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The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

The XXII International Workshop High Energy Physics and Quantum Field Theor y, Samara, June 24-July 1 2015 Theor y of the Lamb shift in muonic helium ions
R.N. Faustov1 , A.A. Krutov2 , A.P. Mar tynenko2,3 , G.A. Mar tynenko
1

2

Dorodnycin Computing Centre RAS, Moscow, Russia, 2 Samara State University, Samara, Russia, 3 Samara State Aerospace University, Samara, Russia

26 June 2015


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Outline

1. Introduction 2. Vacuum polarization effects 3. Relativistic corrections with vacuum polarization effects 4. Nuclear structure and vacuum polarization effects 5. Numerical results and conclusion


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

CREMA(Charge Radius Experiment with Muonic Atoms) collaboration 2010-2015 Task: to measure fine and hyperfine structure in light muonic atoms (muonic hydrogen, muonic deuterium, ions of muonic helium...); to determine charge radii of the proton, deuteron, helion, alpha-par ticle with the accuracy 0.0005 fm.

µ


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

The proton radius puzzle
The proton rms charge radius measured with electrons: 0.8770 ± 0.0045 fm muons: 0.8409 ± 0.0004 fm

7.9
µp 2013 electron avg. scatt. JLab µp 2010 scatt. Mainz H spectroscopy
0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9
ch

Proton charge radius R [fm]


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

If the proton radius puzzle is caused by muon-electron universality breakdown (µHe)+ spectroscopy can reveal it! The transitions in (µ4 He2 )+ and (µ3 He2 )+ are planned to measure with [800, 1000] nm


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

One- and two-loop VP corrections in 1 interaction

a

b

c

d

HB = -

p4 Z Z p4 p2 - + - - 3 3 2µ r 2 8m1 8m2 p2 +
-
Wr 2

I 1 +2 2 m1 m2

(r)- (L 1 ). WrYlm (, ).

Z 2m1 m2 r

r(rp)p r2 1- Wr 2

+

Z r3

1 1 + 2 2m1 m2 4m1 W 3/2 e 26
-
Wr 2

W 3/2 200 (r ) = e 2 2

, 2lm (r ) =

W = µZ .


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

One-loop VP correction to the Lamb shift in 1 interaction
VVP (r ) =
C

3
1



d ( )

-

Z -2me r e r


, ( ) =

2 - 1( 2 2 + 1) 4
2 -x 1+ 2me W

,

EVP (2S ) = - 1 2 5/2 1 2 1 - k1

µ(Z )2 6
1 6



( ) d
0 4

xd x
2

1-
2

x 2

e

=

2 1 - k1

-1 6 8 k1 + 2 7 2 k1 - 4 9 k1 + 6 k1 - 1

2

1 4 k1 + 3 k1 - 2 8

2

+

1 - 8 6 4 2 +3 56k1 - 128k1 + 75k1 + 10k1 - 4 ln EVP (2P ) = - = 1 2 5/2 1 - k1
2 1 - k1

k1

µ(Z )2 72
6

2 1 - k1 = x dxe
3 -x 1+

-2041.9990 meV -2077.2217 meV
2me W

,





( ) d
1 4 2 0

=
2

-1 2 0 k1 + 1 8 4 k1 - 2 3 k1 + 6 k1 - 1

2

2

1 0 k1 + 3 k1 - 3 2

+3

8 6 4 2 4 0 k1 - 8 8 k1 + 4 5 k1 + 1 0 k1 - 4

1 - ln



k1

2 1 - k1 = .

-400.1128 meV -411.4486 meV

,

EVP (2P - 2S ) =

1641.8862 meV 1665.7730 meV


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Two-loop VP correction to the Lamb shift in 1 interaction
1 k2 3 2 9 2


( ) d
1

1 . 2 k 2 + 4me 2 1 ( 2 - 2 ) 3.7207 meV 3.7997 meV .
2 -2me r 2 -2me r - e

VVP -VP (r ) =

C

( ) d
1 1

( ) d

-

Z r

e

.

EVP -VP (2P - 2S ) = 4(Z )2
2 45 2 m1 1

VVP -MVP (r ) = -

()d (r) -

2 me 2 -2me r e r

.

EVP -MVP (2P - 2S ) =

0.0022 meV 0.0023 meV
1 0

.
2me r 1-v 2

V2-loop VP = -

C

2 Z 3r



2

f ( v ) dv (1 - v 2 )

-

e

.

E2-loop VP (2P - 2S ) =

7.6863 meV 7.7696 meV

.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

~ G

~ G

a

b

c


VVP -VP -VP (r ) = -
-2me r

C

Z r

( 3 ) 3

3

( ) d
1 1

( d
1

( ) d â

âe

4 4 4 -2me r -2me r +e +e ( 2 - 2 )( 2 - 2 ) ( 2 - 2 )( 2 - 2 ) ( 2 - 2 )( 2 - 2 ) 4µ3 (Z ) 9 3
1

,

VVP -2-loop VP = -

C

( ) d
1

f ( )d



1 r ( 2 - 2 ) 0.0085 meV 0.0088 meV

e

2 -2me r 2 -2me r - e

,

EVP -VP -VP (2P - 2S ) =

,

EVP -2-loop VP (2P - 2S ) =

0.0359 meV 0.0366 meV

.

There exists also a contribution with three-loop vacuum polarization operator. It was calculated in T. Kinoshita and M. Nio, Phys. Rev. Lett. 62, 3240 (1999); Phys. Rev. D60, 053008 (1999). S.G. Karshenboim, V.G. Ivanov, E.Yu. Korzinin, and V.A. Shelyuto, Pisma v ZheTF 92, 9 (2010); PRA 81, 060501 ( 2010) .


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

One-loop vacuum polarization corrections to the Breit Hamiltonian
VVP (r ) =
B

3
1



4

( ) d
i =1

Vi ,VP (r ),

B

V1,VP =

B

Z 8
B

1
2 m1

+

I
2 m2

4 (r) -

2 4me 2 -2me r e r

,

V2,VP = -
B

2 Z me 2 -2me r e (1 - me r ), m1 m2 r

V3,VP = -
B

Z 2m1 m2

pi

e-2me r r

ij +

ri rj (1 + 2me r ) pj , r2

V4,VP =

Z r3
B

1 1 + 2 4m1 2m1 m2

e

-2me r (1 + 2me r )(L 1 ).

E1,VP (2P - 2S ) =

-0.8670 meV -0.8931 meV 0.0150 meV 0.0116 meV 0.0281 meV 0.0219 meV ,

,

E2,VP (2P - 2S ) =

B

E3,VP (2P - 2S ) =

B

,

E4,VP (2P - 2S ) =

B

-0.0860 meV -0.0876 meV

.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Relativistic and VP corrections in second order per turbation theory
V
B VP

~ G a

VVCP

V

B

~ G b

VVCP

V

C VP

~ G c

VVCP

V

B

~ G d

V

B

~ G e

V

B

~ G f

VP C~ C B~ C ESOPT =< |VVP GVVP | > +2 < |V GVVP | > .

~ G( 2S ) = -
2

Z µ2 - x1 +x2 1 2 e g2S (x1 , x2 ), 4 x1 x2 4
2 3 2 2 2 2

g2S (x1 , x2 ) = 8x< - 4x< + 8x> + 12x< x> - 26x< x> + 2x< x> - 4x> - 26x< x> + 23x< x> - -x< x> + 2 x< x> - x< x> + 4 e ( 1 - x< ) ( x> - 2 ) x> + 4 ( x< - 2 ) x< ( x> - 2 ) x> â â[-2C + Ei (x< ) - ln(x< ) - ln(x> )], ~ G ( 2P ) = -
3 3 3 2 3 2 3 x

Z µ2
22 3 6 x1 x2 3 2

e

x +x -1 2 2

3 ( x1 x2 ) 4 x1 x2

g2P (x1 , x2 ),
2 3 3 3 4 3

g2P (x1 , x2 ) = 24x< + 36x< x> + 36x< x> + 24x> + 36x< x> + 36x< x> + 49x< x> - 3x< x> - -12e< (2 + x< + x< )x> - 3x< x> + 12x< x> [-2C + Ei (x< ) - ln(x< ) - ln(x> )],
x 2 3 3 4 3 3

3

3


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

C~ C First term < |VVP GVVP | >
ESOPT (2S ) = -
VP ,VP

µ2 (Z )2 72 2 x 2
1





( ) d
1

( ) d â

â
0

1-

x 2

e

-x 1-

2me W



dx
0

1-

e

2me -x 1- W

dx g2S (x , x ) =






-1.8640 meV -1.9017 meV

,

µ2 (Z )2 VP ,VP ESOPT (2P ) = - 7776 2




( ) d
1 1

( ) d â -0.1867 meV -0.1942 meV

â
0

e

-x 1+

2me W



dx
0

e

2me -x 1+ W

dx g2P (x , x ) =



,

C ~ Second term < |V B GVVP | >
< | p4 (2µ)2
| m >< m | m

E2 - Em Z r
2

VVP | >=< |(E2 +

C

Z r
C

^ ) ( H0 +

Z r

)

| m >< m | m

E2 - Em

VVP | >=

C

=< |

E2 +

C ~ GVVP | > - < |

Z r

VVP | > + < | 1.4192 meV 1.4682 meV

Z r

| >< |VVP | > .

C

ESOPT (2P - 2S ) =

B ,VP

.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Three-loop vacuum polarization correction in second order per turbation theory

~ G

~ G

~ G

a

b


c


ESOPT
0

VP -VP ,VP

( 2S ) = -

µ3 (Z )2 108 3 1 2 - 2
1

( ) d
1 2 -x (1+

( ) d
1

( ) d
0

dx ( 1 -

x 2

)â

dx ( 1-



x 2

)e

2me -x (1+ ) W

e

2me 2me ) 2 -x (1+ W ) W - e

g2S (x , x ) =



-0.0104 meV -0.0107 meV

,

ESOPT
-x (1+

2-loop VP ,VP

( 2S ) = -

µ3 (Z )2 18 3 x 2
0

1 f ( v ) dv



1 - v2

( ) d â
1



â
0

dx

1-

x 2

e

2me ) 1-v 2 W 0



dx



1-

e

2me -x (1+ ) W g2S (x , x ) =

-0.0168 meV -0.0171 meV

,


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Nuclear structure correction in 1 and 2 interaction

~ G

a

b

c

a

b

Estr (2P - 2S ) = -
Estr (nS ) = -
2

µ3 (Z )4 2 < rN > = 12
dk 0 2

-396.2669 meV . -295.8478 meV
10.28 ± 0.10 meV 6.61 ± 0.07 meV .

µ3 (Z )5 n3

l 0

k

V (k ), EG,str (2P - 2S ) =

V (k ) =

2( F 2 - 1) m1 m2
2 2

+

2 8m1 [-F (0) + 4m2 F (0)] k2 â + 33 m2 (m1 + m2 )k 2m1 m2 2 2 2 2 k 2 + 4m1 3 2 2 2m1 m2 (m1 - m2 )k

â 2(F - 1)(m1 + m2 ) - F m1 +

â

âk

2

2(F - 1)m2 - F m1 + 8m1 F +

2

2

2

2

42

42 16m1 m2 (F 2 - 1)

k2
22

-

-

2 k 2 + 4m2 m1 3 2 2 2m2 (m1 - m2 )k

k

2

2( F - 1) - F

2

2

+ 8m2 F +

4 16m2 (F 2 - 1)

k2

.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Nuclear structure and one-loop VP correction in second order PT
Vstr (r ) =
VP

2 3

Z < rN >

2

3
1



()d (r) -

2 me 2 -2me r e r

.

Estr (2S ) =

VP

2 (Z )4 < rN > µ3



36

( ) d 1 -
1

2 4me 2



W2 ,

xd x ( 1 -
0

x 2 -x (1+ 2me ) W )e 2

=

=
2 2 (Z )4 µ3 < rN > me 108 W2 VP

1.2493 meV 0.9365 meV ( ) d
2

Estr (2P ) = -

VP

1

0

xe

3 -x (1+

2me ) W dx =

-0.0300 meV -0.0225 meV

,

Estr (2P - 2S ) =

-1.2793 ± 0.0130 meV -0.9590 ± 0.0092 meV
2 (Z )4 µ3 < rN >

.

Estr ,SOPT (2P - 2S ) = -
-x (1+

VP

36

( ) d â
1

â
0

dxe

2me x ) 3 2 W (1 - ) 4x (x - 2)(ln x + C ) + x - 13x + 6x + 4

2

=

-2.0083 meV -1.5063 meV

.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Nuclear structure and two-loop VP correction

a

b

c

~ G

~ G

~ G

~ G

a
V P -V P

b

c
2

d


Vstr

(r ) =

2 3

Z < rN >
2 me r ( 2 - 2 )

3 e

2 1



( ) d
1

( ) d â

â (r) -

4 -2me r 4 -2me r - e

,
2me r

Vstr

2-loop VP

(r ) =

4 9

Z < rN >

2



2

- 2 1 f ( v ) dv me e (r) - r (1 - v 2 ) 0 1 - v2



1-v 2



.

Estr

VP ,VP

( 2P - 2S ) =

-0.0102 meV -0.0076 meV

.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.
VP ,VP (1) 2 2 (Z )4 µ3 rN 108 2

Estr ,SOPT (2S ) =


( ) d
1 1 3

( ) d â

1-
0

x 2

dxe

-x 1+

2me W

4x (x - 2)(ln x + C ) + x - 13x + 6x + 4 ,
1

2

Estr ,SOPT (2S ) = -


VP ,VP (2)

22 2 (Z )4 µ3 rN me 2W 2 54

( ) d
1

2

( ) d â

1-
0

x 2

dxe

-x 1+

2me W 0

1-

x 2

dx e

2me -x 1+ W

g2S (x , x ).



Estr ,SOPT (2P - 2S ) =

VP ,VP

-0.0086 meV -0.0065 meV

.

a

b

Estr ,VP (nS ) = -
2

2

2µ3 (Z )5 2 n3
0



kV ( k ) d k

v 2 (1 - v3 )dv , 2 k 2 (1 - v 2 ) + 4me 0
1

2

Estr ,VP (2P - 2S ) =

0.2214 ± 0.0022 meV 0.1270 ± 0.0013 meV

.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Numerical results, comparison with other calculations E. Borie, Ann. Phys. (NY) 72, 052511 (2012). E.Yu. Korzinin, V.G. Ivanov and S.G. Karshenboim, PRD 88, 125019 (2013); S.G. Karshenboim, V.G. Ivanov, E.Yu. Korzinin, and V.A. Shelyuto, PRA 81, 060501 (2010). U.D. Jentschura, Ann.Phys. 326, 500 (2011); U.D. Jentschura, PRA 84, 012505 (2011); U.D. Jentschura, EPJD 61, 7 (2011).


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Our one-loop VP result coincides with the calculation KKIS. KKIS, meV


First order VP: 1665.7729 VP contribution of order (Z )2 in 1 interaction: 1665.7730

Our result, meV



The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Total two-loop contribution from KKIS is equal to


13.2769 meV, (µ4 He)+ 2 13.2789 meV, (µ4 He)+ 2

This agrees with our results


with the accuracy 0.002 meV (a number of two-loop corrections to the Breit Hamiltonian were estimated approximately).


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Our three-loop VP result is also in agreement with the calculation KKIS. KKIS, meV
4 0.074 (µ2 He)+ 4 0.0703 (µ2 He)+

Our result, meV



The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Relativistic corrections with vacuum polarization effects (FOPT, SOPT) in our work coincide with the results of Jentschura. Jentschura, meV


E

vp

= 0.521

Our results, meV


Relativistic-VP correction of order (Z )4 in FOPT: -0.9472 Relativistic-VP correction of order (Z )4 in SOPT: 1.4682



Total: 0.521


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

There exists the only calculation of E. Borie where total results for the Lamb shift in muonic helium ions were obtained. In the 4 case of (µ2 He)+ : Borie, meV


Uehling: 1666.305 VP contribution of order (Z )2 in 1 interaction: 1665.7730 Relativistic-VP contribution of order (Z )4 in FOPT: -0.9472 Relativistic-VP contribution of order (Z )4 in SOPT: 1.4682

Our results, meV






Total: 1666.2940


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Borie, meV


Kallen-Sabr y: 11.573 2-loop VP contribution of order 2 (Z )2 in 1 interaction: 11.5693 Relativistic-2loop VP contribution of order 2 (Z )4 in FOPT: -0.0037 Relativistic-2loop VP contribution of order 2 (Z )4 in SOPT: 0.0058

Our results, meV






Total: 11.5714 The small difference may be related with recoil terms accounted in our calculation.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

We can easily compare our results for nuclear structure corrections with Borie's results. We used the same value for charge radius of -par ticle rHe = 1.676 fm
I. Sick Phys. Lett. B 116, 212 (1982) We also use the same Gaussian parametrization for the formfactors.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Our results, meV


Nuclear structure of order (Z )5 in 2 interaction: 6.605± 0.07 Nuclear structure-VP of order (Z )4 (FOPT): -0.960± 0.0092 Nuclear structure and VP correction of order (Z )4 (SOPT): -1.5063± 0.0092

Nuclear structure of order (Z )4 : -295.848± 2.83







Nuclear structure-2-loop VP correction of order 2 (Z )4 in 1 interaction: -0.0076 Nuclear structure and 2-loop VP correction of order 2 (Z )4 (SOPT): -0.0182 Nuclear structure-VP contribution in 2 interaction: 0.1279± 0.0013 Borie, meV:-292.045





Total: -291.844,


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Comparison between total results for (µ4 He)+ of 2 Borie: E = 1379.2479 meV Our result: E = 1379.1107 meV Discrepancy is equal 0.1 meV.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Thank you for your attention.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

A p p e n d ix


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

The Wichmann-Kroll correction to the Lamb shift

V

WK

(r ) =

(Z )3 r
0

d

4

e

-2me r

-

2 12

2 - 1 ( - 1) +
0



dx

WK 2 - x 2 f (x ) .

E

WK

( 2P - 2S ) =

-0.0197 meV -0.0199 meV

.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Three-loop vacuum polarization correction in third order PT
C~ C~ C C C~~ C E =< 2 |V GV GV |2 > - < 2 |V |2 >< 2 |V GGV |2 > .

ETOPT ,1 (2S ) = -


µZ 2 5 864 3
1









( ) d
1

( ) d
1 dx 0

( ) d
0

1-

x 2

e

-x (1+2me /W ) dx â

1-
0

x 2

e

-x (1+2me /W ) dx

x

e

-x (1+2me /W ) g (x , x )g (x , x ) =

-0.0044 meV -0.0045 meV

,

ETOPT ,2 (2S ) =


2 288 2







( ) d
1 0 1

( ) d
0

1-

x 2

e

-x (1+2me /W ) dx â

1-
0

x 2

e

-x (1+2me /W ) dx

dx g ( x , x ) g ( x , x )

2041.9990 meV 2077.2217 meV

=

0.0037 meV 0.0038 meV

.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

C C C 2 2 Replacement VVP HB , VVP VVP ,VP , HB ,1 = ( Z /2)(1/m1 + I /m2 ) (r)

ESOPT

VP -VP ,HB ,1

( 2S ) =

µ3 (Z )4 2 144 2

1
2 m1

+

I
2 m2 1





( ) d
1

( ) d

1 â 2 - 2



1-
0

x 2

dx [4x (x - 2)(ln x + C ) + x - 13x + 6x + 4] e 0.0050 meV 0.0051 meV µ3 (Z )4 2 24 2

3

2

2 -x (1+

2me 2me ) 2 -x (1+ W ) W - e

=

=

,

ESOPT

2-loop VP ,HB ,1

( 2S ) =

1
2 m1

+

I
2 m2 0

1 f ( v ) dv

1 - v2

â



(1 -
0

x 2

-x (1+

2me W 1-v 2

)dx [4x (x - 2)(ln x + C ) + x - 13x + 6x + 4]e

3

2

=

0.0056 meV 0.0058 meV

.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.
C C C 3 3 Replacement VVP HB , VVP VVP ,VP , HB ,2 = -p4 (1/m1 + 1/m2 ) VP -VP ,HB ,2

ESOPT ,1


( 2S ) = - 1 x 1 8
2

µ4 (Z )4 2 72 2


1
3 m1

+

1
3 m2 1





( ) d
1

( ) d

1 â 2 - 2 =

1-
0

x 2

xd x

-

(1 -
0

x 2

) dx g ( x , x ) e -0.0029 meV -0.0031 meV 1
3 m1



2me 2me 2 -x (1+ W ) 2 -x (1+ W ) - e

=
2-loop VP ,HB ,2

, 1
3 m2 -x (1+ 1 2me W 1-v 2 ) 1 f ( v ) dv 0

ESOPT ,2 x 2

( 2S ) = -

µ4 (Z )4 2 12 2 x 2

+

( ) d

1 - v2

â



(1 -
0

) xd x

1 x

-

1 8

2 0



(1 -

) dx g ( x , x ) e 1
3 m1





=

-0.0045 meV -0.0047 meV 1 â 2 - 2 ,

,

ESOPT ,3

VP -VP ,HB ,2

( 2S ) = - x 2
2

µ4 (Z )4 2 18 2
2 -x (1+

+

1
3 m2 1





( ) d
1

( ) d



1-
0

dx e

2me 2me ) 2 -x (1+ W ) W - e

=

-0.0072 meV -0.0075 meV
f ( v ) dv 1

ESOPT ,4

2-loop VP ,HB ,2

( 2S ) = -

µ4 (Z )4 2 3 2
2me W 1-v 2 )

1
3 m1

+

1
3 m2

1 - v2

â



1-
0 VP ,VP ;V B

x 2

2

-x (1+

dxe

=
B

-0.0083 meV -0.0086 meV

. -0.0066 meV -0.0069 meV

ESOPT

( 2P - 2S ) =

0.0120 meV 0.0127 meV

VP ,V , ESOPT VP (2P - 2S ) =

.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Recoil correction of order (Z )4 Erec (2P - 2S ) =
µ3 (Z )4 2 48m2 µ3 (Z )4 2 12m2

, I = 1 , I = 0

=

0.1265 meV . 0.2952 meV

Recoil correction of order (Z )5
Erec
(Z )5

=

µ3 (Z )5 m1 m2 n3

2 3

l 0 ln

1 Z

-

8 3

ln k0 (n, l ) -

1 9

l 0 -

7 3

an -

2
2 2 m2 - m1

l 0 (m2 ln

2

m1 µ

- m1 ln

2

m2 µ

),

ln k0 (2S ) = 2.811769893120563, ln k0 (2P ) = -0.030016708630213, an = -2 ln 2 n + (1 + 1 2 + ... + 1 n +1- 1 2n l 0 + (1 - l 0 ) l ( l + 1) ( 2l + 1) . .

Erec

(Z )5

( 2P - 2S ) =

-0.5581 meV -0.4330 meV

Recoil correction of order (Z )6 Erec
(Z )6

(2P - 2S ) =

2 (Z )6 m1 8m2

23 - 4 ln 2 6

=

0.0051 meV . 0.0038 meV


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Muon vacuum polarization, muon self-energy correction
EMVP ,MSE (2S ) = (Z )4 µ3 8
2 m1

4 3

ln

m1 4 38 - ln k0 (2S ) + + µ(Z )2 3 45 427 384 ln 2 2 m1 6µ 10.6633 meV 10.9392 meV

+



9 32 10 2 2179 - (3) + ln 2 - - 4 2 27 648 EMVP ,MSE (2P ) = m1 3 µ 3 4 2 2

+ 4 Z 4 3

-

=

,

(Z )4 µ3 8 ln 2 +
2 m1

-

ln k0 (2P ) -

-

-

( 3) -

2 12

+

197 144

=

-0.1653 meV -0.1678 meV

.

Radiative-recoil corrections of orders (Z )5 , (Z 2 )(Z )4
Erad -rec (2S ) = -1.324
2 (Z )5 m1 + 8m2

2 2 9 2
2

-

70 27 1 3

2 (Z )5 m1 + 8 2 m2

+



1 3

ln

(Z )-2 µ

+

11 72

-

1 24

-

7 2
2 32 4m2

+

2 3

2 4m2

-

Erad -rec (2P ) = -

1 3

ln k0 (2P )

4(Z 2 (Z )4 µ3
2 8 m2

ln k0 (2S ) .



4(Z 2 )(Z )4 µ3
2 8 m2

,

Erad -rec (2P - 2S ) =

-0.0656 meV -0.0377 meV

.


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

Nuclear structure corrections of orders (Z )6 , (Z )5
Estr
rel (Z )6

( 2P - 2S ) =

(Z )6 12 1 40 µ
2

µ

3

rN

2

ln µZ r + C -

3 2

-

12 13 rN + r 2 3

1 r

-

-I2 - I3 - µ FNR +

rel

2

r

4

=

2 -0.005064 · rh + 0.11445 = -0.3882 meV 2 -0.00533 · r + 0.07846 = -0.3063 meV

,

Estr

(Z )5

( 2P - 2S ) =

0.0940 meV 0.0702 meV

.

a

b

Erad +VP (nS ) =

µ (Z )
2 m1

3

4

n3
2

Cjl 2 4m1 F1 (0)l 0 + F2 (0) 2l + 1
2 2

, Cjl = l 0 + (1 - l 0 ) m1 me 1 9 395 1296 +3

3 j ( j + 1) - l ( l + 1) - 4 . l ( l + 1)

m1 F1 (0) = F2 (0) =

1 9

ln

2 m1

me - 25 36

-

29 108

ln

+

( 2) + m1 me .

,

1 3

ln

m1 me

+

2 me 4 m1

-4

2 me 2 m1

ln

2 me 2 m1

.

Erad +VP (2P - 2S ) =

-0.0299 meV -0.0307 meV


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

EMSE =

VP


2 3 m1

ln

m1 µ(Z )2

C C~ < n | · VVP |n > +2 < n |VVP G

-

Z r

| n > .

EMSE (2P - 2S ) =

VP

-0.1008 meV -0.1074 meV

.

K. Pachucki, Phys. Rev. A 54, 1994 (1996) U.D. Jentschura and B.J. Wundt, Eur. Phys. Jour. D 65, 357 (2011).


The XXII International Workshop High Energy Physics and Quantum Field Theory, June 24- July 1 2015, Samara, A.A. Krutov et. al.

HVP and nuclear polarizability contributions
E
HV P

=

0.2170 meV 0.2229 meV

.

E. Borie, Z. Phys. A 302, 187 (1981) J.L. Friar, J. Mar torell and D.W.L. Sprung, PRA 59, 4061 (1999). R.N. Faustov and A.P. Mar tynenko, EPJC 6, 1 (1999) E
NP

=

4.9 ± 1.0 meV 2.47 ± 0.15 meV

.

J. Bernabeu and C. Jarlskog, Nucl. Phys. B 75, 59 (1974) C. Ji, N.N. Dinur, S. Bacca and N. Barnea PRL 111, 143402 (2013).