One form of the LC&P sum rule is
2 λ + g2/4 + (g2 + g'2)/4 + yt2/2 ~ 1
... based on their equation 2, and neglecting yukawa couplings for fermions other than the top quark.
As they remark (but I didn't notice until Andrew pointed it out), the contributions from bosons and fermions are almost equal. So we can also say that
2 λ + g2/2 + g'2/4 ~ yt2/2 ~ 1/2
The "fermionic part" of this makes sense, if we recall that yt ~ 1. But the bosonic part
2 λ + g2/2 + g'2/4 ~ 1/2
... just considered by itself, seems to be very notable new numerology, connecting electromagnetic and weak couplings with the Higgs self-coupling λ.
edit: Actually, if I think about it for a moment, I remember that g is small and g' (the weak coupling) is even smaller. So the bosonic part reduces to
2 λ ~ 1/2
i.e. λ ~ 1/4. I noted almost a year ago that this is implied by the fact that the Higgs VEV / electroweak scale is approximately twice the Higgs boson mass.
edit #2: Study of the literature (e.g. PDG 2013 Higgs review) makes it clear that
λ ~ 1/8
is closer to the truth. Apparently there are some factors of √2 that I missed. But now I don't understand why LC&P works.
(Or are we just dealing with different conventions? Remedial study of Higgs-sector basics is in order...)
Friday, January 17, 2014
Tuesday, January 14, 2014
t, H, W, Z and naturalness
The lightness of the Higgs boson is one of the vexing issues in particle physics today. Why isn't it made heavy by virtual particles?
Meanwhile, on this blog I have chronicled a variety of possible relations among the masses of t, H, W, Z. Perhaps the most impressive of these is the sum rule due to Lopez-Castro and Pestieau (anticipated by Garces Doz, and blogged by Andrew Oh-Willeke 1 2 3).
It has a mild resemblance to the "Veltman condition", a t,H,W,Z relation proposed by Martinus Veltman which would imply that the virtual corrections to the Higgs mass cancel out. In its original form, it implies a Higgs mass greater than 300 GeV, which is wrong.
However, the original form of the Veltman condition is specific to the unadorned standard model. Today, Ernest Ma - one of the few theorists to tackle the Koide formula - has told us what a Veltman condition looks like, in a minor extension of the standard model where neutrinos get their mass from dark matter (the "scotogenic" model; skotos means darkness, thus scotogenic, generated from the dark).
The paper is here. The new conditions are equations 8 and 9. With three new free parameters, it may not look so exciting. But it demonstrates that a naturalness condition can deviate a bit from Veltman's original formula, while still retaining a family likeness. (Further examples may be found here.)
This suggests a new interpretation of the LC&P sum rule (and any other valid tHWZ numerology): as a symptom of an underlying, slightly-beyond-standard-model theory, that is natural.
Meanwhile, on this blog I have chronicled a variety of possible relations among the masses of t, H, W, Z. Perhaps the most impressive of these is the sum rule due to Lopez-Castro and Pestieau (anticipated by Garces Doz, and blogged by Andrew Oh-Willeke 1 2 3).
It has a mild resemblance to the "Veltman condition", a t,H,W,Z relation proposed by Martinus Veltman which would imply that the virtual corrections to the Higgs mass cancel out. In its original form, it implies a Higgs mass greater than 300 GeV, which is wrong.
However, the original form of the Veltman condition is specific to the unadorned standard model. Today, Ernest Ma - one of the few theorists to tackle the Koide formula - has told us what a Veltman condition looks like, in a minor extension of the standard model where neutrinos get their mass from dark matter (the "scotogenic" model; skotos means darkness, thus scotogenic, generated from the dark).
The paper is here. The new conditions are equations 8 and 9. With three new free parameters, it may not look so exciting. But it demonstrates that a naturalness condition can deviate a bit from Veltman's original formula, while still retaining a family likeness. (Further examples may be found here.)
This suggests a new interpretation of the LC&P sum rule (and any other valid tHWZ numerology): as a symptom of an underlying, slightly-beyond-standard-model theory, that is natural.
Monday, January 6, 2014
α-numerology from M-theory
The fine-structure constant might be the most popular target of physics numerologists. α numerology has a long history, such as Eddington's efforts and Feynman's remark. It's a recurring topic in this long thread which might be the high point of Internet-era physics numerology.
Today on vixra there is an article which speculates about how to obtain one of the numerological formulas for α, 4π3+π2+π. It's unusual for two reasons. First, the author (Amir Mulic) speaks the technical language of M-theory; he proposes to "interpret... this expression in terms of the volumes of lp-sized three-cycles on G2 holonomy manifolds". (lp would be the Planck length.)
Second, he mentions that the coupling has to "run", i.e. change its value with energy scale. This aspect of quantum field theory is why particle physics professionals tend to ignore even Koide's relation, to say nothing of the more baroque formulae invented by amateur numerologists. The modern paradigm is that simple relations among particle masses and coupling constants exist at ultra-high energies, but that at low energies these relations will be obscured by complicated corrections, e.g. extra terms containing a logarithm of the energy, described by "beta functions" which can be derived from fundamental theory.
I haven't really gone over Mulic's article (I note that he had a similar one on arxiv years ago), and I am apriori skeptical that this particular idea will work out. But what's noteworthy here is just that someone is making this sort of effort - trying to explain the numerological formulas using the full conceptual apparatus of modern mathematical physics.
Before I comment further, it might help to show how things look without such a bridge. On one side, we have the efforts of someone like Angel Garcés Doz, already mentioned several times on this blog. Garcés Doz works hard, and like Mulic, draws inspiration from 7-dimensional geometry. Still, I find his formulas more interesting than his physics.
On the other side, consider this item of F-theory phenomenology (via Lubos). Here we have a genuine example of how a string-theory background geometry might determine a particular value of α: in this case, it's "the number of fuzzy points" in "a non-commutative four-cycle" wrapped by a 7-brane. But the value of α thereby obtained is the high-energy value, the value at the grand unification scale - perhaps 1/24 or 1/25, says Lubos. It only approaches 1/137 at low energies because of those messy correction terms.
Incidentally, this "fuzzy F-theory phenomenology" played a role at the dawn of my own attempts to make sense of what Marni Sheppeard was doing. One day she exhibited a parametrization of the CKM matrix, in terms of circulant matrices, and I was interested in whether this could fit into an existing framework like F-theory. It was very interesting to see that number 24 appearing as one of her parameters, but at the time none of us knew enough to judge whether Brannen and Sheppeard's circulants, and Heckman and H. Verlinde's fuzzy points, could fit into the same theoretical synthesis.
Today on vixra there is an article which speculates about how to obtain one of the numerological formulas for α, 4π3+π2+π. It's unusual for two reasons. First, the author (Amir Mulic) speaks the technical language of M-theory; he proposes to "interpret... this expression in terms of the volumes of lp-sized three-cycles on G2 holonomy manifolds". (lp would be the Planck length.)
Second, he mentions that the coupling has to "run", i.e. change its value with energy scale. This aspect of quantum field theory is why particle physics professionals tend to ignore even Koide's relation, to say nothing of the more baroque formulae invented by amateur numerologists. The modern paradigm is that simple relations among particle masses and coupling constants exist at ultra-high energies, but that at low energies these relations will be obscured by complicated corrections, e.g. extra terms containing a logarithm of the energy, described by "beta functions" which can be derived from fundamental theory.
I haven't really gone over Mulic's article (I note that he had a similar one on arxiv years ago), and I am apriori skeptical that this particular idea will work out. But what's noteworthy here is just that someone is making this sort of effort - trying to explain the numerological formulas using the full conceptual apparatus of modern mathematical physics.
Before I comment further, it might help to show how things look without such a bridge. On one side, we have the efforts of someone like Angel Garcés Doz, already mentioned several times on this blog. Garcés Doz works hard, and like Mulic, draws inspiration from 7-dimensional geometry. Still, I find his formulas more interesting than his physics.
On the other side, consider this item of F-theory phenomenology (via Lubos). Here we have a genuine example of how a string-theory background geometry might determine a particular value of α: in this case, it's "the number of fuzzy points" in "a non-commutative four-cycle" wrapped by a 7-brane. But the value of α thereby obtained is the high-energy value, the value at the grand unification scale - perhaps 1/24 or 1/25, says Lubos. It only approaches 1/137 at low energies because of those messy correction terms.
Incidentally, this "fuzzy F-theory phenomenology" played a role at the dawn of my own attempts to make sense of what Marni Sheppeard was doing. One day she exhibited a parametrization of the CKM matrix, in terms of circulant matrices, and I was interested in whether this could fit into an existing framework like F-theory. It was very interesting to see that number 24 appearing as one of her parameters, but at the time none of us knew enough to judge whether Brannen and Sheppeard's circulants, and Heckman and H. Verlinde's fuzzy points, could fit into the same theoretical synthesis.
Thursday, December 5, 2013
MeVs and GeVs II
I was thinking again, about whether there is some way to explain the Higgs mass as a multiple of the nucleon mass. Not only is the fact that 53=125 surely a red herring, 125 or 126 wouldn't even be the dimensionless quantity of interest. The Higgs is about 126 GeV, but the nucleon mass is somewhat less than 1 GeV...
And then I remembered that other number beloved of physics numerologists, 137. Specifically, I vaguely recalled that there is some instability for a nucleus with atomic number of 137, precisely because the fine structure constant is about 1/137. I was reminded of the recently discovered fact that the value of the Higgs mass (along with the specific value of the top quark mass, and a few other parameters) places the standard model vacuum on the brink of instability.
It is tempting to suppose that some unknown physics has forced the Higgs to a critical value. In the previous post I speculated that "the Higgs field could be a QCD meson condensate weighed down by virtual nucleons". Could it be that the density of these virtual nucleons is bounded by an analogue of this 137-instability?! A crackpot idea, yes; but the first thing is to check what the actual ratio of the masses is.
Let us say that the Higgs mass is 125-126 GeV. The nucleon mass is about 939 MeV. This gives a ratio between 133 and 134. To my mind, this is close enough to 137 that one should persist a while longer with the idea. So what is the mechanism that destabilizes element 137 - which is jocularly known as "feynmanium", because Feynman was the one who noted the instability?
It turns out that the problem emerges first in the Bohr model - the innermost electron would orbit the nucleus faster than light ... and then in a more sophisticated version when using the Dirac equation - a ground-state instability ... and that even more sophisticated analyses push the problem out to atomic number 173, or entirely abolish it. The fact that the "137 instability" appears in different formalisms is mildly encouraging, since it suggests a phenomenon at work that might still exist, even in a wildly different theoretical context.
The next step was to see whether anyone else has had thoughts along these lines. The numerology mentioned by "fzero" in the previous post is getting there, but it's a little back-to-front: it uses the running of the fine-structure constant, to reach a scale where it is approximately 1/125. But as I have already mentioned, 1 GeV is only a ballpark number; 939 MeV is the objectively interesting quantity, and that suggests that we should go with the low-energy value, ~ 1/137.
A search for "feynmanium higgs" turned up a blog comment by "Juan F." which is halfway there. Feynmanium is mentioned, but Juan F. is still using something more like fzero's relation, with the number 126 mooted as significant because it is a "magic number" in nuclear physics.
And then I remembered that other number beloved of physics numerologists, 137. Specifically, I vaguely recalled that there is some instability for a nucleus with atomic number of 137, precisely because the fine structure constant is about 1/137. I was reminded of the recently discovered fact that the value of the Higgs mass (along with the specific value of the top quark mass, and a few other parameters) places the standard model vacuum on the brink of instability.
It is tempting to suppose that some unknown physics has forced the Higgs to a critical value. In the previous post I speculated that "the Higgs field could be a QCD meson condensate weighed down by virtual nucleons". Could it be that the density of these virtual nucleons is bounded by an analogue of this 137-instability?! A crackpot idea, yes; but the first thing is to check what the actual ratio of the masses is.
Let us say that the Higgs mass is 125-126 GeV. The nucleon mass is about 939 MeV. This gives a ratio between 133 and 134. To my mind, this is close enough to 137 that one should persist a while longer with the idea. So what is the mechanism that destabilizes element 137 - which is jocularly known as "feynmanium", because Feynman was the one who noted the instability?
It turns out that the problem emerges first in the Bohr model - the innermost electron would orbit the nucleus faster than light ... and then in a more sophisticated version when using the Dirac equation - a ground-state instability ... and that even more sophisticated analyses push the problem out to atomic number 173, or entirely abolish it. The fact that the "137 instability" appears in different formalisms is mildly encouraging, since it suggests a phenomenon at work that might still exist, even in a wildly different theoretical context.
The next step was to see whether anyone else has had thoughts along these lines. The numerology mentioned by "fzero" in the previous post is getting there, but it's a little back-to-front: it uses the running of the fine-structure constant, to reach a scale where it is approximately 1/125. But as I have already mentioned, 1 GeV is only a ballpark number; 939 MeV is the objectively interesting quantity, and that suggests that we should go with the low-energy value, ~ 1/137.
A search for "feynmanium higgs" turned up a blog comment by "Juan F." which is halfway there. Feynmanium is mentioned, but Juan F. is still using something more like fzero's relation, with the number 126 mooted as significant because it is a "magic number" in nuclear physics.
Sunday, September 15, 2013
String theory and Mulla Sadra
Mulla Sadra is a notable Iranian philosopher, among whose concepts is one that is translated into English as "substantial motion". I do not have a scholar's precise understanding of the notion, but my rough understanding is that the previous notion of change in an entity was that its "accidents" might change but its essence would remain the same, whereas Sadra proposed that it could change in its being. Thus, rather than change just being motion through space, it could also take the form of change "within" a substance.
This reminds me of an elementary duality which appears early in the development of string theory. The string is a one-dimensional entity moving through an n-dimensional space. But it may also be conceived as a one-dimensional space in itself, with a number of fields existing within it. The values of these fields within the string, correspond to the position vectors of the points on the string, in the first picture. Thus this duality exchanges motion of the string through space, for substantial motion within the string. The second perspective is arguably the basis of perturbative string theory, which employs conformal field theory to describe the dynamics of fields within the string (worldsheet fields) corresponding to motion within a particular geometric background.
This reminds me of an elementary duality which appears early in the development of string theory. The string is a one-dimensional entity moving through an n-dimensional space. But it may also be conceived as a one-dimensional space in itself, with a number of fields existing within it. The values of these fields within the string, correspond to the position vectors of the points on the string, in the first picture. Thus this duality exchanges motion of the string through space, for substantial motion within the string. The second perspective is arguably the basis of perturbative string theory, which employs conformal field theory to describe the dynamics of fields within the string (worldsheet fields) corresponding to motion within a particular geometric background.
Monday, September 2, 2013
Proton mass and top quark mass
"In this picture, we discover that Mp ∝ mt2/27: despite the negligible population of virtual top-antitop pairs in the nucleon, the top-quark mass influences the proton mass."
This is not some wild-eyed vixra prophet of numerology speaking, but Chris Quigg, who has had a long and very respectable career, working especially on hadrons.
Koide-ologists ought to take a keen interest in this claim, for several reasons.
First, the "Brannen phases" for the triples u-c-t, d-s-b, e-mu-tau, are all multiples of 2/27.
Second, the centrality of the top is consistent with Rivero-ist ideas such as the waterfall (in which the quark masses can be understood as arising from a chain of Koide triplets starting with t-b-c) and the sbootstrap (in which all the SM fermions are superpartners of quark-(anti)quark composites of the 5 "light" quarks, i.e. everything but the top) and even "84"-ism.
Third, it legitimizes, in a new way, the appearance of QCD scales as "Brannen mass scales" for e-mu-tau and s-c-b.
This is not some wild-eyed vixra prophet of numerology speaking, but Chris Quigg, who has had a long and very respectable career, working especially on hadrons.
Koide-ologists ought to take a keen interest in this claim, for several reasons.
First, the "Brannen phases" for the triples u-c-t, d-s-b, e-mu-tau, are all multiples of 2/27.
Second, the centrality of the top is consistent with Rivero-ist ideas such as the waterfall (in which the quark masses can be understood as arising from a chain of Koide triplets starting with t-b-c) and the sbootstrap (in which all the SM fermions are superpartners of quark-(anti)quark composites of the 5 "light" quarks, i.e. everything but the top) and even "84"-ism.
Third, it legitimizes, in a new way, the appearance of QCD scales as "Brannen mass scales" for e-mu-tau and s-c-b.
Saturday, August 31, 2013
A dark identity
Back in March 2013, the Planck collaboration announced a new estimate for the fraction of today's universe that consists of dark energy; about 68%. As chronicled here, someone soon noticed that this number is approximately 1-1/π. A few months later, it was noticed that this number is also approximately ln(2). If we combine these two observations, we discover what I shall call the "dark identity"
ln(2) + 1/π ≈ 1
The terms on the left-hand side correspond to "dark energy" and "everything else", respectively.
This little gem takes crackpot physics numerology to a new level, because even just as mathematics, it's anomalous; but it also encodes the universe.
One mission of this blog is to look at such relationships and to not immediately dismiss them. But so far, it has been a matter of asking whether particular numerical relationships between quantities from physics could be something other than coincidence.
Here, we first need to ask this of the strictly numerical aspect of the anomaly. Could this near-relationship between π, the number 1, and the natural logarithm of 2, conceivably be something other than a coincidence?
For this to be so, I think it would have to derive from some simple and significant mathematical object. It needs to be simple because numerically, the coincidence is not supernaturally close.
Suppose we learned, for example, that this formula approximates an exact identity concerning the discriminants of an obscure type of number field. This might be unimpressive if there are thousands of equally obscure number fields. But it would be impressive if it was a very special number field, which e.g. played a role in some important construction.
Obviously one should look for mathematical topics where both π and the natural logarithm of 2 play a role.
So much for pure math. What about the physics? Right away a problem presents itself: 68% is the density fraction of dark energy now; but this number changes as the universe expands. It is not impossible that this number means something fundamental; but it will require some unusual assumptions about physics for this to be so, and one must take care, in constructing such unusual models, that one does not violate the more conventional assumptions that the Planck team used to derive the number in the first place.
Similarly, one has to suppose that the hypothesized number field (or whatever) which is mathematically responsible for the dark identity, also plays a role in the mathematics of the physics of the dark sector. This is another constraint on model building; potentially a very stringent constraint, if no mathematical explanation of the identity is forthcoming.
Having spelled out some of the reasons why this may be a barren line of inquiry, now let me say why it's still interesting to think about. Mostly it's interesting because it offers an unexpected perspective on the coincidence problem of dark energy - why the amount of dark energy is about the same as the amount of everything else, rather than being different by orders of magnitude. If you're completely in the dark about how to solve a problem, then even a crazy idea with delirious logic can be a good starting point; in trying to make it work, you may discover some other idea that is similar, but better.
Also, this is a reminder of how one wants physics to work - a harmony in which deep mathematical facts are responsible for deep physical facts.
Finally, let me note that I may not have ever noticed this "dark identity", were it not for L. Edgar Otto's musings on approximations of π, which are apparently part of a larger poetic metaphysics which also includes "dark fluid dynamics".
ln(2) + 1/π ≈ 1
The terms on the left-hand side correspond to "dark energy" and "everything else", respectively.
This little gem takes crackpot physics numerology to a new level, because even just as mathematics, it's anomalous; but it also encodes the universe.
One mission of this blog is to look at such relationships and to not immediately dismiss them. But so far, it has been a matter of asking whether particular numerical relationships between quantities from physics could be something other than coincidence.
Here, we first need to ask this of the strictly numerical aspect of the anomaly. Could this near-relationship between π, the number 1, and the natural logarithm of 2, conceivably be something other than a coincidence?
For this to be so, I think it would have to derive from some simple and significant mathematical object. It needs to be simple because numerically, the coincidence is not supernaturally close.
Suppose we learned, for example, that this formula approximates an exact identity concerning the discriminants of an obscure type of number field. This might be unimpressive if there are thousands of equally obscure number fields. But it would be impressive if it was a very special number field, which e.g. played a role in some important construction.
Obviously one should look for mathematical topics where both π and the natural logarithm of 2 play a role.
So much for pure math. What about the physics? Right away a problem presents itself: 68% is the density fraction of dark energy now; but this number changes as the universe expands. It is not impossible that this number means something fundamental; but it will require some unusual assumptions about physics for this to be so, and one must take care, in constructing such unusual models, that one does not violate the more conventional assumptions that the Planck team used to derive the number in the first place.
Similarly, one has to suppose that the hypothesized number field (or whatever) which is mathematically responsible for the dark identity, also plays a role in the mathematics of the physics of the dark sector. This is another constraint on model building; potentially a very stringent constraint, if no mathematical explanation of the identity is forthcoming.
Having spelled out some of the reasons why this may be a barren line of inquiry, now let me say why it's still interesting to think about. Mostly it's interesting because it offers an unexpected perspective on the coincidence problem of dark energy - why the amount of dark energy is about the same as the amount of everything else, rather than being different by orders of magnitude. If you're completely in the dark about how to solve a problem, then even a crazy idea with delirious logic can be a good starting point; in trying to make it work, you may discover some other idea that is similar, but better.
Also, this is a reminder of how one wants physics to work - a harmony in which deep mathematical facts are responsible for deep physical facts.
Finally, let me note that I may not have ever noticed this "dark identity", were it not for L. Edgar Otto's musings on approximations of π, which are apparently part of a larger poetic metaphysics which also includes "dark fluid dynamics".
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