WHY SHOULD WE CARE ABOUT THE TOP QUARK YUKAWA COUPLING?
F. Bezrukov"■''■'". M. Shaposhnikovd**
a CERN. СH-1211 Geneve 23, Switzerland
bPhysics Department, University of Connecticut, Starrs, CT 06269-3046, USA
'-B1KEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA
d Institut de Théorie des Phénomènes Physiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
Received November 5, 2014
In the cosmological context, for the Standard Model to be valid up to the scale of inflation, the top quark Yukawa coupling y, should not exceed the critical value yf''1, coinciding with good precision (about 0.2 %o) with the requirement of the stability of the electroweak vacuum. So, the exact measurements of yL may give an insight on the possible existence and the energy scale of new physics above 100 GeV, which is extremely sensitive to yt. We overview the most recent theoretical computations of yf'1 and the experimental measurements of yt. Within the theoretical and experimental uncertainties in yt, the required scale of new physics varies from 10' GeV to the Planck scale, urging for precise determination of the top quark Yukawa coupling.
Contribution for the JETP spécial issue in honor of V. A. Rubakov's 60th birthday
Contents
1. Introduction.............................. 389
2. Standard Model and the scale of new physics............................................ 390
3. Vacuum stability and cosmology....... 390
DOI: 10.7868/S0044451015030015
1. INTRODUCTION
In the Spring of 2014, Valory Rubakov was visiting CERN and joined a bunch of theorists for lunch at a CERN canteen. As often happens, the conversation turned to the future of high-energy physics: what kind of questions should be answered and what kind of experiments should be done. Valory was arguing for the high-energy frontier that would allow search-
E-mail: fedor.bezrukov'ffluconn.edu
**E-mail: mildiail.shaposhnikov'&epfl.ch
4. Computation of the critical top-quark Yukawa coupling................................ 393
5. Top Yukawa coupling and experiment . 394
6. Conclusions............................... 396
References................................. 397
ing for new physics, whereas the authors of this article brought attention to the precision measurements of the top-quark Yukawa coupling. We remember Valory asking: "Why should wo care about the top-quark Yukawa coupling?" For some reasons, the interesting discussion was interrupted and we did not have a chance to explain our point of view in detail. We use this opportunity to congratulate Valory on his coming jubilee and give an answer to his question in writing. We apologize to Valory for describing a number of facts well-known to him, which we included to make this essay accessible to a wider audience.
2. STANDARD MODEL AND THE SCALE OF NEW PHYSICS
After the discovery of the Higgs boson at the LHC, the Standard Model (SM) became a complete theory in the sense that all the particle degrees of freedom that it contains theoretically have been found experimentally. Moreover, there are no convincing deviations from the SM in any type of high-energy particle physics experiments. This raises a number of questions: "Have we obtained at last the ultimate theory of Nature?" and "If not, where we should search for the new physics?"
The answer to the first question is well known and is negative. The reasons are coming from the observations of neutrino oscillations, absent in the SM, and from cosmology: the SM cannot accommodate dark matter and baryon asymmetry of the Universe. The last but not the least is the inflation, or, to stay strictly on the experimental evidence side, the flatness and homogeneity of the Universe at large scales and the origin of the initial density perturbations. On a more theoretical side, the list of drawbacks of the SM is quite long and includes incorporation of gravity into a quantum theory, the hierarchy problem, the strong CP problem, the flavor problem, and so on.
The answer to the second question is not known. Theoretically, it is clear that some type of new physics must appear near the Planck energies Mp = = 2.435 • 1018 GeV, where gravity becomes important, but these energies are too high to be probed by any experimental facility. The naturalness arguments put the scale of the new physics close to the scale of elec-troweak symmetry breaking (see, e.g., fl, 2]), but it is important to note that the SM in and of itself is a consistent quantum field theory up to the very high energies exceeding the Planck mass by many orders of magnitude, where it eventually breaks down due to the presence of Landau poles in the scalar self-interaction and in the U(l) gauge coupling.
As for the experimental evidence in favor of the new physics, it does not give any idea of its scale: the neutrino oscillations can be explained by addition of Majorana leptons with the masses ranging from a fraction of electron-volt to 1016 GeV, the IIlclSS of particle candidates for dark matter discussed in the literature varies by at least 30 orders of magnitude, the mass of the in-flat on can be anywhere from hundreds of MeV to the GUT scale, whereas the masses of new particles responsible for baryogenesis can be as small as a few MeV and as large as the Planck scale.
As we argue in this paper, at the present moment the only quantity that can help us to get an idea about
the scale of the new physics is the top Yukawa coupling tji.. It may happen that the situation will change in the future: the signals of new physics may appear at the second stage of the LHC, or the lepton number violation will be discovered, or the anomalous magnetic moment of the muon will convincingly be out of the SM prediction, or something unexpected will show up.
3. VACUUM STABILITY AND COSMOLOGY
In the absence of beyond-the-SM (BSM) signals, the only way to address the question of the scale of the new physics is to define the energy where the SM becomes theoretically inconsistent or contradicts some observations. Because the SM is a renormalizable quantum field theory, the problems can appear only because of the renormalization evolution of some coupling constants, i.e., when they become large (and the model enters strong coupling at that scale), or additional minima of the effective potential develop, changing the vacuum structure. The most dangerous constant1^ turns out to be the Higgs boson self-coupling constant A with the RG evolution at one loop
The right-hand side depends on the interplay between the positive contributions of the bosons and negative contribution from the top quark. Before the discovery of the Higgs boson, it was customary to show the results as a function of the Higgs mass M/, « y/2A(// = M/, ) v, with other parameters of the SM fixed by experiment. The Landau pole in the Higgs self-coupling constant A occurs at energies smaller than the Planck scale for the Higgs mass M/, >175 GeV, and conies closer to the Fermi scale when the Higgs boson mass increases [3 5]. For small Higgs masses, the coupling becomes negative at some scale, and if the Higgs mass is below 113 GeV, the top quark loops give an essential contribution to the Higgs effective potential, making our vacuum unstable with the lifetime smaller than the age of the Universe [6 8]2K
The Higgs boson found at the LHC has
a mass
Mf, « 125.7 ± 0.4 GeV [10], which is well within this
11 The only other problematic parameter is the U(l) hypercharge, which develops a Landau pole, but only at the energy scale significantly exceeding the Planck mass.
2> We note that, strictly speaking, the Universe lifetime depends strongly on the form of the Planck-scale-suppressed higher-dimensional operators in the effective action [9].
A
At, GeV
Fig. 1. Renormalization group running of the Higgs coupling constant A for the Higgs mass M/, = 125.7 GeV and several values of the top-quark Yukawa coupling yt{fi = 173.2 GeV)
GeV4
10T(i
1074
1072
10™
io08
10°° 1064
IO02
1017 1018 1019
y, GeV
Fig. 2. A very small change in the top-quark Yukawa coupling yt (taken at the scale /./ = 173.2 GeV) converts a monotonic behavior of the effective potential for the Higgs field to that with an extra minimum at large values of the Higgs field
interval. This means that the lifetime of our vacuum exceeds that of the Universe by many orders of magnitude (see, e. g., [11]) and that the SM without gravity is a weakly coupled theory even for energies exceeding the Planck scale also by many orders of magnitude. Hence, it looks that we cannot obtain any hint about the scale of the new physics from these considerations. However, this is not true if we include the history of the Universe in analysis, starting from inflation till the present time.
Because we want to gain an insight into the new physics, a way to proceed is to assume that there is none up to the Planck scale and see if we run into any contradiction. We can start from the SM with-
out gravity and consider the effective potential for the Higgs field. The contribution of the top quark to the effective potential is very important, because it has the largest Yukawa coupling to the Higgs boson. Moreover, it conies with the minus sign and is responsible for the appearance of the extra minimum of the effective potential at large values of the Higgs field. We fix all parameters of the SM to their experimental values except the top Yukawa coupling (we see below that presently it is the most uncertain one for the problem under consideration). For definiteness, we use the MS subtraction scheme and take yL at some specific normalization point fi = 173.2 GeV. Then the RG evolution of the Higgs coupling A for various top-quark Yukawa couplings is illustrated by Fig. 1. Close to the "critical" value of the top Yukawa coupling, to be defined exactly momentarily, effective potential (4.2) behaves as shown in Fig. 2. For tjL < yf''11 — 1.2 • 10-6, it increases while the H
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