[Sane-analysis] Target Mass Effects

[Oscar Rondon]

Hi Whit,

Comments follow.

Cheers,

Oscar


Whitney R. Armstrong wrote:
> Hi Oscar,
> 
> Thanks for the reply.
> 
> I have a few comments and questions below
> 
> On Tue, Oct 25, 2016 at 07:59:43PM -0400, Oscar Rondon wrote:
>> Hi Whit,
>>
>> Your conclusion is consistent with Dong's use of experimental
>> parameterizations of SF's (polarized and unpolarized) in his section III
>> to estimate the TMCs, and his discussion after his eq.(20). This is what
>> I also suggested in my tech. note on d_2.
> 
> This statement is true only for M=0. That is if no target mass effects
> are included.

I'm referring to his comparison of C-N vs Nacthmann as a function of
Q^2, for which the experimental parameterizations he uses come directly
from empirical fits to the data (E143 and E155, his refs.
[16],{18},[20]), so they should have all the twists, both the dynamic
HTs a_0, d_2, f_2, and the TMCs or kinematic twists: a_2, a_4, d_4, d_6, ...

By the way I used the indices for the overbar notation, not the
Nachtmann notation: a_0^overbar = Nachtmann's a_1, d_2^overbar = d_3, etc.

>> We also need to keep in mind that, although the first moment of g_1
>> mixes TMCs and HTs (a_2, d_2, f_2 at order M^2/Q^2, and a_4, d_4, etc.
>> at the next order), see Dong's eq.(3), however, the third moment
>> has only TMCs, but no HTs, see eq.(4), or eq.(34) of the Ehrnsperger et
>> al. paper I mentioned in a previous message
>> http://dx.doi.org/10.1016/0370-2693(94)91244-0
>>
>> The integral of x^2 g_1 involves no d_n's or f_n's, only a_2 and
>> higher a_n>2 (the latter are in the O(M^2/Q^2) term on the rhs): If
>> there were any d_2 in the rhs of the x^2 g_1 integral then it would not
>> be possible to isolate d_2 by combining the x^2 g_1 and x^2 g_2
>> integrals.
>>
>> This would mean to me that we would not need to try restoring all the
>> twists (d's, f's etc.) to our g_1 SF calculated from PDFs, only the TMCs
>> would need to be included to get d_2 from Nachtmann moments.
> 
> This is where I get confused. I think g_1(M=Mp) needs all the twists
> restored to properly handle them. However, and here is where I think the
> term "TMC" adds to the confusion, the twist-3 TMC might not play a
> significant role. But I think in the context above, you are using "TMCs"
> to refer to the "Twist-2 TMCs", ie, the a_n?
> 

Looking at the derivation of Dong's eq.(10), which only contains d's, no
a's (Dong's d_3 = d_2^overbar, etc.), the coefficients of the twist-3
d_n>3 terms in that equation would not be the same if g_1 from eq.(6)
only had a's and no d's, which would be the case if only twist-2 TMCs
would be added to g_1.

So the d_n>3 TMCs in g_1 are needed to cancel (or add) to those in the
experimental A_2 we want to use. Fortunately, the factor of 3 in front
of A_2 in the integrand suppresses the contribution of those TMCs if we
were to leave them out.

Including both a's and d's TMCs in g_1 could be done if the starting g_1
is calculated using PDFs which don't have any dynamic HT twists removed,
unlike the LSS PDFs, for example. The TMCs can then be added using
Bluemlein-Tkabladze's eq.(114), which should apply to the LT and any HTs.

The option of explicitly adding twist-3 TMCs using their eq.(127) is not
feasible because the needed massless twist-3 D's of eq. (126) don't exist.

> I will quantify this difference and try to see the impact of using the
> Nachtmann  moment in your note (where A2 and g1_model are used).
> 
> Right now, at Q2 = 1 GeV^2,
> if I use g1(M=Mp) = g1_Twist2(M=Mp) in the Nachtmann moment M_2^3, that
> is I leave out the twist-3 TMC to g1 but leave in the twist-2 and
> twist-3 TMCs for g2) I get an error of about 10%.
> At Q2 = 5 GeV^2 the error is about 5%.
> 
> I suspect using the Nachtmann moment prescription you describe in the
> tech note, this error will be smaller.
> 
> 

10% seems a bit too large a contribution to g_1(1 GeV^2) of terms like
d_5 and higher, if by "leaving out the twist-3 TMCs" you mean you don't
include the d's of Dong's eq.(6). I would think that the
a_3 = a_2^overbar term would dominate the third moment. What are you
using for the d's?

>> Also, it's important to keep track of the notation used in the
>> literature, where the bar over the matrix element symbols means that
>> they represent effective elements, with Wilson coefficients E_2^n
>> absorbed in the definition. Moreover, the conventional sub-indexing like
>> a_2, d_2, etc., based on the power of x in the integrals, is one unit
>> less than the Nachtmann moments index: d_2^bar = d_3 E_2^3
> 
> I agree. This can be confusing.
> 
>> A discussion of these and related items, such as NLO corrections (that
>> are in the implicit Wilson coefficients) is available in the RSS tech.
>> note on moments at this link:
>>
>> http://twist.phys.virginia.edu/~or/rss/index.html
> 
> Thanks. This is very useful.
> 
> By the way, how do you calculate RSS g_2^{WW} (Figure 1)? Is this just
> using the measured RSS g_1?

Since RSS was the first model independent and precision measurement of
g_1 in the resonances, we just integrated our g_1 model fit at each data
point from the x of the point to the pion threshold (g_1 is zero above
the threshold) for free protons or to x = 1 for deuterons. The
integration was at the fixed Q^2 of each point, too, since the fit has
some Q^2 dependence.

> 
> Cheers,
> Whit
> 
> 
>>
>> Cheers,
>>
>> Oscar
>>
>>
>>
>> Whitney R. Armstrong wrote:
>>> Hi Everyone,
>>>
>>> After much debugging, I think I understand the TMCs a little better.
>>>
>>> First, when Y.B.Dong writes (http://inspirehep.net/record/776969):
>>>
>>> "If the spin structure functions are replaced by the target mass
>>> corrected ones, according to eq 6 and 8, one can easily expand the two
>>> Nachtmann momentus up to order M^6/Q^6. The results are:
>>> M_1^n = a_n   and M_2^n = d_n."
>>>
>>> The "corrected ones" means including the target mass effects, not
>>> removing it. In fact it *must* include the target mass effects otherwise
>>> the result is wrong (as I have confirmed from calculation).
>>>
>>> Furthermore if the d2 CN moment is to be calculated from the Nachtmann
>>> moments via Dong's Eq.10, that is up to and including y^6 terms, then at
>>> Q2=2 GeV^2, the error is about 25% while at Q2=1 GeV^2 it becomes nearly
>>> 100%.
>>>
>>> I have attached the output of various calculations starting from very
>>> high Q2 going down to 1 GeV^2. At high Q2 there is little difference as
>>> expected. I used the JAM15 (https://github.com/JeffersonLab/JAMLIB) pdfs
>>> and twist-3 distributions.
>>>
>>> The quantities are:
>>>
>>>    d2_CN        = x^2(2g_1+3g_2) with M=0
>>>                 = (twist-2 part) + (twist-3 part)
>>>    d2_CN_TMC    = x^2(2g_1+3g_2) with M=M_p
>>>                 = (twist-2 part) + (twist-3 part)
>>>   2*d2_D_p_t3   = third moment of twist-3 distribution D_p
>>>   d2_t3         = 3x^2 g_2^{twist-3} with M=0
>>>   d2_t3_TMC     = 3x^2 g_2^{twist-3} with M=Mp   d2    (g2-WW) = 3x^2 [
>>> (2g_1+3g_2) -g_2^{WW} ] with M=0
>>>   d2_TMC(g2-WW) = 3x^2 [ (2g_1+3g_2) -g_2^{WW} ] with M=Mp
>>>   2*M23_p       = 2 times Nachtmann Moment (Y.B.Dong Eq.13 with M=0)
>>>   2*M23_TMC_p   = 2 times Nachtmann Moment (Y.B.Dong Eq.13 with M=Mp)
>>>   d3_nacht      = Nachtmann Moment (Y.B.Dong Eq.13 with M=0)
>>>   d3_nacht_TMC  = Nachtmann Moment (Y.B.Dong Eq.13 with M=Mp)
>>>   d2p_I(no TMC) = d2_CN
>>>   d2p_I(w/ TMC) = d2_CN_TMC
>>>   I_Nacht       = Y.B.Dong Eq. 10 with M=0
>>>                 = 2.0*(M_2^3 + 6 M_2^5 y^2 + 12 M_2^7 y^4 + 20 M_2^9 y^6
>>> ) with M=0
>>>   I_Nacht_TMC   = Y.B.Dong Eq. 10 with M=Mp
>>>                 = 2.0*(M_2^3 + 6 M_2^5 y^2 + 12 M_2^7 y^4 + 20 M_2^9 y^6
>>> ) with M=Mp
>>>
>>> Note that I_Nacht_TMC = d2_CN_TMC for most Q2 except below about 2. It
>>> also worth noting that the JAM15 Q2=3 GeV^2 result for
>>> I_Nacht_TMC/M23_TMC_p also shows a sudden decrease relative to the
>>> increasing trend as Q3 goes from high to low.
>>>
>>> To conclude, I think if we want to use world data on A_1/g_1 then we
>>> have to always include the target mass effects when using various pdf
>>> models.
>>>
>>> Cheers,
>>> Whit
>>>
>>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
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