quantum field theory - No monopoles in the Weinberg-Salam model

I'm reading Chapter 10.4 on the 't Hooft-Polyakov monopoles in Ryder's Quantum Field Theory.

On page 412 he explains why magnetic monopoles cannot appear in the Weinberg-Salam model. I'm I right by saying that he shows that the electromagnetic gauge group $U(1)_{em}$ is not compactly embedded into the $U(1)\times U(1)$ subgroup of $SU(2)\times U(1)$?

He then immediately concludes that the first fundamental group of the unbroken symmetry, which is $H=U(1)_{em}$, $\pi_1(H)$ must be trivial or doesn't exists. Could someone refer me why?

Comment: I know that in the $SU(2)\times U(1)$ ones must consider the second homotopy group from $S^2$ to the orbit $G/H=SU(2)\times U(1)/U(1)$, where $H$ is the isotropy group of a vacuum state, after symmetry breaking. But the second homotopy group of a quotient can be related through a exact series to the kernel of the map from $\pi_1(H)$ into $\pi_1(G)$.

What I do not understand is by which theorem for $H$ having a non-compact covering group $\pi_1(H)$ must be trivial or non-existing (???)?

Answer

I) Consider a Yang-Mills type theory with gauge group $G$. In principle we can consider the same theory with its covering group $\tilde{G}$, with $\pi:\tilde{G}\to G$. The covering group is by definition simply connected: $$\tag{1} \pi_1(\tilde{G})~=~\{\bf 1\}.$$ Any representation $\rho$ of $G$ can naturally be viewed as a representation $\rho\circ \pi$ of $\tilde{G}$. (Warning: the opposite is not true.) Hence the fields of the theory also transform under $\tilde{G}$, and the theory is invariant under $\tilde{G}$. In other words, we could in principle from the very beginning view $\tilde{G}$ as the gauge group of the theory, cf. Section III below.

II) Next assume a spontaneous symmetry breaking of the gauge group $G \to H$ to a subgroup $H$. Define subgroup

$$\tag{2} \tilde{H}~:=~\pi^{-1}(H)~\subseteq~ \tilde{G}.$$

Concretely for the electroweak theory, the gauge group is

$$\tag{3} G~:=~ SU(2)_I\times U(1)_Y ~\supset~ U(1)_I\times U(1)_Y;$$

the covering group

$$\tag{4} \tilde{G}~=~SU(2)_I\times (\mathbb{R},+)_Y$$

is non-compact; and the unbroken electromagnetic subgroup

$$\tag{5} H~:=~U(1)_Q~\subset~ U(1)_I\times U(1)_Y$$

is irregularly/non-compactly/incommensurably/non-topologically embedded, cf. e.g. my Phys.SE answer here. Here we have assume that tangent to the Weinberg angle $\tan\theta_W\in \mathbb{R}\backslash\mathbb{Q}$ is irrational. Now since the non-compact group $H$ is not a subgroup of $SU(2)_I$, the subgroup

$$\tag{6} \tilde{H}~=~(\mathbb{R},+)_Q$$

is also non-compact. It is also simply connected

$$\tag{7} \pi_1(\tilde{H})~\cong~\{\bf 1\}.$$

III) Let us now view the non-compact gauge group $\tilde{G}$ in eq. (4) as the gauge group of electroweak theory. The non-compact gauge group of electromagnetism (5) is then replaced by the non-compact subgroup (6). Standard monopole analysis then shows that there are no magnetic monopoles in the electroweak theory

$$\tag{8} \pi_2(\tilde{G}/\tilde{H})~\cong~{\rm Ker}\left(\pi_1(\tilde{H})\to \pi_1(\tilde{G})\right)~\stackrel{(1)}{\cong}~\pi_1(\tilde{H})~\stackrel{(7)}{\cong}~\{\bf 1\},$$

cf. e.g. Ref. 2.

IV) Now let us return to OP's question. Ref. 1 considers the system from the perspective of $G$ and $H$ rather than $\tilde{G}$ and $\tilde{H}$, and recast the standard monopole analysis in this language. The problem is that a closed loop $\gamma\in G$ that wraps a $U(1)\subseteq G$ has a non-compact lift $\tilde{\gamma}\in \tilde{G}$ which is not a closed loop. Similarly, a closed loop $\gamma\in H$ that wraps a $U(1)\subseteq H$ has a non-compact lift $\tilde{\gamma}\in \tilde{H}$ which is not a closed loop.

Intuitively/heuristically, the relevant notion $\tilde{\pi}_1(\tilde{H})$ (of such non-compact "closed" paths) has more paths than just (7), and for the electroweak theory the relevant quantity

$$\tag{9} \pi_2(G/H)~\cong~\pi_2(\tilde{G}/\tilde{H})~\cong~{\rm Ker}\left(\tilde{\pi}_1(\tilde{H})\to \tilde{\pi}_1(\tilde{G})\right)~\cong~\{\bf 1\}$$

is trivial, in agreement with eq. (8). See Ref. 1 for more details.

References:

1. 
   

   L.H. Ryder, Quantum Field Theory, 2nd eds., 1996; p. 412.

   
   


2. 
   

   S. Coleman, Aspects of Symmetry, 1985; p. 217 & 221.

   
   


3. 
   

   F.A. Bais, To be or not to be? Magnetic monopoles in non-abelian gauge theories, arXiv:hep-th/0407197. (Hat tip: Hunter.)

   
   


4. 
   
   

   S. Weinberg, Quantum Theory of Fields, vol. 2, 1996; p. 443-445.

   
   


5. 
   

   J. Preskill, Magnetic Monopoles, Ann. Rev. Nucl. Part. Sci. 34 (1984) 461-530; Sections 4.2 & 4.3. The PDF file is available here.