Messung des elektrischen Neutron-Formfaktors in der Reaktion 3 He(e,e'n)pp bei Q 2 = 1.58 (GeV/c) 2

Die elektromagnetischen Nukleon-Formfaktoren sind fundamentale Größen, welche eng mit der elektromagnetischen Struktur der Nukleonen zusammenhängen. Der Verlauf der elektrischen und magnetischen Sachs-Formfaktoren G_E und G_M gegen Q^2, das negative Quadrat des Viererimpulsübertrags im elektromagnetischen Streuprozess, steht über die Fouriertransformation in direkter Beziehung zu der räumlichen Ladungs- und Strom-Verteilung in den Nukleonen. Präzise Messungen der Formfaktoren über einen weiten Q^2-Bereich werden daher für ein quantitatives Verständnis der Nukleonstruktur benötigt.rnrnDa es keine freien Neutrontargets gibt, gestaltet sich die Messung der Neutron-Formfaktoren schwierig im Vergleich zu der Messung am Proton. Konsequenz daraus ist, dass die Genauigkeit der vorhandenen Daten von Neutron-Formfaktoren deutlich geringer ist als die von Formfaktoren des Protons; auch der vermessene Q^2-Bereich ist kleiner. Insbesondere der elektrische Sachs-Formfaktor des Neutrons G_E^n ist schwierig zu messen, da er aufgrund der verschwindenden Nettoladung des Neutrons im Verhältnis zu den übrigen Nukleon-Formfaktoren sehr klein ist. G_E^n charakterisiert die Ladungsverteilung des elektrisch neutralen Neutrons und ist damit besonders sensitiv auf die innere Struktur des Neutrons.rnrnIn der hier vorgestellten Arbeit wurde G_E^n aus Strahlhelizitätsasymmetrien in der quasielastischen Streuung vec{3He}(vec{e}, e'n)pp bei einem Impulsübertrag von Q^2 = 1.58 (GeV/c)^2 bestimmt. Die Messung fand in Mainz an der Elektronbeschleunigeranlage Mainzer Mikrotron innerhalb der A1-Kollaboration im Sommer 2008 statt. rnrnLongitudinal polarisierte Elektronen mit einer Energie von 1.508 GeV wurden an einem polarisierten ^3He-Gastarget, das als effektives, polarisiertes Neutrontarget diente, gestreut. Die gestreuten Elektronen wurden in Koinzidenz mit den herausgeschlagenen Neutronen detektiert; die Elektronen wurden in einem magnetischen Spektrometer nachgewiesen, durch den Nachweis der Neutronen in einer Matrix aus Plastikszintillatoren wurde der Beitrag der quasielastischen Streuung am Proton unterdrückt.rnrnAsymmetrien des Wirkungsquerschnitts bezüglich der Elektronhelizität sind bei Orientierung der Targetpolarisation in der Streuebene und senkrecht zum Impulsübertrag sensitiv auf G_E^n / G_M^n; mittels deren Messung kann G_E^n bestimmt werden, da der magnetische Formfaktor G_M^n mit vergleichsweise hoher Präzision bekannt ist. Zusätzliche Messungen der Asymmetrie bei einer Polarisationsorientierung parallel zum Impulsübertrag wurden genutzt, um systematische Fehler zu reduzieren.rnrnFür die Messung inklusive statistischem (stat) und systematischem (sys) Fehler ergab sich G_E^n = 0.0244 +/- 0.0057_stat +/- 0.0016_sys.

Chiral properties of two-flavour QCD at zero and non-zero temperature

Lattice Quantum Chromodynamics (LQCD) is the preferred tool for obtaining non-perturbative results from QCD in the low-energy regime. It has by nowrnentered the era in which high precision calculations for a number of phenomenologically relevant observables at the physical point, with dynamical quark degrees of freedom and controlled systematics, become feasible. Despite these successes there are still quantities where control of systematic effects is insufficient. The subject of this thesis is the exploration of the potential of todays state-of-the-art simulation algorithms for non-perturbativelyrn$\mathcal{O}(a)$-improved Wilson fermions to produce reliable results in thernchiral regime and at the physical point both for zero and non-zero temperature. Important in this context is the control over the chiral extrapolation. Thisrnthesis is concerned with two particular topics, namely the computation of hadronic form factors at zero temperature, and the properties of the phaserntransition in the chiral limit of two-flavour QCD.rnrnThe electromagnetic iso-vector form factor of the pion provides a platform to study systematic effects and the chiral extrapolation for observables connected to the structure of mesons (and baryons). Mesonic form factors are computationally simpler than their baryonic counterparts but share most of the systematic effects. This thesis contains a comprehensive study of the form factor in the regime of low momentum transfer $q^2$, where the form factor is connected to the charge radius of the pion. A particular emphasis is on the region very close to $q^2=0$ which has not been explored so far, neither in experiment nor in LQCD. The results for the form factor close the gap between the smallest spacelike $q^2$-value available so far and $q^2=0$, and reach an unprecedented accuracy at full control over the main systematic effects. This enables the model-independent extraction of the pion charge radius. The results for the form factor and the charge radius are used to test chiral perturbation theory ($\chi$PT) and are thereby extrapolated to the physical point and the continuum. The final result in units of the hadronic radius $r_0$ is rn$$ \left\langle r_\pi^2 \right\rangle^{\rm phys}/r_0^2 = 1.87 \: \left(^{+12}_{-10}\right)\left(^{+\:4}_{-15}\right) \quad \textnormal{or} \quad \left\langle r_\pi^2 \right\rangle^{\rm phys} = 0.473 \: \left(^{+30}_{-26}\right)\left(^{+10}_{-38}\right)(10) \: \textnormal{fm} \;, $$rn which agrees well with the results from other measurements in LQCD and experiment. Note, that this is the first continuum extrapolated result for the charge radius from LQCD which has been extracted from measurements of the form factor in the region of small $q^2$.rnrnThe order of the phase transition in the chiral limit of two-flavour QCD and the associated transition temperature are the last unkown features of the phase diagram at zero chemical potential. The two possible scenarios are a second order transition in the $O(4)$-universality class or a first order transition. Since direct simulations in the chiral limit are not possible the transition can only be investigated by simulating at non-zero quark mass with a subsequent chiral extrapolation, guided by the universal scaling in the vicinity of the critical point. The thesis presents the setup and first results from a study on this topic. The study provides the ideal platform to test the potential and limits of todays simulation algorithms at finite temperature. The results from a first scan at a constant zero-temperature pion mass of about 290~MeV are promising, and it appears that simulations down to physical quark masses are feasible. Of particular relevance for the order of the chiral transition is the strength of the anomalous breaking of the $U_A(1)$ symmetry at the transition point. It can be studied by looking at the degeneracies of the correlation functions in scalar and pseudoscalar channels. For the temperature scan reported in this thesis the breaking is still pronounced in the transition region and the symmetry becomes effectively restored only above $1.16\:T_C$. The thesis also provides an extensive outline of research perspectives and includes a generalisation of the standard multi-histogram method to explicitly $\beta$-dependent fermion actions.

Transport of nanoparticles into polymersomes : a minimal model system of particles passage through biological membranes

This thesis focuses on the design and characterization of a novel, artificial minimal model membrane system with chosen physical parameters to mimic a nanoparticle uptake process driven exclusively by adhesion and softness of the bilayer. The realization is based on polymersomes composed of poly(dimethylsiloxane)-b-poly(2-methyloxazoline) (PMDS-b-PMOXA) and nanoscopic colloidal particles (polystyrene, silica), and the utilization of powerful characterization techniques. rnPDMS-b-PMOXA polymersomes with a radius, Rh ~100 nm, a size polydispersity, PD = 1.1 and a membrane thickness, h = 16 nm, were prepared using the film rehydratation method. Due to the suitable mechanical properties (Young’s modulus of ~17 MPa and a bending modulus of ~7⋅10-8 J) along with the long-term stability and the modifiability, these kind of polymersomes can be used as model membranes to study physical and physicochemical aspects of transmembrane transport of nanoparticles. A combination of photon (PCS) and fluorescence (FCS) correlation spectroscopies optimizes species selectivity, necessary for a unique internalization study encompassing two main efforts. rnFor the proof of concepts, the first effort focused on the interaction of nanoparticles (Rh NP SiO2 = 14 nm, Rh NP PS = 16 nm; cNP = 0.1 gL-1) and polymersomes (Rh P = 112 nm; cP = 0.045 gL-1) with fixed size and concentration. Identification of a modified form factor of the polymersome entities, selectively seen in the PCS experiment, enabled a precise monitor and quantitative description of the incorporation process. Combining PCS and FCS led to the estimation of the incorporated particles per polymersome (about 8 in the examined system) and the development of an appropriate methodology for the kinetics and dynamics of the internalization process. rnThe second effort aimed at the establishment of the necessary phenomenology to facilitate comparison with theories. The size and concentration of the nanoparticles were chosen as the most important system variables (Rh NP = 14 - 57 nm; cNP = 0.05 - 0.2 gL-1). It was revealed that the incorporation process could be controlled to a significant extent by changing the nanoparticles size and concentration. Average number of 7 up to 11 NPs with Rh NP = 14 nm and 3 up to 6 NPs with Rh NP = 25 nm can be internalized into the present polymersomes by changing initial nanoparticles concentration in the range 0.1- 0.2 gL-1. Rapid internalization of the particles by polymersomes is observed only above a critical threshold particles concentration, dependent on the nanoparticle size. rnWith regard possible pathways for the particle uptake, cryogenic transmission electron microscopy (cryo-TEM) has revealed two different incorporation mechanisms depending on the size of the involved nanoparticles: cooperative incorporation of nanoparticles groups or single nanoparticles incorporation. Conditions for nanoparticle uptake and controlled filling of polymersomes were presented. rnIn the framework of this thesis, the experimental observation of transmembrane transport of spherical PS and SiO2 NPs into polymersomes via an internalization process was reported and examined quantitatively for the first time. rnIn a summary the work performed in frames of this thesis might have significant impact on cell model systems’ development and thus improved understanding of transmembrane transport processes. The present experimental findings help create the missing phenomenology necessary for a detailed understanding of a phenomenon with great relevance in transmembrane transport. The fact that transmembrane transport of nanoparticles can be performed by artificial model system without any additional stimuli has a fundamental impact on the understanding, not only of the nanoparticle invagination process but also of the interaction of nanoparticles with biological as well as polymeric membranes. rn

Nucleon electromagnetic structure studies in the spacelike and timelike regions

The thesis investigates the nucleon structure probed by the electromagnetic interaction. One of the most basic observables, reflecting the electromagnetic structure of the nucleon, are the form factors, which have been studied by means of elastic electron-proton scattering with ever increasing precision for several decades. In the timelike region, corresponding with the proton-antiproton annihilation into a electron-positron pair, the present experimental information is much less accurate. However, in the near future high-precision form factor measurements are planned. About 50 years after the first pioneering measurements of the electromagnetic form factors, polarization experiments stirred up the field since the results were found to be in striking contradiction to the findings of previous form factor investigations from unpolarized measurements. Triggered by the conflicting results, a whole new field studying the influence of two-photon exchange corrections to elastic electron-proton scattering emerged, which appeared as the most likely explanation of the discrepancy. The main part of this thesis deals with theoretical studies of two-photon exchange, which is investigated particularly with regard to form factor measurements in the spacelike as well as in the timelike region. An extraction of the two-photon amplitudes in the spacelike region through a combined analysis using the results of unpolarized cross section measurements and polarization experiments is presented. Furthermore, predictions of the two-photon exchange effects on the e+p/e-p cross section ratio are given for several new experiments, which are currently ongoing. The two-photon exchange corrections are also investigated in the timelike region in the process pbar{p} -> e+ e- by means of two factorization approaches. These corrections are found to be smaller than those obtained for the spacelike scattering process. The influence of the two-photon exchange corrections on cross section measurements as well as asymmetries, which allow a direct access of the two-photon exchange contribution, is discussed. Furthermore, one of the factorization approaches is applied for investigating the two-boson exchange effects in parity-violating electron-proton scattering. In the last part of the underlying work, the process pbar{p} -> pi0 e+e- is analyzed with the aim of determining the form factors in the so-called unphysical, timelike region below the two-nucleon production threshold. For this purpose, a phenomenological model is used, which provides a good description of the available data of the real photoproduction process pbar{p} -> pi0 gamma.

Chirale effektive Feldtheorie mit Vektormesonen : vom elektromagnetischen Formfaktor zur virtuellen Compton-Streuung

Die vorliegende Arbeit beschäftigt sich mit der Modellierung niederenergetischer elektromagnetischer und hadronischer Prozesse im Rahmen einer manifest lorentzinvarianten, chiralen effektiven Feldtheorie unter expliziter, dynamischer Berücksichtigung resonanter, das heißt vektormesonischer Freiheitsgrade. Diese effektive Theorie kann daher als Approximation der grundlegenden Quantenchromodynamik bei kleinen Energien verstanden werden. Besonderes Augenmerk wird dabei auf das verwendete Zähl- sowie Renormierungschema gelegt, wodurch eine konsistente Beschreibung mesonischer Prozesse bis zu Energien von etwa 1GeV ermöglicht wird. Das verwendete Zählschema beruht dabei im Wesentlichen auf einem Argument für großes N_c (Anzahl der Farbfreiheitsgrade) und lässt eine äquivalente Behandlung von Goldstonebosonen (Pionen) und Resonanzen (Rho- und Omegamesonen) zu. Als Renormierungsschema wird das für (bezüglich der starken Wechselwirkung) instabile Teilchen besonders geeignete complex-mass scheme als Erweiterung des extended on-mass-shell scheme verwendet, welches in Kombination mit dem BPHZ-Renormierungsverfahren (benannt nach Bogoliubov, Parasiuk, Hepp und Zimmermann) ein leistungsfähiges Konzept zur Berechnung von Quantenkorrekturen in dieser chiralen effektiven Feldtheorie darstellt. Sämtliche vorgenommenen Rechnungen schließen Terme der chiralen Ordnung vier sowie einfache Schleifen in Feynman-Diagrammen ein. Betrachtet werden unter anderem der Vektorformfaktor des Pions im zeitartigen Bereich, die reelle Compton-Streuung (beziehungsweise Photonenfusion) im neutralen und geladenen Kanal sowie die virtuelle Compton-Streuung, eingebettet in die Elektron-Positron-Annihilation. Zur Extraktion der Niederenergiekopplungskonstanten der Theorie wird letztendlich eine Reihe experimenteller Datensätze verschiedenartiger Observablen verwendet. Die hier entwickelten Methoden und Prozeduren - und insbesondere deren technische Implementierung - sind sehr allgemeiner Natur und können daher auch an weitere Problemstellungen aus diesem Gebiet der niederenergetischen Quantenchromodynamik angepasst werden.

Hadronic correlation functions with quark-disconnected contributions in lattice QCD

One of the fundamental interactions in the Standard Model of particle physicsrnis the strong force, which can be formulated as a non-abelian gauge theoryrncalled Quantum Chromodynamics (QCD). rnIn the low-energy regime, where the QCD coupling becomes strong and quarksrnand gluons are confined to hadrons, a perturbativernexpansion in the coupling constant is not possible.rnHowever, the introduction of a four-dimensional Euclidean space-timernlattice allows for an textit{ab initio} treatment of QCD and provides arnpowerful tool to study the low-energy dynamics of hadrons.rnSome hadronic matrix elements of interest receive contributionsrnfrom diagrams including quark-disconnected loops, i.e. disconnected quarkrnlines from one lattice point back to the same point. The calculation of suchrnquark loops is computationally very demanding, because it requires knowledge ofrnthe all-to-all propagator. In this thesis we use stochastic sources and arnhopping parameter expansion to estimate such propagators.rnWe apply this technique to study two problems which relay crucially on therncalculation of quark-disconnected diagrams, namely the scalar form factor ofrnthe pion and the hadronic vacuum polarization contribution to the anomalousrnmagnet moment of the muon.rnThe scalar form factor of the pion describes the coupling of a charged pion torna scalar particle. We calculate the connected and the disconnected contributionrnto the scalar form factor for three different momentum transfers. The scalarrnradius of the pion is extracted from the momentum dependence of the form factor.rnThe use ofrnseveral different pion masses and lattice spacings allows for an extrapolationrnto the physical point. The chiral extrapolation is done using chiralrnperturbation theory ($chi$PT). We find that our pion mass dependence of thernscalar radius is consistent with $chi$PT at next-to-leading order.rnAdditionally, we are able to extract the low energy constant $ell_4$ from thernextrapolation, and ourrnresult is in agreement with results from other lattice determinations.rnFurthermore, our result for the scalar pion radius at the physical point isrnconsistent with a value that was extracted from $pipi$-scattering data. rnThe hadronic vacuum polarization (HVP) is the leading-order hadronicrncontribution to the anomalous magnetic moment $a_mu$ of the muon. The HVP canrnbe estimated from the correlation of two vector currents in the time-momentumrnrepresentation. We explicitly calculate the corresponding disconnectedrncontribution to the vector correlator. We find that the disconnectedrncontribution is consistent with zero within its statistical errors. This resultrncan be converted into an upper limit for the maximum contribution of therndisconnected diagram to $a_mu$ by using the expected time-dependence of therncorrelator and comparing it to the corresponding connected contribution. Wernfind the disconnected contribution to be smaller than $approx5%$ of thernconnected one. This value can be used as an estimate for a systematic errorrnthat arises from neglecting the disconnected contribution.rn

Measurement of the e + e - pi + pi - cross section using initial state radiation at BESIII

The future goal of modern physics is the discovery of physics beyond the Standard Model. One of the most significant hints for New Physics can be seen in the anomalous magnetic moment of the muon - one of the most precise measured variables in modern physics and the main motivation of this work. This variable is associated with the coupling of the muon, an elementary particle, to an external electromagnetic field and is defined as a = (g - 2)/2, whereas g is the gyromagnetic factor of the muon. The muon anomaly has been measured with a relative accuracy of 0.5·10-6. However, a difference between the direct measurement and the Standard Model prediction of 3.6 standard deviations can be observed. This could be a hint for the existence of New Physics. Unfortunately, it is, yet, not significant enough to claim an observation and, thus, more precise measurements and calculations have to be performed.rnThe muon anomaly has three contributions, whereas the ones from quantum electrodynamics and weak interaction can be determined from perturbative calculations. This cannot be done in case of the hadronic contributions at low energies. The leading order contribution - the hadronic vacuum polarization - can be computed via a dispersion integral, which needs as input hadronic cross section measurements from electron-positron annihilations. Hence, it is essential for a precise prediction of the muon anomaly to measure these hadronic cross sections, σ(e+e-→hadrons), with high accuracy. With a contribution of more than 70%, the final state containing two charged pions is the most important one in this context.rnIn this thesis, a new measurement of the σ(e+e-→π+π-) cross section and the pion form factor is performed with an accuracy of 0.9% in the dominant ρ(770) resonance region between 600 and rn900 MeV at the BESIII experiment. The two-pion contribution to the leading-order (LO) hadronic vacuum polarization contribution to (g - 2) from the BESIII result, obtained in this work, is computed to be a(ππ,LO,600-900 MeV) = (368.2±2.5stat±3.3sys)·10-10. With the result presented in this thesis, we make an important contribution on the way to solve the (g - 2) puzzle.

Fabrication methods for creating flexible polymer substrate sensor tags

The authors describe the design, fabrication, and testing of a passive wireless sensor platform utilizing low-cost commercial surface acoustic wave filters and sensors. Polyimide and polyethylene terephthalate sheets are used as substrates to create a flexible sensor tag that can be applied to curved surfaces. A microfabricated antenna is integrated on the substrate in order to create a compact form factor. The sensor tags are fabricated using 315 MHz surface acoustic wave filters and photodiodes and tested with the aid of a fiber-coupled tungsten lamp. Microwave energy transmitted from a network analyzer is used to interrogate the sensor tag. Due to an electrical impedance mismatch at the SAW filter and sensor, energy is reflected at the sensor load and reradiated from the integrated antenna. By selecting sensors that change electrical impedance based on environmental conditions, the sensor state can be inferred through measurement of the reflected energy profile. Testing has shown that a calibrated system utilizing this type of sensor tag can detect distinct light levels wireless and passively. The authors also demonstrate simultaneous operation of two tags with different center passbands that detects light. Ranging tests show that the sensor tags can operate at a distance of at least 3.6 m.

A Dispersive Treatment of Kl4 Decays

K`4 decays are interesting for several reasons: They allow an accurate measurement of a combination of S-wave pp scattering lengths, one form factor of the decay is connected to the chiral anomaly and the decay is the best source for the determination of some low energy constants of ChPT. We present a dispersive approach to K`4 decays, which takes rescattering effects fully into account. Some fits to NA48/2 and E865 measurements and results of the matching to ChPT are shown.

Isospin Breaking Effects in Kℓ4 Decays

In the framework of chiral perturbation theory with photons and leptons, the one-loop isospin-breaking effects in Kℓ4 decays due to both the photonic contribution and the quark and meson mass differences are computed. A comparison with the isospin-breaking corrections applied by recent high statistics Ke4 experiments is performed. The calculation can be used to correct the existing form factor measurements by isospin-breaking effects that have not yet been taken into account in the experimental analysis. Based on the present work, possible forthcoming experiments on Ke4 decays could correct the isospin breaking effects in a more consistent way.

Towards a data-driven analysis of hadronic light-by-light scattering

The hadronic light-by-light contribution to the anomalous magnetic moment of the muon was recently analyzed in the framework of dispersion theory, providing a systematic formalism where all input quantities are expressed in terms of on-shell form factors and scattering amplitudes that are in principle accessible in experiment. We briefly review the main ideas behind this framework and discuss the various experimental ingredients needed for the evaluation of one- and two-pion intermediate states. In particular, we identify processes that in the absence of data for doubly-virtual pion–photon interactions can help constrain parameters in the dispersive reconstruction of the relevant input quantities, the pion transition form factor and the helicity partial waves for γ⁎γ⁎→ππ.

Review of lattice results concerning low-energy particle physics

We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle-physics community. More specifically, we report on the determination of the lightquark masses, the form factor f+(0), arising in semileptonic K → π transition at zero momentum transfer, as well as the decay-constant ratio fK / fπ of decay constants and its consequences for the CKM matrix elements Vus and Vud. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)L × SU(2)R and SU(3)L×SU(3)R Chiral Perturbation Theory and review the determination of the BK parameter of neutral kaon mixing. The inclusion of heavy-quark quantities significantly expands the FLAG scope with respect to the previous review. Therefore, we focus here on D- and B-meson decay constants, form factors, and mixing parameters, since these are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. In addition we review the status of lattice determinations of the strong coupling constant αs.

Introduction to Soft-Collinear Effective Theory

Among resummation techniques for perturbative QCD in the context of collider and flavor physics, soft-collinear effective theory (SCET) has emerged as both a powerful and versatile tool, having been applied to a large variety of processes, from B-meson decays to jet production at the LHC. This book provides a concise, pedagogical introduction to this technique. It discusses the expansion of Feynman diagrams around the high-energy limit, followed by the explicit construction of the effective Lagrangian - first for a scalar theory, then for QCD. The underlying concepts are illustrated with the quark vector form factor at large momentum transfer, and the formalism is applied to compute soft-gluon resummation and to perform transverse-momentum resummation for the Drell-Yan process utilizing renormalization group evolution in SCET. Finally, the infrared structure of n-point gauge-theory amplitudes is analyzed by relating them to effective-theory operators. This text is suitable for graduate students and non-specialist researchers alike as it requires only basic knowledge of perturbative QCD.

A dispersive treatment of K ℓ 4 decays

Kℓ4 decays have several features of interest: they allow an accurate measurement of ππ-scattering lengths; they provide the best source for the determination of some low-energy constants of χPT; one form factor is directly related to the chiral anomaly, which can be measured here. We present a dispersive treatment of Kℓ4 decays that provides a resummation of ππ- and Kπ-rescattering effects. The free parameters of the dispersion relation are fitted to the data of the high-statistics experiments E865 and NA48/2. The matching to χPT at NLO and NNLO enables us to determine the LECs Lr1, Lr2 and Lr3. With recently published data from NA48/2, the LEC Lr9 can be determined as well. In contrast to a pure chiral treatment, the dispersion relation describes the observed curvature of one of the form factors, which we understand as a rescattering effect beyond NNLO.

Finite volume effects in chiral perturbation theory with twisted boundary conditions

We study the effects of a finite cubic volume with twisted boundary conditions on pseudoscalar mesons. We apply Chiral Perturbation Theory in the p-regime and introduce the twist by means of a constant vector field. The corrections of masses, decay constants, pseudoscalar coupling constants and form factors are calculated at next-to-leading order. We detail the derivations and compare with results available in the literature. In some case there is disagreement due to a different treatment of new extra terms generated from the breaking of the cubic invariance. We advocate to treat such terms as renormalization terms of the twisting angles and reabsorb them in the on-shell conditions. We confirm that the corrections of masses, decay constants, pseudoscalar coupling constants are related by means of chiral Ward identities. Furthermore, we show that the matrix elements of the scalar (resp. vector) form factor satisfies the Feynman–Hellman Theorem (resp. the Ward–Takahashi identity). To show the Ward–Takahashi identity we construct an effective field theory for charged pions which is invariant under electromagnetic gauge transformations and which reproduces the results obtained with Chiral Perturbation Theory at a vanishing momentum transfer. This generalizes considerations previously published for periodic boundary conditions to twisted boundary conditions. Another method to estimate the corrections in finite volume are asymptotic formulae. Asymptotic formulae were introduced by Lüscher and relate the corrections of a given physical quantity to an integral of a specific amplitude, evaluated in infinite volume. Here, we revise the original derivation of Lüscher and generalize it to finite volume with twisted boundary conditions. In some cases, the derivation involves complications due to extra terms generated from the breaking of the cubic invariance. We isolate such terms and treat them as renormalization terms just as done before. In that way, we derive asymptotic formulae for masses, decay constants, pseudoscalar coupling constants and scalar form factors. At the same time, we derive also asymptotic formulae for renormalization terms. We apply all these formulae in combination with Chiral Perturbation Theory and estimate the corrections beyond next-to-leading order. We show that asymptotic formulae for masses, decay constants, pseudoscalar coupling constants are related by means of chiral Ward identities. A similar relation connects in an independent way asymptotic formulae for renormalization terms. We check these relations for charged pions through a direct calculation. To conclude, a numerical analysis quantifies the importance of finite volume corrections at next-to-leading order and beyond. We perform a generic Analysis and illustrate two possible applications to real simulations.