Few notes : What is a Pomeron

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Few notes What is a Pomeron

• Po Ju Lin
• August 17, 2004

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Contents

• Life before QCD
• Sommerfeld - Watson transform
• Signature
• Regge poles
• Factroization
• Regge trajectories
• The Pomeron
• Total cross sections

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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.

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1. Life before QCD S-matrix

• S matrix
• Overlap between the in-state and the
• out- state

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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.

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1. Life before QCD Postulates

• Postulate 2. S-matrix is unitary
• This is a natural statement as a consequence
  of conservation of probability.

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1. Life before QCD Postulates

• The scattering amplitude, is related to
  the S-matrix by
• The unitarity of the S-matrix leads to
• This gives us the Cutkosky rules

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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

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2. Sommerfeld Watson Transform

• Consider a two-particle to two-particle
  scattering preocess in t-channel at a center of
  mass energy,
• which is much larger than the masses of
  external particles. The amplitude can be expand
  as a series in Legendre polynomials,
• where is the scattering angle in cms and
  is related to s, t by

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2. Sommerfeld Watson Transform

• Partial wave expansion
• where are called partial wave
  amplitudes.
• In s-channel (interchange s and t)

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2. Sommerfeld Watson Transform

• Sommerfeld, following Watson, rewrote the partial
  wave expansion in terms of a contour integral in
  the complex angular momentum
• plane as
• where the contour C surrounds the positive
  real axis as shown in Fig.1

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2. Sommerfeld Watson Transform

• Fig.1 Sommerfeld Watson Transform

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3. Signature

• Is unique?
• It can be shown that is unique
  provided
• as .
  Unfortunately, the contributions to the partial
  wave amplitude which are proportional to so
  the inequality is violated along the imaginary
  axis. Therefore we need two analytic functions of
  the even and odd partial wave amplitudes
  , .

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3. Signature

• Thus we have
• where takes the values , is called
  the signature of the partial wave and
  and
• are called the even- and
  odd-signature partial wave functions.

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4. Regge Poles

• Next step Deform the contour C to contour C in
  Fig.1. We must encircle any poles or cuts that
  the functions may have at
  . For particular case of simple poles

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4. Regge Poles

• The simple poles are called even- and
  odd-signature Regge Poles.
• In Regge region, i.e. , 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 .

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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 (the leading Regge
  pole). Thus we have

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5. Factorization

• We can view
• as the exchange in the t-channel of an object
  with as object with angular momentum equal to
• . This is of course not a particle
  since the angular momentum is not integer (or
  half-integer) and it is a function of t. It is
  called a Reggeon.

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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 of the Reggeon between
  particle a and c, between b and d and
  a universal contribution from the Reggeon
  exchange.

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5. Factorization

• Fig.2 A Regge Exchange Diagram

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5. Factorization

• Thus we obtain
• For the presence of in the
  denominator, if
• 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.

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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
  and mass ,where
• 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

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6. Regge Trajectories

• Fig. 3 The Chew-Frautschi Plot

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6. Regge Trajectories

• is a linear function of t
• From Fig 3. we obtain the values
• We shall see this linearity continues for
  negative values of t.

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6. Regge Trajectories

• From the amplitude given above we can deduce that
  the asymptotic s-dependence of the differential
  cross-section is proportional to

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6. Regge Trajectories

• Consider a process in which isospin, I 1, is
  exchanged in the t-channel, such as
• 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

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6. Regge Trajectories

• Fig.4 The extrpolation of Fig.3

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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, is proportional to
• For the -trajectory considered in the last
  section, lt 1, which means that the
  cross-section for a process with I 1 exchange
  falls as s increases.

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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.

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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,
• is greater than 1, and which carries
  the quantum number of the vacuum. This trajectory
  is called the Pomeron.

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7. The Pomeron

• Unlike The Regge trajectory, the physical
  particles which would provide the resonances for
  integer values of for positive t have
  not been conclusively identified.
• Particles with the quantum numbers vacuum can
  exist in QCD as bound states of gluons
  (glueballs).

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8. Total Cross-sections

• Fig. 5 shows a compilation of data for total
  cross-sections for and
  scattering, together with a fit due to Donnachie
  Landshoff
• 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.

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8. Total Cross-sections

• Fig.5 Data for and total
  cross-sections.

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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
• and cross-sections at low s.

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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

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8. Total Cross-sections

• Fig.6 Total cross-sections for and
  scattering