Towards a nonsupersymmetric string phenomenology

Over the past three decades, considerable effort has been devoted to studying the rich and diverse phenomenologies of heterotic strings exhibiting spacetime supersymmetry. Unfortunately, during this same period, there has been relatively little work studying the phenomenologies associated with their nonsupersymmetric counterparts. The primary reason for this relative lack of attention is the fact that strings without spacetime supersymmetry are generally unstable, exhibiting large one-loop dilaton tadpoles. In this paper, we demonstrate that this hurdle can be overcome in a class of tachyon-free four-dimensional string models realized through coordinate-dependent compactifications. Moreover, as we shall see, it is possible to construct models in this class whose low-lying states resemble the Standard Model (or even potential unified extensions thereof)—all without any light superpartners, and indeed without supersymmetry at any energy scale. The existence of such models thus opens the door to general studies of nonsupersymmetric string phenomenology, and in this paper we proceed to discuss a variety of theoretical and phenomenological issues associated with such nonsupersymmetric strings. On the theoretical side, we discuss the finiteness properties of such strings, the general characteristics of their mass spectra, the magnitude and behavior of their one-loop cosmological constants, and their interpolation properties. By contrast, on the phenomenological side, the properties we discuss are more model-specific and include their construction techniques, their natural energy scales, their particle and charge assignments, and the magnitudes of their associated Yukawa couplings and scalar masses.

Figure 1

A sketch of the boson/fermion oscillations which are the hallmark of a hidden “misaligned supersymmetry” existing in the spectrum of all tachyon-free nonsupersymmetric closed strings. For each mass level $n$ we plot $\pm \log (|a_{nn}|)$, where $a_{nn}$ is the number of bosonic minus fermionic states at that level. The overall sign is chosen according to the sign of $a_{nn}$; thus positive values indicate surpluses of bosonic states over fermionic states, and negative values indicate the reverse. The points are connected in order of increasing $n$ in order to stress the alternating, oscillatory behavior of the boson and fermion surpluses throughout the string spectrum. These oscillations represent the manner in which nonsupersymmetric string theories continue to exhibit the finiteness for which string theory is famous, even without supersymmetry.

Figure 2

(a) The one-loop zero-point function (cosmological constant) $\mathrm{\Lambda }$. (b) The one-loop one-point dilaton “tadpole” diagram. In general, the value of the dilaton tadpole is always proportional to $\mathrm{\Lambda }$. As a result, a nonzero cosmological constant implies a nonvanishing one-loop dilaton tadpole diagram, which in turn indicates that our effective potential has a linear term in $\varphi$. Thus a nonzero cosmological constant indicates that our string ground state is not stable.

Figure 3

Degeneracies of physical states for the interpolating model in Eq. (3.17) with $a=1$ (upper left), $a=0.33$ (upper right), $a=0.25$ (lower left), $a=0.125$ (lower right). Within each plot, points are connected in order of increasing world-sheet energy $n$. In all cases we see that surpluses of bosonic states alternate with surpluses of fermionic states as we proceed upwards in $n$; this behavior is the signal of an underlying “misaligned supersymmetry” which exists within all modular-invariant nonsupersymmetric tachyon-free string theories and which is ultimately responsible for the finiteness of closed strings—even in the absence of spacetime supersymmetry. For $R=\sqrt{{\alpha }^{\prime }}$ (or $a=1$), we see that this oscillation between bosonic and fermionic surpluses occurs within the exponentially growing envelope function $|a_{nn}|\sim e^{c\sqrt{n}}$ associated with a Hagedorn transition. However, as the compactification radius increases (or equivalently as $a\to 0$), we see that a hierarchy begins to emerge between the oscillator states and their KK excitations; the oscillator states continue to experience densities of states which are exponentially growing as functions of $n$, but their corresponding KK excitations are densely packed within each interval $(n,n+1)$ and, as expected, exhibit constant state degeneracies.

Figure 4

The rescaled cosmological constant $(2/a)\mathrm{\Lambda }$ associated with the model in Eq. (3.17), plotted versus $a\equiv \sqrt{{\alpha }^{\prime }}/R$ (solid line). We see that $(2/a)\mathrm{\Lambda }$ indeed interpolates between $\mathrm{\Lambda }=0$ at $a=0$ and $\mathrm{\Lambda }\approx 725$ as $a\to \infty$, where $\mathrm{\Lambda }\approx 725$ is the cosmological constant associated with the ten-dimensional $SO(16)\times SO(16)$ heterotic string. Note that the factor of $(a/2)$ is the effective (T-dual) “volume” of compactification in the $a\to \infty$ limit; dividing by this factor allows our interpolating nine-dimensional cosmological constant to asymptote to a finite ten-dimensional limit as $a\to \infty$. The dashed line shows the small-$a$ behavior indicated in Eq. (4.10).

Figure 5

Vacuum-energy contributions from massless states (solid line) versus proto-gravitons (dashed line). In this figure, we have plotted $\log (|{\tilde{I}}_{0,0}^{(10)}(a)|)$ and $\log (|{\tilde{I}}_{0,-1}^{(10)}(a)|)$ versus $a$, respectively. We see that the former exceeds the latter in the $a\to 0$ limit, and indeed exceeds the latter for all $a\lesssim 0.54$.

Figure 6

Roadmap illustrating our procedure for constructing semirealistic nonsupersymmetric string models with $N_b=N_f$, as discussed in the text.

Figure 7

The structure of the spectrum of a generic interpolating model with suppressed cosmological constant in the limit of large interpolating radius. States with masses below $M_{\text{string}}$ (or below $n=1$) consist of massless observable states, massless hidden-sector states, their would-be superpartners, and their lightest KK excitations. For these lightest states, the net (bosonic minus fermionic) numbers of degrees of freedom from the hidden sector are exactly equal and opposite to those from the observable sector for all large radii. Note that this cancellation of net physical-state degeneracies between the observable and hidden sectors bears no connection with any supersymmetry, either exact or approximate, in the string spectrum. Nevertheless, it is this conspiracy between the observable and hidden sectors which suppresses the overall cosmological constant and enhances the stability of these strings. For the heavier states, by contrast, the observable and hidden sectors need no longer supply equal and opposite numbers of degrees of freedom. The properties of these sectors are nevertheless governed by misaligned-supersymmetry constraints, and the entire string spectrum continues to satisfy the supertrace relations in Eqs. (2.19) and (2.20). These relations thereby continue to maintain the finiteness of the overall string theory, even without spacetime supersymmetry.

Figure 8

Degeneracies of physical states for our Pati-Salam model with exponentially suppressed cosmological constant. In this figure the inverse radius $a=1/r$ varies from $a=3$ (upper left) to $a=0.1$ (lower right). Comparing with Fig. 3, we see that all of the general features associated with general interpolating models in Fig. 3 survive here as well, including a smoothly growing exponential envelope function for $a\sim \mathcal{O}(1)$ which slowly deforms into a discretely step-wise growing exponential function as $a\to 0$. As discussed in Sec. 3, this reflects the emerging hierarchy between KK states and oscillator states. However, we also observe a critical new feature which reflects the fact that this model has an exponentially suppressed cosmological constant: the removal or “evacuation” of all nonzero net state degeneracies $a_{nn}$ for $n\le 1$ for sufficiently small $a$. Thus, for sufficiently large radius, the spectrum of such models develops an exact boson/fermion degeneracy for all relevant mass levels $n<1$, even though there is no supersymmetry anywhere in the spectrum. Indeed, as illustrated in Fig. 7, this degeneracy does not occur through a pairing of states with their would-be superpartners, but rather as the result of the balancing of nonzero net degeneracies associated with a nonsupersymmetric observable sector against the degeneracies associated with a nonsupersymmetric hidden sector. This is why the evacuation of nonzero state degeneracies for $n<1$ is exact for a sufficiently small (nonzero) inverse radius $a$, even though supersymmetry is only restored in the strict $a\to 0$ limit.

Figure 9

The rescaled cosmological constant $\mathrm{\Lambda }/a^2$ for the Pati-Salam model of Sec. 7, plotted versus $a\equiv \sqrt{{\alpha }^{\prime }}/R$. For large $a$, we find that $\mathrm{\Lambda }/a^2$ tends to a constant, a feature which is similar to that in Fig. 4 and which indicates that the $a\to \infty$ limit of this model is nonsupersymmetric and tachyon-free. We also see that the entire curve is finite, which indicates that no tachyons emerge at any intermediate radii. Thus this model lacks Hagedorn-like instabilities, as we have already noted in the text. However we observe that the small-$a$ behavior of this curve is radically different from that in Fig. 4. First, we see that $\mathrm{\Lambda }$ is not power-law suppressed, as in Fig. 4, but rather exponentially suppressed. Second, and somewhat surprisingly, we observe that $\mathrm{\Lambda }$ has the opposite sign as $a\to 0$ as it does when $a\to \infty$. Indeed, we see that the cosmological constant appears to have a stable minimum near (but not precisely at) the self-dual radius, and moreover that the cosmological constant crosses zero at yet another (slightly higher) radius. It is not clear whether there might exist a hidden unbroken supersymmetry at either of these specific radii.