Grand Unification Dream Kept at Baywhose subtitle – which includes words such as "failed" as well as "limbo" – is much to a greater extent than negative than the available prove suggests. No smoking gun – such as the proton decay – that would "almost" attempt grand unification has been found. But no adept argue to halt loving grand unification has been found, either, as well as that's why lots of particle physicists perish on on thinking most grand unified theories.
Physicists have got failed to divulge disintegrating protons, throwing into limbo the dearest theory that the forces of nature were unified at the initiative of all of time
What is grand unification? The Standard Model has a guess group, \(SU(3)\times SU(2)\times U(1)\), as well as the guess bosons (gluons, photons, W-bosons, Z-bosons) that are almost unavoidable given a guess group. Also, it works alongside fermionic fields that transform as a reducible representation – a collection of separate, irreducible representations – nether the guess group.
Some fermionic fields (quarks) are colored i.e. triplets \({\bf 3}\) nether the \(SU(3)\) group, others (leptons) are color-neutral i.e. singlets \({\bf 1}\) nether it. Some fermionic fields (the left-handed fermions) are doublets \({\bf 2}\) nether the electroweak \(SU(2)\) factor, others (the right-handed ones) are singlets \({\bf 1}\). And all of these bits as well as pieces send diverse values of the hypercharge \(Y\) nether the \(U(1)\) factor of the Standard Model group.
You may experience it's a contrived setup. The guess grouping has iii factors. Why non one? And the quarks as well as leptons check into several separate representations. Why non i or at to the lowest degree fewer? Well, the questions "why" may last legitimized because theories where the spectrum is simpler – to a greater extent than unified – exist.
The oldest instance is the Georgi-Glashow or minimal \(SU(5)\) model. In 1974, when I was finding diapers helpful, they figured out that the Standard Model guess grouping may last embedded into a elementary group. The fermions may last organized into representations of this grouping as well as the Lagrangian may last written inwards such a means that nosotros may obtain the Standard Model subsequently a symmetry breaking that is really analogous to the electroweak symmetry breaking yesteryear the commons Higgs patch inwards the Standard Model.
How does it work?
Well, \(SU(2)\) as well as \(SU(3)\) may last visualized as a grouping of \(2\times 2\) or \(3\times 3\) matrices, respectively. You may have got a \(2\times 2\) matrix as well as a \(3\times 3\) matrix as well as plow them into blocks on the block diagonal of a larger, \(5\times 5\) matrix, cleverly exploiting the ingenious identity \(2+3=5\). The resulting \(5\times 5\) matrix volition yet last unitary. Moreover, if the blocks' determinants are one, the determinant of the \(5\times 5\) matrix volition also last one. So \(SU(2)\times SU(3)\) is isomorphic to a subgroup of \(SU(5)\).
You may embed the hypercharge \(U(1)\) into the \(SU(5)\), too, yesteryear allowing matrices whose blocks' determinants aren't i but whose total determinant is one, namely matrices of the form\[
\begin{pmatrix}
f^{+3}&0&0&0&0\\
0&f^{+3}&0&0&0\\
0&0&f^{-2}&0&0\\
0&0&0&f^{-2}&0\\
0&0&0&0&f^{-2}
\end{pmatrix}, \quad f=\exp(i\alpha)
\] Both blocks are unitary matrices alongside determinants whose absolute value equals one. The blocks belong to \(U(2)\) as well as \(U(3)\), respectively, but non to \(SU(2)\) as well as \(SU(3)\). However, the \(5\times 5\) matrix does have got the determinant \(f^{3+3-2-2-2}=f^0=1\).
Great. The dimension of \(SU(5)\) is \(5^2-1=24\). So it has 24 generators. The Standard Model guess grouping exclusively has \(8+3+1=12\) generators, so at that topographic point are 12 novel generators inwards \(SU(5)\). These generators – as well as their corresponding novel guess bosons – basically transform as the \(2\times 3\) complex rectangles nether \(SU(2)\times SU(3)\) exterior the diagonal. We haven't observed these W-like bosons, so they must last heavy if they exist. Just similar the Standard Model's W-bosons are comparably heavy to the Higgs boson, the novel bosons – X-bosons as well as Y-bosons, as they're sometimes called – should last heavy as the novel generalized Higgs bosons that intermission \(SU(5)\).
This volume scale is probable to last huge, \(10^{13}-10^{16}\GeV\) or so, because alongside this rather extreme choice, i explains the values of the \(SU(3),SU(2),U(1)\) couplings that ask to unify into a unmarried \(SU(5)\) guess coupling at the symmetry breaking scale. Just similar the Standard Model uses the Higgs doubles, the Georgi-Glashow theory may usage some to a greater extent than complicated representations of \(SU(5)\) alongside a potential that wants to intermission the guess grouping from \(SU(5)\) to the Standard Model group.
The fermions may last arranged to \(SU(5)\) representations, too. Each generation of leptons as well as quarks inwards the Standard Model contains fifteen (or sixteen if nosotros include a right-handed neutrino) two-component fermionic fields. They instruct arranged to \({\bf 5}\oplus \bar{\bf 10}\), the amount of the key 5-component representation of \(SU(5)\) as well as the antisymmetric 2-index tensor alongside \(5\times four / 2\times 1 = 10\) components (complex conjugated, thence the bar).
There are really 2 ways how to clump the numerous lepton-and-quark representations into these 5- as well as 10-dimensional representations of \(SU(5)\). Georgi as well as Glashow works life the old way where the lepton doublet teams upward alongside the right-handed up-quark singlets to exercise the 5-dimensional representation, as well as the remaining fermions belong to the 10-dimensional one.
Surprisingly, people had to hold off upward to 1982-1984 when the flipped \(SU(5)\) model was works life yesteryear Stephen Barr, Dimitri Nanopoulos, as well as alongside some of import additions yesteryear Antoniadis, Ellis, as well as Hagelin, the U.S. presidential candidate for the Maharishi Natural Party. In the flipped \(SU(5)\) models, some pairs of representations are, you lot know, flipped. So the 5-dimensional representation arises from the lepton doublet as well as the right-handed up-quark (not down-quark) singlet. The right hypercharges are obtained as long as you lot normalize the \(5\times 5\) matrix displayed to a higher house differently as well as correctly.
These master copy models weren't supersymmetric. You tin strength out add together supersymmetry as well as this gives the models some virtues that perish beyond the virtues of grand unification as well as supersymmetry separately. One of them is the guess coupling unification that seems to function rather good inwards SUSY GUT. Two directly lines (graphs of \(1/g^2\) as a purpose of \(E\) or \(\Lambda\)) almost e'er intersect somewhere inwards the plane. But a 3rd delineate exclusively goes through the 2 lines' intersection if i existent parameter – e.g. the gradient of the 3rd delineate – has the right value. In the minimal non-supersymmetric \(SU(5)\) model, the iii lines almost intersect at i dot but non quite. In the supersymmetric version of this model (and many models that acquit almost indistinguishably), the lines precisely intersect, inside the known fault margins. So at to the lowest degree i existent parameter measured experimentally indicates that the iii guess groups desire to instruct unified, as well as inside a supersymmetric theory.
Also, people works life other grand unified groups that may exercise the job. \(SU(5)\) may last represented as a \(10\times 10\) matrix: retrieve that the complex publish \(a+bi\) basically behaves nether multiplication simply similar the \(2\times 2\) matrix \(((a,b),(-b,a))\). So \(SU(5)\) may last embedded into \(SO(10)\), a somewhat larger group. Its additional wages is that the 5- as well as 10-dimensional representations of \(SU(5)\) may last embedded into a unmarried irreducible representation of \(SO(10)\), the 16-dimensional spinor – a right-handed neutrino becomes mandatory inwards \(SO(10)\).
Alternatively, you lot may pick an fifty-fifty larger elementary group, the exceptional grouping \(E_6\), where the fermions come upward from the key 27-dimensional representation. There be diverse schemes to intermission the symmetries yesteryear Higgs fields inwards diverse large representations of the guess group, or yesteryear stringy Wilson lines that don't acquit as elementary fields inwards point-like particle-based patch theories, as well as there's a lot of fun to study here.
The most well-known experimental prediction of grand unified theories is the proton decay. The conservation of the lepton as well as baryon numbers \(L\) as well as \(B\) no longer holds so alongside an intermediate X-boson or a Y-boson, you lot allow some processes inwards which the proton turns into the (equally charged, every bit spinning) positron – plus pure liberate energy (some particles, similar photons, whose total conserved charges are null as well as the spin is integer).
In Japan, huge tanks alongside purified H2O have got waited for flashes carrying the gospel of the proton decay for some twenty years. They have got works life zip so far. They could have got thrown a light-emitting fish into the H2O tank as well as have the Nobel prize but their scientific pride tells them non to usage Al Gore's methods to win the prize. ;-) If I trust Wolchover's publish which is hopefully up-to-date, the lifetime of the proton is greater than\[
t_{\rm proton} \gt 1.6\times 10^{34}\,{\rm years}.
\] You come across that the proton lifetime is a fleck longer than the average human's life expectancy. Well, fifty-fifty since the Big Bang, exclusively a really tiny fraction of the protons had plenty fourth dimension to decay (not to a greater extent than than 1 proton inwards a gram of affair has decayed, roughly speaking) – if they tin strength out decay at all. The rather stable proton, as indicated yesteryear the experiments, excludes some really specific models based on the grand unification. H5N1 flick from the Quanta Magazine:
You come across some theories that are either killed (red) or stay feasible (green). Supersymmetry tends to prolong the predicted lifetime of the proton which is a adept thing. As you lot tin strength out read inwards Wolchover's article or Brian Greene's The Elegant Universe, Glashow straight off abandoned grand unification inwards all forms in i lawsuit their initiative of all model, minimal \(SU(5)\), was ruled out yesteryear the proton decay experiments.
That's a really unwise reaction, I think, because there's really no adept argue why precisely the master copy Georgi-Glashow model should last the right grand unified model inwards Nature. (In the 1960s, Glashow wasn't lazy as well as tried a unlike model to stand upward for the W-bosons earlier he ended upward alongside the Standard Model representation. In the instance of grand unification, he chose the strategy try-fail-and-relax.) Stephen Barr, the master copy manlike someone bring upward of the flipped \(SU(5)\) model, describes the so far null results of the search for the proton decay as follows. You're waiting for your married adult woman as well as she's already some 10 minutes or 1 hr late. Will you lot already telephone telephone the companionship to organize the funeral? Is Glashow organizing the funerals for his married adult woman this quickly?
There are lots of interesting ideas inwards grand unification – diverse ways to bargain alongside the proton decay, depression neutrino masses, the doublet-triplet splitting work etc. And grand unified theories may last naturally realized inwards several major classes of semi-realistic string compactifications which convey some fifty-fifty ameliorate (and maybe to a greater extent than viable) possibilities to the research.
Grand unification is natural inwards string theory – but string theory may also Pb to models (especially diverse braneworlds) which don't contain the persuasion of grand unification, at to the lowest degree non inwards whatever obvious way. Given the fact that I experience almost sure most the validity of string theory, exercise I believe inwards grand unification? I likely intend it's to a greater extent than probable than not. But I am inwards no means sure that grand unification is realized inwards Nature.
However, I am sure that it's incorrect to tell that the theory was thrown into "limbo". Just human face at the charts to a higher house taken from the Quanta Magazine. H5N1 meaning percentage (about one-half) of the theory bars remains green. The married adult woman hasn't arrived inwards the initiative of all 1/2 of the possible times when she could have got arrived. It's non a terribly strong slice of prove that she is dead. Maybe it's sensible as well as ethical to exclusively innovation the funeral political party in i lawsuit your married adult woman is really falsified as well as you lot come across her dead body. And of course, Nature may prefer variations of the grand unification that brand the proton decay significantly to a greater extent than long-lived, maybe yesteryear some extra discrete symmetries or other structures. Most of these sub-theories' parameter spaces could have got been basically untouched yesteryear the proton decay experiments.
So last sure that phenomenologists perish on on working on grand unified theories. None of them is terribly sure most the right value of the proton lifetime. But the proton's lifetime may last finite. I intend that most phenomenologists believe that it is. If neither the baryon publish nor the lepton publish \(B,L\) is a guess symmetry, as well as they're likely not, as well as so the proton may last considered an ultratiny dark hole which may evaporate – decay – as well as exclusively the genuinely conserved, gauged charges are conserved. So it should last able to decay to the positron plus neutral materials (such as a photon or two), at to the lowest degree yesteryear some (very like shooting fish in a barrel acting) Planck-scale operators.
It would last extremely sloppy to evaluate the electrical flow province of affairs inwards the proton decay experiment yesteryear proverb that the proton is sure stable – or that at that topographic point can't last whatever grand unification inwards Nature.