Scattering Amplitudes

This paper presents a comprehensive theoretical framework for analyzing the stability of tetraquark states in quantum chromodynamics (QCD). We develop a master equation that decomposes tetraquark binding energy into five fundamental contributions: color interactions, flavor-dependent forces, electromagnetic corrections, kinetic energy differences, and confinement potential variations. The framework is calibrated against recent experimental data from LHCb collaborations, including the 2025 confirmation of the T_{cs0}^*(2870)^0 state. Our model successfully reproduces the binding energies of established tetraquark candidates X(3872), T_{cc}^+(3875), and provides predictions for undiscovered states. Three fundamental theorems are formulated regarding color confinement stability, flavor symmetry breaking, and threshold behavior. The theoretical predictions are validated through variational optimization techniques and show excellent agreement with experimental observations. This work provides a unified approach for understanding exotic hadron spectroscopy and guiding future experimental searches at facilities such as the Large Hadron Collider.

We review the status of theoretical predictions for W-pair production at high energies. We discuss a systematic scenario towards a Monte-Carlo generator for e + e -→ 4f (γ), which meets the experimental requirements. In particular we summarize the recent developments in this field.

We further develop the massive constructive theory of the Standard Model and use it to calculate the amplitude and squared amplitude for all 2-body decays, a collection of weak 3-body decays as well as Higgs decay to four neutrinos. We compare our results with those from Feynman diagrams and find complete agreement. We show that in all the cases considered here, the amplitudes of massive constructive theories are significantly simpler than those resulting from Feynman diagrams. In fact, a naive counting of the number of calculations required for a matrix-element generator to compute a phase-space point is orders-of-magnitude smaller for the result coming from the constructive method suggesting that these generators might benefit from this method in the future even in the case of massive weak amplitudes.

We consider the permutation group in n + 1 letters, S n+1 , which corresponds to the finite Weyl group of type A n . S n+1 is generated by the set of simple reflections which corresponds to the set S of transpositions Using the theory of reductive algebraic monoids and representation theory of semisimple algebraic groups we associate to J a projective toric variety denoted by X(J). We are interested in computing the Betti numbers of the projective variety X(J) when it is a rationally smooth variety i.e., X(J) has sufficiently mild singularities. Betti numbers are an important topological invariant of a space which help us measure the number of holes or cuts present in that space and classify spaces with the same topological structure. Moreover we define the Poincaré polynomial of a topological space as the generating function of its Betti numbers, via the polynomial whose coefficients are the Betti numbers. We associate to X(J) a simple polytope P λ obtained as the convex hull of the S n+1 -orbit of a weight λ fixed by only the simple reflections J = {s n , s n-1 , • • • , s n-k+1 } for some k with respect to the A n root lattice. The Betti numbers of the variety X(J) can be computed using the h-polynomial associated to the polytope P λ . It is well-known that the Eulerian polynomials, which count permutations in S n+1 by their number of descents, give the h-polynomial of the simple polytopes known as permutohedra. Therefore the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra. In this thesis we derive a recurrence formula for the h-polynomials of a family of polytopes generalizing this. When J = {s n+k+1 , s n-k+2 , • • • , s n }, 1 ≤ k ≤ n and (W, S) is Weyl group of type A n , we obtain a formula for the Poincaré polynomial of X(J), in terms of Eulerian polynomials. Furthermore, we compute explicitly the Poincaré polynomial of X(J) using the method of descent systems, in the case of a (W, S) finite Weyl group of type A n , and J being a combinatorially smooth of the following forms: (1) J = {s 1 , s 4 , s 5 , • • • , s n } ⊂ S and (2) J = {s 4 , s 5 , • • • , s n } ⊂ S. v Contents CERTIFICATE OF EXAMINATION ii

Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define an equivariant differential operator on G/(L cap K) playing the role of an equivariant Dolbeault Laplacian when restricted to the complex manifold G/L, using a distribution transverse to the fibers and satisfying the Hormander condition. We prove here that this

Exclusive B decays can be factorized as convolutions of hard scattering amplitudes involving the weak interaction with universal hadron distribution amplitudes, thus providing a new QCD-based phenomenology. In addition, semi-leptonic decay amplitudes can be computed exactly in terms of the diagonal and off-diagonal ∆n = 2 overlap of hadronic light-cone wavefunctions. I review these formalisms and the essential QCD ingredients. A canonical form of the light-cone wavefunctions, valid at low values of the transverse momenta, is presented. The existence of intrinsic charm Fock states in the B meson wavefunction can enhance the production of final states of B-decay with three charmed quarks, such as B → J/ψDπ, as well as lead to the breakdown of the CKM hierarchy.

On-shell methods are particularly suited for exploring the scattering of electrically and magnetically charged objects, for which there is no local and Lorentz invariant Lagrangian description. In this paper we show how to construct a Lorentz-invariant S-matrix for the scattering of electrically and magnetically charged particles, without ever having to refer to a Dirac string. A key ingredient is a revision of our fundamental understanding of multi-particle representations of the Poincaré group. Surprisingly, the asymptotic states for electric-magnetic scattering transform with an additional little group phase, associated with pairs of electrically and magnetically charged particles. The corresponding "pairwise helicity" is identified with the quantized "cross product" of charges, e 1 g 2 -e 2 g 1 , for every charge-monopole pair, and represents the extra angular momentum stored in the asymptotic electromagnetic field. We define a new kind of pairwise spinor-helicity variable, which serves as an additional building block for electric-magnetic scattering amplitudes. We then construct the most general 3-point S-matrix elements, as well as the full partial wave decomposition for the 2 → 2 fermion-monopole S-matrix. In particular, we derive the famous helicity flip in the lowest partial wave as a simple consequence of a generalized spin-helicity selection rule, as well as the full angular dependence for the higher partial waves. Our construction provides a significant new achievement for the on-shell program, succeeding where the Lagrangian description has so far failed.

The thin film equation, which is derived from the Navier-Stokes equations, is a degenerate fourth order parabolic equation describing the motion of thin films on a plate. In this talk we shall give an introductory talk on various mathematical results known to this equations. The issue of singularity formation (rupture) will be emphasized. The talk is good for those who have some previous knowledge of parabolic equations. Langlands Functoriality for Classical Groups Dihua Jiang University of Minnesota Abstract The Langlands functoriality conjecture is the main problem in modern theory of automorphic forms, the significance of which has been observed from the intrinsic connections with number theory and arithmetic geometry, including Wiles’ proof of the Fermat Last Theorem. In this talk, I will explain the recent progress for classical groups.The Langlands functoriality conjecture is the main problem in modern theory of automorphic forms, the significance of which has been observed fr...

We analyze numerically the Balitsky-Kovchegov equation with the full impact parameter dependence b. We show that due to the particular b-dependence of the initial condition the amplitude decreases for large dipole sizes r. Thus the region of saturation has a finite extension in the dipole size r, and its width increases with rapidity. We calculate the b-dependent saturation scale and discuss limitations on geometric scaling. We also demonstrate the instant emergence of the power-like tail in impact parameter, which is due to the long range contributions. Thus the resulting cross section violates Froissart bound despite the presence of a nonlinear term responsible for saturation.

We calculate, for nonzero momentum transfer, the dipole formula for the high energy behaviour of elastic and quasielastic scattering of a virtual photon. We obtain an expression of the nonforward photon impact factor and of the nonforward photon wave function, and we give a physical interpretation.

On-shell methods are particularly suited for exploring the scattering of electrically and magnetically charged objects, for which there is no local and Lorentz invariant Lagrangian description. In this paper we show how to construct a Lorentz-invariant Smatrix for the scattering of electrically and magnetically charged particles, without ever having to refer to a Dirac string. A key ingredient is a revision of our fundamental understanding of multi-particle representations of the Poincaré group. Surprisingly, the asymptotic states for electric-magnetic scattering transform with an additional little group phase, associated with pairs of electrically and magnetically charged particles. The corresponding "pairwise helicity" is identified with the quantized "cross product" of charges, e 1 g 2 -e 2 g 1 , for every charge-monopole pair, and represents the extra angular momentum stored in the asymptotic electromagnetic field. We define a new kind of pairwise spinor-helicity variable, which serves as an additional building block for electric-magnetic scattering amplitudes. We then construct the most general 3-point S-matrix elements, as well as the full partial wave decomposition for the 2 → 2 fermion-monopole S-matrix. In particular, we derive the famous helicity flip in the lowest partial wave as a simple consequence of a generalized spin-helicity selection rule, as well as the full angular dependence for the higher partial waves. Our construction provides a significant new achievement for the on-shell program, succeeding where the Lagrangian description has so far failed.

On-shell methods are particularly suited for exploring the scattering of electrically and magnetically charged objects, for which there is no local and Lorentz invariant Lagrangian description. In this paper we show how to construct a Lorentz-invariant S-matrix for the scattering of electrically and magnetically charged particles, without ever having to refer to a Dirac string. A key ingredient is a revision of our fundamental understanding of multi-particle representations of the Poincaré group. Surprisingly, the asymptotic states for electric-magnetic scattering transform with an additional little group phase, associated with pairs of electrically and magnetically charged particles. The corresponding “pairwise helicity” is identified with the quantized “cross product” of charges, e1g2− e2g1, for every charge-monopole pair, and represents the extra angular momentum stored in the asymptotic electromagnetic field. We define a new kind of pairwise spinor-helicity variable, which serve...

We sum up the next-to-next-to-leading logarithmic virtual electroweak corrections to the high energy asymptotics of the neutral current four-fermion processes for light fermions to all orders in the coupling constants using the evolution equation approach. From this all order result we derive finite order expressions through next-to-next-to leading order for the total cross section and various asymmetries. We observe an amazing cancellation between the sizable leading, next-to-leading and next-to-next-toleading logarithmic contributions at TeV energies.

Generalizing the famous Bernstein-Kushnirenko Theorem, Khovanskii proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes. Recently, an analogous combinatorial formula, called the discrete mixed volume, was introduced by Bihan and shown to be nonnegative. By making a footnote of Khovanskii in his paper explicit, we interpret this invariant as the (motivic) arithmetic genus of the non-compact generic complete intersection associated to the family of lattice polytopes.

We present the Vector Equivalence technique. This technique allows a simple and systematic calculating of Feynman diagrams involving massive fermions at the matrix element level. As its name suggests, the technique allows two Lorentz four-vectors to serve as an equivalent of two externai fermions. In further calculations, traces involving these vectors replace the matrix element with the external fermions. The technique can he conveniently used for hot11 symbolic and wmeric calculations.

Hive-state (H(b, 2s, 2p) + P s(1 jt, 2j)) and six-state 2v, 2p) + Ps(h, 2s. 2p)) close coupling melhord have been carried out below the posiironium (Ps) formation threshold to study positron-hydrogen scattering. Ibe r-wavc and p-wave phase-shifts are obuined. The .r-wave phase-shifts are foundno sausfy the bound principle properly.

The purpose of this paper is to establish meromorphy properties of the partial scattering amplitude T (λ, k) associated with physically relevant classes N γ w (ε) ,α of nonlocal potentials in corresponding domains D (δ) γ,α of the space C 2 of the complex angular momentum λ and of the complex momentum k (namely, the square root of the energy). The general expression of T as a quotient Θ(λ, k)/σ(λ, k) of two holomorphic functions in D (δ) γ,α is obtained by using the Fredholm-Smithies theory for complex k, at first for λ = integer, and in a second step for λ complex (Re λ > −1/2). Finally, we justify the "Watson resummation" of the partial wave amplitudes in an angular sector of the λ-plane in terms of the various components of the polar manifold of T with equation σ(λ, k) = 0. While integrating the basic Regge notion of interpolation of resonances in the upper half-plane of λ, this unified representation of the singularities of T also provides an attractive possible description of echoes in the lower half-plane of λ. Such a possibility, which is forbidden in the usual theory of local potentials, represents an enriching alternative to the standard Breit-Wigner hard-sphere picture of echoes.

We extend a recently proposed dipole model which relates the virtual photon-proton cross section to the dipole-proton forward scattering amplitude in momentum space investigating the effects of the gluon number fluctuations. The model interpolates between well known asymptotic behaviours predicted by perturbative QCD from the Balitsky-Kovchegov equation, which describes the rapidity evolution of the dipole-proton scattering amplitude in the mean field approximation. The model was shown to be successful in describing the last HERA data for the case where the strong coupling constant α s is fixed, showing also some important advantages when compared with other dipole models-all of them in coordinate space-in the literature. Based on the fact that the fluctuations may be important in the small-x evolution and on recent results obtained in coordinate space beyond the mean field approximation, we use this model to parametrize the proton structure function and confront it to HERA data using the average (physical) amplitude-then including fluctuations-within the momentum space framework.

We compute a precise value for three more graphs contributing to the g factor of the electron in sixth order. After comparing with other numerical and analytic evaluations, we give an updated "best" theoretical estimate of the g factor, and compare it to the experimental value.

We study the field theory limit of multi-loop (super)string amplitudes, with the aim of clarifying their relationship to Feynman diagrams describing the dynamics of the massless states. We propose an explicit map between string moduli around degeneration points and Schwinger proper-times characterizing individual Feynman diagram topologies. This makes it possible to identify the contribution of each light string state within the full string amplitude and to extract the field theory Feynman rules selected by (covariantly quantized) string theory. The connection between string and field theory amplitudes also provides a concrete tool to clarify ambiguities related to total derivatives over moduli space: in the superstring case, consistency with the field theory results selects a specific prescription for integrating over supermoduli. In this paper, as an example, we focus on open strings supported by parallel D-branes, and we present two-loop examples drawn from bosonic and RNS string theories, highlighting the common features between the two setups.

We study celestial amplitudes in string theory at one-loop. Celestial amplitudes describe scattering processes of boost eigenstates and relate to amplitudes in the more standard basis of momentum eigenstates through a Mellin transform. They are thus sensitive to both the ultraviolet and the infrared, which raises the question of how to appropriately take the field theory limit of string amplitudes in the celestial basis. We address this problem in the context of four-dimensional genus-one scattering processes of gluons in open string theory which reach the two-dimensional celestial sphere at null infinity. We show that the Mellin transform commutes with the adequate limit in the worldsheet moduli space and reproduces the celestial one-loop field theory amplitude expressed in the worldline formalism. The dependence on α′ continues to be a simple overall factor in one-loop celestial amplitudes albeit with a power that is shifted with respect to tree-level, thus making manifest that th...

We show that the asymptotic charges associated with Lorentz symmetries on past and future null infinity match in the limit to spatial infinity in a class of asymptotically-flat spacetimes. These are spacetimes that obey the Ashtekar-Hansen definition of asymptotic flatness at null and spatial infinity and satisfy an additional set of conditions which we lay out explicitly. Combined with earlier results on the matching of supertranslation charges, this shows that all Bondi-Metzner-Sachs (BMS) charges on past and future null infinity match in the limit to spatial infinity in this class of spacetimes, proving a relationship that was conjectured by Strominger. Assuming additional suitable conditions are satisfied at timelike infinities, this proves that the flux of all BMS charges is conserved in any classical gravitational scattering process in these spacetimes.

It is known that 𝒩 = 8 supergravity is dual to 𝒩 = 4 super Yang-Mills (SYM) via the double copy relation. Using the explicit relation between scattering amplitudes in the two theories, we calculate the soft and collinear limits in 𝒩 = 8 supergravity from know results in 𝒩 = 4 SYM. In our application of double copy, a particular self-duality condition is chosen for scalars that allows us to constrain and determine the R-symmetry indices of the supergravity states in the collinear limit.

It is shown that the kernel of the BFKL equation for the octet color state of two Reggeized gluons satisfies the strong bootstrap condition in the next-to-leading order. This condition is much more restrictive than the one obtained from the requirement of Reggeized form for the elastic scattering amplitudes in the next-to-leading approximation. It is necessary, however, for self-consistency of the assumption of Reggeized form of the production amplitudes in multi-Regge kinematics, which are used in the derivation of the BFKL equation. The fulfillment of the strong bootstrap condition for the kernel opens a way to a rigorous proof of the BFKL equation in the next-to-leading approximation.

Dual conformal symmetry and Yangian symmetry are symmetries of amplitudes that have aided the study of scattering amplitudes in highly supersymmetric theories like N = 4 SYM and ABJM. However, in general such symmetries are absent from the theories with lesser or no supersymmetry. In this paper, we show that the tree level 2 → 2 scattering amplitude in the 3d N = 2 Chern-Simons theory coupled to a fundamental chiral multiplet is dual superconformal invariant. In the ’t Hooft large N limit, the 2 → 2 scattering amplitude in this theory has been shown to be tree-level exact in non-anyonic channels, while having only an overall multiplicative coupling dependent renormalisation in the anyonic channel. Therefore, the dual superconformal symmetry that we demonstrate in this paper is all loop exact. This is unlike the previously studied highly supersymmetric theories where dual superconformal symmetry is anomalous at loop levels. Furthermore, we reverse the argument to study the extent to ...

We study the large gauge transformations of massless higher-spin fields in fourdimensional Minkowski space. Upon imposing suitable fall-off conditions, providing higherspin counterparts of the Bondi gauge, we observe the existence of an infinite-dimensional asymptotic symmetry algebra. The corresponding Ward identities can be held responsible for Weinberg's factorisation theorem for amplitudes involving soft particles of spin greater than two.

Recently it has been speculated that the S-matrix elements satisfy the Ward identity associated with the T-duality. This indicates that a group of S-matrix elements is invariant under the linear T-duality transformations on the external states. If one evaluates one component of such T-dual multiplet, then all other components may be found by the simple use of the linear T-duality. The assumption that fields must be independent of the Killing coordinate, however, may cause, in some cases, the T-dual multiplet not to be gauge invariant. In those cases, the S-matrix elements contain more than one T-dual multiplet which are intertwined by the gauge symmetry. In this paper, we apply the T-dual Ward identity on the S-matrix element of one RR (p − 3)-form and two NSNS states on the world volume of a D p-brane to find its corresponding T-dual multiplet. In the case that the RR potential has two transverse indices, the T-dual multiplet is gauge invariant, however, in the case that it has one transverse index the multiplet is not gauge invariant. We find a new T-dual multiplet in this case by imposing the gauge symmetry. We show that the multiplets are reproduced by explicit calculation, and their low energy contact terms at order α ′2 are consistent with the existing couplings in the literature.

Using modern amplitude techniques we compute the leading classical and quantum corrections to the gravitational potential between two massive scalars induced by adding cubic terms to Einstein gravity. We then study the scattering of massless scalars, photons and gravitons off a heavy scalar in the presence of the same R 3 deformations, and determine the bending angle in the three cases from the non-analytic component of the scattering amplitude. Similarly to the Einstein-Hilbert case, we find that the classical contribution to the bending angle is universal, but unlike that case, universality is preserved also by the first quantum correction. Finally we extend our analysis to include a deformation of the form ΦR 2 , where Φ is the dilaton, which arises in the low-energy effective action of the bosonic string in addition to the R 3 term, and compute its effect on the graviton bending.

Form factors of the stress-tensor multiplet operator in $$ \mathcal{N}=4 $$ N = 4 supersymmetric Yang-Mills reveal surprisingly simple structures similar to those appearing in scattering amplitudes. In this paper we show that, as for the case of amplitudes, they also enjoy dual conformal symmetry. We compute the dual conformal anomaly at one loop for an arbitrary number of particles and generic helicities, which matches the expression of the dual conformal anomaly of amplitudes, and perform explicit checks for MHV and NMHV one-loop form factors. In the NMHV case the realisation of dual conformal symmetry requires a delicate cancellation of offending terms arising from three-mass triangles, which we explicitly check in the case of the four-point NMHV form factor.

We derive a compact expression for the three-point MHV form factors of half-BPS operators in N = 4 super Yang-Mills at two loops. The main tools of our calculation are generalised unitarity applied at the form factor level, and the compact expressions for supersymmetric tree-level form factors and amplitudes entering the cuts. We confirm that infrared divergences exponentiate as expected, and that collinear factorisation is entirely captured by an ABDK/BDS ansatz. Next, we construct the two-loop remainder function obtained by subtracting this ansatz from the full two-loop form factor and compute it numerically. Using symbology, combined with various physical constraints and symmetries, we find a unique solution for its symbol. With this input we construct a remarkably compact analytic expression for the remainder function, which contains only classical polylogarithms, and compare it to our numerical results. Furthermore, we make the surprising observation that our remainder is equal to the maximally transcendental piece of a closely related two-loop amplitude in QCD.

We exploit a recently found connection between special triple-cut diagrams and tree-level recursive diagrams to derive a general formula capturing the multi-particle factorisation of arbitrary one-loop amplitudes in the ABJM theory. This formula contains certain anomalous contributions which are reminiscent of the so-called non-factorising contributions appearing in the factorisation of one-loop amplitudes in four-dimensional gauge theory. In the second part of the paper we derive a recursion relation for the supercoefficients of one-loop amplitudes in ABJM theory. By applying this recursion relation, any one-loop supercoefficient can be reduced to special triple-cut diagrams involving at least one four-point tree amplitude. In turn, this implies that any one-loop supercoefficient can be derived from tree-level recursive diagrams.

Having its origin in a successful mapping technique, the Fock-Tani formalism, has produced a corrected 3 P 0 model (C 3 P 0 model), which retains the basic aspects of the 3 P 0 predictions with the inclusion of bound-state corrections. In high energy collisions many new mesons have been discovered in particular the enigmatic D + S0 (2317) and D + S1 (2460). The model is applied for D SJ mesons decay. The amplitude and its respective decay rates are evaluated.

We study Efimov physics of three identical bosons with pairwise multichannel interactions for Feshbach resonances of adjustable width. We find that the two-body multichannel nature of the interaction can affect the universal three-body spectrum, especially for resonances of intermediate width. The shifts in this universal spectrum are caused by trimer states in the closed interaction channels that couple to the universal Efimov states. However, in the narrow resonance limit we find that the Efimov spectrum is set by the resonance width parameter r * only independent of the interaction potential considered. In the broad resonance limit all excited Efimov trimer energies approach the ones from the corresponding single-channel system for the scenarios investigated.

Uncovering a Quantum Phase Transition in Nuclei Simulations predict that the ground states of certain light nuclei lie near a quantum phase transition between a liquid-like phase and a phase involving clusters of alpha particles. by David J. Dean * T hermal phase transitions surround us. Ice melts. Water boils. Triggered by thermal fluctuations, these transitions are typically characterized by a reorganization of matter at a critical temperature from one distinct phase to another. Less obvious in our everyday lives are phase transitions that occur in quantum systems at zero temperature. These quantum phase transitions exist in many condensed matter systems, in which tuning such variables as pressure, atom concentration, or magnetic field brings one quantum phase to a "critical point" with another. Instead of thermal fluctuations, these transitions are driven by the quantum fluctuations dictated by the uncertainty principle, and they often involve competing interactions (for example, repulsive versus attractive) between the phases.

We explore the lattice spacing dependence in Nuclear Lattice Effective Field Theory for few-body systems up to next-to-next-to leading order in chiral effective field theory including all isospin breaking and electromagnetic effects, the complete two-pion-exchange potential and the three-nucleon forces. We calculate phase shifts in the neutron-proton system and proton-proton systems as well as the scattering length in the neutron-neutron system. We then perform a full next-to-next-to-leading order calculation with two-nucleon and three-nucleon forces for the triton and helium-4 and analyse their binding energy correlation. We show how the Tjon band is reached by decreasing the lattice spacing and confirm the continuum observation that a four-body force is not necessary to describe light nuclei.

We explore the lattice spacing dependence in Nuclear Lattice Effective Field Theory for few-body systems up to next-to-next-to leading order in chiral effective field theory including all isospin breaking and electromagnetic effects, the complete two-pion-exchange potential and the three-nucleon forces. We calculate phase shifts in the neutron-proton system and proton-proton systems as well as the scattering length in the neutron-neutron system. We then perform a full next-to-next-to-leading order calculation with two-nucleon and three-nucleon forces for the triton and helium-4 and analyse their binding energy correlation. We show how the Tjon band is reached by decreasing the lattice spacing and confirm the continuum observation that a four-body force is not necessary to describe light nuclei.

Uncovering a Quantum Phase Transition in Nuclei Simulations predict that the ground states of certain light nuclei lie near a quantum phase transition between a liquid-like phase and a phase involving clusters of alpha particles. by David J. Dean * T hermal phase transitions surround us. Ice melts. Water boils. Triggered by thermal fluctuations, these transitions are typically characterized by a reorganization of matter at a critical temperature from one distinct phase to another. Less obvious in our everyday lives are phase transitions that occur in quantum systems at zero temperature. These quantum phase transitions exist in many condensed matter systems, in which tuning such variables as pressure, atom concentration, or magnetic field brings one quantum phase to a "critical point" with another. Instead of thermal fluctuations, these transitions are driven by the quantum fluctuations dictated by the uncertainty principle, and they often involve competing interactions (for example, repulsive versus attractive) between the phases.

Basing on Picard-Vessiot theory of noncommutative differential equations and algebraic combinatorics on noncommutative formal series with holomorphic coefficients, various recursive constructions of sequences of grouplike series converging to solutions of universal differential equation are proposed. These intensively use diagonal series in bialgebra and in a Loday's generalized bialgebra. As application, the unique solution, satisfying asymptotic conditions, of Knizhnik-Zamolodchikov equations is provided by dévissage. Contents 1. Introducing remarks 1 2. Combinatorial frameworks 9 2.1. Algebraic combinatorics on formal power series 9 2.2. More about diagonal series in concatenation-shuffle bialgebra and in a Loday's generalized bialgebra 15 3. Solutions of universal differential equation 18 3.1. Noncommutative differential equations 18 3.2. Knizhnik-Zamolodchikov equations 27 4. Conclusion 32 5. Appendices 34 5.1. Aprroximation solution for KZ 3 and identification the coefficients of log Φ KZ by Drinfel'd 34 5.2. KZ 3 , the simplest non-trial case 35 5.3. KZ 4 , other simplest non-trial case 36 References 38

Processes such as the scattering of alpha particles ((4)He), the triple-alpha reaction, and alpha capture play a major role in stellar nucleosynthesis. In particular, alpha capture on carbon determines the ratio of carbon to oxygen during helium burning, and affects subsequent carbon, neon, oxygen, and silicon burning stages. It also substantially affects models of thermonuclear type Ia supernovae, owing to carbon detonation in accreting carbon-oxygen white-dwarf stars. In these reactions, the accurate calculation of the elastic scattering of alpha particles and alpha-like nuclei--nuclei with even and equal numbers of protons and neutrons--is important for understanding background and resonant scattering contributions. First-principles calculations of processes involving alpha particles and alpha-like nuclei have so far been impractical, owing to the exponential growth of the number of computational operations with the number of particles. Here we describe an ab initio calculation o...

We use tools from conformal representation theory to classify the symmetries associated to conformally soft operators in celestial CFT (CCFT) in general dimensions d. The conformal multiplets in d > 2 take the form of celestial necklaces whose structure is much richer than the celestial diamonds in d = 2, it depends on whether d is even or odd and involves mixed-symmetric tensor representations of SO(d). The existence of primary descendants in CCFT multiplets corresponds to (higher derivative) conservation equations for conformally soft operators. We lay out a unified method for constructing the conserved charges associated to operators with primary descendants. In contrast to the infinite local symmetry enhancement in CCFT2, we find the soft symmetries in CCFTd>2 to be finite-dimensional. The conserved charges that follow directly from soft theorems are trivial in d > 2, while non trivial charges associated to (generalized) currents and stress tensor are obtained from the ...

We consider the application of twistor theory to five-dimensional anti-de Sitter space. The twistor space of AdS 5 is the same as the ambitwistor space of the fourdimensional conformal boundary; the geometry of this correspondence is reviewed for both the bulk and boundary. A Penrose transform allows us to describe free bulk fields, with or without mass, in terms of data on twistor space. Explicit representatives for the bulkto-boundary propagators of scalars and spinors are constructed, along with twistor action functionals for the free theories. Evaluating these twistor actions on bulk-to-boundary propagators is shown to produce the correct two-point functions.

We consider the well-known solution of the Heterotic Superstring effective action to zeroth order in α ′ that describes the intersection of a fundamental string with momentum and a solitonic 5-brane and which gives a 3-charge, static, extremal, supersymmetric black hole in 5 dimensions upon dimensional reduction on T5. We compute explicitly the first-order in α ′ corrections to this solution, including SU(2) Yang-Mills fields which can be used to cancel some of these corrections and we study the main properties of this α ′-corrected solution: supersymmetry, values of the near-horizon and asymptotic charges, behavior under α ′-corrected T-duality, value of the entropy (using Wald formula directly in 10 dimensions), existence of small black holes etc. The value obtained for the entropy agrees, within the limits of approximation, with that obtained by microscopic methods. The α ′ corrections coming from Wald’s formula prove crucial for this result.

We consider supersymmetric theories on a space with compact space-like slices. One can count BPS representations weighted by (−1) F , or, equivalently, study supersymmetric partition functions by compactifying the time direction. A special case of this general construction corresponds to the counting of short representations of the superconformal group. We show that in four-dimensional N = 1 theories the "high temperature" asymptotics of such counting problems is fixed by the anomalies of the theory. Notably, the combination a − c of the trace anomalies plays a crucial role. We also propose similar formulae for six-dimensional (1, 0) theories.