Electron self-energy in a magnetic field

I 8.B. 1~

Nuclear Physics B44 (1972) 288-300. North-tlolland Publishing Company

ELECTRON SELF-ENERGY IN A MAGNETIC FIELD D.H. C O N S T A N T I N E S C U * Scuola Normale Superiore, Pisa, Italy

Received 18 February 1972

Abstract: In the lowest order in tire fine-structure constant, the electron self-energy in an external magnetic field can be written in the form of a double integral representation containing the exact information about the radiative shift and width of the energy levels, without approximation in the field strength. In the low-field expansion of the radiative shift, the leading term is conveniently interpreted in terms of the electron's anomalous magnetic moment, whilst in very strong fields the enhancement of the cyclotron motion makes the shift a positive, slowly increasing function of the field intensity. It follows that, even in superstrong magnetic fields, the electromagnetic interaction cannot give rise to an instability of the electron-positron vacuum.

l. I N T R O D U C T I O N The presumed existence, in n e u t r o n stars, o f superstrong magnetic fields [1] o f the order o f the critical value H c = m2/e ~- 4.4 X 1013 G has stimulated some new interest in the study o f electromagnetic processes in a strong external magnetic field. The presence of such huge intensities could enhance a series o f nonlinear electromagnetic effects that are practically unobservable at the usual laboratory strengths, e.g. vacuum polarization [ 2 , 3 ] , p h o t o n splitting [ 2 , 4 ] , or the radiative shift of energy levels, with possibly i m p o r t a n t astrophysical consequences. It has been suggested [5,6] that the existence of the anomalous magnetic t n o m e n t of the electron might lead, in a very strong magnetic field, to a large negative radiative shift o f the energy levels, with explosive consequences, such as spontaneous pair creation or, at still higher intensities, even spontaneous decay of the electron into a muon. These conclusions have been rightly criticized [7], since they are based on the unjustified e x t r a p o l a t i o n to very high fields o f the low-field radiative correction. An exact calculation of the radiative shift o f the electron ground level in a magnetic field of arbitrary strength has been made by Demeur [8]. Using this result, it was

* On leave of absence from Department of Physics, University of Bucharest, Bucharest, Romania.

D.H. Constantinescu,Electronself energy

289

shown in ref. [7] that even in fields some orders of magnitude higher than H c the radiative correction is still a small fraction of the electron mass, and the conclusions of refs. [5, 6] are unfounded. The purpose of this paper is to extend the calculations of refs. [8, 7] to the excited electron levels. We think this is a good exercise for learning something about self-energy corrections in magnetic field, since it is the simplest possible example but presumably already contains the main general features of the phenomenon. The general expression of the self-energy correction, to lowest order in the finestructure constant, is derived in sect.2. In sect.3 we apply it to the case of an electron in magnetic field. By using the Furry picture, the dependence on the field is taken into account exactly, both in the electron wave functions [9] and propagators [10, 11 ] ; this produces the self-energy correction in the form of a double integral representation. The interpretation of this result is given in sect.4, where such problems as regularization, the low-field limit (including the handling of the spurious infrared divergences), and the leading asymptotic behavior in the field are discussed. In the case of the ground level, a numerical calculation has been performed, the results of which quantitatively confirm the conclusions of ref. [7].

2. RADIATIVE CORRECTIONS TO ELECTRON ENERGY Following Low [ 12], we derive * the expression of the radiative corrections to the electron energy levels E~. We start by expanding both the bare and the electromagnetically dressed propagators in terms of the corresponding eigenfunctions

~(x): e - i E (7.1 - 7.2 )

(1) S'(Xl,X2) =f 2~t. ~ e-iE(rl rz) fc~(E) ~a(Xl ) f~(x2). c~,~ We are interested in finding the poles of the diagonal coefficients faa(E), which give the corrected energy levels. A set of equations for fa~(E) is obtained by introducing the expansions (1)into the integral equation [13]

S'(xvx2)=S(xl,x2)+fdxl dx'zS(xvx])Z(xl,x'z)S'(x'>x2),

(2)

and then using the orthogonality of ~ ( x ) to project the coefficients. The result reads * Four-vectors will be denoted by x = (~, ,0, ~, iz); the metric is x 2 = ~2+ ,02+ ~.2 _ 7.2 The Pauli representation of the gamma matrices is used. The homogeneous external magnetic field is assumed to be oriented along the ~"axis.

290

D.H. Constantinescu, Electron self-energy

E

where

,

(3)

"r

Ha~ (E) = ifd~'e iEr f d x 1dx 2 f a (x I ) ~ (x l, x 2 ; ~) ~t3(x2).

(4)

In the lowest order in the fine-structure constant the self-energy operator is given by the fluctuation diagram in fig. la *, viz. ~(x 1, x2) = e 2 D(x l - x 2 ) 7 u S(x 1, x2) 7u,

(5)

where D(x 1- x2) stands for the photon propagator. In this approximation, the solution of eqs. (3) for the diagonal elements is J ~ ( E ) = [E~ + H~e~(E) - El -1;

(6)

this result holds even if the levels Ec~ are degenerate, since the nondiagonal elements of the self-energy operator contribute only to higher-order corrections. Eq. (6) shows that the pole of J ~ (E) has moved from E~ to a value given by the equation E~ + H ~ (/£) - E = 0. Putting E = E~ + AE~, one gets the radiative correction AE~ = H ~ ( E ) .

(7)

If the set c~ includes some continuous parameters, eqs.(3) contain integrations over these parameters. At the same time, the matrix element H ~ ( E ) factorizes into a reduced matrix element, times the right number of delta functions needed to perform these integrations, the overall result being that the radiative correction is given in this case by the reduced matrix element. We shall encounter this situation in the next section, where to the degenerate levels ENq correspond eigenfunctions of the form

~Napq (X) = CNapq (~) (270-1 e ip~ +iq~,

(a)

(8)

(b)

Fig. 1. Lowest-order electron self-energy graphs; the polarization diagram (b) is zero in a homogeneous magnetic field. * In a homogeneous magnetic field the contribution of the polarization diagram, fig. lb, vanishes; see ref. [8], p.53.

D.H. Constantinescu, Electron self-energy

291

the parameters p and q having a continuous variation between - co and +oo. Assuming that ]~(Xl' X2) = ]~(~1' ~J2' n ' ~','r),

(9)

where x = x 1 - x 2, one has from eq. (4)

HNopq,N,o,p, q , (E) = 8 (p - p ' ) 8 (q - q ' ) H No,N, a, (p,q ;E),

(10)

with

HNo, N,o,(p,q;E ) = i

yd~1 d~2 dr/d~'dr e -ipn-iqf+iEr

(11)

X ~Nopq (~:1) ~ (~1 '~2' r/, ~', "r)dPN'o'pq. Then, the standard manipulations described above yield

AENopq = HNo,N o (19,q ;ENq ).

(12)

3. ELECTRON SELF-ENERGY IN MAGNETIC FIELD Let us apply these considerations to the case of an electron in a homogeneous magnetic field. By using the Furry picture, the external field will be taken into account exactly in the expressions of the wave functions and propagators, and only the electron-photon interaction will be treated as a perturbation. We shall limit ourselves to the lowest order in this interaction, and begin by evaluating the self-energy operator, eq. (5). The photon propagator has the well-known expression D ( x ) = 1 _ ~ _ ? dXexp{liXx2}" 87rZi"0~"

(13)

A similar formula for the electron propagator has been given by G6h~niau and Demeur [10, 11] :

S(x 1, x2) = eief(x l, x ~) SSx l - x2), where

(14)

X2

f(XI,X2) =

f ax.A. XI

(the integral being taken along the straight line from x t to x2) , and

Os)

D.H. Constantinescu,Electron self-energy

292

° d.

2" tan (~-} eH ×exp

{ - - ~,e.( ~

(16)

.2 + V2-) + ½ i . ( ~ ' 2 - r 2 ) - 'm2/ 2 . J"

4 tan ~yfi ] Here B(x) = I(H × x,0), the magnetic field H being oriented along the f axis, and o3 = -i~'172. In both representations (13) and (16) the integration paths are understood to make a small positive angle with the real axis: X -+ X(1 +ie),. ~ .(1 +ie), to ensure a convergence factor. The function f(xl,x2)explicitly depends upon the choice of the potential A(x) used to describe the external magnetic field. However, in eq. (4) the propagator is sandwiched between electron wave functions, and the additional factors brought in by a gauge transformation are cancelled by the corresponding factors coming from the wave functions, which makes the matrix element gauge-invariant. We are free to choose the special gauge

A(x) = (0,H~,0,0),

(17)

f(Xl'X2) = -- 1H (~1 +~2) (f/1 --1"/2)

(18)

which gives

and justifies assumption (9). We come now to the electron energy levels and eigenfunctions [9]. The eigenvalues are

ENq = (2NeH+q2+m2)}, N = 0,1,2 .... ;q E (0%oo).

(19)

The corresponding eigenfunctions have the form (8), with

-(ENq+m)ON-I)P(~)~

~)N°pq(~)=aNq qUN-I'P~~) i(2NeH)~ UN,p(~)

--

I

:

°

+* N + O

<20)

I

0

(ENq+m) ZlN,p(~) #)Napq(~) = GNq [ -i(2NeH)~ UN_l,p(~)

L-q UN,p (~1

, o=

1.

(20')

293

D.H. Constantinescu, Electron self-energy

The normalization factor is given by 1

GNq = [2ENq (ENq+m)]- ~

(21)

and UN,p (~) are harmonic oscillator eigenfunctions: UN, p(~) = C N e - ~P H N ( P ) , p = ( e l l ) 2

+

,

(22)

1

CN=(2NN!)-2

,

pE(_oo,,~,).

The calculation of the self-energy correction is now straight-forward; we briefly sketch the main steps and then give the result. The space-time integrations in eq. (11) are done first. At this stage a partial result is that p disappears from our expression: the corresponding degeneracy is not lifted by the interaction with the photon vacuum. This is a trivial consequence of translational invariance [14] which could have been guessed from the very beginning. Next, we introduce the dimensionless quantities

L -eH

m2 ,

=g--, ~(NL, rD-ENq=(2NL +r/2+l) -~

7? m

-~-

(23)

'

and define NL] B( 1)(NL,77)= ---~1 ( I+NL +NL] A ( 1)(NL,r/)=~-2 ( 1 + ~+1]' ~+1]'

2NL , B(+I)(NL,~)= --~1 (N L ~(~+l)

A(+I)(NL'~)C(NL, n) =

~NL) 1 '

(24)

2NL

We also change the integration variables U=

X X +I2 '

eH 21~ '

D=--

(25)

and define the functions / ( a ; u , v ) = u tan v + (1 + io tan v) (1 - u)v,

(26)

D.H.Constantinescu,Electronself-energy

294

F(N,o,r~;Z;u,v) =[f(-!;u,v) Lf(+ 1;u,v) x IA(+°)v+B(+°)(l~+-l.u~o)itan

exp {2i(1

o) (1

-u)v}] N

-u)v (27)

+

A(-°)v+B ( -

°)(1 + i t a n v ) ( 1 - u ) v

f(-1 ;u, v)

C(1 + tan2o) (1 -

u)v2 t

+ /~-l;u,o,)5+l;~Tv)

j

With these notations, the radiative correction can be written as o~

AENoq =m ~ I(N,o,~;L)

(28)

(we dropped the redundant index p), where

I(N,o,r~;L)= f du o

v)exp

-~

.

(29)

o

The convergence factor needed at the upper limit in u is supplied by the small negative imaginary part of this variable: v ~ v ( 1 - i e ) .

4. DISCUSSION OF THE RESULT In low fields (L ,~ 1) the leading contribution to eq.(29) comes from the region of small u and v. Then, a straightforward approach consists in making an expansion in powers of these variables:

F(N,o,~;L;u,v) = ~

~ Fmn(N,o,r?;L)um(iv) n, (30) m=0 n=max(0,m 1) and then integrating it term by term. This should produce (apart from the L-dependence of the coefficients Fmn) an expansion ofI(N,o, r/;L) in powers of L; however, it comes out that a lot of terms in this expansion have divergent coefficients. This is not unexpected; indeed, if one expands the self-energy loop in fig. la in powers of the external field the individual terms will exhibit ultraviolet as well as infrared divergences. A more careful analysis intended to lead to the removal of these divergences shows that the terms appearing in the expansion (3)fall into three distinct classes, illustrated in the diagram in fig. 2. (a) Terms with n = 0; there are two such terms (m=0,1), leading to a divergent integral in o. They represent the zero-field contribution to I(N,o,~;L), i.e. we have

D.H. Constantinescu Electron self-energy

295

!

°t 3 2

1 0 0

1

2

3

4

m =

Fig. 2. Diagram of the expansion o f F ( N , a, ~;L;u, v) in powers of the variables u, u.

here the ultraviolet divergence of the free electron's self-energy. Of course, due to the L-dependence of the coefficients Fmo this divergent contribution spreads to all orders in L, which is clear enough in a diagrammatic picture. These ultraviolet divergences are eliminated by subtracting the zero-field contribution, which regularizes I(N, o,r/;L) to all orders in L. A simple calculation gives F00 = F10 = l/G; hence the physical radiative correction is obtained by replacing in eq. (29) the function F by 1

F'=F_~(I+u).

(31)

(b) Terms with m - n = 1,0 (n :~ 0). For each n there are two such terms, giving a finite contribution of order L n to I(N, o, ~;L). (c) Terms with m - n < O. For each rn there is an infinite sequence of terms of this kind, leading to divergent integrals in u. It is easy to identify these divergences as being of the infrared type: if one assumes the photon to have a finite mass K, the corresponding propagator, eq. (13), will contain an additional factor exp { - K2/2~}, i.e. in the exponential in eq. (29) u has to be replaced by t~2 t~+-m2

1-u u

this eliminates all the divergences of type (c). However, these infrared divergences are a characteristic not ofI(N,o, r/;L) itself, but only of the individual terms of the expansion in L forced upon it by the method just described. More precisely, their source is the term-by-term integration with respect to u of the expansion of F(N,o, r/;L; u, v) in powers of v (this is seen at once by making in eq. (29) the substitution v -+ Lv). Since after the removal of the zerofield contribution I(N, o, r/;L) is a finite, well-defined quantity, all the terms belonging to the same order in u have to sum up to a finite expression. The pattern emerging from this discussion is the following. Each order m in eq. (30) (i.e. each column in fig.2) gives a finite contribution to I(N, o, r/;L). To compute it, one has to drop the regularization term (n = 0), if present; isolate the finite terms n = m - l , m (n 4: 0); integrate the other terms (n > m) with respect to v and sum up their contributions before integrating in u. This latter operation produces

D.H. Constantinescu, Electron self-energy

296

expressions of the type

urln ( l +-sL (l (r and s are natural numbers) that lead to finite quantities by integration in u. Moreover, when the minus sign is present in the argument of the logarithm its phase jumps from -T( to 0 during the integration in u, giving I(N, o, r~;L) an imaginary part which brings in a finite level width. Keeping only the contribution of the first two rows and columns in fig. 2, this method yields

I(N, o, r/;L)

= ~01

{(½o+O(NL))L

(}+O(NL))L2InL + O(L2)} (32)

iT( G0 {[~(4N -2(1 +o)) +O(NL)] L 2 +O(L3)},

where 1

~0(~) = f(0,,7) = (1 + , ? ) y .

(33)

The leading term in the real part is just what one expects from the well-known value of the anomalous magnetic moment of the electron; the imaginary part, of course, vanishes for the ground state (N=0, o = - 1 ) . The terms O(NL) give higher-order contributions in L; they are also slowly-dependent functions of fT. It is seen from eq. (32) that, for not too highly excited levels (NL ~ i), the basic information about the low-field behavior is already contained in the corresponding formula for the ground level (except for the fact that in this case only the value a = 1 is allowed and the width is zero). This was to be expected, since in the nonrelativistic limit for the cyclotron motion the phenomenon should be dominated by the corrections to the electron mass-times a kinematical factor depending on the free motion along the field. For the ground state a few more terms in the expansion (32) have been calculated by Demeur [8] ; up to terms of order L 2 his result is *

l(O,_l,rl;L)=~O {_½L

4L21nL_ ( T,3g - ~4l n -,;,) L

2+ O(L 3 1n L )}."

(34)

The opposite limiting case (L >> 1) is easier examined by choosing as integration variables x =u,

y = (1

u)u,

(35)

and rewriting eq. (29) as

* In ref. [8] the expansion is pushed up to the term L31nL; however, the corresponding coefficient is incorrect. The leading term, giving the a n o m a l o u s magnetic m o m e n t , had been calcukited previously by Luttinger [15].

D.H. Constantinescu,Electronself-energy I(N'°'rT;L)= / ~x / dy 6(N, o,~;L;u,v)exp{0

L(x-y)},

297

(36)

0

where q5 = F/v. Now, for L >> 1 the integral (36) converges very slowly, and its behavior is dominated by the region y ~ x, x -~ ~ (one has to remember the convergence factor introduced by replacing x ~ x(1 -ie)). In this region the leading dependence ofeq. (36) upon L is concentrated in a factor exp {-e(2N+ 1/L)x}, and one has to discuss separately two distinct cases. (a) For the ground state (N=0, o = - 1 ) the only nonzero coefficients are A ( 1)=_2B~ 1)=2/~0; the result can be expressed in terms of the exponential integral function [16] : I ( 0 , - 1 ,r/;L) ~ ~ 0 Ei2

+ .... L >~ I.

(37)

Since for small values of the argument the exponential integral has the behavior one gets

Ei(ll) ~- In l/J[ + constant,

l(0,-1,r~;L) -~ ~

1

ln2L + ....

L >> 1.

(38)

A more careful evaluation, going beyond the leading term, has been made by Jancovici [7]. (b) The situation is different for the excited levels (N 4: 0); now, the integration yields a constant quantity, the L-dependence coming from the kinematical factors, viz.

2NL l(N,o, rl;L)~-constX-~+ ....

L>> 1, (N:~0).

(39)

The exact evaluation of the proportionality constant is a hard task, but one can show that, even for rather high N and for both values of o, one has Iconst [ -~ 1. Now, the result of eq. (39) can be better understood if one remarks that in superstrong fields the "natural" unit of energy is not the electron mass, but the energy level itself. Therefore, a more suitable form of eq. (28) is

I(N, o, rT;L) AEN°q =ENq 27r ~(NL,rl)'

(40)

whence, by using eq. (39),

A EN~q ~ o~ X const, ENq 21r

L >> 1, (N 4: 0).

(41)

This result could have been obtained by purely dimensional considerations, since the only way of building the dimensionles s quantity 2XENoq/ENq out of the magnetic field only (m and q being irrelevant in this limit) is to make it a constant. The relative importance of the radiative correction with respect to the initial value of the ener. gy is seen to remain small even at very high intensities.

D.H.Constantinescu,Electronself-energy

298

Coming back to the ground level, one would like to see what happens in the intermediate region around the value L = 1. For N = 0 , the integration path in the v-plane in eq. (29) can be rotated by an angle - ½7r, in order to display explicitly the reality o f I ( 0 , - 1 ,r~;L); after regularization, the result reads

1 fl du / do E2v-(1-tanhv)(1 u)v I ( 0 , - 1 ,r/;L) = ~-0 0

0

v [u ~anh v + ( i ~ - a n h ~ ( 1 - - u ) v

(42)

_

A numerical computation of this expression has been made for 0.01 ~
0.10

0.05

2 7i" AEoo

-0.05 0.01

0.1

L

Fig. 3. Graph of the function ~o/(0, I,'o;L) 2nAEOOfor0.01 ~
g?l

299

D.H. Constantinescu, Electron self-energy

1.5

5.0

~x

2.5

0

,~

l

I

10

100

L

Fig. 4. Graph of the function ~o/(0, -1 ,~;L) -

2Tr AEo0 -for 1 ~
l o w - L behavior to high values of this parameter yields both the wrong sign and the wrong order of magnitude. To conclude, let us summarize the results. In a weak magnetic field, the electron self-energy is dominated by the self-mass correction; the leading term is linear in the field and is interpreted in the usual way in terms of the anomalous magnetic moment of the electron. For the excited levels the only difference, in this limit, is the finite width and the presence of corrective terms in the higher-order coefficients. In very strong fields, the self-mass correction becomes a positive, very slowly increasing function of the field; even for intensities some orders of magnitude greater than the critical value H c it is still a small fraction of the electron mass. For the excited levels, the self-energy correction is determined by the enhancement of the cylotron motion, which becomes extremely relativistic; its relative importance with respect to the uncorrected level is still very small. One can safely conclude that, even in arbitrarily strong magnetic fields, the gap between the positive and negative frequencies of the electron cannot be filled by the radiative corrections, and the electron-positron vacuum does not become unstable.

D.tt. Constantinescu, Electron self-energy

300

T h e a u t h o r t h a n k s Professor L.A. Radicati for m a n y i l l u m i n a t i n g discussions, a n d Dr. G. V e n t u r i n i for help w i t h the n u m e r i c a l c a l c u l a t i o n . K i n d h o s p i t a l i t y at Scuola N o r m a l e S u p e r i o r e is gratefully a c k n o w l e d g e d .

REFERENCES [1] T. Gold, Nature 218 (1968) 731; 17. Pacini, Nature 219 (1968) 145; J.E. Gunn and J.P. Ostriker, Nature 221 (1969) 454. [2] Z. Bialynicka-Birula and I. Bialynicki-Birula, Phys. Rev. D2 (1970) 2341. [3] D.H. Constantinescu, Nucl. Phys. B36 (1972) 121. [4] S.L. Adler, J.N. Bahcall, G.C. Callan and M.N. Rosenbluth, Phys. Rev. Letters 25 (1970) 1061 ; S.L. Adler, Ann. of Phys. 67 (1971) 599. [5] H.Y. Chiu, V. Canuto and L. Fassio-Canuto, Phys. Rev. 176 (1968) 1438. [6] R.F. O'Connel, Phys. Rev. Letters 21 (1968) 397; Phys. Letters 27A (1968) 391. [7] B. Jancovici, J. Physique Suppl. 30 (1969)C3. [8] M. Demeur, Acad. Roy. Belgique, Classe des Sciences, Memoires, 1643 (1953). [9] M.H. Johnson and B.A. kippmann, Phys. Rev. 76 (1949) 828. [10] J. G~hdniau, Physica 16 (1950) 822; J. G~h~niau and M. Demeur, Physica 17 (1951) 71. [ 1 l ] G. Kfillen, Encyclopedia of physics (Springer Verlag, Berlin, 1958), vol. 5, part 1, sect. 16. [12] F. Low, Phys. Rcv. 88 (1952) 53. [ 13] A.I. Akhiezer and V.B. Berestetskii, Quantum electrodynamics (Interscience Publishers, New, York, 1965), Sect. 43. [14] H. Bacry, Ph. Combe and J.L. Richard, Nuovo Cimento 67A (1970) 267. [15] J.M. kuttinger, Phys. Rev. 74 (1948) 893. [16] I.S. Gradshtey~ and I.M. Ryzhik, Tables of integrals, series, and products (Academic Press, New York, 1965), formula 4.351 (3).