[Sane-analysis] Target Mass Effects

Wed Oct 26 17:24:48 EDT 2016

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. 

>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?

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.

>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?

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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-- 
Whitney R. Armstrong