Monopole core excitations and the Rubakov-Callan effect

Nuclear Physics B294 (1987) 925-960 North-Holland, Amsterdam

MONOPOLE CORE EXCITATIONS AND THE RUBAKOV-CALLAN EFFECT K. ISLER, C. SCHMID and C.A. TRUGENBERGER

Institut ffir Theoretische Physik, Eidgengssische Technisehe Hochschule, 8093 Zi#ich, Switzerland Received 6 April 1987

We consider the interaction of an SU(2) monopole and its core excitations with massless isodoublet fermions of low energy El. We reduce the theory to the J = 0 sector. The fluctuations are treated in a semiclassical expansion, and the isorotational zero-mode (dyonic excitation) is parametrized by a collective coordinate. We integrate the equations of motion for matrix elements to first order in v2= (e2/8"rr2)ND and Ef/MGu T in the core region and obtain an effective boundary condition for the currents outside the core. We compute the S-matrix and the space-time evolution in the Rubakov-Callan process (catalysis of baryon decay by magnetic monopoles in GUT's), and we study the mechanism of anomalous pair production. The solution shows that the previously neglected dyonic core excitations play an important role unless ~2 In( Mc;uT/lt ) >> 1, where ~ is the constituent mass scale.

1. Introduction

Rubakov [1] and Callan [2] predicted that magnetic monopoles in grand unified theories (GUT) catalyze proton decay with rates of the order of the unitarity limit. They considered the interaction of the J = 0 partial wave of massless fermions with the 't Hooft-Polyakov [3] monopole. This SU(2) monopole is embedded into a GUT

[4]. The mechanism of the catalysis reaction is tied to the following two crucial effects. First, a J = 0 fermion in the background of a 't Hooft-Polyakov monopole must necessarily make charge exchange (baryon-number exchange in GUT) since its radial spin is proportional to its charge (due to the extra angular momentum in the electromagnetic field) and since helicity is conserved (g = 2). The electric charge is transferred to the dyonic core excitation. The second effect is the chiral anomaly, which states that the divergence of the axial current is - e 2 E- B. Both the electric field of the dyonic core excitation and of the fermions produce anomalous (IzaQ51 = 2) pairs. The former makes the decay of the dyonic core excitation possible. These two effects have been discussed by Goldhaber [5] (charge exchange based on conservation law arguments) and Blaer, Christ and Tang [6] (pair production in the field of an external dyon). 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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K. Isler et al. / Rubakov-Callan effect

In ref. [1,2] Rubakov and Callan made the important assumption that dyonic core excitations are frozen for fermionic energies Ef much smaller than the dyon excitation energy e2M ( M is the GUT scale). However dyonic core excitations are not excluded energetically, since the lifetime of a dyon is To= O(e2M) -1 and A M D • ~'D -- 1; and their effect on anomalous pair production cannot be neglected, since AQ5 - e2fdtfrod3rE • B - (e2/rc)zD = O(1), where rc = 1 / M (see ref. [7]). "In this paper we include all core excitations in the J = 0 sector. For simplicity we work with the SU(2) model and massless fermions. Restricting to spherically symmetric configurations results in a U(1) theory on the half line [8]. In this dimensional reduction the original SU(2) gauge field decomposes into the U(1) gauge field and a charged Higgs, the original Higgs translates into a neutral scalar. We use this reduced theory as a starting point. We apply the standard method of soliton quantization [9], expanding fluctuations around the classical monopole solution in h. This corresponds to an expansion in the coupling constant e. To O(e °) the fluctuations obey the linear small oscillation equations. They have a zero mode because the monopole is not invariant under U(1) rotations. These connect static configurations of the same energy since the underlying theory has U(1) invariance. By introducing a collective coordinate a(t), which parametrizes this circle of degenerate configurations, the zero-mode is removed from the small ocillations. The momentum conjugateto a is an electric charge. This is how the monopole develops charged dyonic excitations. They have zero energy to O(e 0). For the other modes we find an energy gap of O(M). They are core vibrations and oscillating charge polarizations. The O(e °) equation for the fermions describes charge-exchange scattering and has been solved by Marciano and Muzinich [10] in the Prasad-Sommerfield [11] limit. Isospin conservation is guaranteed by the collective coordinate. To solve the core dynamics we take matrix elements of the operator equations in the Heisenberg picture in a state corresponding to an incoming fermion. We show that these equations for bosonic fields are still linear to first order in e. The bosonic matrix elements of O(e) determine the O(e 2) terms in the fermionic current. To compute these currents, we replace the Dirac equation by the anomaly equation and the continuity equation for the electric current. This is sufficient to determine currents, since j~' = 2e~"j~ as usual in I + 1 dimensions. The solution to first order in E J M in the core region gives an effective boundary condition for jr outside the core. It represents the integrated core dynamics. It is completely independent of the heavy bosonic modes, O(M), it only depends on the dyonic core excitation, O(e2M). However, the polarization modes contribute a non-negligible part to the total electric current inside the core. Outside the core the dynamics of low energy fermions (E~ << M) decouples from the heavy fluctuations (massive SU(2) gauge bosons). The equations of motion for the currents are linear [12] and are supplemented by the effective boundary condition outside the core, representing the integrated core dynamics. The solution

K. Isler et al. / Rubakov-Callan effect

927

describes the space-time evolution of an incoming pulse and the mechanism of anomalous pair production ( - e2f dt d3rE • B). Two competing contributions on two totally different scales add up to AQ~°t = 2. The first is anomalous pair production via the dyonic electric field and is concentrated in the core region. The second is the contribution to anomalous pair production from the fermionic electric field and is spread between the scales rc and r, = l / g , where g is the mass scale of the fermions. The parameter governing the relative strengths of the two effects is v 2 1 n ( M / g ) , where u 2 = (e2/8~r2)ND and N o is the number of fermion doublets. In the limit of neglecting r 2 1 n ( M / g ) , anomalous pair production is entirely due to the dyonic core excitation. In this case the space-time picture describes charge-exchange scattering followed by the exponential decay ('r o - 1 / e 2 M ) of the dyonic core excitation into a fermion pair. For Efcr> << 1 we do not resolve the scale ~o, and the microscopic mechanism near the core results in an effective charge conserving and helicity flip vertex at the core. By mixing different doublets this effective vertex can violate other fermionic quantum numbers (e.g. baryon number in a GUT). The fermionic contribution to anomalous pair creation becomes important only if v 2 1 n ( M / g ) >_.1. We derive the currents both in first order in r 2 1 n ( M / g ) and in the non-perturbative regime. The dyonic core excitation is frozen only for a pointlike monopole in the sense of u21n(M/tt) --->~ . In grand unified theories the phenomena described above lead to baryon number violating processes. We consider specifically the SU(5) model. The total cross J = o = ~r/k 2 (unitarity limit), and the expectasection for an incoming quark is fftot tion value for baryon number violation is I(AB>I = ½. For a fermion mass scale of /~- 1 GeV the contribution of the dyonic core excitation to the Rubakov-Callan effect is about 50%. Our paper is organized as follows. In sect. 2 we perform the dimensional reduction of the SU(2) model to the J = 0 sector. Sect. 3 is devoted to an analysis of the small fluctuation equations around the monopole solution and to the separation of the dyonic zero mode from the heavy modes. In sect. 4 we analyze the equations of motion to O(e 2) and we derive the effective boundary condition outside the core. In sect. 5 we study the dynamics of low energy fermions outside the core and the Rubakov-CaUan effect. We give a summary of this section at the end.

2. T h e J = 0 reduction

We consider the lagrangian for an SU(2) gauge theory with a Higgs isotriplet @ and isodoublets of massless fermions ~.

•£P= - ½ t r F . ~ F " ~ + t r D . @ D " @ - V ( e o ) + ~ ' y ¢ ( i O . - e A . ) ~ .

(2.1)

K. Islet et al. / Rubakov-Callan effect

928

We use the standard notation

A,

Ta = A a-

~'2'

ya ~ = (pa

D~=O~+ie[A~,qb],

2' F~=O~A~-O,A~+ie[A~,A~].

(2.2)

The symmetry is spontaneously broken down to U(1) by the Higgs potential

V(+)=I~

(

+a+a

__~_],

where M is the mass of the resulting massive gauge boson. This theory has the classical 't Hooft-Polyakov [3] monopole solution of mass O(M/a), a = e2/4~r, and magnetic charge - 1 / e . In the spherical gauge it can be written as

AT(r)

1 - K ( r ) ei~jPj, er

cb~(r) = H(r) P~, er

H(r)=O((Mr)2),

Mr << 1: K ( r ) - 1 = O((Mr)2), Mr >> 1:

P = -r, r

K ( r ) = O(e- M~),

H(r) - Mr = O ( e - M " ' ) .

(2.3)

M H is the mass of the Higgs particle, and we have used ~k / / e 2 = M ~ / 2 M 2. The total angular momentum J of a particle in the presence of the monopole is given in the spherical gauge by J = L + S + T,

(2.4)

where L = r Xp, S is the spin and T the isospin of the particle. In the spherical gauge the generator of unbroken U(1) transformations, which we identify with the electric charge, is given by 1"

q=P~.

(2.5)

Eqs. (2.4) and (2.5) imply that the magnetic charge of the monopole must be negative, since for an up-fermion e

We are interested in the interaction of low energy fermions with the monopole and the bosonic fluctuations of the gauge fields around the monopole solution. We

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K. lsler et al. / Rubakov-Callan effect

will consider only the J = 0 fermionic partial wave, since only fermions in this sector do not feel the centrifugal barrier and reach the core. Correspondingly we will consider the most general form of J = 0 gauge fields in the spherical gauge, neglecting the recoil of the monopole and the corresponding translational zero mode. We use the following abbreviations for invariant tensors 6~ = PiU,

87 = 6 , j _ p p j ,

e7 = gjkp~.

(2.6)

The subscript L and T refers to longitudinal and transverse relative to the radial direction. With these symbols the J = 0 gauge fields can be expressed by 1 + 1 dimensional fields [8]

A "i=e~,' l / e -

q~l(r, t) +6~ i-cp2(r , t) + 6"iaqrL t , t), r

A "°

P

= a°(r, t)P",

,l(r, t) pa.

~a

(2.7)

r

The inversion of the A ai equation is done using the orthogonality relations tr(SLeT) = 0 etc. and the normalization of 1 for ~L and of 2 for 8 T and e T. Under the unbroken U(1) gauge rotation e x p [ - i e O ( r , t)q] the field a~ = (a0, ar) transforms as a 1 + 1 dimensional U(1) gauge field, a~ = a , + 0,0.

(2.8)

It represents the neutral gauge field in the J = 0 sector. The field q0 = ~0a + iqo2 transforms as a 1 + 1 dimensional scalar field of charge + 1,

q~' = e-ie%p.

(2.9)

It represents the charged gauge field in the J 0 sector. The field ~ is a 1 + 1 dimensional neutral (gauge invariant) scalar field; it represents the original Higgs field, which was denoted by a capital ~. We define the 1 + 1 dimensional field strength f,v and the covariant derivative: =

f~, = O~a v - Opa,, D~qg=(O~+iea~)op

(/~= O, r ) .

(2.10)

The 3 + 1 dimensional non-abelian field strengths are expressed by the 1 + 1 dimensional fields as follows: Eai= ~*fOr- ~i

Bai =

( O0q° )2 r

8[,q021 + qo2 - 1 / e 2

r2

+

E~i ( O0qg)l

q- 8~ i

r

(D, qo)l -

r

~, (D, cP)2

-']- ET

r

(2.11)

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K. Isler et aL / Rubakov-Callan effect

The J = 0 fermions will be treated in the formalism of Jackiw and Rebbi [13]. The positive and negative chirality fields X + are

#-+ =

+ vs)

=

l(x+)

(2.12)

+x+

in the Dirac basis (3, 0 diagonal). X + are 2 × 2 matrices, where the first and second indices refer to spin and isospin respectively. After the transformation on the isospin index X ~ = ( M ±~.2)y~

(2.13)

one can drop indices and use matrix notation with • acting from the right. The J = 0 partial wave can be written 1

M ±= r---~[g+-(r,t) + i p ± ( r , t ) P . ' r ] .

(2.14)

One combines the functions g ± and p ± into a new spinor X

gh(r,t) xh(r,t) =

hp~(r, t)

,

h = -4-1,

(2.15)

with h denoting the helicity (equal to chirality for m = 0), and one defines the following set of 2-dimensional y-matrices ~ 0 = ,/.3,

~1 = i , r l ,

7 5 = ~ 0 ~ 1 = _ ,/.2.

(2.16)

Under the unbroken U(1) gauge rotation the new spinor X transforms as

( X h )' = e-ie°qxh, q~

(2.17)

5 -- ±h{ 2

(2.18)

The J = 0 partial wave of the fermionic SU(2) current j~a has a structure analogous to eq. (2.7): jia = ~{i--~

: u~ J1 -~- ~iTa

-~- ~ila

,

r2

_ .¢a jOa = l~]zo

!0 t~ = pa 2 "

(2.19)

The I + 1 dimensional currents and scalar densities can be expressed as bilinears in

K. Isler et al. / Rubakov-Callan effect

931

the spinors X. The neutral part of the vector current, i.e. the U(1) current, reduces for J = 0 to f ' = (j0, jr):

jr= ~_,yCh~*qxh"

(2.20)

h

The total U(1) charge is Q = 4~rfdrj °, because there is no 4~r in eq. (2.19) and in the J = 0 Maxwell equations below (eq. (2.27)). The charge changing parts of the vector current reduce for J = 0 to scalar and pseudoscalar densities 75

J, = ~ y ~ h i T x h , h

J = J1 +/J2 = }-" ~h ( _ ih)(½ + q ) x h .

(2.21)

h

J ( J + ) is an operator of charge +1 ( - 1 ) and therefore transforms under the unbroken U(1) group as J' = e-ie°J.

(2.22)

The axial current in the J = 0 sector reduces to ~o~5~

_

1

o

- - -~-~J5

,

1

~.[i.~5~ = - - "r^i r2J5 r

j~= ( js°, j~) = ~_.,~hh~,"Xh.

(2.23)

h

Using eqs. (2.16) and (2.18) we obtain the typical relation for 1 + 1 dimensional theories: j• = 2e~j~.

(2.24)

The reduced axial current j~, eq. (2.23), looks like a vector current, and the reduced vector current jr, eq. (2.20), if expressed by h and 75 according to eq. (2.18), looks like an axial current. Note that 75 does not refer to helicity (75 does), but to ingoing or outgoing states. Because of charge exchange in the core, q, ~75 and j r are not conserved, while h and j~' are conserved at the 1st quantized level.

K. Isler et al. / Rubakov-Callan effect

932

We will label by u and d the eigenvectors of q with the eigenvalues + ½ and 12" For J = 0 partial waves the extra radial angular momentum 8,r/2 = q must be cancelled out by the radial component of the spin Po/2. Therefore u °~t and d in are left-handed while u i" and d °ut are right-handed. The interaction lagrangian for the fermions &av = _ eA~J,, reduces for J = 0 to the form e

(4~r) - 1 . ~ f l ?

=

__

e a , j ~ + r (~P+J + ~J+)"

(2.25)

The original lagrangian (2.1) reduces for J = 0 to the effective 1 + 1 dimensional lagrangian

e2(1 )2

(4qr)-l~a= _ 1 2r r/Lv arj~j +(D~o)+(D~rp) - 2r 2 72-¢p+q~

e2

•

+ ½0~ O~rl - rTep+~rl2 - ~4r

+ ~h

M2r2 ) rl2 - _ _ e2

eh eh i~_e~q+i__r_~P(½_q)_i~_q~+(½+q)

]

X h.

(2.26)

We have obtained an effective 1 + 1 dimensional abelian gauge theory with a charged Higgs q~ (representing the charged gauge fields), a neutral Higgs ~7 (representing the original Higgs triplet) and fermion doublets. The operator field equations from the above lagrangian are

O~ (r2f~") = ieq~+D"q9 + e-I, ',J , 1 / e -- ep+cp )

DuD"~P - e2ep

2e

r2

e2

(2.27) e

+ 7 g712ep= -J'r

X(

M2r2) = 0

(2.28)

0 , 0 " ~ + -r-g-~+~rl + r--5~/ 7/2 - - e- 2

(2.29)

eh eh + 1 ] O+ie~q+--qg(1-q)---cp (~+q) Xh=0.

(2.30)

r

r

The right-hand side of the inhomogeneous Maxwell equation (2.27) is the conserved total electromagnetic current, which consists of a bosonic and a fermionic part. Different definitions for the electromagnetic field inside the core of a monopole have been proposed: 't Hooft's definition [3], F~~m= t r ( ~ F ~ ) - i t r ( ~ [ D ~ , D ~ ] ) ,

K. Isler et al. / Rubakov-Callaneffect

933

and another definition discussed by Goddard and Olive [14], F ~ = tr(~F~,) • eM-1. If one applies the latter to fluctuating fields, it does not reduce for r >> M -1 and r >> M~ 1 to the electromagnetic field strength of the standard Higgs mechanism. There we should have F~~m= O~A3~- O,A 3 in the string gauge, but this is not the projection of the non-abelian field tensor, F~. We restrict ourselves to the dynamics in the J 0 sector. We start with the J 0 lagrangian (2.26) and we note that it has a local U(1) invariance. According to the gauge principle there is a unique gauge potential and a unique U(1) curvature tensor f~,, which we have identified with the electric field strength. This agrees with the definition of 't Hooft, which reduces for J 0 and for the electric field to E i = ~aEia. With this definition the electromagnetic field equations in the J = 0 sector take the standard form, while any other definition of the electric field leads to much more complicated equations. In the J = 0 sector we have a reduced dynamics with U(1) electric fluctuations but no U(1) magnetic fluctuations. Note that leaving out f , , and a , in the lagrangian (2.26), we obtain a model with a global U(1) invariance, which gives the correct classical ground state, the 't Hooft-Polyakov monopole. The second definition, F ~ - ¢~F~a,, restricted to J = 0 and to the electric field differs from 't Hooft's definition by containing the factor f ( r ) = H ( r ) / M r . This factor is particularly unnatural for M H / M << 1 (and for M H = 0, i.e. for the Prasad-Sommerfield limit), because for r >> rc = M -1 the fermions are free except for electromagnetic forces, but the charges associated with fermions (according to the second definition) are r-dependent and non-local. In 't Hooft's definition the charge of a fermion with definite isospin projection I . ~ is r-independent, q = e l . ~. 't Hoofts definition introduces a singularity at the origin into the three-dimensional electric charge density of the dyon and the magnetic charge density, while the non-abelian currents are regular. For the discussion of regularity properties at the origin a separation into electromagnetic and charged (non-abelian) fields and currents is not useful. On the other hand for the discussion of the dynamics of the magnetic monopole and the fluctuations (with J = 0) the separation according to 't Hooft leads to great simplicity. =

=

=

3. Fluctuations around the monopole Our program is to perform the semiclassical expansion for the 1 + 1 dimensional abelian gauge theory at the end of sect. 2. Since our lagrangian has the scaling property e 2 ~ ( ~ ; e) = . ~ ( e ~ ; 1), the expansion in e is equivalent to the expansion in h. The classical solution is of order e -a. To order e ° we have the linear small oscillation equations. The solutions provide a complete basis in which the full fields are expanded in perturbation theory. In the classical limit (h--, 0 or alternatively e--, 0) the field equations for the boson fields do not contain the fermions, and for time independent fields with a

K. Isleret al. / Rubakov-Callaneffect

934

total electric charge Q = 0, eqs. (2.28) and (2.29) reduce to

K(r) qg(r) = - - ,

H(r) ~(r) =

,

e

e

a t = Xh = 0,

r28~K = KH 2 + K ( K 2 - 1), 1 _~2H(H2_(Mr)2) " r2O]H = 2K2H + -~

(3.1)

The solutions to these equations describe the 't Hooft-Polyakov monopole. In the Prasad-Sommerfield [11] limit, i.e. M ~ / M 2 ~ 0 with the boundary condition H ~ Mr for r ~ oo, the functions K and H are known explicitly Mr

K(r)

H(r)=MrcothMr-1.

sinh Mr '

(3.2)

For Q ~ 0 and time independent fields (Coulomb gauge) Gauss's law (2.27) reduces to the Julia-Zee [15] equation for a dyon

Q J(r) a°

4~r

--

,

r

a = O(1/e),

r2a~J - 2K2J = O.

(3.3)

J ( r ) is the Julia-Zee function and has the following asymptotic behaviour Mr<
lim

2)

Mr>>l:J(r)=Vr-1

J(r)/r =

O(M).

(3.4)

The Prasad-Sommerfield limit gives V = M and J(r) = H(r). To study quantum fluctuations around this classical solution we choose the Coulomb gauge, a r = 0. Since the classical monopole solution is not invariant under a global phase rotation, the small oscillation equation for ~02 has a zero frequency solution. Therefore we make the following ansatz for the fluctuation fields

-4- ¢Pfluct(r, t) , e

H(r) ~ ( r , t) -

e

+ 7/nuct(r, t ) .

(3.5)

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935

We choose the boundary condition for r ~ 0 (3.6)

Cpnu~t= O ( r ) •

The condition q02nm(0) = 0 gives a unique decomposition of ~0 into ~fluct and a. The condition ~01nuct(0)= O(r) is necessary to maintain finite magnetic energy f d r (e2/2r2)(1/e 2 - ~+q~)2. Finiteness of the Higgs gauge interaction energy, i.e. the term f d r ( e Z / r 2 ) ~ + ~ 2, implies the boundary condition */nu~t= O(r) for r ~ 0. The collective coordinate a parametrizes the zero-mode up to a residual time-dependent but r-independent gauge transformation. This is because we do not yet specify a boundary condition for a o. The momentum conjugate to a is:

O~O~+}

OL fdr{O..~Oq~ oT =

o--~ o--g+ o ~

oa

= 4~riefdr~+~9o~ = QB,

(3.7)

which is the total bosonic charge. We now insert the ansatz (3.5) into the equations of motion (2.27)-(2.30) and take the limit e-~ 0. This gives the small oscillation equations for ~, ,/, X where we drop the subscript "fluct": 2KEtio + 2KOocP2,

OrrEOrao

=

OorZOrao

= 2K'0rep

(3.9)

2 ,

o2K1

(Oo'- o2) + T j ~ 2 +

KOo;o= o,

02K 2K 2 ] 2KH (0o2- 02) + ---~- + r----5-jepl + - ~ / = 02H ( 0 0 1 - 0 2 ) + "~

(3.8)

M2

4KH

+-~-5-~-

7/+--~01=0,

0,

[ O-t- K~Se2ieaq]xh = o.

(3.10)

(3.11)

(3.12)

(3.13)

The gauge potential enters the equations only through the combination

d o = a o - Ooa

(3.14)

which is invariant under the residual gauge transformations. Gauss's law (3.8)

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K. Isler et al. / Rubakov-Callan effect

together with the boundary condition (3.6) and finiteness of the electric field at the origin implies fro(0, t) = O.

(3.15)

This will hold to all orders in e because all charge densities are zero at the origin. The equations of motion decouple in three sectors: first the gauge sector (fro, ~02) with Gauss' law, Amp6re's law, and the inhomogeneous Klein-Gordon equation for ~02, which is identical to the continuity equation for the bosonic electric current (divided by 2K). Only two of these three equations are independent; either Amp6re's law or the continuity equation can be derived from the two remaining equations. Second the sector (~1, ~/) with breathing modes of the monopole and Higgs cores. Third the Dirac equation describing the scattering of fermions from a classical monopole. The factor exp(2ieaq) appears in the Dirac equation even in the limit e ~ 0 since (ea) is an angular variable defined on [0, 2~r]. It guarantees the global gauge covariance of the Dirac equation. From now on we shall use the tilde to denote Fourier transformed fields --+oo f(r,~)= J_ f( r ,t)ei'°'dt.

(3.16)

First we briefly discuss the (q01, ~) sector. The equations of this sector describe coupled radial deformations ~01 and ~/ of the monopole and Higgs cores. In the general case M r i ~ 0 one can find lower bounds ~0min= O(M) for the energy eigenvalues of the Cpl and ~/fluctuations by taking the minima of the eigenvalues of the matrix-potential of equations (3.13), and (3.14). On the other hand, in the Prasad-Sommerfield limit, eq. (3.14) for the 71 fluctuation reduces for r >> 1/M to the Klein-Gordon equation for a massless field, i.e. O)min 0. However this is of no importance to us because the (~01, ~/) sector will anyhow completely decouple from the fermions in the order of e and (,o/M) which we will study in the next section. Moreover we shall usually not use the Prasad-Sommerfield limit. We now turn to the gauge sector (do, ep2) and to the separation of the isorotational zero mode and the core polarization modes with ¢0 = O(M). For zero frequency Amp&e's law (3.9) reduces to the equation ff2(r) = 0, while Gauss's law (3.8) reduces to the Julia-Zee equation for d o (eq. (3.3)). Therefore they have the following zero-mode solution =

~2(r, t) = 0,

Q J(r)

do(r, t) . . . . . 40r r

(3.17)

Q is the total charge of the solution. Since we consider fluctuations of O(e °) around

K. Isler et al. / Rubakov-Callaneffect

937

the classical monopole this solution has Q = O(1) and the charge number (radial component of isospin) is Z = Q / e = O(1/e). On the other hand the classical Julia-Zee solution is of O(1/e) with Q = O(1/e) and Z = O(1/e2). The physically interesting dyonic excitations have Z --- O(1), Q = O(e); they are still degenerate in energy with the classical solution to O(e°), and they have a o = 0 to O(e°). Nevertheless they can be labelled by their quantum number Z. To analyze the general case to 4:0 we combine Gauss's law (3.8) and Amp6re's law (3.9) to the following equation for the charge inside a sphere of radius r,

Q( r ) = -4~rr2 0~ao . OrK 2K 2 ] (0 2 - Off)+ 2--~--0~ + - 7 ]Q(r, t ) = 0 .

(3.18)

The boundary condition Q(r)= O(r 2) for r ~ 0 follows from Gauss's law (3.8), which tells us that the charge density o(r) vanishes linearly in r at the origin. Eq. (3.18) reduces asymptotically for r ~ o¢ to

[ O ? - to2 - 20( M)O~]O(r, ¢o) = O

(3.19)

with the solutions to=0: Q ( r , to) = Q + O ( e - 2 g ' ) , to*O: Q(r, to) = O(eXr), = -M+

(M 2-

002)1/2.

(3.20)

From Gauss's law (3.8) and the boundary condition that ~2(r, to) should not diverge at infinity it follows that the + sign in front of the square root in eq. (3.20) for X is excluded for to < M. Eq. (3.20) shows then that the total charge of all solutions with to 4:0 is zero. Only the zero mode to = 0 carries a total charge. This fact follows also directly from total charge conservation

0(to) = 0

(3.21)

and because the charge density to O(e °) is linear in the bosonic fluctuation fields. By introducing the auxiliary field ~p(r, t) = Q(r, t)/K(r), eq. (3.18) becomes (0o2 - 0 / ) ¢ +

2+1

V(r)¢=O,

K. Isler et al. / Rubakov-Callan effect

938

The boundary conditions for the field q~ follow from the asymptotic behaviour of Q(r). r--*0: ~ ( r , ~0) = O(r2), r~:

~ ( r , 0 ) = O(QeMr),

f~(r,~o < M)=O(e-(M2-'~2)X/2rl/2), q~(r, ~0>

Ml=O(e+-i(°~2-~t2)l/2r'/2).

(3.23)

The zero mode is a solution of eq. (3.22) with exponentially increasing 4. For ~0 4: 0, eq. (3.22) with the boundary conditions (3.23) reduces to a radial SchrSdinger problem with potential V(r). In the Prasad-Sommerfield limit the potential V(r) approaches its asymptotic value M 2 like 1/r and therefore admits an infinity of bound states as in the Coulomb case (see also Sonoda [16]). Our numerical analysis showed that the number of bound states decreases with increasing M H. The minimum of the potential is smallest in the Prasad-Sommerfield limit where its value is V(2.609rc) = 0.56M 2. For the lowest energy eigenvalue ~0min of + we get the following lower bound ~0min > 0.78M.

(3.24)

This is a lower bound for the excitation energies of gauge sector modes with ~04: 0. These modes are very heavy compared to physical fermion energies, if one takes the G U T scale for M. Since they have no net charge they describe oscillating electric polarizations of the monopole core. We now determine a 0 for the heavy (H) modes. We set a~(0, t) = 0.

(3.25)

It then follows from (3.14) and (3.15) that the heavy modes have Ooa = 0 and decouple from a. The usual Coulomb gauge boundary condition, a0H(m, t ) = 0, would lead to a coupling, because at least for the lowest mode with ~0 4: 0, d0(m, t) - d0(0, t) g: 0

(3.26)

as can be seen from the auxiliary field +. Each mode q02 has an a0H which is a linear functional

a~(r, t) = where

f dr'G(r,r ' ) 2 g ( r ' )

Oo%(r', t),

(3.27)

G(r, r') is the Green function which solves (--Orr20r + 2 K 2 ) G ( r , r ' ) = 3 ( r - r ' ) ,

r20~G(r,r')[~==O.

(3.28)

The boundary condition follows because the heavy modes t~2 have no charge.

939

K. lsler et al. / Rubakov-Callan effect

The modes ~1, ~2, 'q serve as a basis to expand the full fluctuation fields linearly. But because (3.27) is linear, there is a cancellation in Gauss's law: a part of the full a o which is defined by (3.27)just drops out against 2 K Oo~O2. We therefore separate: a 0 = a0R + a0R .

(3.29)

The remaining part in a 0, a0R, is then defined by inversion of Gauss's law. To do this uniquely we must specify a boundary condition. We choose a R ( ~ , t) = O.

(3.30)

This boundary condition is necessary if the equations of motion are to be derived from a lagrangian. If we would allow variations of a0R(oo, t), the the variation of the term fdr(r~/2)(Srar~) 2 in the lagrangian would imply r28raRI~ = 0, i.e. we would be in a chargeless sector. The boundary condition (3.30) fixes the Coulomb gauge completely and determines a(t). To order e ° this leads to the zero-mode solution (3.17)

( J(r) l aRo(r,t)=Ooa 1-- Vr ]'

Qv O°a= 4rr '

aon = 0 .

(3.31)

With the choice (3.30) the dyon appears as an isorotational zero mode. Note that a ( t ) parametrizes both the dyonic zero mode and the time-independent global gauge rotation mode. To order e ° 80a is proportional to the dyonic charge. This is not true in higher orders in e, where other charges are present and the inversion of Gauss's law for aoR looks different. From the remarks after eq. (3.17) about the order of the dyon charge it is clear that Q is of O(e) in quantum theory. This means that a0R is of O(e)~while a ~ = O(e°). We could now write down the full lagrangian and go to the hamiltonian by expressing a0n with (3.27) completely in terms of 3o¢p2 and a0R by inversion of Gauss's law. Since we are interested in solving operator equations, we shall later work with the total a0, resp. ff0. Finally we turn to the fermionic sector. The Dirac equation in order e ° describes the scattering of massless, isodoublet fermions off the monopole core. The factor exp(2ieaq) guarantees gauge covariance and isospin conservation. Since the commutator [(eSoa),(ea)] = O ( e 2 ) ~ O in order e °, we consider (ea) in a diagonal basis and we treat a as a label

X~a) ~ eieaqxh"

(3.32)

This Dirac equation has been solved explicitly by Marciano and Muzinich [10] in

940

K. Isleret al. / Rubakov-Callaneffect

the Prasad-Sommerfield limit. We are going to solve this equation in the general case to first order in (cor). This means that our solution is a good approximation in the region where (cor) 2 can be neglected compared to 1. The Dirac equation for the upper and lower components X~~) and X~~) of the spinor is (we drop the helicity index):

- (cor)]'(~") + (rOr + K ) ~ ('~)= O,

(3.33)

( cor ) 5"((2") + (tO r -- g ))(~ a) = O.

(3.34)

Regular boundary conditions at the origin for the components ~") and ~(2") of the spinor exclude a cancellation in (rOr + K Jh2 ~'3(-), since this would imply ~ (fl) = O(1/r) for r ~ 0. Therefore eq. (3.33) yields

~(fl)= O(cor)~i ") .

(3.35)

Inserting this in eq. (3.4) and neglecting (cor) 2 compared to 1 we obtain the following solution

~")= A(co)exp[- fr °¢ K(r')

dr']

~(fl)=coA(co)exp [fr ° o Kr'( f ) dr' ]for e x p(- 2

(3.36)

fr °°K(r'') r"

dr" ] dr'.

(3.37)

The Prasad-Sommerfield limit of this solution corresponds to the first order in (cor) of the Marciano and Muzinich solution. Using eqs. (3.35), (2.20) and (2.21) we find the following useful relationships between fermionic currents and scalar densities in order e°:

•(r, co) <~O(cor) f(r, co),

(3.38)

~(")(r, co) <~O(cor)fr(r, ¢o),

(3.39)

J~(")(r, to) ~
(3.4o)

We have introduced the gauge invariant scalar densities

j(~) =

-h i~5 e 2 i e a q x h ,

~X -T

(3.41)

(3.42) h

K. Isler et al. / Rubakov-Callan effect

941

With this notation the interaction lagrangian for the fermions is manifestly gauge invariant (4¢r)-1..~ ( X )

2K

=

--

__j(ot).

(3.43)

r

Note that no index (a) is necessary on j0 and continuity equation for the U(1) current is

jr since they are gauge invariant. The

2K

O.j ~= - - - J ~ ) .

(3.44)

F

4. Low energy core dynamics to first order in e 2

In this section we solve for the fermionic current in the core region to first order in e 2 and ~0rc, where rc - 1/M is the core radius. We consider appropriate matrix elements of the operator equations in the Heisenberg picture. As an initial condition we take an incoming fermion and a monopole. We show that then the equations of motion for the matrix elements of boson fields are still linear to first order in e. They further simplify since we consider only first order in ~0rc. For the fermions we only consider bilinear operators; as equations of motion it is sufficient to consider the anomaly equation and the continuity equation for the current. By integrating these equations in the core to first order in e 2 and 0~rc we obtain an effective boundary condition for the fermionic current outside the core. This is the main result of this section. To study the O(e) of the semiclassical expansion we insert the ansatz (3.5) into the equations of motion (2.27)-(2.30) and we neglect terms of O(e2). The resulting equations are listed in the appendix. The three sectors (gauge sector, (cpa, ~/) sector and fermionic sector) are now coupled to one another and the equations contain bilinear terms where an explicit coupling constant e appears. We now show that the bilinear terms in the matrix elements of eqs. (A.1)-(A.5) can be dropped. In eqs. (A.1)-(A.3) they are of the form e(0]~l(OfP2(0)]0 ) where the subscript "(0)" means zeroth order in e. This is because of our special choice of the initial state. It is a direct product of the vacuum of the bosonic fluctuations with a fermionic state. The latter is a superposition of states with different a such that it corresponds to a neutral monopole. The zeroth order bosonic operators are independent creation and annihilation operators for ~ 4:0 modes, and therefore the above matrix elements vanish. In eqs. (A.4) and (A.5) also bilinears of the form e (0 Icp2(o)lO> appear. These do not vanish but are time-independent. In the Fourier-transformed version they vanish for ~ :# 0. As we shall see, the modes of (A.4) and (A.5) (i.e. Cpl and 7/) do not couple to the fermionic current in next order and are therefore irrelevant. Eqs. (A.1)-(A.5) therefore reduce to (we drop the symbols ( > of matrix

K. Isler et al. / Rubakov-Callan effect

942 elements):

0 : 2 O ~ a o - 2K2ao + 2iogKgp 2 = -ej(~),

_iogr2Orao

2 K'~rq52 = ej(o) 7,-

--

(4.1) (4.2)

,

0~2K1_

e

z,~

(-o92-OrZ ) + --~----jrP2 - iogKao = rJ~(o), ( -

092-°))+

0~2K

2HK

2K 2 ]

+--7- ¢ 1 + -r-2" 71 =

O~2H

M~ H 2 ]

(--o92 -- 02) + ---H-- + ~M- 7 -rT

e . -rJ " 1(0),

(4.3)

(4.4)

HK ~ + 4 - ~ -¢1 = O.

(4.5)

The zeroth order fermionic current and scalar densities act as source for the bosonic matrix elements to first order in e, because ~B(0)) = 0 for a generic boson operator B. We note that eqs. (4.1)-(4.3) are not independent. Since we work in Coulomb gauge (a r = 0), Amprre's law (4.2) should be regarded as derived from (4.1) and (4.3) via the continuity equation (3.44). We now deduce the simplifications in the low energy core dynamics, specificly to first order in o9r¢ where o9 is the frequency of the solution and 0 < o9 ~<0(E 0. For this we analyze eqs. (4.1) and (4.2) in the core region re with o9rc << 1. For the fermionic current we use the normalization of one incoming fermion

j(7~o)(r, o9) = O(1).

(4.6)

The orders of magnitude of the fermionic densities follow from eqs. (3.38)-(3.40).

jL=o(o9rc), &,cO(ogre),

(4.7)

(The equality for f0°) may be checked from conservation of the axial current in order O(e°).) As we have shown in sect. 3 the homogeneous system of Gauss's and Amprre's law has no solution for 0 < o9 << M. Taking account of the boundary condition at r = 0 we deduce from Amprre's law (4.2) the order of magnitude of the solution driven by the fermions

ao(r, o9) = O(e/o9rc) ,

(pr(r, o9) = O(erc).

(4.8)

Combining the first equalities in (4.7) and (4.8) we see that the terms ej(°) and 2io9K~2 may be neglected in Gauss's law (4.1). This reduces in first order in ¢0rc to the Julia-Zee equation

OrrZOrao - 2KZao = 0,

(4.9)

K. Isler et al. / Rubakov-Callan effect

943

with the solution

J(r) a0(r , co) = e), (o~)---~r '

(4.10)

where X(o~) is arbitrary and of order (o~rc)-1. The last step is the derivation of the boundary condition for the fermionic current fr in a boundary region defined by r c <<

rb < < 1 / ~ .

(4.11)

Using eq. (4.10) in Amp~re's law (4.2) we derive j("-~))(rb,t0)=--i6o2~(o~)

r OrG

rb = --i

)t(¢o).

(4.12)

This gives the boundary condition OrJ(o)(rb, ..-r ~ o ) - 0 for the fermionic current in zeroth order in e. It is the same charge exchange (CEX) and helicity conserving boundary condition as the one used by Rubakov and Callan in their work. This is most easily seen by inserting OrJ(o) rr lrb = 0 in eq. (3.44) and using eq. (2.24)

f°)rb= 1L°)r=0. Jio)

(4.13)

2 Jio)

In order O(e °) helicity is conserved for Dirac fermions (g = 2) interacting with the classical 't Hooft-Polyakov monopole. For J = 0 fermions this implies, because of S - P = TP = - 2q, non-conservation of electric charge in the monopole core. In O(e 0) the mass difference for charged isorotational excitations ( - e 2M) is negligible and the fermion energy in the charge-exchange process is conserved. The dyon produced in such a process is stable. This picture is modified in order O(e 2) by the chiral anomaly, which is the second crucial fact in the Rubakov-Callan effect. It causes helicity breaking and therefore modifies the effective boundary condition on .L. The axial current of each massless fermion doublet satisfies separately the following anomaly equation: --e 2

O~(~/y~3,5~k) = l-~2%~ootrF~FO°

e 2 =

4~r2Ea.B a"

(4.14)

Using eqs. (2.11), (2.23) and (2.24) we project this equation onto the J = 0 sector. Summing over N D fermion doublets and taking the lowest order in e we obtain NDe

Ooj r+ Orj °= 8~r2 ( 0r[ (K 2 - 1)ao] + 2 OrKOoeP2}.

(4.15)

944

K. lsler et aL / Rubakov-Callaneffect

The last term in this equation can be neglected in first order in tor~ because of eq. (4.8). The continuity equation for the fermionic current is obtained from the Dirac equation (A.6). 2K

Ooj ° + Orj r + - - J ~ = -2eCPl J~ + 2e~P2J~. r

r

r

(4.16)

Outside the core (r >> re) the two terms on the right-hand side of this equation describe the interaction of the fermions with the heavy charged vector bosons of the theory. The decoupling theorem [17] implies that for ¢0 << M these heavy fields decouple from matrix elements involving only external fermions. For r >> rc we neglect therefore the two terms on the right-hand side of eq. (4.16). The fermionic electric charge is conserved and the continuity equation reduces outside the core to

O~j ~ = O. Subtracting the space derivative of the continuity equation (4.16) from the time derivative of the anomaly equation (4.15) we get a "wave equation" for the current

jr,

(_toz_o~)?=Uoe [2Kr] 8¢r----T Or[-io~(K 2 - l l a 0 ] + Or~--ZJ~ ) + Or[ e--zJ~,o) ) - Or(2e---/-J~,o) ) .

(4.17)

Taking account of the boundary condition fr(0, ¢0) = 0 we neglect ¢02 compared to 0~ in first order in (~0r~) and integrate the remaining equation from 0 to rb; ND e 2

orjr(rb, ~o)

8qr-----7-i¢0X(o~).

(4.18)

Here we have chosen r b outside the 1/r tail of the dyonic gauge potential, i.e. we rc//r b compared to 1. Combining eq. (4.18) with eq. (4.12) gives the boundary condition for fr in O(e 2) and first order in torc.

neglect

OrJr

jr

rb

=/''2V'

/)2 -

NDe2 8~ 2 .

(4.19)

We recall that the Prasad-Sommerfield limit gives V = M. The boundary condition (4.19) determines the dynamics of the fermionic current in the region outside the core (r >> rc). This will be the object of sect. 5. The derivation of this boundary condition shows that the fields ~1 and ~ are negligible for the dynamics of low energy (Ef << M) fermions interacting with the monopole. From eqs. (4.4) and (4.7) we deduce that the deformation ecpl of the core

K. lsler et al. / Rubakov-Callaneffect

945

function induced by the fermions is of order O(e2(¢orc) 2) and therefore negligible to first order in o~rc. To this order the core function is rigid in the radial direction in the ¢plep2-plane. The only relevant degrees of freedom are the tangential ones corresponding to the dyonic core excitation (a) and to oscillating electric polarizations (¢P2)- The bosonic polarization current j~ = 2K 0re& is in fact not negligible in Amp&e's law (4.2). Neglecting it would lead to the following continuity equation for the fermionic current in zeroth order in e and first order in (¢0r~)

•r

- - 2 frj(ro) (r')

0oj(~) + 0r J(0)= z~, J0

7

dr'.

(4.20)

This equation is wrong and this can be checked in the Prasad-Sommerfield limit where the current j(~) is known explicitly. Therefore the field cp2 is necessary inside the core to guarantee the right continuity equation for the fermionic current. However it has no effect on the boundary condition for jr since it drops out in the integration over the core. The dynamics of the fermionic current outside the core is therefore completely unaffected by the fields ~1 and q92. Only the dyon degree of freedom contributes to modify the boundary condition to zeroth order in e. A system of equations consisting of Gauss's law (4.1), Amp~re's law (4.2) and the anomaly equation (4.15) has already been considered by Sonoda [12]. He neglects completely cp2 and has therefore inconsistent equations inside the core. Upon integration over the core however these equations lead to the right boundary condition (4.19). Sonoda performed a numerical analysis of the equations and computed the life time of the dyon. In the last part of this section we construct an effective theory for the interacting monopole fermion system. This is a simplified theory which reproduces to first order in e 2 and E f / M the correct effective boundary condition representing the integrated core dynamics, eq. (4.19). The theory will not describe the right local dynamics of the fermions inside the core. We have shown that to first order in e 2 and E f / M the effective boundary condition for j r outside the core is independent of all heavy fluctuations. This suggests that we can construct the effective lagrangian by neglecting all heavy fluctuations and keeping only the dyon degree of freedom. The result is a lagrangian describing the interaction of massless isodoublet fermions in the background of the monopole with one quantum mechanical degree of freedom, the dyon degree of freedom, (4¢r) 1Aaeff = ½r2(Orao) 2 + K2(a0 - Ooa) 2 K

+ ~h ~(h [ iO-- eao~,°q + i-~Se2ie~q]xh .

(4.21)

By neglecting ¢P2 we give up local charge conservation inside the core. In the following we show that this effective lagrangian leads to the correct boundary

946

K. Isler et al. / Rubakov-Callan effect

condition for fr outside the core, and we construct the effective hamiltonian. We follow the procedure explained in sect. 3. Since we neglect the heavy fluctuation q02 the gauge potential a0H in eq. (3.29) is vanishing and we are left only with a0R with the Coulomb boundary condition a0R(~, t ) = 0. This fixes the Coulomb gauge completely and makes the inversion of Gauss's law unique, -

O,r2Ora~ = epB + e.0 ,,j , epB = -- 2 K 2 ( a ~ -

Ooa) .

(4.22)

The Green function for the inversion of Gauss's law is determined by

(--Orr20r + 2K2)D(r,r ') = 6 ( r - - r'),

(4.23)

D ( ~ , r') = 0.

(4.24)

The solution of eq. (4.23) with the boundary condition (4.24) is given by

, , ,J(r') D(r, r') = O(r- r')g~r) Vr' + 0 ( r ' - r)L_~#g(r,)

(4.25)

J(r)/Vr is the solution of the homogeneous eq. (4.23) which is regular at r = 0 and is normalized to be 1 at infinity, g(r) is a solution of the homogeneous eq. (4.23) which is singular at r = 0 and whose asymptotic form at infinity is l/r, r ~ O: J ( r ) / V r = O(Vr),

g(r) = 0 ( 1 / r 2 ) ,

r --->~: J ( r ) / V r ~ 1 - 1/Vr,

g(r) --->1/r.

(4.26)

Integrating eq. (4.23) and combining the result with eqs. (4.25) and (4.26) leads to

jf Z K 2 t r ,~D ~(r , r , ) d r , = l _

J(r)

-Ff-r

(4.27)

This makes the inversion of Gauss's law (4.22) particularly simple

r ) ) + fD(r,r,)ejO(r,,t)dr," a~(r, t) = Ooa( 1 - J (Vr

(4.28)

The momentum conjugate to a is the dyon charge QD 3L 4¢r Q D = cg---d= -4¢r f 2Kz( aR° - O°a)2 dr= -O°av

+ 4 ~ r f d r (\J ( Vr r ) - 1) ej ° (4.29)

947

K. Isler et aL / Rubakov-Callan effect

and the equation of motion for a expresses total charge conservation

OoQD+ Oofej°dr = O.

(4.30)

In the derivation of eq. (4.30) we have used the continuity equation (3.44) following from the Dirac equation. Compared with the O(e °) case in eq. (3.31) the fermions induce an additional term on the right-hand side of eqs. (4.28) and (4.29). In this case Ooa is no more proportional to the dyonic charge and has no special physical meaning. The relevant quantity which does not change its meaning in different orders in e is the conjugate momentum to a, which is always the total bosonic charge. The dyonic charge density has no more a rigid r-dependence since it depends on the details of the fermionic charge density. However to 1st order in rc the corrections due to fermions can be neglected. This follows in the effective theory from the form of the Green function D(r, r'), eq. (4.25). Neglecting the fermionic charge density in the core region and using the anomaly equation and the continuity equation gives the correct boundary condition (4.19) for the current fr at r b. We derive the effective hamiltonian by expressing a o as a function of the dynamical variables via eq. (4.28):

Heff = ~V

01 +

J(r) ) eQDfdr 1 + Vr j°(r, t)

+ 2~re2ff drdr'j°(r, t)£)(r,r')j°(r't) - fdr~y(h[i~lOr+iK(rr)~seZie"q ] X h , h

D(r,r')=D(r,r')+V

Vr

]~ Vr'

1 .

(4.31)

As expected, in this order in e the isotopic degeneracy is lifted and the isorotational spectrum of the monopole appears. It consists of the charged (dyonic) excitations with QD = O(e). Since a is an angular variable defined on [0, 2~r/e], its conjugate momentum is quantized in steps of e

OD=ne,

n = 0, 5:1, _+2,....

(4.32)

The excitation energy for a dyon is e 2

AE = V - 8qr'

(4.33)

948

K. Isler et al. / Rubakov-Callaneffect

where V = O ( M ) . The moment of inertia for the isorotation is 0 = 41r/e2V. In order O(e °) the moment of inertia becomes infinite and therefore the rotational spectrum collapses to a point, the classical monopole. Since we have neglected qq and ¢P2 we do not see anymore the vibrational levels and the vibrational-vibrational couplings appearing in first order in e. The first three terms in the effective hamiltonian (4.31) are the Coulomb interaction energies. Note that the modified Green function D(r, r') reduces for r, r' >> rc to the usual charged Coulomb Green function r'r'>>rc'

1 L)(r'r')=O(r-r')r+O(r'-r)--'r'

1

(4.34)

The last term in (4.31) is the charge-exchange term. From the canonical commutation relations it follows that [QD,e ±ie~] = +__e+-iea,

(4.35)

i.e. exp(+_ iea) are dyon raising and lowering operators. 5. Fermion dynamics outside the core and the Rubakov-Callan effect

In this section we study the dynamics of the fermionic currents f f and j~ in the region r >> rc = 1 / M . We construct the S-matrix for the Rubakov-Callan effect, compute the life time of the dyonic core excitation and study the mechanism of anomalous pair production. We consider the dynamics of low energy (El << M) fermions interacting with the dyonic core excitation. As we have shown in sect. 4 we can neglect the interaction with the heavy non-abelian fields q01 and q~2 outside the core. Therefore we consider the following system of equations for the matrix elements of the current j r in the region r >> rc. Ooj ° + O~j ~= O,

(5.1) v2

ND e 2

ÙOff+ Orj ° = -- - - O r a o , e

v 2=

Oor2Orao = ej r.

8"F 2 '

(5.2) (5.3)

Subtracting the time derivative of the anomaly equation (5.2) from the space derivative of the continuity equation (5.1) and using Amp6re's law (5.3) leads to the following Bessel equation for the current fr: 3~r + co2 - ~

j>(r, w) = 0.

(5.4)

949

K. lsler et al. / Rubakov-Callan effect

The solution of this equation is expressed in terms of Bessel and N e u m a n n functions of the order ½ + u 2 for I,2 << 1. j ( r , to) =

1 1/2 [a(to)J,2+~2(tor ) + b(oo)N~+~2(tor)] (~tor~)

(5.5)

In the regions tor << 1 and tor >/1, respectively, these functions simplify to (½torqr)l/2jt2+u2(6or)=(o~r)l+u2 ,

sin(tor - 11,2~r),

(5.6)

(½tomr)l/ZN,2 +~2(tor)= -(tor) -~ ,

- cos(~0r- ½g2~r).

(5.7)

The ratio (a/b) is determined by comparing the logarithmic derivative of eq. (5.5) at r b with the effective boundary condition determined from the core dynamics, eq. (4.19). a(to) b(to)

V

= -- P 2 - - (torb)-2v2

(5.8)

to

We work to first order in p2 but we do not neglect corrections of O(u21n torb) since we shall consider incoming pulses centered around o~= 0. Different choices for r b make no difference for (a/b) to first order in u2. In the region rc << r << 1/to the fermionic current and charge density (computed from the current via the continuity equation) reduce to

-

[

_ ~j,~r~_ro_j= _ia(to)(tor)~2[l

(5.9)

_

~1

,

(5.10)

with a(to) arbitrary. Since V = O(M) the contribution of the Neumann function (second term in the square brackets) is negligible in eq. (5.10) for f0 and j~r but not in eq. (5.9) for fr and j~. The chiral charge produced during the whole process inside a sphere of radius r will be denoted by zlQs(r ), AQs(r) =

fdt4crj~(r,t)=4~r~r(r,~o=O).

(5.11)

Since only to = 0 matters, eq. (5.10) is valid for r b < r < o¢. Therefore the r-depen-

K. Isler et al. / Rubakov-Callan effect

950

dence of AQ5 is

AQs(r ) - r

lp 2

.

(5.12)

For AQs(r ) fixed at some r the chiral charge produced over all space, AQ5(r ~ oo), diverges with the power law r ~2. This divergence is cut off at the inverse fermion mass scale r. - 1//~, with/~ - inf. For r >/% we cut off the chiral charge production by setting / , 2 = 0 in the Bessel equation; this gives a free field theory for r >/%. We normalize the currents to one incoming d L-quark for r >/%, 'r 4~ryin(r~,0 ) = ½. fdtn~rdin(r,,t)= .'r

(5.13)

The incoming current for r >/r~ is that combination of jr and j0 which depends only on (r + t), since both j r and j0 are superpositions of free incoming and outgoing waves: jir = l ( j ~ _ j o ) . (5.14) Charge conservation requires f d tj r(r, t) = fr(r, 0) = 0. The normalization condition (5.13) implies therefore f°(%,O) = -1/4~r. Using eq. (5.10) we find i

p2

lim 00"2a(~o) = - -4--~(#)

(5.15)

6o ---~0

With this normalization the chiral charge produced inside r is

AQs(r ) = 2(/xr) "2,

r b ~
(5.16)

and the total production is AQ~°t= 2.

(5.17)

The anomalous pair production is proportional to fdtJd3rE • B. The B field is provided uniquely by the magnetic monopole whereas for the E field we have two sources: the dyonic core excitation and the fermions. We define AQD(r) and z~QV(r) to be the chiral charge produced by the dyonic core excitation and by the fermions respectively. Since the fermionic charge density is negligible compared to the bosonic charge density for r < r b we have

AQs(r ) = AQD(r),

r <

rb.

(5.18)

We choose the boundary radius r b outside the 1/r tail of the dyonic gauge potential. Therefore there is no r-dependence in AQ D for r > r b

AQD(r) = AQ D = 2(/~rb) ~2, AQ~/AQts°t= (/Xrb)u2.

rb<~r , (5.19)

The mechanism of anomalous pair production is governed by the new parameter ~21n(M//x). We consider an expansion of AQ D and AQ v in this parameter, and we

951

K. lsler et al. / Rubakov-Callan effect

neglect contributions of O(v 2) to AQ~ and AQ~. We distinguish the following 3 regimes: (i) ~21n(M/l~) << 1 and neglected. In this case we have (/~r) ~2= 1 for r b < r. Anomalous pair production is entirely due to the dyonic core excitation and is concentrated in the region of the core and its 1/r tail. AQs° = 2 although the currents are of O(p2), since the life time of the dyonic core excitation is rD = O(1/~,2M). This situation has been considered by Callan in ref. [7] and by two of us in ref. [18]. (ii) 1,21n(M / ~t) <_1 and included to 1st order. In this case the contribution of the fermionic electric field to anomalous pair production is uniformly distributed between r b and r~ on a logarithmic scale

OAQ~(r) 0 lnr

= 21' 2 ,

r b < r ~
(5.20)

Because of ~,2 << 1, anomalous pair production via the fermionic electric field is important only for a cutoff radius many orders of magnitude larger than the monopole core radius. (iii) u21n(M/tO arbitrary (e.g. o0). In this case the currents contain arbitrary powers of 1,2. For 1,21n(M/#) >> 1 anomalous pair production is mainly due to the fermionic electric field. The dyonic core excitation is frozen only for the pointlike monopole in the sense of p21n(M/bt) ~ oo. This is the limit considered by Rubakov [1] and in the earlier works of Callan [2]. In regime (i) (neglecting corrections of o(uZln(M/~t))) the space-time evolution of an incoming pulse becomes particularly simple. Since in this case (/xrb) ~z= 1 we can neglect ~,2 in the Bessel and Neumann functions and we have a free field theory (1,2 = 0) already outside the core and its 1/r tail,

? 17r ] = f(~0) [ e - ~ r _+ S(~0)e i'~r]

2J5 ]

r>r b.

(5.21)

The S-matrix, S(o~), is given by 1 - i~:V/o~

S(co)

1 + iueV/co

~2V 1 - 2i o ~ - -ip2V

(5.22)

The response function in time is the Fourier transform S(t) folded with a wave packet satisfying o~ << M.

S(t) = 8(t) + 20(t)FD e-rDt, NOe2 F D =/.p2V--- 8,//.2 V.

(5.23)

TD =/'i~ ~ is the lifetime of the dyonic core excitation. We recall that N o is the number of fermion doublets in our m f = 0 framework.

K. Isler et al. / Rubakov-Callan effect

952

d3L

el+

M

UlR

D-

U2R

M

Fig. 1. The space-time picture (given by eq. (5.24) of the A B = - 1 reaction for an incident d3L in SU(5). This picture corresponds to the regime vEln(M/l~) ~ 0 and to a fermion energy veM << E << M. M = monopole, D = dyonic core excitation, its electric field is represented by the wiggly line.

The solution with the quantum numbers of an incident d L-quark is the function. 1

1

FD

jr(r, t) = - ~ 8 ( t + r) + ~--~8(t - r) - -4--~O(t- r)e -rD(t-r)

(5.24)

folded with a wave packet satisfying w << M. Using the corresponding solution for j°(r, t) (determined via the continuity equation) we read off the space-time picture shown in fig. 1 (the labels correspond to SU(5) as discussed below). The incident pulse with the quantum numbers of a d L-quark (of charge - i in the Georgi-Glashow SU(2) model) makes charge exchange with helicity conservation at the core and emerges instantaneously as a pulse with the quantum numbers of an u L-quark. Because of the continuity equation the monopole gets necessarily a dyonic core excitation of charge - 1 . This dyonic core excitation decays exponentially under emission of a (dRfiR) pair with probability equally shared by all doublets (in our m f = 0 framework). The folding of eq. (5.24) with a wave packet of width AT with TD/AT>> 1 will leave it essentially unchanged. Note however that this picture would be relevant only for a "gedanken experiment" involving fermion energies v2M<< Ef<< M. For To~AT<< 1 instead, the folding of eq. (5.24) with the wave packet will produce a point interaction (on the scale of AT) of 4 pulses with the quantum numbers of 4 ferrnions. The incident pulse with the quantum numbers of a dL-quark reaches the core and emerges instantaneously (on the scale of AT) as a superposition of 3 pulses with the charge and helicity quantum numbers of a dR-quark. This superposition contains however components of different doublets (e.g. ULU~dR). 1-2 2 The dyon charge expectation value during the interaction is of O(To/AT) << 1. However the dyonic core excitation is entirely responsible for the anomalous pair production. In regime (ii) (first order in vZln(M/~) we can study the anomalous pair creation via the fermionic electric field. For this effect to be appreciable the cutoff radius r, has to be many orders of magnitude larger than the monopole core radius,

K. Islet et al. / Rubakov-Callan effect

953

uZln(r,/rb) = O(1). From eq. (5.19) we have

AQ~ = 2~,21n(r,/rb ) , A Q ~ = 2 ( 1 - ~,21n(r,/rb)).

(5.25)

Since we neglect ~,2 in AQf (while keeping v21n(M//x) any one decade in r contributes negligibly to AQ~. For a discussion of anomalous pair production by fermions there is no need to resolve the scale ~'D" In Fourier space this corresponds to considering only pulses with w % << 1. This implies that the microscopic mechanism of anomalous pair creation in the core region gives total electric currents corresponding to helicity flip and charge conservation at the core. For an incident d L-quark the currents correspond to the effective reaction d~ ~ d~ut, ( jr )

1

1

1j;r =

8(t+r-r,)~-frlrS(t-r-r,).

(5.26)

Note however that the outgoing pulse contains components of different doublets. These currents are the source of the fermionic electric field, which generates AQsV. The produced fermion pairs also have an electric field. However this can be neglected to first order in ~,21n(M//z). The current produced by the fermionic electric field is •

p2

jr(r, tl=-fr"dr'[r"+rdt'G(r,r',t,t')g~rr,2, " rb

"r~

(5.27)

r•

where the Green function G(r, r', t, t') satisfying (0 2 - a~)G(r, r', t, t') = a o S ( t - t ' ) 6 ( r - r'), G (0, r', t, t') = 0,

(5.28)

is given by

G(r, r', t, t') = ½0(t- t')5(r- r' + t ' - t) + ½0(t- t')6(r + t - r ' - t') -½0(t-t'-r')8(r-t+t'+r'),

r>~O,

(5.29)

and is shown in fig. 2 (again with the SU(5) labels to be discussed below). The total currents at r = r, are given by

(jr)

l

1 //2

8~-[0(t)-O(t-2r,+

rb) ] •

11] [1

2r,_,

VZ[O(t--2r"--rb)--O(t--4r")]" + 8~r

2r,

t---2r,

2r~

(5.30)

954

K. lsler et a L / Rubakov-Callan effect

eL UlRU2R

dsL

U2RUlLU2L

~IR

Fig. 2. The space-time picture of a ZlB= - 1 reaction included in eq. (5.30). This corresponds to the regime v 2 i n ( M / t t ) ~ l and to a fermion energy ~f~-M<
.r (r/~,t) j~ (r~,,t)=- ~I J5 v 2 zr~, - rb _' 8"/T

0

2r b rp.

2r/~ - r b

zr#* rb

4r~

_v2 2r~. - r b 8w"

2r b rF

Fig. 3. The anomalous currents jr(r~, t) and j~(r,, t) produced by the fermionic electric field in first order in u2. The solid line is given by eq. (5.30), the dashed line indicates the behaviour inside the short distance cutoff.

and are shown in fig. 3. The solution for j~ is determined from j r via the continuity equation (5.1). The scale of the m o n o p o l e core radius, rb, provides an ultraviolet cutoff for the anomalous pair production and prevents the currents from diverging at t = r + r~. T h e space-time picture of eqs. (5.29) and (5.30) is the following. The fermionic electric field outside the incident and reflected pulses produces via the a n o m a l y a pair (5RUR) or (dRdR) with probability equally shared by all doublets. The

K. Isler et al. / Rubakov-Callan effect

955

component with the quantum numbers of an fiR(dR)-quark propagates outward while the partner with the quantum numbers of an U R(dR)-quark reaches the core and emerges again as a superposition of 3 pulses (of eventually different doublets) with the charge and helicity quantum numbers of an UL(dL)-quark. The total contribution of the dyonic core excitation to anomalous pair production is the sum of the two contributions from the core vertices. At the first vertex AQ~ -- 2, while at the second vertex AQ D = -2v21n(M/I*). The time separation between these two effects is of the order of the geometric mean of the two relevant scales for anomalous pair production by the fermions, O(~r,r c ). The folding of eq. (5.30) with a wave packet of width A T with r ~ r c / A T >> 1 will leave it essentially unchanged. This picture would be relevant only for fermions of energies 1/C/~M<< E << g2M. For ~r,r c / A T << 1 instead the folding will produce a point interaction (on the scale ~r,r c ). In regime (iii) (v21n(M/tO arbitrary) we can study non-perturbative anomalous pair production via the fermionic electric field. Like in regime (ii) we do not need to resolve the scale r D for the following discussion. Therefore we consider only pulses with oarD << 1. This gives an effective helicity flip and charge conserving point interaction at the core. Therefore we can neglect the Neumann function N1 +~2 with respect to the Bessel function J12+; in eq. (5.5) for f q The S-matrix can be computed by matching free ingoing and outgoing waves outside r. to the Bessel function ¢c~-j~ + ; ( o a r ) i n s i d e r,

s ( oa) =

- 1 + iR(oa) 1 + iR(oa)

sin x 3.v~J½ + ~ ( x ) - cos x v~-J~ + ~ ( x ) R(oa) =

cos x OxgTJ,2 +; ( x ) + sin x vffJ~2+ ~ ( x )

,

x = oar..

(5.31)

To derive the space-time form of the currents for r ~< r, we use the following trick to include the infrared cutoff for the anomalous pair production in a mathematically convenient way. We choose the initial data at t = 0 such that the solution contains an outgoing u L &pulse in addition to the ingoing d L &pulse, both starting at r = r.. In this way the total electric field is vanishing for all r > t + r. and this region does not contribute to anomalous pair production. The chiral charge produced via the fermionic electric field in a given space-time area is proportional to the typical dimension of the area on the logarithmic scale. Therefore the error on AQ~°t introduced by having the cutoff at r. + t instead of at r. is of O(u 2) (and therefore negligible in our framework) for times t << r 2 / r w The appropriate choice of initial data is 1

jr(t = O, r) = - - ~ 8 ( r - r,),

Oojr(l

=

0, r ) = 0.

(5.32)

K. lsler et al. / Rubakov-Callan effect

956

The corresponding solution [19] of the Bessel equation is

jr(r, t) = OoX(r, r~, t) ,

S5( .r r , t) = 2 O ~ X ( r , r , , t ) ,

(5.33)

X ( r , r~ , t ) = fo d °~ --4-~ J ~ +"~( °~r ) J ~ + "~( ~°r~ ) sin °~t (0;

O
= I ~P;(~)' t ~ 1 sin l ' 2 ~ r Q ~ ( - z ) ,

r+r,
r 2 + r2 -- t 2 z=

(5.34)

2rr~

The space-time form of the currents for r ~
j'(r,t)=

~-~8(t-r,+r)-

8(t-r,-r) 1

-[O(,-

r~ + r ) - O ( t -

r - r~)] ~

t OzP~2(z ) r--~

1 t + 0( t - r - r~) ~r---4 5 sin Try2 0 z Q ; ( - z ) -~r~'

.r r . t ) = Js(

~8(t

-r~+r)+

(5.35)

8(t-r,-r)

1 + O(t-r~+r)-O(t-r-r~)-~OzP;(z

)

r2-r2+t 2r2r,

1 r2t2 2~r----2sin~r~2OzO~2(-z ) r2 + 2r2r~

-O(t-r-r,)

2

(5.36)

The divergence of the currents at t = r + r, is due to the absence of the explicit ultraviolet cutoff r b in this computation. The functions should be cut at t = r + r, + ~D since ~'D is a lower bound for the minimal scale we can still resolve in this framework. The error on AQs(r,) due to our trick for the infrared cutoff is of 0 0 , 2) and therefore negligible. This is because Q~2(x) - x - 1 - ~ for x ~ oo and therefore the contribution to AQs(r,) from the time interval [t. oo] for t >> r, is of 00,2). We now consider the limit p21n(M//~)~ oo, i.e. a pointlike monopole. In this case anomalous pair production is entirely due to the fermionic electric field. AQ~=2/-,i

dt ,ro

•

QD(t)

e

= 2(/~rc)

~

~0.

(5.37)

K. Isler et al. / Rubakov-Callan effect

957

where QD(t) is the expectation value of the dyonic electric charge. For an incoming &pulse, AQ D is a sum of two contributions: first the contribution of zeroth order in v 2 1 n ( M / t t ) A Q D = 2, second the contribution of higher orders in v21n(M/tO, AQ~=2(l~rc);-2~-2. The time separation between the two effects is of O ( ( r j c ). We construct a wave packet of width of O(r~) by superposition of incident &pulses weighted by a function of time. In the limit r ~ r c / r ~ ~ 0 this function is constant on the scale ~r~rc . Therefore in this limit dt

f laD(,)l=0.

(5.38)

We conclude that Q D ( t ) = 0 and the dyonic core excitation is frozen in the limit v21n( M / t t ) ~ oo. A selection rule for anomalous pair production has been proposed by Sen [20] for the case of more than one doublet (labelled by the index i). The difference between the divergences of the currents of two doublets is zero for massless fermions a ~ j ~ 1 - a~,j/~2 = 0.

(5.39)

A [(N L - NR)i= 1 - (N L - NR)i_2] = 0

(5.40)

Sen concluded that

in a physical process on a event-by-event basis. He then concluded that the process d ~ = l ~ U'L-I(dRfiR)i=2 (in SU(2) language) is forbidden and that for an incident d~=1 only the process d~.=1 ~ d~=1 is allowed. According to Sen et al. a baryon-number violating process (in SU(5)) is forbidden for a single incident fermion. Callan (third paper in ref. [2]) reached the opposite conclusion based on the bosonized formulation. (See also Dawson and Schellekens, ref. [21].) Our solution gives a particularly clean counterexample to Sen's argument in the regime v 2 1 n ( M / ~ ) << 1 and for incident wave packets with A T << ~'D, but AT >> re, compare fig. 1. In the first vertex there is necessarily charge exchange and helicity conservation, d~=l+ M ~ U~L=I+ D in SU(2) language, and the U/L=1 escapes as a free particle long before the anomaly has had time to act. This contradicts the conclusion of Sen that the selection rules allow only d L ~ d R for an incident single d L. Also, according to the arguments of Sen, a produced dyon could not decay spontaneously, if more than one doublet is available for anomalous pair emission, D - ~ M + (dR~R)In thiS paper we have only considered expectation values of operator equations and only the massless fermion case. Our solution shows that the process excluded by Sen does happen for any value of Ef and of the infrared cutoff. With the methods used in this paper we cannot make statements on an event-by-event basis nor for massive fermions. Future investigations should elucidate in which way eq. (5.39) is compatible with the non-vanishing expectation value for the process d~-1 ~ U'L-l(d R~ R) ~ 2.

958

K. Isler et al. / Rubakov-Callan effect

We give a review of the main results of this section. There are two basic mechanisms responsible for the Rubakov-Callan effect. The first is the necessity for charge-exchange scattering (with 100% probability) in the background of the monopole (zeroth order in e2). It has been deduced on the basis of conservation laws by Goldhaber, and the corresponding Dirac equation has been solved by Marciano and Muzinich. The second is the anomalous pair production which is proportional to f d t f d 3 r e 2 E • B. Two competing contributions on two totally different scales add up to AQ5 = 2. The first is anomalous pair production via the dyonic electric field which is concentrated on a scale r b (region of the core and its 1 / r tail). The second is the contribution to anomalous pair production from the fermionic electric field and its spread over a scale 1//~ where /~ is the mass scale of the fermions. The parameter governing the different strengths of the two effects is uZln(M/l~). In the case 1,21n(M//~) << 1 anomalous pair production is due entirely to the dyonic core excitation. This situation has been discussed by Blaer, Christ and Tang for the special case of an external dyon. They deduced the chiral flux emitted by a static dyon of charge QD = 47r/e to be F = - V/~r. Therefore the charge flux is d Q v / d t = - e V / 2 ~ r . Using charge conservation the dyon life time is found to be r { l = -O_o/QD= e2V/8~r2 in agreement with our result. Sonoda's numerical value for ~'i~ roughly agrees with this result. In regimes (ii) and (iii), for which the fermionic contribution to anomalous pair production becomes important, we do not resolve anymore the microscopic mechanism. This results in an effective charge conserving and helicity flip vertex at the core. By mixing the doublets, however, this vertex still violates other fermionic quantum numbers (e.g. baryon number in G U T theories). In the limit u21n(M//~) ~ oo of a pointlike monopole the dyonic core excitation is frozen. This case has been analyzed by Rubakov and by Callan in his earlier work. In grand unified theories, the phenomena described in this section lead to baryon-number violating processes. For an SU(2) monopole embedded in SU(5) [4] such that its generators are T = i diag(0, 0, ~', 0), the fermion doublets of the first generation with Qeff = T3 = QE + Yc 4:0 are (e +, d3) and (Ul, fi2) , where the numbers are colour indices. In regime (i), (p21n(M/~) << 1 and neglected) anomalous pair production is due to the dyonic core excitation and is described on the scale of r b by eq. (5.24). Using the SU(5) labels this corresponds to the reactions d3L ~ eL+-UlRU 2R, eL+ eRd - 3R • The AB = --1 reaction is shown in fig. 1 for fermion energies ~,2M << E t << M. In regime (ii) (first order in u21n(M//~) one of the possible AB = - 1 reactions is shown in fig. 2 with the SU(5) labels. In all regimes the total cross section is J = 0 = ~r/k 2 since the incident quark with momentum k necessarily makes an Otot inelastic reaction. In regime (i) there are two possibilities with equal probability, AB = - 1 and AB = 0. Therefore the AB = - 1 cross section is %J~°_ 1 = ~r/2k 2. In regimes (ii) and (iii) there are more possibilities, but we still obtain an expectation value ( A B ) = - ½. In a realistic physical model with 6 fermion doublets coupling to

K. Islet et al. / Rubakov-Callan effect

959

the monopole, with M = 1015 GeV, /, = 1 GeV and a = aou-r = ~ we obtain vZln(M/#) = 0.82 and (/~/M) "2= 0.44. The contribution of the dyonic core excitation to the Rubakov-Callan effect is about 50%. We would like to thank C. Callan, N. Manton, M.B. Paranjape, P. Rossi and V.A. Rubakov for helpful discussions. This work was partially supported by the Schweizerischer Nationalfonds.

Appendix THE EQUATIONS OF MOTION TO O(e)

To O(e) the semiclassical expansion describes the lowest order coupling between the small oscillation modes. To obtain the equations of motion, we insert ansatz (3.5) for the fluctuations, including the collective coordinate, into eqs. (2.27)-(2.30). Keeping only terms up to O(e) we get the following set of equations: Orr20rao-2K2ao=2KOo992+negdo991

+ 2e(9910o992)-ej O,

Oor2Orao = 2K Or992- Ze( 9910r992) + e"r.y,

(0o2 - O~) + ~

(a.2)

992+ K Ooao + 2eao Oo~q+ e9910~oao

2 eK

2 eH

e

+ -7T-991992+ -75-- rt992= - J ~ ,

(A.3)

r

2K ] (ag-

(A.I)

2uK

+ K + - - r~ 1 9 9 ~ + - - =r n - 2eaoOo99=-e99~ Ooao 3K

2

K

2

2eH

- e a 2 K + 7 - e991 + 7 e992 + 7

O~H

M~

( ° ° ~ - ° ~ ) + - - f f - + M -5 7 2ell, 2 + 7

eK 2

e

rl99~ + 7 n = -J~r 1,

4HK

(A.4)

4eK

n+--~--99~+-7--99~n

3 M~ H 2

(991 + 9922)+ 7 - ~ e T g ~ / = 0 ,

(A.5)

[ O+ieao~°q+K~se2ie~q+e99lg'e2ie~q+iehep2 ] r r r e2ieaq Xh = 0.

(A.6)

960

K. Isler et al. / Rubakov-Callan effect

References [1] V. Rubakov, Pis'ma Zh. Eksp. Teor. Fiz. 33 (1981) 658 [JETP Lett. 33 (1981) 644]; Nucl. Phys. B203 (1982) 311 [2] C.G. Callan, Phys. Rev. D25 (1982) 2141; D26 (1982) 2058; Nucl. Phys. B212 (1983) 391 [3] G. 't Hooft, Nucl. Phys. B79 (1974) 276; A.M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. 20 (1974) 430 [JETP Lett. 22 (1974) 194] [4] C. Dokos and T. Tomaras, Phys. Rev. D21 (1980) 2940 [5] A. Goldhaber, Phys. Rev. D16 (1977) 1815 [6] A. Blaer, N. Christ and J.-F. Tang, Phys. Rev. Lett. 47 (1981) 1364; Phys. Rev. D25 (1982) 2128 [7] C.G. Callan, Princeton University report (1984), unpublished [8] E. Witten, Phys. Rev. Lett. 38 (1977) 121; P. Forgacs and N.S. Manton, Commun. Math. Phys. 72 (1980) 15; C.S. Lam and T.-M. Yan, Phys. Rev D31 (1985) 3221 [9] N. Christ and T.D. Lee, Phys. Rev. D12 (1975) 1606; E. Tomboulis and G. Woo, Nucl. Phys. B107 (1976) 221; P. Hasenfratz and D. Ross, Nucl. Phys. B108 (1976) 462; A. Abouelsaood, Nucl. Phys. B226 (1983) 309 [10] W. J. Marciano and I.J. Muzinich, Phys. Rev. Lett. 50 (1983) 1035; Phys. Rev. D28 (1983) 973 [11] M.K. Prasad and C.M. Sommerfield, Phys. Rev. Lett. 35 (1975) 760 [12] H. Sonoda, Nucl. Phys. B238 (1984) 259 [13] R. Jackiw and C. Rebbi, Phys. Rev. D13 (1976) 3398 [14] P. Goddard and D. Olive, Reports Prog. Phys. 41 (1978) 1358 [15] B. Julia and A. Zee, Phys. Rev. D l l (1975) 2227 [16] H. Sonoda, Phys. Lett. 143B (1984) 142 [17] T. Appelquist and J. Carazzone, Phys. Rev. D l l (1975) 2856 [18] C. Schmid and C.A. Trugenberger, ETH Ziirich report (1984), unpublished [19] G.N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1944) [20] A. Sen, Phys. Rev. Lett. 52 (1984) 1755; Phys. Rev. D28 (1983) 876 [21] S. Dawson and A.N. Schellekens, Phys. Rev. D28 (1984) 3125