Non-BPS Monopoles and Dyons via Resurgent Transseries

Semiclassical analysis of quantum field theory is a well-developed method built on computing quantum fluctuations about classical saddle solutions [1][2][3][4]. This process is most explicit in special limits, achieved by tuning parameters in the Lagrangian, for example with various levels of supersymmetry. In such limits there are dramatic simplifications in the classical equations of motion, and correspondingly also in the fluctuation problem. Beyond such special limits, semiclassical analysis is much more difficult to analyze fully. For example, in Yang-Mills-Higgs theory, the classical equations of motion for monopoles and dyons can be reduced by a symmetric static ansatz to a set of coupled nonlinear ODEs which involves a parameter β which is the ratio of the Higgs mass to the W-boson mass [5,6]. When β → 0, the BPS limit, these equations reduce to first order equations with simple explicit solutions [7][8][9], and their fluctuations are also well understood [3,4]. This is an example of a general phenomenon [8]. Away from the BPS limit, the monopole and dyons solutions are considerably more complicated, and even less is known about their fluctuations.

Here we study the ’t Hooft-Polyakov monopole and dyon equations away from the BPS limit using ideas from resurgent asymptotics. Previous work on non-BPS monopole solutions has been numerical [10,11] and/or studying small or large deviations from the BPS limit [12][13][14]. In this paper we apply a method that does not rely on such approximations. This is a general method developed by O. Costin in the context of resurgent asymptotic analysis of coupled nonlinear differential equations [15,16]. The coupled nonlinear ODEs of the ’t Hooft-Polyakov ansatz are naturally solved by a transseries ansatz that expands the solutions in the far field limit as a series of exponentially suppressed terms multiplied by fluctuation expansions. All higher order terms in the transseries are expressed in terms of the leading fluctuation solution. The transseries parameters are simply the coefficients of the leading homogeneous solutions, which then propagate through the transseries in a systematic fashion. This method also applies to the more general dyon solutions [9], with a more general transseries structure.

In Section II we recall some basic notation and results. Section III describes the transseries structure for non-BPS monopoles for β = 1, when the Higgs and W-boson masses are equal. Section IV determines the transseries parameters by matching to a solution from small r. Sections V and VI describe the more general transseries structures that arise for general β, and for dyon solutions.

We consider SU (2) Yang-Mills-Higgs theory with the Higgs field in the adjoint representation, for which the action is [2][3][4][5][6]:

with a = (1, 2, 3). Here, e denotes the gauge coupling constant, with γ the strength of the scalar self-interaction and v the vacuum expectation value of the Higgs field. The mass of the Higgs field is M H = v √ γ, with the W-bosons acquiring a mass M W = ve via the Higgs mechanism. We define the dimensionless mass ratio β ≡ M H /M W . The field strength tensor and covariant derivative are

For monopoles we adopt the ’t Hooft-Polyakov static, purely magnetic, and spherically symmetric ansatz [5,6]

(We discuss dyon solutions in Section VI.) In terms of the radial fields, the monopole mass is:

The classical equations of motion are

Finiteness of the mass requires the leading behavior as r → ∞:

The precise form of the subleading terms is discussed below in Section III. In this paper we apply Costin’s transseries expansion method [15,16] to analyze the coupled nonlinear ODEs in ( 5)- (6).

A. BPS Monopole: β = 0

When β = 0, the monopole mass can be written in factored form

In the β → 0 limit, the classical equations ( 5)-( 6) reduce to first-order (BPS) equations (retaining the boundary condition that H(r) → 1 at infinity) [7,8]:

This leads to the explicit closed-form BPS solutions [7,8]:

As r → ∞ the BPS solutions are expanded in powers of e -r :

These are simple, convergent, transseries expansions, without asymptotic fluctuation factors. The expansions at r = 0 have a finite radius of convergence, r c = π:

The BPS mass is determined by the field values at infinity:

III. NON-BPS MONOPOLES: β = 1

When β > 0, the finiteness of the mass requires the leading asymptotic behavior at infinity to be as in (7), with corrections that are exponentially small. This leads to a transseries expansion as r → ∞, with sums over two independent exponential factors, e -r and e -βr , and with power law fluctuations. To understand more clearly this resurgent transseries structure, we first specialize to the case where the Higgs and W-boson masses are equal: β = 1. This is in some sense ‘intermediate’ between the β → 0 BPS limit [7,8] and the β → ∞ limit studied in [12,14].

A. Transseries Ansatz for β = 1

When β = 1, the classical equations of motion ( 5)-( 6) a

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