{'fraction_non_alphanumeric': 0.0271138625499569, 'fraction_numerical': 0.007209466342763106, 'mean_word_length': 4.8273972602739725, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Continuous simulation of hypothetical physics processes with multiple free parameters

We propose a new approach to simulate hypothetical physics processes which are defined by multiple free parameters. Compared to the conventional grid-scan approach, the new method can produce accurate estimations of the detector acceptance and signal event yields continuously over the parameter space with fewer simulation events. The performance of this method is illustrated with two realistic cases.


Introduction
As data collected by experiments at the Large Hadron Collider (LHC) is rapidly increasing, many efforts have been devoted to searches for various Beyond the Standard Model (BSM) physics at the energy frontier. Many BSM models are defined by multiple free parameters, such as the masses of hypothetical particles and their coupling constants, which are to be constrained by data.
The various cross sections and branching ratios, as well as detector acceptance and signal event yields, depend on the parameters. Consequently, the search results for BSM physics are usually expressed in terms of excluded regions in the multi-dimensional parameter space.
In most BSM searches, the hypothetical signal processes are studied in the "grid-scan" approach, i.e. Monte-Carlo (MC) samples are simulated with specific values of the parameters, corresponding to discrete points on a grid in the parameter space. Each MC sample generally contains a considerable number of events for adequate statistical precision. The signal yields are estimated from MC results of each sample and interpolated between these grid points. However, to produce many samples with full-detector simulation is computationally expensive, especially if sufficient granularity in the multivariate parameter space is desired. It suffers from the so-called "curse of dimensionality", i.e. the number of grid points needed increases exponentially with the number of free parameters. Usually, the grid-scanned parameter space is limited to two dimensions at a time, keeping the other parameters fixed. Moreover, the grid is often constructed sparsely. As a result, the search limits derived are subject to systematic uncertainties from the interpolation which have to be estimated.
In this paper, we propose a new method to achieve continuous prediction of signal yields over the entire parameter space, under the assumption that the acceptance is a smooth function of the parameters. With such an assumption, the simulated events on neighboring space points are no longer treated independently, therefore fewer events are needed in total. This paper is organized as follows. In section 2 we describe the technical procedure of this method. Then we illustrate the performance of the approach in section 3, with a 2-D example which simulates the production of the hypothetical right-handed W boson and heavy Majorana neutrino in the Left-Right Symmetric Model (LRSM). Another 4-D example of a simplified SuperSymmetric (SUSY) model is demonstrated in section 4, to highlight the advantage in higher-dimensional parameter space. In section 5, we discuss potential improvements that can be implemented in various special scenarios.

Continuous simulation
In order to estimate the signal yield, two parts of information are needed from MC simulation. One is the production rate of the considered process (denoted as σ), which includes the cross-section and the branching ratio. This can normally be obtained at the event generation level and is relatively lightweight from a computational point of view. The more relevant part is the detector acceptance ǫ, namely the probability for the events to be successfully reconstructed and selected. This is related to the detector coverage and the reconstruction efficiency, for which many MC events with parton showers and full detector-simulation are needed for a reliable estimation. The signal yield (N) is the product of these two components and the integrated luminosity (L),  [5,6]. This functionality as a non-parametric regression tool has also been exploited by physicists in recent years [7][8][9].
Our usage of BNN to fit the acceptance distribution is also a regression application. It differs from previous regression applications in that the output of our function ǫ(x) is a probability taking values between 0 and 1, and that our input samples are MC events with Boolean target values to indicate whether the event is selected or not. The grid-scan approach needs to accumulate many MC events at each single point before it can evaluate ǫ to a good precision.
Instead, the BNN algorithm we used here [10] estimates the whole probability distribution in an unbinned manner. It can achieve similar precision to the grid-scan method with much fewer MC events.
This unbinned regression is carried out as follows. For each generated MC event, the corresponding parameters of the model x are passed to the BNN as the input variables. In addition, a target value t of 1 or 0 is assigned to each event, depending on whether it survived the selection cuts. Then the BNN is fitted (or "trained") to have its output y(x) approximating the probability distribution ǫ(x) by minimizing the cross-entropy cost function where k loops over all simulated events. The cross-entropy function is derived from Bernoulli likelihood and naturally attributes the meaning of success probability to the output, y.
The trained BNN can then be used to estimate ǫ(x). Given any point x in the parameter space, the BNN will output the estimated value of the acceptance.
Multiplying it to the production rate σ(x) estimated from MC generators and the integrated luminosity, we obtain the corresponding signal yield at any x.

4
The Bayesian implementation of the NN provides not only the fitted value but also the estimation of the uncertainty for each predicted ǫ(x) [10]. This uncertainty estimation reflects the statistical fluctuation of the input events as well as the goodness of BNN fitting. It is based on the same training sample so that no additional simulation and computing resources are needed.

Demo: 2-D Left-Right Symmetric Model
In the following we will demonstrate the application of the continuous MC  only the cross-section of the process, but also the kinematics used in the event selection. In this example, we considered the parameter space region with W R mass between 0.5 TeV and 1.5 TeV, which is being actively studied at the LHC experiments. The N R mass was limited to be smaller than the W R mass in order to accommodate the concerned decay mode.
The Pythia generator [12] is used to simulate this process. For simplicity, detector effects are not included in this demonstration. Only the acceptance ǫ(x) at the generator level is studied. From the above-mentioned mass ranges we picked 100,000 points stochastically, which are dense enough to cover the 2D parameter space. Then 10 MC events were generated at each point so that we have a total of 1 million events simulated for this demonstration of continuous MC. The reason for generating multiple events at each point is to reduce the overhead of generator re-initialization and the computing load of BNN training.
However, we still maintain a relatively low number of events at each point in order to spread over as many space points as possible.
The event selection we assumed requires at least two leptons (e,µ) with transverse momentum greater than 20 GeV and pseudo-rapidity η between -2.5 and 2.5. A 100% fiducial efficiency is assumed for lepton identification. In addition, a common isolation cut for background suppression is applied, requiring the transverse energy flow of all interactive particles inside a cone of size ∆R = (∆η) 2 + (∆φ) 2 = 0.2 around the lepton to be less than 10% of the lepton's transverse momentum.
The acceptance of this event selection is fitted by a BNN with forty neurons in the first hidden layer and ten neurons in the second hidden layer. This topology is arbitrarily chosen, merely to provide redundant complexity within the computing budget. The BNN can automatically constrain the complexity and avoid over-fitting by a regulator mechanism, which is tuned based on the Bayesian evidence framework [1,10]. The fitted distribution of ǫ(x) is given in To test the performance, we also simulated another set of discrete samples by We then compared the ǫ(x) values fitted by the BNN at these grid points to

Demo: 4-D Simplified SuperSymmetric Model
In this section, we will illustrate the performance of the continuous simulation in a 4-D scenario. This is gluino pair production with a two-step cascade decay in one of the Simplified SuperSymmetric Models [13,14]. In this model, the gluino (g) can decay to the Lightest Supersymmetric Particle (LSP), assumed to be theχ The Herwig++ generator [15] was used to simulate this process. For the continuous MC sample, a total number of 1 million events were generated at 100k space points, stochastically distributed in the parameter space region. For simplicity, the study is again limited to generator level acceptance. We looked for final states with at least two leptons (e,µ) with transverse momentum greater than 20 GeV and with pseudo-rapidity η between -2.5 and 2.5. A 100% fiducial efficiency was assumed for lepton identification. In addition, the transverse energy flow of all interactive particles around the lepton within a ∆R = 0.4 cone was required to be below 5 GeV. As a typical SUSY signature, we also required the visible energy imbalance in the transverse direction to be greater than 50 GeV. The acceptance of this di-lepton plus missing transverse energy selection was then parameterized by a BNN of the same topology as in the previous example. The grid-scan suffers more uncertainties of interpolation in a 4-D space.
With the same measurements as in section 3, we estimated the interpolation uncertainties on each of the four parameters as well as their quadratic sum ( fig. 9(a)). Notably, in each direction, due to the constraint of the mass hierarchy, 55 out of the 715 points cannot have this interpolation uncertainty mea- space points ( fig. 9(b)). This indicates the necessity of a grid with finer granularity. However, increasing the number of points in each dimension by a mere factor of two would already mean 16 times larger statistics: a manifestation of the effect of the "curse of dimensionality". give more accurate result for such regions.

Discussion
In case it is impractical to produce additional MC events, an alternative solution is to fit another BNN with the existing sample, but assign higher weights during the fitting to those events in the concerned regions. This is equivalent to repeatedly using these events as additional samples. Although the statistical fluctuation remains, it will drive the fitting to focus more on the numerical feature of these regions and rely less on the interpolation from other regions.
Another potential improvement is to split the parameter space according to different physics processes. The BNN fitting assumes smooth distribution over the parameter space. However, this assumption of smoothness may not hold at boundaries where the physics is different on each side. If such boundaries are known as a priori knowledge, independent BNNs should be applied on each sub-set of the parameter space within which the signal yield is smooth.
Besides fitting the acceptance ǫ(x), the production rate σ(x) can also be fitted by the BNN. This might be necessary if a continuous function of σ(x) is desired or if calculating cross-sections for a large number of parameter space points is impractical. The way of BNN fitting needs to be changed slightly for such a regression application. The NN output y(x) should be allowed to range over all real numbers, while the target value will be the σ(x i ) for each event. The cost function will be the Mean-Square-Error between y(x i ) and σ(x i ), representing a Gaussian likelihood. As a result, the signal expectation N (x) can also be obtained continuously over the parameter space.

Conclusion
In this work, we introduced a new approach to simulate hypothetical processes with multiple free parameters and its use in estimating signal acceptance and yields continuously over the parameter space. In particular, the BNN technique is used for unbinned fitting of the acceptance, which is the key to providing continuous estimation and reducing the required number of simulation events. The two examples we showed both suggest that the acceptance can be well estimated with fewer MC events than needed in the grid-scan approach.
In addition, the uncertainty of the fitted signal yield is estimated by the BNN simultaneously which is a natural consequence of the Bayesian inference.