Possible electric charge non-conservation of a pointlike monopole

Volume 145B, number 5,6 PHYSICS LETTERS 27 September 1984

POSSIBLE ELECTRIC CHARGE NON-CONSERVATION OF A POINTLIKE MONOPOLE

R. JENGO Scuola Internazionale Superiore di Studi A vanzati, SISSA, Trieste, Italy lstituto Nazionale di Fisica Nucleare, Trieste, Italy and International Centre for Theoretical Physics, Trieste, Italy

Received 18 June 1984

We consider a doublet of charged fermions around a 't Hooft-Polyakov monopole in the pointlike limit and we discuss the radiative corrections. We find a sort of spontaneous charge breaking driven by the non-abelian structure and we argue why the electric charge in the core of the pointlike monopole might be unobservable.

Here we propose to discuss again the problem of the fermions in the background of a static monopole [1,2], in the limit where it is considered to be pointlike. The pointlike monopole can be taken to be either an elementary particle, whose quantum field theory has been discussed elsewhere and argued to be a consistent theory [3], or the pointlike limit o fa monopole solution o f an underlying non-abelian theory, whose scale r 0 is much less than that of any length scale ob- servable with present energies. We have in mind in particular the ' t Hoof t -Polyakov monopole [4,5] of the Georgi-Glashow model as the simplest model o f this kind. (We will eventually introduce the discussion of the relevant phenomena for the case o f the SU(5) mo- nopoles.)

Let us begin with two intriguing and wen-known observations:

(a) A fermion can be in a partial wave relative to the monopole such that there is no centrifugal barrier [ 6 -8 ] , contrary to the case ofbosons; call it S-wave.

(b) Even though outside the monopole we have an abelian U(1) gauge theory there is a chiral anomaly since f F P can be ~ 0 [9,10].

As is well known (a) and (b) are at the root of the Callan-Rubakov effect. However there has never been a universal consensus on the interplay between (a) and (b).

From (a) and the analysis of the hamiltonian for a charge doublet S-wave fermion in the monopole background it has been deduced that:

0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(a') If we neglect radiative corrections altogether, the fermion monopole S-wave scattering , t either violates electric charge or chirality conservation [8]. If the pointlike monopole is a pointlike limit of an ex- tended monopole o f the Georgi-Glashow model, then the only allowed channel is I AQI = 2e and chirality is conserved [11,12].

Of course the question arises whether this remains true when radiative corrections are taken into account and, if so, what happens to the charge lost by the fermion. Notice that in the pointlike limit it is impossible to excite the internal degrees of freedom of the monop- oles, which would require an infinite amount of energy, and that the electric charge, unlike e.g. the barionic one, is coupled to a gauge field.

When we switch on the electric charge and include radiative corrections the observation (b) comes into the game. The problem of including the radiative corrections into the S-wave scattering amplitude evalua- tion does not appear so far to be exactly solvable, even though the S-wave scattering is a two dimensional problem (r, t variables). Many discussions have considered the approach [1,2] based on the bosonization method [ 13] where the fermions are represented as the solitons of a sine-Gordon system. This method indeed is very attractive and it has the advantage of incorporating automatically the above point (b). However, when

:t:l We consider in this paper a magnetic charge g and fermions with charge e satisfying the minimal Dirac condition eg = 27r.

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the radiative corrections are included, the problem becomes the one of a sine-Gordon system in a given external background. It can still be exactly solved in the limit of zero fermion masses (since then it reduces to a quadratic functional dependence on the fields): however, since the fermion mass is a factor of the sin( ) term of the sine-Gordon equation, by putting the mass to zero we may lose the very concept of sol- iron. Moreover the zero-mass limit of a two-dimensional system may introduce spurious infrared divergences. A qualitative analysis seems possible, based on classical considerations on the motion of the solitons. Here however one risks losing interference effects and/or overestimating the effects of classical potential bar- tiers which could be penetrated by quantum effects.

Anyhow the discussion based on the bosonization treatment seems to indicate [1,13,14] that the effect of the radiative corrections can be visualized in a Coulomb barrier which prevents in the pointlike limit the IAQ I = 2e reaction. According to the general argu- ment, scattering then occurs in a chirality changing channel. This solution appears to fit the physical in- tuition and the lack of chirality conservation is ac- counted for by the anomaly. Still, we ask ourselves whether the problem can have more than one solution and that besides the one indicated by the bosonization methods there is another one which is triggered by the non-abelian structure of the monopole even in the pointlike limit, and which interpolates smoothly for e-+ 0 the result (a'). We will discuss it by using directly the fermion fields. But then, what happens to the charge which has been lost? We would like to discuss the provocative issue that the electric charge of the monopole (if integer) may be physically unobservable in the pointlike limit. The point is that if we consider Z, the functional integral over the charge doublet Fermi field in a given EM background (i.e. monopole field + ordinary EM field), and we perform a chiral change of variables ff -+ exp(irr~,5) ~ which is unobservable, we get [ 15 ]

( e2 ) Z -+exp i ~-~n f F ff Z . (1)

In our case F is a superposition F = F M + F(A) where Fuv(A ) = ~ ,A u - ~¢4~ is an ordinary EM field and the monopole field is determined by ~, f fM =gjM,jM being the monopole current and J~be~n'g the dual'of F. We can then rewrite (1) as

:exp( i2ey ; ) ,2, and the consistency of the fermion quantization requires (/integer)

e f A ~ J M = ~rl. (3)

Suppose now that our pointlike monopole has also an electric charge which is a multiple 2n of the one of the fermion. The dynamics of this new system is eval- uated by multiplying the exponential of the action of the system by the extra factor

exp(i2 ef ; ): 1 which is of course irrelevant. Therefore this electric charge is unobservable. (The above steps are similar to those of the Witten effect [16,10] but the point of view is different here.)

We are not sure how the requirement (3) or (4) is actually implemented in a quantum-mechanical computation. By analogy with similar problems in non- abelian theories one would guess that the path integral has to be formulated with suitable boundary conditions, equivalent to a suitable space-time topology, such that (3) or (4) comes out automatically.

Here we will consider (3) or (4) as a consistency condition, e.g. when solving the scattering problem by using e.g. perturbation theory or other approxima- tins we will eventually check that it is satisfied, that is we only allow for solutions of the equations of motion which are compatible with it.

For a static monopole at qM = 0 and an S-wave fermion and purely radial EM degrees of freedomA 0(r, t), Ar(r, t), E(r, t) = -OrA 0 + ~tAr, the consistency requirement of eq. (3) is

+ o o oo

e f dt f dr E(r, t) :Trl. (5) - - ~ 0

Notice that this requirement is also manifest in the bosonization treatment where the coupling of the bosonized system to the EM field occurs [1] in the term exp[(ie/rr)ffdt drE(Q + ~)] and the bosonic field Q + q5 is only defined modulo 2~r.

As a consequence of eq. (4) the charge of the monopole in undefined, or better there are an infinite

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number of states with different charges which are degen- erate. Notice that we are not speaking of the Jul ia- Zee [17] dyons which correspond to a given classical field configuration different from the pure monopole one. They are indeed separated by an infinite amount of energy in the limit r 0 -~ 0 and we do not consider them here.

In particular, we can consider a monopole state which is a linear superposition of various charges giving an eigenstate of the global U(1)E M phase (note that if the fermion doublet charge is e =~, the monopole is a superposition of integer charges, in agreement with this picture and eq. (4)).

When we couple the monopole to the fermions we are facing a second quantization problem and we will see that the system undergoes a sort of spontaneous symmetry breaking phenomenon. The symmetry breaking appropriate to the r 0 -+ 0 limit o f the non-abellan monopole corresponds to choosing the U(1)E M phase and defines, as usual, the Hilbert space o f the system. The electric charge is then spontaneously broken, the mechanism being such that we do not foresee the oc- currence of Goldstone bosons.

All that will be illustrated in the following compu- tations.

Here for simplicity we discuss the case where the CP violating vacuum angle O is equal to zero, the gen- eralization to 0 =¢: 0 being obtained by a chiral rotation [18]. Note in any case that due to the condition eq. (4), the degeneracy of the 2ne charge states, and therefore the picture ~ve are discussing, remains also for O q= 0. The 0 angle can be formally interpreted as an additional charge [16], but due to the requirement eq. (3) its effect is a phase exp(ilO) for every "sector" l and it does not appear to have a dynamical role at given l.

The lagrangian describing the interaction of a doublet S-wave fermion with charge z 3 = +1 with the monopole at the origin, and taking into account an ordinary radial electric field giving rise to radiative corrections, is [1] (after rescaling rff ~ ff and eAu ~ A u ) :

f dt dr £= f dt dr[(2rr/e2)r2E 2

- A 0 ~-3 (r 1 ff - A r i f a2

+ ~( io l~ t - r 3 o2~ r - M + If) ~k] . (6)

Here, of course, ~ = ~+O1, the charge operator is T 3 and the chirality operator is a 3 . The term If in eq. (6) is a remnant of a possible non-abelian structure for r -* r 0. In the model we are considering (the detailed dependence on r is not important and we take a 0-function):

W = O/r)(r2cos 3' - Tlsin 3')02" O(ro -- r), (7)

3' being a U(1)E M global phase. In the limit r 0 -+ 0, W acts as a kind of regulator term, to be discussed in the following, similar to an external magnetic field in a ferromagnet. Let us begin by discussing a completely symmetrical situation where If = 0 everywhere.

We find it useful to work in the gauge A.r = 0. By solving for the EM field we get

e 2 1 r - - -- f ~(r', t)r3Olt)(r ', t ) d r ' , (8)

E = 4rr r2 0

i.e. the ususal Coulomb b~w, and substituting in eq. (6) we get

£ = ~ ( io l a t -- r3o2~ r -M)t~

e2 1(0~ ,)2 8rr j ~r3 °1 ~ dr (9)

We want to discuss here the scattering o f one fermion in the initial state to one fermion in the final state. By using Feynman rules, we therefore replace

r r

f d r ' f dri'~(r ', t)7"3cr 1 ~(r ' , t) 0 0

X ~(r", t)~'3Olff(r", t)

r r

~ 2 f dr' f dr"~(r',t)r3o 1 o o

× (¢(r', t)~(r", t))Z3Olff(r" , t ) , (10)

obtaining a term which describes a (non-local) interaction with a background. Of course we then need (ff t~) which we will evaluate using perturbation theory i.e. from (9) for e -+ 0. However, it is well-known that the lagrangian in eq. (9) contains an ambiguity [7,8] for e 2 = 0, due to the fact that a boundary condition at r = 0 is needed in order to make in ir3a3~ r term self- adjoint. This ambiguity appears in the fermion propa-

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gator, which for e = 0 is of the general form (we approach the equal time from spacellke separations)

(0[T(~(r , t)~(r', t))10)e= 0

= N ( i ° l ~t -- r3 o2 ~2 +214) [K0 (Mp) + Ko(M-~)A ],

(11)

where K o is the Bessel funct ion,N a constant,

p = [ ( r - r ' ) 2 - ( t ' t ' ) 211/2,

= [(r + r ' ) 2 - (t - t ' )2 ] 1/2,

and A is a matrix depending on the boundary conditions, which can be shown to be in general of the form

A =r3OlCOSa+(rlCOS3+'r2sin3)sina. (12)

We will then use a self-consistent method: start with a (ff~)e=0, substitute in (10) and (9), then compute (ff~)e and impose lime~ 0 (ff~)e to be equal to the starting one. This will define the zero order in e.

Since the really important effect takes place at small r, we perform the computation for r ~ 0 (we need t = t' in (10)) and it is possible to see that the leading term in (10) is obtained in the approximation for the RHS:

p p

× (~(r', t)~(r", t))e=0r3Ol) ¢(r, t)

giving a local term in the lagrangian. Then only the part proportional to A in eq. (11) contributes and we obtain the following leading lagrangian £2:

£~ = ~{ io l~ t - r3o2~ r

-i(q2/r)[o3cos a + o2sin a(7-2cos t3 - 71sin 3)]

-M}~. (13)

where q2 = (e2/4~r)ln 2. The hamiltonian problem following from the lagrangian (13) is

H ~ E -- {i7-303~ r

+ (q2/r) [o2cos a -- o3sin a(7-2cos 3 -- 7"1 sin 3)]

+ OlM}~ E = E ~ E . (14)

It follows that in general for r ~ 0

~E ~ r+q2 ~+) + r -q2 ~_~ (15) with the condition

U(±) = -+ [r3o 1 cos

- (7-1 cos/3 + r2sin 3)sin a] U(_+). (16)

By choosing the regular behaviour ~E ~ r+q2 U(+) one finds for q2 _+ 0 the following A :

A = r3Ol cos a - (rlCOS/3 + r2sin 3)sin a .

By comparing with eq. (12) we see that self-consistency requires cos a = 1 i.e. the radiative corrections reg- ularize the problem in such a way that the electric charge is conserved and the chiral charge is violated. This solution agrees with what is found by using the bosinization method and a classical analysis_ 2

However, the irregular solution ~E ~ r -q U(_) is still a possibility, the normalizability of the wave function requiring q2 < ½ which is largely satisfied by the physical electric charge. In such a case from the boundary condition (16) we get for q2 ~ 0:

A = - r 3 o 1 cos a + (7" 1 cos/~ + 7"2sin 3)sin a ,

and by comparing with (12) we have self-consistency for sin a = 1 and any 3. This solution, which violates the electric charge and conserves the chiral charge, also gives a self-adjoint definition of the operator ir3o3b r for q2 < ½ and of the hamiltonian, by manag- ing the irregular behaviour r-q z introducing an UV cut-offr > r 0 and letting r 0 ~ 0 at the end.

But this is the way in which the non-abelian structure of the monopole for r < r 0 comes in the pointlike limit. I f we now want to introduce the W term eq. (7) in eq. (9), the solution it requires is ~k - ro with (71cos 3' + r2sin 7)u = v for r < r0, matching at r = r 0

q2 the solution ff ~ r - U(_) for 7 = 3. Therefore in the limit r 0 ~ 0 the W term acts like a vanishingly small external field in a situation of spontaneous magnetiza- tion, selecting the direction of symmetry breaking, i.e. cos a = 0, versus the other possibility cos a = -+ 1, and 3' = 3. Let us also add that no intermediate value of cos a seems to be consistent, in the pointlike r 0 ~ 0 limit. For definiteness we take 3' = 3 = 0 and therefore the b.c.

lim 7-lifE = ~E + o ( r l - q 2 ) • (17) r ~ 0

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In particular in this solution the global U(1)E M in- variance of the lagrangian (9) for W = 0 is broken.

Let us discuss briefly the Ward identities. Consider for instance the matrix element

(01 ~ (x 1)l 1 ) where x 1 = r~ t 1 'and I 1 ) is a one-fermion state. We can derive the following equation, on which we then take the linfit r 0 ~ 0:

+oo o o

ip. f dt f dreipx<OIT(J~(x)d/(Xl))ll) _ o o r 0

= f dt<OlW(Jr(rot)~(Xl))l 1) + r3<0l ff(Xl)l 1).

- ~ ( 1 8 )

In the limit pg ~ 0 we put

p. f dt dr eipx<OIT(J#(x)~(Xl))l 1) = 0 ,

since no Goldstone boson appears to be required. Rather, it is then required that

lira fdt(OIT(Jr(rot)g, (Xl))10) :/: 0 , ro--~O

which we interpret by saying that the breaking of the global U(1)E M is accompanied by a flow of charge to and from the origin [19], according to the picture of the monopole as a coherent superposition of many- charge states.

Finally, we would like to check the general consistency requirement eq. (5).

Let us consider our leading hamiltonian eq. (15) for the relevant case sin a = 1 and t3 = 0, with the b.c. eq. (17). This problem can be solved exactly in terms of Bessel functions. By selecting an incoming wave of definite charge, say positive, we obtain the following asymptotic wave function for r ~ oo [p = (E 2 _M2)1/2 ]

~E(r) - P) exp[- - i (pr + ~-lrq2)]

+ exp [i(pr + ~-lrq2)l , (19)

( E - p)/M/

where the two upper (lower) components of ~k correspond to r 3 = +1 ( - 1 ) and the two entries of both the upper and lower components to 03 = -+1. We see explicitly that I Ar3 [ = 2 and Aa 3 = 0, o 3 of course commutes w i t h H f o r m = 0. We expect this to be a general feature of our solution, independently of the approximation used for obtaining H. Notice that our asymptotic state satisfies r 1 t ~ I,, = ~E I~. But for any r, r 1 ~b~ is again a solution of eq. (14) also satisfying the b.c. eq. (17), therefore r lff~ = qJE everywhere. Therefore the corresponding charge density J0(r) = ~ ( r ) r3ffE(r ) = 0 giving from eq. (8) a zero electric field, consistent with eq. (5) with l = 0. More gener- ally, any wave packet ff (r, t) which is a superposition ofeigenstates fie such that r 1 ~b~ = ~E will give f+_~ dt Jo(r, t) = 0 and then f dr f dt E(r, t) = 0. We may notice that independently of any approximation the lagrangian of eq. (6) (for 3' = 0) is indeed invariant under ~ (r, t) ~ r 1 ~ * (r, - t) , E (r, t) ~ - E (r, -t) , consis- tently with the above calculation.

Let us also remark that the expectation value (ff ~) appearing in eq. (10), which could be called a "conden- sate", is such that < ~k (r, t) ~1 r3 ~ (r, t)} = (Jo(r, t)) = 0 (we imagine a regularization (t~(r + e, t ) a a r 3 ~ (r - e, t)) symmetric in e -* - e ) and therefore ~f dt dr E) = 0, to be compared with the situtation in the non-abelian in- stanton case, where chirality conserving operators have non-vanishing expectation values in the zero topolog- ical number sector.

Finally, the question arises whether the present discussion and its result that a monopole can catalyse a charge violating process, besides indicating that this is in principle possible, can also be directly relevant to a grand unification monopole. In such a case, of course the problem meets the additional difficulty of the non- abelian interactions of SU(3)c and SU(2)w which also are present for r > r0, together with the question of the possibility that also color may disappear. We re- member that the basic monopole [20] and SU(5) has both colored and EM magnetic field. The naive analysis o f a scattering of a fermion on it, ignoring all radiative corrections, would indicate that processes with [ AQE M [ = 5 are possible, together with a corresponding color violation. By construction then, a process (involving various fermions) which conserves color also conserves charge. However this is not the only possible monopole of SU(5) and moreover the physical proper- ties of this object are not clear even before we consider

357


the fermions coupled to it, for instance the long range colored magnetic field is in itself a problem [21 ]. An- other grand unification monopole has also been studied [20,22], which at large distances has just an EM magnetic field and no colored one. Here individual processes like d i ~ e + (i = any color) are the result of the Dirac equation without radiative corrections [22]. We

can imagine in this case a process like d ld2d 2 ~ 3e +, i.e. n -~ 3e ÷ + zr +, which conserves color and violates charge by [AQ [ = 4. Maybe this process could be allowed in the monopole field by a physical effect similar to the one which we have discussed.

We would like to thank Professor G. Calucci, Dr. G. Vedovato and Dr. O. Foda for many discussions.

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Possible electric charge non-conservation of a pointlike monopole