Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qfthep.sinp.msu.ru/proceedings2015/QED_Faustov.pdf Äàòà èçìåíåíèÿ: Sun Nov 29 22:23:56 2015 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:20:03 2016 Êîäèðîâêà:

Relativistic and Vacuum Polarization Corrections to the Lamb Shift in (µ3 H e)+ , (µ4 H e)+ 2 2
A.A. Krutov1 R.N. Faustov, A.P. Martynenko, and G.A. Martynenko Samara State University, 1, Pavlov Str., 443011, Samara, Russia Dorodnicyn Computing Center, RAS, 40, Vavilov Str., 119333 Moscow, Russia Samara State Aerospace University, 34, Moskovskoye Shosse, 443086, Samara, Russia

The Lamb shift (2 P1 tions of orders ,
3

/2 4 ,

- 2S
5

1/2

) in muonic helium ions (µ3 H e)+ , (µ4 H e)+ is calculated taking into account correc2 2

and 6 . Special attention is given to relativistic corrections with the account of vacuum

polarization effects. The obtained shifts 1259.8583 meV ((µ3 H e)+ ) and 1379.1107 meV ((µ4 H e)+ ) can be useful for 2 2 a comparison with experimental data of the CREMA collaboration.

Muonic helium ions are bound states of negative muon and the nucleus (helion or -particle). They are among that simple muonic atoms which attract considerable attention in last years due to the CREMA measurements of fine and hyperfine structure in muonic hydrogen, muonic deuterium and muonic helium ions [1­3]. An analysis of all these experiments will allow to clarify the source of the discrepancies in the description of the electron and muon atoms. For successful realization of the CREMA program and finding more accurate values of charge radii of the proton, deuteron, helion and -particle it is important to perform a precise calculation of the transition frequencies between the levels 2S and 2P [4­6] The aim of this work is to present the Lamb shift (2 P - 2S) calculation in muonic helium ions with the precision 0.001 meV. For a solution of this task we calculate different corrections of orders 4 Â 6 which are determined by relativistic, vacuum polarization effects, nuclear structure corrections in first, second and third orders of perturbation theory. Our approach to the calculation of the Lamb shift corrections in muonic helium ions is based on the quasipotential method in quantum electrodynamics [7­9]. The two-particle bound state is described by the SchrÆdinger equation. The leading order contribution to the particle interaction operator is determined by the Breit Hamiltonian (I = 1(I = 0) for nuclei with a half-integer (integer) spin) [10]: ( ) p2 Z p4 p4 Z 1 I HB = - - - + + 2 (r)- (1) 2µ r 2 m2 m2 8m3 8m3 2 1 1 Z - 2m1 m2 r ( r(rp)p p+ r2
2

)

Z +3 r

(

1 1 + 2m1 m 4m2 1

)
2

(L1 ) = H0 + V B ,

where H0 = p2 /2µ - Z /r; m1 , m2 are the masses of the muon and nucleus (helion or -particle) and µ = m1 m2 /(m1 + m2 ). The wave functions of 2S and 2 P states are the following: 200
1

W 3/2 (r ) = e 2 2

-

Wr 2

(

Wr 1- 2

) , 2l m (r ) =

W 3/2 e 26

-

Wr 2

W rYl m ( , ), W = µ Z .

(2)

aakrutov@rambler.ru

1


XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

The effects of vacuum polarization (VP) lead to different corrections to the Breit Hamiltonian 1. The oneloop VP corrections to the Breit interaction are a subclass of interactions that can be called relativistic corrections taking into account the vacuum polarization. They are determined by expressions obtained in [6, 11]:
B VV P (r ) =

3


1

( )d

i =1



4

ViB P (r ), ,V 4m2 2 e e r ] ,

(3)



B V1,V P

Z = 8

(

1 + I2 2 m1 m2

)[ 4 ( r ) -

-2m e r

(4) (5) (6) (7)

B V3,V

B V4,V

P

Z m 2 2 -2m e r e e (1 - m e r ), m1 m2 r ] [ ri r j Z e -2m e r p i j + 2 (1 + 2me r ) p j , P=- 2m1 m2 i r r ( ) 1 Z 1 + =3 e-2me r (1 + 2me r )(L1 ). 2 2m1 m2 r 4m1
B V2,V P = -

The matrix elements (3) over wave functions (2) give corrections in first order perturbation theory: { -0.8670 meV B E1,V P (2 P - 2S) = , -0.8931 meV {
B E2,V P

(8)

(2 P - 2S ) =
{

0.0150 meV , 0.0116 meV 0.0281 meV , 0.0219 meV

(9)



B E3,V P

(2 P - 2S ) =
{

(10)



B E4,V P

(2 P - 2S ) =

-0.0860 meV . -0.0876 meV

(11)

B B B Terms V2,V P , V3,V P , V4,V P account for recoil effects in the ratio m1 /m2 . Two-loop vacuum polarization 2 ( Z )4 appear as a result of two-loop modification of the Breit Hamiltonian. corrections of order

In second order perturbation theory we have a number of vacuum polarization corrections of orders 2 ( Z )2 and ( Z )4 : VP C~ C C ~ ESO PT =< |VV P G VV P | > +2 < |V B G VV P | >, (12) where the reduced Coulomb Green's function (RCGF) ~ Gn (r, r ) =


l ,m

~ gnl (r, r )Yl m (n)Yl (n ). m

(13)

~ The radial function gnl (r, r ) can be presented in the form of Sturm expansion in Laguerre polynomials. In the calculation of separate matrix elements it is convenient to use reduced Coulomb Green's functions in the form obtained in [12]: Z µ 2 - x1 + x2 1 ~ 2 e g ( x , x2 ), (14) G (2S ) = - 4 x1 x2 4 2S 1 2


XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

2 2 22 g2S ( x1 , x2 ) = 8x< - 4x2 + 8x> + 12x< x> - 26x< x> + 2x3 x> - 4x> - 26x< x2 + 23x< x> - < < >

(15)

-x

32 < x>

+ 2x

3 < x>

-x

23 < x>

+ 4e (1 - x < ) ( x > - 2) x > + 4( x < - 2) x < ( x > - 2) x > â
x

â[-2C + Ei ( x< ) - ln( x< ) - ln( x> )],
Z µ2 ~ G (2 P ) = - e 22 36x1 x2
-
x1 + x2 2

3 ( x1 x2 ) g2 P ( x 1 , x 2 ) , 4 x1 x2

(16) (17)

3 32 23 43 g2P ( x1 , x2 ) = 24x< + 36x3 x> + 36x< x> + 24x3 + 36x< x3 + 36x< x> + 49x3 x3 - 3x< x> - < > > <> x 2 3 33 -12e< (2 + x< + x< ) x> - 3x3 x4 + 12x< x> [-2C + Ei ( x< ) - ln( x< ) - ln( x> )], <>

where x< = mi n( x1 , x2 ), x> = m a x ( x1 , x2 ), C = 0.57721566... is the Euler constant. As a result two-loop VP contribution of first term in (12) can be written in integral form. Subsequent integration over particle coordinates and spectral parameter gives the following result:
(
0 V P ,V ESO PTP )

µ 2 ( Z ) (2S ) = - 72 2
(

2 1

( )d
)


1

( )d â {

(18)

â

x) 1- e 2

( - x 1-

2m e W

dx
0

x 1- 2

) e

( - x 1-

2m e W

d x g2 S ( x , x ) =

1





-1.8640 meV , -1.9017 meV
(19)




V P ,V ESO PTP )

µ 2 ( Z ) (2 P ) = - 7776 2


2 1

( )d

( )d â {

â

e
0

( - x 1+

2m e W

dx
0

e

( - x 1+

2m e W

)

d x g2 P ( x , x ) =





-0.1867 meV , -0.1942 meV

Second term in (12) has a similar structure. A transformation of different matrix elements in it can be carried out by means of following algebraic relations:

< |

p4 (2µ )

2



m

|m >< m | Z ^ Z C VV P | >=< |( E2 + )( H0 + ) E2 - Em r r



m

|m >< m | C VV P | >= E2 - Em

(20)

( ) Z 2 ~ Z Z C C C =< | E2 + G VV P | > - < | VV P | > + < | | >< |VV P | > . r r r Three terms in right part of (20) can be calculated using expressions for wave functions and perturbation potentials. So, for example, in the case of 4 H e their numerical values are equal to (-2.0398) meV, 0.1017 2 meV, (-1.0576) meV, (-0.0600) meV, 0.0201 meV, (-0.0653) meV for 2S and 2P-states. Integral expressions of corrections from three other terms of the Breit potential (1) are:
(
0 V P,(1) ESO PT

( Z )4 µ (2S ) = 48

3

(

1 + I2 2 m1 m2
3

)
1

( )d â {

(21)

x) e 1- 2

- x ( 1 + 2 m e /W )

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

-1.3611 meV , -1.4028 meV

3


XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia



V P,(2) ESO PT

( Z )4 µ (2 P ) = - 648

0

3

(

1 1 + 2 2m1 m 4m1

)
2 1

( )d


0

dx e x2

-x

â

(22)

{ d x e
- x ( 1 + 2 m e /W )

g2 P ( x , x ) =

2 1

-0.0935 meV , -0.0958 meV
dxe
- x ( 1 + 2 m e /W )



0

V P,(3) ESO PT

( Z )4 µ3 (2 P ) = - 2592 m1 m
- x /2

( )d )
2


0

â

(23)

dx g2 P ( x , x ) e x
V P,(3) PT

(

d2 2 1 -- x 4 dx

1

{

xe


- x /2

=

-0.0084 meV , -0.0066 meV
( 1- x) â 2 (24)

ESO

0

(2S ) = -
- x /2

( Z )4 µ3 24 m1 m2

( )d )
2

dxe
0

- x ( 1 + 2 m e /W )

d x g2 S ( x , x ) e





(

2 1 d2 -- x 4 dx

e

- x /2

(

x 1- 2

)

{

=

-0.1064 meV . -0.0829 meV

Omitting further details of the calculation in (20) we present summary numerical contribution to the shift (2 P - 2S) from second term in (12) obtained from all terms of the Breit potential (1): { 1.4192 meV B ,V P ESO PT (2 P - 2S) = . (25) 1.4682 meV Three-loop VP contributions in second order PT can be calculated in a similar way. Corresponding potentials are written explicitly in [13]. Accounting an accuracy of the calculation we present them here only for 2S-state: µ 3 ( Z )2 x V P -V ESO PT P,V P (2S) = - ( ) d ( ) d ( ) d d x (1 - ) â (26) 2 108 3
1 1 1 0


0

x d x (1 - ) e 2


- x (1+

2m e W

)

[ 1 2e 2 - 2

- x (1+

2m e W

)

- e

2 - x (1+

2m e W

)

]

{


g2 S ( x , x ) =

1

-0.0104 meV , -0.0107 meV
(27)



0

2 - l o o p V P ,V P ESO PT

µ 3 ( Z ) (2S ) = - 18 3


1 2 0

f (v)dv 1 - v2
2m e W

( )d â {


â

x) dx 1 - e 2

(

- x (1+

2m e 1- v2 W

)

dx
0



(

x 1- 2

) e

- x (1+

)

g2 S ( x , x ) =

-0.0168 meV , -0.0171 meV

In the evaluation of these and other corrections in SOPT we have two characteristic integrals:


I1 =
0

( x) g2 S ( x , y ) 1 - e 2

- a1 x

dx =

1[ 3 2a1 (y(4(y - 2) + (y - 13)y + 6) + 4)- a5 1

(28)

4


XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

4a2 (y(4(y - 2) + (y - 15)y + 10) + 4) - 4(2( a1 - 2) a1 + 3) a1 (y - 2)y(Ei(-y( a1 - 1)) - ln 1 a1 (y(12(y - 2) + y(3y - 53) + 46) + 12) + 12(y - 2)y + 4a1 e

- y ( a1 -1)

( a1 - 1) y )+ a1 ] ( a1 ( a1 ( y - 2) - y + 4) + 3( y - 1) ) ,
(29)

I2 =
0

g2 P ( x , y ) e

- a1 x

xdx =

[ ( 1 -72 a2 (y(y(y(y + 4 - 9) - 12) - 12) - 8)+ 1 6 ( a1 - 1) a1

4( a1 - 1) a1 y3 (-Ei(y - a1 y) + ln 72a1 e
- y (1- a1 )

) ( a1 - 1) y ) + a1 (y(12 - y(y(y + 4 - 13) - 12)) + 8) - 5y3 - a1 ( ( ) )] a4 y3 + 4 a3 - 1 y2 + 4( a1 - 1) (2 a1 + 1) y + 8( a1 - 1) . 1 1

The contribution of three-loop VP in third order PT is determined by the sum of two terms: ~ ~ ~~ E =< 2 |V C G V C G V C |2 > - < 2 |V C |2 >< 2 |V C G G V C |2 > . Let us write explicitly the matrix elements for 2S state: E
(
0 TO P T ,1

(30)

µ Z 2 5 (2S ) = - 864 3 ) e



1

( )d

0


1

( )d


1

( )d

(
0

1-

x) e 2

- x ( 1 + 2 m e /W )

dxâ

(31)

x 1- 2



- x ( 1 + 2 m e /W )

dx



d x e x

{
- x ( 1 + 2 m e /W )


g ( x, x ) g ( x , x ) = x) e 2

-0.0044 meV , -0.0045 meV
dxâ 0.0037 meV . 0.0038 meV (32)

E
(
0

TO P T ,2

2 (2S ) = 288


2 1

( )d


1

( )d

(
0

1-

- x ( 1 + 2 m e /W )

x 1- 2



) e

- x ( 1 + 2 m e /W )



dx




0

2041.9990 meV d x g ( x, x ) g ( x , x ) = 2077.2217 meV

{

{

In the same way we calculate other two and three-loop VP corrections accounting for recoil effects and nuclear structure corrections. Numerical values of the Lamb shift in muonic helium-3 ion and muonic helium-4 ion are equal to 1259.8583 meV and 1379.1107 meV respectively [13]. These values can be considered as reliable estimates when compared to experimental data of the CREMA collaboration. The reliability of such comparison is provided by the following features of our calculation: 1. Since numerical value of parameter me /µ Z = 0.34 for muonic helium ions, the VP effects give the main contribution to the Lamb shift. We consider one-, two-, three-loop VP corrections in first, second and third orders of perturbation theory. 2. We calculate complicated corrections connected with VP and relativistic effects. 3. The nuclear structure effects are determined in terms of the charge radius of the nucleus and with the help of the electromagnetic form factors. 4. We take into account the results of calculations of a large number of authors [5, 14­16]. Acknowledgments The work is supported by the Russian Foundation for Basic Research (grant 14-0200173), the Ministry of Education and Science of Russia under Competitiveness Enhancement Program 2013-2020. 5


XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

References
[1] R. Pohl, A. Antognini, F. Nez et al., Nature 466, 213 (2010); A. Antognini, F. Biraben, J.M.R. Cardoso et al. Can. J. Phys. 89, 47 (2011); A. Antognini et al., Science 339, 417 (2013). [2] T. Nebel, F. D. Amaro, A. Antognini et al. Hyperfine Int. 212, 195 (2012); A. Antognini, F. Kottmann, F. Biraben, et al., Ann. Phys. 331, 127 (2013). [3] J.J. Krauth, M. Diepold, B. Franke et al., arXiv:1506.01298[physics.atom-ph]. [4] E. Borie, Z. Phys. A 275, 347 (1975); E. Borie and G.A. Rinker, Phys. Rev. A 18, 324 (1978); Rev. Mod. Phys. 54, 67 (1982). [5] E. Borie, Phys. Rev. A 72, 052511 (2005); Phys. Rev. A 71, 032508 (2005); Ann. Phys. (NY) 327, 733 (2012). [6] K. Pachucki, Phys. Rev. A 54, 1994 (1996); Phys. Rev. A 60, 3593 (1999); Phys. Rev. Lett. 106, 193007 (2011); A. Veitia and K. Pachucki, Phys. Rev. A 69, 042501 (2004). [7] A.P. Martynenko and R.N. Faustov, J. Exp. Theor. Phys. 88, 672 (1999); ibid. 98, 39 (2004). [8] A.P. Martynenko, J. Exp. Theor. Phys. 101, 1021 (2005); ibid. 106, 690 (2008). [9] R.N. Faustov, A.P. Martynenko, G.A. Martynenko and V.V. Sorokin, Phys. Lett. B 733, 354 (2014); Phys. Rev. A 90, 012520 (2014); ibid. 92, 052512 (2015). [10] M.I. Eides, H. Grotch and V.A. Shelyuto, Phys. Rep. 342, 62 (2001); Theory of Light Hydrogenic Bound States, Springer Tracts in Modern Physics, V. 222 (Springer, Berlin, Heidelbeg, New York, 2007). [11] A.P. Martynenko, Phys. Rev. A 76, 012505 (2007). [12] H.F. Hameka, Jour. Chem. Phys. 47, 2728 (1967). [13] A.A. Krutov, A.P. Martynenko, G.A. Martynenko, and R.N. Faustov, J. Exp. Theor. Phys. 120, 73 (2015) [Zh. Eksp. Teor. Fiz. 147, 85 (2015)] [14] U.D. Jentschura, Ann. Phys. 326, 500 (2011); Phys. Rev. A 84, 012505 (2011); Eur. Phys. J. D 61, 7 (2011). [15] P. Indelicato, Phys. Rev. A 87, 022501 (2013). [16] E.Yu. Korzinin, V.G. Ivanov and S.G. Karshenboim, Phys. Rev. D 88, 125019 (2013); S.G. Karshenboim, V.G. Ivanov, E.Yu. Korzinin, and V.A. Shelyuto, Phys. Rev. A 81, 060501 (2010); S.G. Karshenboim, E.Yu. Korzinin, V.G. Ivanov, JETP Lett. 88(10), 641 (2008).

6