The gravitational analogue of the Witten effect

Nuclear Physics B256 (1985) 353-364 t'c)North-Holland Publishing Company

THE GRAVITATIONAL ANALOGUE OF THE WrlTEN

EFFECT

Omax FODA International Centre for Theorettcal Physics, Trieste, Italy

Received 5 July 1984

In the presence of massive fermions, and assuming a non-vanishing 0-parameter as the only source of CP violation, the Witten effect (a shift in the electric charge of a magnetic monopole due to CP non-conservation) is shown to follow from an anomalous chiral commutator. Next, given the gravitational contribution to the chiral anomaly, the corresponding anomalous commutator for Dirac fermion currents in a gravitational background is derived. From that, we infer the equivalence of a OAR term in the lagrangian to a shift in the mass parameter of the NUT metric, in proportion to 0. This is interpreted as the gravitational analogue of the Witten effect. Its relevance to certain Kaluza-Klein monopoles is briefly discussed.

1. Introduction A n i m p o r t a n t aspect of m o n o p o l e physics is the W i t t e n effect [1]. T h a t is, in the p r e s e n c e of C P violation, the electric charge Qc of a magnetic m o n o p o l e is shifted with respect to its original integral value by a (generally n o n - r a t i o n a l ) m u l t i p l e of the f u n d a m e n t a l charge e. F u r t h e r m o r e , if the only source of C P violation is a 0-term in the l a g r a n g i a n [2] Oe2 32~r----2 F~,,F ~" , ~

L0

v

(1) then Qc is e x a c t l y calculable: Qc=ne

0e 2o'

(2)

w h e r e n is an integer, a n d e is positive. In other words, in the presence of a m o n o p o l e o f charge he, a OFF term in the lagrangian is equivalent to assigning the m o n o p o l e a c h a r g e Qc as given by (2). T h o u g h (2) was originally derived in pure gauge theory, it is interesting to c o n s i d e r it also f r o m the viewpoint of fermion physics. F o r instance, an u n d e r s t a n d i n g of the 353

354

O. Foda /

Witten effect

effect as a rearrangement of the Fermi sea in the vicinity of the monopole [3], relates it to other phenomena including fermion fractionization (for a review see [4]), and the Callan-Rubakov effect [5]. More relevant to our purposes, the connection to fermions links the effect to anomalous commutators of fermion currents [6], and thus to central extensions of charge algebras in the presence of solitons [7]. In this paper, we use this link to provide an alternative derivation that helps view the effect from a different perspective, and obtain it in the case that the original derivation is difficult to reperform. Next, as an application, and in view of the current interest in gravity, particularly in the context of Kaluza-Klein and supergravity theories, we derive the gravitational analogue of the effect. In sect. 2, the effect is related to the commutator of chiral and electric charges, which is non-vanishing in the presence of a magnetic monopole, and in sect. 3, its gravitational analogue: the commutator of chiral charge and the energy m o m e n t u m of Dirac fermions in a generic gravitational background is derived. In sect. 4, the latter c o m m u t a t o r is shown to be non-vanishing in the background of the N U T solution of the Einstein equations [8]. This metric has a wire singularity on the half-line 0 = ~r, that, following Bonner [9], we interpret as physical: a massless semi-infinite line source of angular momentum, since removing it requires compactifying the time coordinate [10], and would unnecessarily complicate our discussion. Otherwise, the metric is characterized by two parameters: an ordinary, or "electric-type" mass M, and a "dual", or "magnetic-type" mass N. Also, as one may, in retrospect, easily guess, we find that a ORR term in the lagrangian - which may be regarded as the result of a chiral transformation on the path integral - is equivalent to a shift in M, in proportion to 0: 0 Mo ,. o = Mo-o + ]-g-~N.

(3)

We interpret this as the gravitational of the Witten effect. In sect. 5, we discuss the above result, and comment on its relevance to a certain class of Kaluza-Klein monopoles [11], that are based on the euclidean section of the N U T solution.

2. Anomalous commutators and the Witten effect

There are two distinct types of anomalies in renormalized perturbation theory. Firstly, there are violations at the quantum level of Ward identities, associated with exact or approximate symmetries, by hard, i.e. dimension-4 operators [12]. Secondly, there are breakdowns of canonical expectations in the high-energy, or BjorkenJ o h n s o n - L o w (BJL) limit [13]: certain equal-time commutators, evaluated in perturbation theory, differ from their naive canonical counterparts. Though distinct, these two types are related, in the sense that given an anomalous Ward identity, one can derive a corresponding set of anomalous commutators [12].

355

O. Foda / Witten effect

Consider massive spinor QED, described by

L = - ¼F~,~F~'~+ ~ ['/~(i0~,- e0A~) ] ~ - m0~ ~ ,

(4)

eo[m0] is the tree-approximation coupling constant (mass), and the metric signature is ( + - - - ). Defining the vector and chiral currents J ; = e0~Y~,~,

J 5 = ~/y~,y5~ ,

(5)

O~'J¢( X ) = l-~2 ~'~"F~,~(x ) + . . . ,

(6)

the divergence of the latter is 2

where radiative effects are absorbed in the renormalized coupling e, and the dots stand for mass terms, which will be neglected from now on. (However, one has to keep in mind that there are no massless fermions, otherwise the 0-angle could be rotated away.) The appearance of a dimension-4 term on the r.h.s, of (6) signals a violation of chiral symmetry, which can be traced to the familiar triangle diagram [121. Considering the S-matrix element of J~(x) corresponding to the VVA diagram, and applying the LSZ reduction formalism to pull an external photon in, as an operator, then taking the BJL limit, one can derive a set of operator equal-time commutators [14]. (Details of such a derivation are given in sect. 3.) Let us consider the commutator of electric and chiral charge densities [ J05(x), J~( Y)] = ie----~2B,(x)O,y83(x - y ) , 27r 2 8,(x)

=

,jka' Ak(x),

123 = 1.

(7)

Integrating over x and y we obtain

[Qs,Q,]

2~r2ie2~2~ukBidS jk ,

(8)

where the surface integral is over S 2 the two-sphere at space-like infinity. In ordinary QED, the surface term on the r.h.s, of (8) vanishes, and the electric and chiral charges commute. However, in the presence of a "suitable soliton", in this case a magnetic monopole, the surface integral is proportional to the net magnetic charge, and the charges on the l.h.s, of (8) are no longer independent:

ie 2 [Qs,Q~] Q m is the monopole magnetic charge.

2qr2 Om •

(9)

356

O. Foda /

Wttten effect

An obvious consequence of (9) is that the monopole ground state cannot be a simultaneous eigenstate of chirality and electric charge [15]. Here, we would like to go one step further, and remark that (9) relates the ground-state charge to the vacuum angle 0. Our reasoning is as follows: regarding (9) as an operator insertion between suitable monopole states (as will be the case with all commutator relations from now on), Q~ and Q., are c-numbers, and one can use the Dirac quantization condition

eQm = 2~'n,

(10)

where n is an integer, to rewrite (9) conveniently as

[Q~,Q~] = - i e .

(11)

'/T

Next, though Q5 is ill-defined as an operator in Hilbert space, we may formally consider it as the generator of a one-parameter Lie group: U^(l). Then, from (11) we see that Qe is - up to a normalization of ¢ r / e - the conjugate variable to Qs: it parametrizes the UA(1 ) group manifold. On the other hand, we recall from Fujikawa's functional derivation of the chiral anomaly [16], that the response of the fermion functional measure to an infinitesimal chiral transformation is a shift in the coefficient of the F F term:

[d#] ~ [d#]exp[

ie2 f [ d x ] c t ( x ) P " ~ F ~ ( x ) ]

16~.2

(12)

Paying attention to the factor of 1 difference in the normalization of the coefficient -p,V of F F~, between (1) and (12), we see that the conjugate variable to Q5 is nothing but ~0 (in the normalization of (1)):

[Qs, ½0] = ih.

(13)

Setting h = 1 and subtracting (11) from (13) (treated as c-numbers) we obtain 0e

Qe= C-

2-~"

(14)

Fixing the arbitrary constant C from the semi-classical theory of dyons to be an integer multiple of e [17], we end up with Witten's formula (2). Clearly, for any given anomalous commutator one can go through the same exercise as above: evaluate the commutator in the background of a soliton that matches its surface term, and "read off" the analogue of the Witten effect.

357

O. Foda / Witten effect

3. A gravitational anomalous commutator Given the above remarks, we would like to obtain the gravitational analogue of the Witten effect. We consider massive Dirac fermions coupled to a gravitational background, that remains to be specified. Suppressing non-gravitational contributions and mass terms, the divergence of the chiral current is still non-vanishing [18]:

n.J,(x) RR

-'

192~ 2 ='

kR(x),

~ E ~,,~,,8K p.~,pa-t~ " aBP° ,

(15)

where R,,oo is the Riemann curvature tensor. This follows from evaluating the diagrams of figs. 1 and 2 (the second is required for gauge invariance), with an axial-vector vertex and two external gravitons, and enforcing general covariance. (For a review of gravitational as well as other anomalies, see [19] and references therein.) We start by deriving from (15) the anomalous commutator corresponding to (8). Consider the matrix element S, corresponding to figs. 1 and 2:

S = F~ ~*' +*=~(OlJ~(x)lg l, g z ) g ~ / 2 g } / 2 , 1

[2,,]3

f [ d 3 x ] ei, x ,

N,=[2 ]3/212ko]'/2, g, = g,(k,, e,),

(16)

kl+k 2

g~a

goF

Fig. 1. The anomalous triangle diagram.

358

O. Foda /

Witten effect

kl+ k 2

Fig. 2. The anomalous two-sided diagram.

where, without loss of generality, x 0 has been set to zero for later convenience, and g, is the external graviton of 4 - m o m e n t u m k,~ and polarization vector e,"~. Since we are interested in a n o m a l o u s equal-time c o m m u t a t o r s of currents, we wish to extract the O[1/k~0 ] contributions to S, in the BJL limit kl0 ~ oo. We will do this in two steps. Firstly, using the LSZ reduction formula on the external graviton gt, S - [ - ie~'"] F,~{~'+k~'Fyk'D~(OIT[J~(x)ho~(Y)]lgz)N~/2

,

(17)

where h o~ is a graviton field. In a harmonic (Lorentz) gauge [20], it satisfies the linearized field

t3~h~a(x)

= 16~rT~a,

T,a = Tea- ½g~aTVr,

(18)

where we set k = 1, and used unrationalized units. T~a is the fermion energym o m e n t u m tensor, and from now on a barred tensor A-aa will stand for the c o m b i n a t i o n Aaa - - ~1g_a # A ~y" Acting with Uy on the time-ordered operators, and using (18), S = [ - i e ~ '~]

Ff(*l+*')F~k.'(OlT[J~5(x)~(y)]lg2)N1/2 + sea-gulls.

(19)

T h e sea-gull terms ( c o m m u t a t o r s of J~ and the graviton field and its derivatives) are not s h o w n here since they do not contribute O [ 1 / k l o ] terms. Taking the BJL

359

O. Foda / Witten effect

limit of (19) [tr" ] Ff(*t+*2)Fvk'~---~o3(Yo)(Ol[J~5(x). T~,,,( y)]lgz)U}/2

SIkLo~ =

+ O[1/k~0 ] terms.

(20)

Secondly, following [18], S is evaluated explicitly to be

s=

GA =

+k2] aA

4~2~1 ~2 caph'rr~l~2 f01

f01-x

dx

x2y 2

dy [k21y(l_y)+k~x(l_x)+2kl.k2xy+m2]

.

(21)

Taking the k m ---, oo limit in (21), and anticipating that the index will be set to zero in the final expression (since we will be interested in current densities), which helps in dropping many terms, we obtain

SI ktO~o¢ p.-0

t_.

~o%pO~

t.Xl,.'~

144,/r 2~1 G2 CapX.rr~l" 2

® [ S a a k l ' k 2 - k ~ k ~ ] - ~ m 1 + O[1/k~o] "

(22)

Equating (20) and (22) gives r - ( ' , +':)F~L (01 [ J05(x), To.( y)]lg2>Nz ~/2

_ 144qr - t 2 .,O. t'xL"lS,akx'k2-kl~k"l'l ~2 ~oPkr~l~2

1 21

(23)

To extract the commutator from the matrix element (23) we notice that, up to normalization, it acts as an annihilation operator for the external graviton g2:

1

[2-1' _ 28-~ - i ~r2 eoox'k lxk2. [ 3 ~ , k x . k 2 - kBxk~]~oo(k2)N;, , ~,a(k2) = g(k2)e.oa(k2) + g * ( - k 2 ) e~'a(-k2).

(24)

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O. Foda / Witten effect

g* and g are creation and annihilation operators. The creation part is necessary to make the expression hermitian. It is obtained from the matrix element (g~, g2lJ, SI0), hence the extra factor of ½. Fourier transforming (24), and setting o = 0, we obtain ' -

-

288,n-2

-

•

°'

o:o,".]

-'

5761r 2 (25)

Integrating over x and y and using the derivative O~ x to write the result as a surface integral, we obtain [ Q , , b° ] _

i 576~r2 ~ [ d S ° ' ] ®[ 0 " O h ° " -

O'O"OOh°t~- O°r-lh'" + O°Ot~h~j](x ) .

(26)

Notice that upon integrating (25) once, one obtains an ostensibly vanishing expression due to contraction of two indices in t0px~ with a symmetric tensor. This is strictly true only if the metric exists everywhere, which is not true in cases of interest (namely the N U T metric). Our justification for using h~,~ in the intermediary steps is that the same situation holds verbatim in electrodynamics, where the corresponding derivation involves the photon field, though in the presence of monopoles the vector potential is not defined everywhere. (In other words, the non-vanishing result one obtains in either case upon applying Gauss' theorem twice is due to the string singularity. In the presence of a string, "the boundary of the boundary" is not identically vanishing, but is rather the infinitesimal circle that encloses the point where the string punctures the sphere at infinity.) Following the procedure in electrodynamics, what we should do next is reconstruct the full expression, corresponding to (26), in terms of the Riemann tensor and its derivatives, which is defined everywhere, and reduces to (26) in the weak-field approximation. This is done by adding all possible candidates with arbitrary coefficients, then fixing these by taking the weak-field limit and comparing with (26). The result is [Q5, P ~ ] - t ~ d~ [dSOO] 144¢r2 ZS2L - , D t o R o I .

(27)

Since the surface integral on the r.h.s, of (27) is on the 2-sphere at space-like infinity,

O. Foda / Witteneffect

361

one has to treat R~o as an asymptotic field. This requires performing a conformal transformation to "bring the infinity in", add a boundary, define an asymptotic manifold, and an operator that maps covariant tensor fields to asymptotic ones. (For a clear discussion of asymptotes in gravity see [21].) For the moment, we will assume that this can be done, and that R] in (27) can be properly defined as an asymptotic field at space-like infinity. Finally, since we are interested in the properties of the static ground state, one can consider a frame of reference where only ~0 = ½p0 :~ 0, then p0 = m, the groundstate mass, and we end up with

[Q,,m]

=

7~2~s2[dSP°]DlpR° ].

(28)

This is the desired expression, and what remains is to find a gravitational field for which the r.h.s, of (28) is non-vanishing.

4. A gravitational "monopole" Intuitively, if the gravitational analogue of electric charge is mass, then the analogue of a magnetic monopole (a solution of Maxwelrs equations with "dual" charge) would be a solution of the Einstein equations with "dual" o r " magnetic-type" mass. Such solutions do exist: these are the N U T [8] and related metrics [22]. In its non-self-dual lorentzian form, the N U T metric is ds 2 = U[dt + A d ~ ] 2 - U--'dr 2 - [r 2+ N 2 ] [ d 0 2 + sin20d~2], U=I

2[Mr+N2] [ r 2 + N 2] '

A=2N[1-cosO].

(29)

M and N are, respectively, the mass and NUT, or dual-mass parameters. The latter is the analogue of magnetic charge and is related to the non-trivial topology of the N U T metric: its second homology group is non-vanishing [23]. For a precise mathematical definition of dual-mass see [23], and for a more physical discussion see [24]. The above metric exhibits a wire singularity on the half-line 8 = ~ (the analogue of the Dirac string), which is removable, following Misner [10], by using different coordinate charts for the northern and southern hemispheres. However, this can be done consistently only if the time coordinate is compactified. Thus one can no longer discuss globally space-like hypersurfaces, which are necessary for our equaltime commutator to make sense. One way out would be to take advantage of the existence of global null surfaces [23], even in the presence of a compact time coordinate, and calculate light-cone

362

O. Foda / Witten effect

c o m m u t a t o r s and charges instead of the ordinary ones. Or rather, extract the corresponding light-cone commutator by taking the infinite-momentum limit on the intermediate states that are implicitly summed over in the equal-time c o m m u t a t o r (28), and turn ordinary charges into light-cone ones. However, we will not do this here since the light-cone approach is tailored for the compact-time lorentzian metric (29), which is not in itself interesting due to its acausal behavior. In fact, more interesting is the euclidean section of the metric due to its connection with KaluzaKlein monopoles [11], but where the light-cone approach will not apply. Alternatively, we consider a single coordinate patch, and following Bonner [9], regard the wire singularity as physical: a 1-dimensional source. In this framework, the N U T metric describes a mass M at the origin, attached to a semi-infinite rotating massless rod, of angular m o m e n t u m density N. Then, one can define global space-like hypersurfaces, go through the analysis of Ashtekar and Sen [25] and R a m a s w a m y and Sen [23], and transcribe their surface-integral expression for the dual-mass from null infinity to space-like infinity. This is

Nn c= ~----~2[dSP°]DtpS~I, S, = Ro~ - 6' 8 0~R ,

(30)

where n c is a suitable unit vector and all tensors are understood as asymptotic. Note that N in this section is 1 in [23] and the extra minus sign with respect to the expression in [23] is due to notational differences. Setting the index c = 0, and comparing (28) and (30), we obtain

[Q,, m] = ~-~N,

(31)

the c o m m u t a t o r of the "electric-type" mass and Q5 is proportional to the " m a g netic-type" mass. Finally, from (11) and (31) one can " r e a d off" the analogue of the Witten effect. Denoting m in (31) by the more suggestive M o , o, we end up with (3): 0 Mo,,o = Mo_o+ -i-~-~N.

(32)

In the presence of a non-vanishing O, the mass parameter of the N U T metric is shifted in proportion with O. 5. Discussion

One can further motivate the above result by pointing out more analogies with electrodynamics. Recalling the concept of duality rotations in the Q c - Qm plane [26], one can rephrase the Witten effect as an equivalence relation between the

O. Foda / Witten effect

363

0-term and a duality rotation (up to a normalization, since Qm remains invariant). However, a chiral transformation is equivalent to a shift in O. Thus in the presence of a monopole a chiral transformation on the massive fermions is equivalent to a duality rotation in the gauge sector. On the other hand, in gravity, the analogue of a duality rotation is a Geroch transformation [27]: a rotation in the mass-angular momentum potential plane, that in the case of the N U T metric is nothing but the M - N plane. The result of sect. 4 is that a chiral transformation is equivalent to a Geroch duality transformation. Furthermore, the integrand D(pS~I in (30) is proportional to the asymptotic dual Weyl tensor K ~ [21,23], which is the true analogue of the dual electromagnetic field strength tensor F"~. Comparing with (8), the analogy with electrodynamics is very compelling. Finally we turn to the connection with Kaluza-Klein (KK) monopoles. As we mentioned before, the N U T metric as it stands is not physically relevant: in Misner's treatment removing the singularity introduces a periodic time coordinate, while in Bonnet's picture rendering the semi-infinite source unobservable to the matter fields would probably require imposing an equivalent condition on the path integral that only histories periodic in time survive. However in K K theories compact dimensions appear naturally, and one can consider the euclidean section of the metric and identify its compact dimension with one of those. This results in the most studied K K soliton [11]. From the low-energy 4-dimensional viewpoint this soliton appears as a magnetic monopole, and it is interesting to study phenomena associated with ordinary monopoles in this new context. In [28] it was shown that the Callan-Rubakov [CR] effect, in the sense of baryon decay catalysis in the presence of the KK monopole, does not take place. However, the ordinary CR phenomenon is intimately related to the Witten effect, and we expect that the gravitational analogue of the former will be similarly related to that of the latter, and will involve momentum and angular momentum rather than ordinary charges (though, naturally, in the KK approach, these are related). It is my pleasure to thank Drs. A. Abouelsaood, A. Greenspoon, E. Grinberg, C. Mukku, K.S. Narain, R. Percacci and J. Zaneili for discussions and useful remarks, and in particular Prof. R. Jengo for instructive discussions on monopoles, and Dr. Amitabha Sen for patient explanations of his work on the T a u b - N U T solution. Finally, I would like to thank Professor Y.P. Yao for warm hospitality at the University of Michigan, Ann Arbor, where part of this work was done. I also thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO, for hospitality at the International Centre for Theoretical Physics, Trieste. References

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364

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Witten effect

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