High Energy Physics

We have analyzed the γp → pK + K -reaction in the K + K -effective mass region around the mass of the φ(1020) meson. The interference of the S-wave contribution with the P -wave has been studied. Both scalar resonances f 0 (980) and a 0 (980) have been taken into account. We obtained a good description of the available experimental data, in particular the mass distributions and the moments of the kaon angular distribution. Our calculations give values of the integrated S-wave total photoproduction cross section between 4 and 7 nb for the K + K -effective mass range around the φ(1020) mass and at the laboratory photon energy near 5 GeV. These numbers favor lower experimental estimates obtained at DESY.

A summary of the recent ATLAS results at the LHC in Quantum Chromodynamics is given, covering a number of areas that reflect the work of the collaboration on the Bose-Einstein correlations in multi-particle events, the inclusive jet production, the measurements of jet substructure quantities in di-jet events, and the photon-photon scattering exclusive processes.

We have studied the partial wave interference effects to obtain a new information about the contribution of the S-wave to the cross section of the K + K -photoproduction on hydrogen. The K + K -photoproduction channel for the effective masses around 1 GeV is dominated by the φ(1020) resonance with only a small fraction of events coming from decays of scalar resonances f 0 (980) and a 0 (980). However, a careful analysis of angular distributions of the outgoing kaons shows that the S-wave adds an asymmetry to the angular distribution of kaons. A fairly precise estimation of the K + K -photoproduction cross section in the S-wave has been obtained.

We prove that, in a general higher derivative theory of gravity coupled to abelian gauge fields and neutral scalar fields, the entropy and the near horizon background of a rotating extremal black hole is obtained by extremizing an entropy function which depends only on the parameters labeling the near horizon background and the electric and magnetic charges and angular momentum carried by the black hole. If the entropy function has a unique extremum then this extremum must be independent of the asymptotic values of the moduli scalar fields and the solution exhibits attractor behaviour. If the entropy function has flat directions then the near horizon background is not uniquely determined by the extremization equations and could depend on the asymptotic data on the moduli fields, but the value of the entropy is still independent of this asymptotic data. We illustrate these results in the context of two derivative theories of gravity in several examples. These include Kerr black hole, Kerr-Newman black hole, black holes in Kaluza-Klein theory, and black holes in toroidally compactified heterotic string theory.

ACORDE (A COsmic Ray DEtector in ALICE) will be part of the ALICE detector at LHC and its objective is to provide a cosmic ray trigger (level 0). ACORDE will consist of an array of plastic scintillators placed on the top sides of the ALICE magnet. We describe the ...

This study presents empirical evidence of prime resonance structuring in the gravitational wave signals from black hole mergers GW150914, GW151226, and GW170104. Through prime resonance spectral analysis of Fourier-transformed strain data, we identify a consistent pattern: dominant spectral peaks align closely with prime number frequencies (e.g., 73 Hz, 113 Hz, 137 Hz, 173 Hz in GW150914, with analogous alignments in GW151226 and GW170104). The prime resonance strength is notably high (Q ≈ 0.94 for GW150914, with similar values for the others), and entropy reduction post-prime filtering is substantial (∆S ≈ 1.84 bits for GW150914, with comparable reductions in the other events). These results indicate that black holes may function as cosmic prime coherence attractors, organizing spacetime via prime-resonant fields. This discovery offers initial experimental support for a unified theory linking consciousness, prime number theory, and quantum gravity, urging further investigation into additional black hole merger events.

We visited mass spectra and decay constants of pseudoscalar and vector heavy-light mesons (𝐵, 𝐵 푠 , 𝐷, and 𝐷 푆 ) in the framework of QCD sum rule and quark model. The harmonic oscillator wave function was used in quark model while a simple interpolating current was used in QCD sum rule calculation. We obtained good results in accordance with the available experimental data and theoretical studies.

We solved Schrödinger equation with Cornell potential (Coulomb-plus-linear potential) by using neural network approach. Four different cases of Cornell potential for different potential parameters were used without a physical relevance. Besides that charmonium, bottomonium and bottom-charmed spin-averaged spectra were also calculated. Obtained results are in good agreement with the reference studies and available experimental data.

We visited mass spectra and decay constants of pseudoscalar and vector heavy-light mesons (B, Bs, D, and DS) in the framework of QCD sum rule and quark model. The harmonic oscillator wave function was used in quark model while a simple interpolating current was used in QCD sum rule calculation. We obtained good results in accordance with the available experimental data and theoretical studies.

In this paper we revisited phenomenological potentials. We studied S-wave heavy quarkonium spectra by two potential models. The first one is power potential and the second one is logarithmic potential. We calculated spin averaged masses, hyperfine splittings, Regge trajectories of pseudoscalar and vector mesons, decay constants, leptonic decay widths, two-photon and two-gluon decay widths, and some allowed M1 transitions. We studied ground and 4 radially excited S-wave charmonium and bottomonium states via solving nonrelativistic Schrödinger equation. Although the potentials which were studied in this paper are not directly QCD motivated potential, obtained results agree well with experimental data and other theoretical studies.

We calculate the spectrum of glueball masses in non-supersymmetric Yang-Mills theory in three and four dimensions, based on a conjectured duality between supergravity and large N gauge theories. The glueball masses are obtained by solving supergravity wave equations in a black hole geometry. We find that the mass ratios are in good numerical agreement with the available lattice data. We also compute the leading (g~MN)-1 corrections to the glueball masses, by taking into account stringy corrections to the supergravity action and to the black hole metric. We find that the corrections to the masses are negative and of order (g~MN)-3 1 2 • Thus for a fixed ultraviolet cutoff the masses decrease as we decrease the 't Hooft coupling, in accordance with our expectation about the continuum limit of the gauge theories.

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.

Vacuum energy changes during cosmological phase transitions and becomes relatively important at epochs just before phase transitions. For a viable cosmology the vacuum energy just after a phase transition must be set by the critical temperature of the next phase transition, which exposes the cosmological constant problem from a different angle. Here we propose to experimentally test the properties of vacuum energy under circumstances different from our current vacuum. One promising avenue is to consider the effect of high density phases of QCD in neutron stars. Such phases have different vacuum expectation values and a different vacuum energy from the normal phase, which can contribute an order one fraction to the mass of neutron stars. Precise observations of the mass of neutron stars can potentially yield information about the gravitational properties of vacuum energy, which can significantly affect their mass-radius relation. A more direct test of cosmic evolution of vacuum energy could be inferred from a precise observation of the primordial gravitational wave spectrum at frequencies corresponding to phase transitions. While traditional cosmology predicts steps in the spectrum determined by the number of degrees of freedom both for the QCD and electroweak phase transitions, an adjustment mechanism for vacuum energy could significantly change this. In addition, there might be other phase transitions where the effect of vacuum energy could show up as a peak in the spectrum.

We explore how to protect extra dimensional models from large flavor changing neutral currents by using bulk and brane flavor symmetries. We show that a GIM mechanism can be built in to warped space models such as Randall-Sundrum or composite Higgs models if flavor mixing is introduced via UV brane kinetic mixings for right handed quarks. We give a realistic implementation both for a model with minimal flavor violation and one with next-to-minimal flavor violation. The latter does not suffer from a CP problem. We consider some of the existing experimental constraints on these models implied by precision electroweak tests.

We construct a calculable model of low-energy direct gauge mediation making use of the metastable supersymmetry breaking vacua recently discovered by Intriligator, Seiberg and Shih. The standard model gauge group is a subgroup of the global symmetries of the SUSY breaking sector and messengers play an essential role in dynamical SUSY breaking: they are composites of a confining gauge theory, and the holomorphic scalar messenger mass appears as a consequence of the confining dynamics. The SUSY breaking scale is around 100 TeV nevertheless the model is calculable. The minimal non-renormalizable coupling of the Higgs to the DSB sector leads in a simple way to a µ-term, while the B-term arises at two-loop order resulting in a moderately large tan β. A novel feature of this class of models is that some particles from the dynamical SUSY breaking sector may be accessible at the LHC.

We consider Kaplan's domain wall fermions in the presence of an Anti-de Sitter (AdS) background in the extra dimension. Just as in the flat space case, in a completely vector-like gauge theory defined after discretizing this extra dimension, the spectrum contains a very light charged fermion whose chiral components are localized at the ends of the extra dimensional interval. The component on the IR boundary of the AdS space can be given a large mass by coupling it to a neutral fermion via the Higgs mechanism. In this theory, gauge invariance can be restored either by taking the limit of infinite proper length of the extra dimension or by reducing the AdS curvature radius towards zero. In the latter case, the Kaluza-Klein modes stay heavy and the resulting classical theory approaches a chiral gauge theory, as we verify numerically. Potential difficulties for this approach could arise from the coupling of the longitudinal mode of the light gauge boson, which has to be treated non-perturbatively. * See also for other discussions of fermions in extra dimensions.

We present a class of models that contains Randall-Sundrum and Higgsless models as limiting cases. Over a wide range of the parameter space W W scattering is mainly unitarized by Kaluza-Klein partners of the W and Z, and the Higgs particle has suppressed couplings to the gauge bosons. Such a gaugephobic Higgs can be significantly lighter than the 114 GeV LEP bound for a standard Higgs, or heavier than the theoretical upper bound. These models predict a suppressed single top production rate and unconventional Higgs phenomenology at the LHC: the Higgs production rates will be suppressed and the Higgs branching fractions modified. However, the more difficult the Higgs search at the LHC is, the easier the search for other light resonances (like Z ′ , W ′ , t ′ , exotic fermions) will be.

In special relativity, this paper presents the definition of center of mass-energy for a system of particles (massive and non-massive) Consequently, a new principle of conservation is obtained, which depends on the temporal variation of the relativistic energy (power) and the position vector of the particles of the system.

The ALICE experiment at the LHC has studied J/ψ production at mid-rapidity in pp collisions at $ \sqrt{s}=7 $ TeV through its electron pair decay on a data sample corresponding to an integrated luminosity L int = 5.6 nb−1. The fraction of J/ψ from the decay of long-lived beauty hadrons was determined for J/ψ candidates with transverse momentum p t > 1.3 GeV/c and rapidity |y| < 0.9. The cross section for prompt J/ψ mesons, i.e. directly produced J/ψ and prompt decays of heavier charmonium states such as the ψ(2S) and χc resonances, is σprompt J/ψ (p t > 1.3 GeV/c, |y| < 0.9) = 8.3 ± 0.8 (stat.) ± 1.1 (syst.) $ _{-1.4}^{+1.5 } $ (syst. pol.) μb. The cross section for the production of b-hadrons decaying to J/ψ with p t > 1.3 GeV/c and |y| < 0.9 is $ {\sigma_{{{J \left/ {{\psi \leftarrow {{\mathrm{h}}_{\mathrm{B}}}}} \right.}}}} $ (p t > 1.3 GeV/c, |y| < 0.9) = 1.46 ± 0.38 (stat.) $ _{-0.32}^{+0.26 } $ (syst.) μb. The results are compared to QCD model predictio...

We deal with transmission phenomena across highly conductive interfaces, investigating the limit behavior of solutions for second order transmission problems. In particular, we investigate via finite element method the influence of thermal conductivity on the heat flux flowing into an interface.

We examine the boundary behaviour of the gauged N = (2, 0) supergravity in D = 3 coupled to an arbitrary number of scalar supermultiplets which parametrize a Kähler manifold. In addition to the gravitational coupling constant, the model depends on two parameters, namely the cosmological constant and the size of the Kähler manifold. It is shown that regular and irregular boundary conditions can be imposed on the matter fields depending on the size of the sigma model manifold. It is also shown that the super AdS transformations in the bulk produce the transformations of the N = (2, 0) conformal supergravity and scalar multiplets on the boundary, containing fields with nonvanishing Weyl weights determined by the ratio of the sigma model and the gravitational coupling constants. Various types of (2,0) superconformal multiplets are found on the boundary and in one case the superconformal symmetry is shown to be realized in an unconventional way.

We reformulate the Randall-Sundrum (RS) model on the compactified AdS by adding a term proportional to the area of the boundary to the usual gravity action with a negative cosmological constant and show that gravity can still be localized on the boundary without introducing singular brane sources. The boundary conditions now follow from the field equations, which are obtained by letting the induced metric vary on the boundary. This approach gives similar modes that are obtained in and clarifies the complementarity of the RS and the AdS/CFT pictures. Normalizability of these modes is checked by an inner-product in the space of linearized perturbations. The same conclusions hold for a massless scalar field in the bulk.

We derive the necessary and sufficient conditions for the existence of a Killing spinor in N = (1, 0) gauge supergravity in six dimensions coupled to a single tensor multiplet, vector multiplets and hypermultiplets. These are shown to imply most of the field equations and the remaining ones are determined. In this framework, we find a novel 1/8 supersymmetric dyonic string solution with nonvanishing hypermultiplet scalars. The activated scalars parametrize a 4 dimensional submanifold of a quaternionic hyperbolic ball. We employ an identity map between this submanifold and the internal space transverse to the string worldsheet. The internal space forms a 4 dimensional analog of the Gell-Mann-Zwiebach tear-drop which is noncompact with finite volume. While the electric charge carried by the dyonic string is arbitrary, the magnetic charge is fixed in Planckian units, and hence necessarily non-vanishing. The source term needed to balance a delta function type singularity at the origin is determined. The solution is also shown to have 1/4 supersymmetric AdS 3 × S 3 near horizon limit where the radii are proportional to the electric charge.

The classical cross section for low energy absorption of the RRscalar by a stack of noncommutative D3-branes in the large NS B-field limit is calculated. In the spirit of AdS/CFT correspondence, this cross section is related to two point function of a certain operator in noncommutative Yang-Mills theory. Compared at the same gauge coupling, the result agrees with that of obtained from ordinary D3-branes. This is consistent with the expectation that ordinary and noncommutative Yang-Mills theories are equivalent below the noncommutativity scale, but it is a nontrivial prediction above this scale.

The complete spectrum of D = 6, N = 4b supergravity with n tensor multiplets compactified on AdS 3 × S 3 is determined. The D = 6 theory obtained from the K 3 compactification of Type IIB string requires that n = 21, but we let n be arbitrary. The superalgebra that underlies the symmetry of the resulting supergravity theory in AdS 3 coupled to matter is SU (1, 1|2) L × SU (1, 1|2) R . The theory also has an unbroken global SO(4) R × SO(n) symmetry inherited from D = 6. The spectrum of states arranges itself into a tower of spin 2 supermultiplets, a tower of spin 1, SO(n) singlet supermultiplets, a tower of spin 1 supermultiplets in the vector representation of SO(n) and a special spin 1 2 supermultiplet also in the vector representation of SO(n). The SU (2) L × SU (2) R Yang-Mills states reside in the second level of the spin 2 tower and the lowest level of the spin 1, SO(n) singlet tower and the associated field theory exhibits interesting properties.

We investigate oscillating instanton solutions of a self-gravitating scalar field between degenerate vacua. We show that there exist O(4)symmetric oscillating solutions in a de Sitter background. The geometry of this solution is finite and preserves the Z 2 symmetry. The nontrivial solution corresponding to tunneling is possible only if the effect of gravity is taken into account. We present numerical solutions of this instanton, including the phase diagram of solutions in terms of the parameters of the present work and the variation of energy densities. Our solutions can be interpreted as solutions describing an instanton-induced domain wall or braneworld-like object rather than a kink-induced domain wall or braneworld. The oscillating instanton solutions have a thick wall and the solutions can be interpreted as a mechanism providing nucleation of the thick wall for topological inflation. We remark that Z 2 invariant solutions also exist in a flat and anti-de Sitter background, though the physical significance is not clear.

We present a comprehensive theoretical and experimental framework for gravitational modulation using entropy-driven, prime-resonant electromagnetic fields. By embedding symbolic entropy gradients into the dynamics of prime-indexed harmonic oscillators, we demonstrate the emergence of effective curvature in spacetime. This curvature is derived from coherent resonance collapse processes and quantified via a modified gravitational field tensor. We simulate the symbolic curvature field, visualize particle geodesic trajectories, and propose experimental schematics for physical implementation. This framework suggests a novel paradigm for manipulating spacetime geometry through symbolic information dynamics.

We report on measurements of the time-dependent CP violating observables in B 0 s → D ∓ s K ± decays using a dataset corresponding to 1.0 fb -1 of pp collisions recorded with the LHCb detector. We find the CP violating observables C f = 0.53±0.25±0.04, A ∆Γ f = 0.37±0.42±0.20, A ∆Γ f = 0.20±0.41±0.20, S f = -1.09±0.33±0.08, S f = -0.36 ± 0.34 ± 0.08, where the uncertainties are statistical and systematic, respectively. Using these observables together with a recent measurement of the B 0 s mixing phase -2β s leads to the first extraction of the CKM angle γ from B 0 s → D ∓ s K ± decays, finding γ = (115 +28 -43 ) • modulo 180 • at 68% CL, where the error contains both statistical and systematic uncertainties.

We study the first law of black hole thermodynamics in the presence of surrounding gravitational fields and argue that variations of these fields are naturally incorporated in the first law by defining gravitational tension or gravitational binding energy. We demonstrate that this notion can also be applied in Anti-de Sitter spacetime, in which the surrounding gravitational field is sourced by a cosmological fluid, therefore showing that spacetime volume and gravitational tension encode the same physics as spacetime pressure and black hole volume. We furthermore show that it is possible to introduce a definition of spacetime pressure and black hole volume for any spacetime with characteristic length scales which does not necessarily require a cosmological constant sourcing Einstein equations. However, we show that black hole volume is non-universal in the flat spacetime limit, questioning its significance. We illustrate these ideas by studying the resulting black hole volume of Kaluza-Klein black holes and of a toy model for a black hole binary system in five spacetime dimensions (the black saturn solution) as well as of several novel perturbative black hole solutions. These include the higher-dimensional Kerr-Newman solution in Anti-de Sitter spacetime as well as other black holes in plane wave and Lifshitz spacetimes.

Soft and collinear radiation in collider processes can be described in a universal way, that is independent of the underlying process. Recent years have seen a number of approaches for probing whether radiation beyond the leading soft approximation can also be systematically classified. In this paper, we study a formula that captures the leading next-to-soft QCD radiation affecting processes with both final-and initial-state partons, by shifting the momenta in the nonradiative squared amplitude. We first examine W+jet production, and show that a previously derived formula of this type indeed holds in the case in which massive colour singlet particles are present in the final state. Next, we develop a physical understanding of the momentum shifts, showing precisely how they disrupt the well-known angular ordering property of leading soft radiation.

Soft and collinear radiation in collider processes can be described in a universal way, that is independent of the underlying process. Recent years have seen a number of approaches for probing whether radiation beyond the leading soft approximation can also be systematically classified. In this paper, we study a formula that captures the leading next-to-soft QCD radiation affecting processes with both final-and initial-state partons, by shifting the momenta in the nonradiative squared amplitude. We first examine W+jet production, and show that a previously derived formula of this type indeed holds in the case in which massive colour singlet particles are present in the final state. Next, we develop a physical understanding of the momentum shifts, showing precisely how they disrupt the well-known angular ordering property of leading soft radiation.

A search for dijet resonances in events with at least one isolated charged lepton is performed using 139 fb -1 of √ s = 13 TeV proton-proton collision data recorded by the ATLAS detector at the LHC. The dijet invariant-mass (m jj ) distribution constructed from events with at least one isolated electron or muon is searched in the region 0.22 < m jj < 6.3 TeV for excesses above a smoothly falling background from Standard Model processes. Triggering based on the presence of a lepton in the event reduces limitations imposed by minimum transverse momentum thresholds for triggering on jets. This approach allows smaller dijet invariant masses to be probed than in inclusive dijet searches, targeting a variety of new-physics models, for example ones in which a new state is produced in association with a leptonically decaying W or Z boson. No statistically significant deviation from the Standard Model background hypothesis is found. Limits on contributions from generic Gaussian signals with widths ranging from that determined by the detector resolution up to 15% of the resonance mass are obtained for dijet invariant masses ranging from 0.25 TeV to 6 TeV. Limits are set also in the context of several scenarios beyond the Standard Model, such as the Sequential Standard Model, a technicolor model, a charged Higgs boson model and a simplified Dark Matter model.

The production of the prompt charm mesons D0, D+, D*+, and their antiparticles, was measured with the ALICE detector in Pb-Pb collisions at the LHC, at a centre-of-mass energy $ \sqrt{{{s_{\mathrm{NN}}}}}=2.76\;\mathrm{TeV} $ per nucleon-nucleon collision. The p t-differential production yields in the range 2 &lt; p t &lt; 16 GeV/c at central rapidity, |y| &lt; 0.5, were used to calculate the nuclear modification factor R AA with respect to a proton-proton reference obtained from the cross section measured at $ \sqrt{s}=7\;\mathrm{TeV} $ and scaled to $ \sqrt{s}=2.76\;\mathrm{TeV} $ . For the three meson species, R AA shows a suppression by a factor 3–4, for transverse momenta larger than 5 GeV/c in the 20% most central collisions. The suppression is reduced for peripheral collisions.

We propose a new (chiral) description of partially-massless fields in 4d, including the partially-massless graviton, that is similar to the pure connection formulation for gravity and massless higher spin fields, the latter having a clear twistor origin. The new approach allows us to construct complete examples of higher spin gravities with (partially-)massless fields that feature Yang-Mills and current interactions.

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We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical rmatrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the extended monodromy manifold. We show that Fock-Goncharov coordinates are log-canonical for the symplectic form. Using these coordinates we define the symplectic potential on the monodromy manifold and interpret the Jimbo-Miwa-Ueno tau-function as the generating function of the monodromy map. This, in particular, solves a recent conjecture by A. Its, O. Lisovyy and A. Prokhorov.

3 Presentation Highlights 2 3.1 Tau-functions and geometry of moduli spaces. . . . . . . . . . . . . . . . . . . . 2 3.2 Tau-functions and asymptotics of correlation functions . . . . . . . . . . . . . . . 3 3.3 Tau functions, Fredholm and Töplitz determinants . . . . . . . . . . . . . . . . . 3 3.4 Integrable structures of Quantum Field Theory (CFT) . . . . . . . . . . . . . . . . 4 3.5 Tau-functions in the theory of topological recursion and cluster algebras . . . . . . 5 3.6 Tau functions in random matrix theory and random processes. . . . . . . . . . . . 5 3.7 Tau functions, Integrable Systems and vector bundles. . . . . . . . . . . . . . . . . 7

We solve the quantum system with the symmetric Razavy cosine type potential and find that its exact solutions are given by the confluent Heun function. The eigenvalues are calculated numerically. The properties of the wave functions, which depend on the potential parameter a, are illustrated for a given potential parameter ξ. It is shown that the wave functions are shrunk to the origin when the potential parameter a increases. We note that the energy levels ϵi (i∈[1,3]) decrease with the increasing potential parameter a but the energy levels ϵi (i∈[4,7]) first increase and then decrease with the increasing a.

An analysis about the antisymmetric tensor matter fields Avdeev-Chizhov theory in a curved space-time is performed. We show, in a curved spacetime, that the Avdeev-Chizhov theory can be seen as a kind of a $\lambda\phi^{4}$ theory for a &quot;complex self-dual&quot; field. This relationship between Avdeev-Chizhov theory and $\lambda\phi^{4}$ theory simplify the study of tensor matter fields in a curved space-time. The energy-momentum tensor for matter fields is computed.

The black hole area spectrum has been studied in the framework of tunneling mechanism, where a Generalized Uncertainty Principle (GUP) has been considered. The results implies in a non evenly spaced spectrum for the black hole area which becomes increasingly spaced as the black hole evaporates.