aaccardi
commented
2 years ago Nice results. Here is what I read from the plots:
- The $\Omega(\xi_q)$ fit is pretty good, as you said
- For the $\Omega(\xi) + HP(\xi)/Q^2$ fit:
- the observable is not fitted at large $x_B$: the twist-4 fitted term cannot fake the $1+m_q^2/Q^2$ kinematic shift in the $\xi_q$ variable (the observable is not fitted at large $x_B$. Teh reason is as you stated: we would need a $1/Q^4$ term, as well
- BUT, the quark PDF is well fitted, even though with larger uncertainties b/c the PDF parameters correlate a bit with the HT parameters.
- The reason for the good fit is very likely that we have data up to large enough $Q^2$ to fix $\Phi$ with minimal quark mass corrections
It would be interesting to try a fit with only $Q^2 < 30 \text{\ GeV}^2$ or so (may be less than that). Then I would expect a poorer fit also for $\Phi$, on top of a poor fit for $\Omega$.
Summary of Results as of Tuesday, Aug 16th
Fitting the Observables
Data was generated via $F_T^{\bar{x}}(xB,Q^2) = q(\bar{x})|{\bar{x}=\xi_q}$ across the range of $0 < x_B < 1$ and $2 < Q^2 < 66$
A fitquark pdf, qPDF, $\Phi(x)$, is extracted via the fit observable's structure function, OST, $\Omega^{\bar{x}}(x_B,Q^2)$, which is fit to the data: i) $\Omega_1^{\xi_q}(xB,Q^2) = \Phi(\bar{x})| {\bar{x}=\xi_q}$ ii) $\Omega_2^{x}(x_B,Q^2) = \Phi(x)+\frac{1}{Q^2}H(x)$ iii) $\Omega_2^{\xi}(x_B,Q^2) = \Phi(\xi)+\frac{1}{Q^2}H(\xi)$ iv) $\Omega_3^{x}(x_B,Q^2) = \Phi(x)+\frac{1}{Q^2}HP(x)+\frac{1}{Q^4}HHP(x)$
where:
Analysis
At the needed precision to reach high $x_B$ we have found that our provided function $\Phi(x)$ is able to properly fit the qPDF as evidenced by #2i) above. The graph below shows the fits, $\Phi(x)$ and $\Omega(x_B,Q^2)$ verses the quark pdf, $q(x)$ and the observable structure function, $F^{\xi}(xB,Q^2)$.
Top: The difference between the real structure functions and the fits to their data, $q(x)-\Phi(x)$ and $F_T^{\xi}(x_B,Q^2) -\Omega_1^{\xi_q}(x_B,Q^2)$. Middle: The "Residual Distribution Function", RDF between the real vs fit *qPDF $\frac{q-\Phi}{\Delta q}$. Bottom: The RDF between the real structure functions and the fits to their data, $\frac{F_T -\Omega}{\Delta FT}$.
However, when we try to isolate the $Q^2$, with the statistical precision to reach $x_B>0.9$, from our fit qPDF by adding a $Q^2$ dependent "Higher Power" correction term, a single correction term is not able to properly fit the observable and in turn the qPDF suffers. The graph below shows the fits, $\Phi(x)$ and $\Omega(x_B,Q^2)$ verses the quark pdf, $q(x)$ and the observable structure function, $F^{\xi}(xB,Q^2)$.
Top: The difference between the real structure functions and the fits to their data, $q(x)-\Phi(x)$ and $F_T^{\xi}(x_B,Q^2) -\Omega_1^{\xi_q}(x_B,Q^2)$. Middle: The "Residual Distribution Function", RDF between the real vs fit *qPDF $\frac{q-\Phi}{\Delta q}$. Bottom: The RDF between the real structure functions and the fits to their data, $\frac{F_T -\Omega}{\Delta FT}$.
A quick analysis of $F_T^{\xi_q}(x_B,Q^2)$ analytically confirmed what we found. The graph below shows the RDF of the Taylor Series expansion of $F_T^{\xi_q}(x_B,Q^2)$ about $Q^2$ compared to $F_T^{\xi_q}(x_B,Q^{2})$.
The RDF of the observable structure function compared to its Taylor Expansion of up to 1 (no $Q^2$ dependance), 2 ( $Q^2$ dependance), and 3 ( $Q^4$ dependance) terms. The RDF of $q(x)$ is also shown, however, as expected it is the same as the first term in the Taylor Series.
Conclusions
For large $x_B$ we are unable "fake" to the kinematics of the observables via a single $Q^2$ dependent correction term. The statistical power required to reach the high-x requires too many parameters to accurately fit the observable at low $Q^2$ values with such large precision.
Moving Forward