In this paper, we investigate the simplest wormhole solution—the Ellis–Bronnikov one—in the context of the asymptotically safe gravity (ASG) at the Planck scale. We work with three models, which employ the Ricci scalar, Kretschmann scalar, and squared Ricci tensor to improve the field equations by turning the Newton constant into a running coupling constant. For all the cases, we check the radial energy conditions of the wormhole solution and compare them with those that are valid in general relativity (GR). We verified that asymptotic safety guarantees that the Ellis–Bronnikov wormhole can satisfy the radial energy conditions at the throat radius, $r_0$, within an interval of values of the latter, which is quite different from the result found in GR. Following this, we evaluate the effective radial state parameter, $\omega \left(r\right)$, at $r_0$, showing that the quantum gravitational effects modify Einstein’s field equations in such a way that it is necessary to have a very exotic source of matter to generate the wormhole spacetime–phantom or quintessence-like matter. This occurs within some ranges of the throat radii, even though the energy conditions are or are not violated there. Finally, we find that, although at $r_0$ we have a quintessence-like matter, upon growing r, we inevitably came across phantom-like regions. We speculate whether such a phantom fluid must always be present in wormholes in the ASG context or even in more general quantum gravity scenarios.

中文翻译：

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渐近安全重力下的埃利斯-布朗尼科夫虫洞

在本文中，我们在普朗克尺度的渐近安全重力 (ASG) 背景下研究了最简单的虫洞解法——Ellis-Bronnikov 解法。我们使用三个模型，它们采用 Ricci 标量、Kretschmann 标量和平方 Ricci 张量，通过将牛顿常数转换为运行耦合常数来改进场方程。对于所有情况，我们检查虫洞解的径向能量条件，并将它们与广义相对论 (GR) 中有效的条件进行比较。我们验证了渐近安全性保证 Ellis-Bronnikov 虫洞可以满足喉部半径处的径向能量条件，$r_0$, 在后者的值区间内，这与 GR 中发现的结果大不相同。在此之后，我们评估有效径向状态参数，$\omega \left(r\right)$， 在 $r_0$，表明量子引力效应以这样一种方式修改了爱因斯坦的场方程，以至于必须有一个非常奇特的物质来源来产生虫洞时空——幻影或类似精华的物质。这发生在喉部半径的某些范围内，即使能量条件在那里被破坏或没有被破坏。最后，我们发现，虽然在$r_0$我们有一个类似精华的物质，随着r 的增长，我们不可避免地遇到了类似幻影的区域。我们推测这种幻影流体是否必须始终存在于 ASG 背景下的虫洞中，甚至在更一般的量子引力场景中。