Bound state equivalent potentials with the Lagrange mesh method

The Lagrange mesh method is a very simple procedure to accurately solve eigenvalue problems starting from a given nonrelativistic or semirelativistic two-body Hamiltonian with local or nonlocal potential. We show in this work that it can be applied to solve the inverse problem, namely, to find the equivalent local potential starting from a particular bound state wave function and the corresponding energy. In order to check the method, we apply it to several cases which are analytically solvable: the nonrelativistic harmonic oscillator and Coulomb potential, the nonlocal Yamaguchi potential and the semirelativistic harmonic oscillator. The potential is accurately computed in each case. In particular, our procedure deals efficiently with both nonrelativistic and semirelativistic kinematics.Comment: 6 figure

A large-NcN_c PNJL model with explicit ZNc_{N_c} symmetry

A PNJL model is built, in which the Polyakov-loop potential is explicitly Z$_{N_c}$-symmetric in order to mimic a Yang-Mills theory with gauge group SU($N_c$). The physically expected large-$N_c$ and large-$T$ behaviours of the thermodynamic observables computed from the Polyakov-loop potential are used to constrain its free parameters. The effective potential is eventually U(1)-symmetric when $N_c$ is infinite. Light quark flavours are added by using a Nambu-Jona-Lasinio (NJL) model coupled to the Polyakov loop (the PNJL model), and the different phases of the resulting PNJL model are discussed in 't Hooft's large-$N_c$ limit. Three phases are found, in agreement with previous large-$N_c$ studies. When the temperature $T$ is larger than some deconfinement temperature $T_d$, the system is in a deconfined, chirally symmetric, phase for any quark chemical potential $\mu$. When $T<T_d$ however, the system is in a confined phase in which chiral symmetry is either broken or not. The critical line $T_\chi(\mu)$, signalling the restoration of chiral symmetry, has the same qualitative features than what can be obtained within a standard $N_c=3$ PNJL model.Comment: To appear in Phys Rev

Women's experiences of participating in a randomised trial comparing alternative policies for timing of cord clamping at very preterm birth: a questionnaire study

BACKGROUND: The Cord Pilot Trial compared two alternative policies for cord-clamping at very preterm birth at eight UK tertiary maternity units: clamping after at least 2 min and immediate neonatal care with cord intact, or clamping within 20 s and neonatal care after clamping. This paper reports views and experiences of the women who participated in the trial (261 randomised), based on data from two self-completed questionnaires. METHODS: Women were given or posted the first questionnaire between 4 and 8 weeks after birth, and posted a second similar questionnaire at 1 year. Both questionnaires included three questions about experiences of participating in the trial: (1) If time suddenly went backwards and you had to do it all over again, would you agree to participate in the Cord Pilot Trial?; (2) Please tell us if there was anything about the Cord Pilot Trial that you think could have been done better; and (3) Please tell us if there was anything about the Cord Pilot Trial, or your experiences of joining the trial, that you think were particularly good. RESULTS: One hundred and eighty-six women completed the first questionnaire and 133 completed the second. At both time points, 90% responded 'probably' or 'definitely' to participating in the trial again. More women randomised to deferred clamping responded 'definitely yes' than those allocated immediate clamping (78% versus 67% first questionnaire). Women were positive about the level of information and explanations, the friendly and caring staff, and the benefits for their baby and others as a result of participating in the trial. Suggestions for how the trial could be done better included being approached earlier, better staff communication about the trial, more information overall, and better timing of follow-up. CONCLUSIONS: Women were largely positive about participating in the trial. Nevertheless, they had suggestions for how the study could have been improved. These suggestions have implications for the design of future trials. TRIAL REGISTRATION: ISRCTN21456601 . Registered on 28 February 2013

Cord pilot trial, comparing alternative policies for timing of cord clamping before 32 weeks gestation: follow-up for women up to one year.

BACKGROUND: The Cord Pilot Trial compared two alternative policies for cord clamping at very preterm birth at eight UK maternity units: clamping after at least 2 min and immediate neonatal care (if needed) with cord intact, or clamping within 20 s and neonatal care after clamping. This paper reports follow-up of the women by two self-completed questionnaires up to one year after the birth. METHODS: Women were given or posted the first questionnaire between four and eight weeks after birth, usually before their baby was discharged, and were posted a second similar questionnaire at one year. The questionnaire included the Hospital Anxiety and Depression Scale; the Preterm Birth Experience and Satisfaction Scale (P-BESS) and questions about their baby's feeding. RESULTS: Of 261 women randomised (132 clamping ≥2 min, 129 clamping ≤20 s), six were excluded as birth was after 35+ 6 weeks (2, 4 in each group respectively). Six were not sent either questionnaire. The first questionnaire was given/sent to 244 and returned by 186 (76%) (79, 74%). The second, at one year, was sent to 242 and returned by 133 (55%) (66, 43%). On the first questionnaire, 89 (49%) had a score suggestive of an anxiety disorder, and 55 (30%) had a score suggestive of depression. Satisfaction with care at birth was high: median total P-BESS score 77 [interquartile range 68 to 84] (scale 17 to 85). There was no clear difference in anxiety, depression, or satisfaction with care between the two allocated groups. The median number of weeks after birth women breastfed/expressed was 16 (95% confidence interval (CI) 13 to 20, n = 119) for those allocated clamping ≥2 min and 12 (95% CI 11 to 16, n = 103) for those allocated clamping ≤20 s. CONCLUSIONS: The response rate was higher for the earlier questionnaire than at one year. A high proportion of women reported symptoms of anxiety or depression, however there were no clear differences between the allocated groups. Most women reported that they had breastfed or expressed milk and those allocated deferred cord clamping reported continuing this for slightly longer. TRIAL REGISTRATION: ISRCTN 21456601, registered 28th February 2013, http://www.isrctn.com/ISRCTN21456601

Gluons in glueballs: Spin or helicity?

In the last decade, lattice QCD has been able to compute the low-lying glueball spectrum with accuracy. Like other effective approaches of QCD, potential models still have difficulties to cope with gluonic hadrons. Assuming that glueballs are bound states of valence gluons with zero current mass, it is readily understood that the use of a potential model, intrinsically non covariant, could be problematic in this case. The main challenge for this kind of model is actually to find a way to introduce properly the more relevant degree of freedom of the gluon: spin or helicity. In this work, we use the so-called helicity formalism of Jacob and Wick to describe two-gluon glueballs. We show in particular that this helicity formalism exactly reproduces the $J^{PC}$ numbers which are observed in lattice QCD when the constituent gluons have a helicity-1, without introducing extra states as it is the case in most of the potential models. These extra states appear when gluons are seen as spin-1 particles. Using a simple spinless Salpeter model with Cornell potential within the helicity formalism, we obtain a glueball mass spectrum which is in good agreement with lattice QCD predictions for helicity-1 gluons provided instanton-induced interactions are taken into account.Comment: 2 figures, section III.B corrected but conclusions unchange

Auxiliary field method and analytical solutions of the Schr\"{o}dinger equation with exponential potentials

The auxiliary field method is a new and efficient way to compute approximate analytical eigenenergies and eigenvectors of the Schr\"{o}dinger equation. This method has already been successfully applied to the case of central potentials of power-law and logarithmic forms. In the present work, we show that the Schr\"{o}dinger equation with exponential potentials of the form $-\alpha r^\lambda \exp(-\beta r)$ can also be analytically solved by using the auxiliary field method. Formulae giving the critical heights and the energy levels of these potentials are presented. Special attention is drawn on the Yukawa potential and the pure exponential one

Short-range potentials from QCD at order g2g^2

We systematically compute the effective short-range potentials arising from second order QCD-diagrams related to bound states of quarks, antiquarks, and gluons. Our formalism relies on the assumption that the exchanged gluons are massless, while the constituent gluons as well as the lightest quarks acquire a nonvanishing constituent mass because of confinement. The potentials we obtain include the first relativistic corrections, thus spin-spin terms, spin-orbit terms, etc. Such effective potentials are expected to be relevant for the building of accurate potential models describing usual hadrons as well as exotic ones like glueballs and $q\bar q g$ hybrids. In particular, we compute for the first time an effective quark-gluon potential, and show the existence of a quadrupolar interaction term in this case. We also discuss the influence of a possible nonzero mass for the exchanged gluons.Comment: 33 pages, 4 tables and 12 figures ; typos correcte

Glueballs and statistical mechanics of the gluon plasma

We study a pure gluon plasma in the context of quasiparticle models, where the plasma is considered as an ideal gas of massive bosons. In order to reproduce SU(3) gauge field lattice data within such a framework, we review briefly the necessity to use a temperature-dependent gluon mass which accounts for color interactions between the gluons near $T_{\text{c}}$ and agrees with perturbative QCD at large temperatures. Consequently, we discuss the thermodynamics of systems with temperature-dependent Hamiltonians and clarify the situation about the possible solutions proposed in the literature to treat consistently those systems. We then focus our attention to two possible formulations which are thermodynamically consistent and we extract the gluon mass from the equation of state obtained in SU(3) lattice QCD. We find that the thermal gluon mass is similar in both statistical formalisms. Finally, an interpretation of the gluon plasma as an ideal gas made of glueballs and gluons is also presented. The glueball mass is consistently computed within a relativistic formalism using a potential obtained from lattice QCD. We find that the gluon plasma might be a glueball-rich medium for $T\lesssim 1.13 T_{\text{c}}$ and suggest that glueballs could be detected in future experiments dedicated to the quark-gluon plasma.Comment: minor corrections. Accepted for publication in PR

A minimal quasiparticle approach for the QGP and its large-NcN_c limits

We propose a quasiparticle approach allowing to compute the equation of state of a generic gauge theory with gauge group SU($N_c$) and quarks in an arbitrary representation. Our formalism relies on the thermal quasiparticle masses (quarks and gluons) computed from Hard-Thermal-Loop techniques, in which the standard two-loop running coupling constant is used. Our model is minimal in the sense that we do not allow any extra ansatz concerning the temperature-dependence of the running coupling. We first show that it is able to reproduce the most recent equations of state computed on the lattice for temperatures higher than 2 $T_c$. In this range of temperatures, an ideal gas framework is indeed expected to be relevant. Then we study the accuracy of various inequivalent large-$N_c$ limits concerning the description of the QCD results, as well as the equivalence between the QCD$_{AS}$ limit and the ${\cal N}=1$ SUSY Yang-Mills theory. Finally, we estimate the dissociation temperature of the $\Upsilon$-meson and comment on the estimations' stability regarding the different considered large-$N_c$ limits.Comment: 19 pages, 6 figure