2019 article
Doubly virtual (π0,η,η′)→γ*γ* transition form factors in the light-front quark model
Choi, H.-M., Ryu, H.-Y., & Ji, C.-R. (2019, April 19). Physical Review. D/Physical Review. D., Vol. 99.
open access via link.aps.org (publisher PDF)
We report our investigation on the doubly virtual transition form factors (TFFs) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">P</mi><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></math> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">P</mi><mo stretchy="false">→</mo><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">P</mi><mo>=</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>,</mo><mi>η</mi><mo>,</mo><msup><mrow><mi>η</mi></mrow><mrow><mo>'</mo></mrow></msup><mo stretchy="false">)</mo></mrow></math> transitions using the light-front quark model (LFQM). Performing a LF calculation in the exactly solvable manifestly covariant Bethe-Salpeter (BS) model as the first illustration, we use the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>q</mi><mn>1</mn><mo>+</mo></msubsup><mo>=</mo><mn>0</mn></math> frame and find that both LF and manifestly covariant calculations produce exactly the same results for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">P</mi><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></math>. This confirms the absence of the LF zero mode in the doubly virtual TFFs. We then map this covariant BS model to the standard LFQM using the more phenomenologically accessible Gaussian wave function provided by the LFQM analysis of meson mass spectra. For the numerical analyses of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mrow><mi mathvariant="normal">P</mi><msup><mi>γ</mi><mo>*</mo></msup></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mn>1</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi>Q</mi><mn>2</mn><mn>2</mn></msubsup><mo stretchy="false">)</mo></math>, we compare our LFQM results with the available experimental data and the perturbative QCD (pQCD) and vector meson dominance (VMD) model predictions. As <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mn>1</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi>Q</mi><mn>2</mn><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>∞</mi></math>, our LFQM result for doubly virtual TFF is consistent with the pQCD prediction, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">P</mi><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>~</mo><mn>1</mn><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></math>, while it differs greatly from the result of the VMD model, which behaves as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">P</mi><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow><mrow><mi>VMD</mi></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>~</mo><mn>1</mn><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></math>. Our LFQM prediction for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mrow><msup><mi>η</mi><mo>'</mo></msup><msup><mi>γ</mi><mo>*</mo></msup></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mn>1</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi>Q</mi><mn>2</mn><mn>2</mn></msubsup><mo stretchy="false">)</mo></math> shows an agreement with the very recent experimental data obtained from the <i>BABAR</i> Collaboration for the ranges of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>&lt;</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>&lt;</mo><mn>60</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>.