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--> --> -->AdS/CFT [8-10], a conjectured duality between a type IIB string theory in
In this study, we reexamine the jet quenching parameter in a soft wall AdS/QCD model, motivated by the soft wall model of [36]. In particular, we adopt the SW
The paper is organized as follows. In section 2, we briefly review the SW
$ {\rm d}s^2 = \frac{r^2h(r)}{R^2}(-f{\rm d}t^2+{\rm d}\vec{x}^2)+\frac{R^2h(r)}{r^2f}{\rm d}r^2, $ ![]() | (1) |
$ f = 1-(1+Q^2)\left(\frac{r_h}{r}\right)^4+Q^2\left(\frac{r_h}{r}\right)^6,\qquad h(r) = {\rm e}^{\frac{c^2R^4}{r^2}}, $ ![]() | (2) |
Moreover, the chemical potential reads
$ \mu = \frac{\sqrt{3}Qr_h}{R^2}. $ ![]() | (3) |
$ T = \frac{r_h}{\pi R^2}\left(1-\frac{Q^2}{2}\right). $ ![]() | (4) |
$ <W^A[{\cal{C}}]> \approx \exp \left[-\frac{1}{4\sqrt{2}}\hat{q}L_-L^2\right], $ ![]() | (5) |
Using the formulas
$ \hat{q} = 8\sqrt{2}\frac{S_I}{L_-L^2}, $ ![]() | (6) |
To carry out the calculation, one needs to rotate the coordinate to light-cone, e.g.,
$ {\rm d}t = \frac{{\rm d}x^++{\rm d}x^-}{\sqrt{2}},\qquad {\rm d}x_1 = \frac{{\rm d}x^+-{\rm d}x^-}{\sqrt{2}}, $ ![]() | (7) |
$\begin{split} {\rm d}s^2 =& -\frac{r^2h(r)}{R^2}(1+f){\rm d}x^+{\rm d}x^-+\frac{r^2h(r)}{R^2}({\rm d}x_2^2+{\rm d}x_3^2)\\&+\frac{r^2h(r)}{2R^2}(1-f)[({\rm d}x^+)^2+({\rm d}x^-)^2])+\frac{R^2h(r)}{r^2f}{\rm d}r^2. \end{split} $ ![]() | (8) |
$ x^- = \tau, \qquad x_2 = \sigma, $ ![]() | (9) |
$ {\rm d}s^2 = h(r)\left[\frac{1}{2}\left(\frac{r^2}{R^2}-f_1\right){\rm d}\tau^2+\left(\frac{r^2}{R^2}+\frac{\dot{r}^2}{f_1}\right){\rm d}\sigma^2\right], $ ![]() | (10) |
Given that, the induced metric reads
$\begin{split} g_{00} =& \frac{h(r)}{2}\left(\frac{r^2}{R^2}-f_1\right), \qquad g_{01} = g_{10} = 0, \\ g_{11} =& h(r)\left(\frac{r^2}{R^2}+\frac{\dot{r}^2}{f_1}\right). \end{split}$ ![]() | (11) |
$ S = -\frac{1}{2\pi\alpha^\prime}\int {\rm d}\tau {\rm d}\sigma\sqrt{-{\rm det}g_{\alpha\beta}}, $ ![]() | (12) |
$ g_{\alpha\beta} = G_{\mu\nu}\frac{\partial X^\mu}{\partial\sigma^\alpha} \frac{\partial X^\nu}{\partial\sigma^\beta}, $ ![]() | (13) |
Plugging (11) into (12), one has
$ S = \frac{\sqrt{2}L_-}{2\pi\alpha^\prime}\int_0^{\frac{L}{2}}{\rm d}\sigma\sqrt{h^2(r)\left(\frac{r^2}{R^2}-f_1\right)\left(\frac{r^2}{R^2}+\frac{\dot{r}^2}{f_1}\right)}, $ ![]() | (14) |
As action (14) does not depend explicitly on
$ \frac{\partial{\cal{L}}}{\partial\dot{r}}\dot{r}-{\cal{L}} = \frac{-h^2(r)\left(\dfrac{r^2}{R^2}-f_1\right)\dfrac{r^2}{R^2}}{\sqrt{h^2(r)\left(\dfrac{r^2}{R^2}-f_1\right)\left(\dfrac{r^2}{R^2}+\frac{\dot{r}^2}{f_1}\right)}} = C, $ ![]() | (15) |
$ \dot{r}^2 = \frac{f_1r^2}{R^2C^2}\Bigg[\frac{h^2(r) r^2\left(\dfrac{r^2}{R^2}-f_1\right)}{R^2}-C^2\Bigg]. $ ![]() | (16) |
For convenience, we write
$ L = 2R^2\int_{r_t}^\infty {\rm d}r \sqrt{\frac{1}{\left(\dfrac{r^2}{R^2}-f_1\right)Bf_1r^4h^2(r)}}. $ ![]() | (17) |
$\begin{split}S =& \frac{\sqrt{2}L_-}{2\pi\alpha^\prime}\int_{r_h}^\infty {\rm d}r \sqrt{\frac{h^4(r)\left(\dfrac{r^2}{R^2}-f_1\right)^2r^2}{f_1\left[h^2(r) r^2\left(\dfrac{r^2}{R^2}-f_1\right)-R^2C^2\right]}}\\ =& \frac{\sqrt{2}L_-\sqrt{B}}{2\pi\alpha^\prime}\int_{r_h}^\infty {\rm d}r\dfrac{h^2(r)\left(\dfrac{r^2}{R^2}-f_1\right)r}{\sqrt{h^2(r)\left(\dfrac{r^2}{R^2}-f_1\right)Bf_1r^2-f_1R^2}}. \end{split} $ ![]() | (18) |
$\begin{split} S =& \frac{\sqrt{2}L_-}{2\pi\alpha^\prime}\int_{r_h}^\infty {\rm d}r\left[1+\frac{R^2}{2h^2(r)\left(\dfrac{r^2}{R^2}-f_1\right)Br^2}\right]\\&\times \sqrt{\frac{1}{f_1}h^2(r)\left(\frac{r^2}{R^2}-f_1\right)}.\end{split} $ ![]() | (19) |
$ \begin{split} S_0 =& \frac{2L_-}{2\pi\alpha^\prime}\int_{r_h}^\infty {\rm d}r\sqrt{g_{–}g_{rr}}\\ =& \frac{\sqrt{2}L_-}{2\pi\alpha^\prime}\int_{r_h}^\infty {\rm d}r \sqrt{\dfrac{1}{f_1}h^2(r)\left(\dfrac{r^2}{R^2}-f_1\right)}. \end{split} $ ![]() | (20) |
$ S_I = S-S_0 = \frac{\sqrt{2}L_-R^2}{4\pi\alpha^\prime B}\int_{r_h}^\infty {\rm d}r\sqrt{\frac{1}{\left(\dfrac{r^2}{R^2}-f_1\right)f_1r^4h^2(r)}}. $ ![]() | (21) |
$ \hat{q} = \frac{I(q)^{-1}}{\pi\alpha^\prime}, $ ![]() | (22) |
$ I(q) = R^2\int_{r_h}^\infty {\rm d}r\sqrt{\frac{1}{\left(\dfrac{r^2}{R^2}-f_1\right)f_1r^4h^2(r)}}. $ ![]() | (23) |
$ \hat{q}_{\rm SYM} = \frac{\pi^{3/2}\Gamma\left(\dfrac{3}{4}\right)}{\Gamma\left(\dfrac{5}{4}\right)}\sqrt{\lambda}T^3, $ ![]() | (24) |
Let us discuss the results. First, we analyze how
![](http://cpc.ihep.ac.cn/fileZGWLC/journal/article/zgwlc/2020/10/PIC/CPC-2020-0150-1.jpg)
![](http://cpc.ihep.ac.cn/fileZGWLC/journal/article/zgwlc/2020/10/PIC/CPC-2020-0150-2.jpg)
Further, we strive to understand the T dependence of
![](http://cpc.ihep.ac.cn/fileZGWLC/journal/article/zgwlc/2020/10/PIC/CPC-2020-0150-3.jpg)
Finally, we would like to make a comparison to implications of experimental data. In Table 1, we present some typical values of
![]() ![]() | (0, 0) | (0, 0.3) | (0, 0.7) | (0.1, 0) | (0.1, 0.3) | (0.1, 0.7) | (0.3, 0) | (0.3, 0.3) | (0.3, 0.7) |
![]() ![]() | 4.50 | 4.71 | 5.70 | 4.53 | 4.74 | 5.73 | 4.76 | 4.98 | 6.0 |
![]() ![]() | 10.61 | 10.89 | 12.19 | 10.64 | 10.93 | 12.23 | 10.94 | 11.23 | 12.56 |
![]() ![]() | 20.69 | 21.02 | 22.65 | 20.70 | 21.06 | 22.70 | 21.08 | 21.45 | 23.10 |
Table1.Typical values of
Nevertheless, the SW