Journal of Modern Physics, 2013, 4, 904-910
http://dx.doi.org/10.4236/jmp.2013.47122 Published Online July 2013 (http://www.scirp.org/journal/jmp)
The Khuri-Jones Threshold Factor as an Automorphic
Function
B. H. Lavenda
Università Degli Studi, Camerino, Italy
Received February 8, 2013; revised March 10, 2013; accepted April 6, 2013
Copyright © 2013 B. H. Lavenda. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The Khuri-Jones correction to the partial wave scattering amplitude at threshold is an automorphic function for a di-
hedron. An expression for the partial wave amplitude is obtained at the pole which the upper half-plane maps on to the
interior of semi-infinite strip. The Lehmann ellipse exists below threshold for bound states. As the system goes from
below to above threshold, the discrete dihedral (elliptic) group of Type 1 transforms into a Type 3 group, whose
loxodromic elements leave the fixed points 0 and ∞ invariant. The transformation of the indifferent fixed points from −1
and +1 to the source-sink fixed points 0 and ∞ is the result of a finite resonance width in the imaginary component of
the angular momentum. The change in symmetry of the groups, and consequently their tessellations, can be used to dis-
tinguish bound states from resonances.
Keywords: Threshold Factor; Automorphic Function; Elliptic and Loxodromic Elements
1. Introduction and Summary
This paper suggests that the origin of strong interaction
symmetries may be found in functions that are auto-
morphic with respect to a group generated by a fractional
linear transformation. Examples of automorphic func-
tions are the circular functions which are automorphic
with respect to

, and elliptic functions
which are periodic with respect to a group generated by
two translations. Elliptic functions live on tilings that are
parallelograms which fill up the entire plane. By cutting
and pasting, the tessellations can be made into a torus of
genus 1.

2nn
1z
Poincaré and Klein were interested in looking for
automorphic functions of higher genus; that is, complex-
valued functions invariant under specific groups. Klein
had already examples in the shape of regular solids, of
which there are only a finite number. We will show that,
for real angular momenta, the partial wave amplitude is
such an automorphic function corresponding to the dihe-
dral group. According to Khuri [1] and Jones [2], the
partial wave amplitude derives its form from the asymp-
totic large angular momentum limit of the Legendre
function of the second kind which has three singular
points. These singular points are homologues of vertices
of triangles in a conformal mapping, and the group we
will be dealing with is the triangular group.
In the unphysical region the angular momentum be-
comes complex, and, consequently, the vertex angles of
the triangles in the division of the sphere, which would
otherwise have the form of a double pyramid, also be-
come complex.The real part corresponds to a rotation,
while the imaginary part represents a stretching. What
were indifferent fixed points of an elliptic transformation
below threshold, cosz, where



0,z
, become
source-sink fixed points, , of a loxodromic
transformation above threshold. What mathematically
can be obtained by conjugation has a completely diffe-
rent physical explanation: the transition from the physical,


1, 1z



1, 1z 

3SO
0,
2
3 
, to the unphysical, , region as
the system passes through the threshold. This can also be
viewed as a transformation from the discrete, elementary
group of Type 1, consisting of elliptic elements whose
group is conjugate to a subgroup of , to a discrete,
elementary group of Type 3, with loxodromic elements
that leave invariant the fixed points . In this way, it
may be feasible to study the strong interaction sym-
metries through the tessellations of the hyperbolic plane,
, and ball, , that Poincaré used to study the sym-
metries of kleinian groups, or discrete groups of Möbius
transformations with complex coefficients.
C
opyright © 2013 SciRes. JMP

B. H. LAVENDA 905
2. Modified Scattering Amplitude
It is well-known that the scattering amplitude can be cast
into what is now known as the Watson-Sommerfeld
representation [3],






,
ik
k
,
sin
i
ii
A
kz k


P z





k  
 
1Pz
integer
k
(1)
although its history goes back much further to Poincaré,
who used it to study the bending of electromagnetic
waves by a sphere [4]. The poles occur at integer values
of the angular momentum, , and the residue is
written as the product of the Legendre function of the
first kind,  , , which
contains all the angular dependencies, and a factor
, which is a function of the linear momentum, .

Pz

k 
Partial waves can be projected out of (1) by using
  
1 1sin
d,
21





1Pz
Pz z



 (2)
for integer and complex

. The contribution from
one Regge pole to the partial wave scattering amplitude
is
 




.
1
k
kk




2.kk
Ak



The residue has been found to vary as [5,6]

 

 
This gives incorrect threshold behavior by predicting
that
 

20
.
001


 
0
k
A
 

In order to correct the result, Khuri [1] and Jones [2]
modified the residue,



1
cosh ,z 

expkk   (3)
where
2
2
1 1,
2
m
k

20km
0
cosz   (4)
in the unphysical region where , and is the
particle’s mass. In the limit as , they obtained
the correct threshold behavior
2
k
 

1
cosh 2 zkm 


 



22
2
e.
kk
m







The added factor (3) supplies the partial wave with a
branch point at 22
4km because the square root in

2
22
22
1
22
mm
kk





 
 






1
coshln 11z







vanishes there, but it does not give the additional branch
points at 22 2jm k2,3,j


for .
3. The Dihedral Group
Unwittingly, Khuri and Jones transformed the partial
wave scattering amplitude into an automorphic function.
It is the form that the Legendre function of the second
kind takes in the limit as [7], viz.,

1
1cosh
2
1e
z
.Qz




 




(5)
Said differently, the automorphic function,

21wz z



 




(6)


1
2
2
2
1
,
1
zz
zz
 








(7)
is the ratio of two independent solutions of the Fuchsian
differential equation,

2
2
d0,
d
yIzy
z


where





  
2
22
2
2,
111
11
44 2
.
21 21 21
Iz wz
z zz





 

wz

(8)
The curly brackets denote the Schwarzian derivative of
with respect to ,
2
3
,.
2
ww
wz ww

 






It is a projective invariant that was first discovered by
Lagrange in his studies of conformal mapping of a sphere
onto a plane, although the name was coined by Cayley in
favor of Schwarz.

The Schwarzian (8) clearly shows that

Qz
1, 1,z has three singular points at

 w , which are homo-
logues of the vertices of a triangle in the -plane. Since
the sum of angles,

11 ,
22



 



z1, 1,
the triangles are spherical, and there is no real orthogonal
disc which encloses the triangular tessellations. Hence,
the need to project them onto a sphere.
The spherical triangles are conformally represented in
the -half plane by the angular points, . The
plane is then projected onto a sphere stereographically

B. H. LAVENDA 906
[8]. The triangle on the sphere is bounded by arcs of
great circles that cut the equator orthogonally. They form
a double pyramid with the summits at the poles, each
triangle having a vertex angle


, and two right
angles at the base on the equator, as shown in Figure 1.
At the pole,

, the conformal mapping (7) be-
comes



2
2
21
1
1
21
ln1cosh .
w
z





 





 
coshwuiv 
in ,
ln zz
zz
zz
Since ,

cosh coshzw
cosh cosandsinh s
x
uv yuv
0u
0v
the upper half of
the -plane is conformally mapped onto the interior of a
strip in the -plane bounded by the line on the
right, and the lines on the bottom and
zwv


 
on
the top.
The partial wave amplitude, (2), then reduces to
121
 at the pole, so that the contribution from
one Regge pole at


20k
to the partial wave projection,
as , is




2
4ln
1.
21
kmk



1
A 

Inverting the conformal map (7) gives the dihedral
equation [9],

cosh ln,nw 

1.n



11
2
n
n
zw
w



 (9)
where1
(10)
Introducing homogeneous coordinates (9) can be written
as
22
11 2
12
1,
2
nn
nn
w
ww


2
zw
z (11)
which is the arithmetic-geometric mean inequality. The
vanishing of the functional determinant of the numerator
and denominator of (11) [9],

2 2
1 2
40 ,
n n
w
1n
21 21
212 12
12
11
12 12
22
22
nnnn
nn nn
nwnw nw ww
nwwnww



indicates that there are


zeros at
Figure 1. The -sphere for the dihedron where the sphe-
rical triangles consist of hatched and unhatched parts with
bases on the equator. The critical points −1, 0, 1, ∞, corre-
spond to the branch points, −1, +1, and ∞ on the sphere.
w
10w

and
, and another zeros at
20w2n
1
2
1.
n
w
w




Therefore, we can write the ratio of homogeneous
coordinates as either,

11
22
,or0,1, ,1,
rr
ww
gggrn
ww


 (12)
where the first term
2
e, e,
in in
gg

 
w
is the nth root of unity.
Since the values of the homogenous coordinates have
absolute value of unity, they lie on the equator, and (12)
correspond to rotations of the -sphere either through
angles 2rn

, or rn


0w. According to (9), the critical points w

and


z of the w-sphere correspond to

z
r
wg of the -sphere. In addition, the points

correspond to the point on the -sphere,
while those of
1zz
r
wgg

1z all correspond to


1, 1,
on
the same sphere. Hence, there will be three branch points,

 z on the -sphere—the same three singular po-
ints of the Legendre function of the second kind.
4. Complex Angular Momentum
For

close to we can perform a Taylor series ex-
pansion in terms of the energy difference



EEE 
,
,
and retain only linear terms [10],
I
Ei
R

 (13)



Re
where R



, I

 Im

, and the prime denotes
differentiation with respect to the energy, evaluated at the
energy . It is clear from the dihedral equation, (9),
that
E



when both are real. This also follows from
the condition that the fixed energy dispersion relation,
1The most interesting cases for angular momenta are
 
0
0
1d,
z
A
zQzAzz



(14)
converges. For large z,


A
zz


2n
[10, p. 148], while

, and

1

n, although we can choose some multiple
s
of

h
can be made arbitrarily close to an integer whose sole effect is to change
the number of iterates on the elliptic element,
, whic
s
g
g.

B. H. LAVENDA 907


1
Qz z




Pz
  for all values of , and not like
which requires 1 Re 2 
Re Re
, so that (14) will converge only
when


ei
wr
.
Introducing (13) into (9), together with

, give
22
ln 2
cosh1
4
R
Er i
z
E








11
ln
2
.
E r













(15)
The resonance width,
1,
2
I
R




vanishes below threshold, but
R


zxiy
is always positive
definite. Introducing into (15) it becomes
clear that only below threshold do we have a conformal
mapping [11] of circles, rconst

, in the -plane,
onto ellipses,
w

22
2
1,
RR
y
rE


 
z
2
cosh lnsinhln
x
rE (16)
in the -plane, whose semi-major axis and eccentricity
are

coshln rE 

R and

sech lnR rE 
, respec-
tively. In other words,
ln
h ,
R
ri
E
cosz







 (17)
represents an ellipse with semi-major axis

ln rE 
cosh R, and eccentric angle, RE




. In
high energy physics, (17) would be referred to as a Leh-
mann ellipse [12].
Moreover, the straight lines const

w., in the -
plane, are mapped onto hyperbolas,

22
2
1,
RR
y
EE




z
2
cos sin
x

(18)
in the -plane, with semi-axes

cos E 

R and

sin E 

1z
ln r
R. The foci of the hyperbolas are the same
as those of the ellipses (16), viz., . The families
of ellipses, (16) and hyperbolas, (18), that constitute a
system of confocal conics is destroyed above threshold
by a mixing of and

in the real and imaginary
parts of the argument in (15) due to the presence of a
finite resonance width, . 
5. Discontinuous Elementary Groups
Below threshold, the resonance width, , vanishes iden-
tically, and the group of rigid motions are pure rotations,



n.
R
RR
E
E








1


cos si
sin cos
R
Ei
iE




V (19)
The rotation matrix (19) ensures that the fixed points
are at

, instead of the usual rotation matrix which has
indifferent fixed points, i

. The motion on a sphere is
shown in the upper sphere in Figure 2. We are in the
physical region of bound states where


cos 1,1   .
Dynamical resonances for decaying unstable particles
give a finite width above threshold. This introduces de-
formations which are hyperbolic elements,
11
cosh sinh
22
11
sinh cosh
22
RR
RR





 
 





.






U (20)
The fixed points of (20) are the same as those of (19),
and, consequently the trace of their commutator is


,2tr

UV . The combination of stretching and rotation
give rise to loxodromic elements,

11
cosh sinh
22
11
sinh cosh
22
RR
RR
iE iE
iE iE






  
 



 


  

 
 

TUV
(21)
which can be thought of as a rotation through a complex
angle, 1
2
REi







.
The map,
1,
1
z
zz



1 moves

to 0 and 1

to . Mathematically, this is
mere conjugation, but, physically, it represents the trans-

Figure 2. Elliptic (above) and loxodromic (below) motion on
a sphere.

B. H. LAVENDA 908
formation of real (bound states) into complex (dynamical
resonances) angular momenta. The loxodromic motion is
shown in the lower part of Figure 2, which is described
by the scaling map,
,zaz

2
eRiE a



1
1

0

(22)
where the modulus of the multiplier, , is
greater than 1. The fixed point at  (or ) is an
attracting fixed point, or sink, while the fixed point at 0
(or ) is a repelling fixed point, or source. Therefore,
points will spiral out of the source, 0, and spiral into the
sink, , as shown in lower figure in Figure 2. Below
threshold, , and the fixed points are indifferent,
giving rise to the pattern shown in the upper figure in
Figure 2.
For a positive integer, the contour 1 in the de-
finition of the Legendre function of the first kind [13],



1
11
22
Pz i
2
1
1d,
z








z z
11

1

 (23)
is a closed curve, oriented in the positive direction, that
encloses both points . The integrand is real when
is on the real axis between and . This is the phy-
sical region. For non-integer , (23) is a hypergeometric
function with a branch cut going from to 

 in
order to keep it single-valued.
In contrast, Legendre function of the second kind,



2
11
4s in2
Qz i
2
1
1d,
z








1

 (24)
where 2 is a figure 8 encircling the points

and
. The Legendre function of the second kind, (24), is
regular and single-valued in the -plane which has been
cut along the real axis from 1 to . According to (24),
cannot be an integer, and it has logarithmic singulari-
ties at and because the path must cut the
lines joining to and .
1z

1
zz 
U1 U

1
1zz 
1
The isometric circles of and are
1
2
:e0and :
R
IzI z


 

1
2
e 0,
R


 
 
T
T
respectively, which are exterior to one another. Isometric
circles define a complete locus of points in the neigh-
borhood of which lengths and areas remain invariant
under substitutions of the form [14]. It can be shown
that the fundamental region for the group of motions
generated by consists of that part of the plane ex-
terior to
I
and
I
 [14, p. 54]. A point belongs to the
fundamental region if a circle can be drawn about the
point as its center that does not contain an interior of the
isometric circles.
For both real and complex




3

, the groups which
are formed are discrete and elementary. By elementary it
is meant that there exist finite orbits in [15]. In the
case that of real angular momenta, they constitute the
group of finite, cyclic rotations. More specifically, if
s
is number of vertices of the polygon, where each vertex
has
j
n elements, then the relation to the size of the
group, G, and the number of elements of the vertices is
[15, p. 85]
1
11
21 1
s
jj
Gn



 
 





2n
. (25)
Moreover, since j it also follows that the right-
hand side of (25) is bounded by
1
11
1.
2
s
jj
s
n


s






2s Now, for

we have a crescent for by two non
concentric intersecting circles with two vertices, and (25)
reduces to
12
1111
,
2Gnn




which identifies G as the harmonic mean. It can be
satisfied by 12
Gnn


G
2

3s
. By conjugation, the fixed
points, or vertices, can be brought to 0 and . Then
is a finite, cyclic group of rotations in [15].
The case

applies to a dihedron, for which (25)
becomes
1
12
1.
s
jj
nG



n
(26)
Suppose the number of elements at the vertices are
ordered as 123 nn

3n
12n
. The choice 1 is incom-
patible with (26) since the sum on the left will be inferior
to 1. We, therefore, choose , and (26) reduces to
23
1112
.
2nn G

4nn
n
Since 2 will lead to a contradiction, 2 can be
either 2 or 3. In the former case 3 is free to take on any
value from 2 to

so there will be 2Gn
2n

sides of
the various triangles in the division of the sphere, as
shown in Figure 1.
Now another elementary group has , and has
every element of the group leaving 0 and invariant
[15, p. 84]. The group cannot have the Poincaré metric,


2
1dz z
2
2

2

, because loxdromic motion does not pre-
serve the unit disc, . Rather, what is required is
Poincaré’s extension which adds on an additional dimen-
sion so that points on the plane are stereographi-
cally projected onto the sphere . Rotations, like loxo-

B. H. LAVENDA 909
dromic transformations, also do not have an invariant
real disc, , so that both elliptic and loxodromic
generators live on the sphere, .
2
2

1
What appears mathematically as conjugation, by trans-
ferring the fixed points and to 0 and
1

, is
physically quite different for it takes the system from real
to complex angular momenta. In other words, merely by
conjugation, the vertices, , transform the physical
region , or in (4), into 0 and
1, 1
z20

11,k

, of
the unphysical region, , or . The two
types of transformations will give a different tessellation
of hyperbolic space, and so can be studied in much the
same way that Poincaré characterized his kleinian groups,
which are much less well-known than their fuchsian
counterparts [15, Ch. 8].

20k
z

1,1z
1,
20k
1, 1z
6. Strengths and Weaknesses of the Analysis
Different definitions of the Legendre function may be
adopted according to whether is considered to be an
unrestricted complex variable, or a real variable confined
to the open interval . However, there is no
relation between complex values of the degree of a
Legendre function and the unphysical region of
.

1z
According to (4), bound states, , lie in the phy-
sical region. But, what is missing is a conformal map-
ping,


2,k
 

(27)
between the -plane and the

-plane. If we go back to
the Watson-Sommerfeld representation, (1), we can write
it, in the vicinity of , as [3, p. 46] k






 
 
2
222
cos
sin
cos regular function,
cos
k
kP











2
k
kP
kkk



(28)
where

is considered a real function of , the
energy. Thus, if we consider the conformal mapping
2
k


2
sin,wk  
then a semi-infinite strip of width 1 in the

-plane is
mapped onto the upper half -plane where the boun-
daries of the strip are homologues of the points
w
1

and
on the real axis. The poles of (28) will then be at 1
1
2

20k
.
It is assumed that for

, the Regge trajectories
(27) are real, but once , the Regge trajectories
become complex. This being a bound state would lead to
the conclusion that Regge trajectories are the desired
interpolating functions between bound states [16].

20k
20k
For ,

2
k 
Re
is complex, and this describes
resonances. It is argued [16] that

can coincide
with integer 0  for k, while Im remains
small. Then appealing to continuity, is considered
complex where
20

2
k 
2
k

becomes exactly equal to 0. Con-
sequently, we can take


physical with unphysical ,
which is a resonance, or unphysical
2
k

with physical
, which would correspond to a metastable state known
as a shadow state [3, p. 47]. However, we are not at
liberty to consider k2 complex as we originally con-
sidered complex.
2
k
k
Rather, discrete, elementary groups can be employed
to distinguish between bound and resonance states. Only
for maps of the form
2
,1,1,
s
zaz as

Re
will the angular momentum become complex, and so
transform the indifferent fixed points of Type 1 group
into a Type 3 group whose orbits leave invariant the
source-sink fixed points at 0 and [15, p. 84]. What is
still missing is the form of the Regge trajectory (27)
which would connect

with k, and so lead to
physical values in (4), and complex values of
20

with
which result in unphysical values of (4).
20k
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B. H. LAVENDA 910
doi:10.1007/BF02859794
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