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The gluonic bag constant

How do confined gluons alter the results obtained so far? Analogous to the fermionic case the gluonic bag constant 8 x Bg is defined as the vacuum expectation value of the following quadratic boundary condition [6]

\begin{displaymath} B_g=-\frac{1}{4} F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}(\vec{E}^2-\vec{B}^2)\ , \end{displaymath} (5)

where to lowest order in the coupling the field strength tensor $F_{\mu\nu}$ is Abelian, and $\vec{E}$ and $\vec{B}$ denote the electric and magnetic field strength, respectively. Appealing in the sourceless case to the symmetry of Maxwell's equations under the duality transformation $\vec{E}=\vec{B}^D\ ,\ \vec{B}=-\vec{E}^D$, we obtain due to physical transverse polarizations (TE,TM) the following expression for Bg in Feynman gauge
Bg = $\displaystyle \frac{1}{32\pi^{3/2}}\frac{1}{R^3} \int_{1/\lambda^2}^{\infty}dz \ \frac{1}{z^{3/2}}$  
    $\displaystyle {}\times\sum_{n, J\ge 1} \bigg\{(2J+1)\left[({\cal N}_{n,J}^{{\mb... ...ny TM}}}_{n,J}R)\mbox{e}^{-z(\varepsilon ^{{\mbox{\tiny TM}}}_{n,J})^2} \right.$  
    $\displaystyle \phantom{\times\sum_{n, J\ge 1} \Big\{\qquad } {}- \left. ({\cal ... ...M}},D}_{n,J}R) \mbox{e}^{-z(\varepsilon ^{{\mbox{\tiny TM}},D}_{n,J})^2}\right]$  
    $\displaystyle \phantom{\sum} {}+ ({\cal N}_{n,J}^{{\mbox{\tiny TE}}})^2 \, \mbox{e}^{-z(\varepsilon ^{{\mbox{\tiny TE}}}_{n,J})^2}$  
    $\displaystyle \phantom{\sum \quad} {}\times \left[(J\!+\!1)j_{J\!-\!1}^2(\varep... ...TE}}}_{n,J}R) +J\,j_{J\!+\!1}^2(\varepsilon ^{{\mbox{\tiny TE}}}_{n,J}R)\right]$  
    $\displaystyle \phantom{\sum} {}- ({\cal N}_{n,J}^{{\mbox{\tiny TE}},D})^2 \, \mbox{e}^{-z(\varepsilon ^{{\mbox{\tiny TE}},D}_{n,J})^2}$  
    $\displaystyle \phantom{\sum \quad} {}\times \left[(J\!+\!1)j_{J\!-\!1}^2(\varep... ...) +J\,j_{J\!+\!1}^2(\varepsilon ^{{\mbox{\tiny TE}},D}_{n,J}R)\right] \bigg\} .$  

Thereby, the superscript D indicates that the corresponding eigenvalue has been obtained from the linear boundary condition $n_\mu (F^D)^{\mu\nu}=0$ for the dual field strength, and ${\cal N}_{n,J}^{{\mbox{\tiny TE}}}$ ( ${\cal N}_{n,J}^{{\mbox{\tiny TM}}}$) denotes the normalization constant for the corresponding mode. For technicalities concerning Cavity QCD in Feynman gauge see Refs. [26,27]. In Eq.(6), the introduction of the Schwinger parameter z and the subsequent truncation of the z-integration and mode summation due to the subtraction of hard fluctuations in the vacuum is analogous to the fermionic case. Table 3 contains the values for 8 x Bg under variations of R with $\lambda$ adjusted to $\lambda=0.8$ GeV and $\lambda =1.0$ GeV. For radii R less than R=0.7 fm there is no contribution from the mode sum of Eq.(6).
Table: The dependence of the gluonic bag constant on the cutoff $\bar\lambda=\lambda\times R$ with R ranging from 0.4 fm to 1.0 fm. The lower and upper values of $\bar{\lambda}$ correspond to $\lambda = 0.8\ \mbox{GeV}$ and $\lambda =1.0$ GeV, respectively.
R [fm] 0.4 0.5 0.6 0.7 0.8 0.9 1.0
$\bar{\lambda}$ 1.6 1.8 2.0 2.25 2.4 2.7 2.8 3.15 3.2 3.6 3.6 4.05 4.0 4.5
8 x Bg [GeV4] 0 0 0 0 0 0 0.0133 0.0205 0.0128 0.0189 0.0179 0.0302 0.0191 0.0271

We find stability for 8 x Bg under a variation of R at R=0.8 fm with 8 x Bg=0.0128 GeV4 for $\lambda=0.8$ GeV and with 8 x Bg=0.0189 GeV4 for $\lambda =1.0$ GeV. Appealing to the QCD trace anomaly and requiring that the total bag constant $B\equiv 3\times n_f\times B_q+8\times B_g$ produces the central value of the gluon condensate, implies $\lambda$ to be less than $\lambda=0.8$ GeV. As far as the properties of the lowest light-flavor resonances are concerned, which are believed to be strongly correlated with the QCD condensates of lowest mass-dimension, QCD sum rules [28] suggest the onset of the perturbative regime at values of about 1.5-1.8 GeV2 of the spectral continuum threshold s0 [23,29,30,31]. This corresponds to $\lambda$=1.22-1.34 GeV. Hence, our value of $\lambda\approx 1.0$ GeV for the pure quark bag seems already a bit too small which might be due to the mode sum representation of the cavity propagator with implicit spatial correlations, whereas s0 relates to plane-wave states. Nevertheless, it is hard to accept values of $\lambda$ lower than 0.8 GeV for the mixed bag.

next up previous
Next: The deconfinement phase transition Up: Calculation Previous: The fermionic bag constant
Marc Schumann
2000-10-16