few notes : what is a pomeron

Few notes : Few notes : What is a PomeronWhat is a Pomeron

Po – Ju LinPo – Ju Lin

August 17, 2004August 17, 2004

ContentsContents1.1. Life before QCDLife before QCD

2.2. Sommerfeld - Watson transformSommerfeld - Watson transform

3.3. Signature Signature

4.4. Regge polesRegge poles

5.5. FactroizationFactroization

6.6. Regge trajectoriesRegge trajectories

7.7. The PomeronThe Pomeron

8.8. Total cross – sectionsTotal cross – sections

1. Life 1. Life beforebefore QCD QCD

Instead of applying field theory directly, Instead of applying field theory directly, physicists tried to extract as much as physicists tried to extract as much as possible by studying the consequence of a possible by studying the consequence of a reasonable set of postulates about the reasonable set of postulates about the S-matrix. S-matrix.

1. Life 1. Life beforebefore QCD ~ S-matrix QCD ~ S-matrix

S – matrixS – matrix

Overlap between the in-state Overlap between the in-state and the and the

out- stateout- state

inoutab abS

a

b

1. Life before QCD ~ Postulates1. Life before QCD ~ Postulates

Postulate 1. S-matrix is Lorentz invPostulate 1. S-matrix is Lorentz invariantariant

It can be expressed as a function of the It can be expressed as a function of the Lorentz scalar products of the incominLorentz scalar products of the incoming and out going momenta.g and out going momenta.


Postulate 2. S-matrix is unitary:Postulate 2. S-matrix is unitary:

This is a natural statement as a This is a natural statement as a consequence of conservation of consequence of conservation of probability.probability.

1 SSSS


The scattering amplitude, is related The scattering amplitude, is related to the S-matrix by :to the S-matrix by :

The unitarity of the S-matrix leads to :The unitarity of the S-matrix leads to :

This gives us the Cutkosky rulesThis gives us the Cutkosky rules

abA

aba b

baabab AppiS

442

2

cb

cac

a bbaab AAppA 442Im2


Postulate 3. The S-matrix is an analPostulate 3. The S-matrix is an analytic function of Lorentz invariants ytic function of Lorentz invariants (regarded as complex variables), wi(regarded as complex variables), with only those singularities required th only those singularities required by unitarityby unitarity..

It can be shown that this property is a cIt can be shown that this property is a consequence of causality, i.e. that two reonsequence of causality, i.e. that two regions with a space-like separation do nogions with a space-like separation do not influence each other t influence each other

2. Sommerfeld – Watson Transform2. Sommerfeld – Watson Transform

Consider a two-particle to two-particle Consider a two-particle to two-particle scattering preocess in t-channel at a cescattering preocess in t-channel at a center of mass energy, nter of mass energy,

which is much larger than the maswhich is much larger than the masses of external particles. The amplitudses of external particles. The amplitude can be expand as a series in Legendre can be expand as a series in Legendre polynomials,e polynomials,

where is the scattering angle in where is the scattering angle in cmscms and is related to s, t by : and is related to s, t by :

s

coslP

s

t21cos


Partial wave expansionPartial wave expansion::

where are where are called partial wave called partial wave amplitudesamplitudes..

In s-channel (interchange s and t):In s-channel (interchange s and t):

stPsaltsA ll

ldbca/2112,

0

sal

tsPtaltsA ll

lcdab /2112,0


Sommerfeld, following Watson, rewrotSommerfeld, following Watson, rewrote the partial wave expansion in terms e the partial wave expansion in terms of a contour integral in the complex anof a contour integral in the complex angular momentum gular momentum

plane as :plane as :

where the contour C surrounds the poswhere the contour C surrounds the positive real axis as shown in Fig.1itive real axis as shown in Fig.1

tslPl

tlaldl

itsA

C/21,

sin

,12

2

1,

l


Fig.1Fig.1 Sommerfeld – Watson TransformSommerfeld – Watson Transform

3. Signature3. Signature

Is unique?Is unique? It can be shown that is unique It can be shown that is unique

provided provided

as . Unfortunately, the as . Unfortunately, the contributions to the partial wave amplitude contributions to the partial wave amplitude which are proportional to so the which are proportional to so the inequality is violated along the imaginary inequality is violated along the imaginary axis. Therefore we need two analytic axis. Therefore we need two analytic functions of the even and odd partial wave functions of the even and odd partial wave amplitudes , .amplitudes , .

tla ,

tla ,

l ltla exp,

l1

tla ,1 tla ,1

3. Signature3. Signature

Thus we have :Thus we have :

where takes the values , is called twhere takes the values , is called the he signature signature of the partial wave and of the partial wave and and and

are called the even- and odd-sigare called the even- and odd-signature partial wave functions.nature partial wave functions.

tslPtlae

l

ldl

itsA

li

C/21,,

2sin

12

2

1,

1

1 tla ,1

tla ,1

4. Regge Poles4. Regge Poles

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

tstPt

ten

n

n

n

ti n

/21,sin2

~

1

tla , tl n

i

i

li

tslPtle

l

ldl

itsA 2

1

2

11

/21,,2sin

12

2

1,


The simple poles are called even- The simple poles are called even- and odd-signature and odd-signature Regge Poles.Regge Poles.

In Regge region, i.e. , the LegendrIn Regge region, i.e. , the Legendre polynomial is dominated by its leadie polynomial is dominated by its leading term and in this limit the contributing term and in this limit the contribution to the right hand side of the previouon to the right hand side of the previous formula from the integral along the cs formula from the integral along the contour C’ vanishes as .ontour C’ vanishes as .

tn

ts

s


We want to isolate the high energy behWe want to isolate the high energy behavior of the scattering amplitude in the avior of the scattering amplitude in the Regge region. Now in fact we need onlRegge region. Now in fact we need only consider the contribution from the Ry consider the contribution from the Regge pole with the largest value with thegge pole with the largest value with the real part of (the leading Regge pe real part of (the leading Regge pole). Thus we have: ole). Thus we have:

tn

tti

s ste

tsA

2

,

5. Factorization5. Factorization

We can view We can view

as the exchange in the t-channel of an oas the exchange in the t-channel of an object with as object with ‘angular mombject with as object with ‘angular momentum’ equal toentum’ equal to

. This is of course not a particle sinc. This is of course not a particle since the angular momentum is not integer e the angular momentum is not integer (or half-integer) and it is a function of t. (or half-integer) and it is a function of t. It is called a It is called a ReggeonReggeon..

tti

s ste

tsA

2

,

t


We can view a Reggeon exchange ampWe can view a Reggeon exchange amplitude as the superposition of amplitudlitude as the superposition of amplitudes for the exchange of all possible parties for the exchange of all possible particles in t-channel.cles in t-channel.

The amplitude can be factorized as shoThe amplitude can be factorized as shown in Fig.2 into a coupling of the wn in Fig.2 into a coupling of the Reggeon between particle a and c, Reggeon between particle a and c, between b and d and a universal cont between b and d and a universal contribution from the Reggeon exchange.ribution from the Reggeon exchange.

tac

tbd


Fig.2 A Regge Exchange DiagramFig.2 A Regge Exchange Diagram

5. Factorization5. Factorization Thus we obtain : Thus we obtain :

For the presence of in the For the presence of in the denominator, if denominator, if

takes an integer value for some of t takes an integer value for some of t then the amplitude has a pole. For then the amplitude has a pole. For positive integer this can be understood as positive integer this can be understood as a exchange of a resonance particle with a exchange of a resonance particle with integer spin. For negative values they are integer spin. For negative values they are canceled out. canceled out.

tbdacti

s st

tt

t

etsA

sin2,

t

t

6. Regge Trajectories6. Regge Trajectories

Consider t- channel process, with t posConsider t- channel process, with t positive we expect the amplitude to have pitive we expect the amplitude to have poles corresponding to the exchange of oles corresponding to the exchange of physical particles of spin and mass physical particles of spin and mass ,where ,where

Chew & Frautschi plotted the spins of lChew & Frautschi plotted the spins of low lying mesons against square mass aow lying mesons against square mass and noticed that they lie in a straight linnd noticed that they lie in a straight line as shown in Fig.3 e as shown in Fig.3

iJ ii Jm 2im


Fig. 3 The Chew-Frautschi Plot Fig. 3 The Chew-Frautschi Plot


is a linear function of t :is a linear function of t :

From Fig 3. we obtain the values :From Fig 3. we obtain the values :

We shall see this linearity continues We shall see this linearity continues for negative values of t.for negative values of t.

t

tt 0

286.0

55.00

GeV


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

dt

d

2202 ts


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

We expect the Regge trajectory which detWe expect the Regge trajectory which determines the asymptotic s-dependence to ermines the asymptotic s-dependence to be the one containing the I = 1 even paritbe the one containing the I = 1 even parity mesons (the -trajectory). Use the daty mesons (the -trajectory). Use the data acquired in Fig.3, we get Fig.4 a acquired in Fig.3, we get Fig.4

np 0


Fig.4 The extrpolation of Fig.3Fig.4 The extrpolation of Fig.3

7. The Pomeron7. The Pomeron

From the intercept of the Regge trajectoFrom the intercept of the Regge trajectory which dominates a particular scatteriry which dominates a particular scattering process and the optical theorem we ng process and the optical theorem we can obtain the asymptotic behavior of tcan obtain the asymptotic behavior of the total cross-section for that process, nhe total cross-section for that process, namely, is proportional to :amely, is proportional to :

For the -trajectory considered in the lFor the -trajectory considered in the last section, < 1, which means that tast section, < 1, which means that the cross-section for a process with I = 1 he cross-section for a process with I = 1 exchange falls as s increases.exchange falls as s increases.

tot

10 s

0


Pomeronchuck & Okun proved from genPomeronchuck & Okun proved from general assumptions that eral assumptions that in any scattering prin any scattering process in which there is charge exchange tocess in which there is charge exchange the cross-section vanishes asymptoticallyhe cross-section vanishes asymptotically (the Pomeronchuck theorem). (the Pomeronchuck theorem).

Foldy & Peierls noticed the converse : Foldy & Peierls noticed the converse : if fif for a particular scattering process the crosor a particular scattering process the cross-section does not fall as s increases then s-section does not fall as s increases then that process must be dominated by the exthat process must be dominated by the exchange of vacuum quantum numbers.change of vacuum quantum numbers.


Experiments showed that total cross-seExperiments showed that total cross-section do not vanish asymptotically. In fction do not vanish asymptotically. In fact they rise slowly as s increases.act they rise slowly as s increases.

If we are to attribute this rise to the exIf we are to attribute this rise to the exchange of a single Reggeon pole then it change of a single Reggeon pole then it follows that the exchange is that of a Rfollows that the exchange is that of a Reggeon whose intercept, eggeon whose intercept,

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

0P


Unlike The Regge trajectory, the physicUnlike The Regge trajectory, the physical particles which would provide the resal particles which would provide the resonances for integer values of for ponances for integer values of for positive t have not been conclusively ideositive t have not been conclusively identified. ntified.

Particles with the quantum numbers vaParticles with the quantum numbers vacuum can exist in QCD as bound states cuum can exist in QCD as bound states of gluons (glueballs).of gluons (glueballs).

0P

8. Total Cross-sections8. Total Cross-sections

Fig. 5 shows a compilation of data for tFig. 5 shows a compilation of data for total cross-sections for and scaotal cross-sections for and scattering, together with a fit due to Donnttering, together with a fit due to Donnachie & Landshoff :achie & Landshoff :

The first term on the right hand side is The first term on the right hand side is the Pomeron contribution and the secthe Pomeron contribution and the second term is due to the exchange of a Rond term is due to the exchange of a Regge trajectory.egge trajectory.

pp pp

mbss

mbss

pp

pp

45.008.0

45.008.0

4.9817.2

1.5617.2


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

pp pp


The Pomeron couples with the same stThe Pomeron couples with the same strength to the proton and antiproton berength to the proton and antiproton because the Pomeron carries the quantucause the Pomeron carries the quantum numbers of the vacuum.m numbers of the vacuum.

The Regge trajectory can have differenThe Regge trajectory can have different couplings to particles and antiparticlt couplings to particles and antiparticles. This accounts for the difference betes. This accounts for the difference between the ween the

and cross-sections at low s. and cross-sections at low s.

pp

pp


One point of view to argue is that the inteOne point of view to argue is that the intercept 1.08is only an effective intercept anrcept 1.08is only an effective intercept and the underlying mechanism which gives d the underlying mechanism which gives rise to it is not the result of single Pomerrise to it is not the result of single Pomeron exchange but has contributions from on exchange but has contributions from the exchange of two or more Pomerons the exchange of two or more Pomerons (so called (so called Regge cutsRegge cuts). ).

Since the intercepts are universal we exSince the intercepts are universal we expect them to be able to describe other totpect them to be able to describe other total crossections. This is indeed the case, aal crossections. This is indeed the case, as can be seen from Fig.6s can be seen from Fig.6


Fig.6 Total cross-sections for and Fig.6 Total cross-sections for and scatteringscattering

p p

few notes : what is a pomeron

Documents

schannel interchange

regge region

contour c

regge poleswe

oddsignature regge poles

odd partial wave amplitudes

sommerfeld watson transformfig

sommerfeld watson transform3