One-loop running and the monodromy at inﬁnity

4.1 One-loop running and the monodromy at inﬁnity

The pure $S U (2)$ theory contains only an $𝒩 = 2$ vector multiplet for the $S U (2)$ gauge group, with its Lagrangian given by (2.1.11). For reference we reproduce it here:

L = \frac{I m τ}{4 π} \int d^{4} 𝜃 t r Φ^{†} e^{[V, \cdot]} Φ + \int d^{2} 𝜃 \frac{- i}{8 π} τ t r W_{α} W^{α} + c c .

(4.1.1)

A supersymmetric vacuum is classically characterized by the solution to the D-term constraint

[Φ^{†}, Φ] = 0 .

(4.1.2)

This means that $Φ$ can be diagonalized by a gauge rotation. Let

Φ = d i a g (a, - a) .

(4.1.3)

Roughly speaking, the gauge coupling $τ$ runs from a very high energy scale down to the energy scale $a$ according to the one-loop renormalization of the $S U (2)$ theory. Then the vev $a$ breaks the gauge group $S U (2)$ to $U (1)$ . There are massive excitations charged under the unbroken $U (1)$ , but they will soon decouple, and the coupling remains almost constant below the energy scale $a$ . This evolution is shown in Fig. 4.1.

Figure 4.1: Schematic drawing of the running of the coupling.

Let us describe it slightly more quantitatively. Our normalization of the $U (1)$ Lagrangian and the gauge coupling was given in (1.2.7) and (1.2.15), which we reproduce here:

\frac{1}{2 e^{2}} F_{μ ν}^{U (1)} F_{μ ν}^{U (1)} + \frac{𝜃}{16 π^{2}} F_{μ ν}^{U (1)} {\tilde{F}}_{μ ν}^{U (1)}, and τ^{U (1)} = \frac{4 π i}{e^{2}} + \frac{𝜃}{2 π} .

(4.1.4)

In the broken vacuum, the low-energy $U (1)$ and the high-energy $S U (2)$ are related as in (1.3.3), which we also reproduce here

F_{μ ν}^{S U (2)} = d i a g (F_{μ ν}^{U (1)}, - F_{μ ν}^{U (1)}) .

(4.1.5)

Plugging this in to the high-energy Lagrangian (4.1.1) and comparing the deﬁnitions of $τ$ s, we ﬁnd

τ^{U (1)} = 2 τ^{S U (2)} .

(4.1.6)

This relation gets modiﬁed by the quantum corrections.

Let us denote by $τ (a)$ the low-energy coupling of the $U (1)$ gauge ﬁeld when the vev is given by (4.1.3), and by $τ_{U V}$ the high-energy coupling of the $S U (2)$ gauge ﬁeld at the high-energy renormalization point $Λ_{U V}$ . The one-loop running (3.1.6) then gives

\begin{align} τ (a) & = 2 τ_{U V} - \frac{8}{2 π i} log \frac{a}{Λ_{U V}} + \dots & (4.1.7) \\ = - \frac{8}{2 π i} log \frac{a}{Λ} + \dots & (4.1.8) \end{align}

where we deﬁned

Λ^{4} = Λ_{U V}^{4} e^{2 π i τ_{U V}} .

(4.1.9)

The dual variable $a_{D}$ can be obtained by integrating (4.1.8) once, and we ﬁnd

a_{D} = - \frac{8 a}{2 π i} log \frac{a}{Λ} + \dots .

(4.1.10)

As long as we keep $| a | ≫ | Λ |$ , the coupling $τ (a)$ remains weak, and the computation above gives a reliable approximation.

A gauge-invariant way to label the supersymmetric vacua is to use

u = \frac{1}{2} ⟨ t r ϕ^{2} ⟩ = a^{2} + \dots

(4.1.11)

where $\dots$ are quantum corrections. Let us consider adiabatically rotating the phase of $u$ by $2 π$ :

u = e^{i 𝜃} | u |, 𝜃 = 0 \sim 2 π

(4.1.12)

We have $a \mapsto - a$ . From the explicit form of $a_{D}$ we ﬁnd $a_{D} \to - a_{D} + 4 a$ . We denote it as

(a, a_{D}) \to (a, a_{D}) (\begin{matrix} - 1 & 4 \\ 0 & - 1 \end{matrix}) .

(4.1.13)

The mass formula of BPS particles is

M = | n a + m a_{D} | = |(a, a_{D}) (\begin{matrix} n \\ m \end{matrix})| .

(4.1.14)

Therefore, the transformation (4.1.16) can also be ascribed to the transformation of the charges:

(\begin{matrix} n \\ m \end{matrix}) \to (\begin{matrix} - 1 & 4 \\ 0 & - 1 \end{matrix}) (\begin{matrix} n \\ m \end{matrix}) .

(4.1.15)

We call this matrix

M_{\infty} = (\begin{matrix} - 1 & 4 \\ 0 & - 1 \end{matrix})

(4.1.16)

the monodromy at inﬁnity. The situation is schematically shown in Fig 4.2. The space of the supersymmetric vacua, parametrized by $u$ , is often called the $u$ -plane.

Figure 4.2: Monodromy at inﬁnity.

In our argument, the matrix (4.1.13) could have had non-integral entries, as we read the matrix elements oﬀ from an approximate formula of $a$ and $a_{D}$ . However, the transformation (4.1.15) should necessarily map integral vectors to integral vectors, which guarantees that the matrix (4.1.15) is integral. Not only that, this transformation is just a relabeling of the charges and should not change the Dirac pairing

n m^{'} - m n^{'} = det (\begin{matrix} n & n^{'} \\ m & m^{'} \end{matrix})

(4.1.17)

which measure the angular momentum carried in the space when we have two particles with charges $(n, m)$ and $(n^{'}, m^{'})$ , respectively. A transformation given by

(\begin{matrix} n \\ m \end{matrix}) \to M (\begin{matrix} n \\ m \end{matrix})

(4.1.18)

aﬀects the Dirac pairing as

det (\begin{matrix} n & n^{'} \\ m & m^{'} \end{matrix}) \to det M det (\begin{matrix} n & n^{'} \\ m & m^{'} \end{matrix}) .

(4.1.19)

Therefore, $M$ should necessarily has unit determinant. A $2 \times 2$ integral matrix with unit determinant is called an element of $S L (2, ℤ)$ . It is reassuring that the matrix (4.1.16) satisﬁes this condition.

[next] [prev] [prev-tail] [front] [up]