The tensor reduction and master integrals of the two-loop massless crossed box with light-like legs

The class of the two-loop massless crossed boxes, with light-like external legs, is the final unresolved issue in the program of computing the scattering amplitudes of 2 --> 2 massless particles at next-to-next-to-leading order. In this paper, we describe an algorithm for the tensor reduction of such diagrams. After connecting tensor integrals to scalar ones with arbitrary powers of propagators in higher dimensions, we derive recurrence relations from integration-by-parts and Lorentz-invariance identities, that allow us to write the scalar integrals as a combination of two master crossed boxes plus simpler-topology diagrams. We derive the system of differential equations that the two master integrals satisfy using two different methods, and we use one of these equations to express the second master integral as a function of the first one, already known in the literature. We then give the analytic expansion of the second master integral as a function of epsilon=(4-D)/2, where D is the space-time dimension, up to order O(epsilon^0).Comment: 30 pages, 5 figure

Two-Loop Self-Energy Corrections to the Fine-Structure

We investigate two-loop higher-order binding corrections to the fine structure, which contribute to the spin-dependent part of the Lamb shift. Our calculation focuses on the so-called ``two-loop self-energy'' involving two virtual closed photon loops. For bound states, this correction has proven to be notoriously difficult to evaluate. The calculation of the binding corrections to the bound-state two-loop self-energy is simplified by a separate treatment of hard and soft virtual photons. The two photon-energy scales are matched at the end of the calculation. We explain the significance of the mathematical methods employed in the calculation in a more general context, and present results for the fine-structure difference of the two-loop self-energy through the order of $\alpha^8$.Comment: 19 pages, LaTeX, 2 figures; J. Phys. A (in press); added analytic results for two-loop form-factor slopes (by P. Mastrolia and E. Remiddi

Decays of Scalar and Pseudoscalar Higgs Bosons into Fermions: Two-loop QCD Corrections to the Higgs-Quark-Antiquark Amplitude

As a first step in the aim of arriving at a differential description of neutral Higgs boson decays into heavy quarks, $h \to Q {\bar Q}X$, to second order in the QCD coupling $\alpha_S$, we have computed the $hQ{\bar Q}$ amplitude at the two-loop level in QCD for a general neutral Higgs boson which has both scalar and pseudoscalar couplings to quarks. This amplitude is given in terms of a scalar and a pseudoscalar vertex form factor, for which we present closed analytic expressions in terms of one-dimensional harmonic polylogarithms of maximum weight 4. The results hold for arbitrary four-momentum squared, $q^2$, of the Higgs boson and of the heavy quark mass, $m$. Moreover we derive the approximate expressions of these form factors near threshold and in the asymptotic regime $m^2/q^2 \ll 1$.Comment: 56 pages, 2 figure

Two-Loop Bethe Logarithms for Higher Excited S Levels

Processes mediated by two virtual low-energy photons contribute quite significantly to the energy of hydrogenic S states. The corresponding level shift is of the order of (alpha/pi)^2 (Zalpha)^6 m_e c^2 and may be ascribed to a two-loop generalization of the Bethe logarithm. For 1S and 2S states, the correction has recently been evaluated by Pachucki and Jentschura [Phys. Rev. Lett. vol. 91, 113005 (2003)]. Here, we generalize the approach to higher excited S states, which in contrast to the 1S and 2S states can decay to P states via the electric-dipole (E1) channel. The more complex structure of the excited-state wave functions and the necessity to subtract P-state poles lead to additional calculational problems. In addition to the calculation of the excited-state two-loop energy shift, we investigate the ambiguity in the energy level definition due to squared decay rates.Comment: 14 pages, RevTeX, to appear in Phys. Rev.

Caesarean section on maternal request: an Italian comparative study on patients’ characteristics, pregnancy outcomes and guidelines overview

In recent years, the rate of caesarean sections has risen all over the world. Accordingly, efforts are being made worldwide to understand this trend and to counteract it effectively. Several factors have been identified as contributing to the selection of caesarean section (CS), especially an obstetricians’ beliefs, attitudes and clinical practices. However, relatively few studies have been conducted to understand the mechanisms involved, to explore influencing factors and to clearly define the risks associated with the caesarean section on maternal request (CSMR). This comparative study was conducted to elucidate the factors influencing the choice of CSMR, as well as to compare the associated risks of CSMR to CS for breech presentation among Italian women. From 2015 to 2018, a total of 2348 women gave birth by caesarean section, of which 8.60% (202 women) chose a CSMR. We found that high educational attainment, use of assisted reproductive technology, previous operative deliveries and miscarriages within the obstetric history could be positively correlated with the choice of CSMR in a statistically significant way. This trend was not confirmed when the population was stratified based on patients’ characteristics, obstetric complications and gestational age. Finally, no major complications were found in patients that underwent CSMR. We believe that it is essential to evaluate patients on a case-by-case basis. It is essential to understand the personal experience, to explain the knowledge available on the subject and to ensure a full understanding of the risks and benefits of the medical practice to guarantee the patients not only their best scientific preparation but also human understanding

Magnetic dipole operator contributions to the photon energy spectrum in anti-B -> X(s) gamma at O(alpha(s)^2)

We compute the O(\alpha_s^2) contributions to the photon energy spectrum of the inclusive decay \bar{B} -> X_s \gamma associated with the magnetic penguin operator O_7. They are an essential part of the ongoing NNLO calculation of this important decay. We use two different methods to evaluate the master integrals, one based on the differential equation approach and the other on sector decomposition, leading to identical results which in turn agree with those of a recent independent calculation by Melnikov and Mitov. We study the numerical relevance of this set of NNLO contributions in the photon energy spectrum and discuss the change of bottom quark mass scheme.Comment: 18 pages, uses axodraw.st

Feynman Diagrams and Differential Equations

We review in a pedagogical way the method of differential equations for the evaluation of D-dimensionally regulated Feynman integrals. After dealing with the general features of the technique, we discuss its application in the context of one- and two-loop corrections to the photon propagator in QED, by computing the Vacuum Polarization tensor exactly in D. Finally, we treat two cases of less trivial differential equations, respectively associated to a two-loop three-point, and a four-loop two-point integral. These two examples are the playgrounds for showing more technical aspects about: Laurent expansion of the differential equations in D (around D=4); the choice of the boundary conditions; and the link among differential and difference equations for Feynman integrals.Comment: invited review article from Int. J. Mod. Phys.

Analytic Results for Virtual QCD Corrections to Higgs Production and Decay

We consider the production of a Higgs boson via gluon-fusion and its decay into two photons. We compute the NLO virtual QCD corrections to these processes in a general framework in which the coupling of the Higgs boson to the external particles is mediated by a colored fermion and a colored scalar. We present compact analytic results for these two-loop corrections that are expressed in terms of Harmonic Polylogarithms. The expansion of these corrections in the low and high Higgs mass regimes, as well as the expression of the new Master Integrals which appear in the reduction of the two-loop amplitudes, are also provided. For the fermionic contribution, we provide an independent check of the results already present in the literature concerning the Higgs boson and the production and decay of a pseudoscalar particle.Comment: 19 pages, 3 figures, version accepted by JHE

Large scale analytic calculations in quantum field theories

We present a survey on the mathematical structure of zero- and single scale quantities and the associated calculation methods and function spaces in higher order perturbative calculations in relativistic renormalizable quantum field theories.Comment: 25 pages Latex, 1 style fil

Two-Loop Bhabha Scattering in QED

In the context of pure QED, we obtain analytic expressions for the contributions to the Bhabha scattering differential cross section at order alpha^4 which originate from the interference of two-loop photonic vertices with tree-level diagrams and from the interference of one-loop photonic diagrams amongst themselves. The ultraviolet renormalization is carried out. The IR-divergent soft-photon emission corrections are evaluated and added to the virtual cross section. The cross section obtained in this manner is valid for on-shell electrons and positrons of finite mass, and for arbitrary values of the center of mass energy and momentum transfer. We provide the expansion of our results in powers of the electron mass, and we compare them with the corresponding expansion of the complete order alpha^4 photonic cross section, recently obtained in hep-ph/0501120. As a by-product, we obtain the contribution to the Bhabha scattering differential cross section of the interference of the two-loop photonic boxes with the tree-level diagrams, up to terms suppressed by positive powers of the electron mass. We evaluate numerically the various contributions to the cross section, paying particular attention to the comparison between exact and expanded results.Comment: 35 pages, 18 figure