Cascades and collapses, “great walls” and “forbidden cities”: Infinite towers of metastable vacua in supersymmetric field theories

We present a series of supersymmetric models exhibiting an entirely new vacuum structure: towers of metastable vacua with higher and higher energies. As the number of vacua grows towards infinity, the energy of the highest vacuum remains fixed while the energy of the true ground state tends towards zero. We study the instanton-induced tunneling dynamics associated with such vacuum towers and find that many distinct decay patterns along the tower are possible: these include not only regions of vacua experiencing direct collapses and/or tumbling cascades, but also other regions of vacua whose stability is protected by “great walls” as well as regions of vacua populating “forbidden cities” into which tunneling cannot occur. We also discuss possible applications of this setup for the cosmological-constant problem, for studies of the string landscape, for supersymmetry breaking, and for $Z^{\prime }$ phenomenology. Finally, we point out that a limiting case of our setup yields theories with yet another new vacuum structure: infinite numbers of degenerate vacua. As a result, the true ground states of such theories are Bloch waves, with energy eigenvalues approximating a continuum and giving rise to a vacuum “band” structure.

Figure 1

The maximum allowed value of $|\chi |$, plotted as a function of $N$. As $N\to \infty$, we see that $|\chi {|}_{\max }$ asymptotically approaches $1/2$ from above.

Figure 2

The moose diagram for our series of models. The $N$ sites [each representing a $U(1)$ gauge group] are connected by links corresponding to the chiral superfields ${\Phi }_i$ and experience nearest-neighbor kinetic mixing governed by a universal parameter $\chi$.

Figure 3

A sketch of the vacuum structure of the $N=3$ model. For $\lambda >{\lambda }_3^{*}$, the corresponding scalar potential gives rise to two distinct minima: a $\{\mathbf{13}\}$ vacuum which serves as the true ground state of the theory, and a $\{\mathbf{12}\}$ vacuum which serves as an additional, metastable vacuum. These two minima are separated by a saddle-point $\{\mathbf{123}\}$ extremum which reduces to the $\{\mathbf{1}\}$ extremum in the formal $\lambda \to \infty$ limit. Note that this sketch is actually a two-dimensional representation of potential energy contours in a three-dimensional field space parametrized by $\{v_1^2,v_2^2,v_3^2\}$.

Figure 4

The $\{\mathbf{123}\}$ saddle-point solution for $N=3$ and $\chi =1/5$, plotted as a function of $\lambda \ge {\lambda }_3^{*}=5/\sqrt{6}\approx 2.04$. Here we show the nonzero field VEVs $|v_1{|}^2$ (upper left plot), $|v_2{|}^2$ (upper right plot), and $|v_3{|}^2$ (lower left plot), as well as the corresponding scalar potential $V$ (lower right plot). While each of these quantities varies with $\lambda$, they quickly reach formal asymptotes as $\lambda \to \infty$.

Figure 5

Critical $\lambda$ values for the $N=3$ model, plotted as functions of $\chi$. Here ${\lambda }_3^{*}={\lambda }_{3,1}^{*}=1/\sqrt{\chi (\chi +1)}$ and ${\lambda }_{3,2}^{*}=\sqrt{2/(2-\chi )}$. For $\lambda >{\lambda }_{3,1}^{*}$, both the $n=1$ and $n=2$ vacua are stable, while for ${\lambda }_{3,2}^{*}<\lambda \le {\lambda }_{3,1}^{*}$, the $n=1$ vacuum is destabilized and only the $n=2$ vacuum is stable. For $\lambda \le {\lambda }_{3,2}^{*}$, even the $n=2$ vacuum is destabilized; in this range a new $\{\mathbf{123}\}$ solution becomes stable and serves as the ground state of the theory.

Figure 6

A sketch of the vacuum structure of the $N=4$ model. For $\lambda >{\lambda }_4^{*}$, the corresponding scalar potential gives rise to three distinct minima: a $\{\mathbf{134}\}$ vacuum which serves as the true ground state of the theory, and two additional metastable vacua of types $\{\mathbf{123}\}$ and $\{\mathbf{124}\}$ above it. These three vacua are separated by three different saddle-point extrema which are each different solutions of $\{\mathbf{1234}\}$ type; in the formal $\lambda \to \infty$ limit, these reduce to the different two-VEV solutions shown. Note that this sketch is actually a two-dimensional representation of potential energy contours in a four-dimensional field space parametrized by $\{v_1^2,v_2^2,v_3^2,v_4^2\}$.

Figure 7

Critical $\lambda$ values for the $N=4$ model, plotted as functions of $\chi$. For $\lambda >{\lambda }_4^{*}={\lambda }_{4,1}^{*}$, all three of our vacua are stable, while for ${\lambda }_{4,2}^{*}<\lambda \le {\lambda }_{4,1}^{*}$, the $n=1$ vacuum is destabilized and for ${\lambda }_{4,3}^{*}<\lambda \le {\lambda }_{4,2}^{*}$, both the $n=1$ and $n=2$ vacua are destabilized. Finally, for $\lambda \le {\lambda }_{4,3}^{*}$, even the $n=1$ vacuum is destabilized; in this range a new $\{\mathbf{1234}\}$ solution becomes stable and serves as the ground state of the theory.

Figure 8

The vacuum structure of the $N=20$ model, plotted for $\chi =1/5$. Each of the local minimum points corresponds to a (meta)stable vacuum state, while each local maximum point corresponds to the saddle-point configuration which directly connects the nearest-neighbor vacuum configurations on either side. For the purposes of this plot, we have then simply connected these points sequentially with straight lines. The vertical axis indicates the energies of these vacua or saddle points (the latter in their asymptotic limits), while the horizontal axis indicates the cumulative distances in field space along a trajectory which begins at the $n=1$ vacuum and then proceeds along straightline path segments to the (1, 2) saddle point, then to the $n=2$ vacuum, then to the (2, 3) saddle point, and so forth.

Figure 9

The geometry of our metastable vacuum towers in field space, here illustrated for the $N=20$ model with $\chi =1/5$. For each $n$ vacuum ($n=1,\dots ,19$) in the $N=20$ vacuum tower, we have plotted two different notions of corresponding field-space distance. In curve (a), we have plotted the cumulative distance (as used in Fig. 8) that accrues as we wind our way from the top of the tower down to the $n$ vacuum, passing through nearest-neighbor saddle points along the way. By contrast, in curve (b), we have plotted the direct straightline distance between each $n$ vacuum and the reference $n=1$ vacuum at the top of the tower. The dramatic difference between these two curves is a reflection of the fact that the vacua in our vacuum tower actually lie along a spiral or helix in the full 20-dimensional field space.

Figure 10

The upper portions of the $N=100$ vacuum tower with $\chi =1/5$, plotted for saddle-point hops of fixed magnitudes (a) $\Delta n=1$, (b) $\Delta n=3$, and (c) $\Delta n=20$, respectively. Note that the energy of a given vacuum is independent of how it is reached; for example, we see that the $n=4$ vacuum has the same energy whether it is realized as the fourth minimum on the $\Delta n=1$ curve or the second minimum on the $\Delta n=3$ curve. However, the corresponding field-space distances relative to the top of the tower are generally smaller when larger hops are utilized when descending.

Figure 11

Values of ${\lambda }_{N,n}^{*}/{\lambda }_{N,1}^{*}$, plotted as functions of $\chi$ for $N=20$ and $1\le n\le 19$. The curves corresponding to $n=1$, 2, 4, 8, 19 are highlighted and labeled; the rest are not highlighted but proceed in sequential order relative to those which are. For $\chi \approx 0.05$, we see that the $n=2$ vacuum has the highest value of ${\lambda }_{N,n}^{*}$, while it is the $n=4$ vacuum which has this property for $\chi \approx 0.45$. The upper envelope of all of these curves corresponds to the critical stability value ${\lambda }_N^{*}$ for the vacuum tower as a whole. Note that all of these curves converge to ${\lambda }_{N,n}^{*}=\sqrt{2^{N-1}/3}$ at $\chi =1/2$.

Figure 12

The $n$ index of the vacuum with the largest corresponding critical value ${\lambda }_{N,n}^{*}$, plotted as a function of $\chi$. This index specifies which vacuum in the tower is the first to destabilize as $\lambda$ is lowered from infinity. In general, for large $N$, this index shifts from $n=1$ to $n=[N/2]$ (where [$x$] is the greatest integer $\le x$) as $\chi$ increases within the range $0<\chi <1/2$.

Figure 13

The critical values ${\lambda }_N^{*}$, plotted as functions of $\chi$ for (a) $N=10$, (b) $N=30$, (c) $N=50$, and (d) $N=100$. We observe that ${\lambda }_N^{*}$ generally grows with $N$, diverging as $\chi \to 0$ and asymptoting to $\sqrt{2^{N-1}/3}$ as $\chi \to 1/2$. We stress that the $\lambda$ values plotted here are the rescaled, dimensionless versions of our original Wilson-line coefficients. As such, there is no conflict with the perturbativity of our model.

Figure 14

Diagrams contributing to the wave-function renormalization of ${\varphi }_i$ at one loop. Each of these diagrams is proportional to $g^2$. Diagram (a) arises due to the quartic couplings in the $D$-term potential, while diagrams (b), (c), and (d) arise due to gauge-interaction terms for the scalars and the supersymmetrization thereof.

Figure 15

A representative of the class of diagrams which contribute to the renormalization of the effective $2N$-scalar vertex at one loop in the broken phase of the theory, due to quartic $D$-term interactions. The diagram contains $N$ vertices (each proportional to $g^2$) and $N$ scalar propagators (with masses proportional to $g^2\xi$); the overall contribution is thus proportional to $g^2$.

Figure 16

A representative of the class of diagrams which contribute to the renormalization of the effective $2N$-scalar vertex at one loop in the broken phase of the theory, due to $F$-term interactions. This diagram represents a $4N-4$-scalar vertex in which $2N-4$ of the scalars on the external lines are assigned VEVs, and its contribution is proportional to ${\lambda }^4$. At high energies, when supersymmetry is effectively unbroken, this diagram is cancelled by the contribution from the diagram in Fig. 17.

Figure 17

A representative of the class of diagrams which correct the effective $2N$-scalar vertex at one loop. The net contribution from this class of diagrams would exactly cancel the contribution from the class of diagrams depicted in Fig. 16 if supersymmetry were unbroken.

Figure 18

Bounce actions for the $N=4$ model, plotted as functions of $\chi$. In each case we have plotted $B(n,n+\Delta n)$ where (a) $(n,\Delta n)=(1,1)$, (b) $(n,\Delta n)=(1,2)$, and (c) $(n,\Delta n)=(2,1)$. Also shown (dotted line) is the bounce action corresponding to a decay lifetime approximating the age of the Universe. In general, all bounce actions increase with increasing $\chi$, ultimately diverging as $\chi \to 1/2$. However, for each value of $\chi$, we see that the bounce actions corresponding to the greatest $\Delta n$ are smaller than those corresponding to smaller $\Delta n$.

Figure 19

Bounce actions for the $N=20$ theory with $\chi =1/5$. For each vacuum $1\le n\le 18$, we plot $B(n,n+\Delta n)$ where the different “curves” (from top to bottom) correspond to $\Delta n=1$, 2, 3, 9, 14, 17, 18, respectively. Note that each plot is truncated when $n+\Delta n$ would exceed $n_{\max }=19$. We observe that for fixed $n$, the bounce action $B(n,n+\Delta n)$ generally decreases with increasing $\Delta n$. As a result, for fixed $n$, transitions which maximize $\Delta n$ are generally favored.

Figure 20

Bounce actions $B(n,N-1)$ with $\chi =1/5$, plotted as functions of $n$ for (a) $N=20$, (b) $N=50$, and (c) $N=95$. In each case, the vacua which populate the lower portions of each vacuum tower eventually decay by tunneling directly to the true ground state. However, the first metastable vacuum to decay in this manner is not the first excited vacuum; instead, these decays follow a complicated collapse pattern, with different portions of the vacuum tower decaying at different times. By contrast, for sufficiently large $N$, the vacua populating the upper portions of our vacuum towers have lifetimes exceeding the age of the Universe, corresponding to the critical bounce action $B\approx 471$ (dotted line).

Figure 21

Bounce actions $B(n,N-1)$ for $N=100$, plotted as functions of $n$ for (a) $\chi =0.1$, (b) $\chi =0.2$, (c) $\chi =0.3$, and (d) $\chi =0.33$. The critical bounce action (dotted line) corresponds to lifetimes exceeding the age of the Universe. In general, we see that only a narrow band of metastable vacua within the vacuum tower will take part in the collapse pattern and decay to the ground state; by contrast, vacua above or below this band in the vacuum tower will generally remain stable. As the kinetic-mixing parameter $\chi$ increases towards its maximum value $1/2$, this collapse band grows increasingly narrow and ultimately disappears.

Figure 22

Normalized bounce actions $B(1,n_f)$ with $\chi =1/10$, plotted as functions of $n_f$ in the allowed range $2\le n_f\le N-1$. The curves whose lower portions progress from left to right correspond to ordered increasing values of $N\in \{1,2,5\}\times {10}^{\{1,2,3\}}$, ending with $N={10}^4$. We observe that as $N$ increases, the bounce actions $B(1,n_f)$ develop a deepening trough whose minimum is approximately located at $n_f\approx n_f*\equiv \sqrt{N\chi }$. The depth of the trough asymptotes to a finite value as $N\to \infty$ in such a way that the minimum bounce action along any such curve corresponds to $n_f=N-1$.

Figure 23

The normalized bounce actions $B(1,n_f)$, plotted as functions of $n_f$ in the allowed range $2\le n_f\le N-1$, with $N={10}^4$. The different curves progressing from lowest to highest correspond to $\chi ={10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$, and ${10}^{-5}$; note that the lowest curve here is the same as the $N={10}^4$ curve shown in Fig. 22. As $\chi \to 0$, we see that $n_f^{*}\to 0$; thus the bounce actions increasingly tend to grow as functions of $n_f$, at least for sufficiently small $n_f$. Also important is the fact that for sufficiently small $\chi$, the bounce action at $n_f=N-1$ begins to exceed that at $n_f=2$. For each such curve, the minimum bounce action therefore occurs not for $n_f=N-1$, but for $n_f=2$. This triggers the onset of cascade (rather than collapse) behavior for our vacuum towers.

Figure 24

Same as Fig. 22, except plotted for $\chi ={10}^{-5}$ rather than $\chi =1/10$. We observe that as $N$ increases (approaching the true $N\to \infty$ limit), the minimum bounce action begins to occur for relatively small values of $n_f$ rather than for $n_f=N-1$. As in Fig. 23, this triggers the onset of cascade (rather than collapse) behavior for our vacuum towers.

Figure 25

Behavior of $B(1,n_f)$ as a function of $n_f$ in the allowed range $2\le n_f\le N-1$, plotted for different values of $N\chi$. The curves whose end positions progress from left to right correspond to ordered increasing values of $N\chi ={10}^{\{-1,0,1,2,3\}}$, where we have held $\chi ={10}^{-3}$ fixed. As $N\chi$ increases, the bounce actions $B(1,n_f)$ develop a deepening trough whose minimum is approximately located at $n_f\approx n_f^{*}=\sqrt{N\chi }$. As a result, the vacuum-cascade step size increases as a function of $N\chi$.

Figure 26

Lifetimes along the different cascade trajectories, calculated for $N=5000$ and $\chi =2.8\times {10}^{-4}$. For each vacuum $n$, we plot $B(n,n^{\prime })$ for that value of $n^{\prime }$ ($n<n^{\prime }\le N-1$) which minimizes $B(n,n^{\prime })$ and which thereby indicates the next vacuum along the corresponding cascade trajectory. Points are connected according to their sequential trajectories. We also show the great wall (dotted line) signifying the critical bounce action corresponding to the age of the Universe. We see from this plot that in this case the top two vacua are stable, while the $n=3$ through $n=10$ vacua cascade down along four distinct trajectories with decreasing lifetimes until they pass the $n=11$ vacuum. At this stage, each trajectory decays directly to the ground state of our vacuum tower. By contrast, vacua beyond the $n=15$ vacuum populate a forbidden city: it is impossible to tunnel into these vacua from any other locations along the vacuum tower, and the Universe can inhabit such a vacuum (for a brief time only) if and only if it is initially born there.

Figure 27

A schematic of the vacuum structure of the model shown in Fig. 26. Vacua in the stable region to the left of the great wall have lifetimes exceeding the age of the Universe, while vacua in the cascade region decay to other (lower) metastable vacua in the vacuum tower. By contrast, vacua in the collapse region decay directly to the ground state of the vacuum tower. Finally, vacua which populate the forbidden city cannot be reached from outside the forbidden city: such vacua can be populated only as an initial condition at the birth of the vacuum configuration.

Figure 28

The spectrum of massive scalars, plotted as a function of the vacuum index $n$ for $N=20$, $\chi =1/4$, and $\lambda ={\lambda }_{20}^{*}\approx 1.05\times {10}^8$. (a) The solid lines indicate the masses of the two complex scalar couplers, while (b) the dotted lines demarcate the edges of the shaded band within which the masses of the remaining scalars lie. Note that the lighter coupler actually becomes massless for the $n=6$ vacuum; it is in this manner that the $n=6$ vacuum begins to destabilize for $\lambda ={\lambda }_{20}^{*}$, in accordance with the results shown in Fig. 12 for $\chi =1/4$. Note that the masses plotted are dimensionless rescaled masses, as discussed in Sec. 2.

Figure 29

The spectrum of massive fermions, plotted as a function of the vacuum index $n$ for $N=20$, $\chi =1/4$, and $\lambda ={\lambda }_{20}^{*}\approx 1.05\times {10}^8$. (a) The solid line indicates the single fermionic coupler, while (b) the dotted lines demarcate the edges of the shaded band within which the masses of the remaining fermions lie. As in Fig. 28, the masses plotted are dimensionless rescaled masses discussed in Sec. 2.

Figure 30

Gauge-boson mass spectra, plotted for the $n=\{1,5,10,15,19\}$ vacua. For this plot, we have taken $N=20$ and $\chi =1/4$, and the (rescaled) masses are displayed on the vertical axis. Note that when $n$ is small, many of the masses tend to cluster around a particular value. As $n$ increases, however, this value drops; fewer of the masses are clustered around this value, and the masses of the remaining gauge bosons become more widely spaced. By contrast, the mass of the lightest massive gauge boson increases with increasing $n$. Thus, as our system tumbles down the vacuum tower, the lightest nonzero gauge boson tends to become increasingly heavy.

Figure 31

Vacuum structure in the $N=6$ model, plotted for (a) $\chi =0.2$, (b) $\chi =0.4$, (c) $\chi =0.5$, and (d) $\chi =0.54$. As $\chi$ increases from zero, we see that the “slope” of our vacuum staircase decreases, ultimately becoming completely flat at $\chi =0.5$. For $\chi >0.5$, the vacuum staircase inverts, with the lightest states now becoming the heaviest states.