Hyperfine structure of mesomolecules pdμ,

ptμ, dtμ in variational method

The XXIV International Workshop

High Energy Physics and Quantum Field Theory

September 23rd, Sochi

Sorokin V.V.

In collaboration with: A.V. Eskin, A.P.

Martynenko, F.A. Martynenko and O.S.

Sukhorukova

INTRODUCTION AND OVERVIEW

2

The investigation of energy spectra of hydrogen muonic molecules is

important for muonic catalysis of nuclear fusion reactions. A calculation

of fine and hyperfine structure of muonic molecular ions as well as

higher order QED corrections allows us to predict the rates of reactions

of their formation and other parameters of the μCF cycle.

Korobov V.I., Puzynin I.V. and Vinitsky S.I. Physics Letters B 196 (1987)

272-276.

Frolov A.M. and Wardlaw D.M. Eur. Phys. J. D 63 (2011) 339–350.

Aznabayev D.T., Bekbaev A. K., Ishmukhamedov I. S., and Korobov V. I.

Physics of Particles and Nuclei Letters 12 (2015) 689–694.

3

PURPOSE

The aim of this work is to study hyperfine splitting of three-particle tdμ,

dpμ and tpμ muonic molecular ions on the basis of variational approach.

Tasks:

1. Analytical calculation of diagonal and off-diagonal matrix elements of

kinetic energy, potential energy and overlap for basis functions;

2. Writing computer code to solve bound state problem for three particles using

stochastic variational method with correlated Gaussian basis;

3. Calculation of the energy of the ground state of tdμ, dpμ and tpμ muonic

molecular ions and their hyperfine structure.

4

GENERAL FORMALISM

N

ji

ijij

N

i

cm

i

i VVTm

H1

2

2

ptwo-body interaction potentials

EH

Let us consider a system of 3 particles with masses m1, m2 and m3 and charges z1, z2

and z3 respectively. The Schrodinger equation in the Jacobi coordinates has the form:

In variational method the wave function of the system is presented as follows

1

( , )K

i i

i

c A

x

An upper bound for the energy of the ground state is given by the lowest eigenvalue

of the generalized eigenvalue problem:

( , ) | | ( , )

( , ) | ( , )

K

ij i j

ij i j

HC E BC

H A H A

B A A

x x

x x

5

GAUSSIAN BASIS FUNCTIONS

In variational approach with correlated Gaussian basis wave functions have the form:

1 2 12 3 123

/2

1 1

1 1

1 2 3

( , ) ( )

( ) ,

.

.

( ),

( ) [[[ ( ) ( )] ( )] ...]

A

A

A

N N

ij i j

i

L

L l l L LM

j

l L

A G

G e

A A x x

~

x x

~

x x

x

x

x

x x

x

x x

N

N

x

xx ),...,(11

x Jacobi coordinates

center of mass coordinate

The diagonal elements of the (N-1)×(N-1) dimensional symmetric, positive definite

matrix A correspond to the nonlinear parameters of Gaussian expansion, and the off-

diagonal elements connect different relative coordinates thus representing the

correlations between particles.

The angular part of the basis wave function has the following form:

6

We use the following order of particles:

JACOBI COORDINATES

0

3

21

2211

21

R

rrr

λ

rrρ

mm

mm

ρλrrr

ρλrrr

ρrrr

11

2112

21

13223

21

233

mm

m

mm

m

ρ

λ

The Jacobi coordinates are related to the relative

particle coordinates as follows:

For the interparticle coordinates:

. 321321321

μtp and μpd, μdt

GROUND STATE L=0

7

}2

ћ

2

ћ{

det

24|| 00

2

200

1

2

5.2

300

^'

II

BT

'

112222

2

122212122222

2

12

00

221111

2

121112121111

2

12

00

))((2

))((2

ijijij AAB

BBABABAABAI

BBABABAABAI

2 211 22 12

1[ 2 ( )]

200 ( , , )

A A A

A e

ρ λ

ρ λ

Kinetic energy operator:

where:

2

2

22

1

2^

2

ћ

2

ћλρ

T

321

3212

21

211

)(,

mmm

mmm

mm

mm

Matrix elements of kinetic energy:

))((

28

det

28

||

223,13

2

23,13

122

23,13

1

5.200

23,13

22

5.200

12

00

2332

00

1331

00

1221

^'

FFBFI

BBI

IeeIeeIeeV

21

23,13

1,2

2212

23,13

2

21

23,13

1,2

12

2

21

23,13

1,2

2211

23,13

1

2

2)(

mm

mBBF

mm

mB

mm

mBBF

5.1

300'

det

8|

B

GROUND STATE L=0

Potential energy operator:

Matrix elements of potential energy:

Overlap matrix elements:

||||||

21

2

32

21

2

3121

ρλρλρ

mm

m

ee

mm

m

eeeeV

8

HYPERFINE STRUCTURE

9

Hyperfine structure can be described with the following interaction potential:

Where indices 1, 2 denote nuclei of hydrogen isotopes, index 3 denotes muon.

10

MATRIX ELEMENTS FOR HFS

K

ji

ji

K

ji

ji

K

ji

ji

m

mB

m

mBB

cc

m

mB

m

mBB

cc

Bcc

1,2/3

2

12

122

12

11211

2/3

1,2/3

2

12

222

12

21211

2/3

1,2/3

22

2/3

)2(

)2()(

)2(

)2()(

)(

)2()(

23

13

12

r

r

r

ji

ji

ji

AAB

AAB

AAB

mmm

222222

121212

111111

2112

11

AVERAGING OVER SPIN FUNCTIONS

Averaging over spin functions can be performed using the following relations:

1212 12

12 12 1 2 3

12 1 212

12 12 1 1 1 3 3 312 12

12 3 1 21 12

3 12 112

1212 1

, | ( ) | , ( ) ,

, | ( ) | , (2 1)(2 1)(2 1)( 1) (2 1)( 1)

( 1) ,1 ; 1

, | ( ) | , (2

{ }{ }max min

S SS

S S S S S S

S S S S S S

S S S S S S S S S S S S

S S S S S S

S S S S

S S S S S

1 2

1 3

2 3

S S

S S

S S

12 1 2 3

12 2 2 2 3 3 32

12 3 2 12 1 12

3 12 212

1)(2 1)(2 1)( 1) (2 1)( 1)

( 1) .1 1

{ }{ }maxS S S S S

S S S S S S S

S S S S S S

S S S S

12

ENERGY MATRIX

After averaging over spin functions energy matrix takes form:

c

acb

bcba

cba

4

3

4

3

4

3

4

30

4

3

2

1

2

1

4

10

00)(4

1

All particles have spins 1/2

After diagonalization eigenvalues can be obtained:

)(4

1

)(4

1

3

222

2,1

cba

acbcabcbacba

13

cba

cbacb

cbcba

cba

4

1000

012

1

3

1

3

2

3

20

03

2

3

2

12

5

6

5

2

10

0004

1

2

1

2

1

Particles have spins 1, 1/2, 1/2

ENERGY MATRIX

After averaging over spin functions energy matrix takes form:

After diagonalization eigenvalues can be obtained:

cbaacbcabcba

cba

cba

44144994

1

)22(4

1

4

1

222

4,3

2

1

14

ADDITIONAL CORRECTIONS, VACUUM POLARIZATION

To improve the accuracy of our calculations we take into account vacuum polarization

corrections:

Where indices a, b correspond to nuclei of hydrogen isotopes, µ corresponds to muon.

15

ADDITIONAL CORRECTIONS, NUCLEAR STRUCTURE

Nuclear structure corrections of the leading order take the form:

16

THE PROGRAM

For the numerical calculation of energy levels of three-particle Coulomb bound

states the code in MATLAB is written. The program uses stochastic variational

approach with random optimization procedure for nonlinear variational parameters;

The program is based on the Fortran program by K.Varga and Y.Suzuki;

A number of changes is made compared to the Fortran program, including the

ability to calculate states with nonzero L, more convenient generation of variational

parameters and various optimization changes;

The main results of the calculation include energies of ground and excited states

along with variational wave functions for each state. The program is capable of

calculating L=0 and L=1;

We are now working on calculation of L=1 hyperfine structure.

K. Varga, Y. Suzuki// Computer Physics Communications 106 (1997) 157-168

NUMERICAL RESULTS

17

For ground state and hyperfine structure of tdμ, dpμ and tpμ the following numerical results

are obtained:

A. M. Frolov // Eur. Phys. J. D. –– 2012. –– Vol. 66. –– P. 212––223.

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THANK YOU FOR YOUR

ATTENTION