Research


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THESIS TITLE: Transverse structure of proton and azimuthal spin asymmetries in the SIDIS processes

My research works are focussed on the investigation of non-perturbative transverse structure of proton e.g., transverse momentum dependent parton distribution(TMDs), single spin asymmetries, spin-orbital correlations etc. The TMDs provide a three dimensional structure with the probability of finding a quark with longitudinal momentum fraction $x$ and transverse momentum $\bf{p}_\perp$ inside a proton. To investigate these distribution functions we have constructed the Light-front quark-diquark model (LFQDM) where the wave functions are adopted from soft-wall AdS/QCD. In this model, we have presented the single spin asymmetries (SSAs) for SIDIS e.g., Collins asymmetry, Sivers asymmetry, Boer-Mulders asymmetry etc. as well as the double spin asymmetries (DSAs) which involve the leading twist TMDs (T-even and T-odd both). The model results of asymmetries are compared with the COMPASS and HERMES data for $\pi^+$ and  $\pi^-$ channels.  
At first we have developed a quark-diquark model for proton  where the light front wave functions are constructed from the soft-wall AdS/QCD predictions. The model has $SU(4)$ spin-flavor structure and includes the contributions from scalar ($S=0$) and axial vector ($S=1$)  diquarks. The parameters of the model are determined from fitting of the Dirac and Pauli form factor data. The model is consistent with the quark counting rule and Drell-Yan-West relation. The model result for Sachs form factors ratios are presented. The electric and magnetic charge radius for nucleons are found to be in good agreement with the experimental measurement. The evolution of  PDFs are simulated by introducing scale dependent parameters in the unpolarised PDFs. From the fitting of the unpolarised PDFs at different scales, we have given the explicit  scale evolution of each parameter in the model. The helicity and transversity PDFs are calculated as predictions of the model and compared with the phenomenological extractions at the measured scale. In this model the PDFs are shown to satisfy Soffer bound which is a model independent inequality. Our model reproduces the experimental values of axial and tensor charges quite well. 
Then, the T-even TMDs are discussed in this light-front quark-diquark model of the proton. Three dimensional variations of all the leading twist T-even TMDs are shown in Chapter-\ref{Ch4_TMD}. Our model results for transverse moments of the TMDs are compared with the other models. The pretzelocity distributions are found to be negative for $u$ and positive for $d$ quark as shown in other model calculations. But, phenomenologically, opposite polarities are observed with large error corridors.  Experimental data with improved accuracy are required to settle the issue. The TMDs are found to satisfy different inequalities. Similar inequalities are also observed in other models  and are generic to diquark models. The transversity TMD is found to have an upper bound given by the helicity and unpolarised TMDs, we call it Soffer bound for TMDs in this models. In  phenomenological models,  the $\bfp$ dependency is assumed as Gaussian along with the corresponding PDF. In our model, the  $x- \bfp^2$ factorization is not apparent in TMDs. But,  interestingly in our model, the numerical analysis agrees with the Gaussian ansatz considered in phenomenology. First moments in $x$  of TMDs $f_1^\nu(x, \bfp^2)$ and $g_{1T}^\nu(x, \bfp^2)$ are related with the quark densities for different polarizations of the proton and quarks inside it. The quark densities for unpolarized proton are axially symmetric for both $u$ and $d$ quarks. For transversely polarized proton the quark densities are found to be  non-spherical which is consistent with the Lattice QCD results. On $p_\perp$-integration, $f_1^\nu(x, \bfp^2),~ h_1^\nu(x, \bfp^2)$ and $g_{1L}^\nu(x,\bfp^2)$ give the PDFs $f_1^\nu(x),~h_1^\nu(x)$ and $g_1^\nu(x)$ while the other TMDs don't have such collinear interpretations. The TMDs in our model show that certain ratios like $g_{1T}^\nu(x,\bfp^2)/h_{1L}^{\perp \nu}(x,\bfp^2)$ and $h_{1T}^\nu(x,\bfp^2)/f_1^\nu(x,\bfp^2)$ are independent of the evolution scale $\mu$. The first (second) ratio is found to be negative (positive) for $u$ quark and positive (negative) for $d$ quark. It will be interesting to check these results in other models. We have also presented the transverse shape of the  proton which involves transversity and pretzelocity TMDs. For transversely polarized proton, the pretzelosity distribution causes a distortion in the spherical shape for nonzero transverse momentum.
We have presented the model results for both single and double spin asymmetries associated with T-even TMDs in a light front quark-diquark model of the proton in SIDIS processes for both $\pi^+$ and $\pi^-$ channels. The results are compared with COMPASS and HERMES data. The scale dependence of the asymmetries come through the scale evolutions of the TMDs and FFs. We have considered the scale evolution of the unpolarized TMD only  to compare our results with experimental data at different energy scales. We have also compared  the results with different approximate  evolution schemes for polarized TMDs. The single spin asymmetries e.g., $A^{\sin(\phi_h+\phi_S)}_{UT}, A^{\sin(3\phi_h-\phi_S)}_{UT}$ and $ A^{\sin(2\phi_h)}_{UL}$ involve $h^\nu_1, h^{\perp \nu}_{1T}$ and $h^{\perp \nu}_{1T}$ respectively. $A^{\sin(\phi_h)}_{UL}$ and $A^{\sin(2 \phi_h)}_{UL}$ are generated by the same TMD and FF. Our model predictions for  Collin asymmetry $A^{\sin(\phi_h+\phi_S)}_{UT}$ show good agreement with both HERMES and COMPASS data. The  amplitude of the asymmetry $A^{\sin(3\phi_h-\phi_S)}_{UT}$ in our model is found to be suppressed by powers of $P_{h\perp}/M$  compared to the other SSAs and expected to be very small. Experimental data also show that the  average amplitude of the asymmetry $A^{\sin(3\phi_h-\phi_S)}_{UT}$ to be approximately zero. We have also predicted an asymmetry for the future EIC experiment. Our model predicts sizeable Collins asymmetry $A_{UT}^{\sin(\phi_h+\phi_S)}$ for both $\pi^+$  and $\pi^-$ channels for the EIC experiments. 
The double spin asymmetry $A_1^p$ in DIS depends on the PDFs  rather than  TMDs. Our model predictions for $A_1^p$ show excellent agreement with the data.  When the lepton beam is longitudinally polarized but the proton is transversely polarized, both  $A_{LT}^{\cos(\phi_h-\phi_S)}$ and $A_{LT}^{\cos(2\phi_h-\phi_S)}$ in the model are consistent with the experimental data and are found to be almost zero. But, the DSA when both proton  and lepton beams are longitudinally polarized, $A_{LL}$ is quite large for both $\pi^+$ and $\pi^-$ channels which is also predicted in our model. We have explored different relations among the SSAs and DSAs and found an inequality similar to Soffer bound for PDFs. It will be interesting to see if similar relations are also found in other models.
Then we have extended our model calculation to investigate the $T$-odd TMDs namely, the Sivers and Boer-Mulders functions and the spin asymmetries in SIDIS associated with these functions. It is well known that the  final state interaction is responsible to produce the required complex phase in the amplitude which gives rise to the Sivers asymmetries. We have modelled the light-front wave functions to incorporate the effects of FSI. This is done by extending the wave functions in the quark-diquark model to have complex phases consistent with the SIDIS amplitudes.  The complex phases in the light-front wave functions produce the Sivers and Boer-Mulders functions. Both Sivers and Boer Mulders functions and their moments are compared with other models and phenomenological fits. The Sivers asymmetry $A_{UT}^{\sin(\phi_h-\phi_S)}$ for $\pi^+$ channel is found to be a bit smaller than the experimental data; better agreements are observed for Boer-Mulders asymmetry $A_{UU}^{\cos(2\phi_h)}$ for both $\pi^+$ and $\pi^-$ channels. Sivers and Boer-Mulders functions help us to understand the spin structure of the proton at the  parton level. Due to Sivers effect the spin density of an unpolarised quark in a transversely polarized proton is found to be asymmetric in the perpendicular direction to the nuclear spin. The distortions due to Sivers effect in our model for both  $u$ and $d$ quarks are consistent with the results found in other models and lattice QCD. Since the Sivers function is negative for $u$ and positive for $d$ quark, the distortion for $u$ quark  is in opposite direction of the $d$ quark. Similarly, Boer-Mulders function produces the distortion in the spin density of a transversely polarized quark in a transversely polarized proton. Since Boer-Mulders function has the same sign for both $u$ and $d$ quarks, the distortions in the spin densities are also in the same direction. Sivers function integrated over the transverse momentum is related to the anomalous magnetic moment through the lensing function. Our model predicts that the lensing function should go as $(1-x)^{-1}$.
The most important and final goal is to understand the three-dimensional structure of nucleon and its spin and angular momentum distributions. Since model independent study of the these objects are not yet available, different predictions and relations obtained in different models will put constraints on the distributions which help to find more realistic model in future.

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