Callan-Symanzik Function Research Articles

Gell-Mann–Low equation: Relation between the vacuum polarization coefficients up to the eighth order

Gorishny, Kataev, Larin, and Surguladze have obtained the fourth-order Gell-Mann--Low function ${\ensuremath{\psi}}_{4}$ and the Callan-Symanzik function ${\ensuremath{\beta}}_{4}.$ In earlier papers, we used their result to establish relationships between the fifth-, sixth-, and seventh-order vacuum polarization coefficients ${b}_{5},$ ${b}_{6},$ and ${b}_{7}$ and the fifth-, sixth-, and seventh-order Gell-Mann--Low functions ${\ensuremath{\psi}}_{5},$ ${\ensuremath{\psi}}_{6},$ and ${\ensuremath{\psi}}_{7}.$ In this paper, we summarize the results of the previous papers and extend the calculations to include the relationship between the eighth-order ${b}_{8}$ and ${\ensuremath{\psi}}_{8}$ using the Gell-Mann--Low equation and other relations for the Gell-Mann--Low function.

Callan–Symanzik function in QCD

The perturbative expansion of the Callan–Symanzik function β(g) in QCD is extended to orders g7 and g9 using the scheme of the integral equation defining the "running coupling" g(t). PACS Nos.: 11.10Hi, 12.38Bx

Gell-Mann–Low equation: Relation between the seventh-order vacuum polarization coefficients

Gorishny, Kataev, Larin, and Surguladze have obtained the fourth-order Gell-Mann--Low function ${\ensuremath{\psi}}_{4}$ and the Callan-Symanzik function ${\ensuremath{\beta}}_{4}.$ In earlier papers we have used their result to establish relationships between the fifth- and sixth-order vacuum polarization coefficients ${b}_{5}$ and ${b}_{6}$ and the fifth- and sixth-order Gell-Mann--Low functions ${\ensuremath{\psi}}_{5}$ and ${\ensuremath{\psi}}_{6}.$ In this paper, the relationship between the seventh-order ${b}_{7}$ and ${\ensuremath{\psi}}_{7}$ is obtained using the Gell-Mann--Low equation and other relations for the Gell-Mann--Low function.

Gell-Mann-Low equation: Relation between the sixth-order vacuum polarization coefficients.

Gorishny, Kataev, Larin, and Surguladze have obtained the fourth-order Gell-Mann{endash}Low function {psi}{sub 4} and the Callan-Symanzik function {beta}{sub 4}. In earlier papers we have used their result to establish relationships between the fifth- and sixth-order vacuum polarization coefficients b{sub 5} and b{sub 6} and the fifth- and sixth-order Gell-Mann{endash}Low functions {psi}{sub 5} and {psi}{sub 6}. In this paper, the relationship between the seventh-order b{sub 7} and {psi}{sub 7} is obtained using the Gell-Mann{endash}Low equation and other relations for the Gell-Mann{endash}Low function. {copyright} {ital 1997} {ital The American Physical Society}

Finite-size study of the one-dimensional spin-(1/2) dimerized Heisenberg chain.

We present an analysis based on the extrapolation of finite-size results to the infinite one-dimensional spin-(1/2) dimerized isotropic Heisenberg system for the whole range of the dimerization parameter \ensuremath{\delta} (\ensuremath{\Vert}\ensuremath{\delta}\ensuremath{\Vert}\ensuremath{\le}1) at zero temperature. This system undergoes a transition at \ensuremath{\delta}=0, and a gap opens in the spectrum of elementary excitations. The exponent \ensuremath{\nu}, which characterizes the opening of the gap, is estimated with use of the finite-size results (with size up to N=18). We investigate two finite-size-scaling hypotheses, assuming a pure power-law behavior [case (1)], or taking into account logarithmic corrections [case (2)]. To estimate \ensuremath{\nu}, we use the derivative of the reciprocal of the gap, the derivative of the gap, and the Callan-Symanzik function. We show that the first of these is less affected by finite-size corrections than are the other two. Using it, we have obtained, in case (1), \ensuremath{\nu}=0.71\ifmmode\pm\else\textpm\fi{}0.01, in agreement with previous estimates, and in case (2), \ensuremath{\nu}=0.668\ifmmode\pm\else\textpm\fi{}0.001, in very good agreement with the value \ensuremath{\nu}=(2/3) conjectured by den Nijs. We also show that, far from criticality, the ground-state energy per site may be described by the form \ensuremath{\Vert}\ensuremath{\delta}${\ensuremath{\Vert}}^{x}$ with x=1.34\ifmmode\pm\else\textpm\fi{}0.02. However, results for the derivative of this quantity show a different functional dependence upon \ensuremath{\delta}, at least for \ensuremath{\delta}\ensuremath{\gtrsim}0.4. In fact, both the values of the ground-state energy and its derivative agree with the third-order perturbation theory of Harris, with agreement improving as \ensuremath{\delta} approaches 1.

Gell-Mann-Low function and Callan-Symanzik function: the three-lepton case and symmetry under e↔μ↔τ

The Callan-Symanzik functionβ eμτ(α) and the Gell-Mann-Lowψ eμτ are calculated for the three-lepton case up to the sixth order. The symmetry properties of the «asymptotic part» of the photon proper self-energyαπ c ∞ (--k 2/m e 2 k 2/m μ 2 ,k 2/m τ 2 ,α are examined under e↔μ↔τ. This leads to constraints among the sixth-order terms of απ c ∞ which are in agreement with its explicit calculations. An expression for then-leptonβ eμτ...n(α) is also given.

Landau's asymptotic solution of the vertex function in QED

We consider finite massless QED with the electron mass totally dynamical of origin and the fine-structure constant as the (infinite order) zero of the Callan-Symanzik function. In spacelike regions, the electron propagator (in the generalized Landau gauge) and the photon propagator are then proportional to their free counterparts, and the integral equation for the vertex function GAMMA/sub sigma/ may be written solely as a function of GAMMA/sub sigma/ itself. Landau's approximation is used to obtain the asymptotics for GAMMA/sub sigma/(p,p-k;k) for p/sup 2/,k/sup 2/..-->..infinity, p/sup 2//k/sup 2/..-->..0 in spacelike regions.

Modification of the "Coulomb" interaction at small distances in finite quantum electrodynamics

We investigate the ''Coulomb'' interaction in finite QED at small distances. By finite QED it is meant that we sum all photon self-energy subgraphs in renormalized QED and fix ..cap alpha.., the renormalized fine-structure constant, as the (infinite order) zero of the Callan-Symanzik function: ..beta..(..cap alpha..) = 0/sup infinity/. We show that for mcVertical Barx-x 'Vertical Bar/h<<1, the Coulomb interaction between two heavy point test bodies, with renormalized charges e/sub 1/ and e/sub 2/ at x and x ', respectively, is given by V(Vertical Barx-x'Vertical Bar)approx. =(e/sub 1/e/sub 2// 4..pi..Vertical Barx-x'Vertical Bar) (q/sub 1/(..cap alpha..)-q/sub 2/(..cap alpha..)mcVertical Barx-x'Vertical Bar / h+O(m/sup 2/c/sup 2/Vertical Barx-x'Vertical Bar/sup 2//h/sup 2/)), where 1

Callan-Symanzik functions and asymptotic freedom in the Lee model

The Lee model was studied in a paper by Bender and Nash in the context of determining whether the model displays asymptotic freedom. This determination is based on the calculations of the Callan-Symanzik functions. Bender and Nash have renormalized the coupling constant off the mass shell. We have performed on-shell renormalization and evaluated the Callan-Symanzik $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ functions. In agreement with Bender and Nash, we find that the Lee model is asymptotically free for space-time dimensions $N&lt;4$. In addition we find that for $N&gt;4$ (odd $N$) the model is ultraviolet stable and not asymptotically free. Finally, we employ the Weinberg relation to demonstrate the mathematical equivalence between the $\ensuremath{\beta}$ function calculated in this paper and the one presented by Bender and Nash.

Supercurrents, renormalization and anomalies

A renormalized supercurrent is constructed for the massive and massless supersymmetric gφ 3 model of Wess and Zumino. Its pivotal position in the study of the superconformal group is demonstrated. All anomalies occurring in the superconformal currents' Ward identities are shown to be directly related to the Callan-Symanzik functions β and γ. These turn out to be mass independent under suitable off-shell normalization conditions.

Gell-Mann-low function and Callan-Symanzik function: Two-lepton (electron-muon) case

The consistency of the Gell-Mann-Low equation is checked up to order α 2 using the Gell-Mann-Low function to order α 3. The Callan-Symanzik function β e μ ( α,ln( m μ 2 / m e 2)) for the electron-muon case is obtained up to sixth order employing the results of De Rafael and Rosner for the electron Callan-Symanzik funtion β e(α). The corresponding electron-muon Gell-Mann-Low function ψ eμ is derived from β eμ and found to be consistent with the Gell-Mann-Low equation.

Free-field-like behavior of the Dashen-Frishman model and singularity of the Callan-Symanzik function

An example of the Khuri conjecture is provided by considering a special case of the Dashen-Frishman model in which the anomalous dimension of the spinor field vanishes. This leads to a free-field-like theory for nonvanishing couplings. It is concluded that the Callan-Symanzik function $\ensuremath{\beta}(g)$ should be singular at these coupling constants in order to avoid the consequences of the Parisi-Callan-Gross theorem which would require vanishing coupling constant.

Perturbation theory at large order. I. Theφ2Ninteraction

A new method for calculating the large orders of perturbation theory in quantum field theories has been discussed recently by Lipatov. We show that the same method applied to anharmonic oscillators in quantum mechanics allows one to rederive and generalize results previously obtained by Bender and Wu. We have also verified and generalized Lipatov's results to the case of an internal $O(n)$ symmetry. These results show the divergence of the Wilson-Fisher $\ensuremath{\epsilon}$ expansion and indicate its Borel summability which is used for critical exponents. Similarly, the Callan-Symanzik functions for the ${\ensuremath{\varphi}}^{4}$ theory in three dimensions are characterized.

Fixed Points and Anomalous Dimensions in a Thirring-Type Model in 2+ Dimensions

\Ve deal with the U(N) symmetric Thirring-type Lagrangian of .N-component fennions in ~+E dimensions. Applying the 1/N expansion technique we show the existence of a nontrivial ultraviolet fixed point in the Callan-Symanzik function ;3(g) between ~ and 4 space-time dimensions. In 2 dimensions 19 (g) vanishes identically as is characteristic of the Thirring model. vVe calculate an anomalous dimension of the fermion field in 1/ N order and find that it is discontinuous in 2 dimensions, i.e., it is 1/4N if a limit to the~ dimensions is taken from above, while it is a coupling dependent parameter as in the Thirring morld if it is evaluated in ~ dimensions. § l. Introduction and summary It is well-known that the renormalizability of the theory go1·erned by a given Lagrangian is closely related to the dimensionality of a coupling constant O. The dimensionality o is a decreasing function of the space-time dimension and hence, for a gi1·en interaction, the renormalizability requirement o>O sets an upper bound on which ,,-e call a critical dimension of the interaction, D,. Characteristic features of the theory may be described by the Callan-Symanzik function /3(g). At the critical dimension D the behaYior of 13 (g) for small g may :fall into either of the two types sho,vn in Figs. 1 (a) and (b). The latter case is of the so-called asymptotic freedom. For D--;L D the cot1pling constant has a dimension and we must replace g by gpcdn-n,J to make it dimensionless ·where a>O is a numerical constant and ,'1 a

Collective phenomena in gauge theories. II. Renormalization in finite-temperature field theory

A procedure is given for renormalization of finite-temperature and -density Green's functions to all orders of perturbation theory. The Lagrangian counterterms are independent of temperature, density, and all mass parameters of the Lagrangian. Thus renormalization-group equations may be written for Green's functions in a medium for which the Callan-Symanzik functions depend only on the dimensionless coupling constants. By scaling the temperature and/or chemical potential in the renormalization-group equations it is shown that asymptotic freedom in non-Abelian gauge theories occurs for large temperatures (early hot universe) and for large chemical potential (high density of the medium) as well as large momenta.

Asymptotic behavior and symmetry breaking

The asymptotic behavior of Green's functions for large spacelike momenta in the presence of symmetry breaking is considered using the renormalization-group approach. The disappearance of symmetry-breaking effects due to a generalized mass term in the deep Euclidean region allows us to derive a certain inequality satisfied by a function simply related to the Callan-Symanzik function of a "dimensional" coupling constant of the theory. The inequality holds at a fixed point of the renormalization group.

Higher orders in N−1 in the Gross-Neveu model

We study higher-order terms in the N expansion of the two-dimensional model field theory of Gross and Neveu. The quantity M σ 2M F−1 ( M F is the fermion mass, M σ is the scalar bound-state mass), is conjectured to deviate from zero in O( N −4). Order N −1 corrections are found to leave unchanged both the symmetry-breaking, and the absence of zeros of the Callan-Symanzik function β for real non-zero coupling constant. Behavior of β for complex coupling constant is altered considerably by the corrections; consequences are discussed. We obtain the following additional results: proof that the model is renormalizable without the necessity of additional interactions; construction of a perturbation theory in the asymmetric vacuum; proof of a lemma on fermion-antifermion thresholds in two-dimensional space-time; derivation of closed formulae for arbitrary one-loop integrals in two-dimensional space-time.

Ward identities and some clues to the renormalization of gauge-invariant operators

The problem of the renormalization of gauge-invariant operators in the non-Abelian Yang-Mills theory is tackled through the study of a specific example, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{F}}_{\ensuremath{\mu}\ensuremath{\nu}}^{}{}_{}{}^{2}$, for which the explicit solution can be derived from renormalization-group considerations. It is shown that the operator $\stackrel{\ensuremath{\rightarrow}}{\mathrm{F}}_{\ensuremath{\mu}\ensuremath{\nu}}^{}{}_{}{}^{2}$ mixes with non-gauge-invariant operators and that this mixing must be taken into account for the computation of the anomalous dimension of the renormalized gauge-invariant operator. The explicit solution is examined with the help of Ward identities derived from a new type of gauge transformations which appear very convenient from a technical point of view. The multiplicatively renormalizable gauge-invariant operator is shown to satisfy Ward identities and to possess an $\ensuremath{\alpha}$-independent anomalous dimension. As a by-product, we analyze the gauge dependence of the Callan-Symanzik function $\ensuremath{\beta}$.

Renormalization group fixed points in Yukawa theories

The Callan-Symanzik functions of Yukawa theories in which the scalar mesons transform as the regular representation of SU(3), SU(2) or U(1) are calculated in two-loop order. An attractive renormalisation group fixed point away from the origin is found, but at a distance such that perturbation theory can not be considered reliable.

High-energy behavior in Abelian gauge theory, application toγ*decay, and high-energy estimates for form factors

Previously, we have made a study of the high-energy behavior of quantum electrodynamics in the presence of neutral-meson theory with pseudoscalar-pseudoscalar coupling proceeding loopwise. We have seen, by an elementary summation procedure, that the Adler-Baker-Johnson eigenvalue condition for the fine-structure constant $\ensuremath{\alpha}$ remains stable (unaltered) in the presence of the strong dynamics and that the photon self-energy part in the multi-fermion-loop contribution is asymptotically finite at $x=\ensuremath{\alpha}$ independently of the strong-coupling value. We have also learned that the strong dynamics damps out at high energies and no ${\ensuremath{\varphi}}^{4}$ counterterm was required. Conversely, in the present work, we study this dynamics at high energies when the Callan-Symanzik function occurring in pure electrodynamics vanishes at $\ensuremath{\alpha}$ at the outset (with an infinite-order zero) by adding the $\ensuremath{-}\frac{1}{4}{\ensuremath{\lambda}}_{0}{\ensuremath{\varphi}}^{4}$ term to our dynamics. We argue that the effective ${\ensuremath{\gamma}}_{5}$ and ${\ensuremath{\varphi}}^{4}$ couplings in this Abelian gauge theory may become very small at high energies and that then they finally vanish even faster than the presently estimated non-Abelian gauge-theory ones. As an application, the ratio $R({Q}^{2})$ of the cross section associated with the reaction ${\ensuremath{\gamma}}^{*}({Q}^{2})\ensuremath{\rightarrow}\mathrm{anything}$ to the corresponding one with the strong couplings set equal to zero is studied when the momentum squared of the virtual photon ${Q}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}, R({Q}^{2})$ is shown to have a linear increase with ${Q}^{2}$ (broken possibly by logarithmic terms) at truly asymptotic energies, and, contrary to common belief, the leading contribution to it comes from the strong interaction. Some of the statements made in this work are also in pure electrodynamics. For example, the explicit asymptotic form of the photon spectral function in finite quantum electrodynamics at the eigenvalue $\ensuremath{\alpha}$ is given here for the first time. Finally, explicit high-energy estimates are obtained for elastic form factors in quantum electrodynamics in isolation and in the present correlated dynamics. At very large momenta for the external photon, the former are bounded by power-law behavior. Some interesting aspects of this Abelian gauge theory are also mentioned.