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Bound states of non-abelian dyons with fermions and bosons
ANNALS
OF PHYSICS
169, 117-139 (1986)
Bound States of Non-Abelian Dyons with Fermions and Bosons C. M. AJITHKUMAR Department
of Physics,
Universir.v
AND M. SABIR
qf Cochin,
Cochin
682 022, Kerala,
India
Received August 23, 1985
The bound states of spin 4 and spin zero particles in the field of a point SU((2) dyon are studied for isospinor and isovector representations. Energy levels and eigenfunctions for all angular momenta have been calculated for isospinor fermions, isospinor bosons, and isovector bosons. For isovector fermions the bound state spectrum for the lowest angular momentum is determined. The method of separation of angular and radial parts using spherical harmonics adopted in this paper is compared with the alternative method of separation using monopole harmonics and the equivalence of the two procedures is established. A relation connecting monopole harmonics and spherical harmonics is derived. g-1 1986 Academic Press, Inc.
I. INTRODUCTION The study of bound states of magnetic monopoles with fermions and bosons has a long history. Dirac, in his seminal paper on monopoles [ 11, showed that with the usual boundary condition of quantum mechanics, there exist no bound states of monopoles with electrons. The conclusion of Dirac, reinforced by other workers [2], had to be abandoned in the seventies following the theoretical observation of several unusual properties of the charge-pole system. It was shown that if the boundary conditions are chosen to ensure the self-adjointness of the Hamiltonian operator [3, 133, there can exist a spectrum of bound states with the lowest angular momentum value. There are several subtle problems in the charge-pole system which call for careful treatment and the question of the bound state formation between Dirac monopoles and charged particles is far from closed. It may be mentioned that the study of such bound states is also important in the context of experimental searches for monpoles [4]. In recent times, there has been a resurgence of interest in magnetic monopoles largely as a result of the theoretical discovery [5] of monopoles as finite energy, topologically stable solutions to certain spontaneously broken non-abelian gauge theories. The quantum mechanics of fermions and bosons in the background of non-abelian monopoles and dyons [6] has been investigated by several workers. The bound state spectrum of a fermion in the background of a Wu-Yang monopole [7] or dyon [S] of pure gauge theory was determined by Dereli, Swank, and Swank [9]. They showed that while Wu-Yang monopoles have no bound states 117 0003-4916186 $7.50 Copyright 0 1986 by Academic Press, Inc. All rights of reproduction in any form rcservcd.
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with fermions, dyons can have such bound states. For a ‘t Hooft-Polyakov monopole [S], Jackiw and Rebbi [lo] demonstrated the existence of nondegenerate zero energy bound states of monopoles with isospinor or isovector fermions. These solutions, incidentally, imply a doublet of solitons with fractional fermion number. A general analysis of the Dirac equation or Klein-Gordon equation in the background of the ‘t Hooft-Polyakov monopole is not possible at the moment because the regular monopole solution has not been cast in a closed form. In the Prasad-Sommerlield (PS) limit [ 111, where a closed expression is available for the monopole solution, scattering solutions for the lowest partial wave were recently constructed by Marciano and Muzinich [ 121. Bound states have not been obtained in [12], probably because the Higgs-Fermi coupling is neglected in this work. Most of the studies have been done in the point limit of a ‘t Hooft-Polyakov monopole by allowing the size of the monopole core to tend to zero. In this limit the system is essentially abelian, and with special boundary conditions, there exist bound states in the lowest angular momentum channel [ 131. Callias [ 131 has argued that for a regular monopole a finite number of bound states will exist. In the asymptotic (point) limit of the PS monopole it has been shown by Cox and Yildiz [ 143 that the bound states can occur for all values of J. It is the additional - l/r term present in the asymptotic Higgs field which is responsible for the bound states. Cox and Yildiz [ 141, however, determined only the energy eigenvalues and did not construct the eigenfunctions. For PS dyon [17] solutions in the point limit also there exist an infinite number of bound states with all J values. In addition to the regular monopole solution, there exist point singular monopoles and dyons [lS] in the PS limit. In a recent work, Din and Roy [ 163 showed that an isospinor fermion in the background of a singular non-abelian monopole has a well-defined Hamiltonian with ordinary boundary conditions imposed on the wavefunctions at the origin. Monopole fermion bound states were shown to occur for all .Z values. In this paper we study the quantum mechanics of spin j and spinless particles in the background of a point dyon potential. The form of the background dyon potential chosen is such that the asymptotic PS dyon solution arises as a special case. The background may be interpreted as due either to a point singular dyon or to a regular dyon solution with the size of the core neglected. In addition to the isospinor fermions and bosons which have already been discussed in the literature [17, 181 we have also studied isovector fermions and bosons. Exact bound state solutions to the Jackiw-Rebbi equations for all J are obtained for isovector bosons and isospinor bosons and fermions. For isovector fermions a bound state has been obtained for the lowest angular momentum. Furthermore, we have shown that no bound state having Z3= 0 exists for this system. As mentioned above, part of our work concerning isospinor fermions and bosons overlaps that of Tang [17, 181, who considered the same problem with the asymptotic PS dyon as background. However, our method of solution is different. Tang introduced a singular string in the gauge potential by using a singular gauge transformation. The resulting abelianised equations are separated into radial and
BOUND STATESOFNON-ABELIAN
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119
angular parts with the help of monopole harmonics [193. We, on the other hand, follow the method of Jackiw and Rebbi [lo], who used spherical harmonics to separate radial and angular parts. The equivalence of the two procedures is demonstrated by utilising a relationship between monopole harmonics and spherical harmonics which we have deduced. In Section II we review briefly the classical SU(2) gauge theory and obtain the point singular background dyon potential. In Section III we obtain bound state solutions to the relevant Dirac and Klein-Gordon equations. We discuss various results in Section IV. The proof of a relationship between spherical harmonics and monopole harmonics that we have deduced is given in the Appendix.
I. THE BACKGROUND The Lagrangian
DYON POTENTIAL
of the SU(2) Yang-Mills
theory with a Higgs triplet is given by (2.1)
where F”,,=~,A~-~,A”,+~E,~,A~A: D,,@‘=dp&‘+gs,,,A;@.
The spherically symmetric ‘t Hooft-Polyakov-Julia-Zee
Aa,$ J(r) a gay*’
reduces the equations of motion r*Jtr=
A:=$Q.A(r),
(2.2)
to 2JK2
r*H” = 2HK2 - m*r*H -I-L fp g2 r*r
ansatz [ 5, 63
(2.3)
= K(K* - 1 f HZ - J2),
where K(r) = 1 - rA(r) and r is the radial variable. In the limit m2 -+ 0, 1, + 0 with m*/I finite (PS limit), obtained [ 111,
exact dyon solutions were
J(r) = sinh q( - 1 + pr coth /3r) H(r) = cash q( - 1 + /Ir coth j3r) K(r) = /3r/sinh j?r,
where j = gmf,,h.
595/169!1-9
(2.4)
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In this paper we consider the solution K(r) = 0
(A(r)
= l/r)
H(r)=ar+b
(2.5)
J(r) = cr + d,
where a, b, c, and dare arbitrary constants, as the background potential. Unlike the PS solution (2.4), this solution is singular at r = 0 and the classical energy is infinite. But at large distances this solution mimics the behaviour of a regular solution. In fact at large distances r % l//I, this solution coincides with the PS solution, with the identification a = p cash r~, b= -cash rl d= -sinh
c = /I sinh n,
(2.6)
r].
So if the particles do not penetrate the dyon core, then (2.5) with (2.6) will be a good approximation to the regular solution. We can also consider (2.5) as a point dyon [15] solution. The relevance of such a solution is unclear at present, mainly because their classical energy is infinite. To the author’s knowledge the quantum field theory of such objects has not been worked out so far. The fact that the singularity is at the origin seems to be a favourable point since one encounters a similar situation in the case of the electron. The electric charge of this field configuration is given by Q=
+.
(2.7)
To include fermions we add to (2.1) the fermionic yti = v&NJ-W
ICI, -gGvk
Lagrangian
CLGW~~
(2.8)
where QL
=Q,kigT:,,,A;tim
and the T” are SU(2) generators satisfying [T”, Tb] = i&,& T’ T;,,, = $T”,,, = k,,,
for
Z = 4 representation
for
Z= 1 representation.
We will consider fermions in the above two representations background potential (2.4). For bosons instead of (2.8), we consider &=D,,U,*DW,,-M2
IUl’-gGiJ,*T~,qPU,-+h*#*
(2.9)
moving
Iv*.
in the
(2.10)
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III.
DYONS
121
SOLUTIONS
A. Isospinor Fermions In this case the Dirac equation is given by iy”D,$,
- $gGz;,,,1$“1,5~ = MI),.
Putting tin(x) = e-iEftj,(x) and substituting obtain the equation for t,k(x) as
= (E-W)
(3.1)
the gauge and Higgs field ansatz, we
Icl(x),
(3.2)
where a and /? are the Dirac matrices
a=YO.
a = y’y.
We now proceed exactly as in Ref. [lo] to separate the radial and angular parts. But we use a different representation of Dirac matrices,
a=(-oiaz), p=(y A).
(3.3)
It will be seen later that this choice is advatageous when we come to the solution of the radial equations. Splitting the wavefunction into upper and lower components, (3.4) we find that the Dirac equation becomes
(3.5) Here the first index on 1 refers to the spin part and the second one to the isospin part. xi,, is then expressed in terms of two scalar and vector functions, x~~=(g*(x)6,+g’(x).o,)t~,.
(3.6)
The scalar and vector functions are now expressible in terms of ordinary and vector spherical harmonics, respectively,
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g’(x)
= G;(r)
AND
SABIR
Yy(sZ) (3.7)
g:(.)-P:(r)i,Y~(o)+B:(r)~
Y,“(Q)
Here j = Jm and J is the total angular momentum. Total angular momentum is obtained by combining orbital and spin angular momenta and isospin. In this case it takes values 0, 1, 2, etc. B,$ = Cz = 0 by definition. Substituting (3.6) and (3.7) in (3.5) we get eight radial equations, d l-GH(r) dr+;+x-
p+BfTMG;+
(all J)
r
$G:+EP:
(all J) (3.8)
$bT
d I 1+Gff(r) dr
r-
2r
c;-fGj+MB:=
+Ec:
f
,
J>O
,
J>O,
where we have substituted for A(r) from (2.5). These equations, unlike the corresponding equations in Ref. [lo], have a + sign before E and J(r). This is because of our choice of the representation for Dirac matrices. The advantage of this choice is that the equations can now be transformed into a set of four independent coupled pairs of first order differential equations. Each of them can be decoupled and the resulting second order differential equations can be solved exactly. In contrast the decoupling of equations in Ref. [lo] gives a fourth order differential equation [ 161. We now discuss J= 0 and J> 0 solutions separately. J = 0 Solutions.
Setting P,+G$=R’ (3.9)
p$-G$=S’
the J= 0 radial equations become [g+i$j+m+)] [-g+fTf+m-)]P
R* = $+c+) = T(P+E-)
RT
(3.10a)
S’,
(3.10b)
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where we have substituted for J(r) and H(r) from (2.5). Also, B= GM, D=dl, and E+ = elf E. Here we need solve only for the first set. The m * =(aGI)+M, solution of the second set can be obtained by suitable replacements. For solving (3.10a), consider one more dependent variable transformation X’=R+fR-. Equation
(3.11)
(3.10a) now becomes (3.12)
where we have suppressed the symbol over m and E. This equation is exactly the same as that of the hydrogen atom problem in Dirac theory [20] if the B/r term is absent, and it can be solved by a similar technique. For E < m we get discrete bound states and for E > m we get continuum states. Here we give only the bound state solutions. The continuous spectrum can be obtained by suitable replacements [20]. The solution for E
c~/*~Y-~(Q~
+ ~~1,
(3.13)
where
(3.14) Bm-DE
y + (Bm - DE)/~ Q2=
(Dm-BE),3L
,F,
3,
~%+A
P
and 1F,(a, b, p) are Kummer functions. For (3.13) to vanish at the origin, should set B* > D*. The corresponding solutions to (3.10) are R’ =ccP+‘*py+--L(~~
(Q: + Q: &dG
S* =jW~-‘2py-1{,,h~
(Q; + Q;) + d=
where
(Q; -Q:)} (Q; - Q;)},
we
(3.15a) (3.15b)
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and a and p are arbitrary constants (fixed by normalisation). condition after the angular integration becomes
r2 dr[R**R*
The normalisation
+ S**S*].
For the convergence of this integral the Kummer functions should reduce to polynomials. From this requirement we obtain the eigenvalue spectrum. The conditions are y+
Bm, -DE+ 1 +
=-n,
(3.18a)
Y+
Bm--DEA
=-n2,
(3.18b)
where n, and n2 are positive integers. Zero is not possible because in this case y + (Bm - D&)/A = (Dm - B&)/l and Q2 remains divergent. Solving for E+ and E_ we get -=E+
- BD/(n,
+ y)’ k ,/l - (B2 - D2)/(nl 1 + D2/(n, + y)2
- BD/(n,
+ y)2 f Jl - ( B2 - D2)/(n2 + y)’ 1 + D2/(n2 + y)’
mi E-
-= m-
+ y)’
(3.19a) (3.19b)
Inspection of Eqs. (3.19a) and (3.19b) shows that it is not possible to satisfy these two equations simultaneously for the same value of energy. So we can take either (3.19a) and set /I=0 (P,’ =G,‘) or we take (3.19b) and set cr=O (P$ = -G$). The corresponding energy values in terms of the original parameters are E,,=
-;c+
Enz=;c+
M + aG/2
-bdG +\i;m] 1 + d2/4(n, + y)2 4(n, + y)’ -
(3.20a)
(3.20b)
We get different solutions for each sign in (3.20) because p + depends on energy. J > 0 Solutions. As remarked earlier, in this case the eight coupled differential equations can be transformed to a system of four independent coupled pairs of first order differential equations. This is achieved by defining eight new functions in the following way:
BOUND
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DYONS
125
X;=P;+Gf+B:+C: y;=p;+G;-B,"-C,' Z;=pf-G;+B:-CJ
(3.21)
W;=pj-G;-B:-C:.
The radial equations take the form
(3.22)
where
and E* and m + are the same as before. These equations can be solved as in the previous case. We need solve only the first equation. Solutions to the remaining three can be obtained by suitable replacements. We give only the final results (for Ecm), Jf; =
ae-P+/2py
1
IJ;n,+E,(Q:(i,+Q:,~~~(Q:(i)-P:) Y~=ge-p+‘2p’,-1[~~(Q:(-i)+Q:)~~~(Q:(-j)-Q:)1
1
Zj=rle-P~IZpy-1[~~(Q2(j)+Q;)+~~(Q;(j)-Q~)] W~=6eCp-‘2p:-1[~~(Q2(-j)+Q;)k,/~(Q;(-j)-Q;)], (3.23)
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and p + and I + are given by the same expressions as before. Q; ( -j) is obtained by changing the sign of j in Q:(j). As in the earlier case the requirement of normalisability of wavefunctions leads to the conditions y+
Bm, -DE+ i
=-n,
(3.25a)
y+
Brn- -DE~
= -n2.
(3.25b)
+
-
Here n, and n2 can be zero contrary to the previous case. Depending on the sign of (Dm - BE) one normalisable solution exists. Suppose (Dm - BE) > 0. In this case Q2(j) = 0 and Q2( -j) is nonzero and divergent. If (Dm - BE) < 0, Q2( -j) = 0 and Q2(j) is nonzero and divergent. Since E is given by a quadratic equation it can have two roots. For n, = 0 and n2 = 0, Ek is given by E+ =m+ E-
=m
BDlj2 f ,,/l + (B2 - D’)/j’ (1 + B2/‘j2) BDlj2fJ1+(B2-D2)/j2
Hence for both (m + , E,) and (rn-, (Dm-Bc)=m -
(3.26a
(l+B2’j2)
(3.26b
’
E-) we obtain DTJl
+(B’-D2)/j2
(1 + B2/j2) I
’
^
^ Consider the case D > 0. Then if 1DI b J 1 + (B’ - D’)/2
we find (Dm - BE) always positive.’ So solutions containing Q2( -j) should be discarded (by setting /?=S=O). S’imi ‘1ar 1y f or D < 0 we should discard solutions containing Q2( j) (tl = v = 0). If 1D 1< jl + ( B2 - D’)/j~,, , where ji,, = JmaX(JmaX+ 1) is the highest value satisfying this condition (all lower angular momenta will evidently satisfy this), we should again consider two cases. For energy obtained by taking the upper sign in (3.26) we discard solutions containing Q2( j). For the lower sign the solution containing Q2( -j) should be discarded. This is the same for both positive and negative values of D and for all angular momentum J < J,,, . For integers n,, n2 >O the energy levels are the same as given in (3.20a) and (3.20b), except that for the expression for y and for j = 0 we get exactly (3.20). Here again we should consider two separate cases since (3.20a) and (3.20b) cannot both be satisfied simultaneously for the same value of energy. Corresponding to each sign in (3.20a), we get two levels: a#O, /3=~=6=0 and cr=O, /?#O, ~=6=0. In either case we get P: = Gf and C$ = B: . Similarly for each sign in (3.20b) we get two solutions cr=p=O, q#O, 6=0 and cr=p=q=O, 6~0. In this case we get ’ The lowest nonzero value of j2 = J(J+ 1) = 2. If this value momenta will automatically satisfy the same.
satisfies
the condition
all higher
angular
BOUND
STATES
solutions satisfying Pz = -GJ’ will be given in Section IV. B. Isovector
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DYONS
and Cj = -BJ’ . Further discussion of these results
Fermions
In order to facilitate the solution we use a different set of Dirac matrices: a=( With this representation
-9,
P=(i
(3.28)
ul).
the Dirac equation can be reduced to the form
[a. V6,, - A(r)(r^,o, -
E),
- o,i,)]
x,’
J(r) k,,,,
E,,,,?, + A46,, + -
r
i, + Ed,,,
x,’ .
(3.29)
Separation of radial and angular parts is achieved by the use of vector spinor harmonics
(3.30) where $,,,, and +Y/;,,, are spinor harmonics
+%%4(~I= (3.31)
and L, = -is,,iriaj is the angular momentum definition. The radial equations are F,‘_=
-(M+E)F:+
F,‘,=-(M&E)F,‘_
operator. F;+ = F;- = 0 for J= i by
(all J)
(3.32a)
(all J)
(3.32b)
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(all J)
(3.33a)
(all J)
(3.33b)
(3.34a)
(3.34b) where j= J+ $. Here FF+ and FF- do not depend on either the Higgs field or the time component of the gauge field. We will see later that FFp and FF+ correspond to solutions having the third component of isospin I3 = 0. J= 4 Solutions (j = 1). Solutions to (3.32) are readily obtained by directly decoupling them. There are no normalisable solutions corresponding to bound states (E < M); only a continuous spectrum exists. Equations (3.33), however, possess bound state solutions. Energy levels are similar to those of zero angular momentum isospinor fermions. To see this, define X’
= F;- + F3’,
Y’=F$--F;+.
The four mutually equations
coupled equations
now change to two independent
(3.35) coupled
(3.36a) (3.36b) where we have substituted for H(r) and J(r). These equations are similar to (3.12) studied earlier. The solutions to (3.36a) and (3.36b) are not normalisable
BOUND
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DYONS
simultaneously. So the two solutions are X’ ~0, Y’ =0 and X’ =O, Y’ #O. The corresponding energy levels are given by (3.19b) and (3.19a), respectively. J> =jSolutions. As in the J= 4 case there are no normalisable state solutions for Eqs. (3.32); only continuum solutions exists. We have not been able to solve the remaining equations for J> f since decoupling them leads to at least fourth order differential equations. C. lsospinor Bosons In this case the Klein-Gordon D,DW(x)= can be simplified
equation -
M2+g2h2c$2+gG$+P’
U(X)
(3.37)
to
(3.38) where U(x) = eiE’U(x). The angular part can be separated using spinor harmonics: U(x) = F, (r) ??,u,,(s2)+ F;(Y) %&2). F+(r) and F-(r)
(3.39)
obey the radial equations
(3.40) where y*=J(J+
I)+b2h2-;
fi=$2abh2 (3.41) &2= E= + c2/4 m2 = M2 + h2a2. Here we need not consider lowest angular momentum (J = f) and higher angular momenta (J> t) separately, since both equations are valid for all J.
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To solve (3.40) we define two new functions x’
=
F+&Fr
(3.42)
.
The radial equations become d2Xk &2
a-y2 ?.z
+
1x*=0,
p+ +y++mZ+
[
(3.43)
where E; = (ET c/2)2 mZ, = m2 f Ga/2
(3.44)
p,=p+ (F-Ed. ) The solution is given by X’
2y+ 1, p+), -
=e-Pi’2pYi+“2,FI(Y+4-P~/2~~,
(3.45)
where P+=2J*r
(3.46)
A* = (mZ, - E; )"2.
For (3.45) to vanish at the origin we require b2h2 > d2/4. Also for normalisability, the Kummer functions should reduce to polynomials. From this we arrive at the condition y+f-Pk12Ai
= --n,
where n = 0, 1, 2,.... The energy levels are explicitly E,i=kc+
td(Gb-4h’ab)k i
(3.47)
given by
(-G2b2-16a2b2h4+8h2Gab2)
x (4n~)~1+(A4z+h2a2+~)(l+~)]“*}(l+~)~’
(3.48a)
for the upper sign in (3.47) and E; = - f c + k d(Gb + 4h’ab) f
( -G2b2 - 16a2b2h4 - 8h2Gab2)
x (4n~)~1+(M2+h2a2-~)(l+~)]1’2~(l+~)~1
(3.48b)
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DYONS
for the lower sign in (3.47) and ny = 4(n + y + 4). As in the previous cases (3.48a) and (3.48b) cannot be satisfied simultaneously. So we take one solution as X+ # 0, X- = 0 the corresponding energy levels are given by (3.48a). Similarly eigenfunctions with X+ = 0 and X- # 0 will have energy values (3.48b). Since (3.47) can be satisfied only with a positive b, the f sign appearing before the square root in (3.48a) and (3.48b) should be chosen to satisfy this. D. Isovector Bosom The static Klein-Gordon
equation in this case can be written in the form
- AZ(r)[U,+i,i,Uk]+
(3.49) The angular part can be separated using vector spherical harmonics, U, = X,(r)
i, Y,M(Q) + Y,(r) +
z,(r) Y,M(Q) + ij
6dd
WQ),
where Y,M are ordinary spherical harmonics and j= ,/m. J is the total angular momentum and it takes values 0, 1, 2, etc. Y, and Z, are zero by definition. The radial equations are r;+fx;-$x,+
E2-M2-7
hH’(r)
[ YJ+[E2+($+12-(?)‘I =
2EJ(r) - GH(r) r
z J
y
J
1
x J= o
(3SOa) Y,
(J>O)
(3SOb)
(J> 01,
(3SOc)
where we have substituted for A(r). We note that (3SOa) is independent of the electric degree of freedom of the monopole. We will later prove that X, corresponds to the third component of
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isospin zero. Contrary to the case of isospinor fermions there is an interaction with the Higgs field. This is due to the addition of a fourth power boson term (last term in (2.10)) in the Lagrangian. Note that X, does not couple to the linear term in the Higgs field. Since (3SOa) is the only equation for the lowest partial wave (.Z= 0) we conclude that the electric charge of the dyon has no effect on the lowest angular momentum boson. The lowest angular momentum bound state corresponds to a singlet with I, = 0. Solutions to (3SOa) are easily obtained in terms of Kummer functions. Substituting for H(r) we get 1 abh2 v+T+72y+l,p
XJ=LYe-p’2py+3’21F1
where p=2A.r A=,/M2+h2a2-E2
(3.52)
y=&Tmq
and CIis an arbitrary constant. The Kummer function should reduce to polynomials for normalisability. From this we get the condition 1
y+z+T=
abh2
(3.53)
-n,
where n = 0, 1, 2,.... To satisfy this condition, either a or b should be negative. In such a case only we get bound states. The expression for the bound state energy is E,=
f
M2+h2a2-
(n+&+y)2
There are other types of higher angular momentum and (3.50~). To deduce these we set y,+z,=
112
a2b2h4 I
(3.54)
.
states obtainable
from (3.50b)
w;.
(3.55)
Equations (3.50a) and (3.50b) now become d2W~+~dW~+1-j2 dr2 r dr
=
--W~+(E2+(3&‘-(!!9)
W;
(3.56)
BOUND
Substituting
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133
for J(r) and H(r), (3.57)
where y
=
j*
+
h2b2
-
d*
_
3
/I+ =2cd-2h2abT2EdfGb
(3.58)
E,=Efc m2, = M* + h2a2 f Ga.
The solutions are given by W~=a’e-Pf’2p>+3/21F,
.I+;--$,Zy+l,p+ +
For normalisability (3.60) For a bound state p+ should be positive. This fact will be used to calculate the energy levels which follow from (3.60). The energy corresponding to the upper sign solution in (3.59) is E+=c+{d(iGb-h*ab)&[(-y-h’a’b*+Gh*ab*) x (4n~)p1+$(M2+h2a2+Ga)
and that corresponding
(4i+ff nz)]“2][~+fJ’
(3.614
to the lower sign solution is
E,y = -c+{d(iGb+h2ab)$( x (4n~)-1+f(M2+h2a2-Ga)
-F-h4a2b2-Gh’ob’) (4l+c n;)]“2}[;+$]-’
(3.61b)
where nY= 4(n + 4 + y). The sign before the square root is chosen in such a way that /I* given in (3.58) is positive. As in the isospinor case, it is not possible to satisfy the two conditions (3.60) simultaneously. So we should set a+ # 0, a- = 0 for solutions having energy given in (3.61a) and the solutions corresponding to the energy levels (3.61b) have a + = 0, a- # 0.
134
AJITHKUMAR
AND SABIR
VI. RESULTS AND DISCUSSION In order to obtain more information about the solutions we gauge-transform them to the string gauge. For the isospinor fermions the transformation is given by (4.1) where (4.2)
- sin(8/2) e’( Substituting
for u and I,+ we find, explicitly, [-(P’+G’)e-i4sin(8/2) L
L
+ (B’ +C’Lu j
+ (B’-C’)
x (cos(ty2) -&
x (WV2)
- i cosec(B/2) $ (P’
[(P’-G’)cos(8/2)
+ G’) cos(dj2) + (B’+C’) j
j
)I
a as
+ i 9x(8/2) $ (P’ -G*
)I
y34
4).
sin(8/2) e’)
(B’ -C’) .i
e,”
a x ( - sin( O/2) ae
x bmm
a ae
- i sec(8/2) $
a + f cosec( O/2) g (4.3)
As already remarked the wavefunctions corresponding to the energy levels given by (3.25a) satisfy Pf = Gj and Bj = CF. Hence from the explicit form of the wavefunction given above, it is clear that it describes an isospin-up bound state. Similarly the energy levels given by (3.25b) are I, = -$ levels. If we consider the scattering behaviour of the solutions the above property will lead to charge-conserved scattering. This is the same as in the case of fermions in the point monopole background having lowest angular momentum [ 161. But in a regular dyon field the situation is different. Charge exchange scattering occurs in this case [12].
BOUND
STATES
OF NON-ABELIAN
135
DYONS
In the isovector case, the gauge transformation
is given by
where cos 19cots* 4 + sin’ 4 cos f$ sin ~(COS19- 1) cos 4 sin &(cos l3 - 1) cos 8 sin’ f$+ cos2 4 sin 8 cos fj sin 8 sin fj
u=
where the isospin components
-sin 0 cos f$ -sinesin cos
’
(4.4)
8
in $ are written as a 1 x 3 row matrix (F,‘, i, + F;+ d, + F3’-
L,)
YJ,+,
similar matrix with F, c) F,-
After the transformation
we get ayJM
(F$+ sin 4 + iF:- cos 4) ae p’=
- (F;+ sin rj + iF;-
cos 4)
+ (F;+ cos q5- iF:i
ayJM
Xsineag +
sin I#J)
1
similar matrix with F,? CI F,& with Y JM + y/;M ).
(4.6)
From this it is obvious that FF+ and FF- correspond to the solution with Z3= 0. In the isovector boson case we have the row matrix
i
u’=
f(Ycos$-iZsin()z
[f(Ysin~+iZcos~)~
-j&(Ysin4
+]&(Ycos4
ary t
+ izcos(6) - a4
1
ary - iZsin i, -Iad
X(r) YJ”
1
.
(4.7)
Here also the solution with X # 0, Y = 2 = 0 corresponds to a state with Z3 = 0. We shall now compare our solutions with those given by Tang and prove the equivalence of the two methods used to achieve the angular separation. For this it is enough to consider the isospinor bosons. Tang [18] studied this problem by 595/169/l-10
136
AJITHKUMAR
AND
SABIR
applying a singular gauge transformation which creates a singular string in the gauge potential. Then using monopole harmonics the separation of radial angular parts was achieved. Our model reduces to that considered in [18] if we set h = 0 and take the parameters a, b, c, and d as in (2.6). So we can compare the eigenvalues and eigenfunctions in these cases. Since the eigenfunctions are gauge-dependent this comparison must be made in the same gauge. For this we gauge-ransform our solutions to the gauge used in [18].
where u is the unitary matrix given in (4.1). This transformation
yields
(F, + Fp)
u’= (F, -I;-) (4.8)
where use has been made of the recurrence relations of associated Legendre polynomials [21]. This can be further shown to be equal to (see the Appendix)
(4.9)
where Yg,J,Mare monopole harmonics [19]. From the study of radial equations in Section IIIC we have seen that either (F, + Fp) or (F, -F-) is nonzero, but not both. The radial equations obeyed by (F, + F-) and F, -Fe) are the same as those studied in [18]. The angular parts are also the same. From (4.9) we also see that the states having energy values (3.48a) are isospin-up states and those having energy values (3.48b) are isospin-down states. Finally we comment on the degeneracy of the the system. For massless particles in a pure monopole (M = c = 0) field the pair of quantisation conditions for both fermions and bosons, (3.18a), (3.18b), (3.25a), (3.25b), (3.47), (3.60), can be satisfied simultaneously. The degeneracy of the energy levels in this case for fixed n and J is as given in Table I. The inclusion of either a nonzero (M # 0) or dyon degree freedom (c # 0) or both, changes the degeneracy of the spectrum. The pairs of the above-mentioned conditions are not satisfied simultaneously. The degeneracy is half of that given in Table I except for the I, = 0 state of the isovector fermions. Unlike the c = 0, M = 0 case the energy levels are not symmetrically distributed on both sides of zero. We also note that the total number of states for the case of
BOUND STATES OF NON-ABELIAN TABLE
DYONS
137
I Degeneracy of bond states for fixed n and J (n # 0)
Description of particles
Lowest total angular momentum
Isospinor fermions Isospinor bosons Isovector fermions Isovector bosons
O(
Lower angular momentum 2(2J+ 2(2J+ 2(2Jf (2J+
All higher angular momenta
I) 1) 1) 1)
4(2J+ 1) 2(2J+ 1) 84(2J+ 1) Two types of bound states (1) (ZJ+1)(1,=0) (2) 2(2J+ 1)
monopoles with M # 0 agrees with the counting as given in Ref. [ 141. For isovector fermions there are no bound states with I3 = 0 and agreement with Ref. 14 is obtained only if we count the unbound solutions along with the bound states.
APPENDIX
In this section we prove the following results: cos(8/2) Y,M_;12/’+ fi
{ - /z
= Yp112,J.M (Al)
e’( sin(0/2) Yy~rj{~ + /q
cos(0/2) YIM_:1i2} = Y,,2,J,M. (A.2)
Putting x = cos 8, j = J- 4, m = M - $ the left-hand side of (A.1 ) can be written
=(-1)” =(-1)”
(j+m+ (j+m+ ,/G
l)JiXP;“(x)-~P;“+L(x)
l)! (I-x~)~~~JCL 2mm!
m+ 1-j,m+j+2;m+2;T
{,F,(nl-j,m+j+l;m+l;~) l-x
e im4
m-j,m+j+2;m+
l;-
as
138
AJITHKUMAR
(j+m+
=(-1)” [ =(-l)“-“2 x p~P-$12,M
1
+ ll2(x)
&+-
47c(J+ $)!(J-
x P;$M-
I/2).
-CM+
l/2)
I,Z,J,,&
$)!
1 (1_x)‘“1’2
1 i(M
b-1 y-
M)!
1’2 (1 -x)~
4)!
Pl”i”,” ‘(x) eimd
2”
l/2)4
(25+ l)(J-M)!(J+M)!
2M
[
=
“2(1 -.x*)@&x
(2J+ l)(J-M)!(J+ 47c(.r+ i)!(J[
2M
=
l)!(j-Wz)! 4?rj! j!
AND SABIR
l/W7 1 + X)w+ I/2)/2
(M-1/2)/2(1
+*r)-wf+1/2)/2
~ l/Z))
e
4),
(A.3)
where *Fr(cr, p, y, x) are hypergeometric functions and P;fl(x) are Jacobi polynomials [22, 231. For deriving (A.3) we have used the relations [22, 231
p”(x) = (n+ ml! (1- xP2 2F1 n
(n -m)!
PTy(x) =
q(n+a+
2”m!
1)
n! r(a + 1)
(
1-X
m-n,m+n+l;m+l;Z
)
(A.4)
1-X
-n,n+2+fl+l;~+l;~
(A.51
2F~(a,8;~;~)+~~2F~(a+l,B+l;~+l;~)=~F~(a,~+l;y;z)
(-4.6)
and Cl93 l)a(X+
yq,J.M(x~ 4) =ZM r 1 P;~;&(x)=2-“-qx-
w+
lu-~Yv+w!
l’*
(1
l)fl P?P(x)
(A.7)
-x)p’Y+MI/2(1
+X)‘4-M’12
4n(J- q)!(J+ q)!
(‘4.8)
Equation (A.2) can be proved similarly. the relation
In this case instead of (A.6) we should use
(A.9) ACKNOWLEDGMENTS C. M. Research
A. is grateful to the Council of Scientitic and Industrial Research, Fellowship. He is also grateful to K. Babu Joseph for guidance.
New
Delhi,
for a Senior
BOUND STATESOFNON-ABELIAN
DYONS
139
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
P. A. M. DIRAC, Proc. R. Sot. London Ser. A 133 (1931), 60; Phys. Rev. 74 (1948), 817. P. P. BANDERET, Helv. Phys. Aacta 19 (1946), 503; HARISH CHANDRA, Phys. Rev. 74 (1948), 883. A. S. G~LDHABER, Phys. Rev. D 16 (1977). 1815. L. BRACCI AND G. FIORENTINI. Phys. Left. B 124 (1983), 493. G. ‘T HOOFT, Nucl. Phys. B 79 (1974), 276; A. M. POLYAKOV, Pis’ma Zh. Eksp. Teor. Fiz. 20 (1974), 430 [JETP Left. 20 (1974), 1941. B. JULIA AND A. ZEE, Phys. Rev. D 11 (1975), 2227. T. T. Wu AND C. N. YANG, in “Properties of Matter under Unusual Conditions” (H. Mark and S. Fernbach, Eds.), Interscience, New York, 1969. J. P. Hsu AND E. MAC, J. Math. Phys. 18 (1977), 100. T. DERELI, J. H. SWANK, AND L. J. SWANK, Phys. Rev. D 11 (1975), 3541. R. JACKIW AND C. REBBI, Phys. Rev. D 13 (1976). 3398. M. K. PRASAD AND C. M. SOMMERFIELD, Phys. Rev. Left. 47 (1981), 1364; E. B. BOGOMLNY, Yad. Fiz. 24 (1976), 861 [Sov. .I. Nucl. Phys. 24 (1976), 4491. W. J. MARCIANO AND I. J. MUZINICH, Phys. Rev. D 28 (1983). 973; Phys. Rev. Lerr. 50 (1983). 1035. C. J. CALLIAS, Phys. Rev. D 16 (1977), 3068. P. Cox AND A. YILDIZ. Phys. Rev. D 18 (1978). 1211. I. Ju. Phys. Rev. D 17 (1978), 1637. A. M. DIN AND S. M. ROY. Phys. Left. B 129 (1983), 201. J.-F. TANG, Phys. Rev. D 26 (1982). 510. J.-F. TANG, Phys. Lert. B 120 (1983), 364. T. T. Wu AND C. N. YANG, Nucl. Phw. B 107 (1976). 365. V. B. BERESTETSKII, E. M. LIFSHITZ, AND L. P. PITAEVSKII, “Relativistic Quantum Theory,” Part 1, pp. 110-l 15, Pergamon, Oxford, 1971. E. BUTKOV, “Mathematical Physics,” p. 376, Addison-Wesley, Reading, Mass. 1968. W. W. BELL, “Special Functions for Scientists and Engineers,” pp. 203-205. Van Nostrand, London, 1968. I. S. GRADSHTEYN AND 1. M. RYZHIK, “Table of Integrals, Series, and Products.” pp. 1044-1045, Academic Press, New York, 1965.
OF PHYSICS
169, 117-139 (1986)
Bound States of Non-Abelian Dyons with Fermions and Bosons C. M. AJITHKUMAR Department
of Physics,
Universir.v
AND M. SABIR
qf Cochin,
Cochin
682 022, Kerala,
India
Received August 23, 1985
The bound states of spin 4 and spin zero particles in the field of a point SU((2) dyon are studied for isospinor and isovector representations. Energy levels and eigenfunctions for all angular momenta have been calculated for isospinor fermions, isospinor bosons, and isovector bosons. For isovector fermions the bound state spectrum for the lowest angular momentum is determined. The method of separation of angular and radial parts using spherical harmonics adopted in this paper is compared with the alternative method of separation using monopole harmonics and the equivalence of the two procedures is established. A relation connecting monopole harmonics and spherical harmonics is derived. g-1 1986 Academic Press, Inc.
I. INTRODUCTION The study of bound states of magnetic monopoles with fermions and bosons has a long history. Dirac, in his seminal paper on monopoles [ 11, showed that with the usual boundary condition of quantum mechanics, there exist no bound states of monopoles with electrons. The conclusion of Dirac, reinforced by other workers [2], had to be abandoned in the seventies following the theoretical observation of several unusual properties of the charge-pole system. It was shown that if the boundary conditions are chosen to ensure the self-adjointness of the Hamiltonian operator [3, 133, there can exist a spectrum of bound states with the lowest angular momentum value. There are several subtle problems in the charge-pole system which call for careful treatment and the question of the bound state formation between Dirac monopoles and charged particles is far from closed. It may be mentioned that the study of such bound states is also important in the context of experimental searches for monpoles [4]. In recent times, there has been a resurgence of interest in magnetic monopoles largely as a result of the theoretical discovery [5] of monopoles as finite energy, topologically stable solutions to certain spontaneously broken non-abelian gauge theories. The quantum mechanics of fermions and bosons in the background of non-abelian monopoles and dyons [6] has been investigated by several workers. The bound state spectrum of a fermion in the background of a Wu-Yang monopole [7] or dyon [S] of pure gauge theory was determined by Dereli, Swank, and Swank [9]. They showed that while Wu-Yang monopoles have no bound states 117 0003-4916186 $7.50 Copyright 0 1986 by Academic Press, Inc. All rights of reproduction in any form rcservcd.
118
AJITHKUMAR
AND
SABIR
with fermions, dyons can have such bound states. For a ‘t Hooft-Polyakov monopole [S], Jackiw and Rebbi [lo] demonstrated the existence of nondegenerate zero energy bound states of monopoles with isospinor or isovector fermions. These solutions, incidentally, imply a doublet of solitons with fractional fermion number. A general analysis of the Dirac equation or Klein-Gordon equation in the background of the ‘t Hooft-Polyakov monopole is not possible at the moment because the regular monopole solution has not been cast in a closed form. In the Prasad-Sommerlield (PS) limit [ 111, where a closed expression is available for the monopole solution, scattering solutions for the lowest partial wave were recently constructed by Marciano and Muzinich [ 121. Bound states have not been obtained in [12], probably because the Higgs-Fermi coupling is neglected in this work. Most of the studies have been done in the point limit of a ‘t Hooft-Polyakov monopole by allowing the size of the monopole core to tend to zero. In this limit the system is essentially abelian, and with special boundary conditions, there exist bound states in the lowest angular momentum channel [ 131. Callias [ 131 has argued that for a regular monopole a finite number of bound states will exist. In the asymptotic (point) limit of the PS monopole it has been shown by Cox and Yildiz [ 143 that the bound states can occur for all values of J. It is the additional - l/r term present in the asymptotic Higgs field which is responsible for the bound states. Cox and Yildiz [ 141, however, determined only the energy eigenvalues and did not construct the eigenfunctions. For PS dyon [17] solutions in the point limit also there exist an infinite number of bound states with all J values. In addition to the regular monopole solution, there exist point singular monopoles and dyons [lS] in the PS limit. In a recent work, Din and Roy [ 163 showed that an isospinor fermion in the background of a singular non-abelian monopole has a well-defined Hamiltonian with ordinary boundary conditions imposed on the wavefunctions at the origin. Monopole fermion bound states were shown to occur for all .Z values. In this paper we study the quantum mechanics of spin j and spinless particles in the background of a point dyon potential. The form of the background dyon potential chosen is such that the asymptotic PS dyon solution arises as a special case. The background may be interpreted as due either to a point singular dyon or to a regular dyon solution with the size of the core neglected. In addition to the isospinor fermions and bosons which have already been discussed in the literature [17, 181 we have also studied isovector fermions and bosons. Exact bound state solutions to the Jackiw-Rebbi equations for all J are obtained for isovector bosons and isospinor bosons and fermions. For isovector fermions a bound state has been obtained for the lowest angular momentum. Furthermore, we have shown that no bound state having Z3= 0 exists for this system. As mentioned above, part of our work concerning isospinor fermions and bosons overlaps that of Tang [17, 181, who considered the same problem with the asymptotic PS dyon as background. However, our method of solution is different. Tang introduced a singular string in the gauge potential by using a singular gauge transformation. The resulting abelianised equations are separated into radial and
BOUND STATESOFNON-ABELIAN
DYONS
119
angular parts with the help of monopole harmonics [193. We, on the other hand, follow the method of Jackiw and Rebbi [lo], who used spherical harmonics to separate radial and angular parts. The equivalence of the two procedures is demonstrated by utilising a relationship between monopole harmonics and spherical harmonics which we have deduced. In Section II we review briefly the classical SU(2) gauge theory and obtain the point singular background dyon potential. In Section III we obtain bound state solutions to the relevant Dirac and Klein-Gordon equations. We discuss various results in Section IV. The proof of a relationship between spherical harmonics and monopole harmonics that we have deduced is given in the Appendix.
I. THE BACKGROUND The Lagrangian
DYON POTENTIAL
of the SU(2) Yang-Mills
theory with a Higgs triplet is given by (2.1)
where F”,,=~,A~-~,A”,+~E,~,A~A: D,,@‘=dp&‘+gs,,,A;@.
The spherically symmetric ‘t Hooft-Polyakov-Julia-Zee
Aa,$ J(r) a gay*’
reduces the equations of motion r*Jtr=
A:=$Q.A(r),
(2.2)
to 2JK2
r*H” = 2HK2 - m*r*H -I-L fp g2 r*r
ansatz [ 5, 63
(2.3)
= K(K* - 1 f HZ - J2),
where K(r) = 1 - rA(r) and r is the radial variable. In the limit m2 -+ 0, 1, + 0 with m*/I finite (PS limit), obtained [ 111,
exact dyon solutions were
J(r) = sinh q( - 1 + pr coth /3r) H(r) = cash q( - 1 + /Ir coth j3r) K(r) = /3r/sinh j?r,
where j = gmf,,h.
595/169!1-9
(2.4)
120
AJITHKUMAR
AND
SABIR
In this paper we consider the solution K(r) = 0
(A(r)
= l/r)
H(r)=ar+b
(2.5)
J(r) = cr + d,
where a, b, c, and dare arbitrary constants, as the background potential. Unlike the PS solution (2.4), this solution is singular at r = 0 and the classical energy is infinite. But at large distances this solution mimics the behaviour of a regular solution. In fact at large distances r % l//I, this solution coincides with the PS solution, with the identification a = p cash r~, b= -cash rl d= -sinh
c = /I sinh n,
(2.6)
r].
So if the particles do not penetrate the dyon core, then (2.5) with (2.6) will be a good approximation to the regular solution. We can also consider (2.5) as a point dyon [15] solution. The relevance of such a solution is unclear at present, mainly because their classical energy is infinite. To the author’s knowledge the quantum field theory of such objects has not been worked out so far. The fact that the singularity is at the origin seems to be a favourable point since one encounters a similar situation in the case of the electron. The electric charge of this field configuration is given by Q=
+.
(2.7)
To include fermions we add to (2.1) the fermionic yti = v&NJ-W
ICI, -gGvk
Lagrangian
CLGW~~
(2.8)
where QL
=Q,kigT:,,,A;tim
and the T” are SU(2) generators satisfying [T”, Tb] = i&,& T’ T;,,, = $T”,,, = k,,,
for
Z = 4 representation
for
Z= 1 representation.
We will consider fermions in the above two representations background potential (2.4). For bosons instead of (2.8), we consider &=D,,U,*DW,,-M2
IUl’-gGiJ,*T~,qPU,-+h*#*
(2.9)
moving
Iv*.
in the
(2.10)
BOUND STATESOFNON-ABELIAN
III.
DYONS
121
SOLUTIONS
A. Isospinor Fermions In this case the Dirac equation is given by iy”D,$,
- $gGz;,,,1$“1,5~ = MI),.
Putting tin(x) = e-iEftj,(x) and substituting obtain the equation for t,k(x) as
= (E-W)
(3.1)
the gauge and Higgs field ansatz, we
Icl(x),
(3.2)
where a and /? are the Dirac matrices
a=YO.
a = y’y.
We now proceed exactly as in Ref. [lo] to separate the radial and angular parts. But we use a different representation of Dirac matrices,
a=(-oiaz), p=(y A).
(3.3)
It will be seen later that this choice is advatageous when we come to the solution of the radial equations. Splitting the wavefunction into upper and lower components, (3.4) we find that the Dirac equation becomes
(3.5) Here the first index on 1 refers to the spin part and the second one to the isospin part. xi,, is then expressed in terms of two scalar and vector functions, x~~=(g*(x)6,+g’(x).o,)t~,.
(3.6)
The scalar and vector functions are now expressible in terms of ordinary and vector spherical harmonics, respectively,
122
AJITHKUMAR
g’(x)
= G;(r)
AND
SABIR
Yy(sZ) (3.7)
g:(.)-P:(r)i,Y~(o)+B:(r)~
Y,“(Q)
Here j = Jm and J is the total angular momentum. Total angular momentum is obtained by combining orbital and spin angular momenta and isospin. In this case it takes values 0, 1, 2, etc. B,$ = Cz = 0 by definition. Substituting (3.6) and (3.7) in (3.5) we get eight radial equations, d l-GH(r) dr+;+x-
p+BfTMG;+
(all J)
r
$G:+EP:
(all J) (3.8)
$bT
d I 1+Gff(r) dr
r-
2r
c;-fGj+MB:=
+Ec:
f
,
J>O
,
J>O,
where we have substituted for A(r) from (2.5). These equations, unlike the corresponding equations in Ref. [lo], have a + sign before E and J(r). This is because of our choice of the representation for Dirac matrices. The advantage of this choice is that the equations can now be transformed into a set of four independent coupled pairs of first order differential equations. Each of them can be decoupled and the resulting second order differential equations can be solved exactly. In contrast the decoupling of equations in Ref. [lo] gives a fourth order differential equation [ 161. We now discuss J= 0 and J> 0 solutions separately. J = 0 Solutions.
Setting P,+G$=R’ (3.9)
p$-G$=S’
the J= 0 radial equations become [g+i$j+m+)] [-g+fTf+m-)]P
R* = $+c+) = T(P+E-)
RT
(3.10a)
S’,
(3.10b)
BOUND
STATES
OF NON-ABELIAN
123
DYONS
where we have substituted for J(r) and H(r) from (2.5). Also, B= GM, D=dl, and E+ = elf E. Here we need solve only for the first set. The m * =(aGI)+M, solution of the second set can be obtained by suitable replacements. For solving (3.10a), consider one more dependent variable transformation X’=R+fR-. Equation
(3.11)
(3.10a) now becomes (3.12)
where we have suppressed the symbol over m and E. This equation is exactly the same as that of the hydrogen atom problem in Dirac theory [20] if the B/r term is absent, and it can be solved by a similar technique. For E < m we get discrete bound states and for E > m we get continuum states. Here we give only the bound state solutions. The continuous spectrum can be obtained by suitable replacements [20]. The solution for E
c~/*~Y-~(Q~
+ ~~1,
(3.13)
where
(3.14) Bm-DE
y + (Bm - DE)/~ Q2=
(Dm-BE),3L
,F,
3,
~%+A
P
and 1F,(a, b, p) are Kummer functions. For (3.13) to vanish at the origin, should set B* > D*. The corresponding solutions to (3.10) are R’ =ccP+‘*py+--L(~~
(Q: + Q: &dG
S* =jW~-‘2py-1{,,h~
(Q; + Q;) + d=
where
(Q; -Q:)} (Q; - Q;)},
we
(3.15a) (3.15b)
124
AJITHKUMAR
AND
SABIR
and a and p are arbitrary constants (fixed by normalisation). condition after the angular integration becomes
r2 dr[R**R*
The normalisation
+ S**S*].
For the convergence of this integral the Kummer functions should reduce to polynomials. From this requirement we obtain the eigenvalue spectrum. The conditions are y+
Bm, -DE+ 1 +
=-n,
(3.18a)
Y+
Bm--DEA
=-n2,
(3.18b)
where n, and n2 are positive integers. Zero is not possible because in this case y + (Bm - D&)/A = (Dm - B&)/l and Q2 remains divergent. Solving for E+ and E_ we get -=E+
- BD/(n,
+ y)’ k ,/l - (B2 - D2)/(nl 1 + D2/(n, + y)2
- BD/(n,
+ y)2 f Jl - ( B2 - D2)/(n2 + y)’ 1 + D2/(n2 + y)’
mi E-
-= m-
+ y)’
(3.19a) (3.19b)
Inspection of Eqs. (3.19a) and (3.19b) shows that it is not possible to satisfy these two equations simultaneously for the same value of energy. So we can take either (3.19a) and set /I=0 (P,’ =G,‘) or we take (3.19b) and set cr=O (P$ = -G$). The corresponding energy values in terms of the original parameters are E,,=
-;c+
Enz=;c+
M + aG/2
-bdG +\i;m] 1 + d2/4(n, + y)2 4(n, + y)’ -
(3.20a)
(3.20b)
We get different solutions for each sign in (3.20) because p + depends on energy. J > 0 Solutions. As remarked earlier, in this case the eight coupled differential equations can be transformed to a system of four independent coupled pairs of first order differential equations. This is achieved by defining eight new functions in the following way:
BOUND
STATES
OF NON-ABELIAN
DYONS
125
X;=P;+Gf+B:+C: y;=p;+G;-B,"-C,' Z;=pf-G;+B:-CJ
(3.21)
W;=pj-G;-B:-C:.
The radial equations take the form
(3.22)
where
and E* and m + are the same as before. These equations can be solved as in the previous case. We need solve only the first equation. Solutions to the remaining three can be obtained by suitable replacements. We give only the final results (for Ecm), Jf; =
ae-P+/2py
1
IJ;n,+E,(Q:(i,+Q:,~~~(Q:(i)-P:) Y~=ge-p+‘2p’,-1[~~(Q:(-i)+Q:)~~~(Q:(-j)-Q:)1
1
Zj=rle-P~IZpy-1[~~(Q2(j)+Q;)+~~(Q;(j)-Q~)] W~=6eCp-‘2p:-1[~~(Q2(-j)+Q;)k,/~(Q;(-j)-Q;)], (3.23)
126
AJITHKUMAR
AND
SABIR
and p + and I + are given by the same expressions as before. Q; ( -j) is obtained by changing the sign of j in Q:(j). As in the earlier case the requirement of normalisability of wavefunctions leads to the conditions y+
Bm, -DE+ i
=-n,
(3.25a)
y+
Brn- -DE~
= -n2.
(3.25b)
+
-
Here n, and n2 can be zero contrary to the previous case. Depending on the sign of (Dm - BE) one normalisable solution exists. Suppose (Dm - BE) > 0. In this case Q2(j) = 0 and Q2( -j) is nonzero and divergent. If (Dm - BE) < 0, Q2( -j) = 0 and Q2(j) is nonzero and divergent. Since E is given by a quadratic equation it can have two roots. For n, = 0 and n2 = 0, Ek is given by E+ =m+ E-
=m
BDlj2 f ,,/l + (B2 - D’)/j’ (1 + B2/‘j2) BDlj2fJ1+(B2-D2)/j2
Hence for both (m + , E,) and (rn-, (Dm-Bc)=m -
(3.26a
(l+B2’j2)
(3.26b
’
E-) we obtain DTJl
+(B’-D2)/j2
(1 + B2/j2) I
’
^
^ Consider the case D > 0. Then if 1DI b J 1 + (B’ - D’)/2
we find (Dm - BE) always positive.’ So solutions containing Q2( -j) should be discarded (by setting /?=S=O). S’imi ‘1ar 1y f or D < 0 we should discard solutions containing Q2( j) (tl = v = 0). If 1D 1< jl + ( B2 - D’)/j~,, , where ji,, = JmaX(JmaX+ 1) is the highest value satisfying this condition (all lower angular momenta will evidently satisfy this), we should again consider two cases. For energy obtained by taking the upper sign in (3.26) we discard solutions containing Q2( j). For the lower sign the solution containing Q2( -j) should be discarded. This is the same for both positive and negative values of D and for all angular momentum J < J,,, . For integers n,, n2 >O the energy levels are the same as given in (3.20a) and (3.20b), except that for the expression for y and for j = 0 we get exactly (3.20). Here again we should consider two separate cases since (3.20a) and (3.20b) cannot both be satisfied simultaneously for the same value of energy. Corresponding to each sign in (3.20a), we get two levels: a#O, /3=~=6=0 and cr=O, /?#O, ~=6=0. In either case we get P: = Gf and C$ = B: . Similarly for each sign in (3.20b) we get two solutions cr=p=O, q#O, 6=0 and cr=p=q=O, 6~0. In this case we get ’ The lowest nonzero value of j2 = J(J+ 1) = 2. If this value momenta will automatically satisfy the same.
satisfies
the condition
all higher
angular
BOUND
STATES
solutions satisfying Pz = -GJ’ will be given in Section IV. B. Isovector
OF NON-ABELIAN
127
DYONS
and Cj = -BJ’ . Further discussion of these results
Fermions
In order to facilitate the solution we use a different set of Dirac matrices: a=( With this representation
-9,
P=(i
(3.28)
ul).
the Dirac equation can be reduced to the form
[a. V6,, - A(r)(r^,o, -
E),
- o,i,)]
x,’
J(r) k,,,,
E,,,,?, + A46,, + -
r
i, + Ed,,,
x,’ .
(3.29)
Separation of radial and angular parts is achieved by the use of vector spinor harmonics
(3.30) where $,,,, and +Y/;,,, are spinor harmonics
+%%4(~I= (3.31)
and L, = -is,,iriaj is the angular momentum definition. The radial equations are F,‘_=
-(M+E)F:+
F,‘,=-(M&E)F,‘_
operator. F;+ = F;- = 0 for J= i by
(all J)
(3.32a)
(all J)
(3.32b)
128
AJITHKUMAR
AND
SABIR
(all J)
(3.33a)
(all J)
(3.33b)
(3.34a)
(3.34b) where j= J+ $. Here FF+ and FF- do not depend on either the Higgs field or the time component of the gauge field. We will see later that FFp and FF+ correspond to solutions having the third component of isospin I3 = 0. J= 4 Solutions (j = 1). Solutions to (3.32) are readily obtained by directly decoupling them. There are no normalisable solutions corresponding to bound states (E < M); only a continuous spectrum exists. Equations (3.33), however, possess bound state solutions. Energy levels are similar to those of zero angular momentum isospinor fermions. To see this, define X’
= F;- + F3’,
Y’=F$--F;+.
The four mutually equations
coupled equations
now change to two independent
(3.35) coupled
(3.36a) (3.36b) where we have substituted for H(r) and J(r). These equations are similar to (3.12) studied earlier. The solutions to (3.36a) and (3.36b) are not normalisable
BOUND
STATES
OF NON-ABELIAN
129
DYONS
simultaneously. So the two solutions are X’ ~0, Y’ =0 and X’ =O, Y’ #O. The corresponding energy levels are given by (3.19b) and (3.19a), respectively. J> =jSolutions. As in the J= 4 case there are no normalisable state solutions for Eqs. (3.32); only continuum solutions exists. We have not been able to solve the remaining equations for J> f since decoupling them leads to at least fourth order differential equations. C. lsospinor Bosons In this case the Klein-Gordon D,DW(x)= can be simplified
equation -
M2+g2h2c$2+gG$+P’
U(X)
(3.37)
to
(3.38) where U(x) = eiE’U(x). The angular part can be separated using spinor harmonics: U(x) = F, (r) ??,u,,(s2)+ F;(Y) %&2). F+(r) and F-(r)
(3.39)
obey the radial equations
(3.40) where y*=J(J+
I)+b2h2-;
fi=$2abh2 (3.41) &2= E= + c2/4 m2 = M2 + h2a2. Here we need not consider lowest angular momentum (J = f) and higher angular momenta (J> t) separately, since both equations are valid for all J.
130
AJITHKUMAR
AND
SABIR
To solve (3.40) we define two new functions x’
=
F+&Fr
(3.42)
.
The radial equations become d2Xk &2
a-y2 ?.z
+
1x*=0,
p+ +y++mZ+
[
(3.43)
where E; = (ET c/2)2 mZ, = m2 f Ga/2
(3.44)
p,=p+ (F-Ed. ) The solution is given by X’
2y+ 1, p+), -
=e-Pi’2pYi+“2,FI(Y+4-P~/2~~,
(3.45)
where P+=2J*r
(3.46)
A* = (mZ, - E; )"2.
For (3.45) to vanish at the origin we require b2h2 > d2/4. Also for normalisability, the Kummer functions should reduce to polynomials. From this we arrive at the condition y+f-Pk12Ai
= --n,
where n = 0, 1, 2,.... The energy levels are explicitly E,i=kc+
td(Gb-4h’ab)k i
(3.47)
given by
(-G2b2-16a2b2h4+8h2Gab2)
x (4n~)~1+(A4z+h2a2+~)(l+~)]“*}(l+~)~’
(3.48a)
for the upper sign in (3.47) and E; = - f c + k d(Gb + 4h’ab) f
( -G2b2 - 16a2b2h4 - 8h2Gab2)
x (4n~)~1+(M2+h2a2-~)(l+~)]1’2~(l+~)~1
(3.48b)
BOUND
STATES
OF NON-ABELIAN
131
DYONS
for the lower sign in (3.47) and ny = 4(n + y + 4). As in the previous cases (3.48a) and (3.48b) cannot be satisfied simultaneously. So we take one solution as X+ # 0, X- = 0 the corresponding energy levels are given by (3.48a). Similarly eigenfunctions with X+ = 0 and X- # 0 will have energy values (3.48b). Since (3.47) can be satisfied only with a positive b, the f sign appearing before the square root in (3.48a) and (3.48b) should be chosen to satisfy this. D. Isovector Bosom The static Klein-Gordon
equation in this case can be written in the form
- AZ(r)[U,+i,i,Uk]+
(3.49) The angular part can be separated using vector spherical harmonics, U, = X,(r)
i, Y,M(Q) + Y,(r) +
z,(r) Y,M(Q) + ij
6dd
WQ),
where Y,M are ordinary spherical harmonics and j= ,/m. J is the total angular momentum and it takes values 0, 1, 2, etc. Y, and Z, are zero by definition. The radial equations are r;+fx;-$x,+
E2-M2-7
hH’(r)
[ YJ+[E2+($+12-(?)‘I =
2EJ(r) - GH(r) r
z J
y
J
1
x J= o
(3SOa) Y,
(J>O)
(3SOb)
(J> 01,
(3SOc)
where we have substituted for A(r). We note that (3SOa) is independent of the electric degree of freedom of the monopole. We will later prove that X, corresponds to the third component of
132
AJITHKUMAR
AND
SABIR
isospin zero. Contrary to the case of isospinor fermions there is an interaction with the Higgs field. This is due to the addition of a fourth power boson term (last term in (2.10)) in the Lagrangian. Note that X, does not couple to the linear term in the Higgs field. Since (3SOa) is the only equation for the lowest partial wave (.Z= 0) we conclude that the electric charge of the dyon has no effect on the lowest angular momentum boson. The lowest angular momentum bound state corresponds to a singlet with I, = 0. Solutions to (3SOa) are easily obtained in terms of Kummer functions. Substituting for H(r) we get 1 abh2 v+T+72y+l,p
XJ=LYe-p’2py+3’21F1
where p=2A.r A=,/M2+h2a2-E2
(3.52)
y=&Tmq
and CIis an arbitrary constant. The Kummer function should reduce to polynomials for normalisability. From this we get the condition 1
y+z+T=
abh2
(3.53)
-n,
where n = 0, 1, 2,.... To satisfy this condition, either a or b should be negative. In such a case only we get bound states. The expression for the bound state energy is E,=
f
M2+h2a2-
(n+&+y)2
There are other types of higher angular momentum and (3.50~). To deduce these we set y,+z,=
112
a2b2h4 I
(3.54)
.
states obtainable
from (3.50b)
w;.
(3.55)
Equations (3.50a) and (3.50b) now become d2W~+~dW~+1-j2 dr2 r dr
=
--W~+(E2+(3&‘-(!!9)
W;
(3.56)
BOUND
Substituting
STATES
OF NON-ABELIAN
DYONS
133
for J(r) and H(r), (3.57)
where y
=
j*
+
h2b2
-
d*
_
3
/I+ =2cd-2h2abT2EdfGb
(3.58)
E,=Efc m2, = M* + h2a2 f Ga.
The solutions are given by W~=a’e-Pf’2p>+3/21F,
.I+;--$,Zy+l,p+ +
For normalisability (3.60) For a bound state p+ should be positive. This fact will be used to calculate the energy levels which follow from (3.60). The energy corresponding to the upper sign solution in (3.59) is E+=c+{d(iGb-h*ab)&[(-y-h’a’b*+Gh*ab*) x (4n~)p1+$(M2+h2a2+Ga)
and that corresponding
(4i+ff nz)]“2][~+fJ’
(3.614
to the lower sign solution is
E,y = -c+{d(iGb+h2ab)$( x (4n~)-1+f(M2+h2a2-Ga)
-F-h4a2b2-Gh’ob’) (4l+c n;)]“2}[;+$]-’
(3.61b)
where nY= 4(n + 4 + y). The sign before the square root is chosen in such a way that /I* given in (3.58) is positive. As in the isospinor case, it is not possible to satisfy the two conditions (3.60) simultaneously. So we should set a+ # 0, a- = 0 for solutions having energy given in (3.61a) and the solutions corresponding to the energy levels (3.61b) have a + = 0, a- # 0.
134
AJITHKUMAR
AND SABIR
VI. RESULTS AND DISCUSSION In order to obtain more information about the solutions we gauge-transform them to the string gauge. For the isospinor fermions the transformation is given by (4.1) where (4.2)
- sin(8/2) e’( Substituting
for u and I,+ we find, explicitly, [-(P’+G’)e-i4sin(8/2) L
L
+ (B’ +C’Lu j
+ (B’-C’)
x (cos(ty2) -&
x (WV2)
- i cosec(B/2) $ (P’
[(P’-G’)cos(8/2)
+ G’) cos(dj2) + (B’+C’) j
j
)I
a as
+ i 9x(8/2) $ (P’ -G*
)I
y34
4).
sin(8/2) e’)
(B’ -C’) .i
e,”
a x ( - sin( O/2) ae
x bmm
a ae
- i sec(8/2) $
a + f cosec( O/2) g (4.3)
As already remarked the wavefunctions corresponding to the energy levels given by (3.25a) satisfy Pf = Gj and Bj = CF. Hence from the explicit form of the wavefunction given above, it is clear that it describes an isospin-up bound state. Similarly the energy levels given by (3.25b) are I, = -$ levels. If we consider the scattering behaviour of the solutions the above property will lead to charge-conserved scattering. This is the same as in the case of fermions in the point monopole background having lowest angular momentum [ 161. But in a regular dyon field the situation is different. Charge exchange scattering occurs in this case [12].
BOUND
STATES
OF NON-ABELIAN
135
DYONS
In the isovector case, the gauge transformation
is given by
where cos 19cots* 4 + sin’ 4 cos f$ sin ~(COS19- 1) cos 4 sin &(cos l3 - 1) cos 8 sin’ f$+ cos2 4 sin 8 cos fj sin 8 sin fj
u=
where the isospin components
-sin 0 cos f$ -sinesin cos
’
(4.4)
8
in $ are written as a 1 x 3 row matrix (F,‘, i, + F;+ d, + F3’-
L,)
YJ,+,
similar matrix with F, c) F,-
After the transformation
we get ayJM
(F$+ sin 4 + iF:- cos 4) ae p’=
- (F;+ sin rj + iF;-
cos 4)
+ (F;+ cos q5- iF:i
ayJM
Xsineag +
sin I#J)
1
similar matrix with F,? CI F,& with Y JM + y/;M ).
(4.6)
From this it is obvious that FF+ and FF- correspond to the solution with Z3= 0. In the isovector boson case we have the row matrix
i
u’=
f(Ycos$-iZsin()z
[f(Ysin~+iZcos~)~
-j&(Ysin4
+]&(Ycos4
ary t
+ izcos(6) - a4
1
ary - iZsin i, -Iad
X(r) YJ”
1
.
(4.7)
Here also the solution with X # 0, Y = 2 = 0 corresponds to a state with Z3 = 0. We shall now compare our solutions with those given by Tang and prove the equivalence of the two methods used to achieve the angular separation. For this it is enough to consider the isospinor bosons. Tang [18] studied this problem by 595/169/l-10
136
AJITHKUMAR
AND
SABIR
applying a singular gauge transformation which creates a singular string in the gauge potential. Then using monopole harmonics the separation of radial angular parts was achieved. Our model reduces to that considered in [18] if we set h = 0 and take the parameters a, b, c, and d as in (2.6). So we can compare the eigenvalues and eigenfunctions in these cases. Since the eigenfunctions are gauge-dependent this comparison must be made in the same gauge. For this we gauge-ransform our solutions to the gauge used in [18].
where u is the unitary matrix given in (4.1). This transformation
yields
(F, + Fp)
u’= (F, -I;-) (4.8)
where use has been made of the recurrence relations of associated Legendre polynomials [21]. This can be further shown to be equal to (see the Appendix)
(4.9)
where Yg,J,Mare monopole harmonics [19]. From the study of radial equations in Section IIIC we have seen that either (F, + Fp) or (F, -F-) is nonzero, but not both. The radial equations obeyed by (F, + F-) and F, -Fe) are the same as those studied in [18]. The angular parts are also the same. From (4.9) we also see that the states having energy values (3.48a) are isospin-up states and those having energy values (3.48b) are isospin-down states. Finally we comment on the degeneracy of the the system. For massless particles in a pure monopole (M = c = 0) field the pair of quantisation conditions for both fermions and bosons, (3.18a), (3.18b), (3.25a), (3.25b), (3.47), (3.60), can be satisfied simultaneously. The degeneracy of the energy levels in this case for fixed n and J is as given in Table I. The inclusion of either a nonzero (M # 0) or dyon degree freedom (c # 0) or both, changes the degeneracy of the spectrum. The pairs of the above-mentioned conditions are not satisfied simultaneously. The degeneracy is half of that given in Table I except for the I, = 0 state of the isovector fermions. Unlike the c = 0, M = 0 case the energy levels are not symmetrically distributed on both sides of zero. We also note that the total number of states for the case of
BOUND STATES OF NON-ABELIAN TABLE
DYONS
137
I Degeneracy of bond states for fixed n and J (n # 0)
Description of particles
Lowest total angular momentum
Isospinor fermions Isospinor bosons Isovector fermions Isovector bosons
O(
Lower angular momentum 2(2J+ 2(2J+ 2(2Jf (2J+
All higher angular momenta
I) 1) 1) 1)
4(2J+ 1) 2(2J+ 1) 84(2J+ 1) Two types of bound states (1) (ZJ+1)(1,=0) (2) 2(2J+ 1)
monopoles with M # 0 agrees with the counting as given in Ref. [ 141. For isovector fermions there are no bound states with I3 = 0 and agreement with Ref. 14 is obtained only if we count the unbound solutions along with the bound states.
APPENDIX
In this section we prove the following results: cos(8/2) Y,M_;12/’+ fi
{ - /z
= Yp112,J.M (Al)
e’( sin(0/2) Yy~rj{~ + /q
cos(0/2) YIM_:1i2} = Y,,2,J,M. (A.2)
Putting x = cos 8, j = J- 4, m = M - $ the left-hand side of (A.1 ) can be written
=(-1)” =(-1)”
(j+m+ (j+m+ ,/G
l)JiXP;“(x)-~P;“+L(x)
l)! (I-x~)~~~JCL 2mm!
m+ 1-j,m+j+2;m+2;T
{,F,(nl-j,m+j+l;m+l;~) l-x
e im4
m-j,m+j+2;m+
l;-
as
138
AJITHKUMAR
(j+m+
=(-1)” [ =(-l)“-“2 x p~P-$12,M
1
+ ll2(x)
&+-
47c(J+ $)!(J-
x P;$M-
I/2).
-CM+
l/2)
I,Z,J,,&
$)!
1 (1_x)‘“1’2
1 i(M
b-1 y-
M)!
1’2 (1 -x)~
4)!
Pl”i”,” ‘(x) eimd
2”
l/2)4
(25+ l)(J-M)!(J+M)!
2M
[
=
“2(1 -.x*)@&x
(2J+ l)(J-M)!(J+ 47c(.r+ i)!(J[
2M
=
l)!(j-Wz)! 4?rj! j!
AND SABIR
l/W7 1 + X)w+ I/2)/2
(M-1/2)/2(1
+*r)-wf+1/2)/2
~ l/Z))
e
4),
(A.3)
where *Fr(cr, p, y, x) are hypergeometric functions and P;fl(x) are Jacobi polynomials [22, 231. For deriving (A.3) we have used the relations [22, 231
p”(x) = (n+ ml! (1- xP2 2F1 n
(n -m)!
PTy(x) =
q(n+a+
2”m!
1)
n! r(a + 1)
(
1-X
m-n,m+n+l;m+l;Z
)
(A.4)
1-X
-n,n+2+fl+l;~+l;~
(A.51
2F~(a,8;~;~)+~~2F~(a+l,B+l;~+l;~)=~F~(a,~+l;y;z)
(-4.6)
and Cl93 l)a(X+
yq,J.M(x~ 4) =ZM r 1 P;~;&(x)=2-“-qx-
w+
lu-~Yv+w!
l’*
(1
l)fl P?P(x)
(A.7)
-x)p’Y+MI/2(1
+X)‘4-M’12
4n(J- q)!(J+ q)!
(‘4.8)
Equation (A.2) can be proved similarly. the relation
In this case instead of (A.6) we should use
(A.9) ACKNOWLEDGMENTS C. M. Research
A. is grateful to the Council of Scientitic and Industrial Research, Fellowship. He is also grateful to K. Babu Joseph for guidance.
New
Delhi,
for a Senior
BOUND STATESOFNON-ABELIAN
DYONS
139
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
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