Can Particle Exchange Forces Lower Energy in Yukawa's Model?

[da_willem]

In Yukawas model the force between nucleons comes from the exchange of a pion. How precisely does this lower the energy? Has it something to do with confinement energy?

I know an exchange force can be understood in terms of a momentum exchange by the force carrying particle. But there must be a reason this particle is emitted. How does this lower the energy?

 

[marlon]

In Yukawas model the force between nucleons comes from the exchange of a pion. How precisely does this lower the energy? Has it something to do with confinement energy?

I know an exchange force can be understood in terms of a momentum exchange by the force carrying particle. But there must be a reason this particle is emitted. How does this lower the energy?

$$E = \frac{- e^{-mr}}{4 \pi r}$$

This is the expression for the energy that QFT gives us when we consider a massive free field theory (starting from the ordinary Klein Gordon equation) which we will submit to a perturbation expressed by two delta dirac functions. You can think of this as two massive lumps sitting on a matress (the field F to which those massive objects couple). This coupling means that the two massive objects will make the matress vibrate and this vibration (also called fluctuation or perturbation) exactly represents a certain particle of mass m and with integer spin (this will be the pi-meson). Them massive lumps are the nucleons.

This energy is negative, which means that the presence of the two delta dirac functions has lowered the energy of the free field theory. This really means that the two objects attract.

Also dE/dr > 0, two massive lumps sitting on the mattress can lower the energy by getting closer to each other.

The potential drops off exponentially over a distance 1/m : the range of the attractive force generated by the field F, is determined inversely by the mass m of the particle described by that field (ie the pi meson).

These are the reasons for attraction : the lowering of energy. The pi meson (also called Ykawa meson) is only emitted (NOT TO LOWER THE ENERGY) but because of the influence of the two massive objects (ie two nuclei for example) on the field F (of which the fluctuation describes a particle of mass m : the pi-meson.)

regards
marlon

 

[da_willem]

$$E = \frac{- e^{-mr}}{4 \pi r}$$

This is the expression for the energy that QFT gives us when we consider a massive free field theory (starting from the ordinary Klein Gordon equation)

I know the (stationary spherically symmetric) solution to the KG equation is:

$$\psi = C \frac{e^{-\frac{mc}{\hbar} r}}{r}$$

[Only considering the negative signed exponent as the physical one]
What is the interpretation of psi? I know it is not susceptible to a statistical interpretation, but why can you interpret it as a (Yukawa) potential? And the sign of C, isn't that something you put in? So why attraction?

 

[marlon]

Hold on, you are forgetting a very important aspect here : namely the fact that you need to incorporate the nucleons. You just solved the KG equation.

I am going to assume you know the Path integral formalism because this is the easiest way to prove the attraction mediated by a Yukawa meson.

$$W(J) = \frac{-1}{2} \int \int d^4x d^4y J(x) D(x-y) J(y)$$

The J's express the two massive lumps i talked about. The D is the propagator of the free field (coming from the KG equation.) It's equation is
$$D(x-y)= \int \frac{d^4x}{(2 \pi)^4} \frac {e^{ik(x-y)}}{k^2-m^2 + i \epsilon}$$

Now plug this into the expression for W and make a Fouriertransform. Then write each J(x) = J1 + J2 and only look at the contribution of J1 and J2. So we neglect selfcouplings. You will get :

$$W(J) = \frac{-1}{2} \int \frac{d^4k}{(2 \pi)^4} J_2^*(k) \frac {1}{k^2-m^2 + i \epsilon} J_1(k)$$

In order to compute the energy just realize that iW=iET (this comes from the path integral formalism), you will get :

$$- \int \frac{d^3 k}{(2 \pi)^3} \frac{e^{ik.(x_1 -x_2)}}{k^2 +m^2}$$

In the above formula, the x and k are VECTORS where as they are 4-vectors in the first formula's

Now combine this with the insights of my first post and you will have all that you need

marlon

 

[da_willem]

Thanks for the effort, but I am not too familiar with the path integral formalism. I would like to know 2 things:

1) Why can the solution to the KG equation be interpreted as a potential?

2) In the case of the hydrogen molecular ion eg I can understand where the exchange energy comes from. The electron is no longer confined to surrounding one proton but can smear out thus lowering the total energy, constiting the binding force between the two protons. But in the case of a proton and a neutron where does the exchange energy you mention come from?

 

[marlon]

Thanks for the effort, but I am not too familiar with the path integral formalism. I would like to know 2 things:

1) Why can the solution to the KG equation be interpreted as a potential?

Because the fluctuations of this field (generated by two nuclei for example) correspond to a particle of mass m ; ie the Yukawa meson. The interaction between the nuclei is described by the way this KG field vibrates due to the presence of the nuclei. Therefore the KG field really expresses the ongoing interaction between the two nuclei...

2) In the case of the hydrogen molecular ion eg I can understand where the exchange energy comes from. The electron is no longer confined to surrounding one proton but can smear out thus lowering the total energy, constiting the binding force between the two protons. But in the case of a proton and a neutron where does the exchange energy you mention come from?

The example i gave you contains two nuclei expressed by J1 and J2. each of these terms is written as a delta direc function. You need to look at this as two massive lumps "sitting on a KG-mattress". So the interaction really comes from the delta dirac functions expressing the two massive lumps. this has nothing to do with charge...However incorporating charge is not that difficult : just split up the J's in a positive and negative part.

I don't get the example you gave on the hydrogen molecular ion, though ? Two protons do not interact with each other by the exchange of electrons ! Their EM-interactions are mediated by virtual photons. You can prove this by using the eact same way that i described above. However you WILL need to know your introductory QFT/QED in order to both prove and understand what is going on.

marlon

 

[da_willem]

I don't get the example you gave on the hydrogen molecular ion, though ? Two protons do not interact with each other by the exchange of electrons !

A hydrogen molecular contains aside from two protons also an electron. This electron can reside on either of the protons. But when the electron is shared between the two the configuration has a lower energy than a hydrogen molecule and a proton separately. This lowering in energy comes from the fact that the electron is no longer confined to one proton but is smeared out between the two thus lowering its kinetic energy. You can view this as the protons resonating between a proton and a hydrogen molecule.

In Yukawas theory a proton can resonate between being a proton or a neutron and a positive pion. Does the energy lowering have a similar origin to the case of the hydrogen molecular ion?

 

[marlon]

A hydrogen molecular contains aside from two protons also an electron. This electron can reside on either of the protons. But when the electron is shared between the two the configuration has a lower energy than a hydrogen molecule and a proton separately. This lowering in energy comes from the fact that the electron is no longer confined to one proton but is smeared out between the two thus lowering its kinetic energy. You can view this as the protons resonating between a proton and a hydrogen molecule.

Ok, it is clear to me now

In Yukawas theory a proton can resonate between being a proton or a neutron and a positive pion. Does the energy lowering have a similar origin to the case of the hydrogen molecular ion?

No, the above example is a QM-thing. The Yukawa meson is a QFT thing so they are very different in nature and very different in theoretical explanaition

marlon

 

[da_willem]

But can you give me an intuitive physical reason (pls without advanced QFT...?!) why:

1) a particle would continuously emit gauge bosons?

["because of the influence of the two massive objects (ie two nuclei for example) on the field F" doesn't say much to me]

2) why this sometimes gives an attractive and sometimes repulsive force?

[repulsive I can understand by momentumexchange, attractive I can understand because by the uncertainty principle you can't really tell from which direction a gauge boson comes. But why in certain cases atrractive and in certain cases repulsive. If this lowers the energy, why?]

 

[arivero]

But can you give me an intuitive physical reason (pls without advanced QFT...?!) why:
2) why this sometimes gives an attractive and sometimes repulsive force?

Spin of the mediator has a role. Even spin happens to be attractive between identical particles, odd spin happens to be repulsive between identical particles. The proof is done -SHOULD BE DONE, teachers usually ignore the whole point- by comparing Born approximation (convolution of a potential) with tree level feymann diagrams.

Of course identical particles have equal charge.

Electron-Positron, on the other hand, hmm, let me to think, could be seen as two electrons, one coming from the future, other from the past, and then its repulsive force should be seen as attractive. Uff, I can not picture this. Surely it is easier to get a minus sign multiplying the potential got in the previous step.

 

[arivero]

Between electron and quark, I have never seen a calculation of the potential. But I believe I have seen it done between electron and pion, considered both as elementary particles.

 

[arivero]

But can you give me an intuitive physical reason (pls without advanced QFT...?!) why:

1) a particle would continuously emit gauge bosons?

Because it can.

 

[marlon]

But can you give me an intuitive physics al reason (pls without advanced QFT...?!) why:

1) a particle would continuously emit gauge bosons?

Particles emit gauge bosons because they interact with other particles. The energy of this interaction is "used" to generate such gauge-bosons. Don't take this too litterally, it is just a way of speaking. The gauge bosons really express the fact that there is an interaction going on. That is all there is to it.

2) why this sometimes gives an attractive and sometimes repulsive force?

[repulsive I can understand by momentumexchange, attractive I can understand because by the uncertainty principle you can't really tell from which direction a gauge boson comes. But why in certain caskes atrractive and in certain cases repulsive. If this lowers the energy, why?]

Well why do two positive charges repel ?

A more thorough study can be made BUT NOT without QFT, so this is as intuitive as you can get

regards
marlon

 

[marlon]

https://www.physicsforums.com/journal.php?s=&journalid=13790&action=view

Here is another vision on attracion and repulsion. I wrote it my journal

regards
marlon

 

[da_willem]

Spin of the mediator has a role. Even spin happens to be attractive between identical particles, odd spin happens to be repulsive between identical particles. The proof is done -SHOULD BE DONE

Any references on that? Or is the proof very difficult?

Between electron and quark, I have never seen a calculation of the potential. But I believe I have seen it done between electron and pion, considered both as elementary particles.

How is a pion an elementary particle if it is composed of a quark and an antiquark?

Now if you realize that at each point where the foton and the charged particle "meet" you need energy conservation, you can easily see that this picture works.

Does this mean that the charged particle emits as many virtual photons as it absorbes?

 

[arivero]

Any references on that? Or is the proof very difficult?

It is. It appears in old handbooks on advanced quantum mechanics, under "Born approximation". I do not know if there is another way.

How is a pion an elementary particle if it is composed of a quark and an antiquark?

I said "considered". Ie, the mathematics in the argument is the one corresponding to a charged point particle different of the electron (but perhaps excessively different. Beware spins)

Does this mean that the charged particle emits as many virtual photons as it absorbes?

Hmm, it does not mean, nor the contrary. It is an interesting thing to think about. For instance, a fermion loop (vacuum polarisation). can adjust the number of virtual particles. It is a pity that research in this kind of semiclassical approaches is currently deprecated.

 

[arivero]

The proof should be done by reversing calculation of this kind:

http://hrst.mit.edu/hrs/renormalization/dyson51/Dyson-small/034.html

so that instead of getting the scattering from the potential, you work yout the scattering in Feynman style and then you reverse the steps to get the potential, so you will see if it is attractive or repulsive.

 

[marlon]

How is a pion an elementary particle if it is composed of a quark and an antiquark?

Besides, the pion mediates the residual strong force. You know, it holds two hadrons together. The valence quarks of the first hadron feel those of the second hadron. Since the strong force is attractive (and very much so at ground state energies), the two hadrons are "bound together". The energy between those two valence quarks is big enough to create a quark anti-quarkpair out of the vacuum. This is the generated pion. A pion is the lightest possible meson.

Does this mean that the charged particle emits as many virtual photons as it absorbes?

No, it just means the total momentum is conserved.

regards
marlon

 

[da_willem]

As I understand it a proton can exchange a positive pion upon becoming a neutron. Does this mean it can emit pions only at a frequency given by the 'lifetime' of the pion, by conservation of charge?

 

[reilly]

First, the sign of interactions mediated by particle exchange is dependent on spin, isospin, charge conjugation at the minimum. You will find good discusions of Yukawa's work in virtually any book, intermediate level and on, on particle theory, and/or on the history of particle physics. Zee's book on QFT covers this territory.

Some years ago, there was theoretical work called polology, part of the then hot subject of dispersion relations, Mandlestm representations, and bootstrap theory. Polology was primarily about finding whether a particular exchanged particle yielded a positive or negative force -- some of this work flowed from Chew and Low's work on the so-called 3 3 resonance in pion nucleon scattering, one of the first triumphs in strong interaction physics. At the risk of overtooting my own horn, I'll cite a few papers I wrote on polology

Self-Consistency and the Roper Resonance, with E.N. Argyres(one of my students) Phys. Rev. 159, #5 (1967)

B Exchange in pi-omega Scattering and the Spin of the B Meson, Phys. Rev. Vol 142.,#4 (1965)

Goldberger-Treiman Relation for a Composite Pion, Phys. Rev. 162, #5 (1967)

Figuring out the signs of forces can get very tricky, and very tedious.

Regards,
Reilly Atkinson

 

[marlon]

As I understand it a proton can exchange a positive pion upon becoming a neutron. Does this mean it can emit pions only at a frequency given by the 'lifetime' of the pion, by conservation of charge?

I am not really sure what you mean. Keep in mind that the emitted virtual bosons in interactions don't have to respect conservation laws. It is the particles at the begin and end of such interactions that have to respect the conservation laws at hand. For example a particle can emit a virtual force carrier that is more heavy then the emitting particle. This happens in weak interactions...

marlon

 

[da_willem]

First, the sign of interactions mediated by particle exchange is dependent on spin, isospin, charge conjugation at the minimum. You will find good discusions of Yukawa's work in virtually any book, intermediate level and on, on particle theory, and/or on the history of particle physics. Zee's book on QFT covers this territory.

Some years ago, there was theoretical work called polology, part of the then hot subject of dispersion relations, Mandlestm representations, and bootstrap theory. Polology was primarily about finding whether a particular exchanged particle yielded a positive or negative force -- some of this work flowed from Chew and Low's work on the so-called 3 3 resonance in pion nucleon scattering, one of the first triumphs in strong interaction physics. At the risk of overtooting my own horn, I'll cite a few papers I wrote on polology

Self-Consistency and the Roper Resonance, with E.N. Argyres(one of my students) Phys. Rev. 159, #5 (1967)

B Exchange in pi-omega Scattering and the Spin of the B Meson, Phys. Rev. Vol 142.,#4 (1965)

Goldberger-Treiman Relation for a Composite Pion, Phys. Rev. 162, #5 (1967)

Figuring out the signs of forces can get very tricky, and very tedious.

Regards,
Reilly Atkinson

But it is in princible possible from theory to figure out the sign of an interaction, ..joehoe !

 

[arivero]

First, the sign of interactions mediated by particle exchange is dependent on spin, isospin, charge conjugation at the minimum. You will find good discusions of Yukawa's work in virtually any book, intermediate level and on, on particle theory, and/or on the history of particle physics. Zee's book on QFT covers this territory.

Guess you refer to http://theory.itp.ucsb.edu/~zee/QuantumFieldTh.html . It is a very recent book, and sure most graduates and postgraduates haven't read it yet (myself for one). I'd like to hear a bit more on it.

 

[marlon]

What do you mean arivero; QFT in a Nutshell is propably the best book on introductory QFT. I have it and i strongly recommend it...

marlon

 

[arivero]

Marlon, what kind of calculations does it made explicitly? Anomalous magnetic moment? Electroweak currents? QCD? $$\lambda \Phi^4$$?

 

[marlon]

Marlon, what kind of calculations does it made explicitly? Anomalous magnetic moment? Electroweak currents? QCD? $$\lambda \Phi^4$$?

all of those that you mentioned

plus magnetic monopoles, confinement and the whole QCD-picture

marlon

 

[reilly]

da willem -- Theory, now can do quite well, given the Standaard Model, symmetries and all that. But, much of the Standard Model stems from all the experimental work done before the theory was developed, in which case signs and strengths were often determined experimentally.

Regards,
Reilly Atkinson