Study Of Parton Fragmentation Functions In High-energy Electron-positron Collisions Based On Constrained Neural Networks

In the study of high-energy collision physics,it is usually necessary to calculate the cross section for high-energy collision processes.According to the factorization theorem,it is known that the cross-section can be divided into the convolution of perturbative and non-perturbative contributions for the initial and final states involving hadron inclusive processes.The former corresponds to the hard scattering(parton level)process of large transverse momentum transfer,and the latter corresponds to the initial state parton distribution functions(PDFs)and the end-state parton fragmentation functions(FFs).For hard scattering processes,one can use perturbative quantum chromodynamics(p QCD)for calculations,but for the treatment of soft processes such as hadron structure,or parton hadronization,is a problem that has yet to be solved entirely.The PDFs is the physical quantity that describes the hadron structure,while the FFs is the physical quantity that describes the hadronization process.These two physical quantities are essential inputs to high-energy collisions,are universal,and have a pivotal role in current particle physics research,so it is of great importance to study the PDFs and FFs.High-energy lepton-nucleon depth inelastic scattering(DIS)is the main means to study PDFs.The PDFs was first proposed by Feynman et al.Its physical meaning describes the momentum distribution of the parton in a fast-moving hadron.The FFs is a physical quantity similar to the PDFs.Its physical meaning describes the momentum distribution of the parton fragmentation into the final state hadron.However,the current study of the FFs is less detailed than the PDFs.Therefore,in this paper,the extraction of FFs in high-energy electron-positron collisions is carried out by using the method of Physics-informed Neural Network(PINN)with experimental data.The FFs is non-perturbative,which cannot be calculated by first principles and needs to be determined by fitting the relevant experimental data to the whole.After considering the QCD correction,the FFs have scaling violations when the FFs satisfy the DGLAP evolution equation.The usual practice of extracting the FFs is to assume a parametric form of the FFs in some scale and then determine the parameters by evolving the DGLAP evolution equation to the scale of the experimental data and fitting it to the experimental data.With the advent of artificial intelligence,especially the introduction of neural networks in deep learning,they play an essential role in high-energy heavy ion collisions.The first use of neural networks to extract FFs was by the NNPDF collaborative group.The method used was to generate pseudo-data using Monte Carlo and then train it with neural networks.The advantage of this method is that it is the first time a neural network is used to extract FFs.Then it can reduce the difficulty of traditional methods due to the problem of determining too many parameters.The disadvantage is that the neural network is not bounded by prior physical knowledge,resulting in a significant training cost.Secondly,the FFs,after extracting a specific scale,require the additional computation of the DGLAP evolution equation,which makes the neural network not so ”smart.” In this paper,we apply a new technique in deep learning,PINN,a class of neural networks for solving supervised learning tasks.It can learn the distribution of training data samples like traditional neural networks and the physical laws described by mathematical equations.For example,the PINN technique is used in this work to embed the physical constraints of the DGLAP evolution equations in the neural network,which can reduce the training cost and improve the learning efficiency of the neural network.There is no need to assume the parametric form of the FFs.In addition,since the neural network has the constraint of the DGLAP evolution equation,it will make our extracted FFs of any scale satisfy the DGLAP evolution equation.Suppose the particle accelerator generates new experimental data in the future.There is no need to refit,and the FFs can be updated by simply adding the experimental data to the dataset of the neural network.The FFs extracted in this paper based on the constrained neural network(PINN)can better describe the hadron spectrum of proton-proton collisions,which can not only show the correctness of the numerical results of the FFs extracted in this paper but also the universal applicability of the FFs.In addition,we also close the loop to verify that the neural network is satisfying the DGLAP evolution equation.In the future,we expect to add the experimental data of electron-proton(ep)and proton-proton(pp)collisions to the framework of this paper so that the PDFs and FFs can be extracted as a whole.We can also apply the framework of this paper to high-energy heavy-ion collisions,where the modified parton fragmentation functions and the modified parton distribution functions in the medium can be obtained by considering the DGLAP evolution equation for the medium effect,which in turn allows us to study the jet quenching effect.