ANNALS OF PHYSICS 141, iod-115 (1982)

Stability of Einstein-Yang-Mills Monopoles and Dyons

DAKSH LOHIYA

Department of Applies Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9E W, United Kingdom

Received September 1, 198 1; revised January 19, 1982

The stabilities of Yang-Mills magnetic and Coulomb charges have been studied separately before, without laying much emphasis on the singular nature of the effective potential in the resulting radial equation. We investigate an analysis that is valid simultaneously for both Yang-Mills magnetic monopoles and dyons in flat as well as curved spacetime. We find the instability modes with their qualitative behaviour determined completely by the large distance behaviour of the radial equation. One of the results for the dyon is that we could get unstable modes even if there were no such modes for the constituent Yang-Mills “electric” and “magnetic” charges separately.

I. INTRODUCTION AND CONCLUSIONS

Coulomb-like solutions to the classical Yang-Mills theories give rise to long range forces between quarks. We shall study the classical stability of such solutions in this paper. A typical instability would provide a “screening” of the classical source [ 1 ].

The usual way to analyse the stability of a solution is by linearizing small pertur- bations around it and examining the temporal behaviour of the resulting linear system. In the case we encounter, the time and radial dependence of these pertur- bations are separable as !P = exp(--iwt) R(r) g(0, ~0). The (in)stability search reduces to finding the fluctuation eigenfrequencies consistent with the boundary conditions required of a finite localized perturbation. If all the frequencies turn out to be real then the solution is stable and if any of them are complex then some fluctuations would grow exponentially with time-indicating instability. We shall restrict ourselves to the study of Yang-Mills Coulomb-like and monopole-like solutions in flat spacetime and also look for corresponding properties of spherically symmetric solutions of Einstein-Yang-Mills systems in general relativity. In flat spacetime the problem was studied by Mandula [2]. One of the stability arguments was based on the analytic properties of the perturbations as a function of the source strength in an “effective Schrodinger equation.” However, it is known [3] that solutions for a singular potential gV(r) are not analytic functions of the strength parameter g at g = 0. The instability of the Yang-Mills monopoles was studied by Brandt and Neri

104 0003.4916/82/070104-12$05.00/O Copyright Q 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.

MONOPOLE AND DYON STABILITY 105

15 1. The radial solution in their case again satisfies an effective Schriidinger equation and one sees that above a certain critical value of the strength parameter, one gets bound states (in fact a continuum of bound states)-corresponding to pure imaginary frequencies, thereby indicating an instability. It is, however, not immediately obvious whether one should attach a physical meaning to such bound states for attractively singular potentials. Such potentials support a non-unique bound-state spectrum with an infinite number of bound states with no lower bound (3 I. The arbitrariness can be interpreted as being due to the potential being physically cutoff by some unspecified mechanism near the origin. In Section II we specify a cutoff which leads us to a discrete bound-state sectrum with no lower bound above the critical strength of the charge. In Ref. 151 the authors found bound states that are regular at the origin and infinity. On the other hand we have emphasized that for our singular potentials it is not the regularity near the origin but the cutoff which uniquely determines a solution. The qualitative nature of the resulting discrete bound states is determined by the value of the critical “charge.”

For a graviting system Moncrief [4] has shown that solutions to Einstein-Maxwell systems are stable against small perturbations throughout the allowed range of the electric charge of a black hole. Hajicek (61 has further shown that as the dynamical equations for the perturbations are expressible in a form invariant under a duality transformation, a Reisner-Nordstrom solution of mass m carrying an electric charge 4 and a magnetic charge p is classically stable in the range 0 < 4’ + p’ < 111’. when there are no other fields than gravity and electromagnetism. In Section III we shall see that the self-coupling of the Yang-Mills fields and the non-vanishing commutators of their generators completely change the stability issue. We shall derive an effective Schriidinger-type equation for the general case which can be numerically solved to look for bound states. Without going into such a numerical venture, we can make the following statements regarding the classical stability of the solutions in question: (a) Classical Coulomb-like, monopole-like, and dyon solutions to a Yang-Mills theory in flat space are stable for g(‘)* + 2A < { and unstable otherwise. where gcr” and A. as described in the paper, are related to the “electric charge” and the generalized angular momentum of the Yang-Mills dyon. (b) The “pure magnetic” solutions to an Einstein-Yang-Mills system are stable for 2,4 < $ and unstable for 2‘4 > f to small frequency perturbations. (c) The dyon solutions are stable for K ““’ + 211 < $ to low frequency perturbations and unstable for g”“’ + 2A \ $--again to low frequency perturbations. In case of Klein-Gordon and Yang-Mills Coulomb fields in flat space, our results are the same as obtained by Mandula 12 1. Magp / 8 1. and Hey and Mandula [ IO).

II. GENERAL FORMALISM AND STABILITY IN FLAT SPACE

Before tackling the instability of the Yang-Mills solution, we consider a prototype problem that demonstrates the peculiarities one encounters with a singular potential.

106 DAKSH LOHIYA

namely, the fluctuation of a massless Klein-Gordon field in an external Coulomb potential governed by the equation

[ (ao-~)z-vq,=o, Separating the time and angular dependence by v, = (l/r) !P(r) Y?(i) e-l“’ gives

[ 2

-$+0’-- 2wq ,++ Y=O. I

(2.1)

(2.2)

y=q2-l(1+ 1).

The simplest way to analyse the stability of these fluctuations is by rounding off the effective potential in (2.2) to a non-singular value at the origin. Outside some radius r = r0 (independent of w), the fluctuation obeys (2.2). The slope of a solution that is regular for r ( rO, is fixed at r0 by the nature of the cutoff potential. If one rounds off by a constant potential for r < r,,, r0 being chosen small enough for w2 and 2(w/r) q to be negligible in (2.2) in comparison with the l/r2 term, then for y < f the general solution near r = r,, goes as Ar-‘12+m+ Br-“2-m Matching this solution and its slope with the solution for r < r. gives A/B = r;’ v@? Thus in the limit r,, + 0, we see that it is possible to match the solutions only if B = 0. The exact solution of (2.2) with complex w which has such a behaviour near the origin is the Whittaker function of the first kind Miq,my(-2irm) (7) and this diverges at infinity as exp IIm(wr)(. Therefore, the solution with y < f must be stable.

For y > $, the solution bounded at infinity for complex frequencies is the Whit- taker function of the second kind Wiq,im(2iwr), As we go inward from infinity to a radius of order q/[/lwl, the solution increases and then begins to oscillate. The smaller the imaginary part of w, the larger the number of oscillations. The cutoff being w-independent, the solution inside r0 does not vary much with frequency. This means that by choosing Im w to be sufficiently small, we can achieve the matching (of the slope) of the solution when it hits r = r,,. As w + 0 we can pick out an infinite number of w values for a given (small) cutoff radius. Thus w + 0 is a point of accumulation for bound states. We can easily see that the states do not form a continuum. This is readily established by defining m = fi = i2 and performing a series expansion near r = r,, when one finds that the most general solution for small distances goes as W- Cr’12 cos()i In r + B), B being a phase constant. Comparing this with the general solution to (2.2),

W = e’*M,,,,(2iwr) + eeiaMi,, -,(2iwr), (2.3)

which for small r goes as r”* cos[l In r + a + L In(2iw)], implies

a = B - A In (2iw).

MONOPOLE AND DYON STABILITY 107

Requiring the solution to be bounded for large distances gives ] 7 ]

B -A ln(-2iw) - (n + -> 71= arg : L f(l + h)

=(+ + m + is) I

= c (2.4)

which gives

-iw=exp+(B-((n+f)n-ilnZ+o], (2.5)

n being an integer. This gives the discrete system of bound states from -iw = 0 (n = co) to -iw = 00 (n = - ~0 an is indicative of instability as it would give rise ) d to solutions growing exponentially in time. The value of B would depend upon the way one chooses to cut off the singularity. Equation (2.5) shows that -io = 0 is a point of accumulation for the unstable states. All states have the same phase B related to the cutoff parameter, exp(B/A) - l/r o; the magnitudes of binding energy differ from each other by a factor of e”‘,‘. Each frequency is scaled by l/r, but as the infinite sequence of frequencies accumulate at LU = 0, the qualitative picture of instability modes is independent of B, i.e., the cutoff.

The same analysis goes through in the case of a classical Yang-Mills potential. whose stability may be worked out by expanding the field about the classical solution and looking for modes with complex eigenfrequencies. Here one could work in the background field gauge and expand the fluctuation in terms of vector spherical harmonics when they reduce to two coupled second order radial equations. An oscillating condition for small distances for the resulting set would again determine the solution up to a phase. However, it is much simpler to follow the privileged gauges 181. For an SU(3) gauge theory one chooses a frame and a gauge in which the external source has just one (the third) component (T3) in the space spanned by the generators (T”, a = l,..., 8) of the group. In this case the remaining gauge freedom can be used to choose the time components of the perturbation to vanish. Further. as the commutator of T” with T3 is proportional to T” itself; IT”, T”] = ik”T”, one finds that different internal indices do not mix and the perturbation equation is exactly the (uncoupled) radial equation of a massless Klein-Gordon particle with angular momentum (j, m), j > 1. moving in the potential k”@. where

A”‘o’(x) = 6uoT3@(l.u(), -V’@ =p(x)

represents the gauge field, A @, in the presence of a source. Afi(.u. 1) = A (“)u(x) + A“(x, t) represents the perturbation 2” (x, t) around A ‘o’n(x).

.Aj + 1) r’ I

h(r) = 0.

108 DAKSH LOHIYA

Using Q(r) = q/r for a source as before, the previous analysis may be repeated to see that for a total angular momentum j and internal index a, the critical charge is (j + ;)/I k” I. Th is means that for 14) > t we get instability modes and not for weaker sources.

The (in)stability for Yang-Mills monopoles can also be established in a similar way. However, as we shall be studying a slightly more complicated problem (viz., the stability of Einstein-Yang-Mills monopoles) in detail in the next section, we shall just mention the nature of the difftculties that arise in such an analysis. The pertur- bations around a Yang-Mills monopole are separable into angular and radial parts, the latter satisfying a non-relativistic radial equation for a particle in a --/i/r* potential, i.e., Eq. (2.2) with the 2wq/r term missing. Requiring solutions corresponding to distinct energies to be oscillatory and orthogonal again shows that all solutions must approach a constant phase near the origin after a certain number of oscillations.

As before, requiring the solutions to be bounded at infinity gives

-io = exp [ (II - (n + 4) z)/(y - b)“” 1. (2.7)

Thus, we do get unstable modes but not a continuum of unstabie modes as asserted by Brandt and Neri [5].

We end this section by outlining the approach to the stability problem for a Yang- Mills dyon. The stability of Yang-Mills solutions with a sufficiently small magnetic or electric charge has been worked out by Magg and Mandula [2, 81. However, more quantitative results can be achieved by referring to Eq. (A.16) in the Appendix which for a flat spacetime takes the form

R”fjR’f [ (w+q)2+y]R=0

with g@)* + 2A = $ - m*; the general solution to (2.8) is

R = + [eiaMigcp,,,, (2ior) + e-‘“M,,,,,_,(2iwr)].

The solution converging for r-+ co is

R = ?- Wigcp,,,,(2icur). r

(2.9)

For g@)* + 2/1 < 4 there is, as before, no instability. (For g@)* + 2/1 < 0 Eq. (2.8) cannot have any point of inflection for negative w*, and for 0 < g@)* + 2~t < $, the solution with complex frequency and with the right behaviour for matching to a cutoff potential diverges at infinity.) For g@‘* + 2A > i we again get a discrete system of bound states from -icu = 0 to -iw = 00, indicating instability. This is quite interesting as it shows that a dyon could be unstable even though the values of the

MONOPOLE AND DYON STABILITY 109

“electric” and “magnetic” charges were below their individual instability thresholds. To illustrate this with an example let us work with the operator Jt = L”6,j + Sij: L and S are defined in the Appendix as the generalized angular momentum. and spin operators, respectively. In the subspace defined by eigenvalues of J* and J,, we have J= L f 1 or L, with L having a minimum value g”.

For J = / g’“‘) / - 1, L = 1 gem’ 1,

2/1 + gc@)2 = ,Jg’rn’S . r̂ + g”‘2 _ (L* _ g’m”).

For gem) = gee’ = f ,

(2.11)

2/4 + g’e’2 = 41 s . r̂ - (L2 - f ) 1 + g’r’*.

The first term can be positive only when L(L + 1) - f < 1 because (S . ?I B 1. Therefore these dyons shall remain stable.

For J= ( gcm)l, L can be 1 g(“‘) 1 or ( g(“” 1 + 1 and n + g““I/2 is a 2 x 2 matrix. For

g w = g(“‘) = i, this is seen to be (5, 91

There are two eigenvalues Q I-5 + d4=], one of which exceeds 9. Thus the J = 4, L = i state shall be unstable although the “electric” and “magnetic” charges are separately stable.

The state J = : = g’“” + 1, g@ = 4 has no eigenvalue for which 2/1 + g”“’ becomes > i. So this too remains stable, For 1 gem’/ > f , states with J = 1 g”‘1 -- 1. L = ( gm[, have 2/1 + g@)’ = gcrnj2 + g’@’ which exceeds $ for any value or g”“. Hence such dyons shall be unstable. These instabilities are indeed physical as they satisfy all the constraint equations used in deriving Eq. (2.8) 151.

III. STABILITY IN CURVED SPACETIME

Ignoring the back reaction on the metric, we show in the Appendix that the pertur- bation I& around a classical Yang-Mills solution AL”‘” satisfies

-YcoJ2Y; - 2[F;;!, .I’]” + R; !P; = 0 (3.1)

in the background gauge Q~‘!P’ = 0. Here RL is the Ricci tensor and all he operators are defined in the Appendix. Expanding these perturbations in the basis I” of the Lie group G and choosing the “magnetic” and “electric” charges to lie in the Cartan sub-algebra of the commuting elements h0 of the basis reduces (3.1) to the

110 DAKSH LOHIYA

following equation for the Fourier transform of the perturbation about a dyon solution to an Einstein-Yang-Mills system:

Y;‘-+R;Yf=O. (3.2)

Here we have restricted the perturbations to be normal to the radial direction and the background metric is chosen to be

ds2 = -e” dt2 + e-” dr2 + r2 de2 + r2 sin2 Sdp2,

e’ = 1 - 7 + ; (Q”Q, + M,i&).

For a coupled Einstein-Maxwell system the absence of self-coupling of the Maxwell field leads to the vanishing of the g4. The resulting stability analyses would lead us to Moncriefs results [4].

Separating the angular dependence reduces (3.2) to the radial equation (A. 16) in the Appendix. Defining a “tortoise” coordinate r* by dr* = dre-“, and defining R z Q/r, gives

(0 + gce) ) 2

- r

@ + @,+ + ev [

2A - 1 e” 7 +-p @=O.

I (3.4)

This equation can be used to describe super-radiance effects for Re o > -g(@/r+ , where r+ is the larger of the two roots of the equation ep = 0. This follows by integrating (3.4) by parts and taking the imaginary part to give

(3.5)

which implies that we can have solutions with non-vanishing imaginary frequencies only if 0 > Re[w] > -gCe)/r+. For our present purposes, however, we shall just look for the existence of solutions with small imaginary frequencies to look for (in)stability of the classical solutions.

When the “electric” charge of the solution vanishes, then from Eq. (3.5) it follows that the unstable modes can only have pure imaginary frequencies. Now Eq. (3.2) is expressible as

R,,, + G R,re-v

+ e-2”R “” 2A -MaMa + QaQe L 1 r’ rz

MONOPOLE AND DYON STABILITY 111

For g - , (e) - 0 the solution at the horizon goes, with w = ik, as

R - $ [(r - r,)“: (r _ rJt]WW, i--r+ (3.8)

i.e., it is increasing with r. For a solution to be bounded at infinity and increasing at the horizon, we must have a point of inflection, where R # 0 identically, and where R and R,, have opposite signs. This is clearly impossible for 2/1 < 0. For 0 < 2/1 < $. consider the equation

@ ‘r*r* + [W’ + V(r*)l @ = 0 (3.9)

with

V(r*) = g for r$</r*l< co

2A =*z

r o for O<Ir*l<r,*.

For 1 r” 1 > r$ the solution @ - rs has two roots

For Ir* / < r,*, the solution bounded at (r* / + 0 is sin k’r*, -k’* = V(r,) + UJ’, with a logarithmic derivative ik’. Matching this with the log derivative of Ar*S’ + Br*‘l gives B/A = constant x r. *‘I-‘~, Thus in order that the solutions match in the limit rt + 0, we must have B + 0. The solution that goes as r *‘I is not bounded at infinity, being the Whittaker function of the first kind. Now the “effective” potential for Eq. (3.4) with gee) = 0 is greater (“less deep”) than the effective potential in Eq. (3.9) for all r*. Therefore, the absence of bound states for (3.9) implies their absence for Eq. (3.4). Thus the “pure magnetic” Yang-Mills monopoles are stable for 2,4 < $. For 2/1 > d let us consider frequencies that are small enough in the regime or G 1; r+r+, i.e., w wery small enough compared with l/r+. In this regime, we can see that the solution bounded at infinity is again the Whittaker function of the second kind. As it oscillates wildly for wr < 1, it can be easily matched to any solution behaving like Eq. (3.8) near the horizon and integrated out to this region. Thus we again have bound states for small frequencies. The spectrum is given by Eq. (2.7) with large n. The solutions are thus seen to be unstable. The spectrum is obviously bounded from below-i.e., Eq. (2.7) fails for large imaginary cu. The upper bound on IO/ is just the depth of the effective potential of Eq. (3.4). A numerical search for bound states is necessary to determine the exact location of unstable modes for large imaginary frequencies.

When the “electric” charge is non-vanishing, the situation is a little more

595/141/N

112 DAKSH LOHIYA

complicated. The frequency can no longer be pure imaginary (Re w # 0 from 3.5). The solution to Eq. (3.7) behaves like

r* exp(-iw, r*) exp Kr*, o=oR +iK (3.11)

near r = r+ . The instability for sufficiently small frequencies again follows as before, viz., for 2~l + g@)* > $ and for ]wr]< 1, r+r+, the solution well behaved at infinity oscillates in the region ] wr / < 1 (the solution being the Whittaker function again) and can therefore be fitted continuously onto a solution with boundary condition (3.11) integrated outward from r = r+ to this region. For 2A + g(@* < a, the solutions to perturbation equations, for the same 1 wr] region, that are increasing in this region, are the Whittacker functions of the first kind and have the wrong properties at infinity. Thus we get stability for small frequency perturbations. The spectrum of bound states for very small frequencies is thus given by Eq. (2.5) with k = - L/2/1 + gCe12 - i- The details of the horizon, etc., would just effect the phase factor B.

APPENDIX: THE LINEARIZED PERTURBATION EQUATIONS

We derive linearized equations for the perturbation of the Einstein-Yang-Mills field around a dyon solution. Instead of perturbing the Yang-Mills field reinforced with auxiliary fields which smooth out singularities on “strings” when one deals with monopoles, we find it convenient to represent the action in the first order form as [ 51

I = Z,,, + j d4X t/-g[$F”““F;I, - ~F”“‘((V A A”),,, + e(A, A A,,)‘}]

+$S . . i J! da dr Yi’,,YT,l[Eq~F~u(Yi) Mi(U, r> + JqtXir(U, r) g“MiI* (A.11 xi

I EIN is the Einstein action = f fid4x R. The r,(a) are particle trajectories (-co < o < co). i labels particles. Ci are surfaces bounded by ri and by strings parametrized by 0 < t < co. yiw(u, t) is a position on Zi; I - (a/%)~? y’ = (a/&) y. r7 = (a, r). @ is a field associated with the sources. x is a constraint function with ensures that the magnetic charge is constrained to Ci, i.e., y,,,[a + eA(y) n ] M(yi) = 0, where M(u) = M(u, 0) is the particle magnetic charge matrix.

If we define

G,,(X) = - s ,fm dU Jm dZ(.$i A yf),” Mi(U, 7) a4(X - Yi) (A.21 i-co -ccl

MONOPOLE AND DYON STABILITY 113

then varying F,,. gives

F”,,. = 10 A A + eA A A - G” I”,,,. (A.?)

Using this. we can define the second order formulation by

Perturbing around a classical solution:

A”, = A;” + y”,,

Q = Q(O) + 4,

M=M(O)+m. etc..

3 F”,,. -+ F”,,.(A”) + Q~“‘ty,, - C/~!“y, + ec**‘~~ tp:: + g,*, ,

(A.41

(12.5)

where

(A.61 g,,. = 1’ drl (.I$ Y:. - J;,.Y:) m(v) a46 - Y).

Noting that A:’ is a solution to the Yang-Mills equations, the perturbed action can be written as

where we have used %:‘I@’ = 0 as the background gauge and ignored the back reaction on the sources and the metric. The perturbation equation is

MA iy”, = -CY’yf, - 2(F:ql, I$‘]” + R;y; = 0. (A.8)

We now choose No and Q” to lie in the Cartan subalgebra of the commuting elements h” of the basis of The group G. If the basis tB of G [p = (1, a)] is chosen so that [h”, to] = ih”“t4, [a = (1, r)], r and d being the rank and dimension of the Lie group,

h” = la; h”“’ = 0 Va,a’E(l,r),

114

then

DAKSH LOHIYA

&f = 4n C M”h” z- [M, tS] = 4nig”9”; gTrn)6 E r M”h”b. b a a

Q E 411 c Q”h” =c- [Q, tD] = 4nigceb4t4; g@)fl z x Qa@. W) a

Now

Then expanding w,, = Ci=, ab”‘t” reduces (A.8) to

+ 8nig(e)B[6,,6,i - 6,,6,,] &- R;af = 0. (A. 10)

We now specialize to the spherically symmetric solution to Einstein-Yang-Mills equations as given by Perry 191,

ds2 = -e” dt2 + e-” dr2 + r2 d@’ + r2 sin’ (jdq’,

in which we may pick an orthonormal tetrad

go = (eL”’ dt, e-Of2 dr, r dtl, r sin 0 dp).

Then with H s w(‘) A wC3)/r2 Eq. (A.lO) is expressible as -

[(Q{)’ - (@)‘I a{ + 8nig”” & ajD) - RiabO = 0,

(A.ll)

[ (@t)2 - (cP~)‘] as - Rjai4 - 2ig”“jDH @ a4 + 8rig’e’4 --& a0 = 0.

Now in order to have a locally vanishing four divergence of the currents associated with the sources, we must have at = 0 for p # 01. Also, after fixing the background gauge we still have the freedom to change _a -+ _a + _D(‘)&, where D(O) . D”‘L = 0. This freedom can be used to make at = 0 VP and amounts to restricting the perturbations to be normal to r. This reduces the system to the study of one equation, viz.,

W3’ - w)21 a4 - Rjaf - 2ig4[H A a41i = 0. (A.12)

MONOPOLE AND DYON STABILITY 115

Considering the Fourier transform a:(& r) = C, e-‘“‘ci/i(o, r) gives the following form for the component I,v~,

where At is an angular momentum operator defined as

/‘; = g’“‘$, . F- ;[La . Lb - (g-y] &, (A.14a )

LD E r A @b - g’m’bl’, (A.14b)

S, E ieijkfk. (A.14~)

Now as both Qte’ and Q M’ have values in the Cartan subalgebra, both are simultaneously diagonalizable. We may separate the radial and angular dependence in (A.13) by

v$A r) = &.*\.& VI R inb r)- (A. 15)

where ~&d?,.n.o(~~ rp) = A”x~.,,.~ (8, (D). This gives the radial equation used in Section III:

Ri= -~e’.jrze’Ri.,I,,-e”~Ri ++e”RjRi. (A.16)

ACKNOWLEDGMENTS

The author feels strongly indebted to Dr. G. W. Gibbons for pointing out the problem and for help. encouragement, and advice at different stages. Gratefulness is also expressed to Dr. B. Whiting and Dr.

G. Bhattacharya for discussions. Financial support from the Indian government is also acknowledged.

REFERENCES

1. L. I. SCHIFF. H. SNYDER, AND J. WEINBERG, Phys. Rev. 57 (1970). 315.

2. J. E. MANDULA. Phvs. Lett. B 67 (1977), 175: Ph.vs. Reo. D 16 (1977). 937. 3. W. M. FRANK AND D. J. LAND. Rev. Mod. Phvs. 43 (1971), 36.

4. V. MONCRIEF. Phys. Rev. D 9 (1974), 2707; D 12 (1975), 1526. 5. R. A. BRANDT AND F. NERI. Nucl. Phys. B 161 (1974), 253; B 145 (1978). 221.

6. P. HAJICEK. Quantum wormholes. University of Bern, preprint. 1980. 7. M. ABRAM~VICH AND I. A. STEGUN. “Handbook of Mathematical Functions.” Dover. New York.

1964. 8. M. MAGG. Phvs. Left. B 74 (1978). 246. 9. M. J. PERRY, Phys. Left. B 71 (1977). 234,

IO. A. J. G. HEY AND J. E. MANDULA, Phys. Reo. D 19 (1979). 1856.