We elaborate further on the compatibility of the “vacuumon potential” that characterises the inflationary phase of the running vacuum model (RVM) with the swampland criteria. The work is motivated by the fact that, as demonstrated recently by the authors, the RVM framework can be derived as an effective gravitational field theory stemming from underlying microscopic (critical) string theory models with gravitational anomalies, involving condensation of primordial gravitational waves. Although believed to be a classical scalar field description, not representing a fully fledged quantum field, we show here that the vacuumon potential satisfies certain swampland criteria for the relevant regime of parameters and field range. We link the criteria to the Gibbons–Hawking entropy that has been argued to characterise the RVM during the de Sitter phase. These results imply that the vacuumon may, after all, admit under certain conditions, a rôle as a quantum field during the inflationary (almost de Sitter) phase of the running vacuum. The conventional slow-roll interpretation of this field, however, fails just because it satisfies the swampland criteria. The RVM effective theory derived from the low-energy effective action of string theory does, however, successfully describe inflation thanks to the $\sim H^4$ terms induced by the gravitational anomalous condensates. In addition, the stringy version of the RVM involves the Kalb–Ramond (KR) axion field, which, in contrast to the vacuumon, does perfectly satisfy the slow-roll condition. We conclude that the vacuumon description is not fully equivalent to the stringy formulation of the RVM. Our study provides a particularly interesting example of a successful phenomenological theory beyond the $\Lambda$CDM, such as the RVM, in which the fulfilment of the swampland criteria by the associated scalar field potential, along with its compatibility with (an appropriate form of) the weak gravity conjecture, prove to be insufficient conditions for warranting consistency of the scalar vacuum field representation as a faithful ultraviolet complete representation of the RVM at the quantum gravity level.

中文翻译：

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受弦启发的运行真空—“真空”和沼泽标准

我们将进一步详细说明“真空势”的兼容性，“真空势”表征了运行真空模型（RVM）的膨胀阶段与沼泽标准。正如作者最近所证实的那样，这项工作的动机是，RVM框架可以作为有效的引力场理论而派生，该理论源自具有引力异常的底层微观（临界）弦论模型，其中包括原始引力波的凝结。尽管被认为是经典的标量场描述，并不代表完整的量子场，但我们在这里表明，真空子势能满足某些参数和场范围的沼泽标准。我们将标准与Gibbons-Hawking熵联系起来，该熵被认为是在de Sitter阶段表征RVM的特征。这些结果表明，真空子毕竟可以在一定条件下在运行真空的膨胀（几乎是de Sitter）阶段接受作为量子场的作用。但是，该领域的常规慢速滚动解释失败是因为它满足了沼泽地标准。RVM有效理论源于弦理论的低能有效作用，但是由于存在$〜H^4$重力异常凝结引起的术语。此外，RVM的严格版本还涉及Kalb–Ramond（KR）轴突场，与vacuumon相比，它完全满足慢滚动条件。我们得出的结论是，关于vacuumon的描述并不完全等同于RVM的严格表述。我们的研究提供了一个成功的现象学理论之外的特别有趣的例子。$\Lambda$CDM（例如RVM）通过相关的标量场势满足沼泽标准，并与弱重力猜想（一种适当的形式）兼容，不足以保证标量真空的一致性场表示法是量子力学水平上RVM的真实的紫外线完整表示法。