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Few notes :
What is a Pomeron
Po – Ju Lin
August 17, 2004
Contents
1.
2.
3.
4.
5.
6.
7.
8.
Life before QCD
Sommerfeld - Watson transform
Signature
Regge poles
Factroization
Regge trajectories
The Pomeron
Total cross – sections
1. Life before QCD

Instead of applying field theory directly,
physicists tried to extract as much as
possible by studying the consequence of a
reasonable set of postulates about the
S-matrix.
1. Life before QCD ~ S-matrix

S – matrix
Sab  bout ain
Overlap between the in-state a and the
out- state b
1. Life before QCD ~ Postulates

Postulate 1. S-matrix is Lorentz invariant
It can be expressed as a function of the Lorentz
scalar products of the incoming and out going
momenta.
1. Life before QCD ~ Postulates

Postulate 2. S-matrix is unitary:
SS


 S S 1
This is a natural statement as a consequence of
conservation of probability.
1. Life before QCD ~ Postulates
The scattering amplitude, Aab is related to the
S-matrix by :
S ab


  ab  i2     pa   pb  Aab
b

2 a
4
4
The unitarity of the S-matrix leads to :


2 Im Aab  2     pa   pb  Aac Acb
b
 a
 c
4
4
This gives us the Cutkosky rules
1. Life before QCD ~ Postulates

Postulate 3. The S-matrix is an analytic
function of Lorentz invariants (regarded as
complex variables), with only those
singularities required by unitarity.
It can be shown that this property is a
consequence of causality, i.e. that two regions
with a space-like separation do not influence
each other
2. Sommerfeld – Watson Transform

Consider a two-particle to two-particle scattering
preocess in t-channel at a center of mass energy,
s which is much larger than the masses of
external particles. The amplitude can be expand
as a series in Legendre polynomials, Pl cos  
where  is the scattering angle in cms and is
related to s, t by :
2t
cos   1 
s
2. Sommerfeld – Watson Transform

Partial wave expansion:

Aa cbd s, t    2l  1al s Pl 1  2t / s 
l 0
where al s  are called partial wave amplitudes.

In s-channel (interchange s and t):

Aabcd s, t    2l  1al t Pl 1  2s / t 
l 0
2. Sommerfeld – Watson Transform

Sommerfeld, following Watson, rewrote the
partial wave expansion in terms of a contour
integral in the complex angular momentum l 
plane as :
1
al , t 
As, t    dl 2l  1
Pl ,1  2s / t 
2i C
sin l
where the contour C surrounds the positive real
axis as shown in Fig.1
2. Sommerfeld – Watson Transform
Fig.1 Sommerfeld – Watson Transform
3. Signature

Is al , t  unique?
It can be shown that al , t  is unique provided
al , t   exp l  as l   . Unfortunately, the
contributions to the partial wave amplitude
which are proportional to  1 so the inequality is
violated along the imaginary axis. Therefore we
need two analytic functions of the even and odd
1
1


l, t  .
a
l
,
t
a
partial wave amplitudes
,
l
3. Signature

Thus we have :

1
2l  1
As, t  
dl
2i C
sin l
  e a  l, t Pl,1  2s / t 
 2
 il

  1
where  takes the values  1 , is called the
signature of the partial wave and a 1 l, t  and
a 1 l , t  are called the even- and odd-signature
partial wave functions.
4. Regge Poles

Next step: Deform the contour C to contour C’
in Fig.1. We must encircle any poles or cuts that
 
a
the functions l , t  may have at l   n t  . For
particular case of simple poles :

 2l  1   eil  

dl 
 l , t  Pl ,1  2s / t 

 sin l  1 2

1
 i
2
1
 i
2
1
As, t   
2i

 e
 
  1 n
i n t 
2

 n t 
P  n t ,1  2s / t
sin n t 
~


4. Regge Poles


The simple poles  n t  are called even- and oddsignature Regge Poles.
In Regge region, i.e. s  t , the Legendre
polynomial is dominated by its leading term and
in this limit the contribution to the right hand
side of the previous formula from the integral
along the contour C’ vanishes as s   .

4. Regge Poles

We want to isolate the high energy behavior of
the scattering amplitude in the Regge region.
Now in fact we need only consider the
contribution from the Regge pole with the
largest value with the real part of  n t  (the
leading Regge pole). Thus we have:


 e
As, t  

s 
 i t 
2
  t s  
 t
5. Factorization

We can view

 e
As, t  

s 
 i t 
2
  t s  
 t
as the exchange in the t-channel of an object
with as object with ‘angular momentum’ equal to
 t  . This is of course not a particle since the
angular momentum is not integer (or halfinteger) and it is a function of t. It is called a
Reggeon.
5. Factorization


We can view a Reggeon exchange amplitude as
the superposition of amplitudes for the
exchange of all possible particles in t-channel.
The amplitude can be factorized as shown in
Fig.2 into a coupling  ac t  of the Reggeon
between particle a and c,  bd t  between b and d
and a universal contribution from the Reggeon
exchange.
5. Factorization
Fig.2 A Regge Exchange Diagram
5. Factorization

Thus we obtain :

 e
As, t  

s 
 i t 
  t  t  s
ac
bd
 t 
2 sin t   t 
For the presence of  t  in the denominator, if
 t  takes an integer value for some of t then the
amplitude has a pole. For positive integer this can
be understood as a exchange of a resonance
particle with integer spin. For negative values they
are canceled out.
6. Regge Trajectories


Consider t- channel process, with t positive we
expect the amplitude to have poles
corresponding to the exchange of physical
particles of spin J i and mass mi ,where  mi2   J i
Chew & Frautschi plotted the spins of low lying
mesons against square mass and noticed that
they lie in a straight line as shown in Fig.3
6. Regge Trajectories
Fig. 3 The Chew-Frautschi Plot
6. Regge Trajectories
  t 
is a linear function of t :
 t    0   t
From Fig 3. we obtain the values :
 0   0.55
   0.86GeV
2
We shall see this linearity continues for negative
values of t.
6. Regge Trajectories

From the amplitude given above we can deduce
that the asymptotic s-dependence of the
differential cross-section d is proportional
dt
to :
s
 2 0  2 t  2 
6. Regge Trajectories

Consider a process in which isospin, I = 1, is
exchanged in the t-channel, such as :


 p 
0
n
We expect the Regge trajectory which
determines the asymptotic s-dependence to be
the one containing the I = 1 even parity mesons
(the  -trajectory). Use the data acquired in
Fig.3, we get Fig.4
6. Regge Trajectories
Fig.4 The extrpolation of Fig.3
7. The Pomeron

From the intercept of the Regge trajectory
which dominates a particular scattering process
and the optical theorem we can obtain the
asymptotic behavior of the total cross-section
for that process, namely,  tot is proportional to :
s
  0 1
For the  -trajectory considered in the last
section,  0 < 1, which means that the crosssection for a process with I = 1 exchange falls as
s increases.
7. The Pomeron


Pomeronchuck & Okun proved from general
assumptions that in any scattering process in
which there is charge exchange the cross-section
vanishes asymptotically (the Pomeronchuck
theorem).
Foldy & Peierls noticed the converse : if for a
particular scattering process the cross-section
does not fall as s increases then that process
must be dominated by the exchange of vacuum
quantum numbers.
7. The Pomeron


Experiments showed that total cross-section do
not vanish asymptotically. In fact they rise slowly
as s increases.
If we are to attribute this rise to the exchange of
a single Reggeon pole then it follows that the
exchange is that of a Reggeon whose intercept,
 P 0 is greater than 1, and which carries the
quantum number of the vacuum. This trajectory
is called the Pomeron.
7. The Pomeron


Unlike The Regge trajectory, the physical particles
which would provide the resonances for integer
values of  P 0 for positive t have not been
conclusively identified.
Particles with the quantum numbers vacuum can
exist in QCD as bound states of gluons
(glueballs).
8. Total Cross-sections

Fig. 5 shows a compilation of data for total
cross-sections for p  p and p  p scattering,
together with a fit due to Donnachie &
Landshoff :
 pp  2.17 s 0.08  56.1s 0.45mb
 pp  2.17 s 0.08  98.4s 0.45mb

The first term on the right hand side is the
Pomeron contribution and the second term is
due to the exchange of a Regge trajectory.
8. Total Cross-sections
Fig.5 Data for
p p
and
p p
total cross-sections.
8. Total Cross-sections
The Pomeron couples with the same strength to
the proton and antiproton because the Pomeron
carries the quantum numbers of the vacuum.
 The Regge trajectory can have different
couplings to particles and antiparticles. This
accounts for the difference between the p  p
and p  p cross-sections at low s.

8. Total Cross-sections


One point of view to argue is that the intercept
1.08is only an effective intercept and the underlying
mechanism which gives rise to it is not the result of
single Pomeron exchange but has contributions
from the exchange of two or more Pomerons (so
called Regge cuts).
Since the intercepts are universal we expect them
to be able to describe other total crossections. This
is indeed the case, as can be seen from Fig.6
8. Total Cross-sections
Fig.6 Total cross-sections for    p and    p scattering