# Presentation: 594566

## From basic kinematics to Regge poles

- Enrico Predazzi

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Introduction

- Donnachie and Landshoff in 1992 concluded their analysis on total cross sections based on Regge-type fits stating that *Regge theory remains one of the great truths of particle physics*.
- 15 years later this remains even more remarkably true than ever but,
- why?
-    and,
- what was the birth of Regge poles?

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## We shall briefly outline what motivated Regge poles 
why they are a theory more than a model
how they were born
how they made it to become an extremely useful instrument in high energy physics
what made them slowly disappear from the scene 
what triggered their comeback
why they are still with us (and presumably will remain for a long time to come if not forever)   

- why they are a theory more than a model
- how they were born
- how they made it to become an extremely useful instrument in high energy physics
- what made them slowly disappear from the scene
- what triggered their comeback
- why they are still with us (and presumably will remain for a long time to come if not forever)

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Regge poles were born in a brilliant attempt to learn about the properties of the full fledged S-Matrix theory starting from the only really fully manageable scheme, non relativistic potential model. The basic assumption is that we can use the Schrödinger equation to describe the elastic interaction of two (spinless, for simplicity)
              a  +  b  →  a  +  b
 (-h2/2μ) Δψ(r) + V(r) ψ(r)  =  E ψ(r) 

-               a  +  b  →  a  +  b
-  **(-h****2****/2****μ****)** **Δψ(r) + V(r) ψ(r)  =  E ψ(r) **

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## The conventional wisdom when using potential scattering to deal  with particle physics problems is that the realistic underlying dynamics can be mimicked by Yukawa-like spherical (or central) potentials of the type:
            V(r) = ∫ g(α) e-αr dα.
 Using
k = 2μ/ħ2  E  ,     U(r) = 2μ/ħ2 V(r) 
the Schrödinger equation takes the form
            [Δ – U(r) – k2] ψ(r) = 0.

-             V(r) = **∫ **g(α) e-αr dα.
-  Using
- k = 2μ/ħ2  E  ,     U(r) = 2μ/ħ2 V(r)
- the Schrödinger equation takes the form
-             **[Δ – U(r) – k****2****] ψ(r) = 0**.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## The assumption is that an impinging particle comes from (-)infinity as a plane wave to interact with a spherical symmetric potential so that at large (positive) distances, the asymptotic solution of (1.5) becomes the superposition of the incoming plane wave plus a distorted outgoing spherical wave of the form
    ψ(r)  ~ r→∞  eik∙r  + f(k, k’)  eikr/r 

-     **ψ(r)  ~ ****r→∞****  e****ik∙r****  + f(k, k’)  e****ikr****/r**

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## f(k, k’) is known as the scattering amplitude whose squared modulus gives (in potential theory) the differential cross section
       dσ/dΩ  = ׀ f(k, k’)׀2   = ׀ f(k, θ)׀2 
As usual, for a spherical interaction, the scattering amplitude can be expanded in partial waves over all integer positive values of the angular momentum l (from l=0 to l=∞)

      f(k, θ) = ∑l (2l+1) al(k) Pl(cos θ). 

-        dσ/dΩ  = ׀ f(**k, k’**)׀2   = ׀ f(k, θ)׀2
- As usual, for a spherical interaction, the scattering amplitude can be expanded in partial waves over all integer positive values of the angular momentum l (from l=0 to l=∞)
-       **f(k, θ) = ∑****l**** (2l+1) a****l****(k) P****l****(cos θ).**** **

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## al(k) are (complex) quantities known as partial waves related to the phase shifts 
   δl(k) (and to the S-Matrix partial wave amplitudes) by 
 al(k) = (1/2ik) [exp(2i δl(k) ) – 1] =  
            (1/2ik) [Sl(k) -1] 
or
          Sl(k) =  exp[2i δl(k)] 

-    δl(k) (and to the S-Matrix partial wave amplitudes) by
-  **a****l****(k)**** = (1/2ik) [exp(2i ****δ****l****(k) ****) – 1] =  **
**            ****(1/2ik) [S****l****(k) -1]**** **
**or**
**          ****S****l****(k)**** ****=  exp[2i**** ****δ****l****(k)]**

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## For elastic scattering (no inelastic channels open implies no absorption or, within the present scheme, real potentials), δl(k) are real quantities and the elastic unitarity condition reads │Sl(k)│ = 1 or
          Im al(k) = k │al(k)│2 
from the above results the optical theorem 

          Im f(k, 0) = k/π  σtot 

-           **Im a****l****(k) = k │a****l****(k)│****2**
- from the above results the *optical theorem*
-           **Im f(k, 0) = k/****π****  ****σ****tot**** **

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Moving to relativistic kinematics

- In the relativistic scheme, the simple minded two body reaction **a+b**** ****→a+b** is no longer the full story; due to crossing and the other properties in QFT, we have three independent but related reactions. More precisely, if we use the property that an outgoing particle can be viewed as an incoming antiparticle of reversed fourmomentum,

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

##  for a two-body process, we have three channels in which related (but different) reactions occur. Schematically, if we underline a particle to denote the corresponding antiparticle we can relate  a+b→a+b to the annihilation process a+b+a+b and this leads to three channels which are called “s”, “t” and “u” in reminiscence of the values taken by the corresponding Mandelstam variables

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Crossing

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## (2.1)             (s-channel)
a(p1)  +   b(p2)    →    a(p3)    +    b(p4)                     
(2.2)             (t-channel)
a(p1)  +   a(-p3)    →   b(-p2)   +    b(p4)                 
(2.3)             (u-channel)
a(p1)  +   b(-p4)    →    a(p3)    +    b(-p2)                                       
   are, by analytic continuation one and thesame taken in different regions of the complex variables describing the above reactions. 

- a(p1)  +   b(p2)    →    a(p3)    +    b(p4)
- (2.2)             (t-channel)
- a(p1)  +   _a_(-p3)    →   _b_(-p2)   +    b(p4)
- (2.3)             (u-channel)
- a(p1)  +   _b_(-p4)    →    a(p3)    +    _b_(-p2)
-    are, by analytic continuation one and thesame taken in different regions of the complex variables describing the above reactions.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## In (2.1-3), incoming/outgoing particles are viewed as outgoing/incoming antiparticles of reversed momentum (the other three possible reactions are  simply the time reversed of the previous ones). 
Just as an example, take the charge exchange reaction π ̄p→π° n 

- Just as an example, take the charge exchange reaction π ̄p→π° n

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Charge exchange 

- Charge exchange
- reactions

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## To exemplify, the s-channel (charge exchange) reaction
     p(p1)  +   π–(p2)  →    n(p3)   +   π°(p4)
becomes in the t-channel the annihilation reaction 
     p(p1)  +   n(-p3)   →   π+ (-p2)  +   π°(p4) 
and, in the u-channel the hitherto experimentally 
inaccessible charge exchange reaction
     p(p1)  +   π°(-p4)  →  n(p3)  +   π+ (-p2)  
(use has been made of the fact that π+ is the 
antiparticle of  π- and that π° is its own 
antiparticle). 

-      p(p1)  +   π**–**(p2)  →    n(p3)   +   π°(p4)
- becomes in the t-channel the annihilation reaction
-      p(p1)  +   _n_(-p3)   →   π+ (-p2)  +   π°(p4)
- and, in the u-channel the hitherto experimentally
- inaccessible charge exchange reaction
-      p(p1)  +   π°(-p4)  →  n(p3)  +   π+ (-p2)
- (use has been made of the fact that π+ is the
- antiparticle of  π- and that π° is its own
- antiparticle).

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Formal complications arise in the case of realistic particles endowed with so far ignored quantum numbers (like spin etc.) but these need not to concern us here.  

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## These three reactions (2.1-3) are labelled s-, t- and u-channel respectively since for each of them the corresponding invariant Mandelstam variable  
(2.4)                  s = (p1 + p2)2    
(2.5)                  t  = (p1 – p3)2
(2.6)                  u = (p1 – p4)2
 is positive definite while the other two are 
negative being, in essence, the four 
dimensional momentum transfer of the 
corresponding reaction. 

**(2.4)                  s = (p****1**** + p****2****)****2****    **
**(2.5)                  t  = (p****1**** – p****3****)****2**
**(2.6)                  u = (p****1**** – p****4****)****2**
** ****is positive definite while the other two are **
**negative being, in essence, the four **
**dimensional momentum transfer of the **
**corresponding reaction. **

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## For instance, in the s-channel (equal masses case), in the c. m. we have
   s = (p1 + p2)2 = 4 (k2 + m2) > 0
   t  = (p1 – p3)2 = - 2k2 (1 – cos θs) ≤ 0
   u = (p1 – p4)2 = - 2k2 (1 + cos θs) ≤ 0. As a consequence, 
(2.7)   cos θs  =  1 + 2t /(s - 4 m2) 
For on-shell particles only two of these 
variables are independent and, in fact                
                      s + t + u = 4 m2 .

**   ****s = (p****1**** + p****2****)****2**** = 4 (k****2 ****+ m****2****) > 0**
**   ****t  = (p****1**** – p****3****)****2 ****= - 2k****2 ****(1 – cos ****θ****s****) ≤ 0**
**   ****u = (p****1**** – p****4****)****2 ****= - 2k****2 ****(1 + cos ****θ****s****) ≤ 0. As a consequence, **
- (2.7)   **cos θ****s****  =  1 + 2t /(s - 4 m****2****)**
**For on-shell particles ****only two of these **
**variables are independent and, in fact****                **
**                      ****s + t + u = 4 m****2** .

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## This situation may change in case one of the particles is not on shell. Such is the case of an inclusive reaction such as
(2.8)              a  +  b   →   c +  X,
(i.e. a reaction where X stays for an unresolved 
cluster of undetected particles in the final state) 
will be dealt with. In this case, a third variable, for 
instance the so-called missing mass (or any other 
independent variable) 
(2.9)              pX 2 = (p1 + p2 – p3)2
will have to be used to properly describe the 
process.

- (2.8)              a  +  b   →   c +  X,
- (i.e. a reaction where X stays for an unresolved
- cluster of undetected particles in the final state)
- will be dealt with. In this case, a third variable, for
- instance the so-called *missing mass *(or any other
- independent variable)* *
- (2.9)              pX 2 = (p1 + p2 – p3)2
- will have to be used to properly describe the
- process.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## 3. Problems with (high spin) meson exchange. 

- According to the general wisdom going back to the old days of Yukawa (1935), the nuclear forces acting between hadrons are due to virtual particles (mesons) exchanged in the crossed t and/or u channels in strict analogy with e.m. interactions arising from the exchange of virtual photons between electrons.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## This picture becomes inapplicable at high energies (i.e. s → ∞) for the following reason. 
Consider a generic two body reaction
(3.1)             1 + 2   3 + 4
mediated by single particle exchange in the t-
channel. The scattering amplitude for the 
exchange of a particle of mass M and spin J goes as
(3.2) Ames(s, t)  ~ AJ(t)  PJ(cos θt)  = PJ(cos θt) / (t – M2)
where m is the mass of the interacting particle and (3.3)        cos θt  =  1 + 2s /(t - 4 m2) 
is the t-channel scattering angle (compare with eq. 
(2.7)). 

- Consider a generic two body reaction
- (3.1)             1 + 2   3 + 4
- mediated by single particle exchange in the t-
- channel. The scattering amplitude for the
- exchange of a particle of mass M and spin J goes as
- (3.2) Ames(s, t)  ~ AJ(t)  PJ(cos θt)  = PJ(cos θt) / (t – M2)
- where m is the mass of the interacting particle and (3.3)        **cos θ****t****  =  1 + 2s /(t - 4 m****2****)**
- is the t-channel scattering angle (compare with eq.
- (2.7)).

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## If in (3.2) we keep t fixed and let s  ∞, 
using  Pl(z) ~ zl as z  ∞ we find 
(3.4)            Ames(s, t)  ~  sJ
which corresponds to a cross section 
growing like s2J-2  as s  ∞.
This behaviour can be proved to violate (s-
channel) unitarity since it  violates the 
Froissart-Martin bound which, owing to 
unitarity requires σtot  to be bounded by 
ln2s).

- using  Pl(z) ~ zl as z  ∞ we find
- (3.4)            Ames(s, t)  ~  sJ
- which corresponds to a cross section
- growing like s2J-2  as s  ∞.
- This behaviour can be proved to violate (s-
- channel) unitarity since it  violates the
- Froissart-Martin bound which, owing to
- unitarity requires σtot  to be bounded by
- ln2s).

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## As we will see, Regge theory overcomes this difficulty while preserving the notion of crossed channel exchange. Also, according to Regge theory, the strong interaction will turn out to be due not to the exchange of particles of definite spin but to Regge trajectories  i.e. to entire families of resonances. 

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

##  4. Regge poles in non relativistic potential models.

- The starting point is the partial wave s-
- channel expansion of the scattering
- amplitude (1.9) which we rewrite as
- (4.1)    **A(s, z) =** **∑****l**** (2l+1) A****l****(s) P****l****(z)**** **
- where z is the cosine of the physical s-channel scattering angle
- (4.2)    z ≡ cos θs = 1 +  2t /(s - 4 m2)

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## The representation (4.1) of the scattering amplitude is defined in the physical s-channel domain given by
(4.3)    s ≥  4 m2        and   -1 ≤ z ≤ 1.
The question arises, therefore, whether (4.1) converges in a domain of the complex s, t and u variables larger than (4.3) and, more specifically, in a sufficiently large physical domains of the crossed t- and u-channels. As we shall see, this is not the case and the reason is simple to understand qualitatively. 

- (4.3)    s ≥  4 m2        and   -1 ≤ z ≤ 1.
- The question arises, therefore, whether (4.1) converges in a domain of the complex s, t and u variables larger than (4.3) and, more specifically, in a sufficiently large physical domains of the crossed t- and u-channels. As we shall see, this is not the case and the reason is simple to understand qualitatively.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## The s-channel singularities of A(s,t) are contained in the partial wave amplitudes Al(s) but the t-dependence is embodied in the Legendre polynomials and these are entire functions of their argument so that any singularity of the full amplitude must reveal itself from the divergence of the series (4.1) which becomes senseless. 

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## The problem of finding a representation of A(s,t) which can be used to connect the various channels and hence can be used to describe all physical reactions connected by crossing, is solved by introducing the seemingly unphysical concept of  complex angular momenta. Before doing this, let us first investigate the region of convergence of the series (4.1) in the complex θ plane. 

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## The asymptotic expansion of Pl(cos θ) for l real and tending to ∞,
(4.4)      Pl(cos θ)  = O(el│Im θ│ ) 
implies its exponential growth (in l ) for 
complex θ.
As a consequence, the only way the series (4.1) can converge is that, as l  ∞                                  (4.5)             Al(s)  ~ e-lη(s) 
In this case convergence is guaranteed so long as
(4.6)              Im θ  ≤  η(s).

- (4.4)      **P****l****(cos ****θ****)  = O(e****l│Im ****θ****│**** )**
- implies its exponential growth (in l ) for
- complex **θ**.
- As a consequence, the only way the series (4.1) can converge is that, as l  ∞                                  (4.5)             **A****l****(s)  ~ e****-****lη(s)**
- In this case convergence is guaranteed so long as
- (4.6)              **Im θ  ≤  η(s).**

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Thus, convergence is insured in a horizontal strip in the complex θ plane
   symmetric with respect to the real axis of width η(s). Setting χ = ch η(s) (which is always ≥1), the corresponding convergence domain of the partial wave expansion (4.1) in the complex cos θ plane (z= cos θ = x + iy) is 
(4.7)     x2/χ2  +   y2 /( χ2 -1) = 1 
which is an ellipse with foci ±1 and semiaxes χ and √χ2 -1 which is known as the Lehmann ellipse .

-    symmetric with respect to the real axis of width η(s). Setting **χ = ch η(s)** (which is always ≥1), the corresponding convergence domain of the partial wave expansion (4.1) in the complex cos θ plane (**z= cos θ = x + iy**) is
- (4.7)     **x****2****/χ****2****  +   y****2**** /( χ****2**** -1) = 1**
- which is an ellipse with foci ±1 and semiaxes χ and √χ2 -1 which is known as the ***Lehmann ellipse ***.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## The conclusion is that the representation (4.1) converges in a domain which, albeit greater than the physical domain -1 ≤ z ≤ 1 (to which it reduces if η(s)  0), is still finite i.e. never extends to arbitrarily large values of  the complex variable │z= cos θ│. In the language of the s, t, u variables, the domain never extends to asymptotic values of │t│ (or │u│)[recall cos θs = 1 +  2t /(s - 4 m2)]. 
As expected, the representation (4.1) cannot be used to explore the large crossed channels energy domain .

**As expected, the representation (4.1) cannot be used to explore the large *****crossed channels***** energy domain .**

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## As already anticipated, the way to circumvent this difficulty is to continue the expansion (4.1) to complex values of the angular momentum l.
To get a hint as to how to proceed, let us investigate the extreme case when l is purely imaginary. In this case, repeating the previous procedure, provided 
(4.8)            Al(s)  ~ e -│l│δ(s)     ,
convergence will now be insured in a vertical strip parallel and symmetric to the imaginary θ axis i.e. in the strip
(4.9)                │Re θ│ ≤ δ(s) .

- To get a hint as to how to proceed, let us investigate the extreme case when l is purely imaginary. In this case, repeating the previous procedure, provided
- (4.8)            **A****l****(s)  ~ e ****-│****l│δ(s)**     ,
- convergence will now be insured in a vertical strip parallel and symmetric to the imaginary θ axis i.e. in the strip
**(4.9)                ****│Re θ│ ≤ δ(s)** .

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Setting, accordingly  ξ = cos δ  (which is always ≤1), the convergence domain is now given by
(4.19)         x2/ ξ 2  - y2 /(1 - ξ 2) = 1.
Contrary to the previous case, (4.10) defines now an open domain, a hyperbola with foci ±1 and convergence is guaranteed in one of its halves.
In addition, given that it overlaps in part with a portion of the Lehmann ellipse, once we have made an analytic continuation of the amplitude to complex angular momenta, the new representation will define exactly the very same scattering amplitude A(s,t) we started from so that we will be able to safely continue it to domain where the crossed channel energies │t│ and/or │u │ can become arbitrarily large.

- (4.19)         **x****2****/ ξ ****2****  - y****2**** /(1 - ξ ****2****) = 1.**
- Contrary to the previous case, (4.10) defines now an open domain, a hyperbola with foci ±1 and convergence is guaranteed in one of its halves.
- In addition, given that it overlaps in part with a portion of the Lehmann ellipse, once we have made an analytic continuation of the amplitude to complex angular momenta, the new representation will define exactly the very same scattering amplitude A(s,t) we started from so that we will be able to safely continue it to domain where the crossed channel energies │t│ and/or │u │ can become arbitrarily large.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

##  5. Complex Angular Momenta

- We now have to find the condition to continue the partial wave scattering amplitude to complex angular momenta so as to find a representation suitable to make asymptotic expansions.
- First we have to assume that we can continue the partial wave amplitude Al(s) to complex values of l and construct an interpolating function A(l,s)
- which reduces to Al(s) for real integer values of l.
- For this, we suppose that

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- 1-13 July 2007

## A(l,s) has only isolated singularities (poles) in the complex l plane (and, for further simplicity, that they are simple poles);
A(l,s) is holomorphic for Re l ≥ L  (L positive arbitrary but finite)
A(l,s)  0 as │l│∞  for Re l > 0.
An amplitude A(l,s) with the previous properties exists in at least two contexts:

**A(l,s) is holomorphic for Re l ≥ L  (L positive arbitrary but finite)**
**A(l,s) ******** 0 as │l│********∞  for Re l > 0.**
**An amplitude A(l,s) with the previous properties exists in at least two contexts:**

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- 1-13 July 2007

## i) in non relativistic QM when the potential is a superposition of Yukawa potentials of the form 
(1.3)               V(r) =  ∫ g(α) e -αr dα. 
This is supposed to mimic particle exchange in the t-channel of a general two-body reaction (3.1)  
ii) in the relativistic case when further requirements are satisfied (such as the validity of dispersion relations).

- (1.3)               **V(r) =  ∫ g(α) e ****-αr**** dα****.**
- This is supposed to mimic particle exchange in the t-channel of a general two-body reaction (3.1)
- ii) in the relativistic case when further requirements are satisfied (such as the validity of dispersion relations).

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## If such an A(l,s) exists, then it also unique thanks to a theorem by Carlson and the partial wave expansion (4.1) can be rewritten as
(5.1)  A(s, z) = ∑l (2l+1)  Al(s) Pl(z) 
   – 1/2i  ∫C (2l+1) A(l,s) Pl(-z)/sin πl   dl
where the discrete sum runs up to N-1 (N being the smallest integer larger than L) and C is the contour parallel to the real axis to the right of all singularities of  A(l,s) (see figure where λ=l+1⁄2). 

- (5.1)  A(s, z) = ∑l (2l+1)  Al(s) Pl(z)
-    – 1/2i  ∫C (2l+1) A(l,s) Pl(-z)/sin πl   dl
- where the discrete sum runs up to N-1 (N being the smallest integer larger than L) and C is the contour parallel to the real axis to the right of all singularities of  A(l,s) (see figure where λ=l+1⁄2).

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Slide 38

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## The validity of (5.1) and its equivalence with (4.1) are a direct consequence of Cauchy residue’s theorem (the integrand f(l) in (5.1) has simple poles at all integers l=n with residues 2i (2n+1) An(s) Pn(z)). 
Given that there are no singularities of A(l,s) at the right of l=L and due to the asymptotic properties of A(l,s) and of 
   Pl(-z)/sin πl, we can now deform the integration contour along the imaginary l axis at the right of  Re l>L. 

- Given that there are no singularities of A(l,s) at the right of l=L and due to the asymptotic properties of A(l,s) and of
-    Pl(-z)/sin πl, we can now deform the integration contour along the imaginary l axis at the right of  Re l>L.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Next, if the only singularities of A(l,s) are simple poles, we can further move the integration contour to a parallel of the imaginary axis (see figure ...) collecting the residues of A(l,s) (at the same time, the residues of sin πl cancel the terms of the sum in (5.1)) so that, finally, we get the so called (Watson-Sommerfeld-Regge representation of the scattering amplitude whose asymptotically dominant contribution (we neglect the integral along a parallel to the left of the imaginary l axis which vanishes as z goes to ∞) turns out to be
(5.2) A(s, z) ~ -∑i [2αi(s)+1] βi(s) Pαi(-z)/sin παi(s).

- (5.2) A(s, z) ~ -∑i [2αi(s)+1] βi(s) Pαi(-z)/sin παi(s).

- QCD, low x, Saturation and Diffraction, Copanello

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## Slide 41

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## The representation is now convergent in the half of the  hyperbola (4.10) and we can now take the asymptotic z limit (i.e. the high energy limit in the crossed t/u) channel). 
If we call α(s) the pole with the largest real part, owing to Pα(s)(z) ~ zα(s) and using (4.2) we get the asymptotic behaviour
(5.3) A(s,t) ~t→∞ - β(s) tα(s) /sin πα(s) 
where nothing is known about the residue β(s).

- If we call α(s) the pole with the largest real part, owing to **P****α(s)****(z) ~ z****α(s)** and using (4.2) we get the asymptotic behaviour
- (5.3) **A(s,t) ~****t→∞**** - β(s) t****α(s)**** /sin πα(s)**
- where nothing is known about the residue β(s).

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Several complications can occur that muddle this simple result like, for instance higher order poles in the complex angular momentum plane, cuts etc. We shall not dig further on this matter in these notes and we refer the interested reader to the existing literature.
One additional important point comes in the relativistic case due again to crossing. 

- One additional important point comes in the relativistic case due again to crossing.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Invoking now the validity of crossing (....) we come, finally, to the amazingly simple asymptotic Regge behaviour as dominated by the singularity with the largest real part in the complex angular momentum plane of the crossed t-channel (beware, we are going to interchange s with t)
(5.4) At(s,t) ~s→∞ -β(t) sα(t)/sin πα(t)

- (5.4) **A****t****(s,t) ~****s→∞**** -****β****(t) s****α****(t)****/sin ****πα****(t)**

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## When s→∞ at fixed t, also u→-∞ (recall that s+t+u= 4m2). Consequently, if (5.4) is the asymptotic behaviour generated by the t-channel, we will have also a contribution from the u channel singularity (-u)α(t) which, 
after some technical refinement, is written as
   Au(s, t) ~s→∞ β(t) ξ e-iπα(t)/sin πα(t)
where ξ is called signature and can take only the two possible values ±1.
For future references, we shall introduce the signature factor  η(t)
    η(t) = - (1+ξ e-iπα(t))/sin πα(t)

- after some technical refinement, is written as
**   ****A****u****(s, t) ~****s→∞**** ****β****(t)**** ξ**** e****-i****πα****(t)****/sin ****πα****(t)**
- where ξ is called *signature* and can take only the two possible values ±1.
- For future references, we shall introduce the *signature factor * η(t)
-     **η****(t) = - (1+****ξ**** e****-i****πα****(t)****)/sin ****πα****(t)**

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## so that, in conclusion, the complete asymptotic behaviour of the scattering amplitude will be written as
(5.5)   A(s,t) ~s→∞- β(t) sα(t) 
           (1+ξe-iπα(t)) /sin πα(t)
The first remarkable observations concerning Regge poles are:
The energy dependence of the asymptotic amplitude is predicted
its phase is fixed

- (5.5)   **A(s,t) ~****s→∞****- ****β****(t) s****α****(t)**** **
**           ****(1+****ξ****e****-i****πα****(t)****) /sin ****πα****(t)**
**The first remarkable observations concerning Regge poles are:**
**The energy dependence of the asymptotic amplitude is predicted**
**its phase is fixed**

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

##  6. Regge trajectories.

- Near one of its poles in the complex angular momentum α(t), the partial wave amplitude A(l, t) can be approximated by
- (6.1) A(l, t) ~ l→α(t)  β(t) / (l - α(t))
- For t physical in the s-channel (t ≤ 0), the l plane singularities occur, in general at complex values of α(t); these can take on integer real values at unphysical (t≥0)
-    t-values.  In this case, Regge poles correspond to resonances or bound states.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Suppose that for some real t0 value, we have
                       α(t0) = l + i ε
where l is some integer and ε some real number
(which we suppose much smaller than one). Expanding α(t0) around t0, 
we find
        α(t) = l + i ε + α'(t0) (t - t0) + ....
so that the denominator in (6.1) can be written as
(6.4)          1/ (l - α(t)) ≈ 1/(t - t0 + iΓ) 
where Γ = Im α(t0)/ α'(t0) = ε/ α'(t0).
This is the typical structure of a Breit-Wigner resonance of mass M=√t0 and width Γ which will be real iff
d Im α(t)/dt│t0  << d Re α(t)/dt│t0

-                        α(t0) = l + i ε
- where l is some integer and ε some real number
- (which we suppose much smaller than one). Expanding α(t0) around t0,
- we find
-         α(t) = l + i ε + α'(t0) (t - t0) + ....
- so that the denominator in (6.1) can be written as
- (6.4)          1/ (l - α(t)) ≈ 1/(t - t0 + iΓ)
- where Γ = Im α(t0)/ α'(t0) = ε/ α'(t0).
- This is the typical structure of a Breit-Wigner resonance of mass M=√t0 and width Γ which will be real iff
- d Im α(t)/dt│t0  << d Re α(t)/dt│t0

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Notice, however that while the vanishing of the denominator sin πα(t) signals the possible presence of a resonant state at every integer value of  α(t), owing to the signature factor (1+ξ e-iπα(t)) induced by the crossed term in (5.5), actual bound states will be interpolated by a Regge trajectory at even values of the angular momentum (spin) if the signature ξ = +1 while negative signature if ξ = -1 will interpolate odd spin particle.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## The crucial message is that the asymptotic s-channel behaviour is due to the exchange of families of resonances in the crossed channels which amplifies the message contained in the Yukawa message about the relevance of the exchange of particles extending its role to the determination of the asymptotic behaviour of the scattering amplitude.
Different processes will, in general, receive contribution from different trajectories according to their quantum numbers.

**Different processes will, in general, receive contribution from different trajectories according to their quantum numbers.**

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## It is interesting to note that around t=0 one can expand α(t) in powers of t; for small enough t we can use the linear approximation
                     α(t) = α(0) + α' t
where α(0) and α' are known as the intercept and the slope respectively of the trajectory. According to (5.5) and all the previous discussion, it will be the trajectory with the highest intercept (whose real part lies higher in the complex angular momentum plane) which will determine the asymptotic behaviour of scattering amplitudes and cross sections.

-                      α(t) = α(0) + α' t
- where α(0) and α' are known as the *i****ntercept**** *and the* ****slope**** *respectively of the trajectory. According to (5.5) and all the previous discussion, it will be the trajectory with the highest intercept (whose **real part lies higher in the complex angular momentum plane**) which will **determine the asymptotic behaviour** of scattering amplitudes and cross sections.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Quite unexpectedly, the linear approximation (6.6) turns out to provide a reasonably good account of the physical situation up to considerably higher values of t than one would a priori have guessed. The figure shows an example of the extent to which all dominant mesons lie on the same straight line up to |t| values of the order of 6 ÷ 7 (GeV)2 irrespective of their spin being even or odd (exchange degeneracy). All leading meson trajectories, ρ, f2 ,a2 , and ω appear basically superimposed. Just as an example, we list the quantum numbers of the leading mesonic trajectories whose names come from the first resonance they interpolate.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Quantum numbers of leading mesonic tr.
f2
  P = +1, C = +1 , G = +1 ,  I = 0,  ξ = +1
ρ       
  P = -1,  C = -1 , G = +1 ,   I = 1,  ξ = -1
ω       
  P = -1, C = -1 ,  G = -1 ,   I = 0,  ξ = -1
a2         
  P = +1, C = +1 , G = -1 ,   I = 1,  ξ = +1

- f2
-   P = +1, C = +1 , G = +1 ,  I = 0,  ξ = +1
- ρ
-   P = -1,  C = -1 , G = +1 ,   I = 1,  ξ = -1
- ω
-   P = -1, C = -1 ,  G = -1 ,   I = 0,  ξ = -1
- a2
-   P = +1, C = +1 , G = -1 ,   I = 1,  ξ = +1

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Mesonic Regge trajectory

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Note also that, among the above trajectories, f2 has the quantum numbers of the vacuum. As it is well known, a special trajectory exists with the q. n. of the vacuum called Pomeron, whose existence determines the properties of the high energy cross sections (see  Landshoff). In fact, all mesonic trajectories appear basically exchange degenerate and their intercept is very close to 
(6.7)              α(0)  ≈  1⁄2,

- (6.7)              α(0)  ≈  1⁄2,

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Needless to say, this exchange degeneracy of the Regge trajectories is slightly broken if one plots them more accurately. It is, in fact, impossible for a trajectory to rise linearly indefinitely. It can be proved that analyticity forces a trajectory to bend asymptotically to a lesser growth than linear (power or logarithmic as the case may be).
Similar conclusions hold for fermionic trajectories. They, however, lie lower in intercept than the mesonic ones and their role is correspondingly  less relevant to account for the gross features of the cross sections at high energies (but essential for subdominant features such as the backward behaviour or other).

- Similar conclusions hold for fermionic trajectories. They, however, lie lower in intercept than the mesonic ones and their role is correspondingly  less relevant to account for the gross features of the cross sections at high energies (but essential for subdominant features such as the backward behaviour or other).

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## A most important exception in the above description is represented by the Pomeron trajectory will turn out to have an intercept very near 1 and will, therefore, provide the dominant behaviour of the cross sections
(6.8)   αP(0)  ≈  1. 
We will not, however, spend much time on this subject which has been covered in great detail in another course.

- (6.8)   αP(0)  ≈  1.
- We will not, however, spend much time on this subject which has been covered in great detail in another course.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## A last feature to be noticed is the almost universal slope of the mesonic (and also of the fermionic trajectories) which turns out to be very closely
(6.9)               α' ≈ 1 (GeV)-2          
and is related to the so-called string tension in the realm of string theories. The noticeable exception to this almost universal property is, once again, the Pomeron whose slope turns out to be much smaller or close to zero so that many authors consider it on a different footing than the other Regge trajectories.

- (6.9)               α' ≈ 1 (GeV)-2
- and is related to the so-called ***string tension**** *in the realm of string theories. The noticeable exception to this almost universal property is, once again, the **Pomeron** whose slope turns out to be much smaller or close to zero so that many authors consider it on a different footing than the other Regge trajectories.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

##  7. A brief account of Regge phenomenology.

- We will not spend much time on Regge phenomenology. We just wish to stress one feature which makes the Regge pole approach unique in the description of high energy physics phenomena. In fact, **Regge poles determine the asymptotic behaviour of the scattering amplitudes** (and, therefore, cross sections) in the s-channel (i.e. as s→∞) **when t is negative** and, at the same time, as we have just discussed, **they provide the basic information on resonant states when t is positive.**

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## A further remarkable property is that the phase of the dominant contribution is predicted and turns out to be imaginary in the forward direction in perfect agreement with what high energy data demand.
Rewriting the signature factor (5.6) for positive signature ξ =+1 as
(7.1)    η(t)  = - e-i1⁄2πα(t)/sin 1⁄2πα 
And using (6.6) with α(0) ≈ 1, near t=0 
                  η(0) = i

- Rewriting the signature factor (5.6) for positive signature ξ =+1 as
- (7.1)    **η****(t)  = - e****-i1⁄2****πα****(t)****/sin 1⁄2πα**
- And using (6.6) with α(0) ≈ 1, near t=0
-                   **η(0) = i**

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## We also expand near t ≈ 0 the residue function β(t) which we assume to have an exponential form (this is unessential)
(7.2)             β(t) ≈ β(0) exp[1⁄2B0t].
Putting everything together, the asymptotic (s → ∞) near-forward scattering amplitude reads
(7.3)    A(s,t) ~s→∞  i β(0) sα(0) eB(s)t 
where the slope B(s) is thus predicted to have a ln s  growth 
(7.4)      B(s) ≈ 1⁄2B0  + α' (ln s – i π/2).

- (7.2)             **β(t) ≈ β(0) exp[1⁄2B****0****t].**
- Putting everything together, the asymptotic (s → ∞) near-forward scattering amplitude reads
- (7.3)    **A(s,t) ~****s→∞****  i β(0) s****α(0)**** e****B(s)t**
- where the slope B(s) is thus predicted to have a ln s  growth
- (7.4)      **B(s) ≈ 1⁄2B****0****  + α' (ln s – i π/2).**

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## The result (7.3, 4) contains an unexpectedly simple and interesting set of properties. First of all, at high energies a diffraction peak is predicted to appear in the near forward direction. Furthermore, the slope is predicted grows logarithmically or the peak is  predicted to shrink as the energy increases.
Notice that it is not the assumption on β(t) which accounts for these properties. The assumption on β(t) provides a constant term in the slope (as the data seem to require). 

- Notice that it is not the assumption on β(t) which accounts for these properties. The assumption on β(t) provides a constant term in the slope (as the data seem to require).

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## The shrinkage of the forward peak (energy increase of the slope) is purely consequence of the exchange of a Regge trajectory with the vacuum quantum numbers and this seems indeed a general property supported by the data (see figure of angular pp distributions near t=0). 
In turn, the shrinkage of the diffraction peak is often interpreted as the increase of the effective interaction radius of the hadrons (we will not elaborate on this point here). 

- In turn, the shrinkage of the diffraction peak is often interpreted **as the increase of the effective interaction radius** of the hadrons (we will not elaborate on this point here).

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## Slide 64

- QCD, low x, Saturation and Diffraction, Copanello

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## On the other hand, using the general form (5.5) or directly (7.3), we can now analyze the Regge pole prediction for what concerns high energy total cross sections. Due to the optical theorem which in the present case reads
 (7.5) σtot ~s→∞ 1/s Im A(s, t=0) ~s→∞ sα(0)-1
we  see that the total cross section σtot grows as a power if  α(0) > 1. This conflicts potentially with a bound on such a growth put by unitarity due to Froissart, Lukaszuk & Martin which restricts this growth to be at most 
 (7.6)        σtot ~s→∞ O(ln2s). 

-  (7.5) σtot ~s→∞ 1/s Im A(s, t=0) ~s→∞ sα(0)-1
- we  see that the total cross section σtot grows as a power if  α(0) > 1. This conflicts potentially with a bound on such a growth put by unitarity due to Froissart, Lukaszuk & Martin which restricts this growth to be at most
-  (7.6)        **σ****tot**** ~****s→∞**** *****O*****(ln****2****s).**** **

- QCD, low x, Saturation and Diffraction, Copanello

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## Slide 66

- QCD, low x, Saturation and Diffraction, Copanello

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## In actual terms, the rate at which the Pomeron intercept αP(0) would be required to exceed unity is so mild (αP(0)  ≈  1.08) that the violation of the unitarity bound would occur at energies well beyond reach at any foreseeable future. 
This is true but:
       i) the predicted growth of total c.s. is at risk of being exceeded at Tevatron energies
       ii) at the same energies b-unitarity is near violation.

- This is true but:
-        i) the predicted growth of total c.s. is at risk of being exceeded at Tevatron energies
-        ii) at the same energies b-unitarity is near violation.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## 8. Conclusion 

- Regge pole phenomenology has been extremely popular some decades ago when it was shown to describe successfully the gross features of a large class of reactions (basically all reactions – it was also extended to production amplitudes) with a rather limited number of adjustable parameters.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## At some point, however, people lost interest in the model when it became apparent that it was unable to reproduce a number of delicate points such as polarization data, charge exchange reactions and, in general, most subleading features of high energy processes. 
In addition, as soon as one tries to go beyond the simple notion of poles in the complex angular momentum plane, the complication and the arbitrariness grow fast and get rapidly out of control. 

- In addition, as soon as one tries to go beyond the simple notion of poles in the complex angular momentum plane, the complication and the arbitrariness grow fast and get rapidly out of control.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## A further reason of loss of interest in Regge poles was the difficulty of extending their notion and derivation to the realm of field theory beyond the original framework of non relativistic potential theory and beyond few manageable and simple exchange model.

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007

## A last (psychological) reason for the diminution of interest in Regge models lies in the explosion of interest in Deep Inelastic Scattering (DIS) which in the Seventies gave rise to the new “Rutherford” experiment where it was shown that inside the proton exist seemingly point-like particles, the quarks (very much like Rutherford proved that inside the nuclei seemingly point-like particles, the nucleons, exist).

- QCD, low x, Saturation and Diffraction, Copanello

- 1-13 July 2007