[b1_ana] b1 and Pzz

[O. A. Rondon]

Hi JP,

If b1 results from coherent scattering, then I agree that it is like
measuring 3He structure functions versus nucleon structure functions in
nuclear targets.

But if it depends on coherent scattering, it would only be finite for
either elastic e-d scattering or at very low x, where, as Edelmann and
others point out, it could be due to double scattering or shadowing.
This process seems to be dominated by S-D interference, like both
Edelmann (just after eq. 36) and Jaffe (sec. 5.3) say.

At other values of x, between > ~0.1 and < 1, DIS is incoherent
scattering on the nucleons, and what we usually call g1d, etc. is just
the incoherent sum of g1p and g1n. There would be no b1 to measure.

So for intermediate x, I assume that what works is the b1 definition of
Kumano, eq. (1), in terms of PDF's, and my understanding is that the
hadron spin state lambda he mentions, would be the total angular
momentum of the nucleons in the D-state, which can be 1, following the
usual angular momentum composition rules, illustrated at the bottom of
p. 10 of Jaffe's talk on b1,
http://twist.phys.virginia.edu/~or/b1/b1_jaffe-talk.pdf

This would imply that at mid x we would be trying to measure a b1
resulting not from coherent low x or elastic scattering, but from
nucleons in the D-state, with total angular momentum 1. The nucleon
total angular momentum would have two orientations: along the photon
(nucleon m_j= +/-1), or transverse (m_j=0). And, somehow, DIS on the
(tensor) combination m_j = 1 + m_j = -1 - 2 m_j = 0  (which naturally
happens in the deuteron, no need to have Pzz, just Pz to have a target
helicity) would be non-zero.

If this is the b1 that we would be trying to see at mid x, it would be a
nucleon SF that can only be measured on bound nucleons, to have
something other than spin 1/2. Then Nd would matter. But, as I said
before, I'm not sure this is the same b1 as the b1 at low x, or at high
x that Frankfurt and Strikman discuss in their fig. 3.

But if the b1 is only a coherent deuteron SF, then a significant
non-zero result at mid x would be a major surprise, since it would imply
that all the other nucleon SF's measured on nuclear targets, especially
g1 would need substantial corrections, beyond just the usual nuclear
corrections (D-state, etc.). And, of course, for coherent b1, we need
large Pzz.

So, I'm not sure exactly what b1 we are looking for at mid-x.

Cheers,

Oscar


J. P. Chen wrote:
> Oscar,
> 
> Since you used N and He as example. it is clear to me that
> this is not the same: in NH3, the polarization never include the N
> in there while in deuteron tensor polarization, D and S states are
> already both counted.
> 
> As in our polarized 3He, when we talk about He3 asymmetry, the S, D or
> S' state does not matter, only He3 polarization matters. But if you want
> to extract neutron asymmetry (note: not He3 asymmetry) then the two
> unpolarized protons (which from S state) becomes the dilution or with
> slightly opposite polarization (from S' and D state).
> 
> So I believe in the tensor asymmetry (which is coherent to the deuteron)
> polarization should already include the effects you used D-state Dilution
> to take into account.
> 
> Jian-ping
> 
> On 3/14/2013 6:44 PM, O. A. Rondon wrote:
>> Hi Patricia and all,
>>
>> Thank you for looking into the rates and times for Azz.
>>
>> I totally agree that if Nd needs to be there, then the experiment does
>> not seem feasible. On the other hand, the fact remains that we'll count
>> events that scatter off the S-state and the D-state, but those in the
>> S-state are a dilution, just like scattering on N or He.
>>
>> To me, it seems that in an asymmetry measurement, we would need to
>> correct for this dilution, just like we correct for the events on N or
>> He. The total rate from the ammonia and LHe is always higher than the
>> rate from the protons or the deuterons. Since the HERMES target has no
>> dilution, they may not have considered this other one. But I may be on
>> the wrong track about this.
>>
>> In any case, let me update all about progress here. We met with
>> Simonetta and her doctoral student Kunal Kathuria, who is a co-author in
>> their paper on testing the angular momentum sum rule on spin-1 systems.
>>
>> Simonetta and Kunal will work with Dustin in trying to resolve the
>> issues about the connection between the definition of b1 and the
>> observables that could be used to extract it: polarized parallel and
>> perpendicular DIS cross sections, as proposed by Jaffe et al., or some
>> asymmetry like Azz, and what kind of target polarization would be
>> required. They believe they can report back at our meeting following the
>> one we have planned for next week.
>>
>> In the mean time, I should remark that, since our original proposal was
>> based on the polarized cross sections of Jaffe et al.'s section 6
>> depending on Pz, not on Pzz, the new method I proposed on measuring the
>> parallel polarized minus unpolarized difference with coaxial target cups
>> gets a boost, as follows:
>>
>> >From eq. (2) and eq. (1) of the method, we can estimate the difference
>> Delta N (eq. 12) of polarized minus unpolarized counts to be
>>
>> Delta N = -Pz/3 (b1/F1) N_U
>>
>> where N_U is the number of unpolarized counts. With Pz = 0.45, and
>> b1/F1 = 0.01, Delta N = -1.5E-3 N_U.
>>
>> We would like to measure Delta N with a relative statistical error of
>> 20%, which propagates directly to b1. Since Delta N <<1, we have
>>
>> N_U = N_|| = N = Delta N/1.5E-3,
>>
>> The error d(Delta N) = sqrt(2N), so the relative error
>>
>> d(Delta N)/Delta N  = sqrt(2*N)/(1.5E-3*N) = 0.2, or N = =2.22E7.
>>
>> Using Narbe's total rates ~ 55 Hz, this represents ~ 110 h for one
>> point. Keeping in mind that the method does include the presence of
>> unpolarized material, it looks like it could be done this way, no need
>> to figure out how to form an Azz. One advantage of cross section
>> differences is that no dilutions are involved. Another is that it is
>> defined in terms of polarization along the electron beam, not along q.
>>
>> Link to the F1 - b1 method
>> http://twist.phys.virginia.edu/~or/b1/b1_method.pdf
>>
>> Cheers,
>>
>> Oscar
>>
>>
>>
>>
>>
>> Patricia SOLVIGNON wrote:
>>> Hi Oscar.
>>>
>>> If Nd has to be in the equation to get Azz from A_measured, I don't
>>> see how we can do the measurement (even if it is Pz instead of Pzz):
>>>
>>> Azz ~ 1E-2
>>> Even with delta_Azz = Azz
>>> time = (1/f/Nd/Pzz)^2 *(1/delta_Azz)^2 * 1/RD = (8230 days)/RD
>>> For x=0.45 and Q2=2.67, R~80Hz --> time=102 days
>>>
>>> I think that Nd is taken into account in the measurement of Azz and
>>> it is because the D-component is small that Azz is so small. If I
>>> follow your argument, it would mean that HERMES should have applied
>>> it when they extracted Azz from Ameas and therefore their asymmetries
>>> would be 20 times smaller, which will lead us to the same impossible
>>> measurement.
>>>
>>> I started a note (see attachment) for the rates and error extractions
>>> (combining Oscar's email and more). I am going to pass on the
>>> rates/kinematics task to Ellie.
>>>
>>>
>>>
>>> ------------------------------------------------------------------------
>>>
>>>
>>>
>>> Patricia
>>>
>>> On Mar 14, 2013, at 1:40 PM, O. A. Rondon wrote:
>>>
>>>> Hi,
>>>>
>>>> Thinking about Patricia's question on why we would need to include a
>>>> factor Nd representing the probability that the nucleons are in the
>>>> D-state, I believe there is some confusion between b1 and Pzz.
>>>>
>>>> As we know, with J = 1 hbar, the projection of the deuteron spin along
>>>> the magnetic field direction takes values M_J = +/-1, 0. The different
>>>> populations of deuterons for each substate give rise to Pz and Pzz.
>>>>
>>>> This applies to the nuclear spin, but it has nothing to do with b1,
>>>> which is supposed to be a polarized PDF involving, as Kumano discusses,
>>>> unpolarized quark and anti-quark distributions in NUCLEONS in substates
>>>> m_J = +/-1, 0 of nucleon (not nuclear) total angular momentum j = 1.
>>>>
>>>> Only nucleons in the D-state can be in nucleon substate m_J=0, which
>>>> results from a combination of L=2 = L_proton + L_neutron and S=1 =
>>>> s_proton + s_neutron, but this is unrelated to Pzz, which involves
>>>> M_J=0, but does not care where M_J comes from.
>>>>
>>>> In other words, Pzz results from how many deuterons have M_J =+1,-1, 0,
>>>> but M_J = +/-1 here just means how many deuteron spins point
>>>> parallel or
>>>> anti-parallel to the target field, and M_J=0 how many point
>>>> perpendicular to the field. But it does not involve how J = 1 was
>>>> formed: the deuterons in M_J = 0 still have their n and p in the
>>>> S-state
>>>> 95% of the time, so those must have b1 = 0.
>>>>
>>>> Another way of looking at this is that in the S-state, with J = 1
>>>> coming
>>>> only from s_p + s_n, the deuteron must be in a "cigar" shaped
>>>> configuration. This configuration points along one of the three
>>>> +/-1, 0,
>>>> orientations with respect to the field. There may be a significant
>>>> difference in the number of deuterons with M_J = 0 vs those with
>>>> |M_J| =
>>>> 1, i.e. a big  Pzz, but this does not enhance the measurement of b1.
>>>>
>>>> To me, this totally decouples Pzz from b1. Namely, even in vector
>>>> polarized deuterons, the only nucleons that could contribute to b1 are
>>>> those in the D-state. I believe this is what Jaffe et al. mean in their
>>>> section 6.
>>>>
>>>> Note also, that in sec. 5.3, Jaffe and Co. clearly explain that b1 gets
>>>> TWO contributions, one from the D-state, another from D-S interference.
>>>> And the mixing of both depends on an angle alpha, where sin^2(alpha) =
>>>> D-state probability, etc.
>>>>
>>>> The point of Jaffe et al. in considering parallel vs perpendicular
>>>> vector polarizations is that F1 cancels in the difference, but the
>>>> D-state effect that gives rise to b1 stays.
>>>>
>>>> In any case, at the low Bjorken x < 0.5 where the b1 effects seem to be
>>>> present, there only is incoherent DIS off nucleons. These nucleons
>>>> could
>>>> have total angular momentum 1 from D-state orbital plus spin, and there
>>>> are other effects like shadowing and double scattering, as mentioned by
>>>> Edelmann et al., which could contribute to b1. But I don't think Pzz
>>>> plays any role here.
>>>>
>>>> It's all a question of D-state, with a quantization axis defined by the
>>>> target field. One could argue that for M_J=0 deuterons, the nucleon
>>>> total angular momentum in the D-state is normal to that of the D-state
>>>> nucleons in the deuterons with M_J=+/- 1.
>>>>
>>>> But this is basically the parallel vs perpendicular cross section
>>>> differences of Jaffe et al.: the M_J=0 deuterons for parallel
>>>> correspond
>>>> to the M_J=+/- 1 of perpendicular, and vice versa, but in both
>>>> cases, b1
>>>> is the same, with an extra kinematic factor of -2 for parallel. This
>>>> same argument is given in Edelmann et al. after their eq. (36), where
>>>> they explicitly mention transverse and longitudinal polarizations.
>>>>
>>>> So, I don't see any connection between Azz and Pzz. Azz is related to
>>>> b1/F1, per Edelmann et al. eq. (47), but this asymmetry is defined in
>>>> terms of photon-NUCLEON helicities, not of deuteron spin orientations.
>>>> No tensor polarization is ever mentioned, only tensor FORCES, but those
>>>> would be internal to the nucleus, nothing to do with Pzz.
>>>>
>>>> Only HERMES came up with the idea that Pzz is involved. But it is not a
>>>> question of preparing the deuteron in M_J = +/-1 states vs 0 state,
>>>> rather of scattering off the D-state not the S-state.
>>>>
>>>> I have posted the slides of two talks about b1, one by Jaffe, and the
>>>> other I think by Manohar. They make clear some of the concepts I
>>>> discuss above. For example how the nucleon j = 1 comes from nucleon
>>>> L > 0 and spin 1/2, leading to nuclear spin is illustrated at the
>>>> bottom of p. 10 of Jaffe's talk.
>>>>
>>>> In conclusion:
>>>>
>>>> 1. I don't think we need Pzz, just Pz.
>>>>
>>>> 2. This makes the F1-b1 method with field along the beam about four
>>>> times easier than if we depended on Pzz.
>>>>
>>>> 3. I think that just as one could measure
>>>>
>>>> A1 = (sigma_1/2 - sigma_3/2)/Sum
>>>>
>>>> directly by aligning the target along the q_vector, instead of solving
>>>> for it from A_para and A_perp, we could measure b1 directly by
>>>> measuring
>>>> Azz along the q_vector, for opposite Pz's. The opposite Pz's should
>>>> cancel the S-state scattering, leaving only any effects due to the
>>>> D-state.
>>>>
>>>> In the case of field parallel to the beam, I think that the subtraction
>>>> of F1 from the measured cross section does the job of canceling the
>>>> S-state. But since both Jaffe and Edelmann speak of S-D interference,
>>>> the latter saying it's the dominant effect, I'm not entirely sure.
>>>>
>>>> 4. The above would imply that there should be an Nd factor so, for Pb =
>>>> 0, A_measured = f*Nd*Azz*Pz (not Pzz). Pz comes in to account for the
>>>> fraction of nucleons with helicity along q.
>>>>
>>>> But I may be missing something. Further thinking by all is needed. My
>>>> point is that we need to be as rigorous as possible, so we don't go
>>>> after a red herring.
>>>>
>>>> Cheers,
>>>>
>>>> Oscar
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> <b1_jaffe-talk.pdf><b1_talk.pdf>
>>
>>
>>
>>
>>
> 
>