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Next: The gluonic bag constant Up: Calculation Previous: The canonical vacuum energy

The fermionic bag constant

Due to the vacuum expectation value of the quadratic boundary condition, the fermionic bag constant Bq reads
Bq = $\displaystyle -\frac{1}{4\pi^{3/2}} \int_{1/\lambda^2}^{\infty}dz \, \frac{1}{z^{1/2}} \sum_{\kappa}(2J+1)$  
    $\displaystyle {}\times \sum^{n_\lambda}_{\atop{n>0}}\, \frac{1}{R^3} {\cal N}^2_{n,\kappa}\, \varepsilon _{n,\kappa}^2 \mbox{e}^{-z\varepsilon _{n,\kappa}^2}$  
    $\displaystyle \phantom{\sum^{n_\lambda}_{\atop{n>0}}}\times \Big[\frac{j_{l}(\v... ...\vert R)} {2l+1}\, \left(l\ j_{l-1}(\vert\varepsilon _{n,\kappa}\vert R)\right.$  
    $\displaystyle \phantom{\sum^{n_\lambda}_{\atop{n>0}}\times \Big[\frac{j_{l}(\ve... ...{2l+1}\,} {}\left. -(l+1)\, j_{l+1}(\vert\varepsilon _{n,\kappa}\vert R)\right)$  
    $\displaystyle \phantom{\times\ \times \Big[} -\frac{j_{\bar{l}}(\vert\varepsilo... ...+1}\, \left(\bar{l}\,j_{\bar{l}-1}(\vert\varepsilon _{n,\kappa}\vert R) \right.$  
    $\displaystyle \phantom{\sum^{n_\lambda}_{\atop{n>0}}\times \Big[\frac{j_{l}(\ve... ...- (\bar{l}+1)j_{\bar{l}+1}(\vert\varepsilon _{n,\kappa}\vert R)\right)\Big]\, .$ (2)

Figure 1: The canonical part of the one-flavor, one-color vacuum energy in dependence on the cutoff. Both quantities are given in units of R-1.
\begin{figure} \epsfxsize 8cm \hfil\epsfbox{fig.1}\hfil \end{figure}

Figure 1 shows the result of the calculation of $\bar{E}\equiv R\times E$ as a function of $\bar{\lambda}\equiv R\times \lambda$. The discontinuous behavior is due to the fact that mode eigenvalues at low energies are not spaced equidistantly. To smooth the ``nervous'' behavior, we use a quadratic regression as indicated by the solid line. In Fig. 2 the $\bar{\lambda}$ dependence of $\bar{B}_q\equiv R^4\times B_q$ is depicted. Again, a quadratic fit is used to average over discontinuities. Tables 1 and 2 contain a list of values for 3 x nf x Bq, -3 x nf x E under variation of R, where $\bar{\lambda}$ is adjusted to $\lambda=1.2$ GeV, $\lambda =1.6$ GeV and $\lambda=0.8$ GeV, $\lambda =1.0$ GeV, respectively. Thereby, nf=2 stands for the light-flavor multiplicity, and the factor three is the number of colors.

Figure 2: The one-flavor, one-color fermionic bag constant in dependence on the cutoff. The bag constant and the cutoff are given in units of R-4 and R-1, respectively.
\begin{figure} \epsfxsize 8cm \hfil\epsfbox{fig.2}\hfil\end{figure}


Table: The dependence of the fermionic bag constant and the canonical part of the fermionic vacuum energy on the cutoff $\bar\lambda=\lambda\times R$ for two light-quark flavors with R ranging from 0.4 fm to 1.0 fm. The lower and upper values of $\bar{\lambda}$ correspond to $\lambda = 1.2\ \mbox{GeV}$ and $\lambda =1.6$ GeV, respectively.
R [fm] 0.4 0.5 0.6 0.7 0.8 0.9 1.0
$\bar{\lambda}$ 2.4 3.2 3.0 4.1 3.6 4.9 4.3 5.7 4.9 6.5 5.5 7.3 6.1 8.1
3 x nf x Bq[GeV4] 0.032 0.053 0.024 0.079 0.021 0.089 0.026 0.088 0.028 0.082 0.028 0.075 0.027 0.068
-3 x nf x E [GeV] 0.450 1.060 0.716 1.439 0.942 1.779 1.145 2.095 1.334 2.398 1.513 2.690 1.686 2.976


Table: Same as in Table 1. 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 2.0 2.0 2.5 2.4 3.0 2.8 3.5 3.2 4.1 3.6 4.6 4.1 5.1
3 x nf x Bq [GeV4] 0.129 0.065 0.027 0.011 0.006 0.007 0.003 0.010 0.003 0.012 0.004 0.013 0.005 0.014
-3 x nf x E [GeV] -0.031 0.193 0.154 0.415 0.300 0.597 0.422 0.755 0.530 0.900 0.628 1.034 0.720 1.162

Appealing to the one-loop trace-anomaly [22] of the QCD energy-momentum tensor $\theta^{\mu\nu}$

\begin{displaymath} \left\langle \theta^\mu_\mu \right\rangle = -\frac{1}{8}\lef... ...lpha_s}{\pi} F_{\kappa\nu}^a F^{\kappa\nu}_a\right\rangle \ , \end{displaymath} (3)

we assume for the moment that only quark fluctuations contribute to the bag constant. Using the fact that the canonical part of $\theta^{\mu\nu}$ is traceless in the mixed MIT bag model, we obtain (apart from a sign) the relation

\begin{displaymath} 3\times n_f\times B_q=0.302\times \left\langle \frac{\alpha_s}{\pi} F_{\kappa\nu}^a F^{\kappa\nu}_a\right\rangle \ . \end{displaymath} (4)

Thereby, the value of the (renormalization-scale independent) gluon condensate [23] is $\left\langle \frac{\alpha_s}{\pi} F^a_{\mu\nu}F^{\mu\nu}_a \right\rangle =0.024\pm 0.012\ \mbox{GeV}^4$. Comparing by means of Eq.(4) the central value of the gluon condensate with the values of 3 x nf x Bq (Tables 1, 2), which are stable under variation of R, we obtain agreement for $\lambda =1.0$ GeV and a bag radius R of 0.6 fm. Given these values of $\lambda$ and R, the results of Table 2 indicate that -3 x nf x E is close to phenomenologically obtained values: In Ref.[9] Z0 parametrizes the Casimir energy as -Z0/R. Fits to the hadron spectrum yield values of about Z0=2 [9]. The effect of the center-of-mass contribution to Z0 was found to be of the order of 40% in Refs.[24,25]. In comparison, our value of -3 x nf x E=0.597 GeV at R=0.6 fm corresponds to Z0=1.79 with no center-of-mass contribution.

next up previous
Next: The gluonic bag constant Up: Calculation Previous: The canonical vacuum energy
Marc Schumann
2000-10-16