String Production in the Abelian Higgs Vacuum

TL;DR

This paper numerically investigates how classical Abelian-Higgs gauge field wavepackets produce strings, identifying conditions for prompt string formation from single or colliding wavepackets.

Contribution

It introduces a numerical framework to study string production from wavepackets and maps the parameter space for string formation in Abelian-Higgs models.

Findings

Single wavepackets can produce strings promptly under certain conditions.

Colliding sub-critical wavepackets can generate strings in specific parameter regions.

The study delineates parameter space regions conducive to string formation.

Abstract

We numerically study string production by evolving classical Abelian-Higgs gauge field wavepackets. Initial conditions are constructed for the propagation of a single wavepacket and for the collision of two wavepackets. We identify regions of parameter space that lead to prompt production of strings from a single wavepacket. The collision of two sub-critical wavepackets can also lead to the production of strings in certain regions of parameter space.

Equations

$$\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\frac{1}{2}|D_{\mu }\varphi {|}^2-\frac{\lambda }{4}{\left(|\varphi {|}^2-{\eta }^2\right)}^2$$

$$E=\frac{1}{2}|D_0\varphi {|}^2+\frac{1}{2}|D_i\varphi {|}^2+\frac{1}{2}({\mathbf{E}}^2+{\mathbf{B}}^2)+\frac{\lambda }{4}{\left(|\varphi {|}^2-{\eta }^2\right)}^2$$

$$\varphi =\eta f(r)e^{i\theta },A_i=v(r){ϵ}_{ij}\frac{x^j}{r^2}(i,j=1,2)$$

$$\mu =\pi {\eta }^{2}F(\beta )$$

$$n=\frac{-i}{2\pi {\eta }^2}\oint d{x}^i{\varphi }^{\ast }{\partial }_i\varphi =\frac{1}{2\pi }\oint \frac{d\theta }{dl}dl$$

$${\mathcal{A}}_x$$

$${\mathcal{A}}_y$$

$${\mathcal{A}}_z$$

$$A_i(t=0,\mathbf{x})$$

$${\partial }_t{A}_i(t=0,\mathbf{x})$$

$${\mathcal{A}}_x^{\prime }$$

$${\mathcal{A}}_y^{\prime }$$

$${\mathcal{A}}_z^{\prime }$$

$$f_1(x,y)=a\exp \left[-\frac{x^2+y^2}{2w^2}\right]$$

$$f_2(t+z-z_0)=\exp \left[-\frac{(t+z-z_0{)}^2}{2w^2}\right]$$

$$f_3(t-z-z_0))=\exp \left[-\frac{(t-z-z_0{)}^2}{2w^2}\right]$$

$$\varphi (t=0,\mathbf{x})=\eta ,[{\partial }_t\varphi (t,\mathbf{x}){]}_{t=0}=0.$$

$$z_0=\frac{{\bar{z}}_0}{\eta },a=\frac{\bar{a}}{\eta },w=\frac{\bar{w}}{\eta },\omega =\bar{\omega }\eta ,{\omega }^{\prime }={\bar{\omega }}^{\prime }\eta$$

$${\partial }_t^2{\varphi }_a$$

$$-\lambda ({\varphi }_b{\varphi }_b-{\eta }^2){\varphi }_a$$

$${\partial }_t{F}_{0i}$$

$${\partial }_t\Gamma$$

$$\Gamma (t=0,\mathbf{x})=0.$$

$$d\theta \to \Delta \theta \equiv {\theta }_{i+1}-{\theta }_i+2\pi k$$

$$\overline{z_0}=1.8,\bar{w}=0.6,{\bar{\omega }}^{\prime }=\bar{\omega }.$$

$$\beta \in [0.08,8.0],\bar{a}\in [0.6,7.0],\bar{\omega }\in [0.2,8.0].$$

$$E_{\mathrm{Single}}=\frac{{\bar{a}}^2{\pi }^{\frac{3}{2}}\eta (9+10{\bar{w}}^2{\bar{\omega }}^2+4{\bar{w}}^4{\bar{\omega }}^4+2e^2({\bar{w}}^2+{\bar{w}}^4{\bar{\omega }}^2))}{8{\bar{w}}^3}$$

$$E_{\mathrm{Double}}$$

Full text

Topological defects such as kinks, strings and magnetic monopoles are classical solutions in a wide range of field theories. In quantum theory, topological defects can be viewed as a bound state of a large number of quanta. The interpretation of solitons as particles is most explicitly known in the sine-Gordon model. In that case, the operators that create and destroy solitons (which are fermions), can be written in terms of particle quanta that are bosons. The question of interest in this paper is if it is possible to assemble particles to make strings? And if so, can we say something about the initial conditions necessary to produce strings?

The transition from particles to solitons is difficult to treat because particles are described by quantum field theory whereas solitons are described by classical field theory. However, from a practical standpoint, we often produce high occupation number states of quantum particles that behave quasi-classically. For example, by sending currents into a light bulb we produce light that can be described as classical radiation using Maxwell’s equations. Thus it is relevant to consider the production of solitons in the scattering of classical waves or wavepackets. We will restrict our attention to this situation and ask what classical initial conditions lead to the production of solitons in the final stage.

These questions were addressed in Ref. Vachaspati (2016) for an SO(3) field theory, where incoming wavepackets led to the production of magnetic monopoles. While the possible production of magnetic monopoles is exciting, it is in the realm of speculative physics because we don’t know if Grand Unified Theories are correct. On the other hand, strings are closer to reality since we do have superconductors in which (gauge) strings exist. In this paper we focus on the production of gauge U(1) strings, where the class of initial conditions we use are motivated by the initial conditions of Ref. Vachaspati (2016).

There are several aspects of the string creation problem that differ from the monopole creation problem. In the latter, once monopole-antimonopole pairs are created with enough energy, they fly apart and survive indefinitely. On the other hand, only closed loops of string can be created. These oscillate, radiate, collapse, and survive only for a finite amount of time. If some of the loops are produced with large angular momentum, they live for longer but eventually decay. A second difference is that the properties of the string network that is produced change with time because the strings interact with each other and intercommute to form smaller loops.

We introduce the field theory and string solution in Sec. I followed by our choice of initial conditions in Sec. II. The computational methods used in our analysis are described in Sec. III and then we present our results in Sec. IV. We conclude in Sec. V.

I Abelian Higgs Model and Strings

We consider the Abelian Higgs model given by the Lagrangian,

[TABLE]

where $\phi=\phi_{1}+i\phi_{2}$ is a complex scalar field, $D_{\mu}=\partial_{\mu}+ieA_{\mu}$, $A_{\mu}$ is the gauge field with field strength tensor $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and $\lambda$ and $e$ are coupling constants. The energy density in the fields is,

[TABLE]

where $E_{i}=F_{0i}$ is the electric field and $B_{i}=\epsilon_{ijk}F_{jk}/2$ is the magnetic field.

Topological string solutions in the Abelian Higgs model are well known. The solution for a straight string along the $z-$axis is,

[TABLE]

where we work in cylindrical coordinates $r=\sqrt{x^{2}+y^{2}}$, $\theta=\tan^{-1}(y/x)$, $f(r)$ and $v(r)$ are profile functions that vanish at the origin and go to 1 asymptotically. The energy per unit length (also the tension) of the string is given by

[TABLE]

where $\beta\equiv 2\lambda/e^{2}$. The function $F(\beta)$ is known numerically and is a smooth, slowly varying function. We also have $F(1)=1$ in the so-called BPS limit when the scalar mass in the model, $m_{S}=\sqrt{2\lambda}\eta$ equals the vector mass, $m_{V}=e\eta$. For $\beta$ not too large, the thickness of the scalar fields in the string is $\sim m_{S}^{-1}$ and of the vector fields is $\sim m_{V}^{-1}$.

The string is characterized by a topological winding number that is defined by

[TABLE]

where $\theta$ is the phase of the scalar field at a given point on the contour and $l$ denotes the parameter along the integration curve.

II Initial Conditions

We base the initial conditions for our simulations on those used for monopole-antimonopole production Vachaspati (2016). We adopt the temporal gauge for all our simulations, that is $A_{0}=0$, and construct circularly polarized gauge wavepacket configurations (not solutions) that propagate along the $\pm z-$axis. Consider the form of a wavepacket propagating in the $-z$ direction,

[TABLE]

where $f_{1}=f_{1}(x,y)$, $f_{2}=f_{2}(t+z-z_{0})$ will be specified below, and $z_{0}$ determines the initial ($t=0$) location of the wavepacket along the $z-$axis. Now the initial conditions for the gauge fields and their time derivatives are,

[TABLE]

This form for the gauge fields satisfies $\nabla\cdot{\bm{A}}=0$ which will be useful later when we discuss Gauss constraints.

We can also construct a wavepacket traveling in the $+z$ direction in a similar manner. To do this, we write the formulae in terms of $f_{3}(t-(z+z_{0}))$:

[TABLE]

And these can be used to construct initial conditions for a wavepacket that propagates in the $+z$ direction as above.

We choose profile functions in a manner that localizes the gauge wavepacket in all directions;

[TABLE]

[TABLE]

[TABLE]

where $a$ is the amplitude and $w$ is the width of the wavepacket.

The initial conditions for the scalar field are “trivial”,

[TABLE]

The free parameters in the initial conditions are $z_{0}$, $a$, $w$, $\omega$ and $\omega^{\prime}$. For our simulations, we will rescale these parameters as follows:

[TABLE]

The dimensionless parameters $\bar{z}_{0}$, $\bar{a}$, $\bar{w}$, $\bar{\omega}$, and $\bar{\omega}^{\prime}$ above are varied in our code. In addition, the Abelian Higgs model has the parameters $e$, $\lambda$ and $\eta$. However, by field and coordinate rescalings, there is only one model parameter given by the ratio of scalar and vector masses, $\beta=m_{S}^{2}/m_{V}^{2}=2\lambda/e^{2}$.

III Computational Techniques

Following the numerical relativity based approach developed in Vachaspati (2016), we introduce a new dynamical variable $\Gamma=\partial_{i}A_{i}$. Then the field variables are: $\phi$, $A_{i}$ and $\Gamma$, altogether 6 functions. The equations of motion for these variables are,

[TABLE]

where $a=1,2$, $\epsilon_{ab}$ is the Levi-Civita tensor with $\epsilon_{12}=1$, $F_{0i}=\partial_{t}A_{i}$ in the temporal gauge, and $g_{p}^{2}$ is a new parameter introduced for numerical stability. The idea is that the square bracket in Eq. (15) vanishes in the continuum because of the Gauss constraints. However, it may not vanish upon discretization. By writing the equations in the above form with the auxiliary function $\Gamma$, we obtain improved numerical stability Baumgarte and Shapiro (2010). The value of the parameter $g_{p}^{2}$ is chosen by numerical experimentation; we have set $g_{p}^{2}=0.75$ in our simulations. The initial conditions for the auxiliary function $\Gamma$ follow from the choice of initial conditions for the gauge field,

[TABLE]

For our analysis, we discretized these equations on a $256^{3}$ lattice with lattice spacing $\Delta x=0.05$ and time step size $\Delta t=\Delta x/4$. The difference equations were solved using the explicit Crank-Nicholson method with two iterations. To reduce computation times, we parallelized our numerical code. As a check of our evolution code, we find that the total energy inside the box is conserved to within 1% during the entire evolution period.

In addition to the evolution of equations, we developed a tracking code which detects strings and calculates the number of loops that are present in the simulation domain at any given time. The program calculates the phase winding as defined in Eq. (5) on every plaquette of the lattice. A non-zero winding on a plaquette implies that a string passes through the plaquette and enters/exits the corresponding cells. The program then connects the strings and records the properties of the loops.

The string tracking algorithm is the same as used in earlier work  Vachaspati and Vilenkin (1984); Pogosian and Vachaspati (1998) but with one subtlety. In calculating the winding as in Eq. (5), we have to find the discretized value of $d\theta$ along the links of the lattice. Generally one uses the “geodesic rule” and the phase difference between lattice sites $i$ and $i+1$ is

[TABLE]

where $k=0,\pm 1$ is chosen to minimize $|\Delta\theta|$. However, this rule ignores the case when $|\Delta\theta|=\pi$. The justification in earlier works has been that this possibility is of zero measure. In our case, however, this situation arises quite frequently. The reason can be seen from the equations of motion and the initial conditions. We start out with $\phi_{1}=\eta$ and $\phi_{2}=0$ i.e. $\theta=0$ throughout the lattice. The equations of motion are such that they tend to preserve $\phi_{2}=0$, and all the non-trivial dynamics is in the $\phi_{1}$ variable, at least at early times. Now $\phi_{1}$ can become negative. When $\phi_{1}$ differs in sign at neighboring lattice sites, this gives a phase difference of exactly $\pm\pi$ and the geodesic rule is ambiguous. In evaluating the winding number, we choose $+\pi$ or $-\pi$ with equal probability.

IV Results

As we have discussed above, the problem contains 1 model parameter, namely $\beta$, and 5 initial condition parameters. We will fix some of these parameters and scan over a range of a few parameters. We set

[TABLE]

We have explored,

[TABLE]

We did not see any qualitative changes as we varied $\beta$ (see below) and so for most of our runs we set $\beta=1$, equivalently $e=0.5$, $\lambda=0.125$. We also chose $\eta=1$ and this sets the length scale in the simulation.

From our initial runs, we found that energy is condensed into strings even from a single wavepacket, i.e. without scattering two gauge wavepackets. We will call this “prompt string production” and it is reminiscent of the discovery in Hindmarsh and Rajantie (2000) that strings may be formed due to purely gauge field fluctuations during a phase transition. In the next subsection, we will explore prompt string production and find that there are regions of parameter space where prompt production does not occur. We will then move on to explore this region of parameter space and find a sub-region where strings are produced when wavepackets collide.

IV.1 Prompt string production

For the single pulse case, after fixing the parameters of the theory, we managed to analytically find expression for total energy in the simulation domain for the wavepacket profiles shown in the previous section. It is as follows

[TABLE]

where $\gamma\equiv w\omega$. With this expression, we can trade one of the parameters for the total energy.

In Fig. 1 we show the prompt production of strings at various times during the evolution. In the first frame, there is energy density of the wavepacket but no strings. Some time steps later, the scalar field has adjusted to the gauge wavepacket and strings, as detected by topological winding, are produced. As the system evolves further, the dense network of strings chops itself up and decays.

We have examined prompt production for several different values of the model parameter $\lambda$ (equivalently $\beta$ since we fix $e=0.5$). Fig. 2 shows how the length in strings – evaluated by counting the plaquettes that contain non-trivial topological winding – changes with time. The figure shows that the outcome is not very sensitive to the value of $\lambda$ and hence we set $\lambda=0.125$ ($\beta=1$) in the runs described below.

In contrast, as seen in Fig. 3, the prompt production of strings depends sensitively on the parameter $\bar{\omega}$. The general trend is that less length is produced for larger $\bar{\omega}$ but the strings that are produced survive for a longer time. This can happen if larger $\bar{\omega}$ leads to larger loops or to loops with higher angular momentum.

In Fig. 4 we plot the energy density integrated over $\bar{x}$ and $\bar{y}$ coordinates, as a function of $\bar{z}$. Prompt string production occurs at the initial location of the wavepacket ($\bar{z}_{0}=1.8$ or 36 lattice spacings away from the center of the lattice). Then the string cluster moves towards the left and also decays.

We have calculated the length of strings at any given time by counting the number of plaquettes with non-trivial winding. We can also estimate the energy in the string network by adding up the field energies in all the cells within $m_{S}^{-1}$ or $m_{V}^{-1}$ (whichever is larger) of the string network. However, the plot in Fig. 6 of the energy vs. time shows reasonable correspondence with the length versus time plot in Fig. 3 for $\bar{\omega}=2.0$, indicating that the strings do not have significant kinetic energy at formation.

As expected, greater initial energy produces more strings. However, our analysis indicates some subtleties in the process of string production. From Eq. (20), it can be seen that, for fixed energy, amplitude becomes smaller as we increase the frequency and vice-versa. (The wavepacket width $\bar{w}$ is fixed in all our runs.) After experimenting with different values of amplitude and frequency at fixed energy, we noticed that there is a minimum/critical amplitude below which we do not produce any strings, as seen in Fig. 6). The parameter space under the critical curve, for which strings are not produced, gets smaller as the energy increases. In the opposite limit of small $\bar{\omega}$ (large amplitude), we see that the total length of strings is far greater (also seen in Fig. 3).

IV.2 Wavepacket collisions

We now consider the case when two wavepackets collide. The parameters are chosen so that there is no prompt string production. However, strings are produced when the wavepackets collide. So now we have two wavepackets in the initial conditions that are headed towards a collision. The initial energy is,

[TABLE]

We again use Eq. (20) for fixing kinematic parameters. For the simulation, we chose $\bar{a}=0.578$ and $\bar{\omega}=9.0$ for the individual wavepackets. With this choice prompt production of strings does not occur, that is, the parameters lie below the critical curve for the single pulse case shown in Fig. 6.

Fig. 7 shows the evolution of the wavepackets and string formation after collision. Very few short-lived strings are produced even though the total input energy is much higher ($\approx 8000$) compared to the single pulse run presented in the previous subsection. The fractional energy in strings as a function of time is shown in Fig. 9. By scanning over different amplitudes, ${\bar{a}}$, for the same total energy, we find the critical curve for string formation when wavepackets collide. The critical curve is plotted in Fig. 9.

V Conclusions

We have explored the formation of U(1) gauge strings due to wavepackets of of gauge fields in two settings: (i) the prompt formation of strings from gauge fields, and (ii) the formation of strings when gauge wavepackets collide. We have restricted our attention to a class of wavepackets with certain parameters, and found critical curves in parameter space that demarcate string formation regions.

It is interesting to contrast string production with magnetic monopole production. Unlike the case of magnetic monopoles, the string loops that are formed are short-lived as they collapse and produce radiation. The loops may live longer if we could find initial conditions that provide them with greater angular momentum but these too will not live indefinitely. On the other hand, once a magnetic monopole and antimonopole pair are produced with sufficient velocity, they will move apart and survive indefinitely. Furthermore, magnetic monopoles are localized objects and so the colliding wavpackets need not be very extended. For strings, the wavepackets have to extend over a region that is the size of the string loop that is to be produced, and only relatively small loops can be produced. In these respects it appears that magnetic monopoles are easier to produce than strings.

The flip side is that we know systems that contain gauge strings while the existence of magnetic monopoles is still speculative. Gauge strings are known to exist in superconductors and, in that setting, our gauge field wavepackets correspond to photon wavepackets. This suggests that by shining light on superconductors we could produce strings within the superconductor. However, a realistic superconductor is described by a different set of equations that take into account the dependence of the model parameters on the temperature Kennes and Millis (2017). It will be interesting to adapt our analysis to study string production in superconductors.

VI Acknowledgment

We are grateful to Onur Erten for remarks. AS thanks the MCFP, University of Maryland for hospitality. The computations were done on the A2C2 Saguaro and Agave clusters at ASU. This work is supported by the U.S. Department of Energy, Office of High Energy Physics, under Award No. DE-SC0018330 at ASU.