Chemistry - Extended Yukawa potential

In particle physics, it is roughly speaking the case that exchange of virtual pions determines nuclear forces .

Usually the Lagrangian for the pionic pseudoscalar field

p (x)

is taken as

$L\sim (Dp)^2+{p^2\over a^2}$

where $a$ is a scale of length.

For a statical spherically symmetric field we have

$p\sim {a\over R} \exp \left(-{R\over a}\right).\,$

It is the Yukawa potential which describes short range forces in the nucleus. Because the self-energy of this field is infinite, the Yukawa potential is not valid as

$R\rightarrow 0.\,$

It is possible to extend Yukawa potential to the region as $R\rightarrow 0$ . The pionic cloud in a nucleon has non-zero mass density, and so the pionic field has a parameter

u (x)

which is the local four-velocity vector of the field. Then the source of the pionic field

${\delta L_\mathrm{int}\over \delta p}$

in the general case is a linear function of the four velocity vector, and the simplest Lagrangian for the pionic field, when the electromagnetic interaction is switched off, is

$L\sim {(Dp)^2\over 2}+{p^2\over {2a^2}} +cp(u\cdot D)p +{k^2\over 2}(Du)^2$

where the last term is the Lagrangian of a free w-field

For virtual pions in a nucleon, the bound energy is larger compared with energy of a free pion, hence

$u^2 <0.\,$

For statical spherically symmetrical states

$p=p(z)\,$

$u\gamma_0=u(z)\vec e_R\,$

$z={R\over a }.$

In the simplest case, the w-field is in a vacuum state. That means

$D^2 u=0\,$

with coherence condition

$\int_{0}^{\infty} p\dot p\,dz=0.$

From energy limitations, the potential of the w-field in vacuum state is

$u(z)\sim {1\over z^2}.$

The nontrivial equation for pionic potential is

$\ddot p +{2\dot p\over z}=p+\beta {\dot p\over z^2}$

where the constant $β$ is the relative scale of fields. Then the pionic potential is

$p=C \prod_{n=1}^N(R-R_n) \exp\left(-{R\over a}-\beta {a\over R}\right)$

and it is an 'extended Yukawa potential.

The constants

$z_n={R_n\over a}$

are determined by equations

$\sum_{k}{1\over z_n-z_k}+{1\over z_n}+\beta {1\over{2z_n^2}}=1,k\ne n$

with restriction

$\sum {1\over {x_n}}=-1$

so the constant $β$ run line of numbers.

The coherence condition

$\int_{0}^{\infty}p\delta \dot p\,dz=0$

is automatically fulfilled, because only the integration constant $C$ is unknown and

$\delta p\sim \dot p\delta C.\,$

The extended Yukawa potential rapidly decreases as $R\rightarrow \infty$ and $R\rightarrow 0$ . The unexpected thing is that there exist series of cluster states , with resonances in each cluster.

This solution has only necessary condition of existence. When it is subset into equation then it is not hard to see that this algebraic solution must be equal zero. Physical source of this puzzle is clear. In area $R\rightarrow 0$ vacuum w-state may exist if always it is constant.

Then for finding spherical symmetrical potentials may use iterations calculations. Potential of w-field is

$u(R)={c_1\over R^2}+{c_2\over R^2}\int_{0}^{R} p^2 R^2\,dR$

Then in first approach $p 0 = c o n s t$ , $u_0={c_1\over R^2}+c^2R$ . There are not physical solutions. At next approach

$\ddot p_1+{2\over R}\dot p_1=-sp_1+\left({d\over R^2}+kR \right)\dot p_1$

and simplest pionic potential is

$p=C exp \left(-{s\over 6}R^2-{d\over R}\right)$

s > 0

d > 0

$k={s\over 3}$

where constants $d$ , $s$ are connect by coherence condition. This simplest extended Yukava potential have essential different futures compared with previous potentials.As example see Qantization pionic interaction

Categories: Theoretical physics