Running Quark Masses Research Articles

Quark running mass and vacuum energy density in truncated Coulomb gauge QCD for five orders of magnitude of current masses

We study in detail the effect of the finite current quark mass on chiral symmetry breaking, in the framework of truncated Coulomb gauge QCD with a linear confining quark-antiquark potential. In the chiral limit of massless current quarks, the breaking of chiral symmetry is spontaneous. But for a finite current quark mass, some dynamical symmetry breaking continues to add to the explicit breaking caused by the quark mass. Moreover, using as order parameter the mass gap, i. e. the quark mass at vanishing moment or the quark condensate, a finite quark mass transforms the chiral symmetry breaking from a phase transition into a crossover. For the study of the QCD phase diagram it thus is relevant to determine how the current quark mass affects chiral symmetry breaking. Since the current quark masses of the six standard flavours u, d, s, c, b, t span over five orders of magnitude from 1.5 MeV to 171 GeV, we develop an accurate numerical method to study the running quark mass gap and the quark vacuum energy density from very small to very large current quark masses.

Entropy, confinement, and chiral symmetry breaking

This paper studies the way in which confinement leads to chiral symmetry breaking (CSB) through a gap equation. We argue that entropic effects cut off infrared singularities in the standard confining effective propagator $1/p^4$, which should be replaced by $1/(p^2+m^2)^2$ for a finite mass $m\sim K_F/M(0)$ [$M(0)$ is the zero-momentum value of the running quark mass]. Extension of an old calculation of the author yields a specific estimate for $m$. This cutoff propagator shows semi-quantitatively two critical properties of confinement: 1) a negative contribution to the confining potential coming from entropic forces; 2) an infrared cutoff required by gauge invariance and CSB itself. Entropic effects lead to a proliferation of pion branches and a $\bar{q}q$ condensate, and contribute a negative term $\sim -K_F/M(0)$ to the effective pion Hamiltonian allowing for a massless pion in the presence of positive kinetic energy and string energy. The resulting gap equation leads to a well-behaved running constituent quark mass $M(p^2)$ with $M^2(0)\approx K_F/\pi$. We include one-gluon terms to get the correct renormalization-group ultraviolet behavior, with the improvement that the prefactor (related to $<\bar{q}q>$) can be calculated from the confining solution. We discuss an integrability condition that guarantees the absence of IR singularities at $m=0$ in Minkowski space through use of a principal-part propagator.

Using highly excited baryons to catch the quark mass

Chiral symmetry in QCD can be simultaneously in Wigner and Goldstone modes, depending on the part of the spectrum examined. The transition regime between both, exploiting for example the onset of parity doubling in the high baryon spectrum, can be used to probe the running quark mass in the mid-IR power-law regime. In passing we also argue that three-quark states naturally group into same-flavor quartets, split into two parity doublets, all splittings decreasing high in the spectrum. We propose that a measurement of masses of high-partial wave Delta* resonances should be sufficient to unambiguously establish the approximate degeneracy and see the quark mass running. We test these concepts with the first computation of the spectrum of high-J excited baryons in a chiral-invariant quark model.

Probing the Infrared Quark Mass from Highly Excited Baryons

We argue that three-quark excited states naturally group into quartets, split into two parity doublets, and that the mass splittings between these parity partners decrease higher up in the baryon spectrum. This decreasing mass difference can be used to probe the running quark mass in the midinfrared power-law regime. A measurement of masses of high-partial-wave Delta* resonances should be sufficient to unambiguously establish the approximate degeneracy. We test this concept with the first computation of excited high-j baryon masses in a chirally invariant quark model.

Quark and lepton masses at the GUT scale including supersymmetric threshold corrections

We investigate the effect of supersymmetric (SUSY) threshold corrections on the values of the running quark and charged lepton masses at the grand unified theory (GUT) scale within the large $\mathrm{tan}\ensuremath{\beta}$ regime of the minimal supersymmetric standard model. In addition to the typically dominant SUSY QCD contributions for the quarks, we also include the electroweak contributions for quarks and leptons and show that they can have significant effects. We provide the GUT scale ranges of quark and charged lepton Yukawa couplings as well as of the ratios ${m}_{\ensuremath{\mu}}/{m}_{s}$, ${m}_{e}/{m}_{d}$, ${y}_{\ensuremath{\tau}}/{y}_{b}$ and ${y}_{t}/{y}_{b}$ for three example ranges of SUSY parameters. We discuss how the enlarged ranges due to threshold effects might open up new possibilities for constructing GUT models of fermion masses and mixings.

Updated values of running quark and lepton masses

Reliable values of quark and lepton masses are important for model building at a fundamental energy scale, such as the Fermi scale M-Z approximate to 91.2 GeV and the would-be GUT scale Lambda(GUT) similar to 2 x 10(16) GeV. Using the latest data given by the Particle Data Group, we update the running quark and charged-lepton masses at a number of interesting energy scales below and above M-Z. In particular, we take into account the possible new physics scale (mu similar to 1 TeV) to be explored by the CERN LHC and the typical seesaw scales (mu similar to 10(9) GeV and mu similar to 10(12) GeV) which might be relevant to the generation of neutrino masses. For illustration, the running masses of three light Majorana neutrinos are also calculated. Our up-to-date tables of running fermion masses are expected to be very useful for the study of flavor dynamics at various energy scales.

Non-perturbative quark mass renormalization in two-flavor QCD

The running of renormalized quark masses is computed in lattice QCD with two flavors of massless O ( a ) improved Wilson quarks. The regularization and flavor independent factor that relates running quark masses to the renormalization group invariant ones is evaluated in the Schrödinger functional scheme. Using existing data for the scale r 0 and the pseudoscalar meson masses, we define a reference quark mass in QCD with two degenerate quark flavors. We then compute the renormalization group invariant reference quark mass at three different lattice spacings. Our estimate for the continuum value is converted to the strange quark mass with the help of chiral perturbation theory.

Infrared and ultraviolet properties of the Landau gauge quark propagator

We present a current summary of a program to study the quark propagator using lattice QCD. We use the Overlap and ``Asqtad'' quark actions on a number of lattice ensembles to assess systematic errors. We comment on the place of this work amongst studies of QCD Green's functions in other formulations. A preliminary calculation of the running quark mass is presented.

Running quark mass in two flavor QCD

We present first results for the step scaling function σ P of the renormalization factor Z P of the pseudoscalar density. The simulations are performed within the framework of the Schro¨dinger functional with two flavors of O( a) improved Wilson fermions. The knowledge ofσ P is required to compute the renormalization group invariant quark masses. We also study the performance of a variant of the HMC algorithm using two pseudofermion fields.

Supersymmetric electroweak corrections to heavier top squark decay into a lighter top squark and neutral Higgs boson

We calculate the Yukawa corrections of order ${\cal O}(\alpha_{ew}m_{t(b)}^2/m_W^2)$, ${\cal O}(\alpha_{ew}m_{t(b)}^3/m_W^3)$ and ${\cal O}(\alpha_{ew}m_{t(b)}^4/m_W^4)$ to the widths of the decays $\tilde t_2\to \tilde t_1 + (h^0,H^0,A^0)$ in the Minimal Supersymmetric Standard Model, and perform a detailed numerical analysis. We also compare the results with the ones presented in an earlier literature, where the ${\cal O}(\alpha_{s})$ SUSY-QCD corrections to the same three decay processes have been calculated. Our numerical results show that for the decays $\tilde t_2\to \tilde t_1 + h^{0}$, $\tilde t_2\to \tilde t_1 + H^{0}$, the Yukawa corrections are significant in most of the parameter range, which can reach a few ten percent, and for the decay $\tilde t_2\to\tilde t_1 + A^{0}$, the Yukawa corrections are relatively smaller, which are only a few percent. The numerical calculations also show that using the running quark masses and the running trilinear coupling $A_t$, which include the QCD, SUSY-QCD, SUSY-Electroweak effects and resume all high order ($\tan\beta$)-enhanced effects, can vastly improve the convergence of the perturbation expansion. We also discuss the effects of the running of the higgsino mass parameter $\mu$ on the corrections, and find that they are significant, too, especially for large $\tan\beta$.

QCD CORRECTIONS IN Γsl(B)

Short-distance expansion of the total semileptonic B widths is reviewed for the OPE-conformable scheme employing low-scale running quark masses. The third- and fourth-order BLM corrections are given and the complete resummation of the BLM series presented. The effect of higher perturbative orders with running quark masses is found to be very small. Numerical consequences for |Vcb| are addressed.

C,b quark masses and f,f decay constants from pseudoscalar sum rules in full QCD to order α2

Abstract The pseudoscalar sum rules of the heavy–light quark systems are used for extracting simultaneously the c,b quark masses and the decay constants fD(s),fB(s) of the D(s),B(s) mesons. To order αs2, one obtains the running quark masses: m c (m c )=(1.10±0.04) GeV, m b (m b )=(4.05±0.06) GeV implying m b (M Z )=(2.65±0.05) GeV; the perturbative pole masses: Mc=(1.46±0.04) GeV, Mb=(4.69±0.06) GeV; and the decay constants: fD=(205±20) MeV, fB=(203±23) MeV and fDs=(235±24) MeV, fBs=(236±30) MeV, in the normalization where fπ=130.56 MeV. The fitted values of the pole and running masses satisfy quite well their three-loop perturbative relation. The value fD≃fB confirms earlier findings from the sum rule that the 1/ M P heavy quark symmetry scaling law is affected by large 1/MP corrections.

Pion form factor and quark mass evolution in a light-front Bethe-Salpeter model

We discuss the soft contribution to the elastic pion form factor with the mass evolution from current to constituent quark being taken into account using a light-front Bethe-Salpeter (LFBS) model, which is a light-front quark model (LFQM) with a running mass. It is shown that partial conservation of the axial-vector current (PCAC) is satisfied with a running quark mass. We examine the sensitivity of the pion form factor using two different functional forms of the quark propagator. The Ball-Chiu ansatz is used to maintain local gauge invariance of the quark-photon vertex. The extension of our model to the hard contribution is also discussed.

Improved supersymmetric QCD corrections to Higgs boson decays into quarks and squarks

We improve the calculation of the supersymmetric $O(\alpha_s)$ QCD corrections to the decays of Higgs bosons into quarks and squarks in the Minimal Supersymmetric Standard Model. In the on-shell renormalization scheme these corrections can be very large, which makes the perturbative expansion unreliable. This is especially serious for decays into bottom quarks and squarks for large $\tan\beta$. Their corrected widths can even become negative. We show that this problem can be solved by a careful choice of the tree-level Higgs boson couplings to quarks and squarks, in terms of the QCD and SUSY QCD running quark masses, running trilinear couplings $A_q$, and on-shell left-right mixing angles of squarks. We also present numerical results for the corrected partial decay widths for the large $\tan\beta$ case.

Loop corrections to the neutral Higgs boson sector of the MSSM with explicit CP violation

We compute one-loop corrections to the mass matrix of the neutral Higgs bosons of the Minimal Supersymmetric Standard Model with explicit CP violation. We use the effective potential method, allowing for arbitrary splitting between squark masses. We include terms O(g 2h 2) , where g and h stand for electroweak gauge and Yukawa couplings, respectively. Leading two-loop corrections are taken into account by means of appropriately defined running quark masses.

Precision computation of the strange quark's mass in quenched QCD

We determine the renormalization group invariant quark mass corresponding to the sum of the strange and the average light quark mass in the quenched approximation of QCD, using as essential input the mass of the K-mesons. In the continuum limit we find (M s + M ̂ )/F K =0.874(29) , which includes systematic errors. Translating this non-perturbative result into the running quark masses in the MS scheme at μ=2 GeV and using the quark mass ratios from chiral perturbation theory, we obtain m s (2 GeV)=97(4) MeV . With the help of recent results by the CP-PACS Collaboration, we estimate that a 10% higher value would be obtained if one replaced F K by the nucleon mass to set the scale. This is a typical ambiguity in the quenched approximation.

The running quark mass in the SF scheme and its two-loop anomalous dimension

The non-perturbatively defined running quark mass introduced by the ALPHA collaboration is based on the PCAC relation between correlation functions derived from the Schrödinger functional (SF). In order to complete its definition it remains to specify a number of parameters, including the ratio of time to spatial extent, T/ L, and the angle ø which appears in the spatial boundary conditions for the quark fields. We investigate the running mass in perturbation theory and propose a choice of parameters which attains two desired properties: firstly the two-loop anomalous dimension d 1 SF is reasonably small. This is needed in order to ease matching with the non-perturbative computations and to achieve a precise determination of the renormalization group invariant quark mass. Secondly, to one-loop order of perturbation theory, cut-off effects in the step-scaling function are small in O( a) improved lattice QCD.

Padé improvement of QCD running coupling constants, running masses, Higgs decay rates, and scalar channel sum rules

We discuss Pad\'e-improvement of known four-loop order results based upon an asymptotic three-parameter error formula for Pad\'e-approximants. We derive an explicit formula estimating the next-order coefficient $R_4$ from the previous coefficients in a series $1+R_1 x + R_2x^2 + R_3x^3$. We show that such an estimate is within 0.18% of the known five-loop order term in the O(1) $\beta$-function, and within 10% of the known five-loop term in the O(1) anomalous mass-dimension function $\gamma_m(g)$. We apply the same formula to generate a [2$|$2] Pad\'e-summation of the QCD $\beta$-function and anomalous mass dimension in order to demonstrate both the relative insensitivity of the evolution of $\alpha_s(\mu)$ and the running quark masses to higher order corrections, as well as a somewhat increased compatibility of the present empirical range for $\alpha_s(m_\tau)$ with the range anticipated via evolution from the present empirical range for $\alpha_s(M_z)$. For $3 \leq n_f \leq 6$ we demonstrate that positive zeros of any [2$|$2] Pad\'e-summation estimate of the all-orders $\beta$-function which incorporates known two-, three-, and four-loop contributions necessarily correspond to ultraviolet fixed points, regardless of the unknown five-loop term. Pad\'e-improvement of higher-order perturbative expressions is presented for the decay rates of the Higgs into two gluons and into a $b \bar{b}$ pair, and is used to show the relative insensitivity of these rates to higher order effects. However, Pad\'e-improvement of the purely-perturbative component of scalar/pseudoscalar current correlation functions is indicative of large theoretical uncertainties in QCD sum rules for these channels, particularly if the continuum-threshold parameter $s_0$ is near 1 GeV$^2$.

Renormalons and the heavy quark effective theory

We propose a non-perturbative method for defining the higher dimensional operators which appear in the Heavy Quark Effective Theory (HQET), such that their matrix elements are free of renormalon singularities, and diverge at most logarithmically with the ultra-violet cut-off. Matrix elements of these operators can be computed numerically in lattice simulations of the HQET. We illustrate our procedures by presenting physical definitions of the binding energy ( Λ and of the kinetic energy ( −λ 1 2m Q ) of the heavy quark in a hadron. This allows us to define a “subtracted pole mass”, whose inverse can be used as the expansion parameter in applications of the HQET. We also discuss the determination of the Wilson coefficients of the subtracted operators, necessary for predictions of physical quantities, such as the running quark mass m Q in the MS scheme.

Quartic mass corrections to Rhad

The influence of nonvanishing quark masses on the total cross section in electron positron collisions and on the Z decay rate is calculated. The corrections are expanded in m 2/ s and α s. Methods similar to those applied for the quadratic mass terms allow us to derive the corrections of order α s m 4/ s 2 and α s 2 m 4/ s 2. Coefficients which depend logarithmically on m 2/ s and which cannot be absorbed in a running quark mass arise in order α s 2. The implications of these results on electron positron annihilation cross sections at LEP and at lower energies particular between the charm and the bottom threshold and for energies several GeV above the b b threshold are discussed.