Re: [BLAST_ANAWARE] t20 thoughts

From: Tancredi Botto (tancredi@lns.mit.edu)
Date: Sun Nov 07 2004 - 15:56:22 EST


Hi aron,
I welcome the discussion. I altough you only focus on T20, this discussion
has an immediate relation with the tensor e,e'p

> 1) because of the tensor component of the N-N interaction, deuterium
> has a density that depends on m_z, the projection of the total spin
> along the quantization axis.

also, not being in a central potential anymore, protons and neutrons need
not to be in a L=0 eigenstate anymore and thus they can be in D-waves. But
we know they are because of the quadrupole moment.
 
> 2) the m_z = 0 density is toroidally shaped, like a donut lying in the
> x-y plane. the donut has a certain thickness, t.

this is the state with Pzz=-2

> 3) the m_z = +1 or -1 densities are dumbbell shaped, like two balls
> centered at z = +d/2 and -d/2.

This is the state with Pzz = +- 1

You z axis is the spin direction. That's the key of what zz mentioned in
the last email (but this was not discussed for the first time and is
re-disccued here.)

So now you can look at pm|| and pm_perp to that spin direction. In the
first case you expect to find higher density and more protons in the
subtate mz +1. So if you build the asymmetry as Pzz+=1 - Pzz=-2 it will
be positive. In the second case the opposite, since the nucleons (that
include the proton too..) are mostly pushed outside of the spin direction
for m_z=0 (Pzz=-2).

But this is now up to a certain shell thickness "t" which corresponds up
to a max missing momentum and/or a given inter-nucleon distance. Now when
you vary the missing momentum magnitude you varying the weight of the tensor
force with respect to -e.g.- the 1/r long-range part of the pion exchange
potential and the very short-distance repulsive core (which involves
heavier mesons, very short distnaces and is ths forced in S-wave).

In other words you should see this asymmetry rise, peak and then go away
again as pmiss increases. In fact we are talking about the
region where S and D wave are comparable in strenght.
I can't help but point out to ZZ thesis and the region around p~1.5 fm^-1
in fig 2.5 and 2.6
But you should also get a similar result at fixed pmiss by scanning Q2
which has the same effect of probing shorter internucleon distances.

> t20 analysis can help determine the values of t and d. here's how:

Yes, we all know T20 is very sensitive to the spin dependent deuteron
structure. YOu have the Gc Gq form factors and their relative strenghts
tell you about S and D waves.

But so should also be the case ee'p, with the added benifit that now you
have an extra handle, pmiss, to scan the interplay between these
components. This is important since you want to study deuteron structure,
and you want to do that at a fixed Q2 before falling back into the
discussion about the discussion about "what d.o.f at Q2 of 0.1 and 2.0 ?"

You -or maybe only me - hope can do it all with "one theory", even if that
works only on a limited range of Q2. You are after the deuteron. Not Q2
which per se is only a tool and maybe not the best in this case.

So you exploit the fact that now you scan the nucleon inter-distance because
the nucleons are already moving for you and all you have to do is
observe that in the eep channel. YOu do not need to have to smash it with
a probe exposed to different d.o.f. (e.g. partonic d.o.f ??) at different
Q. I believe deuteron structure would still exist then

> 4) the charge form factors for deuterium, F^C_{m_z}(q), are related to
> the fourier transform of the deuteron density, rho^{m_z}(\vec{r}):
>
> F^C_{m_z}(q) = (1/2)*int{rho^{m_z}(\vec{r})*exp(i*q*z)*d^3\vec{r}}
>
> 5) figure 7 in the paper shows that F^C_{+1 or -1}(q) has minima at q =
> 3.6fm-1 and 12.6fm-1. one can argue that, at its minima,
> F^C_{+1 or -1} is highly determined by the dumbbell-ness of the m_z
> = +1 or -1 state. thus, in this way, F^C_{+1 or -1} is sensitive
> to the value of d.
>
> 6) similarly, the minima of F^C_{0}(q) is sensitive to the value of t.
>
> 7) t20 also has minima and maxima. at its extrema, t20 has a simple
> mathematical form:
>
> [(F^C_{0}(q))^2 - (F^C_{1}(q))^2]
> t20 = -sqrt(2) * ---------------------------------
> [(F^C_{0}(q))^2 + (F^C_{1}(q))^2]
>
> 8) the minima of t20 occur at F^C_{1}(q) = minimum; the maxima of t20
> occur at F^C_{0}(q) = maximum. thus, by determining the extrema of
> t20, you are ultimately measuring the d and t values for deuterium.
>
> specifically, the first minimum of t20 provides information on d. the
> first maximum of t20 provides information on t. d, in turn, is directly
> related to the diameter of the maximum-density torus (see figure 3). one
> can see from the figure that there is not much model uncertainty at
> maximum density. thus, it would seem to me that determining the actual
> value of the first t20 minimum is not of too much theoretical
> significance. (additionally, in a lecture that donnelly gave to us
> graduate students, i seem to remember him saying that the minima isn't of
> much interest, but maybe i'm wrong here; one of the other students at the
> lecture can help here).
>
> on the other hand, the first maximum of t20 has never been measured (at
> least not that i am aware of). as it will give information on the value
> of the thickness of the density for the m_z = 0 deuteron, there is
> definite merit to aiming for it.
>
> i realize that, in some sense, the data is what it is, and we cannot "get
> higher Q2 values just because we want to". thus, i know that we will
> not be able to get high-enough Q2 data to reach the maximum. but maybe
> this will help focus the analysis some.
>
> or maybe all of my thoughts are wrong. someone more knowledgable can
> decide.
>
> aaron
>



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