On the n-p Mass Difference in QCD

Charge symmetry breaking (CSB) effects associated with the $u$ and $d$ quark mass difference are investigated in the quark distribution functions and spacelike electromagnetic form factors of the pion and kaon. We use a confining version of the Nambu--Jona-Lasinio model, where CSB effects at the infrared scale associated with the model are driven by the dressed $u$ and $d$ quark mass ratio, which because of dynamical chiral symmetry breaking is much closer to unity than the associated current quark mass ratio. The pion and kaon are given as bound states of a dressed quark and a dressed antiquark governed by the Bethe-Salpeter equation, and exhibit the properties of Goldstone bosons, with a pion mass difference given by $m_{\pi^+}^2 - m_{\pi^0}^2 \propto (m_u - m_d)^2$ as demanded by dynamical chiral symmetry breaking. We find significant CSB effects for realistic current quark mass ratios ($m_u/m_d \sim 0.5$) in the quark flavor-sector electromagnetic form factors of both the pion and kaon. For example, the difference between the $u$ and $d$ quark contributions to the $\pi^+$ electromagnetic form factors is about 8\% at a momentum transfer of $Q^2 \simeq 10\,$GeV$^2$, while the analogous effect for the light quark sector form factors in the $K^+$ and $K^0$ is about twice as large. For the Parton distribution functions, we find CSB effects which are considerably smaller than those found in the electromagnetic form factors.

A modification of perturbation theory, known as the delta expansion (variationally improved perturbation), gave rigorously convergent series in some D=1 models (oscillator energy levels) with factorially divergent ordinary perturbative expansions. In a generalization of the variationally improved perturbation technique appropriate to renormalizable asymptotically free theories, we show that the large expansion orders of certain physical quantities are similarly improved, and prove the Borel convergence of the corresponding series for $m_{\mathrm {v}} \lessim 0$ , with $m_{\mathrm {v}}$ the new (arbitrary) mass perturbation parameter. We argue that non-ambiguous estimates of quantities relevant to dynamical (chiral) symmetry breaking in QCD are possible in this resummation framework.

We address the problem of accounting for light neutrino masses in theories with dynamical electroweak symmetry breaking. We discuss this in the context of a class of extended technicolor (ETC) models. As a possible solution, we propose a combination of suppressed Dirac masses and a seesaw involving dynamically generated condensates of standard‐model singlet, ETC‐nonsinglet fermions. We show how this can be realized in an explicit ETC model. An important feature of this proposal is that, because of the suppression of Dirac neutrino mass terms, a seesaw yielding realistic neutrino masses does not require superheavy Majorana masses. We then generalize this mechanism to theories with extended strong‐electroweak gauge symmetries.

Adding a fourth generation to the standard model and assuming it to be valid up to some cutoff $\ensuremath{\Lambda}$, we show that electroweak symmetry is broken by radiative corrections due to the fourth generation. The effects of the fourth generation are isolated using a Lagrangian with a genuine scalar without self-interactions at the classical level. For masses of the fourth generation consistent with electroweak precision data (including the $B\ensuremath{\rightarrow}K\ensuremath{\pi}$ $CP$ asymmetries), we obtain a Higgs mass of the order of a few hundreds GeV and a cutoff $\ensuremath{\Lambda}$ around 1--2 TeV. We study the reliability of the perturbative treatment used to obtain these results taking into account the running of the Yukawa couplings of the fourth quark generation with the aid of the renormalization group equations, finding similar allowed values for the Higgs mass but a slightly lower cutoff due to the breaking of the perturbative regime. Such low cutoff means that the effects of new physics needed to describe electroweak interactions at energy above $\ensuremath{\Lambda}$ should be measurable at the LHC. We use the minimal supersymmetric extension of the standard model with four generations as an explicit example of models realizing the dynamical electroweak symmetry breaking by radiative corrections and containing new physics. Here, the cutoff is replaced by the masses of the squarks and electroweak symmetry breaking by radiative corrections requires the squark masses to be of the order of 1 TeV.

Dynamical symmetry breaking can happen through a Higgs mechanism but also spontaneously within a strongly-enough coupled theory. We treat a field-theoretical quark model of QCD based on a linear+Coulomb Cornell potential (to account for the longitudinal interaction), together with a transverse interaction (to account for Coulomb-gauge gluons) in BCS approximation. After extracting the well-known BCS ground state on which abundant hadron phenomenology implementing dynamical chiral symmetry breaking has been built, we find two excited replica vacuum like states shown in figure 1. At the BCS level they are classically unstable, but second quantization blocks transitions between them in volumes much larger than a hadron size. Mesons built over the replica vacua have relative masses similar to normal mesons on the ordinary vacuum; we find no negative-mass mode, confirming their stability found earlier with a harmonic oscillator potential.

COL~A~ and GLASttOW have discussed on the SUs (and SU2) symmetry breaking (1). According to their discussions the mass splitting within a multiplet is caused by the two symmetry-breaking types, i.e. nontadpole and tadpole (2). In this note we discuss on the basis of their discussions the problem of the extraordinary mass splitting (m E ran), especially the nontadpole contribution to mE--too. DAS et al. (3) discussed the pion electromagnetic-mass difference, which includes only the nontadpole contribution because of the AI = 2 electromagnetic-mass difference. They obtained the excellent value m~+-m~o=5MeV (experimental value 4.6MeV) by means of Weinberg's two sum rules for the spectral functions of S U e x SU2. Their method seems to be good for obtaining the nontadpole contribution to the mass difference within a multiplet. We intend to estimate the nontadpole contribution to the mass splitting (mE--too) in the same manner as theirs. Then we assume Weinberg's two sum rules for the spectral functions of the chiral S U 3 • 3 symmetry (4), Ne'eman's fifth interaction (5). NE'EI~IAN introduced the field Z which causes the S U 3 symmetry breaking. The interaction Hamiltonian density is gz Vs~ (x) Xs(x), where gz is the medium strong coupling constant gz/4S= 0.1--0.3 and V~(x) the hypereharge hadronic current. The mass splitting Am(~-mK--m~) i s expressed by the summation of the nontadpole term and the tadpole term, A m = Am(nontad)-~ qAm(tad).

We study the full infrared dynamics of 2+1 flavor quantum chromodynamics (QCD) with the functional renormalization group approach. We resolve self-consistently the glue dynamics as well as the dynamics of chiral symmetry breaking. The computation hosts no phenomenological parameter or external input. The only ultraviolet input parameters are the physical ones in QCD: the light and strange quark masses. They are adjusted to the physical ratios of the pion and kaon masses, divided by the pion decay constant. The results for other observables of current first-principles computations are in quantitative agreement with the physical ones. This work completes the series of papers, initiated and furthered in [; ], on dynamical chiral symmetry breaking and the emergence of mesonic bound states within the functional renormalization group. As a first application we discuss the formation of light mesonic bound states. Among other applications such as the phase structure of QCD, the current work paves the way for studying QCD parton distribution functions within the functional renormalization group approach to first-principles QCD.

Modeling finite-volume effects and chiral symmetry breaking in two-flavor QCD thermodynamics

As a toy model of quantum chromodynamics (QCD), we investigate the influence of the vacuum polarization on the photon propagator, in massless quantum electrodynamics in 2+1 dimensions (QED3), using the coupled Dyson–Schwinger equations for the fermion and photon propagators with a range of fermion–photon vertices. It is found that the form of the photon propagators in the Nambu–Goldstone and Wigner phases exhibit distinctly different behaviors and it is expected that similar things happen in QCD. Hence in the phenomenological application of QCD one should choose different forms of gluon propagator as input to solve the quark self-energy functions in the two different phases, especially in the study of QCD phase transitions. In addition, employing these results, we discuss the causation of dynamical chiral symmetry breaking (DCSB) in QED3 and compare it with that in QCD.

Isospin symmetry breaking is discussed as a tool for studying the nature and production mechanisms of light scalar mesons. We are concerned with isospin breaking effects with an amplitude (instead of the usual ∼ md − mu), where mu and md are the u and d quark masses, whose magnitude and phase vary with energy in a resonance-like way characteristic of the threshold region. We consider a variety of reactions that can experimentally reveal (or have revealed) the mixing of and f0(980) resonances that breaks the isotopic invariance due to the mass difference between K+ and K0 mesons. Experimental results on the search for mixing in f1(1285) → f0(980) π0 → π+ π− π0 and η(1405) → f0(980) π0 → π+ π− π0 decays suggest a broader perspective on the isotopic symmetry breaking effects due to the K+−K0 mass difference. It has become clear that not only the mixing but also any mechanism producing pairs with a definite isospin in an S wave gives rise to such effects, thus suggesting a new tool for studying the nature and production mechanisms of light scalars. Of particular interest is the case of a large isotopic symmetry breaking in the η(1405) → f0(980) π0 → π+ π− π0 decay due to the occurrence of anomalous Landau thresholds (logarithmic triangle singularities), i.e., due to the transition (where it is of fundamental importance that the K* meson has a finite width).

The fermion propagator is studied in the whole Minkowski space with the help of the Schwinger-Dyson equations. Various integral representations are employed to get solutions for the dynamical breaking of chiral symmetry in different regimes of the coupling constant. In particular, in the case of massive boson, we extend the singularity structure of the fermion propagator to the two real pole Ansatz.

We study dynamical chiral symmetry breaking in massless QCD by the use of the generalized Hartree-Fock method. As the order parameter of chiral symmetry we choose the dynamical quark mass in the zero momentum limit which we call low energy quark mass. We calculate the low energy mass to the second order of diagrammatic expansion around shifted perturbative vacuum. We then show that the mass is finite and renormalization group invariant. After the improvement of the result by the method of effective charges we estimate the mass in the true vacuum under the gap and stationarity conditions and demonstrate that both of them produce non-zero mass proportional to a conventional scale, which breaks down the chiral symmetry.

Using lattice QCD, we examine several key features associated with dynamical chiral symmetry breaking and quark confinement. Calculations are performed in pure SU(3) gauge theory using the chiral overlap fermion action. Vortex-only and vortex-removed configurations are derived from the the original Monte Carlo gauge fields via a centre vortex identification procedure. The topological charge density, static quark potential, quark mass function and hadron spectrum are compared across the three ensembles. Remarkably, in each instance, the vortex-only degrees of freedom (after some smoothing) are able to encapsulate the pertinent features of the original gauge fields. The removal of centre vortices is consistently associated with a loss of dynamical chiral symmetry breaking. In particular, preliminary results for the vortex-removed meson spectrum hint at the restoration of chiral symmetry at light quark masses.

Pion valence, glue and sea distributions are calculated using a continuum approach to the two valence-body bound-state problem. Since the framework is symmetry preserving, physical features of the distributions are properly expressed. The analysis reveals that the emergent phenomenon of dynamical chiral symmetry breaking causes a hardening of the valence-quark distribution function, ${q}^\pi(x)$. Nevertheless, this distribution exhibits the $x\simeq 1$ behaviour predicted by quantum chromodynamics (QCD). At the scale $\zeta_2:=2\,$GeV, the following momentum fractions are predicted: $\langle x_{\rm valence} \rangle = 0.48(3)$, $\langle x_{\rm glue} \rangle = 0.41(2)$, $\langle x_{\rm sea} \rangle = 0.11(2)$. Evolving to $\zeta=5.2\,$GeV, the result for ${q}^\pi(x)$ agrees with that computed using lattice QCD. These outcomes should both spur improved analyses of existing experiments and stimulate efforts to obtain new data on the pion distribution functions using available and envisioned facilities.

We construct domain-wall brane models based on the grand-unification group SO(10), generalizing the SU(5) model of Davies, George, and Volkas. Motivated by the Dvali-Shifman proposal for the dynamical localization of gauge bosons, the SO(10) symmetry is spontaneously broken inside the wall. We present two scenarios: in the first, the unbroken subgroup inside the wall is $\mathrm{SU}(5)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1{)}_{X}$, and in the second it is the left-right symmetry group $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2{)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2{)}_{R}\ifmmode\times\else\texttimes\fi{}U(1{)}_{B\ensuremath{-}L}$. In both cases we demonstrate that the phenomenologically-correct fermion zero modes can be localized to the wall, and we briefly discuss how the symmetry-breaking dynamics may be extended to induce breaking to the standard model group with subsequent electroweak breaking. Dynamically localized gravity is realized through the type 2 Randall-Sundrum mechanism.