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Pion Form Factor in QCD at Intermediate Momentum Transfers

V. M. Braun, A. Khodjamirian, M. Maul

Abstract

We present a quantitative analysis of the electromagnetic pion form factor in the light-cone sum rule approach, including radiative corrections and higher-twist effects. The comparison to the existing data favors the asymptotic profile of the pion distribution amplitude and allows to estimate the deviation: $(\int du/u φ_π(u))/ (\int du/u φ^{\rm as}_π(u))=$ 1.1$\pm$ 0.1 at the scale 1 GeV. Special attention is payed to the precise definition and interplay of soft and hard contributions at intermediate momentum transfer, and to matching of the sum rule to the perturbative QCD prediction. We observe a strong numerical cancellation between the soft (end point) contribution and power suppressed hard contributions of higher twist, so that the total nonperturbative correction to the usual pQCD result turns out to be of order 30% for $Q^2\sim 1$ GeV$^2$.

Pion Form Factor in QCD at Intermediate Momentum Transfers

Abstract

We present a quantitative analysis of the electromagnetic pion form factor in the light-cone sum rule approach, including radiative corrections and higher-twist effects. The comparison to the existing data favors the asymptotic profile of the pion distribution amplitude and allows to estimate the deviation: 1.1 0.1 at the scale 1 GeV. Special attention is payed to the precise definition and interplay of soft and hard contributions at intermediate momentum transfer, and to matching of the sum rule to the perturbative QCD prediction. We observe a strong numerical cancellation between the soft (end point) contribution and power suppressed hard contributions of higher twist, so that the total nonperturbative correction to the usual pQCD result turns out to be of order 30% for GeV.

Paper Structure

This paper contains 13 sections, 88 equations, 12 figures.

Table of Contents

  1. Introduction
  2. The method of light-cone sum rules
  3. Radiative corrections
  4. General case
  5. Study case: asymptotic distribution amplitude
  6. The $1/Q^2$ expansion in general case
  7. Numerical estimates
  8. Higher-twist corrections
  9. Twist 4 contributions
  10. Factorizable twist 6 contributions
  11. Numerical analysis
  12. Matching with the NLO perturbative calculation
  13. Conclusions

Figures (12)

  • Figure 1: The tree-level contribution to the correlation function in Eq. (\ref{['2:cor']}).
  • Figure 2: The transverse-distance separation between the quark and the antiquark in the leading-order light-cone sum rule (\ref{['2:SR2']}) in the large $Q^2$ limit for typical values of the sum rule parameters $s_0=0.7$ GeV$^2$ and $M^2= 1.0$ GeV$^2$.
  • Figure 3: The radiative corrections to the correlation function in Eq. (\ref{['2:cor']}). Dashed lines denote virtual gluons.
  • Figure 4: The relative contributions of hard (dashed) and soft (dotted) regions in the leading-twist light-cone sum rule (\ref{['3:NLO']}) for the pion form factor (the sum: solid lines). The upper figures correspond to asymptotic and the lower figures to CZ pion distribution amplitude. The three figures from left to right correspond to the choice of factorization scale $\mu^2 = Q^2$, $\mu^2= s_0$ and $\mu^2= \bar{u} Q^2 + u M^2$, respectively.
  • Figure 5: Average momentum fraction $u$ as a function of $Q^2$ for the asymptotic (solid) and CZ (dashed) distribution amplitudes at $M^2= 1.0$ GeV$^2$. The scale is $\mu^2= (1-u)Q^2 + uM^2.$
  • ...and 7 more figures