Tuning to the Edge of the Abyss in SU(5)

I show that if a dimensionless parameter is tuned to be close to the boundary of the positivity domain and symmetry breaking is driven by a cubic term in the Lagrangian, the scale of the physics of symmetry breaking in a quantum field theory as measured by the Higgs mass can be much greater than the dimensional scales in the classical Lagrangian. Radiative corrections produce large and physically important corrections, helping to stabilize the large VEV. The resulting picture contrasts sharply with the"modern"view of QFT as an effective field theory. I describe how this mechanism might produce the GUT scale in an SU(5) model in which the dimensional parameters in the Lagrangian are at the low scale.

I show that if a dimensionless parameter is tuned to be close to the boundary of the positivity domain and symmetry breaking is driven by a cubic term in the Lagrangian, the scale of the physics of symmetry breaking in a quantum field theory as measured by the Higgs mass can be much greater than the dimensional scales in the classical Lagrangian.Radiative corrections produce large and physically important corrections, helping to stabilize the large VEV.The resulting picture contrasts sharply with the "modern" view of QFT as an effective field theory.I describe how this mechanism might produce the GUT scale in an SU (5) model in which the dimensional parameters in the Lagrangian are at the low scale.
Dimensional analysis is one of our simplest and most powerful tools.But there are situations in which it can be confusing if used uncritically.In this note, I show that in a field theory in which a dimensionless parameter can be tuned close to the boundary of the positivity domain, a cubic term in the Lagrangian can produce symmetry breaking in which the scale of the physics of symmetry breaking is parametrically larger than the dimensional scales in the classical Lagrangian.This is not exactly a failure of dimensional analysis because there is a small dimensionless parameter measuring the distance to the boundary.However, it is an outlandish suggestion because it does not fit with the modern picture of quantum field theory as an effective theory.We have gotten used to thinking about quantum field theory as a low-energy approximation to some other high-energy physics.It is hard to argue with this if we are looking at a field theory of quasiparticles in condensed matter physics.But for fundamental physics, until we understand quantum gravity or string theory or whatever comes next, I believe that this jury is still out and we should consider unconventional possibilities like the one proposed here.
For a renormalizable Lagrangian, the classical positivity domain is the region of the space of dimensionless parameters for which the quartic potential of scalar fields ϕ is positive definite so that the potential is positive for large |ϕ|.On the boundary of the positivity domain, some homogeneous combination of components of ϕ vanishes as a coupling λ → 0. If we are very near that boundary, the quartic potential depends very weakly on that combination.Then cubic terms in the potential may push the minimum to very negative values for large |ϕ| so that some components of the VEV are parametrically large compared to the dimensional parameters, with the Higgs mass and other physical parameters going to infinity as we go to the boundary.Cubic terms are crucial.
The application I will describe is to a toy GUT model.But before I describe the model, it may be helpful to explain why the standard model is not an example of the mechanism.In an ungauged doublet Higgs model with quartic coupling λ and negative mass squared −m 2 , the VEV squared is v 2 ∝ |m 2 |/λ which goes to ∞ as λ → 0. However the Higgs mass squared, which is the scale of the physics of symmetry breaking, is ∝ λv 2 ∝ |m 2 | which remains fixed and of the order of the dimensional parameter in the Lagrangian as λ gets small.Of course, if we gauge the electroweak symmetry to get the standard model, the gauge boson masses get large as λ → 0, but at least at tree level, their masses are just proportional to the gauge couplings which have nothing to do with the physics that produces the VEV.[1] In the mechanism I will describe, the symmetry breaking is driven not by a negative mass-squared term (m 2 can be positive or zero), but by a cubic term in the Lagrangian with coefficient κ with dimensions of mass rather than mass squared.As a coupling λ goes to zero, the VEV of a field goes like v ∝ κ/λ and the tree-level Higgs mass goes to ∞ as λ → 0 like κ/ √ λ for fixed κ.And there are even heavier particles with mass ∝ |κ|/λ, which are important in the radiative corrections.[2] Thus there are large physical scales in the theory that are independent (at tree level) of the gauge couplings without any large scales in the Lagrangian.
When I decided to investigate this mechanism, I rather expected that quantum corrections would drastically change things.And indeed, I find that the leading quantum corrections are large.But they actually stabilize the vacuum with the large VEV.
I believe that this mechanism can be applied to many models with trilinear couplings and multiple quartic couplings.A relatively simple example where a mechanism to produce a large VEV might be physically interesting is an SU(5) gauge theory [3] with a 24 of scalars [4].We consider a toy GUT with only a 24 of scalars.It is not crazy to think of this by itself because the 24 cannot couple directly to the standard model fermions, so it makes some sense to ignore the fermions in the physics of the GUT symmetry breaking.In the conventional picture, the large VEV of the 24 is generated by a conventional Higgs mechanism with a negative mass-squared term of the order of the square of the GUT scale.But in fact, we can tune close to the abyss and let the vacuum do this job with no large dimensional parameters in the Lagrangian.Take the 24 scalar field, ϕ, to be a hermitian, traceless 5×5 matrix, and write the most general invariant potential This has the required cubic interaction and λ 14 and λ 23 define the domain of positivity.The coefficients of λ 14 and λ 23 are both positive semi-definite.The λ 14 term vanishes when for some unitary U .The λ 23 term vanishes when Thus positivity requires λ 14 > 0 and λ 23 > 0 and if one of these two couplings becomes very small, the κ term drives the corresponding component of ϕ to a very large VEV.For very small coupling, the effect of the mass term is very small, and a non-zero mass complicates the formulas enormously, so for pedagogical reasons we will take m = 0 in the rest of the paper.This is not a renormalizable constraint because m 2 depends on the renormalization scale which we will discuss later.But it simplifies the formulas without making any essential difference for small λ.
In this limit, the VEV is either (2) or (3).The results are summarized below.
In the application to GUTs we would like to have λ 23 < 2 2/3 λ 14 /3 so that the β 23 vacuum is picked out and the symmetry breaks to SU(3)×SU(2)×U(1).We will discuss this case in detail.The potential V (ϕ 23 β 23 ) has an inflection point at ϕ 23 = 0 and for κ > 0 and small λ 23 , there is a deep minimum at κ/λ 23 as shown in figure 1.For non-zero real m, the inflection point splits (barely noticeably for small λ 23 ) into a shallow local minimum at ϕ 23 = 0 and a weak local maximum for ϕ 23 of order m 2 /κ which stays fixed and irrelevant when λ 23 → 0 as the deep minimum moves out to infinity.Then the massive particles in the theory are the (3, 2) of leptoquark gauge bosons, a scalar (1, 1) which I will refer to as the GUT-Higgs to avoid confusion with the standard model Higgs, and the adjoint scalars transforming like (8, 1) and (1, 3) under SU(3)×SU(2).Their tree-level masses for small λ 23 are where g is the gauge coupling.All have masses that go to infinity as λ 23 → 0 for fixed κ, but at different rates!The smaller mass for the GUT-Higgs would be very interesting if it persisted beyond tree level because it would be an intermediate mass scale, but I will argue that the one loop corrections modify this quantitative detail while leaving intact the important qualitative feature of large physical masses in the small λ 23 limit.
We can address the quantitative issue using the background field method a la Coleman-Weinberg (CW) [5] and constructing the CW potential as a function of ϕ 23 .The SU(3)×SU(2) ensures that the other fields in (6) will be eigenstates of the ϕ 23 dependent "mass" that appears in the CW calculation.
The CW contribution to the potential is where µ is an arbitrary renormalization scale and n j is the degeneracy of the multiplet.The M j and n j are shown in Table I.
*There are 12 color-flavor states, each with 3 spins.Taking a cue from [5], we can simplify the analysis by choosing µ so that the CW term leaves the VEV at its classical value.This requires (8) In the small λ 23 limit, this gives a µ of the order of the mass of the heaviest particles (scalar or gauge boson).
The resulting CW contribution to the effective potential can be much larger than the tree level contribution but it further stabilizes the tree-level vacuum.For some typical parameters, an example is shown in figure 2 general shape of the corrections is easy to understand from (7) and Table I.All of the M j (ϕ 23 ) 2 are proportional to ϕ 23 (this is exact in our approximation of m 2 = 0, but in general the M j (ϕ 23 ) 2 are all small at ϕ 23 = 0).And they are ∝ ϕ 2 23 for large ϕ 23 .This means that the CW contribution has at least an approximate quadratic zero at the origin.This is a local maximum because the logs are negative.By construction the CW contribution has an extremum at ϕ 23 = κ/λ 23 .And because of the two ϕ 23 s in the mass squares, it grows like ϕ 4 23 at large ϕ 23 .The minimum scales like κ 4 /λ 4 23 , so for sufficiently small λ 23 , it dominates the tree-level contribution.One can think of figure 2 approximately as a symmetric CW potential tilted by the effect of the cubic term.This also means that the CW contribution to the GUT-Higgs mass squared scales like κ 2 /λ 2 23 and for sufficiently small λ 23 will overwhelm the tree level contribution that scales like κ 2 /λ 23 , so the scale of the physics of symmetry breaking is proportional to 1/λ 23 like the scalar and massive gauge boson masses.The CW contribution to the GUT-Higgs mass squared for small λ 23 is 9(25g 4 + 56λ 2 14 )κ 2 50π 2 λ 2 23 (10) Because the CW contribution to the GUT-Higgs mass squared is so large, one might worry that higher order contributions will continue to grow and the loop expansion will be useless.I do not expect this to happen.The point is that dimensional analysis implies that large physical masses are proportional to the large VEV which is proportional to 1/λ 23 .The tree level GUT-Higgs mass squared is proportional to 1/λ 2 23 from the VEV squared times a coupling factor of λ 23 .The point is that the 1loop corrections are not anomalously large.The tree-level result is anomalously small because of the small coupling factor λ 23 (which is not present for the other heavy particles in (6).The only way the small λ 23 coupling can get into a denominator is through the VEV, (4), so dimensional analysis guarantees that squared masses beyond tree level will be proportional to 1/λ 2 23 time a power series in the larger coupling, λ 14 , as usual in perturbation theory.
It is a long way from this toy SU(5) model to a realistic GUT.Because the symmetry-breaking mechanism is different, some model-building issues will have to be solved in new and different ways.But I hope that I have convinced the reader that this mechanism for symmetry breaking with cubic terms and small couplings at the edge of the positivity domain is worth exploring as a possible alternative to the conventional scheme.I don't believe that this solves the tuning problems in GUTs, but it might replace the usual tuning problems with very different ones.One might even dream of a theory in which the large scale of gravity arises in this way.And I somehow just like the idea that some of the puzzles in fundamental physics might arise because our world is perched precariously near the edge of oblivion.

TABLE I .
The ϕ23-dependent masses that appear in(7)