Unitarity Method Research Articles

Analysis of bridge structure selection using a hybrid decision-making technique

Implementation of sustainability principles in civil engineering has increased the substantive range of bridge engineering. The consideration of additional criteria, in particular ecological and social ones, requires the design process to be supported by appropriate tools. It refers especially to large bridge elements which affect their load-bearing capacity, e.g. girders or decks. The aim of this paper is to develop an original MCDA (Multi-Criteria Decision Analysis) method as a potential tool to support a decision-making process in the selection of material and design alternatives for bridge main girders. Therefore, an advanced hybrid algorithm was created consisting of the following methods: EA FAHP+FDEMATEL+ZUM (Extent Analysis Fuzzy Analytic Hierarchy Process + Fuzzy Decision-Making Trial and Evaluation Laboratory + Zero Unitarization Method), applied at the structure design stage. The pre-dimensioned alternatives selected for analysis were then subjected to an evaluation process based on a complex set of criteria arranged in a hierarchical control structure (HCS). The algorithm has been applied based on the example of a medium span slab-and-girder bridge, assuming 6 alternative concepts of girders, different in material and dimensions. The hybrid method was compared with the EA FAHP method. Analysis results obtained based on judgments from 3 teams of Decision Makers (DM) indicate effectiveness of the proposed algorithm and its practical aspect, which may contribute to improved quality and safety of bridge structures.

Double-virtual NNLO QCD corrections for five-parton scattering. II. The quark channels

We complete the computation of two-loop helicity amplitudes required to obtain next-to-next-to-leading order QCD corrections for three-jet production at hadron colliders, including all contributions beyond the leading-color approximation. The analytic expressions are reconstructed from finite-field samples obtained with the numerical unitarity method. We find that the reconstruction is significantly facilitated by exploiting the overlaps between rational coefficient functions of quark and gluon processes, and we display their compact generating sets. We implement our results in a public code, and demonstrate its suitability for phenomenological applications. Published by the American Physical Society 2024

Double-virtual NNLO QCD corrections for five-parton scattering. I. The gluon channel

We compute the two-loop helicity amplitudes for the scattering of five gluons, including all contributions beyond the leading-color approximation. The analytic expressions are represented as linear combinations of transcendental functions with rational coefficients, which we reconstruct from finite-field samples obtained with the numerical unitarity method. Guided by the requirement of removing unphysical singularities, we find a remarkably compact generating set of rational coefficients, which we display entirely in the manuscript. We implement our results in a public code, which provides efficient and reliable numerical evaluations for phenomenological applications. Published by the American Physical Society 2024

Two-loop QCD corrections for three-photon production at hadron colliders

We complete the computation of the two-loop helicity amplitudes for the production of three photons at hadron colliders, including all contributions beyond the leading-color approximation. We reconstruct the analytic form of the amplitudes from numerical finite-field samples obtained with the numerical unitarity method. This method requires as input surface terms for all relevant five-point non-planar integral topologies, which we obtain by solving the associated syzygy problem in embedding space. The numerical samples are used to constrain compact spinor-helicity ansätze, which are optimized by taking advantage of the known one-loop analytic structure. We make our analytic results available in a public C++ library, which is suitable for immediate phenomenological applications. We estimate that the inclusion of the subleading-color contributions will decrease the size of the two-loop corrections by about 30% to 50%, and the NNLO cross sections by a few percent, compared to the results in the leading-color approximation.

Green growth in agriculture―New measurement concept and its empirical verification

AbstractThis paper proposes a new method for evaluating green growth in agriculture (GGA) based on a set of key performance indicators and a synthetic index. To this end, we used the zero unitarisation method. The objects of study were seven countries/groups of countries with the largest share in agricultural production in 2020. This paper attempts to answer the following questions: (1) Has there been any progress in fulfilling the assumptions of the green growth concept implemented in the agricultural sector of the examined countries? (2) Which countries do better, and which do worse in that respect? (3) What are the challenges respective countries need to face? In the analysed years, the mean GGA index increased from 0.3287 to 0.3342, which means that the overall level of ‘greening’ in agriculture grew insignificantly. Out of the seven analysed countries/regions, China did the best in handling the challenges of GGA, while India did the worst. The paper discusses the reasons for this state of affairs, simultaneously making recommendations for improving the existing situation.

Classical gravitational observables from the Eikonal operator

We propose two possible eikonal operators encoding the effects of classical radiation as coherent states of gravitons and show how to compute from them different classical observables. In the first proposal, only genuinely propagating gravitons are included, while, in the second, zero-frequency modes are added in order to recover the effects of a static gravitational field. We first calculate the radiated energy-momentum and the change in each particle's momentum, or impulse, to 3PM order finding agreement with the literature. We then calculate the angular momentum of the gravitational field after the collision. In order to do so, we adapt the method of reverse unitarity to the presence of derivatives in the operators describing the angular momentum and reproduce the result of [1] obtained by resumming the small-velocity expansion. As a new application, we derive also the variation in each particle's angular momentum up to 3PM: calculating separately field and particle contributions allows us to check the balance laws explicitly. We also show how the eikonal operator encodes the linear-response formula of Bini-Damour by deriving the linear radiation-reaction contribution to the transverse impulse at 4PM.

Pre- and During-Pandemic Financial Standing of Companies in Poland Ranked According to the Type of Business Activity

Abstract Subject and purpose of work This paper seeks to examine how the COVID-19 pandemic affected the financial standing of companies classified according to the type of their business activity. Materials and methods This paper is based on aggregate data on businesses operating in Poland in the period 2017–2021, grouped according to the type of business activity. The data was published by the Central Statistical Office GUS in Poland. The company status was investigated using the calculated values of diagnostic indicators (return on assets, current financial liquidity, level of debt, stock rotation, and growth in sales). These indicators were normalized with the zero unitarization method to construct a synthetic indicator. The synthetic indicator was then used to generate business rankings. Results The vast majority of the examined companies proved to be profitable. In 2020, the profitability and sales revenue of the businesses concerned were found to deteriorate slightly. There has been a downward trend in the rankings for companies operating in the accommodation and catering sector and the manufacturers of petroleum products. Pharmaceutical companies were the businesses that improved and topped the rankings. Conclusions The pandemic had a strong negative impact on businesses operating in the market sector of accommodation and catering. The producers of refined petroleum products were also affected. The producers of pharmaceuticals benefited most during the pandemic. There are no clear indicators as to the impact of the pandemic on other types of businesses.

Regional Discrepancies in the Level of Knowledge-Based Economy in Poland

AbstractSubject and purpose of workThis paper analyzes the level of knowledge-based economy (KBE) in terms of sustainable development across the provinces (voivodships) of Poland in 2020. The underlying assumption of this study is that knowledge and skills are one of the fundamental preconditions for successfully implementing the concept of sustainable development.Materials and methodsIn order to calculate the rankings of Polish voivodships a synthetic indicator developed on the basis of the zeroed unitarization method was used.ResultsThere are vast disproportions in the levels of KBE across the voivodeships of Poland. The most advanced KBE was identified in the Mazowieckie Voivodship, and the least developed KBE was found in the Lubuskie voivodship.ConclusionsThe capacity to develop a knowledge-based economy is inextricably interrelated with the quality of the social capital, which is directly attributable to the levels of education. In order to offset the interregional disproportions spending on research and development needs to be increased and the investments in human capital (e.g. through career planning, training, improving the professional qualifications of employees, discretionary bonuses) and new technologies should be expanded.

Gluonic evanescent operators: two-loop anomalous dimensions

Evanescent operators are a special class of operators that vanish in four-dimensional spacetime but are non-zero in d = 4−2ϵ dimensions. In this paper, we continue our systematic study of the evanescent operators in the pure Yang-Mills theory and focus on their two-loop renormalization. We develop an efficient strategy to compute the two-loop divergences of form factors of high-dimensional and high-length operators by combining the d-dimensional unitarity method and the improved tensor reduction techniques. Two-loop anomalous dimensions are obtained for the dimension-10 basis in the planar YM theory, for which both the overline{textrm{MS}} scheme and the finite-renormalization scheme are used. We verify that the two-loop anomalous dimensions are the same in these two schemes at the Wilson-Fisher conformal fixed point. Our computation shows that the evanescent operators are indispensable in order to obtain the correct two-loop anomalous dimensions. This work provides a first computation of the two-loop anomalous dimensions of the complete set of dimension-10 operators. The method we use is also expected to provide an efficient strategy for the two-loop renormalization of general high-dimensional operators.

Generalized Unitarity Method for Unstable Particles

In theories with unstable particles, unitarity is satisfied by the inclusion of only stable states in unitarity sums. Therefore, unitarity cuts are not to be taken through unstable particles. This presents a challenge to the generalized unitarity method, whose aim is to reconstruct amplitudes by analyzing sets of unitarity cuts. Nevertheless, under some general physical conditions, and perhaps some methodological modifications, we prove that the method is still reliable for one-loop amplitudes containing resonances. We discuss some simple examples which illustrate these features.

Assessment of the level of economic development of Polish voivodships by means of linear ordering methods in the dynamic approach

The research on the economic development of voivodships is important in the context of the policy of levelling regional disparities and has been necessitated by the fact the disproportions in the level of the economic development of Polish voivodships have persisted for years. The main aim of this study was to determine the direction and scope of changes affecting the level of economic development of particular voivodships. To this end, the study used linear ordering methods in the dynamic approach. The additional aim, of the methodological nature, was to compare two methods of the normalisation of variables – the Strahl quotient transformation and the zero unitarisation method. The research was conducted on the basis of relevant data from Statistics Poland’s Local Data Bank database for the years 2010–2020. The results of the study include the calculation of the synthetic measure of the level of development of Polish voivodships in the studied period, the compilation of the relevant ranking of the voivodships, and the comparison of changes thereof. The performed analysis demonstrated that all the voivodships developed economically in the examined period, but the changes took place at a different pace and the disparities between them remained. The indicator of economic development grew to the largest extent in the voivodships which were already the most developed in economic terms, i.e. in the Mazowieckie, Dolnośląskie, Wielkopolskie and Małopolskie voivodships. Both of the applied methods of the variable normalisation led to similar conclusions. The differences between them concerned the value of the development level indicator, and affected to some degree the assessment of the scope and direction of changes, along with the assessment of the distance between the objects.

Economic Activity of Residents and Revenue Autonomy of Municipalities in the Context of the COVID-19 Pandemic

Theoretical background: The impact of the COVID-19 pandemic has had far reaching effects on different strata of socio-economic life, including the financial well-being of residents and the financial situation of local government units. Generally, research papers either approach “local development” conceptually or quantify data regarding a certain locality: demographic structure, residents’ education, condition of the labour market, residents’ income and their technological literacy. This paper is an attempt to arrive at a larger picture and identify the interdependence between economic activity of residents and financial autonomy of the local government, and secondly, to pinpoint the impact the COVID-19 pandemic had on these categories. Purpose of the article: The purpose of the paper is to identify residents’ income sources and the dependence between them and the development potential of the municipality they live in. The paper examines residents’ revenue, income and income tax advance payments. The survey covers all municipalities in Poland according to their formal administrative status: urban, rural, urban-rural and cities with powiat rights. The research covers the period of 2015–2020, as per data availability. Research methods: The paper applied monograph methodology to review the literature on the subject, and comparative methodology to process quantitative data, which allowed the authors to calculate all deviations. Next, the economic phenomena were assessed, i.e. their nature (positive or negative) and frequency of occurrence were determined according to an assessment scale where the base is the starting point. The structure of the number of taxpayers, their revenues and incomes was compared by sources of income, in result of which hierarchy of income sources was created for each municipality. Following that, the main income sources were analysed and correlated with the municipality’s own (tax) revenue. To enable comparability of diagnostic variables, the zero unitarisation method was applied. Main findings: The findings reveal that the main sources of residents’ income include salaries, pensions and disability pensions, non-agricultural economic activity, wages and contractual income. Revenue streams generated from residents’ economic activity show a variance across municipalities. The results obtained confirm the dependence between residents’ economic activity and own revenue sources of local government units. Based on the findings, it can be concluded that the relationship between taxpayers’ income from particular sources and revenue autonomy of municipalities and cities with powiat rights varies by the type of municipality. Thus, the findings clearly demonstrate that residents’ economic activity is an endogenous resource that definitely affects municipal development. For local authorities, understanding economic realities of residents may be a game changer and an insight that will help them create environments conducive to socioeconomic growth.

Unitarizing non-relativistic Coulomb scattering

We compare the exactly solvable nonrelativistic Coulomb scattering with two recent unitarization methods for infinite-range forces. These methods require to calculate perturbatively the corresponding partial-wave amplitudes, which are then unitarized. We calculate the Coulomb partial-wave amplitudes up to the one-loop order. On the one hand, the unitarization method developed by Refs. [1,2] reproduces properly the exact solution, with an accuracy improving as the order in the perturbative calculation of the input perturbative partial-wave amplitudes increases. This is also shown to be the case for the pole position of the ground state. On the other hand, the method developed by the more recent Ref. [3] gives rise to partial-wave amplitudes that do not reproduce the known solvable solution, and gives rise to a pole position with zero binding energy.

Renewable energy production capacity and consumption in Europe

Faced with the prospect of achieving 32 % renewable energy (RE) consumption by 2030 and climate neutrality by 2050, Europe has embarked on a path of promoting the use of clean energy through policies and investments essential to its development. Against this backdrop, the aim of the present article is to analyse both RE production capacity and the determinants of RE consumption in a sample of 30 European countries for the period 2010–2019. Through the construction of a synthetic measure by the zero unitarization method, the four types of RE (hydro, wind, solar and biofuels) are jointly assessed, yielding a ranking of countries categorized as being in a favourable, fairly favourable, rather unfavourable, or unfavourable situation. Furthermore, the possible relationship between RE production capacity and consumption is estimated with a sample of panel data (fixed effects and random effects). The results show that Sweden (SM: 0.419), Austria (SM: 0.384) and the Netherlands (SM: 0.381) are at the top of the ranking, demonstrating their leadership in RE generation and providing a model to follow for Eastern European countries whose production levels are still well below average. Moreover, the study provides evidence that the RE production capacity (coefficient: 0.253***), low gas emissions (coefficient: −0.479***), energy productivity (coefficient: 0.333***), and the size of the country (coefficient: 0.016***) foster the use of RE. Overall, European countries are on the right track to achieve the goals set, although they may be progressing at different speeds.

Gluonic evanescent operators: classification and one-loop renormalization

Evanescent operators are a special class of operators that vanish classically in four-dimensional spacetime, while in general dimensions they are non-zero and are expected to have non-trivial physical effects at the quantum loop level in dimensional regularization. In this paper we initiate the study of evanescent operators in pure Yang-Mills theory. We develop a systematic method for classifying and constructing the d-dimensional Lorentz invariant evanescent operators, which start to appear at mass dimension ten. We also compute one-loop form factors for the dimension-ten operators via the d-dimensional unitarity method and obtain their one-loop anomalous dimensions. These operators are necessary ingredients in the study of high dimensional operators in effective field theories involving a Yang-Mills sector.

Analytical evaluation of AdS4 Witten diagrams as flat space multi-loop Feynman integrals

We describe a systematic approach for the evaluation of Witten diagrams for multi-loop scattering amplitudes of a conformally coupled scalar ϕ4-theory in Euclidean AdS4, by recasting the Witten diagrams as flat space Feynman integrals. We derive closed form expressions for the anomalous dimensions for all double-trace operators up to the second order in the coupling constant. We explain the relation between the flat space unitarity methods and the discontinuities of the short distance expansion on the boundary of Witten diagrams.

Feynman diagrams in terms of on-shell propagators

It is shown that the usual expression for a Feynman diagram in terms of the Feynman propagator $\Delta_F(x-y)$ can be replaced by an equivalent expression involving the positive-energy on-shell propagator $\Delta^+(x-y)$, supplemented by appropriate functions associated with time-ordering. When this alternate way of expressing a Feynman diagram is Fourier transformed into momentum space, the momentum associated with each function $\Delta^+(x-y)$ is on-shell, and is only conserved at each vertex if an energy is attributed to the contributions of the time-ordering functions. The resulting expression is analogous to what Kadyshevsky had obtained by deriving an alternate expansion for the $S$--matrix. A detailed explanation of how this alternate expansion is derived is given, and it is shown how it provides a straightforward way of determining the imaginary part of a Feynman diagram, which makes it useful when using unitarity methods for computing a Feynman diagram. By considering a number of specific Feynman diagrams in self-interacting scalar models and in QED, we show how this alternate approach can be related to the old perturbation theory and can simplify direct calculations of Feynman diagrams.

Holographic S-fold theories at one loop

A common feature of tree-level holography is that a correlator in one theory can serve as a generating function for correlators in another theory with less continuous symmetry. This is the case for a family of 4d CFTs with eight supercharges which have protected operators dual to gluons in the bulk. The most recent additions to this family were defined using S-folds which combine a spatial identification with an action of the S-duality group in type IIB string theory. Differences between these CFTs which have a dynamical origin first become manifest at one loop. To explore this phenomenon at the level of anomalous dimensions, we use the AdS unitarity method to bootstrap a one-loop double discontinuity. Compared to previous studies, the subsequent analysis is performed without any assumption about which special functions are allowed. Instead, the Casimir singular and Casimir regular terms are extracted iteratively in order to move from one Regge trajectory to the next. Our results show that anomalous dimensions in the presence of an S-fold are no longer rational functions of the spin.

Sustainable Development of Agriculture in Member States of the European Union

The main aim of this study was to evaluate the sustainable development of agriculture in the member states of the European Union (EU). Sustainable development is the main objective of Polish agriculture. Sustainable development encompasses three pillars: economic, social and environmental. In recent years, Polish agriculture has undergone a considerable change to integrate and adapt the agricultural sector to the EU standards. The achievement of the sustainable development goals in European agriculture was evaluated based on the Eurostat data as well as the analysis of the literature. The discussion on sustainable development of agriculture is rich. However, little attention is paid to the measurement of sustainable development. Our intention was to fill in the gap in the literature and provide a method to evaluate sustainable development. The paper contributes to the measurement of sustainable development based on Hellwig’s method. The changes in sustainable agricultural development were assessed with an index that was normalized with the zero unitarization method. In the first step, descriptive statistics for the variables applied in the process of calculating the sustainable development index were analyzed. The taxonomic measure of development (TMD) was then calculated for the economic, social and environmental dimensions of sustainable development. In the following stage, 27 EU member states were divided into sustainable development classes based on the mean values of the TMD for each of the three pillars of sustainability. The conducted research revealed changes between the European Union countries in terms of sustainable development. In 2018, the highest values of the sustainable development index were noted in the Netherlands, Germany, France and the United Kingdom.

Smearing of causality by compositeness divides dispersive approaches into exact ones and precision-limited ones

Scattering off the edge of a composite particle or finite-range interaction can precede that off its center. An effective theory treatment with pointlike particles and contact interactions must find that the scattered experimental wave is slightly advanced, in violation of causality (the fundamental underlying theory being causal). In practice, partial-wave or other projections of multivariate amplitudes exponentially grow with Im(E), so that analyticity is not sufficient to obtain a dispersion relation for them, but only for a slightly modified function (the modified relations additionally connect different J). This can limit the precision of certain dispersive approaches to compositeness based on Cauchy's theorem. Awareness of this may be of interest to some dispersive tests of the Standard Model with hadrons, and to unitarization methods used to extend electroweak effective theories. Interestingly, the Inverse Amplitude Method is safe (as the inverse amplitude has the opposite, convergent behavior allowing contour closure). Generically, one-dimensional sum rules such as for the photon vacuum polarization, form factors or the Adler function are not affected by this uncertainty; nor are fixed-t dispersion relations, cleverly constructed to avoid it and whose consequences are solid.