The electron magnetic moment at high temperature

Volume 114B, number 5

PHYSICS LETTERS

5 August 1982

THE ELECTRON MAGNETIC MOMENT AT HIGH TEMPERATURE Yasushi FUJIMOTO and Jae HYUNG YEE 1

International Centre for Theoretical Physics, Trieste, Italy Received 27 January 1982

The one-loop QED correction to the electron magnetic moment is computed at high temperature. We find that the correction reduces the magnetic moment.

The physics at high temperature is assuming an increasing importance since the appearance of the unified field theories, particularly GUTs [1]. These models provide interesting possibilities concerning the evolution of the universe. We may be able to explain, for example, the baryon number asymmetry [2] and the missing mass [3]. To put it very briefly our expectation is that the structure of the present universe is essentially determined by the physics at very high temperature. We are here talking about the temperature corresponding to ~ 102 GeV for the weak and electromagnetic interaction and ~ 1014 GeV or even larger for GUTs. The physics at such high temperatures would be very different from that of the present world which can be described with good approximation by the zero temperature formalism. Regarding to the temperature effects there exist many works on the effective potential [4]. The temperature corrected effective potential has been used to discuss the phase transitions. However, we have to do more than that and eventually should calculate the scattering matrix elements at finite temperature. Otherwise we will never know the physics at high temperature in its entirety. Several years ago Takahashi and Umezawa [5] developed a formalism called "Thermo Field Dynamics" which allows the interpretation and calculation of the statistical average of a given physical quantity as the vacuum expectation value of the associated quantum operator. In the present paper we make use of their formalism and the results of the subsequent works by Matsumoto [6] and Ojima [7] and calculate the QED one-loop correction to the magnetic moment of electron at high temperature as one example beyond the existing calculation of the finite temperature effective potential. We shall not repeat the details of their formalism since the coherent presentations have been given but only state the essentials below and proceed directly to the calculation. Takahashi and Umezawa found that it is possible to define a suitable temperature dependent vacuum IO(fl)) which satisfies, for a given operator F,

(O(fl)lFlO(fl)) = z - l ( f l ) ~ ( n J F l n ) e x p ( - f l E n ) ,

(1)

n

where

HIn)=Enln) ,

(nlm)=6nm .

(2,3)

It is made possible by the introduction of the fictitious fields (denoted with tilde below) which are in a sense a copy of the set of the original fields but commute (or anti-commute in case of fermions) with the original fields. They choose IO(fl)) = Z-1/2(fl) ~-J exp(--flEn/2)Jn, n) .

(4)

1l

i On leave of absence from Department of Physics, Yonsei University, Seoul, Korea. 359 0 031-9163/82/0000 0000/$02.75 © 1982 North-Holland

Volume 114B, number 5

PHYSICS LETTERS

5 August 1982

Then it is noted that this temperature dependent vacuum is obtained from the original one IO) (with the fictitious fields included) by the Bogoliubov transformation IO(/3)) = e x p ( - i G ) [ O ) .

(5)

For instance in the case of the free fermion system G = -i0(/3)(3a - a ~ 3 ~ ) ,

(6)

where cos 0(/3) = (1 + e-3Co)-1/2

(7)

and {ff~} = (aa'~} = 1 ,

(a, 3 } = ( a t , 3 } = 0 .

(8,9)

Since IO(/3)) is expressed as the Bogoliubov transformation of IO) the appropriate temperature operators a(/3), 3(/3) should also be connected with the original fields by the Bogoliubov transformation. For the fermion case, we find a (/3) = e - i G a e iG = u (/3) a -- o (13) 3"~ ,

tl'(/3) = e - iG 3 e iG = u (/3) 3 + v (/3) a'~ ,

(10,11)

where o = (1 + e/~t°)-l/2 ,

u 2 + 02 = 1 .

(12,13)

A consequence particularly relevant for our calculation is that the temperature propagators we are now going to use are actually given by the Bogoliubov transformation of those for the zero temperature case. In the case of QED Ojima [7] obtains for the photon and electron

(TAu(x)Av(Y))=-ifd4pa (27r)4 exp[-ip(x-Y)](~+ --27ri~(p2-m2)guuie e&-I

1) '

(14)

1 ), e~e - 1

(15)

(T~u(x)~u(y))=_if d4P.exp[_ip(x_y)]( -guy 27ri6(p2 m2)guv (TAu(x)Av(y))= - i f

a(27r)4

\ p 2 _ ie

d4p. e x p [ - i p ( x - y ) ] " (2rr) 4

(-27ri)guv 6(p 2 - m 2) e-&/~--~-2 e&-I '

(TrY(x)~0,)) = i f

d 4 p e x p [ - i p ( x - y ) ] i(/k + m ) ( 1 +2zd a(2zr)4 p2 _ m 2 + ie

(T~(x) t~(y))= i af

8(p2 - m

(16)

2) e - - ~ e ~ ) 1 + e-& '

(17)

d4p e x p [ - i p ( x - y ) ] i($ + m) p2 _ m 12 _ ie +2zriS(p 2 - r n 2) e - & ~ (2704 1 + e-a* 1'

(18)

(T¢(x) ~(y)) = i/"-d4p a(27r)4

e x p [ - i p ( x - y ) ] i(Lb + m)(-ZTri) 6(p 2 -

m 2) e -¢e/2 e#e + 1 6 ( P 0 )

(19)

where e(p)-= (p 2 + m2)1/2 ,

6(Po)-O(Po)-O(-Po).

From here on we calculate the well known triangle diagram to estimate the one-loop correction to the electron magnetic moment. In this calculation the tilde field and ordinary field do not mix and thus we can use only eq. (14) and eq. (17) for propagators. And the vertex is the standard one. Thus the calculation is relatively simple for this diagram. Also we note that this diagram allows the high temperature expansion. 360

Volume 114B, number 5

PHYSICS LETTERS

5 August 1982

The full vertex FU(p, p ' ) can be expressed as the sum of the zero and finite temperature parts, /a

F U ( p , p ' ) = r ~ ( p , p ' ) + Ft~ ( p , p ) ,

(20)

where u ' 2/" d4k ( 1 1 1 F~ ( p , p ) = e J (2rr) 3 [~Cou +pU') - (~ + m ) kU] - 8(k 2) k 2 _ 2p • k k 2 - 2p' • k exp(13Y~p) - 1

1 2 _ 21p " k exp(13~p,) 1 + 6(k 2 - 2p • k) k-~k + 1 + 6(k 2 - 2 p ' . k)

12

1 1 ) k 2 - 2p" k exp(/3k 0 + 1 '

(21)

where ~p = [07 - k ) 2 + m 2 ] 1/2 and ~p, = [07' - k) 2 + m 2] 1/2. t In eq. (2 1)Pu and Pu are respectively the initial and t'mal momenta of electron and we set them on shell, i.e. p2 = p,2 = m 2./t denotes the Lorentz index. If we employ the high temperature expansion (/5-1 >> m) (see the appendix of Jackiw and Dolan [4]), I'~ reduces effectively to p O _ e2

1

~f

(dr2 r 0

e2

1

1

(

P~ - 327r PoP'o f12 i d a 2

l+~-/c0"/c

_

/ci( 0"/~ + 0"/~)

I:'/c(0+0')'/~

-(o,

) ,

i'/~/ci (1 +/c" 0/~" fi') I

r 0 {1 - (0-/~)2} {1 - (0"/~)2} - (1 - ~ - ~ { - 1

(22)

(23)

-- (0 ~ :/~)2~-}] '

where dr2 = sin 0 dO dtp,

0 =p/p0,

b' =p'/p'o,

[c = k / l k l .

(24-27)

After the angular integration we arrive at u ' - e2 1 . ( 2m2 _ ) 132PoP~'i\PoP'o Ft3(P'P)-~ 1 ru

i e2 1 m2 e2 1 2m 12 2PoPo' PoPo' °UVqv + 12 flZpop'0 r0f0u + O(1/fl).

(28)

We observe that in eq. (28) the firs~ and the second terms take the form of the familiar expression for the magnetic form factors at zero temperature F ou ( P , P ' ) = r u g 0 + (i/2m) oUVqvFO

(29)

We also observe that the third term seems to break the gauge invariance since /.t

t

t

qu T~ (p, p ) = (e2]12)(1/132pOP'O) qo ro = (e2/12132)(ro/Po - rO/PO)

(30)

and is nonvanishing. However the separate calculation of the electron self-energy gives us: 2;t~07) = (e2/24) f l - Z p ~ 2 [ r " p (1 + 0 2) + 2r0P0] + O(1/t3, 03),

(3 1)

where N# 07) denotes the temperature dependent part of the self-energy. Note that the second term of Zt~(p ) satisfies the Ward-Takahashi identity quT~( p, P ' ) = ~;#(P) - ~t3(P') •

(32)

(In fact one can show that the zero- and finite temperature parts independently satisfy the Ward-Takahashi identity.) Therefore the last terms of eq. (28) and eq. (31) can be amalgamated into the renormalization constants Z 1 and Z 2 and thus do not affect the physically observable quantity. Then the final result for the magnetic form factor is F2( p = p ' ) = FO(p = p ' ) + F~(p = p ' ) = ct/ZTr - (arr/3)(m2/p 2) p~213- 2 ,

(33)

and thus the electron magnetic moment Ix in the high temperature approximation is given by 361

Volume 114B, number 5

PHYSICS LETTERS

It = (e/2m)e [1 + a/2rr - ~azr(m2/p 2) /3-2p~2] .

5 August 1982 (34)

It is noted that from the above procedure it is also possible to obtain the low temperature correction to It. In this limit (1//3 "~ m ) the last two terms o f eq. (21) are o f order e x p ( - / 3 m ) . Thus the low temperature contribution to Fff comes solely from the first term which is proportional to 1//32. After careful computation one can show that the numerical factor for this contribution constitutes 2/3 o f the whole contribution computed in the high temperature approximation. Therefore the electron magnetic m o m e n t in this limit is

It= (e/2m)e(1 + a/2~ - a ~2T r / 3 - 2 m - 2 ) ,

1//3 ~ m .

(35)

Now to the discussion o f our result. As it is the result is not covariant. The non-covariance is inherent to the present formalism since the energy and momenta are treated differently. If we naively apply the result to the small energy case, say P0 "~ m, then at some temperature which is high compared with the electron mass the magnetic moment vanishes and from then on becomes negative. We feel however that the case in which/3 - 1 ~ P0 would be relevant since it is the energy o f the typical electron in the statistical system with temperature/3-1 (>>m). In such case the temperature correction is small compared with the zero temperature one but it still retains the negative sign. We believe that negative sign is the reflection o f the effect o f the thermal agitation on the motion o f the electron whose induced randomness should reduce the magnetic moment. Some stars are known to be in the temperature o f keV order. Thus eq. (35) must have appreciable effects on the events happening in such stars, although the high temperature result may only have to do with the structure of the early universe. We also note that the formalism up to this order is identical to the real time formalism [4]. We hope our sample calculation would stimulate more work in this direction, for example the problem of covariance and renormalization, so that we m a y achieve a better and sure understanding o f the high temperature system.

Note added in proof. After completion o f the work we received a preprint [8] in which the same quantity is calculated with the low temperature approximation. Our high temperature case is complementary to theirs. We note the difference in the sign o f the temperature corrections between ours and theirs. We also note the existence o f a review paper [9] on a different approach to the finite temperature system. We wish to thank Professor J. Strathdee and Professor Y. Takahashi for discussions and correspondence. We would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Center for Theoretical Physics, Trieste. One of us (J.H.Y.) is partially supported by Korea Science and Engineering Foundation. [1] J.C. Pati and A. Salam, Phys. Rev. D8 (1973) 775; H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438; A.J. Buras, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B135 (1978) 66. [2] M. Yoshimura, Phys. Rev. Lett. 41 (1978) 208; B. Toussant, S.B. Treiman, F. Wilczek and A. Zee, Phys. Rev. D19 (1979) 1036; J. Ellis, M.K. GaiUard and D.V. Nanopoulos, Phys. Lett. 80B (1979) 360; S. Dimopoulos and L. Susskind, Phys. Lett. 81B (1979) 416; S. Weinberg, Phys. Rev. Lett. 42 (1979) 850. [3] E. Witten, invited talk, unpublished; S.M. Bilenky and B. Pontecorvo, Phys. Rep. 41C (1978) 226; A.D. Dolgov and Ya.B. Zeldovich, Rev. Mod. Phys. 53 (1981) 1. [41 D.A. Kirzhnits and A.D. Linde, Phys. Lett. 42B (1972) 471; L. Dolan and R. Jackiw, Phys. Rev. D9 (1974) 1686; S. Weinberg, Phys. Rev. D9 (1974) 3357. [51 Y. Takahashi and H. Umezawa, Collective phenomena, Vol. 12 (1975) pp. 55-80. [6] H. Matsumoto, Fortschr. Phys. 25 (1977) 1. [7] I. Ojima, Ann. Phys. 137, no. 1 (1981) 1. [8] G. Peressuti and B.S. Skagerstam, Goteborg preprint (1981). [9] P.D. Morley and M.D. Kislinger, Phys. Rep. 51 (1979) 63. 362