Nuclear Physics B272 (1986) 301-321 North-Holland, Amsterdam
THE E V A P O R A T I O N OF Q - B A L L S Andrew COHEN, Sidney COLEMAN, Howard GEORGI and Aneesh MANOHAR* l~vman LaboratoD, of Physics, Harvard University, Cambridge, MA 02138, USA Received 21 January 1986
In extended electroweak models with scalar fields carrying lepton number, it is possible for these fields to form extended objects of the Q-ball sort. In general, these are destabilised by the Yukawa coupling of the scalar to neutrinos. This decay process takes place only on the surface of the object, not in the interior. Thus the Q-balls evaporate away. We set up the general theory of this process, find an absolute upper bound for the evaporation rate, and explicitly compute the rate in a simple case.
1. Introduction and conclusions C e r t a i n field theories i n f o u r s p a c e - t i m e d i m e n s i o n s with u n b r o k e n c o n t i n u o u s g l o b a l s y m m e t r i e s a d m i t a r e m a r k a b l e class of e x t e n d e d objects*. T h e s e objects, c a l l e d Q - b a l l s , are spherically s y m m e t r i c n o n - d i s s i p a t i v e s o l u t i o n s of the classical field e q u a t i o n s t h a t are absolute m i n i m a of the e n e r g y for a fixed v a l u e of the c o n s e r v e d c h a r g e Q. T h e s i m p l e s t t h e o r y d i s p l a y i n g this p h e n o m e n a is the U(1) i n v a r i a n t t h e o r y of a single c o m p l e x scalar field of c h a r g e n w i t h n o n - d e r i v a t i v e i n t e r a c t i o n s . T h e t h e o r y is d e f i n e d b y the L a g r a n g e d e n s i t y L = a~q,* 3 ~ ¢ -
U(I¢I).
(1.1)
T h e U ( 1 ) s y m m e t r y is
,/, --, ein°4. T h e associated conserved current is
L = in ( 4" 9~,¢ - 8~,~* ~), and the conserved charge is Q =
f
d3xj0 "
* Junior Fellow, Harvard Society of Fellows. * We follow here the terminology and use the results of a recent paper by one of us [1]. There is a large literature on a variety of closely related objects, of which we can cite only a portion here [2]. 0550-3213/86/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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A. Cohen et aL / The evaporation of Q-balls
We are interested in the case where the U(1) symmetry is unbroken, and hence assume the absolute minimum of the potential occurs at ~ = 0. By convention, we take U(0) = 0. The condition for the existence of Q-balls was found in ref. [1] to be
rain
,2 _ ½U " ( 0 ) .
(1.2)
We take ~0 to be the field value for which this minimum occurs. These Q-ball solutions are of the form q~ = e i ~ t ~ ( r ) ,
(1.3)
where if(r) is a monotone decreasing function of r, going to zero at infinity, and co is a constant. That is, this object rotates in internal U(1) space with constant angular velocity. The charge density in the Q-ball is given by P = 2n colqffr)l 2. As Q goes to infinity, q~ resembles a smoothed-out step-function. For r less than a certain radius R, ~ = q~0; outside this radius, q~= 0. The transition region between these two regimes has a thickness of order ~-1. In this limit co approaches coo = ~/U(q~0)/Iq~0[ 2 , and the values of local quantities inside the Q-ball become independent of the Q-ball size. In particular the energy per unit charge of the homogeneous state that exists in the Q-ball interior is just coo~n, where n is the charge of the scalar field. For this reason these objects are analogous to lumps of ordinary matter, and following ref. [1] we call the state that exists inside the Q-ball Q-matter. Although the only renormalisable interaction allowed for our theory never satisfies the condition (1.2) for the existence of Q-balls, we need not worry. If we were to complicate the theory by increasing the size of the internal symmetry group we could easily obtain renormalisable interactions that yield a Q-ball spectrum. More importantly U(ff) could be an effective potential obtained by integrating over heavy degrees of freedom. Since we will never go beyond the leading semi-classical approximation for the scalar field, the difference between an effective interaction and a fundamental interaction is irrelevant. Several authors have considered the addition to the standard model of scalars carrying lepton number [3], which presents the possibility of forming classically stable solutions of the type described above. In such models the scalar field will couple to neutrinos. We wish to consider the influence of this coupling on the Q-ball properties. Thus we modify the Lagrange density (1.1) by the addition of L' = +*(i 0 0 + io- V ) + - ig4+*a2~b* + ig4*+To2~b •
(1.4)
Here q~ is the neutrino field, a two-component Weyl spinor, and the o's are the
A. Cohen et al. / The evaporation of Q-balls
303
usual Pauli matrices. The neutrino field has lepton number L = 1 and the scalar field has L = 2*. The most obvious change in the properties of L-balls with the addition of this neutrino coupling is that they become unstable: they can decay into pairs of (massless) neutrinos. However, the decay process occurs only at the surface of the L-ball, not in its interior; for a large L-ball, the neutrino production rate is proportional to the area of the L-ball, not its volume. In other words, large L-balls evaporate. This is a surprising result. (At least, it was a surprise to us.) In sect. 2 we explain how evaporation arises as a result of the fact that neutrinos are fermions; if we were to consider an alternative theory in which the L-ball coupled to pairs of bosons, we would find a volume-dependent production rate. In the course of this analysis, we derive an absolute upper bound on the rate of pair production per unit area, dN ~ dt d ~ ~< 192v ~ "
(1.5)
This result may be expressed in terms of the velocity of recession of the L-ball surface. The L-ball interior has a lepton number density of 4c00q}, while each pair produced carries a lepton number of 2h; thus dr dN - 4¢00~A ~ = 2h d~-'
(1.6)
where r is the L-ball radius and A its surface area. Thus, dr
h~oo2
dt ~< 2384~r ~
"
(1.7)
Note that, at least in the semiclassical (small h) limit, this is a highly nontrivial bound, in that it is much smaller than the elementary causality bound, the speed of light. In sect. 3 we develop the detailed formalism for computing the evaporation rate, and apply it to a simple limiting case, an L-ball with a step-function boundary. Even in this limit, we have not been able to evaluate the rate analytically, and have had to resort to numerical integration. Fig. 1 shows the result of the integration, in the form of a plot of the evaporation rate as a function of gq~o/~o, the only independent dimensionless variable in the problem. We can see from this plot that the evaporation rate can come quite close to the upper bound; indeed, it is greater than half the upper bound for gq~o/O~ogreater than roughly 0.1. * Units: We set c = 1, but keep explicit factors of h. Thus, the neutrinos, the quanta of the neutrino field, carry lepton number h.
A. Cohen et al. / The evaporation of Q-balls
304
Evaporation 1.0
I
I
I
Rate I
I
I
10 0
101
I
0.8
'13 e~
0 ~Q
0.6
r~
0.4 r~ o
"~
0.2
u,4
0"100-5
1
10-4
10-3
10-2
10-1
I
I
10 2
10 8
g¢o/~o Fig. 1. Evaporation rate per unit area as a function of g~o/COo.
As one would expect, the evaporation rate vanishes when g goes to zero. We have been able to evaluate this limit analytically; we find
dN --=37r dtdA
gq~0
~°03
~oo 192~r2"
(1.S)
In sect. 4 we derive this result and show that it is universal; it is independent of our assumption of a step-function boundary, and even of the spherical shape of the L-ball. We have no idea whether objects like L-balls actually exist in the world. It is straightforward to incorporate the neutral scalar meson q~ into a model with electroweak SU(2) x U(1) as the neutral component of a complex SU(2) triplet. The model then resembles a class of "majoron" models, except that the B - L symmetry is not spontaneously broken. If these objects actually exist in the world, they might have interesting effects on cosmology. One can imagine, for example, that large L-balls produced in the big bang might have provided the inhomogeneities necessary to initiate galaxy formation, and subsequently evaporated away, leaving cosmologists to wonder how the galaxies developed. We do not know whether L-balls with the appropriate properties can actually be produced in a realistic cosmology.
305
A. Cohen et al. / The et,aporation of Q-balls 2. The stability of L-matter
It is instructive to begin by considering an alternative theory in which the neutrino field is replaced by a massive scalar field, L' = a,~b* O~b -
m2~*~
-- gdp*~b 2 --
gq~hb.2 .
We shall treat ~ as a classical background field, the L-ball solution, with which the q u a n t u m field q, interacts. As we shall see shortly, this is equivalent to calculating the scalar production rate to lowest non-trivial order in an expansion in powers of h. We know the production rate if we know the amplitude for the ground state of the theory to stay the ground state. This is given by
(2.2)
where W is the sum of all connected vacuum-to-vacuum Feynman graphs. In the case at hand, this is given by the series shown in fig. 2. If we were to treat ~ as a q u a n t u m field, we would have many additional graphs, with internal q~ lines; some of these are shown in fig. 3. However, all of these involve more loops, and therefore m o r e powers of h, than the graphs of fig. 2. Although we will later return to the general case, for most of this section we will restrict ourselves to L-matter, the homogeneous state that exists in the interior of an L-ball of infinite radius. For L-matter, = ~0e-i'~°' .
(2.3)
This background field is invariant under space translations and under simultaneous time translations and L rotations. This enables us to define a + propagator, . d4k T(Ol~P*(x)~p(y)[O) = - i e i'°o(x°-y°)/2 i - - e i k ( x - Y ) D ( k J (29) 4
]4,
),
14' I
+
,
.....
14,
Fig. 2. Diagrams contributing to W and to the evaporation rate for a background ~ field.
(2.4)
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A. Cohen et aL / The evaporation of Q-balls
14,
14' +
14,"
!4,"
Fig. 3. Higher-ordercontributions to W.
from which we can construct W in the standard way r d 4k W= - ~ VT J ~71n
D (k ) .
(2.5)
where V T is the volume of space-time. For fixed k, D is a rational function of k °, behaving at infinity like the free propagator, that is to say, like -(k°) -2. Thus, if all the singularities of D are on the real axis, we can rotate the k 0 integration contour to the imaginary axis. In this way we obtain a manifestly pure imaginary integral for W; there is zero probability for the ground state to decay. However, if there are singularities of D off the real axis, these can produce a real part of W and a nonzero probability of decay. Note that the real part of W is automatically proportional to T, as it should be for the exponential decay law to hold, and also proportional to V, as it should be for an extended system, which can decay anywhere within the volume it occupies. The singularities of D occur at the characteristic frequencies of the scalar field. Thus, to have a nonzero decay probability, there must be, for at least some values of k, normal modes that display exponential growth or decay. We shall now search for such modes. The field equations are (0 2 + m 2 ) ~ + 2g~x = 0,
(2.6a)
(0 2 + m 2 ) x + 2gO*+ = 0.
(2.6b)
Here, X-= +*, but we ignore this fact when finding normal modes. (It becomes relevant only when we sum the modes to form the general solution; then conjugate modes must occur with conjugate expansion coefficients.) The symmetries of the background field (2.3) imply that the normal modes must be of the form = e-i°o°t/2e-ikxqj(O),
X = ei'°°t/2e-ik:'x(O).
(2.7)
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A. Cohen et al. / The evaporation of Q-balls
Inserting this in the field equations, we find 1 2 60 - (k°+ 60o) 2gq'o where 602 =
k 2
2gq~o
60 _
] [qJ(0)
60o12]1x(0)) =°'
(2.8)
-I- m z. The zeroes of the determinant are given by ( k ° ) 2=~600q_1 2 6°2~ ~/60060k2 2q_4g2q52 .
(2.9)
Thus, for arbitrarily small g, there are solutions with imaginary k ° for 60k suffi1 ciently close to 560o. Of course, this is only possible if m is itself less than ½60o; this is thus a necessary condition for decay at small g. All of this is in perfect agreement with naive expectation. The surprise comes when we attempt to extend this analysis to the theory of neutrinos defined by (1.4). Until we reach the field equations, the analysis is identical to that in the scalar case, except for occasional numerical factors and sums over spinor indices. The field equations are ( iO o + i•. V ) ~ b - gq~x = O,
(2.10)
(iO o -- i o . V ) X - g~*ap = O,
where X = irr24'*- Performing the substitution (2.7) we find
ko+ 60o-O.k
-g o
-g ,o
](+(0) 1 ]
(2.11)
From this expression it is immediate that k ° is always real, for it is an eigenvalue of a hermitian matrix. In contradiction to naive expectation, pair creation does not occur. In the leading semiclassical approximation, L-matter is stable. We can gain insight into the origin of this stability by considering an alternative decay process, one which involves tunneling and thus does not occur to any finite order in an expansion in powers of h. This is spontaneous cavitation: a cavity, a region of vanishing if, might appear inside L-matter; the lepton number of the missing L-matter would appear as massless neutrinos inside the cavity. This process, if it occurred, would be much like the decay of a false vacuum through the appearance of a bubble of true vacuum. We shall now show that cavitation cannot occur. The argument rests only on counting powers of h, so we will not bother to keep track of purely numerical factors. Let R be the linear size of the cavity. The energy of the L-matter formerly in the cavity is given by E L - 60~R 3 ,
(2.12)
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A. Cohen et al. / The evaporation of Q balls
while its lepton number is given by AL - ~0¢2R 3 .
(2.13)
From the theory of a free Fermi gas, N massless neutrinos in the cavity have an energy E,, - h N 4 / 3 / R
- h ( A L/h )4/3/8
.
(2.14)
Thus, g L / g t , _ ( h~2/~2 )1/3.
(2.15)
In the semiclassical (small h) limit, cavitation is always energetically forbidden, no matter what the size of the cavity. Once again, things would be very different if neutrinos were bosons. In this case, the factor of N 4/3 in eq. (2.14) would be replaced by a factor of N (that is to say, all the neutrinos would condense in the lowest mode of the cavity), and eq. (2.15) would become E L / E ~ - o~oR.
(2.16)
Cavitation would always be energetically favored for sufficiently large cavities. It is the exclusion principle that keeps L-matter stable. Indeed, the exclusion principle is the invisible hand at work in our computation of Im W. There we tacitly assumed that the k ° integration contour (before rotation) passed slightly below the negative real axis and slightly above the positive real axis. But in the quantization of a Fermi field, the point at which the contour crosses the axis is the point to which we fill the Dirac sea. Vanishing k ° corresponds to energy 1 5h~0 o, neutrino production makes the Dirac sea overflow until Fermi pressure prevents further production. This observation is at the root of the absolute upper bound (1.5). Let us consider some object that is producing neutrino pairs of total energy he0o and that is enclosed by some surface of area A. Each pair contains one neutrino with energy less than 5h~o i o,. thus we can bound the average pair production rate if we can bound the expectation value of the normal component of the neutrino current density for neutrinos in this frequency range. But this expectation value is maximized in the state in which every energy level with outward-moving neutrinos is occupied and every level with inward-moving neutrinos is empty. Thus, - d3k
1
(2.17)
where n is the outward-pointing normal. (Since we are dealing with Weyl neutrinos,
309
A. Cohen et al. / The evaporation of Q-balls
there is only one helicity state for each k.) The integral is easily done in polar coordinates:
1 {n "J)~'2~o ~< ~
~003 fo~°/2k2 d k folCOS0 dcos Ofo2~dq~
192~r2 .
(2.18)
Integrating over the enclosing surface immediately yields eq. (1.5). This bound is rather like the space-charge limitation on current density in a vacuum tube, with Fermi pressure taking the place of Coulomb repulsion. It explains why our earlier computations predicted stability of L-matter; as the size of the system grows to infinity, the decay probability per unit volume goes to zero. Of course, a finite L-ball is allowed to decay. We now turn to a computation of this decay rate.
3. The computation of evaporation rates We wish to study neutrino production by a large but finite L-ball in the leading semiclassical approximation*. As we explained at the beginning of the preceding section, this is equivalent to studying the theory of a quantized neutrino field in a classical L-ball background. Our method will be to construct the quantum field as a superposition, with operator-valued coefficients, of the classical solutions to the neutrino field equations (2.10). It is straightforward to construct these solutions in both the interior of the L-ball, where q, = ~0, and the exterior, where ~ = 0. Matching the interior and exterior solutions requires knowledge of the shape of the L-ball boundary. We have chosen to do the matching for the simplest possible case, a step-function boundary: ~) = d~oe i~°ot, = o,
r <~ R r > R.
(3.1)
We emphasize that simplicity is our only reason for choosing this case; it would require only a small additional amount of computer time to do the computation for any given boundary shape. We shall use the rotational invariance of the L-ball to write the general solution to the equations of motion as a superposition of partial waves (rotational eigenfunctions). This will unfortunately but inevitably involve some of the more sleep-inducing portions of the theory of higher transcendental functions, and the reader may well wonder if this is necessary. After all, if pair production does take place only at * Since from here on we shall be working to a fixed order in an expansionin powers of h, there is no point in keeping track of the h's, and we will use units in which h = 1.
A. Cohen et al. / The evaporation of Q-balls
310
the L-ball surface, one might as well go immediately to the limit of a semi-infinite L-ball, one that occupies a half-space, for which the surface is not a sphere but a plane• In this limit, the rotation group contracts to the two-dimensional euclidean group, and complicated spherical harmonics become simple plane waves. We shall indeed go to this limit, but only at the end of our computation• For reasons which will become clear as we proceed, a direct attack on the semi-infinite system is frustrated by ambiguities that can only be resolved by considering it as the limit of a finite L-ball. The L-ball is invariant not only under spatial rotations but also under simultaneous time translations and L rotations. This symmetry allows us to choose normal modes such that ~ is proportional to exp[-i(0a + ½%)t] and X is proportional to e x p [ - i(~0- ½~%)t]. Pair production occurs in those modes which mix annihilation operators (positive frequencies) with creation operators (negative frequencies); thus for our purposes we need only concern ourselves with modes for which ~ is in the interval ( - ~oa 1 0, ~%). 1 In this interval, all the annihilation operators are in q, and all the creation operators are in X; as expected, we have only to deal with neutrinos, not antineutrinos. We begin by studying the equations of motion for the free theory, that is to say, for vanishing q,. In this case the ~b and X equations decouple: 1 - i o - vq, = (~,% + o~) + = k ++
-io'VX=(½~o-°a)X =k X. io. V, + and X are afortiori eigenfunctions of
(3.2a) (3.2b)
As eigenfunctions of - 172 = ( - io • V )2. Also, with no loss of generality, we can choose them to be eigenfunctions of j 2 and 4 , with eigenvalues j(j + 1) and m, respectively• It is then straightforward to construct the solutions from the well-known spinor solutions to the free SchrSdinger equation:
=u(k+,j,m;r)e ik+,,
(3.3a)
X = u(k , j, m; r)e +ik t,
(3.3b)
where
u(k,j,m;r) ~/ k
x ~l+-m+½ l_+rn+7 Jt (kr)Yt m_(O,q') - i 2l_+1 -27++i J'+(kr)Yt+m (e, ep)
~/l_-m+½ jt+(kr)Yl+m+(O,~) ~]_-_+-~ Jt (kr)Yt m+(O,q~)+i i l + 2+m+½ l++1 (3 •4)
A. Cohen et al. / The evaporation of Q-balls
311
where l + = j + 12 and m + = m _+ ~. We have normalized these functions such that
f d3,'u*( k, j, m; ,')u( k ', j ', m " r) = 8( k - k')Sj/6.,,,,,
(3.5)
(If you're bothered by the ugliness of eq. (3.4), don't worry. We have written things out in full detail just for completeness; nothing we do will depend in any significant way on those terrible Clebsch-Gordan coefficients.) F r o m these functions, we construct the free quantum field, =
j•,,fo
'~°dk+
e-ik+tu(k+,
j, m; r ) a ( k + , j, m) + • • •
(3.6)
where the triple dots indicate contributions with frequencies outside the range of interest. The canonical commutation relations and the normalization condition (3.5) imply that
{ a( k, j, m ), at( k ', j', rn')} = 8( k
-
k')~jj,t~mm,
.
(3.7)
The a's and a t's have the usual interpretation as free-neutrino annihilation and creation operators. We now turn to the theory in the presence of the L-ball background. Outside the L-ball, the field must still satisfy the free equations of motion. (This is true even if we do not have a step-function boundary, if we interpret "outside" as meaning r >> R.) Of course, the solutions in this region do not necessarily continue to nonsingular functions at r = 0; in general, they are linear combinations of the singular solutions u (1) and u ¢~), the functions obtained by substituting the spherical Hankel functions, h (1) and h ~2), for the spherical Bessel functions in the definition of u. We choose one of our independent solutions to have no incoming X wave. Thus for r >> R,
~b = e-ik+t[u(Z)(k+, j, m; r) + R ( k + , j)u(1)(k+, j, m; r ) ] , X = eik-tT(k+, J)u(2)( k - , J, m; r),
(3.8)
where R and T are coefficients (independent of m by rotational invariance) fixed by the equations of motion inside the L-ball. These coefficients can be given a meaning in the context of scattering theory. If we form a wave packet of these solutions, then in the far past only the contribution of the incoming wave, the first term in eq. (3.8), survives, while in the far future only the contributions of the two outgoing waves survive. Thus the wave packet describes
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312
the scattering of an incoming +-wave off the L-ball, with R the amplitude for reflection and T the amplitude for transmutation. From this we can construct a second solution by using the symmetry of the equations of motion, X ° io2+*, and the identity
iozu(l'Z)( k, j, m; r)* = (-1)m+ u(Z'l)( k, j, - m ; r) .
(3.9)
We thus obtain, for r >> R, another solution: + = - ( - 1) m+e -ik tT(k+, j ) * u ° ) ( k ,
j, - m; r),
X = (--1)m'eik+t[uO)(k+, J , - - m ; r) + R(k+, j)*u(2'(k+, j, -rn; r ) ] .
(3.10)
When we construct the quantum field, these two solutions must occur with adjoint coefficients, so X will equal io2+*. Thus, for r >> R, +=
E fo'%dk+{ai,(k+,j,m)e
ik+,
× [u(Z)(k+, j, m; r) + R(k+, j)uCt)(k+, j, m; r)]
+ati,(k_, j , m ) e
ik+'(-1)m+T(k
+ ... .
, j)*u(1)(k+, j , m ; r)} (3.11)
This equation needs some explanation: (i) At this stage, ain should be thought of as merely an expansion coefficient. We will explain its physical meaning in the next paragraph. (ii) We have made the change of variables, m o - m , k + ~ k ~= co0 - k +, in the last term. The matrix element of the field between normalisable states is a wavepacket solution of the field equations. Thus, in the far past, only the incoming wave survives, and, in this limit, eq. (3.11) becomes identical with the free-field expansion, eq. (3.6). This gives the physical meaning of ain: it is the annihilation operator for an incoming neutrino. This also implies that the ain'S and their adjoints obey the free-field anticommutation algebra, eq. (3.7). Likewise, if we go to the far future, only the outgoing waves survive, and we can identify their coefficients as aout'S, annihilation operators for outgoing neutrinos. Thus, aout(k+,j,m)=R(k+,j)ain(k
+, j , m ) + ( - 1 ) ' ~ T ( k
, j)*atm(k_, j , - m ) . (3.12)
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313
Taking the adjoint, and making the change of variables m ~ - m , k+ ~ k , we find atout(k, j,-rn)
=
R(k_, j)a~in(k_, j , - m ) - (-1)m+T(k+, j)ain(k+, j, m). (3.13)
These equations define a Bogoliubov transformation, a linear transformation between incoming and outgoing creation and annihilation operators. Because these two sets of operators obey the same anticommutation algebra, the Bogoliubov transformation must be unitary. From this it follows that
IT(K+, J)l =IT( k ,J)l ~<1.
(3.14)
These relations will be useful to us in the sequel. We are now in a position to discuss neutrino production. Initially, we have no neutrinos; that is to say, the state of the system is 10)in, the incoming vacuum. We wish to compute the average number of outgoing neutrinos in this state. From eq. (3.12), in{01atout(k,
j, m)aout(k', j',
m')10)i n =
IT(k, j)12a(k
-
k')~jj,am,,,.
(3.15)
If we attempt to compute the neutrino density in the final state directly, by setting k = k', j = j ' , and m = m', we obtain a result proportional to 8(0) - that is to say, infinity. This is a familiar phenomenon. There are an infinite number of neutrinos produced because the neutrino-producing interaction acts for an infinite time. If we alter the theory in such a way that the interaction acts only for some finite time, T, then delta-functions of frequency are replaced by smooth functions; in particular, 8(0) is replaced by T/2~r. Thus,
~_. fo wOdkin@lat(k, j, m)a(k, j,
m)10)i n = ~T j~. fo'°° d k ( 2 j + a)lT(k, j ) l 2
. ] , ,'7"1
(3.16) The number of pairs is half the number of neutrinos. By eq. (3.14), we halve our 1 0. Hence, result by replacing the upper limit of integration by 5w
dN s~ ".'o/2 d k 12. dt = Jo ~ ( 2 j + l)lT(k,j)
(3.17)
So far, our formalism has been independent of the size of the L-ball. Things simplify considerably when the L-ball radius, R, becomes large. A normalisable
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314
wave-packet scattering off a large L-ball should not be able to tell it is encountering a curved surface rather than a plane one. In this transition from sphere to plane, the angular momentum variable, j, is traded for a linear momentum variable. We shall now work out in detail how this happens. We shall begin in position space and then pass to the conjugate variables. Let us consider an ordinary cartesian coordinate system with center of coordinates at the north pole of a sphere of radius R. If we stay close to the center of this coordinate system, we can approximate the sphere by the x-y plane. The zcoordinate, in this approximation, is connected to the r-coordinate in our original coordinate system, r = R + z.
(3.18)
The three components of J are treated asymmetrically: ~ remains a rotation generator, but Jx and Jv become translation generators in the approximating plane,
J~,v = RPx,y-
(3.19)
j ( j + 1) = m 2 + R2k~,
(3.20)
Thus
where kll is the magnitude of the momentum parallel to the plane. It will also be convenient for us to introduce the variable k z, defined by
k 2 = k 2+ k~.
(3.21)
We wish to study the limit of large R with m, k z, and kll fixed. In this limit
j = Rkll.
(3.22)
Let us assume that T has a smooth limit as R goes to infinity. (In fact, as we shall see shortly, this assumption is wrong. Had it been correct, we could have worked directly in a planar formalism and avoided the horrors of spinor spherical harmonics. However, let us make the assumption now, for simplicity; it will be easy to correct for our error later.) Under this assumption, we can replace the sum in eq. (3.17) by an integral,
dN dt
R2fo,%/2dkfokklldkll[T(k, Rkll)]2 ¢r
(3.23)
Since the area of a sphere is a 4~rR 2, this can also be written as dN
dA dt
_
1 Jo Jo[~°/2dk[*k'ldklllT[ 2
4~r2
(3.24)
A. Cohenet al. / The evaporationof Q-balls
315
If we replace ITI by its maximum value (one, by eq. (3.14)), we obtain once more the absolute upper bound (1.5). Eq. (3.24) gives the evaporation rate in terms of the T's. To compute these we need the solutions of the equations of motion in the interior of the L-ball. For r far from the surface, the equations of motion are
( ½O:o+ O:+ io " V ) f - gOox = O, - gO0q~+ (~: -1~0:0 - io. ~7)X = 0.
(3.25)
Just as in the exterior case, the solutions of these equations can be chosen to be eigenfunctions of -io.~7, j2, and J~. Once these choices are made, eqs. (3.25) become algebraic equations and it is trivial to find the solutions. They are (up to an irrelevant normalization)
+=(o:+~2-g20~)u(k'+_,j,m;r)e X = gepou(k'+, J, m; r)e ik-t ,
-~k*', (3.26)
where k'+_ =17O:o +_ ~o:2_ g2q~g .
(3.27)
To complete the computation, we need the form of these functions near z = 0 in the large-R limit. We can then match the interior and exterior solutions and find the transmutation amplitudes. For a general boundary, matching involves solving the equations of motion across the boundary region; for a step-function boundary, matching simply means setting the solutions equal to each other at z = 0. The limiting form of the interior solutions depends critically on whether I,~1 is greater or less than g~o. If it is less than g~o, k'_+ have imaginary parts, and the interior solution is the sum of increasing and decreasing exponentials. In the large-R limit, only the increasing (with r) exponential survives; we don't need to compute its coefficient, since this is irrelevant to the matching problem, where it is absorbed in the normalization of the solution. In this regime we could indeed have gone directly to the semi-infinite L-ball, and avoided spherical harmonics altogether. The situation is more complicated if I~01 is greater than gq~0- In this case, the interior solution is a sum of oscillating terms, and we must know their relative magnitudes and phases to compute the transmutation amplitudes. Standard formulas [4] give the asymptotic form of the u's:
1,/ u( k, j, m; r) = ~
V Rkz
( ei'- *J,,_ ( kllrll)[cos( kzz - a) - i cos( kzz - a - fl )] ) , (3.28) x eim+q'Jm+(kllrll)[cOs(kzz or)+ i c o s ( k z z - O t - ~ ) ]
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A. Cohen et al. / The evaporation of Q-balls
where
=R(klltan
lk"°-k~ ]
k,,
]
+¼~,
k~
fi = tan 1 k,'
(3.29)
and rll = ~ y2. Similar formulas give the asymptotic forms of u (1) and u (2), with the cosines replaced by exponentials. All the R-dependence of u is through the phase a. The exterior solution is constructed from u's with two different values of k, and thus involves two different c~'s. However, these are not relevant to our computation; they can be absorbed into the phases of the reflection and transmutation amplitudes. The interior solution also involves two a's, a_+, associated with k'+, and these are relevant. Matching the interior and exterior solutions and computing T involves solving a set of linear equations whose coefficients are linear functions of the sines and cosines of the a's. Thus, I TI 2 will be a rational function of these sines and cosines and afortiori strongly dependent on R. This would cause our computation to collapse, were it not that a+ are also strongly dependent on kll; going from one partial wave to the next increases kllR by one, and thus increases a+ by cos l( kll/k'+ ). These are in general two incommensurate angles. Thus, summing over the many partial waves that exist for large R in any fixed interval of kll is equivalent to averaging over ~_+, treated as two independent uniformly distributed random variables. This averaging removes all dependence on the radius R from I Th 2 and enables us to go smoothly to the large-R limit and evaluate the integral (3.25). This is the sequence of steps (matching, computing I TI 2, averaging over phase angles, and integrating) that we have performed, with the aid of a computer, for the step-function boundary, to obtain the plot of fig. 1.
4. The weak coupling limit The only result announced in sect. 1 which we have not yet explained is the analytic expression for small coupling, eq. (1.8). We shall now give a derivation of this formula. Let us consider a ~ wave packet, with support in some small region of momentum space, scattering off an L-ball. For a sufficiently large L-ball, the packet will contain arbitrarily many partial waves. Thus, by computing the probability of
A. Cohen et al. / The evaporation of Q-balls
317
transmutation of the wave packet in the limit of infinite L-ball radius, we automatically compute ]TI 2 averaged over phase angles. We can think of the scattering process as a sequence of events, each of which, for large radius, can be well approximated by scattering off a planar boundary. In the first scattering, part of the initial wave packet is reflected at the L-ball boundary, and part is transmitted into the interior. The transmitted portion is in general the sum of a k'~ packet and a k2 packet; these propagate through the interior with different velocities. When each of these hits the boundary, part is transmitted to the exterior and part is reflected back into the interior; again, each reflected part is the sum of two packets which propagate with different velocities. Iteration of these steps leads to an infinite sequence of scatterings; we must sum the corresponding infinite sequence of amplitudes to obtain the transmutation probability. This is an intolerably clumsy way of approaching the problem, except for small g, where, as we shall see, it becomes surprisingly simple. We begin by analyzing a very special case, a plane wave moving in the z-direction impinging on a planar step-function boundary at z = 0. It will turn out that for small g we will be able to express everything in terms of the solution to this problem. Neutrino spin is conserved in this process; that is to say, we can choose the solutions to be eigenfunctions of o 3. We shall work things out in detail for the case in which the eigenvalue is + 1. In this case all spinors are proportional to (1). We will simplify our subsequent equations by suppressing this factor. Once we are done with this case we can obtain the answer for spin - 1 by rotation about the y-axis. Both inside and outside the L-ball, we can read off the solutions from eq. (2.11). Outside the L-ball, we have a + wave moving in the positive z direction, /
%
and a X wave moving in the negative z direction,
Inside the L-ball, we have two possibilities. If ]o~] is less than g~0, there are no propagating solutions at all, just increasing and decreasing exponentials. Thus, all of the incident flux must be reflected from the boundary. Since any reflected wave must be a X wave, the probability of transmutation is unity. Hence,
I r l 2 = 1,
1601 ~ gq~o.
(4.3)
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A. Cohen et al. / The evaporation of Q-balls
If I~01 is greater than gq~0, inside the L-ball we again have a wave moving in the positive z direction,
and one moving in the negative direction,
where tanh 1~ =
g'/'0
(4.6)
(We have normalised these solutions so they carry unit (positive or negative) values of the conserved current density, J3 = q~to3~b- X*o3X.) The solutions for negative spin are of the same form with z replaced by - z . To do the multiple scattering summation, we must have the amplitudes for a variety of processes: a wave of either spin may approach the boundary, either from inside or outside the L-ball. It is straightforward to match solutions at z = 0 and verify that in all cases the probability of reflection is ereflection = tanhZ/3.
(4.7)
By the conservation of probability, e t . . . . mission = sech2B.
(4.8)
If we let g go to zero for fixed w, the probability of reflection is O(g2), as one would expect. However, if we scale w down with g, the probability is O(1). This is a little surprising - how can a perturbation of O(g) produce a transition probability of O(1)? This can best be answered by going to position space and considering a small-w wave packet moving toward the L-ball. Small w means the support of the packet in k-space is restricted to a shell of O(w) around the sphere [k[ = ~o~0 . 1 Thus, although in the transverse directions the wave packet may have a width of O(~0ol), in its direction of propagation it must necessarily be spread out over a length of O(~0 -1) = O(g 1). A perturbation of O(g) acting over a length of O(g -1) may well produce an effect of O(1). This argument is crucial, because it shows that any change in the L-ball that does not act over a depth of O(g -1) will only make a change of O(g) in the transition amplitude. In particular, such irrelevant changes include replacing the step-function
319
A. Cohen et al. / The evaporation of Q-balls
b o u n d a r y b y a b o u n d a r y of another shape, and changing the angle between the L-ball surface and the incident m o m e n t u m . Eq. (4.7) is universal; it is the leading t e r m for small g and ~0, no matter what*. W e can n o w sum the multiple-scattering series. A ~p-wave incident on an L-ball will be t r a n s m u t e d to a x-wave if either (i) it is reflected on its first scattering, or (ii) it is transmitted into the interior, undergoes an odd n u m b e r of internal reflections, a n d is then transmitted to the exterior. Hence,
]r[2 = tanh2/7 + (sech2 fl )2
~.~(tanh2/7)2r +1 r=0
2 tanh2fl 1 + tanh2fl '
]0~] > gq'0.
(4.9)
T h e integral of eqs. (4.3) and (4.9) give the leading contribution to the evaporation rate for small g. The integrand is large only in a region of O ( g ) near ]k I = 7~00. Thus, to leading order, we can make the substitution (4.10)
~kl~< ~°o dSk I TI 2 = rroa02fo°'O/2 d k IT 12
If we introduce the dimensionless variable y = ~o/g4~o, we find dN dAdt
321r 2
-t
y2+yy2~_
1
"
(4.11)
F o r small g, we can replace the upper limit in the last integral by infinity; the integrals then b e c o m e elementary, and we obtain eq. (1.8). Michael Peskin [5] has pointed out that this analysis must break down for very large L-balls. We have found an evaporation rate proportional to g for small g. This is because neutrinos are produced significantly only in a spherical shell in k-space of radius 7~00 1 and thickness of order g, and thus can only evaporate from these levels. H o w e v e r , in fact, this shell is depopulated not just by evaporation at the surface but also b y neutrino-neutrino scattering in the interior. This process is higher order in h * A skeptical reader might doubt this reasoning, on the basis of the following argument: in reflection, the incident ray and the reflected ray go in quite different directions; they are parallel only for normal incidence. This is manifestly an effect or order uni(v that depends on the angle of incidence. The error in
the argument is that the stated property holds for reflection without transmutation, like the reflection of an electromagnetic wave from a conducting surface. But in the case at hand, there is no reflection without transmutation. The transmuted wave moves away from the surface because its frequency has changed sign, not because anything has happened to k, and the transmuted ray is always parallel to the incident ray, whatever the angle of incidence.
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A. Cohen et al. / The et,aporation of Q-balls
than the processes we have considered; nevertheless, since its probability is proportional to the L-ball volume, it will dominate over evaporation for sufficiently large radius. Thus for very large L-balls, even though neutrinos are originally produced in a thin shell, they scatter into all the energetically allowed levels, and the evaporation rate is on the order of the absolute upper bound [5]. To estimate the radius at which this mechanism becomes important, we must compute the rate at which the shell is depopulated through scattering. (We already know, from our earlier computation, the rate at which it is depopulated by evaporation.) As long as the scattering rate is significantly smaller than the evaporation rate, eq. (1.8) is safe. We compute the neutrino scattering amplitude in empty space, ignoring the effects of the 0-field background. Since the neutrino-neutrino interaction has a range on the order of the meson Compton wavelength, much smaller than (g~0) -1, this should be a good approximation as long as we are not concerned about which states the final neutrinos end up in. In the theory at hand, for small g, the scattering amplitude is given by one-meson exchange. An elementary Feynman computation yields 16g4s 2 IAI 2 -
(~2--S)2
(4.12)
~
where A is the invariant neutrino-neutrino scattering amplitude (evaluated between relativistically normalized states) and s is the square of the c.m. energy. We now fold this formula into an initial state in which all the occupied levels are in a thin spherical shell of radius 7~0. We find scattering rate
gS~poRh2
evaporation rate
12qr2
X2
--]ldx . % (/~2/~02- x)2
(4.13)
Everything in this expression, other than the 12e 2, is easy to understand: The trick is to think of computing the ratio of the two rates per occupied level. Thus the factor of R comes from the surface-to-volume ratio in the evaporation rate. The integral and a factor of g4 comes from the scattering probability, averaged over all possible values of s. A factor of g~0 comes from the thickness of the occupied shell, which gives the incident flux. The factor of h 2 and the (absent) powers of ¢% are fixed by dimensional analysis. The value of R at which eq. (1.8) breaks down is quite large for weak coupling, being proportional to g-Sh-2. Just to get some definite numbers, we have chosen the parameters in our theory to equal their closest analogs in the standard electroweak theory. Thus, we have chosen g to be the electron-Higgs Yukawa coupling, ~0 to be the Higgs expectation value, and % to be the intermediate vector boson mass.
A. Cohen et al. / The evaporation of Q-balls
321
We have also approximated the integral in eq. (4.13) by one. These choices yield a pedestrian evaporation velocity of around one m/sec, and predict that our weakcoupling formulas, valid for L-ball radii larger than 1/gq~ o ~ 3 x 10 -7 cm, break down for radii larger than 2 × 10 s km, somewhat larger than the radius of the earth's orbit around the sun. One of us (H.G.) would like to thank S. Dimopoulos for useful discussions. This research is supported in part by the National Science Foundation under grant number PHY-82-15249. References [1] S. Coleman, Nucl. Phys. B262 (1985) 263 [2] G. Rosen, J. Math. Phys. 9 (1968) 996; T.D. Lee and G.C. Wick, Phys. Rev. D9 (1974) 2291: R. Friedberg, T.D. Lee and A. Sirlin, Phys. Rev. D13 (1976) 2739; Nucl. Phys. B115 (i976) 1, 32; R. Friedberg and T. D. Lee, Phys. Rev. D15 (1977) 1694; TD. Lee, m Proc. Symp. on Frontier problems in high-energy physics, eds. L. Foa and L. A. Radicatti (Scuola Normale Superiore, Pisa, 1976) 47; Particle physics and introduction to field theo~, (Harwood, London, 1981) 141; J. Werle, Phys. Lett. 71B (1977) 367; T.F. Morris, Phys. Lett. 76B (1978) 337, 78B (1978) 87 [3] G. Gelmini and M. Roncadelli, Phys. Lett. 99B (1981) 411; G. Gelmini, S. Nussinov, and M. Roncadelli, Nucl. Phys. B209 (1982) 157 [4] Handbook of mathematical functions, ed. M. Abramowitz and I.A. Stegun, NBS Applied Math Series 55, eqs. (9.1.71), (9.3.3) [5] M. Peskin, private communication