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

纯粹的R^2引力 *（只有R^2引力，没有附加的R项，其中R是里奇标量）已经被证明等价于具有非零宇宙常数 * 和一个额外的无质量标量场的爱因斯坦引力。这种等价是通过尺度对称性的自发破缺而发生的（它实际上拥有一个更大的对称性，称为限制外尔对称性）。无质量标量场被确定为 * 破碎扇区的戈德斯通玻色子。非零的宇宙学常数意味着在 * 爱因斯坦作用量中，真空或背景时空具有非零的里奇标量，可以是德西特（dS）或反德西特（AdS），但不是闵可夫斯基 * 时空。在不间断扇区中，真空解R=0，闵可夫斯基 * 时空是背景解。然而，先前的作者已经证明了关于闵可夫斯基时空的线性化R^2引力并不产生引力子，而只是一个传播标量。简单地说，闵可夫斯基在这个理论中的线性化水平上并没有“引力”。** 我们计划的目标之一是研究在原始作用量中加入一个与里奇标量非最小耦合的希格斯场的影响。希格斯场最初必须是无质量的，以保持受限的外尔对称性，但在对称性自发破缺后，它在爱因斯坦作用量中获得了质量。加入希格斯场有两个关键的后果。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产生了一个新的相互作用项。第二，虽然没有希格斯场的原始纯R^2理论没有可行的闵可夫斯基背景，但当希格斯场获得非零真空期望值时，它确实有一个。如果希格斯玻色子是一个三重标量场，人们就可以在爱因斯坦引力下研究磁单极子。我的 * 合作者和我已经获得了数值爱因斯坦-杨-米尔斯-希格斯 *（EYMH）磁单极子在闵可夫斯基，dS和AdS背景下的非最小耦合。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。* 我们可以把 * 所有标准模型场都包含在我们最初的作用量中，并带有一个无质量的希格斯粒子 * 双重态。在自发对称破缺之后，我们将恢复通常的标准模型，其中有一个大质量的希格斯粒子，但在标量场和希格斯偶线之间有一种新的相互作用。这将是非常有趣的研究 * 作为物理学的一部分，超越标准模型。

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.