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

15

QFTHEP’2019

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:

16

QFTHEP’2019

.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:

17

QFTHEP’2019

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

QFTHEP’2019

19

Norm

aliz

ed p

robabili

ty

km ,L

QFTHEP’2019

20

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

QFTHEP’2019

21

Norm

aliz

ed p

robabili

ty

km ,L

QFTHEP’2019

22

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(

QFTHEP’2019

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

QFTHEP’2019

The detection probability reads

and can be written in the form

which is in accordance with the prediction of the

standard approach.

24

QFTHEP’2019

25

The results of numerical integration with the same parameters:

Norm

aliz

ed p

robabili

ty

km ,L

QFTHEP’2019

26

Norm

aliz

ed p

robabili

ty

km ,L

QFTHEP’2019

27

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.

QFTHEP’2019

28

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.

QFTHEP’2019

Thank you for your attention!

QFTHEP’2019