SCA NG F FERMIONS FROM A COSMIC STRING 1. Introduction Some years ago, Callan and

34 Pages·2002·2.04 MB·English

Nuclear Physics B353(1991) 237-270
North-Holland
SCA NG F FERMIONS FROMA COSMIC STRING
W.B.PERKINS', L. PERIVOLAROPOULOS2,A.-C. DAVIS1.3,
R.H. BRANDENBERGER2 and A. MATHESON2
'Department ofAppliedMathematics and TheoreticalPhysics, UniversityofCambridge,
Cambridge, CB3 9EW, UK
2DepartmentofPhysics, Brown University, Providence, RI02912, USA
3KingsCollege, UniversityofCambridge, Cambridge, UK
Received 28August 1990
(Revised 19November 1990)
The Dirac equation issolved for a fermion in the background fieldof acosmic string. We
focus on the amplitude of the spinor at the core radius and study its dependence on the flux
alongthe string and on the core model used. Our results show that there is an amplification of
the wave function at the core radius for some values ofthe flux. The implications for baryon
decay catalyzed by cosmic strings are discussed. We show that, for certain values ofthe string
flux, the catalysis cross section is a strong interaction cross section. The elastic and inelastic
scattering cross sections are calculatedand the model dependence ofthe results discussed.
1. Introduction
Some years ago, Callan and Rubakov [1,2] showed that it was possible for a
grand unified monopole to catalyze baryon decay with a strong interaction cross
section rather than the very much smaller geometric cross section. This enhance-
ment can be understood classically as the capture cross section for a spin-'-,
particle by the monopole [3] or quantum mechanically as being due to the
amplification of the fermionic wave function at the monopole core [4]. Both of
these explanations stress the importance of the long-range magnetic field of the
monopole.
Given that the monopole catalysed decay cross section is so large, the obvious
question to ask is: does a similar enhancement of the cross section occur for cosmic
strings? Again we have a region whose thickness is given by the inverse GUTmass
within which baryon number violating processes can occur. Thus the naive cross
section is related to the inverse GUT mass, M-'. Since strings are linear defects,
we will always be concerned with the cross section per unit length. Classically the
string has no long range fields, so we would expect no enhancement. Quantum
0550-3213/91/$03.50©1991 - Elsevier Science PublishersB.V. (North-Holland)
238 W.B. Perkinset nl. / Cosmic string
mechanically the long-range gauge field of the string may cause enhancement, but
this was not thought to be the case for strings with integer magnetic fluxes [4].
It has recently been shown [5,6] that the cross section for cosmic strings with
non-integer magnetic fluxes can be enhanced from its naive geometrical value to
the much larger inverse baryon mass O(m -'). The analysis of ref. [6] was based on
a first quantized treatment, in contrast to the perturbative second quantized
methods of ref. [4].
In this paper, we investigate the quantum mechanical scattering of fermions
from a cosmic string allowing for fractional magnetic flux, using the methods of ref.
[4]. We demonstrate that the results depend crucially on the model for the core of
the string. We calculate elastic and inelastic cross sections.
We obtain the same results using the first quantized method of ref. [6] and the
second quantized prescriptions of ref. [4].
By using physically reasonable approximations for the fields in the core of the
string we avoid the mathematical problems associated with infinitesimally thin flux
lines encountered in refs. [7,81. Considering the string in the wire approximation
led to problems with the boundary condition on the wave function at the origin. It
was pointed out that the wave function had to be square integrable, and not
necessarily regular, at the origin for the hamiltonian to be well defined. (This was
obtained by considering self-adjoint extensions to the hamiltonian.) However, by
considering a string of finite thickness, the interior solution is both square
integrable and regular at the origin. Consequently, the cross-sections we derive are
physically more reasonable, being continuous around integer flux values.
Our main result is that the enhancement of the elastic and inelastic cross
sections depends both on the value of the fractional flux of the string and on the
core model. In our models baryon number violation is mediated either by Higgs or
gauge fields. In both cases maximal enhancement results in a strong interaction
cross section. For the latter case we find that maximal enhancement occurs for
integer flux, in the former for half integer flux. Our results are purely quantum
mechanical in nature and can be traced to a "spin-flux coupling" that is, a direct
coupling of the fermion spin to the flux of the string.
We use both first and second quantized methods to calculate cross sections. The
field theoretic calculations are given in sects. 2-6 while sect. 7 contains the
quantum mechanical calculations. As expected, the results agree. For both meth-
ods we have verified unitarity, using both the optical theorem and flux conserva-
tion.
In sect. 2 we review the scheme of ref. [4] to compute the catalysis cross section.
This approach is motivated by first-order perturbation theory in a second-quan-
tized analysis.
In sect. 3 we consider the Dirac equation for a fermion in the field of a
Nielsen-Olesen vortex [9]. The fermion couples only to the gauge field ("pure
gauge case"). Using a partial wave analysis, we determine the amplitude of the
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W.B. Perkins et al. / Cosmic string 239
fermion spinor at the string core radius. Forsimplicity, we consider a toy model for
the string core in which the gauge field vanishes. A more realistic model of the
core is analyzed in appendix A. While the toy calculation is of interest in its own
right, it also highlights the essential features of the more realistic calculation of
appendix A and can easily be generalized to other cases of interest.
In sect. 4 it is shown that this calculation is readily generalized to allow for the
inclusion of scalar fields which couple quarks and leptons in the string core. We
then have a self-contained model of a string which can catalyze baryon decay.
In sect. 5 we consider a model in which extra gauge fields excited in the string
core mediate baryon decay. This model is motivated by grand unification.
Having calculated the wave function amplification factors at the core radius for
the various models considered in sects. 3-5, in sect. 6 we use these results to
calculate the scattering and catalysis cross sections using the second-quantized
method of sect. 2.
In sect. 7, first-quantized elastic and inelastic scattering cross sections are
calculated for the various models already introduced. The pure gauge model of
sect. 3 is found to give an elastic scattering cross-section which interpolates, as the
fractional charge varies. between the Aharonov-Bohm cross section [10] and that
found by Everett [11]. The first-quantized cross sections are found to agree with
the catalysis cross sections determined in sect. 6.
There are five appendices. Appendix A contains a detailed derivation of the
fermion wave function amplification factor for a core model in which the gauge
field increases linearly with radius. The results agree with those derived using the
toy model of sect. 3. Appendix B is a discussion of the matching conditions at the
core radius. The normalization and asymptotic behavior of the wave functions are
discussed in appendix C. Appendix D contains the technical details of the first-
quantized calculations of sect. 7. Finally, appendix E is a brief analysis of unitarity.
2. The proton decay catalysis cross section
Here we briefly review the prescription of ref. [4] to calculate the proton decay
catalysis cross section. We consider free quarks and hence our analysis applies at
energies larger than the confinement scale. At smaller energies we would have to
consider the inelastic scattering of skyrmions by a cosmic string [12]. We focus on
interactions catalyzed by fields excited in the core of the string. For nonabelian
strings there are also direct Aharonov-Bohm type processes which lead to baryon-
number violation [13].
We assume that the baryon-number violating processes in the core are due to an
interaction lagrangian _f', which couples the quark fields 41 to gauge and scalar
fields AI and 0 of the string. 0 is dynamical whereas A,_, and 0 are treated as
_L
static background fields.
(cid:9)(cid:9)(cid:9)(cid:9)
2-x0 W.B. Perkins etal. / Cosmicstring
According to first order perturbation theory in a quantum field theory, the
transition matrix element between an initial state 141 > and a final state 14Y> is given
by
= J411i0-x - (411 d4x_Vj(x)iq1i
1f
Since Yj is suppressed outside the core, the integration over x reduces to an
integration over the world-tube of the string.
We divide the computation into two steps. In the first we evaluate (2.1) using
free fermions spinors, resulting in what we define as the geometrical cross section
(du/df2)geom- In the second step, we solve the Dirac equation for 0 with
scattering boundary conditions to determine the amplitude of the spinor at the
string core radius R. We define the amplification factor A as
A= (2.2)
I free(R) I .
'V2
Since the cross section is proportional to and since W involves two fermion
spinors, the catalysis cross section is enhanced by a factor A4 over the geometrical
cross section,
du - 4 du
-A (2.3)
df2 df2 ~geom
This prescription has been applied in ref. [4] to proton decay by monopoles
(rederiving the results of Callan and Rubakov [1, 2] and cosmic strings with integer
flux (for a review see ref. [14]). Here, we shall generalize the analysis to strings
with fractional flux.
3. The irae equation in the vortex fields
To be specific, we consider a Nielsen-Olesen vortex [9]. This represents an
infinitely long, straight, static cosmic string. The gauge field of the vortex has only
an azimuthal component, A.. Beyond the string core, A, - 1/r and within the
core it falls to zero. There are no analytic solutions of the field equations for the
vortex, so we are forced to model the core. The obvious form to take is Ae a r for
and A ey I11r for r > R. This corresponds to a tube of uniform magnetic flux
of radius R. We will refer to R as the core radius. A second model results if we
take A. = 0 inside the core. This model is less physical but easier to analyze
mathematically, and hence it will be considered first. In appendix A we shall
summarize the computations for the first model. Since both models give the same
results, we conclude that the cross section does not sensitively depend on the
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W.B. Perkinsetal. /Cosmicstring 241
detailed model of the gauge field inside the core. Note, however, that the result
sensitively depends on the coupling to and model for the scalar field (see sect. 4).
We wish to solve the Dirac equation for fermions which are only coupled to the
vortex gauge field and not to the scalar field [151
(y'(id.-eA,j-th)4i4 =0 with (0, - yf, xf,0) , (3.1)
A,_L=
where m is the mass of the fermion outside the string. We can write the four
component spinor `Y4 as two two-component spinors 412 b, which are in general
given by two coupled equations. Using the basis
o °~3 0 I '0»2 0 2 -'el 0 3 0 1
y = 0 , y = 0 , y = , y = -1
-473 -iu2 0 if, 0),
(3.2)
the coupling terms involve only z-derivatives. Thus, if we consider wave functions
independent of z, the equations decouple. Making the ansatz
°` Fa,h( r) e"r"e
02,b
(3.3)
r)e~e
!1= - oc GiTa,h(
we find that the components of h are related by the equations
42'
d n
-i(w+m)G +l ---ere F =0,
~dr r
( d n +1
-i(w-m)F + + +er~ G =
dr r
where m = m for the upper two-spinor and m = -rn for the lower.
To model the gauge field of a cosmic string as discussed above, we should take
the following approximate form for ~:
~= ~ 1/g r<R
(3.5)
1/hr2 r>R'
where for continuity hR2 =g. The total magnetic flux in the core is 2T/h. For this
model, the solutions of the Dirac equation for r < R are not given in closed form.
We will relegate their discussion to appendix A and instead discuss the simpler
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242 W.B. Perkisset al. / Cosinicstring
core model given by
0 r<R
(3.6)
`' 1Ihr2 r>R'
which corresponds to a flux ring of radius R.
With (3.6), the first-order equations are
d v
-i(w +m)G + - )F 0, (3.7)
( dr r
d v+1
i(w - m)F + + G 0, (3.8)
( dr r° )
where
v=n +a0(r-R), (3.9)
and a, the flux on the string, equals e/h and Or -R) is the step function. It is
convenient to rescale the radial variable
2
x= (w -m2)1/2r=kr . (3.l0)
By combining eqs. (3.7) and (3.8), we obtain the following decoupled second-order
differential equations for F,, and G,,:
d2 1 d a v2
dx2 + x dx &(x -xo) - x2 + 1 F = 0,
x0
d2 1 d a (v +
1)2
S(x-x + 1 G = 0. (3.12)
x= + ddxx + x, -"°) - x2
0
In the above, xO, =kR.
We shall solve eqs. (3.11) and (3.12) by finding the general solutions for x <xO~
and x >xO and matching at the boundary. The matching conditions are discussed
in appendix . For the model given by (3.6), they become
(cid:9)G+(F+(xxo) =F-(oxo) , )= G-(xo) , (3.l3)
(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)
W.B. Perkins et al. / Cosr»iicstring 243
or equivalently
a
F+(xo) - F~(x() _ -F(xo)
xO
a
G',(x,» - G'(xo) = - -G(xo) - (3.14)
xO
For convenience, we suppress the index n. The subscripts + and - stand for the
exterior and interior solutions respectively, and the prime stands for d/dx.
It is important to keep in mind that F and G are not independent; they are
related by the first-order equations. Hence, we shall determine F by solving the
second-order equation (3.11) and then infer G from (3.7). Both in the interior and
exterior regions, eq. (3.11) is a Bessel equation. The interior solution which is
squared integrable (and regular) at the origin is
x)
FT,-( =cJ(x)  (3.15)
where J(x) are the Bessel functions. The exterior solution can be written in terms
of any two linearly independent solutions Zti(x) and Z,,(x) of the Bessel equation,
e.g. J,,(x) and J_,,(x):
F, +(x) =aZ,l(x) +bZL2(x) . (3.16)
The coefficients a,t, b and c are determined by matching at x = xo, and by
demanding that for x ~, F, +(x) approach a plane wave plus cylindrical out-
going wave.
Using the Bessel function identities
v
Z ± (3.17)
-xZV= +Z",-+11
the lower spinor components G can be determined from (3.7). The resulting
spinor has the form
x Zi Z2
412 = n--x a,~,i ±1BZ(1vV+ 1) eiloB + b~1 _j ~ 2V 1)ei9 e"ne- (3.18)
for x > xo, and
c J
e~ne-«t~t
(3.19)
" iBJ(n+ )e'e
x
(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)
244 WB. Perkinset al. / Cosmic string
for x <xo. In the above, B = k/(w + ni), and the sign takes the value + 1 for
Z,, = J, N,, and H,,, and -1 for Z, = J_, N_, H_, where N and H are
Neumann and Hankel functions respectively.
Inserting eqs. (3.15) and (3.16) into the matching conditions(3.13) and (3.14), we
obtain
_b,, __ Z,',J,, - J,,Z," + («lxo)Z~J,, __ Z,',+ 1J,, -Z,',J,,+1
(3.20)
t
a,: JZ-' -J~1Zv -(«/x0)ZVJn Zvjn+1 +Zv+IJn
The second equality follows from eq. (3.17). Similarly,
cit Zv1Zv2' - ZIvtz2v ZvZv+1 +ZvZv+1
(3.21)
2+1
a JrtZ21r -JirtZv2- -JttZ2v JrrZv + ZVJft+1
x{)
Obviously, all functions are to be evaluated at x0).
We shall choose the particular basis Zv =J and Z2 =.j_ (This choice is valid
provided v is not an integer.) In this case, the boundary conditions at x ---) 00
demand that the larger of the coefficients a and b be 1 (up to a phase; see
appendix C).
Typically, the fermion momentum k will be much smaller than the mass scale
ltd = R- ' of the cosmic string. Hence, x0) << 1. From the small argument expansion
of the Besse! functions, it follows that for n > 0
-b1t 20,+1) (3.22)
'xo
a
while for n < 0
bit
2v
. -xo . (3.23)
a
Hence, for n > 0 we must set a = 1 for v > -1 and b = 1 for v < -1, while for
n < 0 we set a,,=1 for v> 0 and b,,=1 for v.<0
Inserting the above values for the coefficients a and b into the partial wave
expansion of 02 [see eq. (3.18)] and evaluating the result at x1) using the small
argument expansion of the Bessel functions, we obtain the order of magnitude of
4 at the core radius. For n > 0 and v < -1 the result is
-v-2
x()
02,n ti _v_1 (3.24)
x()
(cid:9)
(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)
W.B. Perkinsetal. /Cosmicstring 245
while for v> -1
v
X0
v2,n 3.25
~ Xv+1
0 ~'
which are to be compared with 412 - 1 for the free spinors. From (3.24) and (3.25),
we conclude that the upper component of is amplified while the lower one is
`12
suppressed.
For n < 0 and v> 0, the dominant term in the partial wave expansion of is
'A2
412,n (3.26)
while for v < 0
412,n (3.27)
XO
In this case, the lower component of the spinor is amplified and the upper one
suppressed.
The amplification factor A defined in eq. (2.2) is
A -Xj - k (3.28)
( M)
where the power p is the inverse of the exponent in (3.24)-(3.27). In fig. 1 we plot
p(v) as a function of v for n > 0 and n <0. Finally, we want the amplification
P(v )
2
e
-4 -3 ,--2 e'-I 0 I °e 2 3 4
v v
`v ° P
v
Fig. 1. The inverse amplification exponent p plotted as a function of v=n+a for n>0 and rt<0.
Nocouplingwithscalarfield isassumed.
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246 W.B. Pet-hillsetnl. / Cosmicstring
P(a)
2
n=2 n=1 n=0 n=-I n=-2 n=-3
v I-
v
v
-4 -3 -2 -I 0 I 2 3 4
Fig. 2. The maximum inverse amplification exponent p plotted as a function of a (the fractional flux
ofthestring). Nocouplingwith the scalarfield.
factor p as a function of the fractional flux a in the string. For fixed a we find the
p(v)
partial wave n which gives the largest value of ). The result is sketched in fig. 2.
The peak amplification factors in this model arise for a = -n for n = - l,
-2,. . . and for a = -ti- 1 for n=0,1,2. . . . . For a < 0 amplification occurs in
the upper component of 0,, for a > 0 in the lower component.
In the case of integer flux our chosen basis is not complete. We have done a
separate calculation using a complete basis and find, for integer flux, the result
(3.28) modified by the logarithmic factor (see appendix D, eq. (13.21)).
In an attempt to gain a heuristic understanding of these results, we can view eqs.
(3.11) and (3.12) as Schrödinger type equations with potentials
(cid:9)a(n +a)2
V+(a,x) = x2 + -(x-xo), (3.29)
x0
(n+a+1)2 a
V-(a,x) - 2 - 5(x -X(» . (3.30)
x x0
The first term is the centrifugal barrier term. Note that the string gives rise to a
shift in the angular momentum modes. The second term in eqs. (3.29) and (3.30) is
a spin-flux coupling which couples the spin of the fermion directly to the flux of
the string. For a > 0, this force is repulsive for the upper component of the spinor
and attractive for the lower, for a < 0 it is attractive for the upper and repulsive
for the lower component. Thus, for a > 0, amplification of the fermion wave
function is due to the lower component, for a < 0 it is due to amplification of the
upper component.