[b1_ana] Follow up to Re: b1 and Pzz

Tue Mar 19 01:22:52 EDT 2013

Applying the coherence length defined in Edelmann et al., eq. (19),

lambda = 2\nu/(MX^2 +Q^2)

where MX is the mass of the intermediate state that is produced in the
double scattering, dominated by MX^2 ~ Q^2 (fig. 2 of Edelmann), I get
that only the lowest three HERMES x-Q^2 points have coherence lengths
comparable to the deuteron radius, which is the coherence condition

lambda >~ r_d = 4 fm.

For the points we have been considering for an experiment at JLab,
lambda << 4fm.

Detailed numbers are posted in my spradsheet here (last columns of the
tables on pages "JLab" and "HERMES", numbers are in fm)
https://twist.phys.virginia.edu/~or/b1/db1-rate.ods

If b1 depends on coherent scattering, it seems very unlikely to be
finite for x > 0.1. But it could all be model dependent.

Oscar

O. A. Rondon wrote:
> 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>
>>>
>>>
>>>
>>>
>>
> 
> 
> 
> 
> 
> 
> 
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