NEW PHYSICS AND THE POSSIBLE DETECTION OF THE FIFTH DIMENSION

In 2010, just before the first collisions at the LHC, the experimental expectations of the particle physics community were impressive. The LHC could detect the Higgs particle, supersymmetric particles, dark matter particles, hidden dimensions, mini-black holes, new fundamental fields, new symmetries, and new particles that respond to new laws of physics. Unfortunately, nature refused to reveal its deepest secrets to physicists eager for knowledge, and only one of those possibilities became a reality: the Higgs particle. In subsequent years, all attempts to detect new physics (physics beyond the established Standard Model of particle physics) have failed. Physicists know that the current Standard Model (SM) cannot be the definitive theory because physical phenomena such as dark matter and dark energy exist in our Universe that are not included in it.

Recently, a new and exciting possibility has been attracting the attention of the theoretical physics community. The main attraction of this proposal is that it is based on very general ideas from string theory and allows for a comprehensive answer to many of the problems not explained by the Standard Model, such as the nature of dark matter and dark energy, as well as an explanation for the so-called "cosmic coincidence problem" and the "naturalness" problem. Furthermore, its predictions are quite concise and are right at the limits of the detection capacity of current measuring equipment. This could be the opportunity physicists have been waiting for. Will nature finally reveal its deepest secrets to us?

New physics and the outstanding problems of the Standard Model

The Standard Model of particle physics is one of the most important achievements of human knowledge and contains everything we know about the behavior of fundamental fields and particles. The experimental achievements of the Standard Model are truly impressive, and all experiments conducted so far have conclusively confirmed its predictions. However, there are natural phenomena not included in the Standard Model, such as dark matter and dark energy, which is why the vast majority of physicists believe the Standard Model should be modified or replaced with a more complete theory. Furthermore, there are serious theoretical indications that the Standard Model is incomplete; among these indications, we highlight the "naturalness" problem and the "cosmic coincidence" problem.

The problem of naturalness lies in the enormous and inexplicable energy difference involved in the physical phenomena of the SM. In particle physics, the "natural" scale is the Planck scale. However, the SM incorporates scales trillions of times smaller. For example, the electroweak unification scale is 16 orders of magnitude (100,000 trillion) times smaller than the Planck scale, and the SM prediction for the value of the cosmological constant varies between 44 and 52 orders of magnitude larger than the experimentally measured value.

The problem of "cosmic coincidence" refers to a "double coincidence" that seems almost unbelievable. In the early days of our Universe, energy was concentrated in the form of radiation since matter did not exist. As the Universe cooled and atoms formed, the energy density of matter began to increase, although the energy density of radiation remained dominant. Experimental measurements show us a coincidence that is difficult to believe: throughout the history of the Universe, radiation has dominated over matter, but right now, at the moment humans are taking measurements, the two energy densities are equal! Furthermore, this density is practically equal to the density of dark matter. As if this were not intriguing enough, there is another suspicious coincidence: cosmological measurements indicate that the Universe has just begun to expand at an accelerated rate; that is, the consequences of dark energy have not been felt until our present time. Leaving aside the anthropic principle, this double coincidence is very difficult to explain.

The "Swampland" string theory program

Quantum field theory (QFT) explains the behavior of the fields that make up our current low-energy Universe. Every QFT has a validity range, that is, an energy interval outside of which it ceases to be valid. Therefore, depending on the energy scale we use, we must use one field theory or another; this is why these QFTs are often called effective field theories (EFTs). Our best-established theory of quantum gravity is string theory, and thanks to it and several quantum phenomena related to black holes, we have certain general principles that every theory of quantum gravity must fulfill to be consistent. The key question is: Which EFT theories in our low-energy Universe are consistent with these principles of quantum gravity at high energy?

The so-called "Swampland" program of string theory seeks to answer this question. Surprisingly, the answer is that very few EFTs are compatible with quantum gravity. We say that those EFTs that are not compatible with our theories of quantum gravity are in "the swamp."

A recreation of Yoda reflecting on the fact that most known EFTs are incompatible with what we know about quantum gravity. These EFTs are said to be in Swampland and cannot be representative of our real world.

In this article, we will focus on two of the best-established "Swampland" conjectures: the distance conjecture and the "trans-Planck mode censoring" conjecture. We briefly explain these two conjectures below.

The distance conjecture is based on the fact we discussed in the previous section on TFEs: they are only valid in a range of energies. If we take any TFE and move to extreme values of the field (toward infinity or zero), an "anomaly" appears, and we obtain new states. This anomaly consists of a very characteristic "tower" of particles. In string theory, this phenomenon is universal, and the mass of the resulting tower of particles decays exponentially as follows:

Where alpha is a constant and psi represents the value of the field.

As we approach the limits of validity of EFT, we obtain a tower of particles whose mass decays exponentially.

The trans-Planck censorship conjecture is based on a phenomenon that must occur during cosmic inflation. During inflation, the Universe expands exponentially, causing very small distances in space to be greatly amplified. The conjecture tells us that distances smaller than the Planck distance cannot be amplified and become part of the Hubble horizon (the causal zone) during inflation, since these "trans-Planck" modes have no physical reality. Similar to the previous conjecture, this implies that potentials must decay exponentially when we consider large values of the field. Specifically, we have:

Where d is the characteristic exponent of the exponential decay process.

These two conjectures concern generic behaviors of quantum fields and are based on principles whose non-compliance would produce serious physical inconsistencies (1). As we will see in the next section, these two conjectures allow us to unify and explain dark energy and dark matter in our Universe.

Dark energy and dark matter in our Universe

The first Swampland conjecture we saw earlier applies to any field parameterized by a scalar. In string theory, the cosmological constant of the Universe is governed by a scalar field called the "quintessence." The current cosmological constant of our Universe, in Planck units, has a tiny value very close to zero:

Therefore, our conjecture implies the existence of a tower of very weakly interacting light particles of mass:

This tower of particles is the perfect candidate to constitute dark matter!

Since these particles originate from the tiny value of the cosmological constant, this allows us to unify dark matter and dark energy. This would also shed light on the problem of naturalness: tiny values of certain fields, such as the cosmological constant, would be linked to much larger values associated with the particle tower.

But what would be the energy associated with these particles? These weakly interacting particles contribute an energy of m d to the vacuum energy. Therefore, in our four-dimensional Universe, "a" must be greater than or equal to 1/d, otherwise their contribution to the vacuum energy would be too large and would have been detected. This implies that:

Right in the range expected for dark matter particles!

As we saw in the previous section, the second conjecture implies that:

This restriction on field potentials has an astonishing consequence:

A de-Sitter (dS) spacetime with a positive cosmological constant like ours has an upper limit on the estimated mean lifetime. This time is given by the time associated with the Hubble horizon:

Where we have used the fact that the Hubble horizon scale is:

This means that the maximum expected time for a space-time dS is:

This is just the lifetime of our Universe!

This explains the problem of cosmic coincidence: we are in a typical Universe, which has the maximum time allowed for an expanding Universe. After this time, dS space-time is expected to begin to decay.

The size of the new mesoscopic dimension

As we saw in the previous section, the first conjecture states that the value of the cosmological constant must be less than:

This mass corresponds to an energy of:

This "tower" of particles could originate from string vibrations or from extra compacted dimensions. The first option is ruled out since strings with the above energy would be strongly coupled in the gravitational sector, producing mini-black holes that have not been detected. Therefore, the tower of particles must originate from compacted dimensions. The key question is: How large would these extra dimensions be? The answer is surprising: the value of the above energy implies that the total radius of the compacted dimensions must be greater than 88 micrometers! This scale is often referred to as the mesoscopic scale.

The above expression has an uncertainty of approximately one order of magnitude, which implies that the radius of the compacted dimensions must be on the order of a micrometer . This value is right at the limit of the sensitivity of our current experiments!

We know that at scales of the order of the radius of the compacted dimensions, gravity becomes weaker because gravitons can propagate through the compacted dimensions. This implies that at distances smaller than the compacted radius, the inverse square law of distance no longer holds.

At scales smaller than R the force of gravity fulfills:

Current experiments have confirmed that gravity follows Newton's formula up to distances of about 30 micrometers. This implies that if the Swampland conjectures are correct , we should be very close to detecting the fifth dimension!

As we have seen, the total radius of the compacted dimensions must be on the order of a micrometer. This radius could contain several large compact dimensions. The next question we ask is: How many large compact dimensions could exist within this radius?

Calculating the number of mesoscopic dimensions

The relationship between the usual 4-dimensional Planck mass Mp and the multidimensional Planck mass ^Mp is:

Where n is the number of compacted dimensions and Vn is the total volume of these dimensions. For dimensions on the order of a micrometer, Vn can be considered the total volume of these large dimensions (the volume of much smaller dimensions can be neglected).

For n>2 the Planck mass is less than 1TeV which is experimentally ruled out since the LHC would produce mini black holes of that energy.

Therefore, the only possibilities are one or two large compact dimensions. Calculating for n = 1 and n = 2, we obtain:

The n=2 option is right at the limit of what the LHC has explored, but it can be ruled out for the following reason: cosmological measurements allow us to determine the temperature of neutron stars and their cooling rate once formed. The resulting Planck mass of n=2 would produce a reduction in photon emission that would prevent these objects from cooling at the observed rate.

Therefore the only option is for n to be equal to 1. For this case the radius of the compacted dimensions that we obtain is: R<44 micrometers.

Right in the range suggested by our Swampland conjectures! The conclusion, then, is that there must exist a single dimension on the order of a micrometer.

The last question we have left to answer is: What is the nature of dark matter according to the Swampland conjectures?

The nature of dark matter

Before answering this key question, we must ask ourselves: in the context of extra dimensions, where do the SM fields and particles we observe reside? There are two possibilities: either they reside in the 4 macroscopic dimensions, or they reside on a P-dimensional brane located in a subset of the total D-dimensional spacetime. The first option is ruled out since we would have to observe, for each SM particle, a "tower" of KK particles with the same quantum numbers, and this clearly has not been detected. So we will analyze the second option. The SM fields must then reside on a 3+1-dimensional brane located in a (3+n)+1-dimensional spacetime, where n is the number of extra dimensions of mesoscopic size. Since n=1, the SM 4-brane is located in the 5-dimensional spacetime composed of the 4 macroscopic dimensions and the 5th mesoscopic dimension.

Gravity permeates freely in all dimensions of full spacetime, so the SM brane fields must couple to freely propagating gravitons. The original Kaluza-Klein theory tells us that an observer in the 4 macroscopic dimensions will observe a tower of 1/R energy particles pouring in from the 5th dimension. These two phenomena give us the answer we are looking for: dark matter is due to graviton excitations leaking from the mesoscopic 5th dimension into the 4 macroscopic dimensions .

To calculate the approximate density of filtered gravitons, we must analyze how this process occurred in our early Universe. It is known that during primordial nucleosynthesis after the Big Bang, the SM fields were in thermal equilibrium. Since these fields were located on a brane in the fifth dimension, we can ask whether they were also in equilibrium with the rest of the 4D macroscopic dimensions. The answer must be negative, since if this were true, we would observe different properties in the SM fields. Therefore, there must exist a temperature Ti at which only the SM fields were in equilibrium in the mesoscopic fifth dimension.

At temperature Ti, only the SM brane located in the fifth dimension was in thermal equilibrium. Gravitons, which can travel freely through the 10 dimensions of spacetime, interact with the SM fields. A portion of these excited gravitons leak into the four macroscopic dimensions, producing the tower of particles that constitute the dark matter of our Universe.

The coupling between the SM fields and gravitons is given by:

Where y is the number of extra dimensions. Typically, one would expect:

Therefore we have:

Therefore, at the equilibrium temperature Ti, the density of KK modes (graviton excitations) filtered from the fifth dimension is given by:

Including the values of n and ^Mp we have:

In the radiation period we have that t=1/ T 2 and taking into account that:

The above expression is transformed into:

Integrating between t0 and Ti we obtain

But now, how can we obtain the value of Ti? This value must be less than the mass of the particle tower so as not to excite the fields of the inner dimensions, therefore:

It is logical to assume that these modes were excited before equilibrating at Ti and then settled to the equilibrium value. However, following the Swampland conjectures, this process must occur at a rate at least:

The fields couple to the gravitons with an amplitude of 1/Mp, so we have:

Therefore:

Therefore our natural solution is:

Including this value for Ti in the expression for y DM that we obtained earlier we have:

This is the dark matter density value we detected in our Universe!

Furthermore, this would explain the problem of "cosmic coincidence," that is, the problem of why the energy densities of radiation and matter are equal right now, and why this value coincides with the value at which dark energy begins to dominate the gravity of matter. The explanation is that we have:

And so, using the expression for y DM that we obtained earlier:

Where TMR is the temperature at which the density of radiation and matter become equal and TDE is the temperature at which dark energy begins to dominate.

Finally, it should be noted that this tower of particles is not stable and disintegrates into other, lighter particles, primarily photons. Detecting these photons would be another sign pointing to the existence of the fifth dimension.

The initial energy tower of particles around 1 GeV would disintegrate into photons that could be detected by our experimental equipment.

Conclusions

The detection of new hidden dimensions of space-time would be one of the most important discoveries of all time. Our current understanding of how quantum gravity works tells us that one of the compact dimensions must lie in the micrometer range. Performing calculations with an extra dimension of this size leads us to the correct value for the density of dark matter, to an explanation of the nature of dark matter, to a unified origin of dark matter and dark energy, and to an explanation of the naturalness problem and the cosmic coincidence problem. Furthermore, this extra mesoscopic dimension is right at the limit of what we can achieve in our current experiments.

Too good to be true? We'll find out soon enough!

Grades:

(1) The distance conjecture has been exhaustively verified for AdS spacetimes. For dS spacetimes like ours, there is fairly strong evidence that the conjecture still holds.

Sources:

Swamplandish Unification of the Dark Sector