Studying Deeply Virtual Compton Scattering with Neural Networks

Deeply virtual Compton Scattering (DVCS) is recognized as the theoretically cleanest process for accessing generalized parton distributions (GPDs) [1][2][3], which describe the three-dimensional structure of nucleon in terms of partonic degrees of freedom. Determination of GPDs, beside improving our general understanding of QCD dynamics, allows to address important questions such as the partonic decomposition of the nucleon spin [4] and characterization of multiple-hard reactions in proton-proton collisions at LHC collider [5]. Concerning the latter, GPD-describable non-trivial transversal structure of proton, such as the correlation between parton's longitudinal momentum fraction and its transversal distance is already finding its way in popular event generators, such as PYTHIA [6].

Similarly to extraction of normal parton distribution functions (PDFs), extraction of GPDs can be performed by global model or local fits to available data [7][8][9][10][11][12]. However, compared to global PDF fits, extracting GPDs from data is a much more intricate task and model ambiguities are much larger. Due to the facts that GPDs cannot be fully constrained from data and that they depend at the input scale on three variables, the space of possible functions, although restricted by GPD constraints, is huge. As a result, the theoretical systematic error induced by the choice of the fitting model is much more serious than in the PDF case, where the model functions depend at the input scale only on one variable, namely, the longitudinal momentum fraction x.

Here we report on some results obtained using alternative approach [13], in which neural networks are used in place of specific models. This essentially eliminates the problem of model dependence and, as an additional advantage, facilitates a convenient method to propagate uncertainties from experimental measurements to the final result. Our approach is mostly similar to the one already employed for F 2 structure function and PDF extraction [14][15][16] and will be shortly described in the next section. To reduce the mathematical complexity of the problem we have fitted not the GPDs itself, but the dominant Compton form factor (CFF) H(x B , t), depending on Bjorken variable x B and momentum transfer t. At leading order, the imaginary part of this CFF is related to the corresponding GPD H(x, ξ, t) at the cross-over line x = ξ:

so knowledge of CFF H provides us with direct information about the proton structure. The neural network type used in this work, known as multilayer perceptron, is a mathematical structure consisting of a number of interconnected “neurons” organized in several layers. It is schematically shown on Figure 1, where each blob symbolizes a single neuron. Each neuron has several inputs and one output. The value at the output is given as a function f ( j w j x j ) of a sum of values at inputs x 1 , x 2 , • • • , each weighted by a certain number w j . For activation function f (x) we employed logistic sigmoid function

for neurons in inner (“hidden”) layer, while for input and output layers we used the identity function.

By iterating over the following steps the network is trained, i.e., it “learns” how to describe a certain set of data points:

(i) Kinematic values (two in our case: x B and t) of the first data point are presented to two input-layer neurons (ii) These values are then propagated through the network according to values of weights w j .

In the first iteration, weights are set to some random value.

(iii) As a result, the network produces some resulting values of output in its output-layer neurons.

Here we have two: Im H and Re H. Obviously, after the first iteration, these will be some random functions of the input kinematic values: Im H(x B , t) and Re H(x B , t). (iv) Using these values of CFF(s), the observable corresponding to the first data point is calculated and it is compared to actually measured value, with squared error used for building the standard χ2 function. (v) The obtained error is then used to modify the network: It is propagated backwards through the layers of the network and each weight is adjusted such that this error is decreased. (vi) This procedure is then repeated with the next data point, until the whole training set is exhausted.

This sequence of steps 2 is repeated until the network is capable to describe experimental data with a sufficient accuracy. To guard against overfitting the data (“fitting to the noise”), one (randomly chosen) subset of data is not used for training but only for monitoring the progress and stopping the training when error of network description of this data starts to increase significantly. This ensures that resulting neural network represents a function which is not too complex and which is thus expected to provide a reasonable estimate of the actual underlying physical law.

To propagate experimental uncertainties into the final result we used the “Monte Carlo” method [17] where ne

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