# Pp and pBARp scattering amplitudes

• A
Is A_pp(s,t)=A_pBARp(t,s) true based on crossing symmetry?
Consider pp and pBARp elastic colissions (p + p -> p + p and p + BAR(p) -> p + BAR(p)). The scattering amplitudes are related by crossing in the following way:

1) A_pp(s,t)=A_pBARp(u,t) \simeq A_pBARp(-s-t,t) (energy large compared to 4m^2)
A, scattering amplitude
s,t,u Mandelstam variables
p proton
BARp antiproton

I don´t have any problem with this.

However, unless I made a huge mistake, crossing should also impose:

2) A_pp(s,t)=A_pBARp(t,s) which is very, very difficult for me to accept because of the limit t=0 (pp scattering amplitude equal to pBARp pure annihilation plus later pBARp pair creation?? How on earth can they interact if they're not approaching?? Is this a pBARp resonance?? Moreover, there's no energy for them to scatter off. t=0 makes no sense to me on the right hand side of the equation but it does on the left hand side. Unless they're moving in the same direction, instead of head-on. Could this be it? Would this explain a 0 C.O.M s, but a huge t? I think it does. In the rest frame p and BAR(p) would resonate, annihilate and then be created again moving in opposite directions. They will follow the COM trajectory, so that the total 4-momentum is conserved.

Can anybody tell me if this latter relationship is wrong?

By the way:

1) has a very interesting implication in the t=0 limit, that could, perhaps, be easily checked with the existing models:

tg-1(1/rho^pp(s,t=0))-tg-1(1/rho^pBARp(-s,t=0))=|2*n*pi|, where:

n is a non-specified natural number.

rho:=Re(A)/Im(A), A scattering amplitude.

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My question is probably too technical to get any answers.

I provide a slightly outdated (it does not include the odderon) presentation on this subject, just in case someone is interested.

https://indico2.riken.jp/event/2729/attachments/7480/8729/PomeronRIKEN.pdf

It would be interesting to check if A_pp(4m^2,epsilon)\simeq A_pBARp(epsilon,4m^2) , epsilon very small compared to 4m^2. This data must be available somewhere.
m proton mass.

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Is A_pp(s,t)=A_pBARp(t,s) true based on crossing symmetry?
Consider pp and pBARp elastic colissions (p + p -> p + p and p + BAR(p) -> p + BAR(p)). The scattering amplitudes are related by crossing in the following way:

1) A_pp(s,t)=A_pBARp(u,t) \simeq A_pBARp(-s-t,t) (energy large compared to 4m^2)
A, scattering amplitude
s,t,u Mandelstam variables
p proton
BARp antiproton

I don´t have any problem with this.

However, unless I made a huge mistake, crossing should also impose:

2) A_pp(s,t)=A_pBARp(t,s) which is very, very difficult for me to accept because of the limit t=0 (pp scattering amplitude equal to pBARp pure annihilation plus later pBARp pair creation?? How on earth can they interact if they're not approaching?? Is this a pBARp resonance?? Moreover, there's no energy for them to scatter off. t=0 makes no sense to me on the right hand side of the equation but it does on the left hand side. Unless they're moving in the same direction, instead of head-on. Could this be it? Would this explain a 0 C.O.M s, but a huge t? I think it does. In the rest frame p and BAR(p) would resonate, annihilate and then be created again moving in opposite directions. They will follow the COM trajectory, so that the total 4-momentum is conserved.

Can anybody tell me if this latter relationship is wrong?

By the way:

1) has a very interesting implication in the t=0 limit, that could, perhaps, be easily checked with the existing models:

tg-1(1/rho^pp(s,t=0))-tg-1(1/rho^pBARp(-s,t=0))=|2*n*pi|, where:

n is a non-specified natural number.

rho:=Re(A)/Im(A), A scattering amplitude.

I will elaborate a little further, assuming that both 1) and 2) are right (I don't know if 2) is):

From 1) and 2) it is immediate to deduce that:

3) A_pBARp(u,t)=A_pBARp(t,s) => A_pBARp(4m^2-s-t,t)=A_pBARp(t,s) and making t=0,

4)A_pBARp(4m^2-s,0)=A_pBARp(0,s) and something qualitatively new must happen at s^(1/2)=2m\simeq 2 GeV and,

It really does!!

a) p + p total cross-section has a minimum at that energy!! (about 20 mb).

b) There is a resonance that suddenly rises the total cossection up to 50 mb!!

c) Non elestic pp collission do have a 2 GeV threshold!!

I guess that if you already know nuclear physics is not surprising, but this is a purely mathematical result.

From the point of view of p + BAR(p) things look very messy, however. A_pBARp(0,0)=A_pBARp(0, 2 GeV). Well, not really, a p + BAR(p) pair that transfer q<=2 GeV, being both at rest, is just impossible (partial annhihilation is just not an option) and a p + BAR(p) whose C.O.M frame energy is less than 2 GeV is also impossible.

Well, it looks like 1) and 2) are telling us, at the very least, things that are known to be true on physical grounds using pure mathematical considerations. Maybe 2) is right after all.

I will elaborate a little further because I think there might be experimental evidence that supports/discards (2):

A_pp(s,t)=A_pBARp(4m^2-s-t,t) => (t=0)

=> A_pp(s,0)=A_pBARp(4m^2-s,0).

If the (global) minimum of the total pp cross-section happens, more or less, at s^(1/2)=2m+protonium binding energy\simeq 2Da+0.102Da (theoretical estimate)=1.97 GeV then (2) should be OK.

http://pdg.lbl.gov/2013/reviews/rpp2013-rev-cross-section-plots.pdf

Page 11, up left. I'm right!

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I did not realize that I could use Latex in this forum, so I've re-formatted everything:

Is ##A_{pp}(s,t)=A_{p\bar p}(t,s)## true based on crossing symmetry?

Consider ##pp## and ##p\bar p## elastic colissions (##p + p \rightarrow p + p## and ##p + \bar p \rightarrow p + \bar p##). The scattering amplitudes are related by crossing in the following way:

1) ##A_{pp}(s,t)=A_{p\bar p}(u,t) \simeq A_{p\bar p}(-s-t,t)## (energy large compared to ##4m^2##)

where:

##A##, scattering amplitude.

##s,t,u## Mandelstam variables.

##p## proton ##\bar p## antiproton.

I don't have any problem with this.

However, unless I made a huge mistake (the more I look at my graphs the more convinced I am that I haven't), crossing should also impose:

2) ##A_{pp}(s,t)=A_{p\bar p}(t,s)## which is very, very difficult for me to accept because of the limit t=0 (##pp## scattering amplitude equal to ##p\bar p## pure annihilation plus later ##p\bar p## pair creation??

How on earth can they interact if they're not approaching?? Is this a ##p\bar p## resonance?? Moreover, there's no energy for them to scatter off.

##t=0## makes no sense to me on the right hand side of the equation but it does on the left hand side. Unless ##p## and ##\bar p## are moving in the same direction, instead of head-on. Could this be it? Would this explain a 0 C.O.M frame ##s##, but a huge ##t##? I think this could be the explanation.

It would be interesting to check if ##A_{pp}(4m^2,\epsilon)\simeq A_{p\bar p}(\epsilon,4m^2)## , ##\epsilon## very small compared to ##4m^2##.

This data must be avialable somewhere.

##m## proton mass.

Can anybody tell me if this latter relationship is wrong?

By the way:

1) has a very interesting implication in the ##t=0## limit, that could, perhaps, be easily checked with the existing models:

##tg^{-1}\frac{1}{\rho^{pp}}(s,t=0)-tg^{-1}\frac{1}{\rho^{p\bar p}}(-s,t=0)=|2n\pi|##, where:

##n## is a non-specified natural number.

##\rho:=\frac{Re(A)}{Im(A)}##.

##A##, scattering amplitude.

I will elaborate a little further, assuming that both 1) and 2) are right (I don't know if 2) is):

From 1) and 2) it is immediate to deduce that:

3) ##A_{p\bar p}(u,t)=A_{p\bar p}(t,s) \Rightarrow A_{p\bar p}(4m^2-s-t,t)=A_{p\bar p}(t,s)##.

and making ##t=0##,

4) ##A_{p\bar p}(4m^2-s,0)=A_{p\bar p}(0,s)## and something qualitatively new must happen at ##s^\frac{1}{2}=2m\simeq 2 GeV## and,

It really does!!

a) ##p + p## total cross-section has a minimum at that energy!! (about ##20 mb##).

b) There is a resonance that suddenly rises the total cossection up to ##50 mb##!!

c) Non elestic ##pp## collission do have a ##2 GeV## threshold!!

I guess that if you already know nuclear physics this is hardly surprising, but this is a purely mathematical result.

From the point of view of ##p + \bar p## things look very messy, however. ##A_{p\bar p}(0,0)=A_{p\bar p}(0, 2 GeV)##.

Well, not really, a ##p + \bar p## pair that transfers ##q\leq 2 GeV##, being both at rest, is just impossible (partial annhihilation is just not an option) and a ##p + \bar p## whose ##C.O.M## frame energy is less than ##2 GeV## is also impossible.

Well, it looks like 1) and 2) are telling us, at the very least, things that are known to be true on physical grounds using pure mathematical considerations. Maybe 2) is right after all.

I will elaborate a little further because I think there might be experimental evidence that supports/discards (2):

##A_{pp}(s,t)=A_{p\bar p}(4m^2-s-t,t) \Rightarrow## (t=0)

##\Rightarrow A_{pp}(s,0)=A_{p\bar p}(4m^2-s,0)##.

If the (global) minimum of the total ##pp## cross-section happens, more or less, at ##s^\frac{1}{2}##=##2m## + protonium binding energy##\simeq## ##2 Da + 0.102 Da## (theoretical estimate)=##1.97 GeV## then (2) should be OK.

http://pdg.lbl.gov/2013/reviews/rpp2013-rev-cross-section-plots.pdf

Page 11, up left. I'm right!

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By the way:

1) has a very interesting implication in the ##t=0## limit, that could, perhaps, be easily checked with the existing models:

##tg^{−1}\rho_{pp}(s,t=0)−tg^{−1}\rho_{p\bar p}(−s,t=0)=|2n\pi|##, where:

##n## is a non-specified natural number.

##\rho:=\frac{Re(A)}{Im(A)}##.

##A##, scattering amplitude.

Oops! what I meant is:

By the way:

1) has a very interesting implication in the ##t=0## limit, that could, perhaps, be easily checked with the existing models:

##tg^{−1}\rho_{pp}(s,t=0)−tg^{−1}\rho_{p\bar p}(4m^2-s,t=0)=|2n\pi|##, where:

##n## is a non-specified natural number.

##\rho:=\frac{Re(A)}{Im(A)}##.

##A##, scattering amplitude.

##pp## and ##p\bar p## scattering can be approximately described (in the Regge limit, that is, when ##s \gg m \gt |t|##) by the exchange of Reggeons defined by the following Regge trajectory (low ##s##):

##\alpha_R(t)=\alpha_R(0)+\frac{d\alpha_R}{dt}t##

and the exchange of one (or several) Pomerons (whose quantum number must be equal to those of the vacuum) that are defined by the following Regge trajectory (high ##s##):

##\alpha_P(t)=\alpha_P(0)+\frac{d\alpha_P}{dt}t##

where:

##s## is the COM frame energy,

##t## is the 4-momentum transfer,

##\alpha_R(0)\simeq 0.55##,

##\frac{d\alpha_R}{dt}\simeq 0.86##,

##\alpha_P(0)\simeq 1.08##,

##\frac{d\alpha_P}{dt}\simeq 0.25##.

Surprisingly, the critical exponents of the 3d directed percolation theory, that happens to be described by a Reggeon Field Theory, have the following critical exponents:

##\eta_\bot\simeq 0.581##,

##\beta\simeq 0.81##,

##\eta_\|\simeq 1.105##.

I'd love to take a deeper look at this apparent coincidence myself (because I don't think it is a coincidence), but it's going to take a while because my knowledge of Regge Field Theory is very limited. If anybody else has that knowledge and is interested in this apparent "accident", please, be my guest.

The question is:

Are hadron interaction ##s## and ##t## exponents related to the critical exponents of the 3d directed percolation model critical exponents?

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Hadron interaction ##s## and ##t## exponents seem to be related to the critical exponents of the 3d directed percolation model. Their numerical values have a good degree of agreement and both problems are described by a Regge Field Theory, so it makes sense think that they might be the same numbers.

The specific question is:

Why are these critical exponents values so closed?

This question is relevant because the nature of Reggeons and Pomerons (and the Odderon, that is not included here) is unknown.

They need to be explained in terms of the Standar Model of particle physics (this is one of the unsolved problemas in physics, see, for example https://en.wikipedia.org/wiki/Regge_theory) but, although there are some speculations (glueballs?), nothing is known for sure.

Any link between this problem (Reggeon and Pomeron as interaction mediators) and a well known problem (the directed percolation model of Critical Phenomena) may provide useful information.

jedishrfu
Mentor
Since the OP has found the answer to his question, its time to close this thread.

Thanks to all who participated here.