In perturbation theory, the spectral densities of two-point functions develop non-integrable threshold singularities at higher orders. In QCD, such singularities emerge when calculating the diagrams in terms of the pole quark mass, and they become stronger when one rearranges the perturbative expansion in terms of the running quark mass. In this paper we discuss the proper way to handle such singularities.
Certain perturbation graphs in the calculation of the effects of the medium on neutrino scattering in supernova matter have a nonintegrable singularity in a physical region. A number of papers have addressed the apparent pathology through an ansatz that invokes higher order (rescattering) effects. Taking the Gamow-Teller terms as an example, we display an expression for the spin-spin correlation function that determines the cross-sections. It is clear from the form that there are no pathologies in the order by order perturbation expansion. Explicit formulae are given for a simple case, leading…
We consider the problem of propagation of an unstable particle in the framework of Quantum Field Theory. Using unitarity, we show that a real renormalization constant free of threshold singularities naturally arises.
We present a method to evaluate numerically Feynman diagrams directly from their Feynman parameters representation. We first disentangle overlapping singularities using sector decomposition. Threshold singularities are treated with an appropriate contour deformation. We have validated our technique comparing with recent analytic results for the gg->h two-loop amplitudes with heavy quarks and scalar quarks.
In a previous communication by two of us, Phys. Rev. Lett. 81, 1373 (1998), the gauge-dependent deviations of the on-shell mass and total decay width from their gauge-independent pole counterparts were investigated at leading order for the Higgs boson of the Standard Model. In the case of the widths, the deviation was found to diverge at unphysical thresholds, m_H = 2 root{xi_V} m_V (V = W,Z), in the R_xi gauge. In this Brief Report, we demonstrate that these unphysical threshold singularities are properly eliminated if a recently proposed definition of wave-function renormalization for unstab…
We show how to correctly treat threshold singularities in fixed-order perturbative calculations of the electron anomalous magnetic moment and hadronic pair production processes such as top pair production. With respect to the former, we demonstrate the equivalence of the "non-perturbative", resummed treatment of the vacuum polarization contribution, whose spectral function exhibits bound state poles, with the fixed-order calculation by identifying a threshold localized term in the four-loop spectral function. In general, we find that a modification of the dispersion relation by threshold subtr…
We present the first comprehensive analysis of the unitarity thresholds and anomalous thresholds of scattering amplitudes at two loops and beyond based on the loop-tree duality, and show how non-causal unphysical thresholds are locally cancelled in an efficient way when the forest of all the dual on-shell cuts is considered as one. We also prove that soft and collinear singularities at two loops and beyond are restricted to a compact region of the loop three-momenta, which is a necessary condition for implementing a local cancellation of loop infrared singularities with the ones appearing in r…
The effect of threshold singularities induced by unstable particles on two-loop observables is investigated and it is shown how to cure them working in the complex-mass scheme. The impact on radiative corrections around thresholds is thoroughly analyzed and shown to be relevant for two selected LHC and ILC applications: Higgs production via gluon fusion and decay into two photons at two loops in the Standard Model. Concerning Higgs production, it is essential to understand possible sources of large corrections in addition to the well-known QCD effects. It is shown that NLO electroweak correcti…
On the analytic continuation and matching of threshold singularities from Vacuum Amplitudes
Universidad Autónoma de Sinaloa（西诺阿自治大学）；Universität Zürich（苏黎世大学）；Universidad de Salamanca（萨拉曼卡大学）；Universitat de València – Consejo Superior de Investigaciones Científicas（瓦伦西亚大学–西班牙国家研究委员会）
We propose a new approach that allows for the separate numerical calculation of the real and imaginary parts of finite loop integrals. We find that at one-loop the real part is given by the Loop-Tree Duality integral supplemented with suitable counterterms and the imaginary part is a sum of two-body phase space integrals, constituting a locally finite representation of the generalised optical theorem. These expressions are integrals in momentum space, whose integrands were specially designed to feature local cancellations of threshold singularities. Such a representation is well suited for Mon…
The spectrum of hadrons is the manifestation of color confinement of quantum chromodynamics. Hadronic resonances correspond to poles of the S-matrix. Since 2003, lots of new hadron resonant structures were discovered in the mass regions from light mesons to hadrons containing a pair of a heavy quark and an antiquark. Many of them are candidates of exotic hadrons, and they are usually observed as peaks in invariant mass distributions. However, the S-matrix also has kinematical singularities due to the on-shellness of intermediate particles for a process, such as two-body thresholds and triangle…
The knowledge of hadron spectrum is based on experimental observations of hadronic resonances. The resonances are normally observed as peaks in certain invariant mass distributions. However, neither is a peak necessarily due to the presence of a resonance, nor does a resonance necessarily lead to a peak. Kinematic singularities can also produce peaks. Here, we discuss such possibilities and methods distinguishing genuine resonances from kinematic effects.
We demonstrate that triangle singularity (TS) and box singularity (BS) mechanisms can produce unique narrow enhancements at the $Λ_c\bar{D}$ and $Λ_c\bar{D}^*$ thresholds in the invariant mass spectra of $J/ψp$ and $J/ψpπ$, respectively. Taking into account that such mechanisms only depend on the initial $Σ_c^{(*)}\bar{D}^{(*)}$ interactions near threshold within the TS or BS kinematic regimes, the $Λ_c\bar{D}$ and $Λ_c\bar{D}^*$ threshold enhancements can be regarded as a feed-down phenomenon originated from both the heavier pentaquark decays and the $Σ_c^{(*)}\bar{D}^{(*)}$ scatterings from …
At one loop in the conventional on-mass-shell renormalization scheme, the production and decay rates of the Higgs boson H exhibit singularities proportional to (2 M_V - M)^{-1/2} as the Higgs-boson mass M approaches from below the pair-production threshold of a vector boson V with mass M_V. This problem is of phenomenological interest because the values 2 M_W and 2 M_Z, corresponding to the W- and Z-boson thresholds, lie within the M range presently favoured by electroweak precision data. We demonstrate how these threshold singularities are eliminated when the definitions of mass and total dec…
We have analyzed the singularities of a triangle loop integral in detail and derived a formula for an easy evaluation of the triangle singularity on the physical boundary. It is applied to the $Λ_b \rightarrow J/ψK^{-} p$ process via $Λ^*$-charmonium-proton intermediate states. Although the evaluation of absolute rates is not possible, we identify the $χ_{c1}$ and the $ψ(2S)$ as the relatively most relevant states among all possible charmonia up to the $ψ(2S)$. The $Λ(1890)\, χ_{c1}\, p$ loop is very special as its normal threshold and triangle singularities merge at about 4.45 GeV, generating…
Motivated by the existence of candidates for exotic hadrons whose masses are close to both of two-body and three-body hadronic thresholds lying close to each other, we study degenerate two-body and three-body coupled-channel systems. We first formulate the scattering problem of non-degenerate two-body and three-body coupled-channels as an effective three-body problem, i.e.\ effective Alt-Grassberger-Sandhas (AGS) equations. We next investigate the behavior of $S$-matrix poles near the threshold when two-body and three-body thresholds are degenerate. We solve the eigenvalue equations of the ker…
We discuss analytical properties of partial waves derived from projection of a 4-legged amplitude with crossed-channel exchanges in the kinematic region of the direct channel that corresponds to the XYZ peaks in charmonium and bottomonium. We show that in general partial waves can develop anomalous branch points in the vicinity of the direct channel physical region. In a specific case, when these branch points lie on the opposite side of the unitary cut they pinch the integration contour in a dispersion relation and if the pinch happens close to threshold, the normal threshold cusp is enhanced…
Threshold and infrared divergences are studied as possible mechanisms of particle production and compared to the usual decay process in a model quantum field theory from which generalizations are obtained. A spectral representation of the propagator of the decaying particle suggests that decay, threshold and infrared singularities while seemingly different phenomena are qualitatively related. We implement a non-perturbative dynamical resummation method to study the time evolution of an initial state. It is manifestly unitary and yields the asymptotic state and the distribution function of prod…
We analyze ill-defined pinch singularities characteristic of out of equilibrium thermal field theories. We identify two mechanisms that eliminate pinching even at the single self-energy insertion approximation to the propagator: the first is based on the vanishing of phase space at the singular point (threshold effect). It is effective in QED with a massive electron and a massless photon. In massless QCD, this mechanism fails, but the pinches cancel owing to the second mechanism, i.e., owing to the spinor/tensor structure of the single self-energy insertion contribution to the propagator. The …
The $q_{\mathrm{T}}$ and $Δϕ_{t\bar{t}}$ spectra in top-antitop hadroproduction at NNLL+NNLO: the interplay of soft-collinear resummation and Coulomb singularities
The gauge invariant quark Green's function with a path-ordered phase factor along a straight-line is studied in two-dimensional QCD in the large-Nc limit by means of an exact integrodifferential equation. Its spectral functions are analytically determined. They are infra-red finite and lie on the positive real axis of the complex plane of the momentum squared variable, corresponding to momenta in the forward light cone. Their singularities are represented by an infinite number of threshold type branch points with power-law -3/2, starting at positive mass values, characterized by an integer num…
We investigate the double logarithmic corrections $α_s\ln^2 x$, x being a parton momentum fraction, in two-body nonleptonic B meson decays in collinear factorization theorem of perturbative QCD (PQCD). It is found that these corrections are universal for factorizable amplitudes, {\it i.e.}, at the leading power, and that their threshold resummation smears the end-point singularities from $x\to 0$. The double logarithmic corrections, depending on the topologies of nonfactorizable amplitudes at the subleading power, are negligible due to the color-transparency argument and to the overlap of thre…
We present the analytic expressions for the two-loop form factors for the production or decay of pseudo-scalar quarkonia, in a scheme where the quarks are produced at threshold. We consider the two-loop amplitude for the process $γγ\leftrightarrow {^1S_0^{[1]}}$, that was previously known only numerically, as well as for the processes $gg \leftrightarrow {^1S_0^{[1]}}$, $γg \leftrightarrow {^1S_0^{[8]}}$ and $gg \leftrightarrow {^1S_0^{[8]}}$, which have not been computed before. The two-loop corrections to $gg \leftrightarrow {^1S_0^{[1]}}$ are the last missing ingredients for a full NNLO cal…
Analytic decomposition of two-body electroweak processes with left-hand cuts
Ruhr-Universität Bochum（波鸿鲁尔大学）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；CSIC‐Universitat de Valéncia（西班牙国家研究委员会-瓦伦西亚大学）
