the spin 12 gravitational anomalous magnetic moment

ANNALS OF PHYSICS 139, 48-67 (1982)

The Spin 1 Gravitational Anomalous Magnetic Moment

WALTER WILCOX*

Department of Physics,

University of California, Los Angeles, CaliJornia 90024

Received February 11, 1981; revised May 27, 1981

The gravitational anomalous magnetic moment of a spin f particle is calculated and compared with an earlier dimensional regularization result. The answers disagree although

both agree the answer is finite. It is argued that this quantity is not uniquely determined by physical requirements. The methods used are shown to respect the gravitational and elec-

tromagnetic gauge invariances. In addition, a physical interpretation of the regularization method employed is offered.

I. INTRODUCTION

Quantum gravity is known to produce finite first order corrections to the spin f magnetic moment [ 1 ] as well as to the magnetic and electric properties of a Yang-Mills vector field [2). These results are especially interesting since they are in stark contrast to the non-renormalizability of quantum gravity when interacting with matter [3]. They are also significant since these results represent some of the very few finite part calculations done in either quantum gravity or supergravity.

Although the corrections mentioned are finite, they may not be unique. The original, computer aided, calculation of the gravitational spin i anomalous magnetic moment used dimensional regularization. Certainly the results, if unique, should be obtainable from other points of view. A simpler calculation of his effect using source theory techniques has therefore been undertaken. The magnetic moment found is finite as expected, but disagrees with the dimensional regularization value.

A critical point considered is the question of whether the method of regularization preserves the gravitational (and electromagnetic) gauge invariance. The method used to regulate divergent integrals resembles analytic regularization [4, 5, 121, except that no appeal to analyticity is made. It is found that this scheme preserves the gauge invariances.

Finally, it is argued that the answer obained for the spin f gravitational anomalous magnetic moment is not unique. In addition, some speculations concerning the regularization method used and the results obtained are presented. The parameter used to control the ultraviolet divergences is viewed as the coupling constant for some unknown regularizing interaction. The apparent non-uniqueness of gravitational corrections to physical processes is discussed in the light of supergravity theories.

* Present address: Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078.

48 0003.4916/82/030048-20%05.00/O Copyright B 1981 by Academic Press, Inc. All rights of reproduction in any form reserved.

THE GRAVITATIONAL ANOMALOUS MAGNETIC MOMENT 49

II. THE CALCULATION

In this section will be presented the mathematics responsible for producing a result for the gravitational spin $ magnetic moment. An analysis of the method used and other issues, including the question of gauge invariance, are taken up in the next sections.

The diag,rams that contribute as gravitational sources are shown in Fig. 1. One may view both diagrams as contributing to the initially causal probability amplitude

(0, IO-) = itc j dx dx’ t,,,(x) D+(x - x’) ~?‘“*‘~t~~~(x’), (1)

where h&L bdx’) are the graviton detection and emission sources, respectively. Here K = %cG (G is Newton’s gravitational constant in It = c = 1 units), and D + (x - x’) 7YJK is the causal graviton propagation function where

D+ (x - x’) = i I

dw, eik(x-x’), x0 > x0’. (3)

Figure la is associated with the energy momentum tensor

(4)

In the presence of an electromagnetic field, this becomes

fu,(X) = $ v(x) YO(Yr T. + Y”71,) w(x)7 (5)

where

71, = f 3, - ieqA,(x) (6)

FIG. 1. Diagrams that contribute as graviton emission or detection sources. q(x) is an extended

electron source shown emitting virtual electrons. The thin solid lines are real electrons. The wavy lines are photons, one of which is virtual and originates from the x-designated source, and the dotted lines are real gravitons.

50 WALTER WILCOX

is the covariant derivative. Similarly Fig. lb is associated with

C,“(X) = +J dx’ dx” ly(x’) yOD+(X -x”) yQ7rm,p”(x -x”) v(x’), (7)

where

&,r”(x-4 = p,,a, +FHoa, +gJ%,aA +g,J%,a” -g,.l;,,aA1 4x-x”). (8)

FA, is being treated as a constant external magnetic field and g,, is the flat space Minkowski metric with signature (- + t +).

After the causal analysis on the spin 4 fields, the graviton detection and emission sources, when placed in the context of Eq. (l), become

t,,,(x) = v,(x) Y0 [

ik) f (271- k), y, t f (27~ - k), Y, - ieq ‘a’~‘;’ ] wz(4, (9)

fz,&‘> = Vl(X’> Y0 [ +(2n- k>n y, ++(2n-k),y, + ieq

Y”% AK(k) vI (x’) iz ] z ,

(10)

with

%z,,~(k) =F&, +F&, + g,&%, + gvnk%,, - g,mk%,. (11)

Equation (11) is the momentum version of (8) except for a factor of i. The rc, in (9) and (10) operate on the spin 1 fields, and the k, are the result of differentiation on the eiktxPX’) in the graviton propagator when in momentum space.

One may verify that the sources (9) and (10) are conserved, i.e., kwtl,,(x) = k*t2,,,Jx’) = 0, by using the free particle field equations

VI(X) YO(P + m> = 03 bb-k)+ml v,(x)=O, vl(x’> y” [y(n - k) t ml = 0, (y7c t m) v/*(x') = 0.

(12)

(13)

Conservation of the sources is necessary for gravitational gauge invariance to hold, as seen later.

The effective field product

v&) vdx’) Y’ leff = -iG(x, ~‘1~~ (14)

(G(x, x’)’ is the spin $ propagation function in the presence of an electromagnetic field) is now supplied. This is the last step in a causal derivation, the one condition being the space-time non-overlapping of the fields v,(x) and w2(x’). This gives rise to the diagrams of Fig. 2. Now, however, the causal restrictions on the sources are

THE GRAVITATIONAL ANOMALOUS MAGNETIC MOMENT 51 n,(d n,(x’ 1 n,(x’ ) I’ I I m + \ i + : \ \ n,(x) n,(x) n&x I n,h.‘) n,(x’) \ + E E + 1’ n,(x) n,(xl FIG. 2. The causal first order diagrams that contribute to the gravitational electron magnetic

moment. The exchanged particles become virtual when the causal restriction on the sources is removed.

removed bfy supplying the full non-causal forms of the graviton and spin 4 propagatons. The result is the symbolic non-causal amplitude:

I dk

K @)4 thy0 + (2= - k), y, + f (27r - k), y, - ieq y(L’=$v W

]

&4K 1

’ k= y(n - k) + m

x f (2~ - k), y, + $ (271- k), yA + ieq C

Y”q? a (k) v/2 k2X 1 . (15)

The fields r,u, and v2 in (15) are not required to be causally arrayed now, but must still not overlap in space-time.

In order to deal with the upcoming divergent integrals (the k-integral in (15) is quadratically ultraviolet divergent, as seen from power counting), the following gauge invariant method will be used. The momentum space part of the graviton propagator along with its representation

are changed to

(16)

(17)

where 6 is a small, positive quantity and m in (17) is a mass parameter, taken for convenience as the electron’s. Intuitively, we see that the inserted factor helps the

52 WALTER WILCOX

integrals converge. It is understood that the limit a--+0 is taken at the end of the calculation. Mathematically, a better behaved theory results. Physically, one must be wary that the underlying theory has not been damaged by the introduction of this device.

The end result of this prescription, as far as the mathematics is concerned, is the use of the representation

m2 s 1 c-1 1 k’ k” y(~ - k) + m

= -i(m - y(n - k)) joa ds s*(im’s)’

x ‘du I

Cl- 41+8 e-isJp 0 r(1 +S) (18)

in Eq. (15), where

1

OF = 2 u’uF 110’ (3 fi” = + [JP”, y”], (19)

H’ = (k - wr)’ + u( 1 - u) x2 + u(m’ - eqoF). (20)

The standard techniques displayed involving the integral representation of propagators may be found in the sources listed [6, 71.

Only the terms linear in the magnetic field are required. Putting the pieces together yields the unified expression (the fields v, y”, w2 are understood)

with

(I) = ((2~ -k), y,[m - y(n - k)] e-isH’k2?‘v*AK(2n - k), y,), (22)

(II) = ((27~ - k), y,[m - y(7c - k)] e-isH’~u”,‘cieqy4~~,~~(k))

- (ieqy% ,,,,(k)[m - y(n - k)] e-isH’+‘“*A”(2n - k), y,), (23)

where the brackets mean the integral j dk/(2n)4, which is viewed as an expectation value. The second term in (23) just doubles the contribution from the first since these terms represent Fig. lb taking place both at the emission and absorption ends. The function f(S) appearing in front of the integral is

f(d)= 1 + C6+ *a*, (24)

where C is an arbitrary constant. That such an arbitrary multiplicative factor inevitably appears with this method is shown later. This translates into the arbitrariness associated with renormalization constants in the usual formulation of field theory. In source theory, Eq. (24) will generate terms that vanish because of the non-overlapping field condition. In either case, physical results are not affected.


The appearance of yn to the extreme left or right in (I) or (II), i.e., next to either v/, y” or v/:r, produces -m from the field equations since the spin f fields are considered Ion-shell for the purposes of this evaluation. The commutator

[n“, n”] = ieq FM” =s- (yn)’ = --71* + eqaF (25)

must also be used to reduce these expressions. From inspection of (22) and (23), it is seen that evaluations of expectation values of the product of up to five kp’s must be performed. The necessary evaluations are carried out according to methods developed by Schwinger [8,9]. These evaluations yield, to first order in the space-time constant magnetic field FM”:

. -isx e (ecisH) = ++7 [ 1 + isu(u - 1) eqoF], (26)

. -isx e (e-iSHk,) = -@-$T [m, + isu*(u - 1) eqoFn, + u’( 1 - u) eqsF,.r”], (27)

. -isx e (eisHkU k,) = @-$T

+ + u(u - 1) eqaFg,, + isu3(u - 1) eqaFn,n,

+ u3(1 - u) eqs(F,,n%, + F,,dn,) 1 ,

. -is.x e (eiSHk2k,,) = $-$ s2 u’q, n* - 3 T n, + u4( 1 - u) eqsF,,, nAz2

+ iu*eqF,, n*(-3 + 224) + isu4(u - 1) eqaFr*n,

+ 3u2(u - 1) eqaFn, , 1

(28)

(29)

where

H = H’ + uequF, (30)

x = --u(l - u)(p)’ + urn*. (31)

From (22) we see that expressions like the above, but with an extra factor of k* are necessary. Rather than working out these expressions in the same fashion, much labor is saved if the relation

i (i-+$) e-isH = pe-iSH (32)

54 WALTER WILCOX

is used. The additional pieces involving k* can be worked out with (26~(32) and are not recorded here.

These results are all that are necessary to carry out the calculation (besides a great deal of care). The organization of the calculation is as follows. The expressions (I) and (II) are written out in full. The expansion

e -iSH _ - eeisH( 1 + isuequF) = (1 + isuequF) e CisH, (33)

produces more linear terms in the magnetic field. The evaluations (26)-(29) are then used. After the reductions involving the field equations are used, the following expressions for (I) and (II) are found. (Notice e-iSx -+ e-ism2u2 only after all other reductions have been made.)

((24 - 12u - 36~’ + 20~~) + (-28~~ + 34u4 - 10~~) inn2

+ [4u4(1 -u)” + u6(1 - u)] s2m4} imequF,

-i e ~ isdd

(II)-t~ s3 {-12(1 -u) + [4u2 + 4u2(1 - u)] inn*} imeqaF.

We have exhibited only the terms linear in the magnetic field. Referring back to (21), our result may now be written

(O+IO-)=Jijdxty,(x)yOEy/,(x),

J=~J~$(im*~)'J'~u 'Li"+'G' e-iSmZU2 {-12~+36~*-20~~ 0

+ (-16~~ + 36u3 - 34u4 + 10~~) ism*

+ (-4u4(1 -u)’ - u6(1 -u)] s2m4}.

(34)

(35)

(36)

(37)

The result obtained is finite. That is, all the l/S terms, infinite as 6 + 0, are seen to cancel in the magnetic moment. Upon the trivial space-time extrapolation of (36) and comparison with the form

(+- l)~~~xV(x)yo~yr(x), (38)

the final result is obtained

(39)


This disagrees with the result of a calculation using dimensional regularization [ 11. The earlier result is

(40)

The reasons for this disagreement and other matters will now be discussed.

III. THE METHOD

Entirely apart from the actual calculation performed, there is the question of whether the method respects the gravitational gauge invariance. After all, a possibly unphysical parameter (6) has now been incorporated into the theory. Although this helps the formal aspects of the problem, we must make sure that the physics has not been damaged.

The object in Eq. (15) that refers to the graviton is the graviton propagator, ““*‘IC/k2. This non-causal propagator was deduced from the causal form that

tppears in IEq. (1); the effective replacement has been made in going from (1) to (15):

& (gk(X-X’) lr P'q'Ki dokeik(x-x'),xO > xO', 7cyv.I~

(2n)4 k2-ic ’ (41)

However, because the graviton sources are conserved, a greater freedom is available in our choice of non-causal propagation functions. Terms proportional to graviton momentum will not contribute because of the conserved nature of the sources. Thus, the most general covariant form of the momentum space non-causal graviton propagator is expressed by:

(42)

where symmetry in p, v and A, K and their interchange has been maintained. The xUSAK transform as tensors, and are symmetric in the latter two indices. The freedom in the choice of non-causal propagation functions is just the freedom of gravitational gauge transformations. Equation (42) is analogous to the electromagnetic situation:

(43)

where the 11” may be chosen to depend on the values of k” and k in certain reference frames in order to give simple forms in those frames [lo]. The choice 2“ = - i(l - r) k’/k4 results in the propagator

g ‘” - (1 - <) k”k”/k2 k2 -iie (44)

56 WALTER WILCOX

and is just the class of gauges singled out when a gauge-fixing term

q= - f~-‘(mJ* (45)

is added to the usual Lagrange function

4p,, = - @54” - a”A”)(a,A” - a”A,). (46)

Similarly, the choice x“*“‘ = - $(l - <)(k’g”” + k”guA)/k4 in (42) results in the propagator singled out by the gravitational gauge-fixing term

Pti= - p(Phpu - $,h”,)(a*h*” - fa”hQ, (47)

which is

7c P”qaK _ i!$ (k’kAg”” + kUk”gua + k”kAgW + k”kKg,‘a)

k*-ic (48)

The use of the gauge terms in the graviton propagator must not influence the vacuum expectation value, (15). If the extra propagator terms are substituted in (15) and use of the field equations

‘c/l YO(P + m) = 0, (m+ m> w* = 0, (49)

are made, the remaining integral is completely local and thus vanishes under the non- overlapping field condition. “Local” here means a quantity expressable as a finite expansion in the covariant derivative z L(. For example, a term proportional to yn is local, but a term such as l/(r(n - k) + m) is not. Thus, before the introduction of the 6 parameter the vacuum amplitude is explicitly gravitationally gauge invariant. What must be verified now is that the introduction of the parameter 6 does not spoil this reduction. Substitution of the gauge terms from (42) into an equation analogous to (21) generates an expression for the extra pieces of the vacuum amplitude’

(I - ‘)‘+’ (FIA(k,x) r(l + s)

x [(n - k)’ + m* - eqaF] episH’Fzn(k,x)), (W

’ The representation

1

(n-k)Z+m2-eqaF e-iW

is used here to produce enough factors of k* in the denominator to incorporate the gauge terms of (48). for example.


where use has been made of the field equations (49). It is understood that the right- hand side of (50) stands between the fields vi y”, I,v*. The exact form of the vectors E;iA(k,x), I;‘,,(k, x) are unimportant for this discussion except they have no s or u dependence. The factor [(x - k)* + mz - eqoF] is just what is needed to cancel the non-local spin f propagator denominator

- y(7c - k) + m y(n-k)+m = (TC-k)*+m*-equF’ (51)

In the form (15), without the 6 parameter factor, the non-local dependence on rt“ would now have been removed by this factor. How is this locality to be shown in (50) where the non-locality is associated with an exponential instead of a denominator? First, it is noticed that the exponential satisfies the differential equation

. a (

(1-u) a I as+sz 1 e

-‘SN’=[(~-k)2+m*_eqaF]e-‘~~‘.

Second, after inserting (52) into (50), a partial integration is performed on the a/as, a/au terms to yield

(0, lo_),,,,, - ire om ds s3 ;(;;;) ) (1 - u)~+” E(u, s)l: i

+ (4 + S) j’ du(1 - u)~+~ E(u, s)l 0

+ ire i

’ du (1 - u)3+8 r(l + 6)

s4(im2s)’ E(u, s)lr 0

- (4 + 6) lom ds s3(im2 s)’ E(u, s) 1 , (53)

where

E(u, s) = (F,‘(k, x) e isH’F2n(k, x)). (54)

The terms proportional to (4 + S) in (53), which are the non-local ones (i.e., involve the full exponential eeisH’), are seen to cancel, leaving two boundary terms which either vanish or are local since

(1 - u)4+6 w, s)lu=, = 0,

E(O, s) = (F,A(k x) e-iSk*F2A(k,x)),

lim E(u, s) = 0, s-+cc

-W 0) = (f’,%k x> F,,t(k xl>.

(55)

(56)

(57)

(58)

58 WALTER WILCOX

The result (57) holds if the understood -ic term in the graviton propagator denominator is recalled, which becomes a convergence factor here. Thus, the extra part of the vacuum expectation value vanishes. This general argument has been supplemented by an explicit calculation, in the class of gauges described by (48), of the resulting extra terms linear in the field F’“. The calculation shows that the terms proportional to (1 - r) vanish and the answer (39) for the gravitational magnetic moment is not influenced.

This satisfactory state of affairs is upset a bit when it is realized that the local terms in (53) are not finite. This is to be expected since somewhere must be encoun- tered the formal quadratic divergences that exist in the starting point, (15), as well as the extra part of the vacuum amplitude. The method adopted, with 6 a small positive quantity, is sufficient to render convergent logarithmic ultraviolet divergences at best. Let us, therefore, briefly consider an alternate procedure in which the 6 parameter is allowed to take on values that make the starting point ultraviolet convergent. This is the main idea of the “analytic continuation” regularization method. When it is attemped to verify gravitational gauge invariance using this altered method, one temporarily lets 6 + I+, which disposes of the second boundary term in Eq. (53). (See Eq. (60), below.) The remaining term in (53) is

(0+/O-),,,,, - itc om ds s3 ~(~~~) E(0, s). I (59)

In the gauge model given by (48), one easily shows

E(O,s)--$e-sE, (60)

where the understood convergence factor associated with the -is factor in the graviton propagator has been explicitly brought out. Then

If it is understood that the E limit is taken after 6 has been extended downward to some small fixed value, this term now vanishes. What has happened is the formal suppression of such singular integrals as

C(F) I dk 1

7c 4’ @-ij-v

just as in the dimensional regularization scheme. While it is reassuring to know that a method exists that suppresses the divergent

local terms and results in a finite expression (if we remember to control the infrared divergences also), it is unnecessary to use it since the local terms, whether divergent or not, vanish under the non-overlapping field condition. Thus, it is preferred here to


view 6 as a small fixed positive quantity and not to appeal to analytic continuation. In the usual formulation of field theory where it is not attempted to enforce a non- overlapping condition on the fields, it is appropriate to use some variation of the analytic continuation method discussed above to suppress the divergences that now appear. The restriction to small positive 6 values also allows an interesting physical interpretation of the method which will be discussed in the next section. No matter which meth(od is preferred, the same answer for the gravitational magnetic moment results, given by Eq. (39).

Electromagnetic gauge invariance of the vacuum amplitude may also be ascer- tained. Figure lb contains an exchanged photon which contributes the kb2 factor in the sources (9) and (10). If the photon propagator is changed in the manner of (43), and the field equations (49) are used, it is easily shown that the gauge terms vanish in the same manner as the gravitational gauge terms.

The mathematical reason for the appearance of the function S(S), which first appeared in (21), will now be pointed out. Before, it was stated that it represented the arbitrariness associated with so-called renormalization constants, which are not observable. This is just the common sense observation that any identification of “divergent” parts by introduction of cutoffs or other mathematical devices must be arbitrary up to an additive constant.

To see how thef(6) function arises, insert a factor 1 = k2/k2 in (15) and use

8 1 1 k” y(7c - k) + m

= + (m - y(n - k)) jam d~(irn~s)~

3 (62)

in place of the representation (18). This gives the equally valid starting point in place of (21), ignoringf(6) for the moment:

1 --K .!‘“: ds s3(im2s)” i,’ du 8 o

(’ - 202+8 [(Ik’) + 2 (IIk2)]. r(l + s)

Using the relation pointed out in (32) gives

[(Ik’) + 2(IIk2)] = i g-3@+2(II)l.

Then, after performing partial integrations on the a/&, a/au terms, it is found that (63) becomes

-fK (1 +;)/om ds s2(im2s)s 1; du (l- uY+s [(I) + 2(11)], r(l + 6) (65)

which is just (21) with f(8) = (1 + a/2). Obviously, this particular form of the

60 WALTER WILCOX

function f(s) is completely arbitrary and can be changed by changing the representation. The realization that the representations (18) and (62) are equally applicable shows that the emergence of a multiplicative function f(s) of the form (24) is inevitable in this method.

The appearance off(d) suggests that the introduction of the arbitrary parameter 6 into the theory must be made in as general a manner as possible, while still main- taining the physical principles intact (i.e., gravitational and electromagnetic gauge invariance, conservation of sources, unitarity, etc.) In the “analytic continuation” model, one would say that the continuation in 6 of the expression (15) must be as general as possible, consistent with the physics. From either viewpoint, the possibility that the sources themselves become dependent on the continuation variable, i.e., f&+ tJx, 6) cannot a priori be excluded if one maintains the conservation condition (which automatically assures gravitational gauge invariance)

kUtw,(x, S) = 0. (66)

This extra dependence on the arbitrary parameter introduced could change calculated quantities such as the gravitational anomalous magnetic moment. Thus, from an entirely mathematical point of view the answer (39) is not the only one consistent with the physical principles involved.

The appearance of J(S) also emphasizes that all diagrams contributing to a physical process must be handled exactly the same when introducing the integral representation of propagators. Otherwise, one may be implicitly supplying different f(S) functions for the different pieces rather than a single f(s) that multiplies all diagrams. Such a procedure can be shown to destroy the gravitational gauge invariance in this problem.

With regard to the question of unitarity, the following observations are made. A necessary aspect of unitarity is the identification of the causal propagation function as the imaginary part of the noncausal form, i.e.,

eik”S(k2), (67)

so that

Im D+(~ -x/) = j do, eik~x-ikOlxO-xO’l~ (68)

This makes the absorptive part of noncausal diagrams equal to the corresponding causal amplitude and ensures the preservation of unitarity. The right-hand side of (68) has a definite interpretation in terms of a sum over real intermediate states. In contrast it is clear that if the expression

dk e Dy’(x-xx’)= jw ik(x-x’)(1722)6

(k’ - i&)‘+’ (69)

THEGRAVlTATlONALANOMAtOUS MAGNETICMOMENT 61

is adopted as the modified non-causal propagator, this simple interpretation in terms of a sum over real intermediate states must be modified. Taking the imaginary part of this expression no longer enforces the real particle condition k’ = 0 as in (67). Thus, acknowledging the imaginary part of (69) as the modified causal propagation function formally preserves what is meant by unitarity here, but sacrifices the real particle condition kZ = 0 when 6 # 0.

Another important point to be made concerns Ward identities.’ Any expression containing covariant derivatives is invariant under the combined gauge transfor- mation:

A,-,A, +a,,A

P, -+ e - iesApu e it+. (70)

The immediate consequence of this invariance for the mass operator identified from (21) is

k, JM@) ” dA,,(k’)

= eq(M(7r) - M(~c’)),

which is derived by using the infinitesimal forms of (70) on M(Z). This is the statement of the electromagnetic Ward identity when a classical external field takes the place of the usual quantized electromagnetic field. The use of (69) in M(n) does nothing to modify this situation, and so the maintenance of the Ward identity in this case is immediately verified.

As a final remark on the regularization procedure used in this paper, it is emphasized that the method as employed here is not just a different form of analytic regularization. It is a conceptually different development used in a logically different context. As stated before, it is appropriate to use such a procedure only with the non- overlapping field condition present in source theory. In the absence of such a condition, an extra hypothesis requiring analytic continuation in 6 must be assumed.

IV.PHYSICAL INTERPRETATION

So far the regularization method chosen has been dealt with in an entirely mathematical way. It will now be attempted to supplement these mathematical arguments with a specific physical model.

The interpretation of 6 as a small positive quantity is kept. The observation is made that only the first two terms in the expansion of the vacuum amplitude in 6 are ever neceslsary in any calculation, suggesting that there might be some significance in keeping just the first two terms in the original parameterization that introduced 6. Now

as = eslna = 1 + 6lna + . . . . (72)

’ This derivation of the Ward identity is due to Professor Schwinger.

62 WALTER WILCOX

Therefore

1 (73)

This is naively compared with the result of vacuum polarization on the photon propagator :

D(k) 5: , k2 P m2. (74)

There are two flaws in this comparison. The first is that (73) represents a change to a graviton, not photon, propagator. Second, the 6 identified is negative, not positive. Obviously, electrodynamics is not the interaction responsible for producing the modification (73). However, if the parameter 6 is viewed as the coupling constant of some unidentified but physically possible interaction, one is provided with a regularization method that is much more physically motivated than simply viewing 6 as a continuation variable. One may view (73) as a statement that there exist interactions that modify the graviton propagator in such a manner as to counteract the theoretically unlimited strength of the gravitational interaction at high energies, just as at very low energies adding an effective photon mass to the photon propagator represents the modifications due to screening on the theoretically unlimited range of the Coulomb force. However, Coulomb screening proceeds by understood mechanisms, while the situation here is only a possible scenario.

In terms of this physical model it is again clear why the gravitational magnetic moment calculated in Section II is not uniquely determined. If a first order effect of the a-interaction is to produce a modified graviton propagator, then to be physically consistent the effects of this interaction must be included to the same order on the vertices also. This means in general t,“(x) + tru(x, S) as stated before, with the restriction (66). The expansion

L”(X, 4 = f,“(x) + qL”(X) (75)

represents the first order effect on the sources. Besides the conservation condition (66), two additional assumptions on t;,(x) are necessary to assure a viable regularization method; these conditions are given in the Appendix. Under these assumptions, (75) will not add new divergences to the theory but will produce finite additions to originally divergent (by power counting) quantities. Since the gravitational spin 4 magnetic moment contains logarithmically divergent pieces, a realistic change in the sources as in (75) should in general change its value.

It is briefly mentioned that this method may be applied to electrodynamics with


complete success. If a situation is considered where virtual photons are present, their sources must be changed in accordance with the above ideas:

j,(x) +jp(x) + qA4. (76)

The effect of the extra terms in the vacuum amplitude will again be to add finite terms to originally divergent quantities. Thus, completely convergent quantities like magnetic moments are not affected. The extra finite terms will add to off-shell quantities where their effect is removed by the non-overlap of the fields. In the usual field theoretic la.nguage this means that (76) changes only unobservables like renormalization constants. This change in sources can therefore be consistently ignored. An example of such a QED calculation is given in the Appendix.

V. CONCLUSIONS

One of the few calculable quantities in quantum gravity, the anomalous self-gravity induced spin f magnetic moment, has been found to be finite, confirming an earlier result. The gauge invariant method used for regularizing the integrals strongly suggests from both mathematical and physical points of view that the answer is not unique. The earlier result, also obtained from a gauge invariant method, differs from the one here, and thus supports this view.

This arb’itrariness is also suggested from the point of view of dimensional regularization. Within this method one may perform both the integrals and Dirac algebra in r’2 dimensions or perform the Dirac algebra in four dimensions and only continue the final integrals to n dimensions [3]. Either method yields gravitationally gauge inva.riant results in this problem since the conservation condition on the sources is still maintained in it # 4 dimensions. These two methods will in general produce different finite parts, leading again to the non-uniqueness already found. Both dimensional regularization and the method employed here fail to produce an unam- biguous answer essentially because the full dependence of the expressions on the regularization parameters (6 or n - 4) is not uniquely determined by physical requirements.

The fact that this particular quantity in the context of a nonrenormalizable theory of gravitation is not unique is not surprising. It illustrates a problem, however, which may persist in more viable theories, such as supergravity. The new physical requirement that emerges in these theories is the maintenance of the supersymmetry throughout the calculation. Perhaps this extra condition will be enough to force uniqueness. The problem is to find a regularization procedure that does not spoil the supersymmetry. The search for such schemes (if they exist) has concentrated so far on different forms of dimensional regularization; for example, dimensional regularization via dimensional reduction, where all momenta are continued to n < 4 dimensions while the Dirac algebra is performed in four dimensions [ 111. The successful use of a different scheme here, although not in a supergravity context,

595/139/1-s

64 WALTER WILCOX

points out other methods that may be candidates also. Whether the answers are unique in supergravity may very well depend on the agreement reached between several such schemes.

APPENDIX

A straightforward example of the mechanics of the regularization method used, in the context of source theory, is now given. Consider the exchange of a virtual electron-photon pair originating from extended electron sources in the presence of a uniform magnetic field (i.e., the QED analogy to the gravitational calculation in Section 11). The symbolic vacuum amplitude is

e* I m - y(n - k)

(r-k)’ +m2-eqaF yMv2’

The representation

1 (r-k)’ + m2 - equF

=-- (I - ‘1” -iSH’ j* dss(im2s)’ I,’ du r(l e ,

0

is used, where

(A.11

(A.2)

H’ = (k - UX)’ + ~(1 - U) x2 + u(m2 - equF) + (I - u)p’. 64.3)

P is a small photon mass introduced to control infrared divergences, and 6 retains its meaning as a small positive quantity. The mass operator in the absence of an electromagnetic field implied by (A.l) is identified when comparing with the vacuum amplitude mass term:

-i i dx ~~(4 y0w2(x). (A.4)

This gives the mass operator

M(yn)=Kf(d)jm~(im2s)d ji dus,-Jg (pm + (1 _ u) Y7Z)e-i~(m*~*+dn, 0-W

with

8%“= u(l - u)(m’ - (~7r)~) + (1 - u)~L’. (A4


The s integral is done and yields

M(v) = $+-f(S) j; du(2m + (1 - U) y7r) . (A.7)

This expression is now expanded in a power series in 6 about 6 = 0. The first term is

$$ (2m+&n). G4.8)

It is completely local and so does not contribute with non-overlapping fields. The next term ia

du(2m + (1 - u) y7r) In

(A.9)

where C is the arbitrary constant in the expansion off(S) in (24). The term proportional to C in (A.9) is local again and does not contribute. The physical normalization conditions are now imposed:

(A. 10)

by adding local terms. This yields

(A.ll)

where it is understood the photon mass has been suppressed for simplicity. The correctness of this expression is verified when contact is made with Ref. [5] by performing a partial integration and letting

(2m+(l -u)p)+m(l +u>, (A.12)

which is appropriate in the application of (A.1 1) to situations where the spin 4 fields are real.

66 WALTER WILCOX

We must ascertain that there is no ambiguity in this regularization scheme. That is, under a physically allowable change in the sources (76), we do not change the result (A. 11). This will be the case if two conditions on the new source term are met:

1. The extra term contributed to the vacuum amplitude does not have a worse high energy dependence than the original expression.

2. The locality in the currents is preserved.

The first condition assures that no new divergences will be incorporated into the theory, i.e., convergent quantities before (76) is introduced remain convergent afterward, and that the only modification will be the addition of finite terms to originally divergent quantities. The second condition makes sure that the extra finite terms in QED will be local and will not contribute when the fields do not overlap.

For example, the first condition rules out a current of the form

&(x) = t j dx’v(x’) yOw,S(x - x’) Wh (A.13)

f(k2) = [ d(x -x’) eikcxpx’)f(x -x’) k2rx, k2, (A.14)

because it would introduce a quadratic divergence in the expression (A.l) which is only logarithmically divergent at high energies. As another example, the second condition rules out a piece of the current of the form

J&4 = 5 j ~‘~4-4 yOew,f(x - x’> VW (A.15)

where f(x - x’) is a non-local expression (i.e., not a Dirac delta function or derivative of one).

The extra piece contributed to the mass operator by a change (76) in the sources must be of the form given by (A.5):

where F(yn, U) is some unknown function of yn and u, and the logarithmic divergence is reflected in the s integral. (If the extra terms had better than logarithmic behavior, they would automatically vanish in the limit 6 + 0.) The s integral is done and the result expanded about 6 = 0:

du F(yz, u) + O(6). (A.17)

THEGRAVITATIONALANOMALOUS MAGNETICMOMENT 67

The non-local behavior of the exponential in (A.16) is not present here. Now using the second condition above, this is revealed as a completely local term. Thus, it is seen that a change in the sources of the form

assuming continued current conservation and the maintenance of the two conditions above, does not change the result (A. 11).

ACKNOWLEDGMENTS

The elegant methods and encouragement of Dr. Julian Schwinger as well as the suggestions and help of Dr. Kimball Milton are very gratefully acknowledged. Conversations with Dr. Lester DeRaad were also extremely helpful.

REFERENCES

1. F. BERENDS AND R. GASTMANS, Phvs. Left. B 55 (1975), 311. 2. V. VAN PROEYEN, Ph-vs. Rev. D 15 (1977), 2144. 3. M. T. GRISARU. P. VAN NIEUWENHUIZEN, AND C. C. Wu, Phys. Rec. D 12 (1975). 1813. 4. B. BREITENLOHNER AND H. MITTER. Nucl. Phys. B 7 (1968). 443. 5. C. G. BOLLINI AND J. J. GIAMBIAGI. Nuoco Cimento 31 (1964). 550. 6. J. SCH\C~FIGER, “Particles. Sources and Fields” Vol. II, (Addison-Wesley. Reading, Mass., 1973). 7. J. SCHWINGER, Phw. Reu. 82 (1951). 664. 8. J. SCHWINGER, Phys. Rev. D 7 (1973). 1696. 9. L. L. DERAAD. N. D. HARI DASS, AND K. MILTON, Phm. Rec. D 9 (1974). 1041.

10. V. B. BEF:ESTETSKI ef al.. “Relativistic Quantum Theory,” Part I, Pergamon. Oxford, 1979. II. W. SIEGEL, Phys. Left. B 84 (1979), 193; 94 (1980). 37; H. NICOLAI AND P. K. TOWNSEND,

“Supersymmetric Regularization and Superconformal Anomalies,” CERN Report TH 2840, 1980: P. K. TOWNSEND AND P. VAN NIEUWENHUIZEN, Phys. Reu. D 20 (1979). 1832; D. M. CAPPER er al.. Nut. .Phys. B 167 (1980), 1479; L. V. AVDEEV. G. A. CHOCHIA, AND A. A. VLADIMIROV, Phys. Ler~. B 105 (1981), 272.

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the spin 12 gravitational anomalous magnetic moment

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