Perturbative width of open rigid strings

We present a perturbative calculation of the energy profile width of rigid strings up to two loops in $D$ dimensions. The perturbative expansion of the extrinsic curvature term, signifying the rigidity/smoothness of the string in Polyakov-Kleinert action, is taken around the free Nambu-Goto string. The mean-square width of the string field is derived for open strings with the Dirichlet boundary condition. We compare the broadening of the smooth Polyakov-Kleinert string to the lattice Monte Carlo data of the QCD flux tube just before the deconfinement point and find a good match at the intermediate and large color source separation.

Figure 1

Feynman diagrams corresponding to disconnected loops [92] of the first two terms of NLO expansion, Eq. (28).

Figure 2

Connected and disconnected loops corresponding to the interaction terms $⟨{\mathit{X}}^2{S}_{(c)}^{(2)}⟩$, $⟨{\mathit{X}}^2{S}_{(d)}^{(2)}⟩$, and $⟨{\mathit{X}}^2{S}_{(e)}^{(2)}⟩$ in Eq. (28) with dimensionless couplings ${\alpha }_r$.

Figure 3

The mean-square width of the string $W^2(\frac{R}{2})$ at the middle plane versus the string length $R$. The temperature scales $T/T_c$ mimics a Yang-Mills theory at $\beta =6.0$ [56]. (a) The plot compares $W^2=W_{{\mathrm{NG}}_{(\ell \mathrm{o})}}^2$ of the free NG string, Eq. (21), and the corresponding square width of PK string $W^2=W_{{\mathrm{NG}}_{(\ell \mathrm{o})}}^2+W_{{\mathrm{NG}}_{(\mathrm{n}\ell \mathrm{o})}}^2+W_{{\mathrm{Ext}}_{(\ell \mathrm{o})}}^2$ given by Eq. (24) at one loop in extrinsic curvature, at the standard value [116] of string tension $\sqrt{\sigma }=440\mathrm{MeV}$. (b) Two-loop mean-square width of pure NG string, Eqs. (21) and (38), plotted versus the string length for each depicted value of $\sigma T^{-2}$.

Figure 4

(a) The plot compares the profile of the rigid string for the depicted values of the rigidity parameter ${\alpha }_r$ at next-to-leading order ($W^2=W_{{\mathrm{NG}}_{(\ell \mathrm{o})}}^2+W_{{\mathrm{NG}}_{(\mathrm{n}\ell \mathrm{o})}}^2+W_{{\mathrm{Ext}}_{(\ell \mathrm{o})}}^2+W_{{\mathrm{Ext}}_{(\mathrm{n}\ell \mathrm{o})}}^2$) and high temperature $T/T_c=0.9$ in the middle plane. (b) The plot shows the profile of the rigid string with respect to the different temperatures. The dashed lines are for the middle plane’s width ($W^2=W_{{\mathrm{NG}}_{(\mathrm{n}\ell \mathrm{o})}}^2+W_{{\mathrm{Ext}}_{(\ell \mathrm{o})}}^2$) at one loop in the rigidity, Eqs. (21) and (24); whereas the solid lines are the same as (a) and represent the width at NLO of the rigid string given by Eq. (53).

Figure 5

The mean-square width of the string $W^2(R/2)$ versus $Q\overline{Q}$ separations measured in the middle plane $R/2$ at $T/T_c=0.9$. (a) The fits of the LO width square of NG string, Eq. (21), and combination of rigidity contributions given by PK string models Eqs. (70)–(72). (b) The fits of the NLO width of NG string, Eq. (38), and combination of rigidity contributions given by PK string models, Eqs. (70)–(72). The Monte Carlo lattice data of the square width of the Euclidean action density width are shown with boxes. The lattice data have been fitted in the interval $R\in [0.5,0.7]\mathrm{fm}$.

Figure 6

Same as Fig. 5, the mean-square width of the string $W^2(R/2)$ versus $Q\overline{Q}$ separations measured at the middle plane $R=2$ at $T/T_c=0.9$, however, the fit range involves larger source separation $R\in [0.7,1.2]\mathrm{fm}$. (a) The fits of the LO width square of NG string, Eq. (21), and combination of rigidity contributions given by PK string models Eqs. (70)–(72). (b) The fits of the NLO width of NG string, Eq. (38), and combination of rigidity contributions given by PK string models, Eqs. (70)–(72).