[Sane-analysis] Target Mass Effects

[Oscar Rondon]

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.

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.

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

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

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
> 
>