Few notes : What is a Pomeron

1. Few notes : What is a Pomeron Po – Ju Lin August 17, 2004

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

3. 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.

4. 1. Life before QCD ~ S-matrix • S – matrix Overlap between the in-state and the out- state

5. 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.

6. 1. Life before QCD ~ Postulates • Postulate 2. S-matrix is unitary: This is a natural statement as a consequence of conservation of probability.

7. 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

8. 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

9. 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 :

10. 2. Sommerfeld – Watson Transform • Partial wave expansion: where are called partial wave amplitudes. • In s-channel (interchange s and t):

11. 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

12. Fig.1Sommerfeld – Watson Transform 2. Sommerfeld – Watson Transform

13. 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 , .

14. 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.

15. 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 :

16. 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 .

17. 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:

18. 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.

19. 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.

20. 5. Factorization Fig.2 A Regge Exchange Diagram

21. 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.

22. 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

23. 6. Regge Trajectories Fig. 3 The Chew-Frautschi Plot

24. 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.

25. 6. Regge Trajectories • From the amplitude given above we can deduce that the asymptotic s-dependence of the differential cross-section is proportional to :

26. 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

27. 6. Regge Trajectories Fig.4 The extrpolation of Fig.3

28. 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, < 1, which means that the cross-section for a process with I = 1 exchange falls as s increases.

29. 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.

30. 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.

31. 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).

32. 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.

33. 8. Total Cross-sections Fig.5 Data for and total cross-sections.

34. 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.

35. 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

36. 8. Total Cross-sections Fig.6 Total cross-sections for and scattering