[Sane-analysis] Target Mass Effects
Whitney R. Armstrong
whit at jlab.org
Tue Oct 25 16:19:40 EDT 2016
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
--
Whitney R. Armstrong
-------------- next part --------------
Q2 : 1000.000000 GeV^2
d2_CN = 0.00078
= 0.00027 (t2) + 0.00051 (t3)
d2_CN_TMC = 0.00078
= 0.00027 (t2) + 0.00051 (t3)
2*d2_D_p_t3 = 0.00052
d2_t3 = 0.00051
d2_t3_TMC = 0.00051
d2 (g2-WW) = 0.00051
d2_TMC(g2-WW) = 0.00050
2*M23_p = 0.00078
2*M23_TMC_p = 0.00078
d3_nacht = 0.00039
d3_nacht_TMC = 0.00039
d2p_I(no TMC) = 0.00078
d2p_I(w/ TMC) = 0.00078
I_Nacht = 0.00078
= 2.0*(0.00039 + 6.0* 0.00004*y2 + 12.0* 0.00001*y2*y2 + 20.0* 0.00000*y2*y2*y2 )
I_Nacht_TMC = 0.00078
= 2.0*(0.00039 + 6.0* 0.00004*y2 + 12.0* 0.00001*y2*y2 + 20.0* 0.00000*y2*y2*y2 )
Q2 : 100.000000 GeV^2
d2_CN = 0.00180
= 0.00030 (t2) + 0.00150 (t3)
d2_CN_TMC = 0.00180
= 0.00030 (t2) + 0.00151 (t3)
2*d2_D_p_t3 = 0.00153
d2_t3 = 0.00150
d2_t3_TMC = 0.00150
d2 (g2-WW) = 0.00150
d2_TMC(g2-WW) = 0.00146
2*M23_p = 0.00179
2*M23_TMC_p = 0.00180
d3_nacht = 0.00090
d3_nacht_TMC = 0.00090
d2p_I(no TMC) = 0.00180
d2p_I(w/ TMC) = 0.00180
I_Nacht = 0.00181
= 2.0*(0.00090 + 6.0* 0.00012*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00180
= 2.0*(0.00090 + 6.0* 0.00012*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
Q2 : 50.000000 GeV^2
d2_CN = 0.00189
= 0.00030 (t2) + 0.00158 (t3)
d2_CN_TMC = 0.00190
= 0.00030 (t2) + 0.00160 (t3)
2*d2_D_p_t3 = 0.00162
d2_t3 = 0.00158
d2_t3_TMC = 0.00159
d2 (g2-WW) = 0.00158
d2_TMC(g2-WW) = 0.00150
2*M23_p = 0.00188
2*M23_TMC_p = 0.00189
d3_nacht = 0.00094
d3_nacht_TMC = 0.00094
d2p_I(no TMC) = 0.00189
d2p_I(w/ TMC) = 0.00190
I_Nacht = 0.00192
= 2.0*(0.00094 + 6.0* 0.00013*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00190
= 2.0*(0.00094 + 6.0* 0.00013*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
Q2 : 20.000000 GeV^2
d2_CN = 0.00198
= 0.00032 (t2) + 0.00167 (t3)
d2_CN_TMC = 0.00202
= 0.00032 (t2) + 0.00170 (t3)
2*d2_D_p_t3 = 0.00170
d2_t3 = 0.00167
d2_t3_TMC = 0.00168
d2 (g2-WW) = 0.00167
d2_TMC(g2-WW) = 0.00142
2*M23_p = 0.00195
2*M23_TMC_p = 0.00198
d3_nacht = 0.00097
d3_nacht_TMC = 0.00099
d2p_I(no TMC) = 0.00198
d2p_I(w/ TMC) = 0.00202
I_Nacht = 0.00206
= 2.0*(0.00097 + 6.0* 0.00014*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00202
= 2.0*(0.00099 + 6.0* 0.00014*y2 + 12.0* 0.00004*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
Q2 : 15.000000 GeV^2
d2_CN = 0.00198
= 0.00032 (t2) + 0.00166 (t3)
d2_CN_TMC = 0.00203
= 0.00032 (t2) + 0.00171 (t3)
2*d2_D_p_t3 = 0.00169
d2_t3 = 0.00166
d2_t3_TMC = 0.00168
d2 (g2-WW) = 0.00166
d2_TMC(g2-WW) = 0.00132
2*M23_p = 0.00193
2*M23_TMC_p = 0.00197
d3_nacht = 0.00096
d3_nacht_TMC = 0.00099
d2p_I(no TMC) = 0.00198
d2p_I(w/ TMC) = 0.00203
I_Nacht = 0.00208
= 2.0*(0.00096 + 6.0* 0.00014*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00203
= 2.0*(0.00099 + 6.0* 0.00015*y2 + 12.0* 0.00004*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
Q2 : 10.000000 GeV^2
d2_CN = 0.00205
= 0.00033 (t2) + 0.00172 (t3)
d2_CN_TMC = 0.00213
= 0.00033 (t2) + 0.00180 (t3)
2*d2_D_p_t3 = 0.00175
d2_t3 = 0.00172
d2_t3_TMC = 0.00175
d2 (g2-WW) = 0.00172
d2_TMC(g2-WW) = 0.00117
2*M23_p = 0.00197
2*M23_TMC_p = 0.00204
d3_nacht = 0.00098
d3_nacht_TMC = 0.00102
d2p_I(no TMC) = 0.00205
d2p_I(w/ TMC) = 0.00213
I_Nacht = 0.00221
= 2.0*(0.00098 + 6.0* 0.00014*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00212
= 2.0*(0.00102 + 6.0* 0.00016*y2 + 12.0* 0.00004*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
Q2 : 9.000000 GeV^2
d2_CN = 0.00207
= 0.00033 (t2) + 0.00173 (t3)
d2_CN_TMC = 0.00216
= 0.00033 (t2) + 0.00182 (t3)
2*d2_D_p_t3 = 0.00176
d2_t3 = 0.00173
d2_t3_TMC = 0.00177
d2 (g2-WW) = 0.00173
d2_TMC(g2-WW) = 0.00111
2*M23_p = 0.00198
2*M23_TMC_p = 0.00206
d3_nacht = 0.00099
d3_nacht_TMC = 0.00103
d2p_I(no TMC) = 0.00207
d2p_I(w/ TMC) = 0.00216
I_Nacht = 0.00225
= 2.0*(0.00099 + 6.0* 0.00014*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00215
= 2.0*(0.00103 + 6.0* 0.00016*y2 + 12.0* 0.00004*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
Q2 : 8.000000 GeV^2
d2_CN = 0.00208
= 0.00033 (t2) + 0.00175 (t3)
d2_CN_TMC = 0.00219
= 0.00034 (t2) + 0.00185 (t3)
2*d2_D_p_t3 = 0.00178
d2_t3 = 0.00175
d2_t3_TMC = 0.00179
d2 (g2-WW) = 0.00175
d2_TMC(g2-WW) = 0.00103
2*M23_p = 0.00198
2*M23_TMC_p = 0.00208
d3_nacht = 0.00099
d3_nacht_TMC = 0.00104
d2p_I(no TMC) = 0.00208
d2p_I(w/ TMC) = 0.00219
I_Nacht = 0.00230
= 2.0*(0.00099 + 6.0* 0.00014*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00218
= 2.0*(0.00104 + 6.0* 0.00016*y2 + 12.0* 0.00004*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
Q2 : 7.000000 GeV^2
d2_CN = 0.00210
= 0.00034 (t2) + 0.00177 (t3)
d2_CN_TMC = 0.00223
= 0.00034 (t2) + 0.00189 (t3)
2*d2_D_p_t3 = 0.00180
d2_t3 = 0.00177
d2_t3_TMC = 0.00181
d2 (g2-WW) = 0.00177
d2_TMC(g2-WW) = 0.00093
2*M23_p = 0.00199
2*M23_TMC_p = 0.00210
d3_nacht = 0.00099
d3_nacht_TMC = 0.00105
d2p_I(no TMC) = 0.00210
d2p_I(w/ TMC) = 0.00223
I_Nacht = 0.00236
= 2.0*(0.00099 + 6.0* 0.00014*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00221
= 2.0*(0.00105 + 6.0* 0.00016*y2 + 12.0* 0.00004*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
Q2 : 6.000000 GeV^2
d2_CN = 0.00214
= 0.00034 (t2) + 0.00180 (t3)
d2_CN_TMC = 0.00230
= 0.00035 (t2) + 0.00195 (t3)
2*d2_D_p_t3 = 0.00183
d2_t3 = 0.00180
d2_t3_TMC = 0.00186
d2 (g2-WW) = 0.00180
d2_TMC(g2-WW) = 0.00080
2*M23_p = 0.00201
2*M23_TMC_p = 0.00214
d3_nacht = 0.00100
d3_nacht_TMC = 0.00107
d2p_I(no TMC) = 0.00214
d2p_I(w/ TMC) = 0.00230
I_Nacht = 0.00246
= 2.0*(0.00100 + 6.0* 0.00014*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00227
= 2.0*(0.00107 + 6.0* 0.00017*y2 + 12.0* 0.00004*y2*y2 + 20.0* 0.00002*y2*y2*y2 )
Q2 : 5.000000 GeV^2
d2_CN = 0.00216
= 0.00034 (t2) + 0.00181 (t3)
d2_CN_TMC = 0.00236
= 0.00037 (t2) + 0.00199 (t3)
2*d2_D_p_t3 = 0.00185
d2_t3 = 0.00181
d2_t3_TMC = 0.00188
d2 (g2-WW) = 0.00181
d2_TMC(g2-WW) = 0.00057
2*M23_p = 0.00200
2*M23_TMC_p = 0.00216
d3_nacht = 0.00100
d3_nacht_TMC = 0.00108
d2p_I(no TMC) = 0.00216
d2p_I(w/ TMC) = 0.00236
I_Nacht = 0.00256
= 2.0*(0.00100 + 6.0* 0.00014*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00232
= 2.0*(0.00108 + 6.0* 0.00017*y2 + 12.0* 0.00005*y2*y2 + 20.0* 0.00002*y2*y2*y2 )
Q2 : 4.000000 GeV^2
d2_CN = 0.00222
= 0.00035 (t2) + 0.00187 (t3)
d2_CN_TMC = 0.00252
= 0.00041 (t2) + 0.00211 (t3)
2*d2_D_p_t3 = 0.00190
d2_t3 = 0.00187
d2_t3_TMC = 0.00195
d2 (g2-WW) = 0.00187
d2_TMC(g2-WW) = 0.00028
2*M23_p = 0.00202
2*M23_TMC_p = 0.00224
d3_nacht = 0.00101
d3_nacht_TMC = 0.00112
d2p_I(no TMC) = 0.00222
d2p_I(w/ TMC) = 0.00251
I_Nacht = 0.00278
= 2.0*(0.00101 + 6.0* 0.00014*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00242
= 2.0*(0.00112 + 6.0* 0.00018*y2 + 12.0* 0.00005*y2*y2 + 20.0* 0.00002*y2*y2*y2 )
Q2 : 3.000000 GeV^2
d2_CN = 0.00212
= 0.00036 (t2) + 0.00176 (t3)
d2_CN_TMC = 0.00258
= 0.00051 (t2) + 0.00207 (t3)
2*d2_D_p_t3 = 0.00179
d2_t3 = 0.00176
d2_t3_TMC = 0.00186
d2 (g2-WW) = 0.00176
d2_TMC(g2-WW) = 0.00038
2*M23_p = 0.00187
2*M23_TMC_p = 0.00218
d3_nacht = 0.00093
d3_nacht_TMC = 0.00109
d2p_I(no TMC) = 0.00211
d2p_I(w/ TMC) = 0.00257
I_Nacht = 0.00295
= 2.0*(0.00093 + 6.0* 0.00012*y2 + 12.0* 0.00003*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00236
= 2.0*(0.00109 + 6.0* 0.00018*y2 + 12.0* 0.00005*y2*y2 + 20.0* 0.00002*y2*y2*y2 )
Q2 : 2.000000 GeV^2
d2_CN = 0.00211
= 0.00037 (t2) + 0.00174 (t3)
d2_CN_TMC = 0.00308
= 0.00087 (t2) + 0.00222 (t3)
2*d2_D_p_t3 = 0.00177
d2_t3 = 0.00174
d2_t3_TMC = 0.00188
d2 (g2-WW) = 0.00174
d2_TMC(g2-WW) = 0.00126
2*M23_p = 0.00175
2*M23_TMC_p = 0.00233
d3_nacht = 0.00088
d3_nacht_TMC = 0.00117
d2p_I(no TMC) = 0.00210
d2p_I(w/ TMC) = 0.00307
I_Nacht = 0.00378
= 2.0*(0.00088 + 6.0* 0.00011*y2 + 12.0* 0.00002*y2*y2 + 20.0* 0.00001*y2*y2*y2 )
I_Nacht_TMC = 0.00245
= 2.0*(0.00117 + 6.0* 0.00021*y2 + 12.0* 0.00006*y2*y2 + 20.0* 0.00002*y2*y2*y2 )
Q2 : 1.000000 GeV^2
d2_CN = 0.00246
= 0.00040 (t2) + 0.00206 (t3)
d2_CN_TMC = 0.00619
= 0.00306 (t2) + 0.00313 (t3)
2*d2_D_p_t3 = 0.00210
d2_t3 = 0.00206
d2_t3_TMC = 0.00230
d2 (g2-WW) = 0.00206
d2_TMC(g2-WW) = 0.00132
2*M23_p = 0.00174
2*M23_TMC_p = 0.00341
d3_nacht = 0.00087
d3_nacht_TMC = 0.00171
d2p_I(no TMC) = 0.00244
d2p_I(w/ TMC) = 0.00615
I_Nacht = 0.00850
= 2.0*(0.00087 + 6.0* 0.00009*y2 + 12.0* 0.00002*y2*y2 + 20.0* 0.00000*y2*y2*y2 )
I_Nacht_TMC = 0.00310
= 2.0*(0.00170 + 6.0* 0.00030*y2 + 12.0* 0.00007*y2*y2 + 20.0* 0.00002*y2*y2*y2 )
I_CN/I_Nacht = 1.98347
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