Mini Black Hole Signatures in the Standard Model – Instantons at the LHC

I was recently very lucky and got an ERC grant to search for axion-like particles at colliders. I might write about this topic in the future, but today I want to write about a QCD Instantons. Those have been the topic of my ERC grant proposal two years ago, which was not funded. I don’t take that personal (one never should, since there is a huge portion of luck involved in  all funding decisions), however, I still think that QCD Instantons are a super interesting topic. Hence Instantons will be the first topic of this blog.

I guess most of you have never heard of instantons. Funnily, Instanton processes are known already in non-relativistic quantum mechanics, where they describe tunneling transitions of finite action. As an illustrative example, one can calculate the tunnel processes in a double-well potential shown in Figure 1. These processes are forbidden in classical physics, as they represent solutions to the equation of motion with negative kinetic energy. However, by rotating the real time t to imaginary (Euclidean) time 𝛕=i·t, the potential well changes into a hump, also shown in Figure 1 and a classical solution can be derived. The consequence of this rotation of the time-coordinate becomes clear when interpreting it in the Feynman path integral formalism: Here, every possible path is weighted by 

Although any path is allowed in quantum mechanics, the dominant contribution comes from paths which maximise the weight factor and thereby minimise the classical action. When calculating the solution in Euclidean time, one finds

which is illustrated in Figure 2 along with its imaginary time derivative. These solutions are called (anti)-instanton, depending on the sign. As can be seen, the transition is localized around the time τ0, i.e. the system changes its state rapidly, therefore the name instanton. The corresponding action of this classical solution yields

where λ is the height of the potential well and x0 corresponds to one of its minima. The action is finite, does not depend on τ and therefore non-trivial solutions have been found with a finite transition probability.

Figure 1: Double well potential with two classical states (left); transition for potential to τ = i · t into the double hump potential (right).	Figure 2: Instanton solution in imaginary time (left) and its derivative, illustrating the localisation in time.

The concept of Instanton solutions can be extended to Yang-Mills theories. The non-Abelian nature of Yang-Mills theories implies non-contractable loops in the space of its gauge fields, leading to a non-trivial vacuum structure. The topology of a Yang-Mills vacuum is depicted in Figure 3 by the energy density of the gauge field as a function of the Chern-Simons or winding number NCS, describing the topological charge of a system. In analogy to the quantum-mechanical example, instantons describe tunneling transitions in Minkowski spacetime between classically degenerate vacua, which only differ by their winding number by one unit, i.e. ΔNCS =1. Here, an Instanton solution is not only localized in time, but also in space, i.e. it has a certain spatial extension. There is also a second class of classical solutions, known as Sphalerons, corresponding to a transition from one vacuum by a half-integer winding number on top of the energy barrier (also shown in Figure 3), where its static energy corresponds to the barrier height.

Figure 3: Instanton and Sphaleron processes in the topology of a Yang-Mills vacuum; energy density of the gauge field (y-axis) vs. winding number NCS (x-axis).	Figure 4: Production and decay of an Instanton pseudo-particle

   

These solutions differ significantly from the solutions known from ordinary perturbation theory, where only those field configurations are accessible which correspond to small changes of the vacuum field at NCS = 0, while other minima, which are not accessible by continuous transformation of the gauge field, are ignored. Clearly, this approximation holds only as long as the energy barrier between the vacua is sufficiently large. 

Let’s take a short break and recap: The vacuum structure of the Standard Model – both the QCD but also the electroweak part – is not trivial: there is an infinite number of vacua. The reason, why you probably never heard of these Instanton processes is simple: in perturbation theory, we just choose one specific vacuum, expand around our coupling constants and ignore all other vacua around it, as we assume they are separated by the potential wall in between. 

Nevertheless, Instanton and Sphaleron solutions provide crucial ingredients for an understanding of several aspects in the Standard Model: On the one hand, Instanton and Sphaleron processes in the electroweak sector are associated to baryon+lepton number violation. These become important at high temperatures, as the system has then enough energy to “move” from one vacuum to another [1]. This has a crucial consequence on the evolution of the baryon and lepton asymmetries of the universe (see [2] for a review). On the other hand, such topological fluctuations of the gauge fields in QCD have been argued to play an important role in various long-distance aspects of QCD, and as such provide a possible solution to the axial U(1) problem [3] or are at work in chiral symmetry breaking.

The actual height of the energy barrier between two vacua, sometimes called Sphaleron mass MSp, depends on the type of the underlying Yang-Mills theory. For the electroweak sector, the height is in the order of 10 TeV. For QCD, on the other hand, the barrier height between two vacua is defined by the energy scale parameter Q of the underlying process. It will be large for high energy processes, but it will be small when we reach low energies, e.g. ΛQCD=218 MeV. In fact, the barrier is inversely proportional to the fine-structure constant of the relevant gauge theory. This also means, that Instanton processes play a role in low energy phenomena, which is the reason why they are indeed observed in lattice QCD calculations. 

The question for me as experimental physicists is: Can we find experimental evidence for Instanton (or Sphaleron) processes in the lab? To answer this question, we have to answer two subsequent questions: first, what is the cross-section, i.e. the probability that these processes occur and second, what is their experimental signature. Unfortunately, the answer to both questions is not very pleasant. 

Naively, the cross section of Instanton processes in particle collisions are exponentially suppressed by the height of barrier: for Instanton processes in the electroweak sector, one expects cross sections in the order of 0.001 fb for a 200 TeV proton-proton collider. If you are not familiar with cross sections: This essentially means you will need to measure for many many years at a collider that isn’t even planned by the most ambitious among my colleagues. So, what about QCD? Here things are more promising in proton-proton collisions at 13 TeV, i.e. the energies of the LHC: We expect cross sections in the order of millibarns for processes with energies in the 10 GeV region and still a few pico-barns for processes with a relevant energy scale of 200 GeV. That sounds cool – no? Well, here comes the second question into the game: What are the experimental signatures of an Instanton process? For a reason, which I am happy to admit that I don’t fully understand, an Instanton tunnel processes can be seen as the creation and decay of a pseudo-particle with a certain mass, the latter determined by the barrier height [5].

The leading Feynman diagram for the production and the decay of a QCD Instanton pseudo-particle (let’s call it from now on Instanton for simplicity) in proton-proton collisions is shown in Figure 4. The experimental signature is hence given by the isotropic decay of the pseudo-particle into all accessible quarks in addition to some gluons - very similar to the expected decay of mini black holes. The available energy of these decay processes is given by the associated Instanton mass. The energies of the decay particles are therefore expected to lie in the range between a few hundred MeV and up to several GeV. As an experimentalist you expect therefore lots of charged particle tracks and/or jets in your events – unfortunately that looks pretty much like normal QCD processes, typically known as underlying event (for the low energy regime) or multi-jet processes (for the high energy regime). So, some serious thinking has to be done, how to find ways to discriminate QCD Instantons from all the rest. I think it is totally worth it, as it would be a real breakthrough to proof the non-trivial vacuum structure of the Standard Model and all its associated effects, which are predicted since decades. It might not be as fundamental as the Higgs Boson, but equally spectacular. 

So far, no dedicated search efforts for Instanton processes in proton-proton collisions have never been conducted. I think the main reason for that is, that until very recently, no predictions of their signatures have been available for the LHC. This changed recently, but several concerns on the validity of these predictions have been made [6]. In any case: I think Instantons are cool and certainly among the most promising candidates to observe new effects at the LHC.

What you should take away? Instantons are a fundamental prediction by the Standard Model but have never been observed. While it is extremely challenging to the search for them, I am convinced that it is totally worth the effort. And last but not least: Getting rejections from funding agencies shouldn’t demotivate you.

[1] On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe, V.A. Kuzmin, V.A. Rubakov (Moscow, INR), M.E. Shaposhnikov (ICTP, Trieste), Jan 1985. 7 pp., Published in Phys.Lett. 155B (1985) 36 
[2] Electroweak baryon number nonconservation in the early universe and in high-energy collisions, V.A. Rubakov (Moscow, INR), M.E. Shaposhnikov (CERN & Moscow, INR). Mar 1996. 123 pp. Published in Usp.Fiz.Nauk 166 (1996) 493-537, Phys.Usp. 39 (1996) 461-502 
[3] How Instantons Solve the U(1) Problem, Gerard 't Hooft (Utrecht U.). Apr 1986. 50 pp. Published in Phys.Rept. 142 (1986) 357-387 
[4] A Theory of Light Quarks in the Instanton Vacuum, Dmitri Diakonov, V.Yu. Petrov (St. Petersburg, INP). Apr 1985. 33 pp. Published in Nucl.Phys. B272 (1986) 457-489 
[5] Zooming in on instantons at HERA, A. Ringwald, F. Schrempp (DESY). Dec 2000. 13 pp. Published in Phys.Lett. B503 (2001) 331-340 

[6] Large Effects from Small QCD Instantons: Making Soft Bombs at Hadron Colliders, Valentin V. Khoze, Frank Krauss (Durham U., IPPP), Matthias Schott (University Coll. London & Mainz U.). Nov 21, 2019. 29 pp, e-Print: arXiv:1911.09726 [hep-ph]