I published a paper on my nonlocal electroweak theory in Modern Physics Letters A in 1991,17 extending the work published earlier that same year with Richard Woodard on QED. Later, I co-opted my graduate student Michael Clayton into this project, and we did further calculations to try to justify this new electroweak theory.18 We included a prediction for the mass of the as-yet-undiscovered top quark. We published this second paper in the same year, 1991, in the same journal, Modern Physics Letters A.
However, the idea of spontaneous symmetry breaking and the associated prediction of the Higgs boson had, since the early 1970s, become so entrenched in the minds of physicists that an alternative to the Weinberg–Salam model was considered irrelevant, and my published papers on a nonlocal electroweak theory were mostly ignored by the particle physics community. I realized at the time that we were probably 20 to 25 years away from experiments that could decide whether the Higgs boson existed, so it seemed rather futile to continue these speculations and I abandoned my efforts to develop Higgsless models.
BUILDING A NEW PARTICLE PHYSICS THEORY WITHOUT A HIGGS BOSON
Fifteen years later, during the time of the construction of the LHC, the major goal of which was to find or exclude the Higgs boson, I returned to the investigation of whether one could construct an electroweak theory without a Higgs boson. Clearly, the issue of whether the Higgs particle existed would eventually be decided by the LHC experiments. I became curious again about how robust the standard-model electroweak theory was. If the Higgs boson was confirmed definitely to exist, then there would be no need for any alternative electroweak theory. On the other hand, if the Higgs boson were excluded experimentally, then it would be necessary to redesign the Weinberg–Salam model.
The experimentalists at the Tevatron accelerator at Fermilab had devoted serious attention to seeking the elusive Higgs particle before the LHC became operational. So far, the Tevatron experiments had not succeeded in finding a Higgs boson. Now, it seemed to me, was the time to rethink the electroweak theory. Since then, as the data continue to accumulate at an astonishing rate—no longer at the Tevatron, but at the LHC—the possibility that the Higgs boson does exist is becoming real. By July 2012, the hints of its existence were growing stronger, with the announcement of a new boson at 125 GeV. But during the construction of the LHC, things looked quite different.
In 2008, Viktor Toth and I investigated further my early ideas from 1991 on how to generate particle masses from a nonlocal quantum field theory.19 At that time, there were no experimental hints of the existence of the Higgs boson—and if the current evidence for a Higgs boson turns out to be false, there should be renewed interest in the physics community in alternative, non-Higgs models.
Our paper, like the standard model, was based on the premise that you start with a phase in the early universe in which all the elementary particles are massless, including the W and Z bosons. In our model, as in my earlier papers, the particle masses were generated from the quantum field theory self-energy mechanism, in contrast to the standard Higgs model, in which the masses were generated by the interactions of the particles with the Higgs field in the vacuum.
Later, though, I began worrying about the widely held assumption that the universe starts with an unbroken symmetry phase in which the quarks, leptons, and W and Z bosons were massless. Why did this have to be so? A reason for making this assumption in the standard model is that it invokes the basic symmetry of the theory, in this case the group symmetry described by SU(2) × U(1). The idea is that some mass-generating mechanism breaks this symmetry, which is necessary because the different masses of the elementary particles demand that the symmetry be broken. Yet, I wondered whether the whole idea of generating masses for the elementary particles from either a Higgs–ether field or from complicated quantum field theory calculations was unnecessary. Maybe the particle masses had their origin in some other fundamental physical process, such as gravity or the mysterious origin of the inertial mass of particles?
I decided to rethink the whole electroweak theory from a different angle. I considered the possibility that there never was a phase in the universe in which particles were massless, except for the photon and the gluon. To agree with experimental data, I assumed that the basic nonabelian symmetry group SU(2) was always intrinsically broken in the universe and, apart from possible quantum field theory corrections, the particle masses were simply represented in the calculations by their experimental values. That is, perhaps it was a meaningless question to ask how the leopard got his spots; he simply always had them.
The idea that particles always had their masses may not seem so surprising to a nonphysicist, but since the 1960s, particle physicists have clung persistently to the belief that there exists an explanation for the origin of particle masses. This belief was only reinforced by the publication of the idea that spontaneous symmetry breaking and a Higgs field could explain the origin of masses. Although this explanation could be justified for the W and Z bosons, it was not as convincing for the quarks and leptons because their masses could not be predicted by their interactions with the Higgs field. Their masses were simply adjusted to fit the experimental data by means of free coupling–constant parameters. One may wonder why particle physicists were so focused on discovering the origin of the masses of elementary particles, while not even considering the origin of, for example, electric charge or the color charge of quarks. Or why were attempts not made to explain the origin of the fine-structure constant, which measures the strength of electromagnetic interactions? In the past, physicists such as Wolfgang Pauli and Arthur Eddington had speculated on the origin of the fine-structure constant, but later generations of physicists showed little concern about the beginnings of anything beyond mass.
I wrote a new paper on electroweak theory that started with the premise that there does not exist a massless phase of fermions and bosons in the early universe, except for the photon and gluon. This paper was published in 2011.20
BACK TO LOCAL QUANTUM FIELD THEORY
One of the challenges in doing cutting-edge research in particle physics is to make sure that you have explored all possible avenues when approaching a fundamental problem. I began to have the nagging feeling that I had not sufficiently justified giving up local quantum field theory and its retention of microcausality. So I then posed a new question: Is it possible to develop a renormalizable theory of electroweak interactions using local quantum field theory without a Higgs particle? This required a complete mental turnaround from my nonlocal quantum field theory ideas back to the standard local quantum field theory. Had we missed some key element in our investigations of weak interactions in electroweak theory since the early 1960s? Was there some important feature of local quantum field theory that was missed from the beginning that could avoid the infinities produced by the massive W and Z particles in calculating scattering amplitudes? I appreciated the fact that the odds of reaching this goal were small, considering that many of the best brains in physics had contemplated this problem for more than half a century. Nevertheless, I forged ahead to investigate this question just to satisfy my own curiosity.
The key barrier to finding a successful finite, local electroweak theory was the issue of gauge invariance. I posed the question: Is there some way of reintroducing gauge invariance with massive W and Z particles? Recall that because the W and Z are massive, they break the required gauge invariance of electroweak theory, rendering the theory unrenormalizable and violating unitarity.
Back in 1938 Swiss physicist Ernst Stuekelberg proposed a way of maintaining the gauge symmetry of a quantum field theory even though the masses of the particles interacting in the theory were put in “by hand.”21 Normally, if you simply put the particle masses into the theory by hand, this breaks the gauge invariance of the theory and it becomes unrenormalizable and violates unitarity. Stuekelberg published his paper in French in a European journal that was not widely known to quantum field theorists at the time. He recognized in his 1938 paper that mass
ive electrodynamics contains a hidden scalar field, and he formulated a version of what would become known as the abelian Higgs mechanism.
His method of retaining gauge invariance even with masses added in by hand works without renormalizability problems for electrically neutral boson interactions such as a hypothetical massive photon or the neutral Z particle of weak interactions; however, when we try to extend this Stuekelberg technique to massive charged vector bosons, like the positively or negatively charged W boson, we again run into difficulties with renormalizability.
In 1962, Lee and Yang published a paper22 that attempted to unify electromagnetism with the weak interactions. This paper was published a year after Sheldon Glashow’s seminal paper introducing the idea that we need a weak neutral current, with its associated electrically neutral Z boson, to complete a unified theory of electromagnetic and weak interactions. Lee and Yang based their quantum field theory arguments on a charged massive vector boson, W, and on the electromagnetic photon fields. We recall that this was some 20 years before the discovery of the W and Z bosons at CERN in 1983. Lee and Yang cleverly used a form of Stuekelberg’s gauge invariance without actually referring to his paper, and discovered that under certain constraints the theory could be made renormalizable and not destroy the conservation of probabilities (unitarity). This was two years before the Group of Six discovered the significance of spontaneous symmetry breaking in gauge field theory, and five years before the publication of Steven Weinberg’s paper on electroweak unification based on the Higgs mechanism and spontaneous symmetry breaking of the vacuum.
I reviewed the many publications alluding to the problems of renormalizability with the W particle, and despite these problems, in 2011 I attempted to construct an electroweak theory using the Stuekelberg formalism to maintain gauge invariance and renormalizability. I titled the paper “Can Electroweak Theory without a Higgs Particle be Renormalizable?” The idea was to discard spontaneous symmetry breaking, the Higgs mechanism, and the existence of a Higgs particle, and begin with a theory that has intrinsic SU(2) nonabelian symmetry breaking, which means that the universe did not start with a massless phase. The masses of the particles are their experimental masses, which are never zero except for the photon and gluon. The theory was based on the minimal model of the observed 12 quarks and leptons, and the W, Z, photon, and gluon bosons.
The Stuekelberg formalism came with some baggage: extra scalar spin-0 bosons, which have not been detected experimentally as elementary particles. These scalar bosons, like the scalar spin-0 Higgs boson, were an essential part of the Stuekelberg formalism. They were not a problem for interactions involving the neutral Z boson, which is associated with an abelian U(1) symmetry like the photon, because these scalar bosons did not interact with the Z boson or the photon. This theory based on Stuekelberg’s gauge invariant formalism was potentially a renormalizable and unitary local field theory.
However, for the W particle in my theory, things were not so easy. The unwanted scalar bosons did couple with the W bosons, causing difficulties with renormalizability and conservation of probabilities (unitarity) in scattering amplitudes. Yet the W “propagator” could be realized within the Stueckelberg formalism, allowing the theory to be renormalizable.23 The conservation of probabilities was in danger in the Stueckelberg formalism of my electroweak theory because of the unavoidable elementary scalar particles interacting with the charged W boson. But it turned out that the masses of Stueckelberg’s scalar bosons could be made large enough that they would be undetected by the LHC even when it is running at its maximum energy of 14 TeV. Above that mass–energy, my electroweak theory breaks down and is no longer renormalizable and unitary. So for energies below this maximum value for the mass of a scalar boson, the theory was fully renormalizable and conserved unitarity. This means that such a theory was only an “effective” electroweak theory, valid up to a high energy somewhere above 14 TeV, at which point the nasty scalar bosons start interacting with the W.
To me this was a disappointment, because I was not able to construct a fully complete, renormalizable electroweak theory based on the standard local quantum field theory, which has been accepted by physicists as the gospel truth for the past seven decades. This was in contrast to my electroweak theory based on nonlocal quantum field theory, for which I was able to construct an ultraviolet complete quantum field theory valid to infinite energy. At this stage, would I prefer the nonlocal electroweak theory over the effective local electroweak theory? I had to answer this question by investigating carefully the experimental consequences and predictions of both theories.
The problem with an effective theory of the kind I was able to construct using standard local quantum field theory is that it raises the problem of how to “cure” the theory when it breaks down at high energies. The solution would require new physics, presumably in the form of new particles at these higher energies—for example, Stueckelberg’s scalar bosons. However, if these energies are above the attainable energy of the LHC (14 TeV), then the whole subject would be cast into limbo until a new accelerator is built that can go to much higher energies.
Let us return to the standard Higgs boson model of electroweak interactions. Is this theory valid to all energies? I believe the answer is a firm no. There are issues of instability of the vacuum with a light Higgs boson, such as the 125-GeV Higgs-like boson discovered at the LHC. Moreover, a scalar field such as the Higgs field suffers from the pathology of what is called the Landau singularity, or Moscow zero, discovered in QED by Landau during the 1950s. This singularity is intrinsic to a scalar field and will appear at some energy reached eventually by future accelerators. It destroys the consistency of the theory at these high energies.
Perhaps the higher-energy accelerator now being planned at CERN, the ILC (international linear collider), will be able to decide these issues of the breakdown of theories at very high energies. This accelerator will collide positrons and electrons at much higher energies than LEP2, making it easier to discover new particles. This is a different process than at the LHC, which is a ring collider that smashes protons and protons. However, in the event that no new particles are detected at the LHC beyond this current new Higgs-like boson at 125 GeV, then the government funding for higher energy new machines like the ILC may be in jeopardy.
If the new 125-GeV boson turns out to be a light Higgs boson, but the hierarchy problems remain, the particle physics community would still have to be convinced to give up the standard local quantum field theory and consider alternative theories such as my nonlocal quantum field theory. Currently, this seems unlikely. Physicists, on the whole, are a conservative lot and do not give up on their cherished theories easily, which have taken decades to develop. What would be required if the LHC comes up empty-handed beyond the new boson is a “changing of the guard.” The older physicists who have devoted their careers to developing standard quantum field theory will not want to sacrifice these ideas, even though they do not provide a complete and successful description of particle physics. To young physicists, on the other hand, this situation would be a rare opportunity to come up with new ideas.
ANOTHER POSSIBLE LOCAL ELECTROWEAK THEORY
I made another attempt to build an electroweak theory based on local quantum field theory. I asked myself the question: Can we generate the masses of the elementary particles from a dynamical field theory symmetry breaking? In contrast to my previous attempt, in which the elementary particles had intrinsic mass from the beginning, this time I began by assuming that there was a massless symmetric phase in the early universe. This phase would contain an exact SU(2) × U(1) group symmetry, as in the standard model with the Higgs boson.
I studied the 1962 paper about the renormalizability of gauge theories by Abdus Salam, my former professor at Cambridge.24 This paper came out before the spontaneous symmetry-breaking epiphany by the Group of Six in 1964. In it, Salam cleverly formulated three conditions under which a nonabelian gauge theory could be renormalizable. The first c
ondition required that the “bare” mass—a particle’s mass without interactions with other particles and fields—of all the elementary particles in a formalism using perturbation theory was zero. Recall that, in a renormalizable theory based on perturbative expansions of the field quantities, the measured mass of a particle is equal to its bare mass plus its self-energy mass. The second and third conditions, involving some complicated mathematics, could not be met for charged W intermediate vector bosons with a nonzero bare mass in a nonabelian gauge theory. However, they could be met for the massive neutral Z particle, which obeys an abelian gauge invariance.
I formulated my local electroweak theory starting with the premise that the bare masses of the elementary particles, and in particular the masses of the W and Z bosons, are zero. Now I invoked the idea that I had investigated previously—that all the masses of the elementary particles were generated by their self-energies instead of by a spontaneous symmetry breaking with a Higgs boson. Following Salam’s paper, I used a method of approximating the electroweak equations based on a technique borrowed from atomic physics and condensed matter physics called the Hartree–Fock self-consistent procedure. In the perturbation solution, I chose a new vacuum state that allowed me to calculate the self-energies of the particles through a step-by-step iterative method. I was able to show that, within this scheme, and using the self-energies of the particles and a symmetry breaking of the group SU(2) × U(1) that incorporated the unification of electromagnetism and the weak interactions, the theory was renormalizable. In addition, a self-energy calculation was able to fit the masses of the W and Z bosons while keeping the photon massless. Moreover, my theory achieved unitarity successfully for the scattering of the W bosons at energies between 1 TeV and 2 TeV.
Cracking the Particle Code of the Universe Page 19
