Implications of a Higgs field for the spontaneously broken pure R^2 gravity: magnetic monopoles, new interactions and dark radiation

纯R^2引力(没有附加R项的R^2引力，其中R是Ricci标量)已经被证明等同于具有非零宇宙常数和额外的无质量标量场的爱因斯坦引力。这种等价性是通过自发地打破其标度和对称性而发生的(它实际上拥有一个更大的对称性，称为限制性韦尔对称性)。无质量标量场被确定为破碎扇区的Goldstone玻色子。非零的宇宙常数意味着，在爱因斯坦作用下，真空或背景时空具有非零的Ricci标量，可以是De Sitter(DS)或反De Sitter(ADS)，但不是Minkowski时空。在连续扇区，真空解的R=0，Minkowski时空是背景解。然而，前人的研究表明，将Minkowski时空的R^2引力线性化并不会产生引力子，而只会产生一个传播标量。简而言之，在这个理论中，明可夫斯基在线性化的水平上并不“引力”。我们计划的目标之一是研究在原始作用中添加一个非最小耦合到Ricci标量的希格斯场的含义。希格斯场最初必须是无质量的，才能保持受限的Weyl对称性，但在对称性自发破坏后，它在爱因斯坦作用下获得了质量。加入希格斯场有两个至关重要的后果。首先，在爱因斯坦作用量中，希格斯粒子和额外的无质量标量场之间出现了一个新的相互作用项。其次，尽管没有希格斯场的原始纯R^2理论没有可行的Minkowski背景，但当希格斯场获得非零负真空期望值时，它确实有一个。如果希格斯粒子是三重标量场，人们就可以在爱因斯坦引力下研究磁单极子。我和我的合作者已经在Minkowski、DS和ADS背景下数值获得了具有非最小耦合的爱因斯坦-杨-米尔斯-希格斯(EYMH)磁单极子。我们现在想用相互作用项来研究EYMH磁单极子，这是迄今为止从未进行过的事情。我们可以用一个无质量的希格斯二重态把所有的标准模型场都包含在我们的原始动作中。在自发对称性破缺后，我们将用质量希格斯粒子恢复通常的标准模型，但在标量场和希格斯双重态之间有一种新颖的相互作用。作为超越标准模型的物理学的一部分，这将是非常有兴趣的研究。

Pure R^2 gravity (R^2 gravity alone with no additional R term where R is the Ricci scalar) has been shown to be equivalent to Einstein gravity with non-zero cosmological constant and an extra massless scalar field. This equivalence occurs via the spontaneous breaking of its scale symmetry (it actually possesses a larger symmetry called restricted Weyl symmetry). The massless scalar field is identified as the Goldstone boson of the broken sector. The non-zero cosmological constant implies that in the Einstein action the vacuum or background spacetime have non-zero Ricci scalar and can be either de Sitter (dS) or anti-de Sitter (AdS) but not Minkowski spacetime. In the unbroken sector, the vacuum solution has R=0 and Minkowski spacetime is a background solution. However, it was shown by previous authors that linearized R^2 gravity about Minkowski spacetime does not yield gravitons but only a propagating scalar. Simply put, Minkowski does not "gravitate" at the linearized level in this theory. One of the objectives of our program is to study the implications of adding a Higgs field that is non-minimally coupled to the Ricci scalar in the original action. The Higgs field must be originally massless to preserve the restricted Weyl symmetry but it acquires a mass in the Einstein action after the symmetry is spontaneously broken. There are two crucial consequences to adding the Higgs field. First, in the Einstein action, there is a new interaction term that arises between the Higgs and the extra massless scalar field. Secondly, though the original pure R^2 theory without Higgs field has no viable Minkowski background, it does have one when the Higgs field acquires a non-zero vacuum expectation value. If the Higgs is a triplet of scalar fields, one can study magnetic monopoles under Einstein gravity. My collaborator and I already obtained numerically Einstein-Yang-Mills-Higgs (EYMH) magnetic monopoles with non-minimal coupling in Minkowski, dS and AdS backgrounds. We would now like to study the EYMH magnetic monopoles with the interaction term, something which has never been carried out to date. We can include all the Standard Model fields in our original action with a massless Higgs doublet. After spontaneous symmetry breaking we would recover the usual Standard Model with a massive Higgs but with a novel interaction between the scalar field and the Higgs doublet. This would be of great interest to study as part of physics beyond the Standard Model.