A celebration of scalar fields in string theory
Because the Higgs boson became such a standard player in mainstream particle physics in the recent 50 years, people underestimate the importance of the looming discovery of the first spinless elementary particle we know in Nature as a sign that string theory is on the right track.
The God particle may look boring: it seems like just another elementary particle in Nature. One may understand the Higgs mechanism – one with the Mexican hat potential sketched above – which gives masses to all particles and that's it. However, this intellectual content is limited only because the people's way of looking at the field is so narrow-minded and skewed, based purely on the philosophy of elementary courses in quantum field theory.
However, string theory tells us that there are many ways how to think different, many new angles from which we should look at similar particles and related mechanisms. Also, the Higgs boson has many close cousins in almost all vacua of string/M-theory and they play several vital theories in Nature as described by string/M-theory.
I would like to look at all these questions – look at elementary scalar fields from a stringy perspective.
First, we are going to ask whether there is a reason to believe that particles with no spin, \(j=0\), should be less real than others.
Why the spin \(j=0\) is at least as good as others
Most textbooks of quantum field theory begin with the scalar field, the Klein-Gordon field. They have a good reason to do so: the scalar field \(\phi(x,y,z,t)\) carries no extra indices so one is not distracted by technicalities such as the Dirac spinors or gauge invariance. These technicalities may be added later.
Despite this role of scalar fields in the teaching process, there has existed a remarkably strong movement claiming that the spinless fields are special: they shouldn't exist! Before the Higgs discovery, we haven't known a single spinless elementary field in Nature. Recall that leptons and quarks have \(j=1/2\), photons, gluons, and other gauge bosons have \(j=1\), and gravitons have \(j=2\). There is still a whole school claiming that scalar fields are "obliged" to be composite because it would be fun if Nature were not allowed to start with spinless particles. You may have encountered people defending such ideas. You may personally be one of its champions.
This historical coincidence – the fact that the scalar field is going to be discovered as the last one – was the only semi-justification of this attitude. There's nothing wrong with the scalar fields that are as fundamental as the electromagnetic field or the electron's Dirac field. Well, to be more precise, there exists one reason why scalar fields could be less real – more precisely, less accessible – than the fields with a nonzero spin.
Why the scalar fields behave a bit differently: they don't like a diet
The reason is actually related to the very "lack of internal structure" that makes the scalar field an attractive toy model for textbooks. The point is that this extra structure is capable of naturally explaining why various fields with a nonzero spin remain massless or light.
Consider \(j=1/2\) Dirac fields. We may have the Dirac gamma matrices to act upon such fields. For example, we may change the phase of such fields according to the rule\[
\Psi \to \exp(i\alpha\gamma_5) \Psi.
\] The appearance of the matrix \(\gamma_5\) implies that the two "Weyl spinors" that team up to produce the Dirac spinor, namely the left-handed and the right-handed "Weyl spinor", are acquiring phases that are inverse to one another. Such a transformation may be either a symmetry of the full quantum field theory; or it may be at least a symmetry of an important and justifiable approximation to a quantum field theory.
Such a symmetry is important because the mass term in the Lagrangian\[
\LL_{\rm mass} = -m\cdot \bar \Psi \Psi
\] isn't invariant under the \(\gamma_5\)-based transformation above, the so-called chiral transformation. So if you impose this chiral symmetry (and imposing a new symmetry is always "natural" as long as the resulting models aren't excluded), the corresponding fermion (or fermions) have to stay massless. In reality, we're often interested in theories in which the symmetry is broken but we may still find arguments why the breaking is weak. These arguments automatically imply that the corresponding fermionic particle is light.
Similarly, we may consider gauge fields \(A_\mu\). They have a negative-norm time-like component (which would guarantee that some probabilities are negative if these modes remained physical). So there has to be a gauge symmetry of the kind\[
A_\mu\to A_\mu + \partial_\mu \lambda
\] that "decouples" the wrong polarizations from the rest of physics. Probabilities of any processes involving the "healthy" particles and polarizations remain positive; the gauge invariance surgically removes (decouples) the negative-probability tumors. However, the gauge invariance also has another consequence. The mass term for the gauge boson such as\[
\LL_\text{gauge boson mass} = -\frac{m^2}{2} A_\mu A^\mu
\] are not invariant under the gauge transformation above. That's why photons and gluons stay massless (well, gluons are also confined so the masslessness is less visible). The W-bosons and Z-bosons are massive. That's because their associated gauge invariance is broken (although it's a misleading description of the process from a rigorous physics perspective; but I don't want to go into this here because this is a blog entry on scalar fields). However, the mechanism of breaking – the Higgs mechanism – may be argued to be "mild" which is why these bosons remain "light". Well, it is mild if the Higgs boson itself is light – a condition that seems hard to satisfy, as I will discuss later.
Since 1993, it's been said that the Higgs boson is just like Margaret Thatcher because other party members like to be closer to her and they get slower as a result but I am not sure that you may learn a lot from this metaphor.
Now, we have no symmetry that could explain why scalar fields remain massless or light. They don't have separate left-handed and right-handed pieces. They can't be associated with any gauge symmetry because such a gauge symmetry acting on a scalar field would destroy all the "beef" in it; the total number of physical degrees of freedom would be zero.
That's why the scalar field masses are said to be "unprotected" against large corrections. Any new particle that interacts with the scalar fields has a capacity to adjust the scalar particle's mass by any constant corresponding to any energy scale that may appear in your theory. This is manifested by various quadratic divergences in loop corrections to the propagators and similar annoying features of Feynman diagrams.
Because of this fact, it may be reasonable to expect that there is something different about the \(j=0\) fields, after all. Their low mass is unprotected which is why they may naturally get heavy and inaccessible. That's the theoretical expectation for the Higgs boson itself. The very fact that simple enough theories naturally predict that the Higgs should be as heavy as the highest energy scales in physics – while experiments make it totally clear that the Higgs boson is much lighter (and this fact was pretty much known decades before the Higgs discovery) – is known as the "hierarchy problem".
Various solutions to the hierarchy problem have been proposed in literature. Supersymmetry is the most well-motivated one. In low-energy supersymmetry, the Higgs boson stays light because supersymmetry – valid up to pretty low energy scales – implies that the Higgs mass can't differ too much from the mass of its superpartner, the higgsino. The masses would be totally equal if SUSY were unbroken; the mass difference is still small if the breaking is sufficiently soft. And the higgsino is a \(j=1/2\) spinor particle whose mass has rather good reasons to stay light. Equivalently, we may say that the corrections to the Higgs boson mass are canceled or mostly canceled in between bosons and fermions, particles interacting with the Higgs that are naturally arranged to pairs of superpartners.
I should mention that there exists a potential reason why some scalar fields may be forced to remain massless or light, too. The Goldstone bosons arising from a broken symmetry may be proved by the Goldstone theorem to be massless. (This can't be applied to the physical Higgs boson which is not a Goldstone boson linked to a broken symmetry generator.) And in some cases, we may find and prove the existence of "moduli spaces" parameterized by the vacuum expectation values of scalar fields that have a vanishing potential energy. Such scalar fields look like massless particles at each point of these "moduli spaces". To actually prove that the potential energy is zero, we usually need supersymmetry, anyway.
Moduli spaces in string theory: scalar fields instead of parameters
The moduli spaces – possible spaces parameterized by allowed values of many scalar fields so that each point of the moduli space may be considered a new vacuum that is physically different from all others – are omnipresent in string theory, especially in its most symmetric (and not quite realistic) vacua.
For example, consider any perturbative string theory. It has a "string coupling constant" that tells you how the importance of a process (of splitting and merging strings) decreases every time you add some loops and virtual particle/string pairs into it. Similar coupling constants in quantum field theory are parameters that must be extracted from experiment. Each value of a coupling constant yields a non-equivalent theory.
Things are completely different in string theory. All values of the string coupling constant actually correspond to the same, single, unique theory. The value of \(g\) may be shown to be linked to the expectation value of a new field, the dilaton:\[
g = \langle \exp(\phi_\text{dilaton} ) \rangle
\] In different regions, the value of the dilaton may be different so the interactions in this region look like interactions with different values of the coupling constant. We know that this variability of the environment is absent in Nature around us (and we know that there are no new long-range forces that would result from the exchange of a new massless scalar particle) so the potential for this dilaton can't be zero; the degeneracy must be "lifted". Some processes must generate a potential that is nonzero and that has local minima.
Nature's vacuum is only ready to sit in one of these local minima – we say that the dilaton gets stabilized (all scalar fields have to be stabilized in realistic vacua!) – and in principle, the vacuum expectation value of the dilaton (and therefore also the string coupling constant) is calculable. In certain string compactifications, this talk "in principle" has been largely transformed to a "success in practice"; people know how to calculate lots of effective potentials and similar things resulting from individual well-defined string compactifications although our mastery isn't complete yet (and one may say that we don't even possess the completely universal definition of string theory that would allow us to calculate all such things at least in principle: so our ignorance about some precise corrections isn't just due to the technical difficulties of a well-known calculation but partly due to our ignorance about the very rules of such a calculation).
In the example above, I implicitly discussed a simple moduli space with one parameter only:\[
g\in \RR^+,\quad \phi_\text{dilaton}\in \RR.
\] The real axis is too simple a moduli space. In string theory, we usually work with much more complicated, multi-dimensional moduli spaces that require several or very many scalar fields to be parameterized. For example, the "dilaton only" moduli space is relevant for type IIA string theory; in that theory, the string coupling (which may also be interpreted as a function of the radius of the circular 11th coordinate of M-theory) is the only scalar field in 10 dimensions.
Things are different in type IIB string theory which has a dilaton as well as the Ramond-Ramond axion \(a\), a periodic scalar field with period \(2\pi\). One may combine them into\[
\tau = \frac{a}{2\pi} + \frac{i}{g}, \quad \tau\in \CC
\] the complexified dilaton-axion field. A funny thing is that not all complex values are inequivalent. In fact, if we replace \(\tau\) by \[
\tau' = \frac{a\tau+b}{c\tau + d}, \quad \{a,b,c,d\}\subseteq\ZZ, \quad ad-bc=1,
\] we obtain exactly equivalent laws of physics. We say that that type IIB string theory is invariant under the \(SL(2,\ZZ)\) S-duality group, a discrete symmetry inherited by the \(\NNN=4\) gauge theory and some other theories. The origin of this symmetry may be visualized by a 12-dimensional master theory, the F(ather)-theory, that is compactified on a two-dimensional torus.
Calabi-Yau spaces have their moduli spaces, too. The scalar fields parameterize various sizes of 2-cycles in these manifolds and various other numbers encoding the exact shape. Much like the Calabi-Yau spaces themselves, these moduli spaces are pretty complicated, symmetry-less complex manifolds. (In fact, the moduli space of possible states of a D0-brane on a Calabi-Yau manifold is essentially the same Calabi-Yau manifold itself.)
But I want to mention one example of a stringy/M moduli space that is more complicated than the moduli space of the dilaton-axion \(\tau\) in type IIB string theory but less complicated than the Calabi-Yau spaces' moduli spaces. And this example may be described in many very different but physically totally equivalent ways.
Take heterotic string theory and compactify it on a 3-torus. A heterotic string theory may be interpreted as a hybrid of the 26-dimensional "bosonic string theory" employed for the left-moving degrees of freedom on the world sheet; and the 10-dimensional "superstring theory" employed for the right-movers. There are 16 extra dimensions on the left-moving side; for modular invariance, they have to be compactified on a 16-dimensional even self-dual lattice.
This is actually true even if we compactify 3 extra left-moving and 3-extra right-moving coordinates. We need an even self-dual lattice in signature\[
(16,0)+ (3,3) =(19,3).
\] Up to isometries, there's actually a unique lattice of this kind, \(\Gamma_{19,3}\). If you work a little bit, you will find out that space of possible heterotic compactifications on a 3-dimensional torus is\[
{\mathcal M} = SO(19,3,\ZZ) \backslash SO(19,3) / SO(19)\times SO(3).
\] The discrete group in front of the backslash is a T-duality group that imposes various identifications in the \(19\times 3 = 57\)-dimensional moduli space described as the quotient in the rest of the line. These 57 fields may be described as 3 radii and 3 angles of the 3-torus; 3 components of the B-field two-form on the three-torus; and \(16\times 3\) Wilson lines for the \(U(1)^{16}\) maximally broken subgroup of the 10-dimensional \(SO(32)\) or \(E_8\times E_8\) gauge group. Check that \(3+3+3+48 = 57\).
However, exactly the same moduli space \({\mathcal M}\) actually appears at yet another place in mathematics: it's the space of all K3 manifolds. K3 is the simplest 4-real-dimensional manifold after the 4-dimensional torus. All the 57 parameters have a geometric interpretation here: they describe the shape of the 4-real-dimensional curved manifold that just happens to preserve 1/2 of the supersymmetries (its holonomy group is \(SU(2)\), a natural subgroup of the generic potato's holonomy group \(SO(4)\)).
It's actually not an accident that we found the same space in heterotic string theory and the geometry of the K3 manifolds. The reason is that there exist two totally equivalent – we say "dual" – descriptions of the very same physics in string/M-theory. Heterotic strings compactified on a 3-torus produce 57 scalar fields in the remaining 6+1 large dimensions and lots of other physical objects and interactions. They are actually physically indistinguishable from the 57 scalar fields that you obtain by compactifying M-theory on a 4-real-dimensional K3 manifold – which also leaves 11-4=7 large spacetime dimensions. The equivalence is exact.
Lots of these dualities were discovered during the "Duality Revolution", a part of the "Second Superstring Revolution", in the mid 1990s. In most cases, the appearance of the same moduli spaces on both sides may be considered the first "consistency check" of the equivalence. The interpretations of the scalar fields are different on both sides; we use different words to describe what the scalar fields indicate. However, these are just words, not physics. You may write down a dictionary to replace words by other words. If you only list the set of all objects, their spectrum (masses), and the scattering amplitudes for any combinations of such objects, you will find out that the two theories in a dual pair are identical, regardless of the different terminologies of the starting points.
Without scalar fields, the Duality Revolution in string theory would have been impossible.
Other scalar fields in string theory: tachyons, axions, and inflatons
But there are many other scalar fields in string theory. The oldest one is the tachyon in the 26-dimensional "old" bosonic string theory, one that started the field in the late 1960s. As the Greek name related to "speed" indicates, a tachyon is naively a particle that is obliged to move faster than light. This is actually a very naive perspective, indeed. If you upgrade your description to a quantum field theory in which particles are excitations of quantum fields, the defining feature of tachyons is that their squared mass is negative:\[
\LL = \dots - \frac{m^2}{2} T^2
\] is the mass term and tachyons have \(m^2\lt 0\) so the coefficient of \(T^2\) in the Lagrangian density is positive for them. For this reason, the point \(T=0\) is a local maximum of the potential. That's an unstable point; the center of the Mexican hat potential at the picture above is another example. In the Standard Model, the symmetry-preserving point for the Higgs, \(h=(0,0)\), actually contains tachyonic Higgs excitations! It's an unstable point so the marble rolls down into a local minimum (in a random direction: all of them are actually gauge-equivalent at the beginning) and the symmetry gets broken. This is what the BEH/Higgs mechanism is all about.
In the old bosonic string theory, the tachyon was one of the pathological properties of the theory. While the naive interpretation of the problem is that there are superluminal particles that contradict special relativity, a more sophisticated interpretation is an unstable vacuum; or, and this is related, nasty infrared (long-distance-related) divergences in loop diagrams. The bosonic string theory also predicts that there should be no fermions; it's unrealistic for many reasons.
The Standard Model Higgs boson has a local minimum – away from the center of the Mexican hat – where the marble may sit comfortably. Is there such a final destination in which the bosonic string theory's tachyon wants to sit? Well, it depends on whether you talk about the closed string tachyon or the open string tachyon.
The closed string tachyon, the "lightest" (lighter than massless!) excitation of a closed string is a tachyon that has a local maximum of its (spacetime) potential energy but doesn't seem to have any local minimum. Once the marble starts rolling, it will never stop, destroying the spacetime ever more brutally along the way.
However, it was realized around 2000 that the story is much more interesting for the open string tachyons, the ground states of a single open bosonic string. In that case, the open string excitations may be said to "live at a D-brane". The condensation of the tachyon corresponds to the self-destruction of the (unstable) D-brane that supports the tachyon (as well as all other open-string excitations). Ashoke Sen conjectured that there is a local minimum corresponding to the complete evaporation of the D-brane. This interpretation allowed him to deduce that the energy difference between the local minimum and the local maximum must be equal to the D-brane tension. This prediction was confirmed in various informal ways and, most impressively, by Martin Schnabl's analytic solution to the open cubic string field theory.
Before 1999 or so, people would simply dismiss any theory with tachyons as a pathological, not really consistent toy model. The situation changed in 1999 or 2000. Tachyons were carefully reclassified as "instabilities", and not "downright inconsistencies", and people got used to asking "where the instability may lead", what is the stable point at which the process of tachyon condensation (the marble rolling down from the local maximum) may end.
They understood the answer in the case of open-string tachyons – instabilities of some D-branes etc. Adams, Polchinski, and Silverstein also gave a totally convincing answer in the case of some closed-string tachyons living in twisted sectors of orbifolds; their condensation is interpreted as the erosion of initially sharp orbifold singularities. The closed-string tachyons outside twisted sectors are believed to have no canonical minima where they want to end.
Tachyons are no longer the particles you should "never touch", the ultimate taboos; they're particles of scalar fields that tell us stories about some instabilities, stories we may try to understand in much more detail – despite the fact that the existence of tachyons usually makes a point of the configuration space (or landscape) unrealistic as a theory of our Cosmos.
At the beginning, I described the scalar fields in string theory that parameterize moduli spaces; by definition, they want to be massless (or light if the description using a moduli space is approximate). Now, I've discussed tachyons whose squared mass is nonzero, namely negative. But there are at least two more important classes of scalar fields arising from string theory: axions and inflatons.
Axions generalize the Ramond-Ramond axions \(a\) I mentioned in the context of the 10-dimensional type IIB string theory. These scalar fields should be periodic; they are helpful as solutions to the strong CP-problem (why the coefficient of the \(F\wedge F\) term in QCD seems to be much smaller than one even though nothing would apparently be wrong if it were of order one); and some string theorists believe that string theory pretty clearly predicts that there should be lots of types of axions in the world around us – a whole "axiverse" of axions.
Note that axions are viable dark matter candidates – candidates that differ from WIMP (which is most typically the LSP, the lightest superpartner, most typically a neutralino or gravitino in supersymmetry). So that's another reason to think that Nature may like axions. The people who believe in the "axiverse" point out that typical string compactifications employ rather complicated Calabi-Yau-like manifolds and for each "cycle" of (or "hole" in) this manifold, one may find an axion. The spectrum of the axions' masses may densely cover a very wide interval of energies on the log scale.
Another crucial type of a scalar field in fundamental physics is the inflaton. It is responsible for the cosmic inflation – the probably correct explanation why our Universe is so exponentially huge, why the spatial sections are so flat and uniform (including the constant cosmic microwave background temperature in all directions), and why it has such a small density of exotic objects such as the magnetic monopoles. An inflaton wants to sit near the local maximum – similar to the center of the Mexican hat or the point where it looks like a tachyon – and resemble a huge positive cosmological constant that makes the Universe expand in an accelerated way. Eventually it rolls down closer to a local minimum, inflation stops, and the energy is converted to the matter that later gives rise to early galaxies.
For some conceptual reasons related to the weak gravity conjecture and other things, it actually seems like string theory doesn't like – and may be strictly incompatible with – the slow-roll inflation, also known as the new inflation. It seems that there is a general inequality in string theory that is pretty much equivalent to saying that the inflaton's potential can't have a "slow roll". If that's true, string theory could prefer the "old inflation" from the original paper by Alan Guth.
String theory's diverse geometric ways to look at Higgs bosons
Finally, I want to spend some time with the discussion of "genuine Higgses" in string theory; I mean scalar fields whose nonzero values break a gauge symmetry down to a smaller subgroup. Realistic and even unrealistic vacua contain these scalar fields, too. They're so natural that string theory apparently indicates it would be foolish and uneconomic to try to construct a Higgs-like object out of smaller point-like particles. As far as I know, there doesn't exist a single prominent enough model of technicolor within string theory. To a large extent, these are mutually excluding approaches to the physics of the electroweak symmetry breaking.
Realistic stringy scenarios such as heterotic strings on Calabi-Yau spaces, M-theory in singular \(G_2\) holonomy manifolds, intersecting type IIA braneworlds, and F-theory (whether you talk about the KKLT-like flux vacua or the bottom-up Vafa-style models) naturally predict Higgs fields, i.e. new scalar matter fields ready to break some gauge symmetries.
It's interesting to look at three examples of the clever ways in which the Higgs fields are "geometrized" in string theory.
Distance between D-branes
My first example is concerned with the D-branes' gauge group. If you consider a stack of coincident (and therefore parallel) D-branes of the same dimension at a generic point in space, you will find out that they carry a \(U(N)\) gauge group. It's the same group we know from our discussions of Matrix theory: the positions \(\vec x_i\) of \(N\) particles, the D0-branes (and similarly for higher-dimensional branes), are promoted to the diagonal entries \(\vec x_{ii}\) (no summation) but these matrices also naturally have the off-diagonal matrix elements.
So the coordinates in the directions transverse to the D-branes (all spatial coordinates are transverse in the case of D0-branes, of course) are described by \(N\times N\) matrices, i.e. the adjoint representation of the \(U(N)\) group. If you separate the stack of \(N\) D-branes to a smaller stack of \(K\) D-branes and \((N-K)\) D-branes by assigning different transverse positions to the two smaller stacks, you will break the gauge symmetry in this way:\[
U(N)\to U(K)\times U(N-K).
\] The transverse position of the D-branes is nothing else than a Higgs field that typically breaks the gauge symmetry to a smaller group. This Higgs field has to transform in the adjoint representation of a \(U(N)\) group – or, in the case of orientifolds, \(SO(N)\) or \(USp(2N)\). So by this group-theoretical property, it differs from the Standard Model Higgs that transforms as the fundamental 2-dimensional complex (well, pseudoreal) representation of \(SU(2)\). However, aside from these different representations, the physical logic is the same. The transverse positions of the D-branes are exactly described as Higgs fields in any effective field theory.
Deviation from the self-dual radius
My second example will deal with the "enhanced gauge symmetry" at the self-dual radius in bosonic and heterotic string theory. Recall that in 1919, Theodor Kaluza proposed a unification of electromagnetism and gravity in terms of 5-dimensional general relativity. Oskar Klein later clarified that the fifth dimension should be a circle. The \(U(1)\) isometry of the circle (at each point in the remaining 3+1 dimensions) is interpreted as the gauge symmetry of electromagnetism; \(g_{\mu 5}\), an off-diagonal element of the metric tensor, is related to the electromagnetic potential \(A_\mu\). The momentum in the 5th dimension (which is quantized because the dimension is periodic) is interpreted as the electric charge (in the right minimal units).
Whenever you deal with compactifications of gravitational theories (especially string theory) on a circle, this Kaluza-Klein mechanism is there. Consider, for example, bosonic string theory. Compactify one of the 26 spacetime coordinates on a circle. So in the remaining 24+1 coordinates, you will get a \(U(1)\) gauge symmetry. In fact, in bosonic string theory, you will get \(U(1)\times U(1)\). You obtain two new \(U(1)\) groups. Much like the momentum in the circular dimension gives you the electric charge under the first \(U(1)\), there is another charge: the fundamental strings can wind around the extra circle, giving you one more charge (the winding number which is interchanged with the momentum under T-duality). The vector potential for this winding number arises from the B-field (a two-form field under which the strings are electrically charged) of string theory, \(B_{\mu,25}\).
String theory not only doubles the Kaluza-Klein gauge symmetry in this situation. It actually extends it to a non-Abelian, Yang-Mills symmetry \(SU(2)\times SU(2)\) if you choose the radius of the circle to be at the "self-dual value" (so that the minimum-momentum particles moving in the 5th dimension have the same mass/energy as the minimum-winding strings wrapped around the circular dimension). One needs to know some \(d=2\) conformal field theory to prove it (one needs to know different ways to describe the \(SU(2)\) current algebra CFT) but this symmetry is actually exact.
If you're away from the self-dual radius, you see that the momentum modes and the winding modes have different masses. That's just one way to see that the symmetry (that was originally able to interchange the momentum and winding modes: the very T-duality was interpreted as a particular rotation by 180 degrees in a full-fledged \(SU(2)\times SU(2)\) group of symmetries) gets broken. The breaking may be once again described as a Higgs mechanism. The corresponding Higgs field is essentially the difference of the actual radius of the circle and the self-dual radius. If you look at the spectrum carefully, you will see that the spectrum of these Higgs scalars transforms as \(({\bf 3},{\bf 3})\) under the \(SU(2)\times SU(2)\) gauge group. If you give it a generic nonzero vev, the gauge group is broken to \(U(1)\times U(1)\) I have already mentioned: it's the group under which the Higgs vevs are invariant. Note that this is another "extra story" behind the Higgs mechanism and it's group-theoretically inequivalent to the D-brane case.
In heterotic string theory, there's only one enhanced \(SU(2)\); only the left-moving, bosonic-string-theory-based portion of the degrees of freedom give you the enhanced symmetry and its breaking.
Smoothening of ADE singularities
Finally, string theory cures problems and ambiguities of short-distance physics. In particular, it resolves physics at the singularities. This statement doesn't necessarily mean that all singularities are replaced by manifestly smooth manifolds although it's sometimes the case, too. However, in some cases, the singularities in the spacetime stay singular but string theory is capable of making finite, non-singular predictions about the physical phenomena, anyway. This ability partly boils down to the existence of new light states that wouldn't exist in point-like-particle-based field theories (e.g. wound strings or wrapped branes) confined to the vicinity of the singularities; however, it's also important that strings can't probe arbitrarily tight neighborhoods of the singular loci because they're extended objects with a finite resolution.
For example, string theory allows you to study physics in the presence of ADE singularities. They may be strictly singular and even if they are singular, string theory produces totally healthy predictions. These singularities are special places in the spacetime and a gauge field corresponding to the "same" gauge group in the ADE classification lives there (according to the miraculous ADE dictionary between Lie groups and particular singular manifolds).
However, these singularities may also be deformed (or "resolved") to less singular or totally smooth manifolds. When you do so, the gauge symmetry is broken to a smaller one. The scalar fields describing the deviation of the shape away from the singular ADE manifold are Higgs fields and their nonzero vev breaks the original ADE gauge group down to a smaller subgroup according to the normal rules of the Higgs mechanism.
Summary
This text has already become pretty long and I have written something under all the sections I had originally prepared. So let me end at this point.
My main message is that the \(j=0\) scalar fields (and their corresponding spinless particles – each of which is composed of a single tiny string much like the photon and other particles are composed of a single string) are totally natural in string theory, they play many roles, and they allow us to look at the physical phenomena from many perspectives and use many intellectual templates to visualize what's going on.
String theory has always predicted that the electroweak symmetry should be broken by objects that look like elementary, structureless scalar fields, and if the Higgs boson is indeed discovered soon, it will be a confirmation of the victory of string theory in its battle against some competing models (e.g. technicolor and other modes largely based on the phobia from scalar fields) which have never been comparably influential as string theory and string-theory-friendly phenomenology but they have existed, anyway.
An apparently elementary Higgs field is a modest but nonzero confirmation of string theory's being on the right track and due to some extra arguments sketched above and due to the SUSY-friendly (light enough) mass \(125\GeV\) of the discovered Higgs boson, it is also a signal that we may (and maybe we should) see some other discoveries envisioned by string theory such as supersymmetry.
And that's the memo.
Because the Higgs boson became such a standard player in mainstream particle physics in the recent 50 years, people underestimate the importance of the looming discovery of the first spinless elementary particle we know in Nature as a sign that string theory is on the right track.
The God particle may look boring: it seems like just another elementary particle in Nature. One may understand the Higgs mechanism – one with the Mexican hat potential sketched above – which gives masses to all particles and that's it. However, this intellectual content is limited only because the people's way of looking at the field is so narrow-minded and skewed, based purely on the philosophy of elementary courses in quantum field theory.
However, string theory tells us that there are many ways how to think different, many new angles from which we should look at similar particles and related mechanisms. Also, the Higgs boson has many close cousins in almost all vacua of string/M-theory and they play several vital theories in Nature as described by string/M-theory.
I would like to look at all these questions – look at elementary scalar fields from a stringy perspective.
First, we are going to ask whether there is a reason to believe that particles with no spin, \(j=0\), should be less real than others.
Why the spin \(j=0\) is at least as good as others
Most textbooks of quantum field theory begin with the scalar field, the Klein-Gordon field. They have a good reason to do so: the scalar field \(\phi(x,y,z,t)\) carries no extra indices so one is not distracted by technicalities such as the Dirac spinors or gauge invariance. These technicalities may be added later.
Despite this role of scalar fields in the teaching process, there has existed a remarkably strong movement claiming that the spinless fields are special: they shouldn't exist! Before the Higgs discovery, we haven't known a single spinless elementary field in Nature. Recall that leptons and quarks have \(j=1/2\), photons, gluons, and other gauge bosons have \(j=1\), and gravitons have \(j=2\). There is still a whole school claiming that scalar fields are "obliged" to be composite because it would be fun if Nature were not allowed to start with spinless particles. You may have encountered people defending such ideas. You may personally be one of its champions.
This historical coincidence – the fact that the scalar field is going to be discovered as the last one – was the only semi-justification of this attitude. There's nothing wrong with the scalar fields that are as fundamental as the electromagnetic field or the electron's Dirac field. Well, to be more precise, there exists one reason why scalar fields could be less real – more precisely, less accessible – than the fields with a nonzero spin.
Why the scalar fields behave a bit differently: they don't like a diet
The reason is actually related to the very "lack of internal structure" that makes the scalar field an attractive toy model for textbooks. The point is that this extra structure is capable of naturally explaining why various fields with a nonzero spin remain massless or light.
Consider \(j=1/2\) Dirac fields. We may have the Dirac gamma matrices to act upon such fields. For example, we may change the phase of such fields according to the rule\[
\Psi \to \exp(i\alpha\gamma_5) \Psi.
\] The appearance of the matrix \(\gamma_5\) implies that the two "Weyl spinors" that team up to produce the Dirac spinor, namely the left-handed and the right-handed "Weyl spinor", are acquiring phases that are inverse to one another. Such a transformation may be either a symmetry of the full quantum field theory; or it may be at least a symmetry of an important and justifiable approximation to a quantum field theory.
Such a symmetry is important because the mass term in the Lagrangian\[
\LL_{\rm mass} = -m\cdot \bar \Psi \Psi
\] isn't invariant under the \(\gamma_5\)-based transformation above, the so-called chiral transformation. So if you impose this chiral symmetry (and imposing a new symmetry is always "natural" as long as the resulting models aren't excluded), the corresponding fermion (or fermions) have to stay massless. In reality, we're often interested in theories in which the symmetry is broken but we may still find arguments why the breaking is weak. These arguments automatically imply that the corresponding fermionic particle is light.
Similarly, we may consider gauge fields \(A_\mu\). They have a negative-norm time-like component (which would guarantee that some probabilities are negative if these modes remained physical). So there has to be a gauge symmetry of the kind\[
A_\mu\to A_\mu + \partial_\mu \lambda
\] that "decouples" the wrong polarizations from the rest of physics. Probabilities of any processes involving the "healthy" particles and polarizations remain positive; the gauge invariance surgically removes (decouples) the negative-probability tumors. However, the gauge invariance also has another consequence. The mass term for the gauge boson such as\[
\LL_\text{gauge boson mass} = -\frac{m^2}{2} A_\mu A^\mu
\] are not invariant under the gauge transformation above. That's why photons and gluons stay massless (well, gluons are also confined so the masslessness is less visible). The W-bosons and Z-bosons are massive. That's because their associated gauge invariance is broken (although it's a misleading description of the process from a rigorous physics perspective; but I don't want to go into this here because this is a blog entry on scalar fields). However, the mechanism of breaking – the Higgs mechanism – may be argued to be "mild" which is why these bosons remain "light". Well, it is mild if the Higgs boson itself is light – a condition that seems hard to satisfy, as I will discuss later.
Since 1993, it's been said that the Higgs boson is just like Margaret Thatcher because other party members like to be closer to her and they get slower as a result but I am not sure that you may learn a lot from this metaphor.
Now, we have no symmetry that could explain why scalar fields remain massless or light. They don't have separate left-handed and right-handed pieces. They can't be associated with any gauge symmetry because such a gauge symmetry acting on a scalar field would destroy all the "beef" in it; the total number of physical degrees of freedom would be zero.
That's why the scalar field masses are said to be "unprotected" against large corrections. Any new particle that interacts with the scalar fields has a capacity to adjust the scalar particle's mass by any constant corresponding to any energy scale that may appear in your theory. This is manifested by various quadratic divergences in loop corrections to the propagators and similar annoying features of Feynman diagrams.
Because of this fact, it may be reasonable to expect that there is something different about the \(j=0\) fields, after all. Their low mass is unprotected which is why they may naturally get heavy and inaccessible. That's the theoretical expectation for the Higgs boson itself. The very fact that simple enough theories naturally predict that the Higgs should be as heavy as the highest energy scales in physics – while experiments make it totally clear that the Higgs boson is much lighter (and this fact was pretty much known decades before the Higgs discovery) – is known as the "hierarchy problem".
Various solutions to the hierarchy problem have been proposed in literature. Supersymmetry is the most well-motivated one. In low-energy supersymmetry, the Higgs boson stays light because supersymmetry – valid up to pretty low energy scales – implies that the Higgs mass can't differ too much from the mass of its superpartner, the higgsino. The masses would be totally equal if SUSY were unbroken; the mass difference is still small if the breaking is sufficiently soft. And the higgsino is a \(j=1/2\) spinor particle whose mass has rather good reasons to stay light. Equivalently, we may say that the corrections to the Higgs boson mass are canceled or mostly canceled in between bosons and fermions, particles interacting with the Higgs that are naturally arranged to pairs of superpartners.
I should mention that there exists a potential reason why some scalar fields may be forced to remain massless or light, too. The Goldstone bosons arising from a broken symmetry may be proved by the Goldstone theorem to be massless. (This can't be applied to the physical Higgs boson which is not a Goldstone boson linked to a broken symmetry generator.) And in some cases, we may find and prove the existence of "moduli spaces" parameterized by the vacuum expectation values of scalar fields that have a vanishing potential energy. Such scalar fields look like massless particles at each point of these "moduli spaces". To actually prove that the potential energy is zero, we usually need supersymmetry, anyway.
Moduli spaces in string theory: scalar fields instead of parameters
The moduli spaces – possible spaces parameterized by allowed values of many scalar fields so that each point of the moduli space may be considered a new vacuum that is physically different from all others – are omnipresent in string theory, especially in its most symmetric (and not quite realistic) vacua.
For example, consider any perturbative string theory. It has a "string coupling constant" that tells you how the importance of a process (of splitting and merging strings) decreases every time you add some loops and virtual particle/string pairs into it. Similar coupling constants in quantum field theory are parameters that must be extracted from experiment. Each value of a coupling constant yields a non-equivalent theory.
Things are completely different in string theory. All values of the string coupling constant actually correspond to the same, single, unique theory. The value of \(g\) may be shown to be linked to the expectation value of a new field, the dilaton:\[
g = \langle \exp(\phi_\text{dilaton} ) \rangle
\] In different regions, the value of the dilaton may be different so the interactions in this region look like interactions with different values of the coupling constant. We know that this variability of the environment is absent in Nature around us (and we know that there are no new long-range forces that would result from the exchange of a new massless scalar particle) so the potential for this dilaton can't be zero; the degeneracy must be "lifted". Some processes must generate a potential that is nonzero and that has local minima.
Nature's vacuum is only ready to sit in one of these local minima – we say that the dilaton gets stabilized (all scalar fields have to be stabilized in realistic vacua!) – and in principle, the vacuum expectation value of the dilaton (and therefore also the string coupling constant) is calculable. In certain string compactifications, this talk "in principle" has been largely transformed to a "success in practice"; people know how to calculate lots of effective potentials and similar things resulting from individual well-defined string compactifications although our mastery isn't complete yet (and one may say that we don't even possess the completely universal definition of string theory that would allow us to calculate all such things at least in principle: so our ignorance about some precise corrections isn't just due to the technical difficulties of a well-known calculation but partly due to our ignorance about the very rules of such a calculation).
In the example above, I implicitly discussed a simple moduli space with one parameter only:\[
g\in \RR^+,\quad \phi_\text{dilaton}\in \RR.
\] The real axis is too simple a moduli space. In string theory, we usually work with much more complicated, multi-dimensional moduli spaces that require several or very many scalar fields to be parameterized. For example, the "dilaton only" moduli space is relevant for type IIA string theory; in that theory, the string coupling (which may also be interpreted as a function of the radius of the circular 11th coordinate of M-theory) is the only scalar field in 10 dimensions.
Things are different in type IIB string theory which has a dilaton as well as the Ramond-Ramond axion \(a\), a periodic scalar field with period \(2\pi\). One may combine them into\[
\tau = \frac{a}{2\pi} + \frac{i}{g}, \quad \tau\in \CC
\] the complexified dilaton-axion field. A funny thing is that not all complex values are inequivalent. In fact, if we replace \(\tau\) by \[
\tau' = \frac{a\tau+b}{c\tau + d}, \quad \{a,b,c,d\}\subseteq\ZZ, \quad ad-bc=1,
\] we obtain exactly equivalent laws of physics. We say that that type IIB string theory is invariant under the \(SL(2,\ZZ)\) S-duality group, a discrete symmetry inherited by the \(\NNN=4\) gauge theory and some other theories. The origin of this symmetry may be visualized by a 12-dimensional master theory, the F(ather)-theory, that is compactified on a two-dimensional torus.
Calabi-Yau spaces have their moduli spaces, too. The scalar fields parameterize various sizes of 2-cycles in these manifolds and various other numbers encoding the exact shape. Much like the Calabi-Yau spaces themselves, these moduli spaces are pretty complicated, symmetry-less complex manifolds. (In fact, the moduli space of possible states of a D0-brane on a Calabi-Yau manifold is essentially the same Calabi-Yau manifold itself.)
But I want to mention one example of a stringy/M moduli space that is more complicated than the moduli space of the dilaton-axion \(\tau\) in type IIB string theory but less complicated than the Calabi-Yau spaces' moduli spaces. And this example may be described in many very different but physically totally equivalent ways.
Take heterotic string theory and compactify it on a 3-torus. A heterotic string theory may be interpreted as a hybrid of the 26-dimensional "bosonic string theory" employed for the left-moving degrees of freedom on the world sheet; and the 10-dimensional "superstring theory" employed for the right-movers. There are 16 extra dimensions on the left-moving side; for modular invariance, they have to be compactified on a 16-dimensional even self-dual lattice.
This is actually true even if we compactify 3 extra left-moving and 3-extra right-moving coordinates. We need an even self-dual lattice in signature\[
(16,0)+ (3,3) =(19,3).
\] Up to isometries, there's actually a unique lattice of this kind, \(\Gamma_{19,3}\). If you work a little bit, you will find out that space of possible heterotic compactifications on a 3-dimensional torus is\[
{\mathcal M} = SO(19,3,\ZZ) \backslash SO(19,3) / SO(19)\times SO(3).
\] The discrete group in front of the backslash is a T-duality group that imposes various identifications in the \(19\times 3 = 57\)-dimensional moduli space described as the quotient in the rest of the line. These 57 fields may be described as 3 radii and 3 angles of the 3-torus; 3 components of the B-field two-form on the three-torus; and \(16\times 3\) Wilson lines for the \(U(1)^{16}\) maximally broken subgroup of the 10-dimensional \(SO(32)\) or \(E_8\times E_8\) gauge group. Check that \(3+3+3+48 = 57\).
However, exactly the same moduli space \({\mathcal M}\) actually appears at yet another place in mathematics: it's the space of all K3 manifolds. K3 is the simplest 4-real-dimensional manifold after the 4-dimensional torus. All the 57 parameters have a geometric interpretation here: they describe the shape of the 4-real-dimensional curved manifold that just happens to preserve 1/2 of the supersymmetries (its holonomy group is \(SU(2)\), a natural subgroup of the generic potato's holonomy group \(SO(4)\)).
It's actually not an accident that we found the same space in heterotic string theory and the geometry of the K3 manifolds. The reason is that there exist two totally equivalent – we say "dual" – descriptions of the very same physics in string/M-theory. Heterotic strings compactified on a 3-torus produce 57 scalar fields in the remaining 6+1 large dimensions and lots of other physical objects and interactions. They are actually physically indistinguishable from the 57 scalar fields that you obtain by compactifying M-theory on a 4-real-dimensional K3 manifold – which also leaves 11-4=7 large spacetime dimensions. The equivalence is exact.
Lots of these dualities were discovered during the "Duality Revolution", a part of the "Second Superstring Revolution", in the mid 1990s. In most cases, the appearance of the same moduli spaces on both sides may be considered the first "consistency check" of the equivalence. The interpretations of the scalar fields are different on both sides; we use different words to describe what the scalar fields indicate. However, these are just words, not physics. You may write down a dictionary to replace words by other words. If you only list the set of all objects, their spectrum (masses), and the scattering amplitudes for any combinations of such objects, you will find out that the two theories in a dual pair are identical, regardless of the different terminologies of the starting points.
Without scalar fields, the Duality Revolution in string theory would have been impossible.
Other scalar fields in string theory: tachyons, axions, and inflatons
But there are many other scalar fields in string theory. The oldest one is the tachyon in the 26-dimensional "old" bosonic string theory, one that started the field in the late 1960s. As the Greek name related to "speed" indicates, a tachyon is naively a particle that is obliged to move faster than light. This is actually a very naive perspective, indeed. If you upgrade your description to a quantum field theory in which particles are excitations of quantum fields, the defining feature of tachyons is that their squared mass is negative:\[
\LL = \dots - \frac{m^2}{2} T^2
\] is the mass term and tachyons have \(m^2\lt 0\) so the coefficient of \(T^2\) in the Lagrangian density is positive for them. For this reason, the point \(T=0\) is a local maximum of the potential. That's an unstable point; the center of the Mexican hat potential at the picture above is another example. In the Standard Model, the symmetry-preserving point for the Higgs, \(h=(0,0)\), actually contains tachyonic Higgs excitations! It's an unstable point so the marble rolls down into a local minimum (in a random direction: all of them are actually gauge-equivalent at the beginning) and the symmetry gets broken. This is what the BEH/Higgs mechanism is all about.
In the old bosonic string theory, the tachyon was one of the pathological properties of the theory. While the naive interpretation of the problem is that there are superluminal particles that contradict special relativity, a more sophisticated interpretation is an unstable vacuum; or, and this is related, nasty infrared (long-distance-related) divergences in loop diagrams. The bosonic string theory also predicts that there should be no fermions; it's unrealistic for many reasons.
The Standard Model Higgs boson has a local minimum – away from the center of the Mexican hat – where the marble may sit comfortably. Is there such a final destination in which the bosonic string theory's tachyon wants to sit? Well, it depends on whether you talk about the closed string tachyon or the open string tachyon.
The closed string tachyon, the "lightest" (lighter than massless!) excitation of a closed string is a tachyon that has a local maximum of its (spacetime) potential energy but doesn't seem to have any local minimum. Once the marble starts rolling, it will never stop, destroying the spacetime ever more brutally along the way.
However, it was realized around 2000 that the story is much more interesting for the open string tachyons, the ground states of a single open bosonic string. In that case, the open string excitations may be said to "live at a D-brane". The condensation of the tachyon corresponds to the self-destruction of the (unstable) D-brane that supports the tachyon (as well as all other open-string excitations). Ashoke Sen conjectured that there is a local minimum corresponding to the complete evaporation of the D-brane. This interpretation allowed him to deduce that the energy difference between the local minimum and the local maximum must be equal to the D-brane tension. This prediction was confirmed in various informal ways and, most impressively, by Martin Schnabl's analytic solution to the open cubic string field theory.
Before 1999 or so, people would simply dismiss any theory with tachyons as a pathological, not really consistent toy model. The situation changed in 1999 or 2000. Tachyons were carefully reclassified as "instabilities", and not "downright inconsistencies", and people got used to asking "where the instability may lead", what is the stable point at which the process of tachyon condensation (the marble rolling down from the local maximum) may end.
They understood the answer in the case of open-string tachyons – instabilities of some D-branes etc. Adams, Polchinski, and Silverstein also gave a totally convincing answer in the case of some closed-string tachyons living in twisted sectors of orbifolds; their condensation is interpreted as the erosion of initially sharp orbifold singularities. The closed-string tachyons outside twisted sectors are believed to have no canonical minima where they want to end.
Tachyons are no longer the particles you should "never touch", the ultimate taboos; they're particles of scalar fields that tell us stories about some instabilities, stories we may try to understand in much more detail – despite the fact that the existence of tachyons usually makes a point of the configuration space (or landscape) unrealistic as a theory of our Cosmos.
At the beginning, I described the scalar fields in string theory that parameterize moduli spaces; by definition, they want to be massless (or light if the description using a moduli space is approximate). Now, I've discussed tachyons whose squared mass is nonzero, namely negative. But there are at least two more important classes of scalar fields arising from string theory: axions and inflatons.
Axions generalize the Ramond-Ramond axions \(a\) I mentioned in the context of the 10-dimensional type IIB string theory. These scalar fields should be periodic; they are helpful as solutions to the strong CP-problem (why the coefficient of the \(F\wedge F\) term in QCD seems to be much smaller than one even though nothing would apparently be wrong if it were of order one); and some string theorists believe that string theory pretty clearly predicts that there should be lots of types of axions in the world around us – a whole "axiverse" of axions.
Note that axions are viable dark matter candidates – candidates that differ from WIMP (which is most typically the LSP, the lightest superpartner, most typically a neutralino or gravitino in supersymmetry). So that's another reason to think that Nature may like axions. The people who believe in the "axiverse" point out that typical string compactifications employ rather complicated Calabi-Yau-like manifolds and for each "cycle" of (or "hole" in) this manifold, one may find an axion. The spectrum of the axions' masses may densely cover a very wide interval of energies on the log scale.
Another crucial type of a scalar field in fundamental physics is the inflaton. It is responsible for the cosmic inflation – the probably correct explanation why our Universe is so exponentially huge, why the spatial sections are so flat and uniform (including the constant cosmic microwave background temperature in all directions), and why it has such a small density of exotic objects such as the magnetic monopoles. An inflaton wants to sit near the local maximum – similar to the center of the Mexican hat or the point where it looks like a tachyon – and resemble a huge positive cosmological constant that makes the Universe expand in an accelerated way. Eventually it rolls down closer to a local minimum, inflation stops, and the energy is converted to the matter that later gives rise to early galaxies.
For some conceptual reasons related to the weak gravity conjecture and other things, it actually seems like string theory doesn't like – and may be strictly incompatible with – the slow-roll inflation, also known as the new inflation. It seems that there is a general inequality in string theory that is pretty much equivalent to saying that the inflaton's potential can't have a "slow roll". If that's true, string theory could prefer the "old inflation" from the original paper by Alan Guth.
String theory's diverse geometric ways to look at Higgs bosons
Finally, I want to spend some time with the discussion of "genuine Higgses" in string theory; I mean scalar fields whose nonzero values break a gauge symmetry down to a smaller subgroup. Realistic and even unrealistic vacua contain these scalar fields, too. They're so natural that string theory apparently indicates it would be foolish and uneconomic to try to construct a Higgs-like object out of smaller point-like particles. As far as I know, there doesn't exist a single prominent enough model of technicolor within string theory. To a large extent, these are mutually excluding approaches to the physics of the electroweak symmetry breaking.
Realistic stringy scenarios such as heterotic strings on Calabi-Yau spaces, M-theory in singular \(G_2\) holonomy manifolds, intersecting type IIA braneworlds, and F-theory (whether you talk about the KKLT-like flux vacua or the bottom-up Vafa-style models) naturally predict Higgs fields, i.e. new scalar matter fields ready to break some gauge symmetries.
It's interesting to look at three examples of the clever ways in which the Higgs fields are "geometrized" in string theory.
Distance between D-branes
My first example is concerned with the D-branes' gauge group. If you consider a stack of coincident (and therefore parallel) D-branes of the same dimension at a generic point in space, you will find out that they carry a \(U(N)\) gauge group. It's the same group we know from our discussions of Matrix theory: the positions \(\vec x_i\) of \(N\) particles, the D0-branes (and similarly for higher-dimensional branes), are promoted to the diagonal entries \(\vec x_{ii}\) (no summation) but these matrices also naturally have the off-diagonal matrix elements.
So the coordinates in the directions transverse to the D-branes (all spatial coordinates are transverse in the case of D0-branes, of course) are described by \(N\times N\) matrices, i.e. the adjoint representation of the \(U(N)\) group. If you separate the stack of \(N\) D-branes to a smaller stack of \(K\) D-branes and \((N-K)\) D-branes by assigning different transverse positions to the two smaller stacks, you will break the gauge symmetry in this way:\[
U(N)\to U(K)\times U(N-K).
\] The transverse position of the D-branes is nothing else than a Higgs field that typically breaks the gauge symmetry to a smaller group. This Higgs field has to transform in the adjoint representation of a \(U(N)\) group – or, in the case of orientifolds, \(SO(N)\) or \(USp(2N)\). So by this group-theoretical property, it differs from the Standard Model Higgs that transforms as the fundamental 2-dimensional complex (well, pseudoreal) representation of \(SU(2)\). However, aside from these different representations, the physical logic is the same. The transverse positions of the D-branes are exactly described as Higgs fields in any effective field theory.
Deviation from the self-dual radius
My second example will deal with the "enhanced gauge symmetry" at the self-dual radius in bosonic and heterotic string theory. Recall that in 1919, Theodor Kaluza proposed a unification of electromagnetism and gravity in terms of 5-dimensional general relativity. Oskar Klein later clarified that the fifth dimension should be a circle. The \(U(1)\) isometry of the circle (at each point in the remaining 3+1 dimensions) is interpreted as the gauge symmetry of electromagnetism; \(g_{\mu 5}\), an off-diagonal element of the metric tensor, is related to the electromagnetic potential \(A_\mu\). The momentum in the 5th dimension (which is quantized because the dimension is periodic) is interpreted as the electric charge (in the right minimal units).
Whenever you deal with compactifications of gravitational theories (especially string theory) on a circle, this Kaluza-Klein mechanism is there. Consider, for example, bosonic string theory. Compactify one of the 26 spacetime coordinates on a circle. So in the remaining 24+1 coordinates, you will get a \(U(1)\) gauge symmetry. In fact, in bosonic string theory, you will get \(U(1)\times U(1)\). You obtain two new \(U(1)\) groups. Much like the momentum in the circular dimension gives you the electric charge under the first \(U(1)\), there is another charge: the fundamental strings can wind around the extra circle, giving you one more charge (the winding number which is interchanged with the momentum under T-duality). The vector potential for this winding number arises from the B-field (a two-form field under which the strings are electrically charged) of string theory, \(B_{\mu,25}\).
String theory not only doubles the Kaluza-Klein gauge symmetry in this situation. It actually extends it to a non-Abelian, Yang-Mills symmetry \(SU(2)\times SU(2)\) if you choose the radius of the circle to be at the "self-dual value" (so that the minimum-momentum particles moving in the 5th dimension have the same mass/energy as the minimum-winding strings wrapped around the circular dimension). One needs to know some \(d=2\) conformal field theory to prove it (one needs to know different ways to describe the \(SU(2)\) current algebra CFT) but this symmetry is actually exact.
If you're away from the self-dual radius, you see that the momentum modes and the winding modes have different masses. That's just one way to see that the symmetry (that was originally able to interchange the momentum and winding modes: the very T-duality was interpreted as a particular rotation by 180 degrees in a full-fledged \(SU(2)\times SU(2)\) group of symmetries) gets broken. The breaking may be once again described as a Higgs mechanism. The corresponding Higgs field is essentially the difference of the actual radius of the circle and the self-dual radius. If you look at the spectrum carefully, you will see that the spectrum of these Higgs scalars transforms as \(({\bf 3},{\bf 3})\) under the \(SU(2)\times SU(2)\) gauge group. If you give it a generic nonzero vev, the gauge group is broken to \(U(1)\times U(1)\) I have already mentioned: it's the group under which the Higgs vevs are invariant. Note that this is another "extra story" behind the Higgs mechanism and it's group-theoretically inequivalent to the D-brane case.
In heterotic string theory, there's only one enhanced \(SU(2)\); only the left-moving, bosonic-string-theory-based portion of the degrees of freedom give you the enhanced symmetry and its breaking.
Smoothening of ADE singularities
Finally, string theory cures problems and ambiguities of short-distance physics. In particular, it resolves physics at the singularities. This statement doesn't necessarily mean that all singularities are replaced by manifestly smooth manifolds although it's sometimes the case, too. However, in some cases, the singularities in the spacetime stay singular but string theory is capable of making finite, non-singular predictions about the physical phenomena, anyway. This ability partly boils down to the existence of new light states that wouldn't exist in point-like-particle-based field theories (e.g. wound strings or wrapped branes) confined to the vicinity of the singularities; however, it's also important that strings can't probe arbitrarily tight neighborhoods of the singular loci because they're extended objects with a finite resolution.
For example, string theory allows you to study physics in the presence of ADE singularities. They may be strictly singular and even if they are singular, string theory produces totally healthy predictions. These singularities are special places in the spacetime and a gauge field corresponding to the "same" gauge group in the ADE classification lives there (according to the miraculous ADE dictionary between Lie groups and particular singular manifolds).
However, these singularities may also be deformed (or "resolved") to less singular or totally smooth manifolds. When you do so, the gauge symmetry is broken to a smaller one. The scalar fields describing the deviation of the shape away from the singular ADE manifold are Higgs fields and their nonzero vev breaks the original ADE gauge group down to a smaller subgroup according to the normal rules of the Higgs mechanism.
Summary
This text has already become pretty long and I have written something under all the sections I had originally prepared. So let me end at this point.
My main message is that the \(j=0\) scalar fields (and their corresponding spinless particles – each of which is composed of a single tiny string much like the photon and other particles are composed of a single string) are totally natural in string theory, they play many roles, and they allow us to look at the physical phenomena from many perspectives and use many intellectual templates to visualize what's going on.
String theory has always predicted that the electroweak symmetry should be broken by objects that look like elementary, structureless scalar fields, and if the Higgs boson is indeed discovered soon, it will be a confirmation of the victory of string theory in its battle against some competing models (e.g. technicolor and other modes largely based on the phobia from scalar fields) which have never been comparably influential as string theory and string-theory-friendly phenomenology but they have existed, anyway.
An apparently elementary Higgs field is a modest but nonzero confirmation of string theory's being on the right track and due to some extra arguments sketched above and due to the SUSY-friendly (light enough) mass \(125\GeV\) of the discovered Higgs boson, it is also a signal that we may (and maybe we should) see some other discoveries envisioned by string theory such as supersymmetry.
And that's the memo.
Higgs boson, scalar fields, and a victory for string theory
Reviewed by DAL
on
June 24, 2012
Rating:
Reviewed by DAL
on
June 24, 2012
Rating:
No comments: