In quantum physics, Regge theory () is the
study of the analytic properties of scattering
as a function of angular momentum, where the
angular momentum is not restricted to be an
integer multiple of ħ but is allowed to take
any complex value. The nonrelativistic theory
was developed by Tullio Regge in 1959.
== Details ==
The simplest example of Regge poles is provided
by the quantum mechanical treatment of the
Coulomb potential
V
(
r
)
=
−
e
2
/
(
4
π
ϵ
0
r
)
{\displaystyle V(r)=-e^{2}/(4\pi \epsilon
_{0}r)}
or, phrased differently, by the quantum mechanical
treatment of the binding or scattering of
an electron of mass
m
{\displaystyle m}
and electric charge
−
e
{\displaystyle -e}
off a proton of mass
M
{\displaystyle M}
and charge
+
e
{\displaystyle +e}
. The energy
E
{\displaystyle E}
of the binding of the electron to the proton
is negative whereas for scattering the energy
is positive. The formula for the binding energy
is the well-known expression
E
→
E
N
=
−
2
m
′
π
2
e
4
h
2
N
2
(
4
π
ϵ
0
)
2
=
−
13.6
e
V
N
2
,
m
′
=
m
M
M
+
m
,
{\displaystyle E\rightarrow E_{N}=-{\frac
{2m'\pi ^{2}e^{4}}{h^{2}N^{2}(4\pi \epsilon
_{0})^{2}}}=-{\frac {13.6eV}{N^{2}}},\;\;m^{'}={\frac
{mM}{M+m}},}
where
N
=
1
,
2
,
3
,
.
.
.
{\displaystyle N=1,2,3,...}
,
h
{\displaystyle h}
is the Planck constant, and
ϵ
0
{\displaystyle \epsilon _{0}}
is the permittivity of the vacuum. The principal
quantum number
N
{\displaystyle N}
is in quantum mechanics (by solution of the
radial Schrödinger equation) found to be
given by
N
=
n
+
l
+
1
{\displaystyle N=n+l+1}
, where
n
=
0
,
1
,
2
,
.
.
.
{\displaystyle n=0,1,2,...}
is the radial quantum number and
l
=
0
,
1
,
2
,
3
,
.
.
.
{\displaystyle l=0,1,2,3,...}
the quantum number of the orbital angular
momentum. Solving the above equation for
l
{\displaystyle l}
, one obtains the equation
l
→
l
(
E
)
=
−
n
+
g
(
E
)
,
g
(
E
)
=
−
1
+
i
π
e
2
4
π
ϵ
0
h
(
2
m
′
/
E
)
1
/
2
.
{\displaystyle l\rightarrow l(E)=-n+g(E),\;\;g(E)=-1+i{\frac
{\pi e^{2}}{4\pi \epsilon _{0}h}}(2m'/E)^{1/2}.}
Considered as a complex function of
E
{\displaystyle E}
this expression describes in the complex
l
{\displaystyle l}
-plane a path which is called a Regge trajectory.
Thus in this consideration the orbital
momentum can assume complex values.
Regge trajectories can be obtained for many
other potentials, in particular also for the
Yukawa potential.Regge trajectories appear
as poles of the scattering amplitude or in
the related
S
{\displaystyle S}
-matrix. In the case of the Coulomb-potential
considered above this
S
{\displaystyle S}
-matrix is given by the following expression
as can be checked by reference to any textbook
on quantum mechanics:
S
=
Γ
(
l
−
g
(
E
)
)
Γ
(
l
+
g
(
E
)
)
e
−
i
π
l
,
{\displaystyle S={\frac {\Gamma (l-g(E))}{\Gamma
(l+g(E))}}e^{-i\pi l},}
where
Γ
(
x
)
{\displaystyle \Gamma (x)}
is the gamma function, a generalization of
factorial
(
x
−
1
)
!
{\displaystyle (x-1)!}
. This gamma function is a meromorphic function
of its argument with simple poles at
x
=
−
n
,
n
=
0
,
1
,
2
,
.
.
.
{\displaystyle x=-n,n=0,1,2,...}
. Thus the expression for
S
{\displaystyle S}
(the gamma function in the numerator) possesses
poles at precisely those points which are
given by the above expression for the Regge
trajectories; hence the name Regge poles.
== History and implications ==
The main result of the theory is that the
scattering amplitude for potential scattering
grows as a function of the cosine
z
{\displaystyle z}
of the scattering angle as a power that changes
as the scattering energy changes:
A
(
z
)
∝
z
l
(
E
2
)
{\displaystyle A(z)\propto z^{l(E^{2})}}
where
l
(
E
2
)
{\displaystyle l(E^{2})}
is the noninteger value of the angular momentum
of a would-be bound state with energy
E
{\displaystyle E}
. It is determined by solving the radial Schrödinger
equation and it smoothly interpolates the
energy of wavefunctions with different angular
momentum but with the same radial excitation
number. The trajectory function is a function
of
s
=
E
2
{\displaystyle s=E^{2}}
for relativistic generalization. The expression
l
(
s
)
{\displaystyle l(s)}
is known as the Regge trajectory function,
and when it is an integer, the particles form
an actual bound state with this angular momentum.
The asymptotic form applies when
z
{\displaystyle z}
is much greater than one, which is not a physical
limit in nonrelativistic scattering.
Shortly afterwards, Stanley Mandelstam noted
that in relativity the purely formal limit
of
z
{\displaystyle z}
large is near to a physical limit — the
limit of large
t
{\displaystyle t}
. Large
t
{\displaystyle t}
means large energy in the crossed channel,
where one of the incoming particles has an
energy momentum that makes it an energetic
outgoing antiparticle. This observation turned
Regge theory from a mathematical curiosity
into a physical theory: it demands that the
function that determines the falloff rate
of the scattering amplitude for particle-particle
scattering at large energies is the same as
the function that determines the bound state
energies for a particle-antiparticle system
as a function of angular momentum.The switch
required swapping the Mandelstam variable
s
{\displaystyle s}
, which is the square of the energy, for
t
{\displaystyle t}
, which is the squared momentum transfer,
which for elastic soft collisions of identical
particles is s times one minus the cosine
of the scattering angle. The relation in the
crossed channel becomes
A
(
z
)
∝
s
l
(
t
)
{\displaystyle A(z)\propto s^{l(t)}}
which says that the amplitude has a different
power law falloff as a function of energy
at different corresponding angles, where corresponding
angles are those with the same value of
t
{\displaystyle t}
. It predicts that the function that determines
the power law is the same function that interpolates
the energies where the resonances appear.
The range of angles where scattering can be
productively described by Regge theory shrinks
into a narrow cone around the beam-line at
large energies.
In 1960 Geoffrey Chew and Steven Frautschi
conjectured from limited data that the strongly
interacting particles had a very simple dependence
of the squared-mass on the angular momentum:
the particles fall into families where the
Regge trajectory functions were straight lines:
l
(
s
)
=
k
s
{\displaystyle l(s)=ks}
with the same constant
k
{\displaystyle k}
for all the trajectories. The straight-line
Regge trajectories were later understood as
arising from massless endpoints on rotating
relativistic strings. Since a Regge description
implied that the particles were bound states,
Chew and Frautschi concluded that none of
the strongly interacting particles were elementary.
Experimentally, the near-beam behavior of
scattering did fall off with angle as explained
by Regge theory, leading many to accept that
the particles in the strong interactions were
composite. Much of the scattering was diffractive,
meaning that the particles hardly scatter
at all — staying close to the beam line
after the collision. Vladimir Gribov noted
that the Froissart bound combined with the
assumption of maximum possible scattering
implied there was a Regge trajectory that
would lead to logarithmically rising cross
sections, a trajectory nowadays known as the
pomeron. He went on to formulate a quantitative
perturbation theory for near beam line scattering
dominated by multi-pomeron exchange.
From the fundamental observation that hadrons
are composite, there grew two points of view.
Some correctly advocated that there were elementary
particles, nowadays called quarks and gluons,
which made a quantum field theory in which
the hadrons were bound states. Others also
correctly believed that it was possible to
formulate a theory without elementary particles
— where all the particles were bound states
lying on Regge trajectories and scatter self-consistently.
This was called S-matrix theory.
The most successful S-matrix approach centered
on the narrow-resonance approximation, the
idea that there is a consistent expansion
starting from stable particles on straight-line
Regge trajectories. After many false starts,
Dolen Horn and Schmidt understood a crucial
property that led Gabriele Veneziano to formulate
a self-consistent scattering amplitude, the
first string theory. Mandelstam noted that
the limit where the Regge trajectories are
straight is also the limit where the lifetime
of the states is long.
As a fundamental theory of strong interactions
at high energies, Regge theory enjoyed a period
of interest in the 1960s, but it was largely
succeeded by quantum chromodynamics. As a
phenomenological theory, it is still an indispensable
tool for understanding near-beam line scattering
and scattering at very large energies. Modern
research focuses both on the connection to
perturbation theory and to string theory.
== See also ==
Faber–Jackson relation
Quark–gluon plasma
Dual resonance model
