tag:blogger.com,1999:blog-4167378123314341202017-03-08T09:06:56.212-08:00theorynot endorsed by snarxivMitchellnoreply@blogger.comBlogger71125tag:blogger.com,1999:blog-416737812331434120.post-51253310029942511872017-03-07T03:00:00.000-08:002017-03-07T03:17:13.802-08:00Two problemsThere was unexpected progress, posted at Physics Stack Exchange, on two problems that were low on my list.<br /><br />First, <a href="http://snarxivblog.blogspot.com/2016/05/proton-charge-radius.html">numerology of the charge radius</a>. See my <a href="http://physics.stackexchange.com/a/254114/1486">2017 update</a>: I ran across a model of the nucleon in which the radius is 4 natural units, divided by the mass. That doesn't explain why the radius comes out a little different for muonic hydrogen compared to electronic hydrogen; but it can explain why dandb's ratio is approximately 4 in both cases.<br /><br />Second, <a href="http://snarxivblog.blogspot.com/2013/08/weak-interaction-bootstrap.html">mystery of the Z0 decay width</a> - that it lies on the same curve as a number of mesons. It's one of @arivero's minor observations, and not one that I spent any time on. I was just going through the motions of investigating it, when to my surprise, <a href="http://physics.stackexchange.com/a/316947/1486">something turned up</a>. Mitchellnoreply@blogger.com0tag:blogger.com,1999:blog-416737812331434120.post-51081388516329976342017-02-10T02:45:00.000-08:002017-02-10T02:45:12.959-08:00tHWZ - latest formulationDuring a discussion at PF, I found the following interesting way to think of these quantities:<br /><br />m<sub>H</sub> ~ √2 m<sub>Z</sub><br />m<sub>t</sub> ~ 2 m<sub>Z</sub><br />H<sub>vev</sub> ~ 2 √2 m<sub>Z</sub><br />m<sub>W</sub> ~ √7 / 3 m<sub>Z</sub><br /><br />The last one may look a little odd, but it allows us to approximate sin<sup>2</sup> of the Weinberg angle as 2/9.<br /><br />The impetus was <a href="http://www.physicsforums.com/threads/top-higgs-higgs-vev-relation-from-conformal-symmetry.768598/#post-5684753">a comment by @arivero</a> in which he pointed out that a tHWZ mass estimate due to Hans de Vries implies<br /><br />(m<sub>W</sub><sup>2</sup> - m<sub>H</sub><sup>2</sup>) / (m<sub>Z</sub><sup>2</sup> - m<sub>t</sub><sup>2</sup>) = 3/8<br /><br />Now in many GUTs, at the GUT scale, we have that<br /><br />m<sub>W</sub><sup>2</sup> / m<sub>Z</sub><sup>2</sup> = 3/8<br /><br />So it's as if (m<sub>W</sub><sup>2</sup> - m<sub>H</sub><sup>2</sup>) / (m<sub>Z</sub><sup>2</sup> - m<sub>t</sub><sup>2</sup>) is almost invariant under renormalization group flow, with m<sub>H</sub> = m<sub>t</sub> = 0 at the GUT scale.<br /><br />We could even speculate that my set of four approximations above is an infrared fixed point. (The approximations are not exact, but one could think of these as valid at tree level.)<br /><br />Unfortunately I don't see how any of this makes sense in terms of Hans de Vries's original physical hypothesis.<br /><br />Anyway, I find that the LC&P formula also works neatly using the four approximations. And I would remark again that m<sub>Z</sub> is very close to <a href="http://snarxivblog.blogspot.com/2016/01/h-z-susy.html">the standard model's μ parameter</a>. Mitchellnoreply@blogger.com6tag:blogger.com,1999:blog-416737812331434120.post-71955084038430274202016-08-24T13:25:00.001-07:002016-08-24T13:25:53.440-07:00Multifractal worldsheetThe opinion is spreading that the real discovery of the LHC was that the Higgs boson mass is special. The most impressive prediction was Shaposhnikov and Wetterich 2006, which got the right value from the assumption that gravity is asymptotically safe.<br /><br />This creates cognitive dissonance for anyone who appreciates the string-theoretic model of quantum gravity. Asymptotic safety isn't even consistent with the holographic principle, is it?<br /><br />Well, asymptotic safety is one of several heterodox approaches to quantum gravity in which the dimension of spacetime seems to change from 4 to 2 at the smallest scales. Sabine Hossenfelder <a href="http://backreaction.blogspot.com/2016/08/what-if-universe-was-like-pile-of.html">lists a few others</a> and says, "It is difficult to visualize what is happening with the dimensionality of space if it goes down continuously, rather than in discrete steps".<br /><br />Fractals can have non-integer dimensionality. But they are typically embedded in a larger space. Meanwhile, in string theory, one has a 2d worldsheet embedded in a "target space" that usually has more than two dimensions. So what if the world sheet embeds in the target space as a <a href="http://snarxivblog.blogspot.com/2014/06/goldfain-on-lc.html">multifractal</a> surface that is 4d on large scales but 2d on small scales?Mitchellnoreply@blogger.com0tag:blogger.com,1999:blog-416737812331434120.post-5226969130429911132016-07-09T02:46:00.002-07:002016-07-09T02:46:32.236-07:00A formula for αOn page 4 of <a href="http://arxiv.org/abs/0801.2562">"Naturally Speaking"</a> by G.F. Giudice, after a short list of numerological formulas for the fine-structure constant α, one finds a formula for α according to physical orthodoxy, i.e. grand unification. I reproduce it here for the edification of passing numerologists: <br /><br />α = { α<sub>s</sub> sin<sup>2</sup>θ<sub>W</sub> (b<sub>1</sub>−b<sub>3</sub>)+3/5 cos<sup>2</sup>θ<sub>W</sub> (b<sub>3</sub>−b<sub>2</sub>) } / (b<sub>1</sub>−b<sub>2</sub>) + higher-order terms. <br /><br />"Here, the fine-structure constant α, the strong coupling constant α<sub>s</sub> and the weak mixing angle θ<sub>W</sub> are evaluated at the same renormalization scale and b<sub>1,2,3</sub> are the gauge β-function coefficients. Higher-order terms cannot be neglected to achieve a prediction that matches the experimental accuracy."Mitchellnoreply@blogger.com4tag:blogger.com,1999:blog-416737812331434120.post-51609747650472327162016-05-14T04:02:00.000-07:002016-05-14T04:02:32.231-07:00Higgs, top, 750 GeV IIThe idea that the 750 GeV anomaly might be a top-antitop bound state has been taken to a new level by <a href="http://arxiv.org/abs/1605.03909">Froggatt & Nielsen</a>, who have sketched a whole phenomenology for their particle. The reasoning is "crude" (their word), but still on a much higher plane than any mere numerology of masses.<br /><br />So things may be about to get serious there. Meanwhile, I want to enumerate a few relationships which <i>are</i> still just numerology, but have the potential to be part of a genuine theoretical synthesis.<br /><br />m<sub>H</sub> ~ m<sub>t</sub>/<span class="st">√2</span><br /><span class="st">H<sub>vev</sub> ~ 2 m<sub>H</sub></span><br /><span class="st">m<sub>375</sub> ~ 3 m<sub>H</sub></span><br /><span class="st">m<sub>750</sub> ~ 6 m<sub>H</sub></span><br /><span class="st"><br /></span><span class="st">m<sub>H</sub> is the Higgs boson mass, m<sub>t</sub> is the top quark mass, H<sub>vev</sub> is the Higgs field vev. m<sub>750</sub> is the mass of the 750 GeV particle. m<sub>375</sub> is the mass of a 375 GeV particle that Lubos may have found in the data. </span><br /><br /><span class="st">The picture I get is that the Higgs field is a top quark condensate, the 750 is a sort of loose bound state of 6 Higgs bosons (that is a "1S" toponium when analyzed at the level of quarks), and the 375 is like the 750 but with only half of the available top states occupied. </span>Mitchellnoreply@blogger.com0tag:blogger.com,1999:blog-416737812331434120.post-16786207231056419092016-05-11T05:36:00.000-07:002016-05-11T05:36:43.951-07:00Higgs, top, 750 GeVIt is a long-standing idea that the Higgs might be a top-antitop bound state. (I have proposed a <a href="http://www.physicsoverflow.org/35142/higgs-as-toponium-bootstrap">bootstrap version</a> of this idea.)<br /><br />My credence for that idea has just gone way up, now that I have discovered another long-standing proposal, that there might be <a href="http://www.physicsforums.com/threads/higgs-top-resonances.856586/#post-5470425">a light bound state of 6 tops and 6 anti-tops</a>. The number 6 appears because the top quark has two spin states and three color states, so this is the maximum number of tops in the same wavefunction that is allowed by the Pauli exclusion principle.<br /><br />I had already wondered if the <a href="http://snarxivblog.blogspot.com/2015/12/lhc-bump.html">LHC bump</a> at 750 GeV was somehow 6 Higgs bosons bound by top loops, since 750 GeV = 6 x 125 GeV, the Higgs mass. But if the Higgs is already a top-antitop bound state...Mitchellnoreply@blogger.com4tag:blogger.com,1999:blog-416737812331434120.post-36603124122853004662016-05-06T07:30:00.001-07:002016-05-06T07:31:52.976-07:00Proton charge radiusA new user at Physics Stack Exchange, "dandb", has <a href="http://physics.stackexchange.com/questions/254050/the-mass-of-the-proton-times-its-charge-radius-is-very-close-to-4%C4%A7-c-is-this-a">made an observation</a> which I express as follows:<br /><br />"The charge radius of the proton (in muonic hydrogen) is almost exactly four times the reduced Compton wavelength of the proton." Mitchellnoreply@blogger.com4tag:blogger.com,1999:blog-416737812331434120.post-26088143860450835592016-01-22T05:05:00.000-08:002016-01-22T05:05:01.504-08:00Susy IIIn some ways, the MSSM seems promising as a framework for tHWZ numerology. The reason may be seen at the end of section 3.4 in <a href="http://arxiv.org/abs/hep-ph/9709356">Stephen Martin's primer</a>: susy determines the couplings of the scalar potential, in terms of coupling constants elsewhere in the theory. If I am interpreting that passage correctly, the quadratic coupling will be set by the top yukawa, and the quartic by the gauge couplings.<br /><br />However, the MSSM has a lot of annoying particles like gauginos and sfermions which get in the way. Here last decade's ideas about split susy are useful. In particular, in section 2.3.3 of <a href="http://arxiv.org/abs/hep-th/0501082">"Predictive Landscapes..."</a> we read about a framework midway between split and supersplit susy, in which only the Higgsino is light. That sounds worth exploring.Mitchellnoreply@blogger.com1tag:blogger.com,1999:blog-416737812331434120.post-61022143959541313362016-01-18T07:09:00.001-08:002016-01-18T07:11:27.845-08:00H, Z, susyI finally noticed that the Higgs mass parameter μ, 89 GeV, is very close to the Z boson rest mass, 91 GeV (and the width of the Z is a few GeV). <p>In the standard model, these quantities should be independent. But in the MSSM, the Z boson is the upper bound on the tree-level mass of the Higgs. <p>I am too tired to develop an interpretation. But tomorrow is another day. Mitchellnoreply@blogger.com0tag:blogger.com,1999:blog-416737812331434120.post-52080109730051671092016-01-14T02:58:00.002-08:002016-01-14T02:58:36.906-08:00t, H, W, Z in 2016Recently I have been puzzling again, over the <a href="http://snarxivblog.blogspot.com/2012/01/dharwadker-and-khachatryans-prediction.html">Dharwadker-Khachatryan sum rule</a><br /><br />m<sub>H</sub> = m<sub>W</sub> + 1/2 m<sub>Z</sub><br /><br />The problem being that it works quite well; but theory tends to favor relations among the <i>squares</i> of these masses (e.g. the "Veltman condition").<br /><br />The primary purpose of this post is just to observe that you can get such a relation by squaring both sides of the D-K equation.<br /><br />You do also get a term m<sub>W</sub> m<sub>Z</sub>. Perhaps it could result from a geometric mean, as in <a href="http://arxiv.org/abs/1209.0474">Torrente-Lujan</a>.<br /><br />Another simple thing that I want to observe, is that you might obtain something like D-K, by taking the square root of a Veltman-like sum rule. In other words, it could be approximately true, not by chance and not because it is directly implied by a fundamental theory, but as an algebraic side-effect of the truly fundamental relationship.<br /><br />(The same applies to the Lopez-Castro - Pestieau - Garces-Doz sum rule, <a href="http://snarxivblog.blogspot.com/2014/09/viks-relation.html">previously discussed here</a>, which does involve masses squared, and therefore even more closely resembles Veltman's condition.)<br /><br />P.S. Dharwadker also has a numerology for the ratio baryonic matter : dark matter : dark energy, which he deduces to be <a href="http://www.dharwadker.org/cosmology/main.html">1:5:18</a>.Mitchellnoreply@blogger.com3tag:blogger.com,1999:blog-416737812331434120.post-18985054022778000802015-12-16T14:49:00.001-08:002015-12-16T14:49:20.404-08:00LHC bumpThere were some events at 750 GeV, resembling a second Higgs boson. As we all know, the first Higgs boson, with a mass of 125 GeV, saturates a theoretical bound (vacuum stability), and so it is encouraging to read <a href="http://arxiv.org/abs/1305.0002">here</a> of a scenario in which 750 GeV is the theoretical upper bound on the mass of a second Higgs. However, within that scenario, the upper limit is correlated with another parameter in a way that is ruled out empirically. Therefore, by numero-logic, this was a false alarm and we can all go back to sleep.Mitchellnoreply@blogger.com1tag:blogger.com,1999:blog-416737812331434120.post-18822126915388003012015-10-28T16:31:00.002-07:002015-10-28T18:58:34.801-07:00Today's vixra weatherThe number-one reason I monitor vixra is physics numerology. However, a lot of other stuff happens there, and a few times I have remarked upon it here.<br /><br />Today has an unusual surge of interesting activity. Most of this is due to a document dump by <a href="http://vixra.org/author/sai_venkatesh_balasubramanian">an independent researcher in Bangalore</a>, who appears to combine some genuine expertise and originality in signal processing and device physics, with a commitment to a traditional cultural synthesis embracing music, metaphysics, and language, and then rounded out with general polymathic speculation.<br /><br />But there's also just a higher-than-usual density of papers combining geographically diverse origins and professionalism above the vixra average. In particular, this <a href="http://vixra.org/abs/1510.0446">paper on neutrino mixing</a> from China has a quite professional look and exposition - until the author wants to motivate their ansatz for the mixing matrix, and then suddenly there's crazy talk [*] about fractals, Tsallis statistics, and M theory. Also notable is that this newly uploaded paper immediately acquired a comment by "Critic" purporting to explain its error. It makes me wonder if "Critic" is some colleague of the author's, who promised to publicly explain their criticism.<br /><br />Finally, someone is using vixra to lampoon a notoriously self-promoting critic of modern physics, by writing preprints which assert that his work is vindicated by the most avantgarde trends of the mainstream. The <a href="http://vixra.org/abs/1509.0295">first such paper</a> was just silliness, but <a href="http://vixra.org/abs/1510.0444">today's paper</a> is sophisticated silliness, in that it shows a fluency in contemporary mathematical physics beyond what can be obtained by just copy-pasting snarxiv output. My guess is that the author is at PhD level.<br /><br />[*] I may write a separate post about this paper. Mitchellnoreply@blogger.com0tag:blogger.com,1999:blog-416737812331434120.post-57744536362224247592015-08-21T12:18:00.004-07:002015-08-21T12:18:52.828-07:00Constituent pionsI have stumbled upon an idea regarding the structure of a "constituent quark": a bare quark plus three pions. Now to see if it makes sense and if it would explain anything.Mitchellnoreply@blogger.com1tag:blogger.com,1999:blog-416737812331434120.post-25055074677701515052015-07-31T04:23:00.000-07:002015-07-31T04:23:38.874-07:00Today's crackpot synthesis1. If one assumes that QCD embeds in a GUT group, then one can show that the proton mass m<sub>p</sub> is proportional to a power of the top mass, m<sub>t</sub><sup>2/27</sup>. <p>Today I realized that Brannen's reformulation of the Koide relation can be described as follows: the Brannen mass scale of e,mu,tau is m<sub>p</sub>/3, and the Brannen phase is 3 * 2/27. <p>2. If the quark families (b,s,d and u,c,t) are treated as Koide triples, their phases are also (arguably) multiples of 2/27. <p>There is also a "waterfall" of Koide triples, descending from the top, which <i>alternate</i> between the families. One of these triples - s,c,b - has a Brannen mass of m<sub>p</sub>, and a Brannen phase of 2/3. <p>3. In Rivero's sBootstrap, the leptons are superpartners of mesons made of the five light quark flavors, and the quarks themselves are superpartners of diquarks made of those five flavors. <p>I proposed to interpret this as similar to a Seiberg duality. The primordial theory is like six-flavor QCD with one heavy quark and five massless quarks, and N=1 supersymmetry. The other is the standard model, with the light quark masses, the leptons, and the electroweak sector all emerging from the duality. <p>The leptons would then be the mesinos of the primordial theory, and the phenomenological quarks would be a mixture of the primordial quarks and the diquarkinos. <p>4. This suggests a way of thinking about the numerology in 1 & 2. <p>The primordial fact would be that m<sub>p</sub> ∝ m<sub>t</sub><sup>2/27</sup> is already true in the QCD-like theory on one side of the duality. The appearance of m<sub>p</sub> and 2/27 in standard model numerology is then to be attributed to the duality. 1 comes from the "lepton-mesino duality", and everything in 2 from the "quark-diquarkino self-duality". Mitchellnoreply@blogger.com4tag:blogger.com,1999:blog-416737812331434120.post-51820815672899718392015-03-30T05:00:00.000-07:002015-03-30T05:00:57.983-07:00hep-dada #2Another transparently fake paper, <a href="http://vixra.org/abs/1503.0244">posted to vixra</a>. Maybe they are coming from David Simmons-Duffin, the inventor of snarxiv, or even from Andrew Bulhak, who wrote the original Postmodernism Generator. Anyway, it gets old fast.Mitchellnoreply@blogger.com1tag:blogger.com,1999:blog-416737812331434120.post-8297560736263946972015-03-22T17:44:00.000-07:002015-03-25T17:02:19.115-07:00hep-dada<a href="http://snarxivblog.blogspot.com/2011/06/another-bad-idea.html"></a>This blog was originally inspired by the parodies of arxiv abstracts produced by snarxiv. snarxiv in turn reminded me of the Postmodernism Server, which generated whole essays written in bland postmodernese. Now the wheel has turned further and someone has posted to vixra <a href="http://vixra.org/abs/1503.0163">a whole "paper", expanding on a snarxiv-like abstract</a>. <br /><br />Coincidentally, the Italian net.artist <a href="http://twitter.com/Stalagmathron/">Roberta Betti</a> recently took to tweeting out the text of imaginary papers in mathematics, complete with the typographic mangling produced e.g. by viewing Google's cached copy of such a document. Behold <a href="https://twitter.com/Stalagmathron/status/574672870804054016">"Canonically Stochastic Fibonacci Spaces Over Matrices"</a>, <a href="https://twitter.com/Stalagmathron/status/573729934549479424">"Existence In Real Logic"</a>, and <a href="https://twitter.com/Stalagmathron/status/572860936790257664">"On the Integrability of Universally Parabolic Lines"</a>. Could Betti (aka Stalagmathron) be behind the new vixra paper by "Baruch Seiberg and Claude Witten"? Has someone written a script which generates full-length snarxiv papers? Will vixra now be spammed with cheap machine dada? Time will tell.<br /><br /><b>edit</b>: @Stalagmathron has disappeared, but some of Betti's earlier work lives on at <a href="https://archive.org/search.php?query=creator%3A%22Roberta+Betti%22">archive.org</a>. Mitchellnoreply@blogger.com0tag:blogger.com,1999:blog-416737812331434120.post-54128431656559234342014-09-19T00:09:00.002-07:002014-09-19T00:15:26.497-07:00Vik's relationOver two years ago, a series of posts on <i>tHWZ</i> relations was <a href="http://snarxivblog.blogspot.com/2012/04/t-h-w-z.html">launched</a> here, starting with the observation that<br /><br /><<span class="st">ϕ</span>> ~ <span class="st">√2</span> m<sub>t</sub>, m<sub>t</sub> ~ <span class="st">√2</span> m<sub>H</sub><br /><br />which I was prompted to record, when Andrew Oh-Willeke remarked that<br /><br /><<span class="st">ϕ</span>> ~ 2<span class="st"></span> m<sub>H</sub><br /><br />(Where <<span class="st">ϕ</span>>, also often written as v, is the Higgs field "vacuum expectation value".)<br /><br />Recently a <a href="http://arxiv.org/abs/1409.0492">paper</a> appeared on arxiv, noting that first relation, in the form<br /><br />4 m<sub>H</sub><sup>2</sup> = 2 m<sub>t</sub><sup>2</sup> = v<sup>2</sup><br /><br />and Andrew <a href="http://www.physicsforums.com/showthread.php?t=768598#3">commented</a> that<br /><br />"I think it is more likely that the observed relationship is really an approximation of the relationships<br /><br />sum((Fi(^2)=v^2/2 and sum((Bj)^2)=v^2/2 for all fundamental fermion rest masses Fi and fundamental boson rest masses Bj"<br /><br />which is an aspect of the <a href="http://arxiv.org/abs/1305.4208">LC&P sum rule</a>, of which he also says that it is<br /><br />"quite a bit more profound than the fact that the heaviest fermion by itself accounts for about half of the Higgs vev squared, or that the Higgs mass square accounts for about a quarter of the Higgs vev squared."<br /><br />I agree that the LC&P sum rule looks to be the fundamental thing here. But there is an interesting final twist which he didn't note.<br /><br />To recapitulate:<br /><br />1. The sum of the squares of all the fundamental particle masses, is approximately the square of the Higgs VEV.<br /><br />2. The contributions to this total from bosons and fermions are approximately equal. (Given the love of supersymmetry in the particle physics community, it really is remarkable that this isn't visibly being talked about.)<br /><br />3. The top quark is responsible for the great majority of the fermion contribution, and thus about half of the total.<br /><br />4. The Higgs boson is responsible for about half the bosonic contribution, and thus about a quarter of the total.<br /><br />So where does the rest of the bosonic contribution come from? It comes from the W and Z bosons. So we have a fifth fact:<br /><br />5. The W and Z bosons are responsible for the other half of the bosonic contribution, and thus for the remaining quarter of the total.<br /><br />If we write this up as an equation, we get<br /><br />m<sub>H</sub><sup>2</sup> ~ m<sub>W</sub><sup>2</sup> + m<sub>Z</sub><sup>2</sup> ~ 1/2 m<sub>t</sub><sup>2</sup><br /><br />The first part of this equation appeared as a <a href="http://resonaances.blogspot.com/2012/07/h-day-morning-after.html?showComment=1344223302471#c7974718000473295304">blog comment</a> by S. Vik, who is apparently a retired physicist from Wilfrid Laurier University in Canada. At the time I gave it a low probability of being meaningful, but I did record it. It would be ironic if it is yet another genuine clue to what lies beneath the standard model. Mitchellnoreply@blogger.com4tag:blogger.com,1999:blog-416737812331434120.post-90026702326632926722014-09-12T19:35:00.003-07:002014-09-12T19:35:43.173-07:00vixra watchMany times on this blog I have cited papers from vixra, the alternative to arxiv. Today I just want to note two surprising new additions to the vixra user base, <a href="http://vixra.org/author/simon_plouffe">Simon Plouffe</a> and <a href="http://vixra.org/author/jacob_barnett">Jacob Barnett</a>. They both have biographies at Wikipedia: <a href="http://en.wikipedia.org/wiki/Simon_Plouffe">Plouffe</a> is, I guess, a computational number theorist, and <a href="http://en.wikipedia.org/wiki/Jacob_Barnett">Barnett</a> is a teenage theoretical physicist who has been in the media since he was 12 (he's 16 now). Mitchellnoreply@blogger.com0tag:blogger.com,1999:blog-416737812331434120.post-22746755517528619522014-06-27T14:25:00.001-07:002014-06-27T14:25:03.531-07:00Goldfain on LC&PI record here the existence of two papers by Ervin Goldfain <a href="http://vixra.org/abs/1310.0133">1</a> <a href="http://vixra.org/abs/1406.0168">2</a> claiming to derive the LC&P sum rule. <br /><br />His concept seems to be that the effective dimension of space-time varies with energy scale, that the masses of SM particles define special scales, and that the LC&P formula follows from a "closure relation" that must connect these different scales.<br /><br />Incidentally, he is not just talking about spaces with an integer number of dimensions, as in Kaluza-Klein theories or string theories, where e.g. the number of dimensions may increase from 4 to 10 or 11, at energies above the compactification scale. Instead he talks of there being 4+<span class="st">ε dimensions, reminiscent of <a href="http://arxiv.org/abs/1211.1741">dimensional regularization</a>... but the modified concept of dimensionality that he really emphasizes is that of <a href="http://en.wikipedia.org/wiki/Fractal_dimension">fractals</a>. </span><br /><span class="st"><br /></span><span class="st">Informally, one might say that Goldfain's concept is that space is crinkled or creased in a fractal way, so that e.g. the volume of space inside a cube doesn't simply vary as the third power of the side of the cube. Instead, the exponent describing the change in volume is non-integer, and also varies with the size of the cube (length of its side). If we take a cube and shrink it, we might find that as the side shrinks to one millimeter, volume is proportional to size^3.1, but by the time we have shrunk to one micrometer, volume is proportional to size^3.3. Apparently in the world of fractals, such behavior is called multifractal. </span><br /><span class="st"><br /></span><span class="st">The references to millimeter and micrometer above are purely illustrative. Goldfain seems to believe that the first significant deviations from integer dimensionality (4 space-time dimensions) only begin to occur above the electroweak energy scale, which would correspond to distances less than 10^-18 meters. </span><br /><span class="st"><br /></span><span class="st">Goldfain is an independent investigator who publishes at vixra and in various web "journals", but the concept of multifractal space-time isn't just some whimsy of his, it has seen some <a href="http://arxiv.org/abs/1209.1110">mathematical development</a>. The real problem I am having with his work so far, is that I don't understand where the "closure relation" comes from - and that's the crucial step towards obtaining the LC&P formula. </span><br /><span class="st"><br /></span><span class="st">See for example equation 5 in paper "1". The "r"s are the different scales, and the "D" is a fractal dimension. The LC&P formula is a sum of squares, and so if scales were associated with masses, and if D was equal to 2, then we might be able to obtain it from equation 5. </span><br /><span class="st"><br /></span><span class="st">Goldfain has written other papers trying to obtain SM mass ratios from fractal dimensional flow. A skeptical reading might say that all we have here is a conceptual framework in which multiple length scales can assume a special significance, and since masses can be mapped to length scales in physics, this multiscale conceptual framework can be a playground for a physics numerologist trying to explain particle masses. </span><br /><span class="st"><br /></span><span class="st">I am skeptical, but <a href="http://www.physics.ntua.gr/cosmo11/Naxos2011/09-16%20Friday%20Talks/Scalisi.pdf">dimensional flow</a> is not a bad thing to think about. I will make a follow-up post if I have anything more concrete to add. </span>Mitchellnoreply@blogger.com0tag:blogger.com,1999:blog-416737812331434120.post-72766806467893509522014-05-08T02:10:00.000-07:002014-05-08T02:20:02.953-07:00BICEP2 numerologyIt's been a while since I've posted. It's been a while since I talked cosmology. And meanwhile BICEP2 came out with what may be the big measurement of the decade, along with LHC's 2012 determination of the Higgs boson mass.<br /><br />Specifically, BICEP2 has estimated the cosmological "r" parameter, which quantifies the relative magnitude of tensor perturbations and scalar perturbations of the cosmic microwave background, as 0.2. I'll confess that I'm still working out the basic meaning of this quantity. It seems to be a ratio of energies-squared - the square of the energy in the tensor perturbations, divided by the square of the energy in the scalar perturbations. And the physical meaning of squaring the energy may be, that it corresponds to the "work done" by that type of perturbation. So perhaps it would mean that the fluctuations of the inflaton field (which supposedly caused the scalar perturbations) did five times as much work on the CMB photons, as was done by the fluctuations of the gravitational field (which supposedly caused the tensor perturbations). But you should probably ask someone better informed, before believing me about this. <br /><br />Now there are all sorts of complicated models out there - Higgs inflation... axion monodromy inflation from string theory... - in which people are trying to get an "r" near 0.2. Meanwhile, what are physics numerologists saying? So far, I have spotted two examples of BICEP2 numerology.<br /><br />First was a <a href="http://vixra.org/abs/1403.0300">vixra paper by Tony Smith</a>, in which Tony estimates "r" as 7/28 = 0.25. 7 and 28 are the dimensions of different algebras which he associates with the tensor and scalar perturbations, respectively, in the context of an octonionic theory of inflation. Of course I don't understand Tony's logic, but an important part is probably the proposition, a few pages along, that "Cl(64) is the smallest Real Clifford algebra for which we can reflexively identify each component Cl(8) with a vector in the Cl(8) vector space". So it all has something to do with space-time qubits and Bott periodicity and self-embeddings.<br /><br />Then there was a <a href="http://arcadianomegafunctor.blogspot.com/2014/03/supersymmetry-in-cmb.html">characteristically laconic post by Marni Sheppeard</a>, in which the idea is that "r" is about 1/5, and that this would be a ratio of... dimensions of certain Hilbert spaces, I think, that are relevant for her theory of mass generation in quantum gravity. In her paradigm, space-time is something like a big concatenation of morphisms between these vector spaces. For more, see her papers at vixra.<br /><br />My "contribution" to BICEP2 numerology is not going to be based on advanced math - though it does build on the observation that 0.2 = 1/5. My thought is just that this is also the ratio of baryonic matter to dark matter densities in the present-day universe. (I'd also like to acknowledge that work by A. Hattawi helped to fix this fact in my mind - that the OM/DM ratio is about 1/5.) So my question is, is there some theory in which this is not just a coincidence? Mitchellnoreply@blogger.com6tag:blogger.com,1999:blog-416737812331434120.post-1122070523707614112014-03-10T19:10:00.000-07:002014-03-10T19:10:17.307-07:00Various developmentsEmilio Torrente-Lujan has updated a <a href="http://arxiv.org/abs/1209.0474">tHWZ numerology paper</a> to include a number of new relations, and Stephen Adler has put out <a href="http://arxiv.org/abs/1403.2099">"SU(8) unification with boson-fermion balance"</a>, sketching a theory that would resemble N=8 supergravity, but without actually being supersymmetric. Further comments to come. Mitchellnoreply@blogger.com1tag:blogger.com,1999:blog-416737812331434120.post-24393304518021797652014-01-17T03:05:00.002-08:002014-01-21T02:04:16.369-08:00Coupling constants IIOne form of the <a href="http://arxiv.org/abs/1305.4208">LC&P sum rule</a> is<br /><br />2 λ + g<sup>2</sup>/4 + (g<sup>2</sup> + g'<sup>2</sup>)/4 + y<sub>t</sub><sup>2</sup>/2 ~ 1<br /><br />... based on their equation 2, and neglecting yukawa couplings for fermions other than the top quark.<br /><br />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 <br /><br />2 λ + g<sup>2</sup>/2 + g'<sup>2</sup>/4 ~ y<sub>t</sub><sup>2</sup>/2 ~ 1/2<br /><br />The "fermionic part" of this makes sense, if we recall that y<sub>t</sub> ~ 1. But the bosonic part<br /><br />2 λ + g<sup>2</sup>/2 + g'<sup>2</sup>/4 ~ 1/2<br /><br />... just considered by itself, seems to be very notable new numerology, connecting electromagnetic and weak couplings with the Higgs self-coupling λ.<br /><br /><b>edit</b>: 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<br /><br />2 λ ~ 1/2<br /><br />i.e. λ ~ 1/4. I noted <a href="http://snarxivblog.blogspot.com/2012/04/t-h-w-z.html?showComment=1364246790902#c556978571775223659">almost a year ago</a> that this is implied by the fact that the Higgs VEV / <a href="http://en.wikipedia.org/wiki/Electroweak_scale">electroweak scale</a> is approximately twice the Higgs boson mass.<br /><br /><b>edit #2:</b> Study of the literature (e.g. <a href="http://pdg.lbl.gov/2013/reviews/rpp2013-rev-higgs-boson.pdf">PDG 2013 Higgs review</a>) makes it clear that<br /><br />λ ~ 1/8<br /><br />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.<br /><br />(Or are we just dealing with different conventions? Remedial study of Higgs-sector basics is in order...)<br /><br />Mitchellnoreply@blogger.com2tag:blogger.com,1999:blog-416737812331434120.post-90890629047001823872014-01-14T18:46:00.000-08:002014-01-14T18:46:01.970-08:00t, H, W, Z and naturalnessThe lightness of the Higgs boson is one of the vexing issues in particle physics today. Why isn't it made heavy by virtual particles?<br /><br />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 <a href="http://arxiv.org/abs/1305.4208">the sum rule due to Lopez-Castro and Pestieau</a> (anticipated by <a href="http://snarxivblog.blogspot.com/2011/07/numerology-of-mr-nothing.html">Garces Doz</a>, and blogged by Andrew Oh-Willeke <a href="http://dispatchesfromturtleisland.blogspot.com/2013/08/higgs-numerology-from-lp-c-paper-and.html">1</a> <a href="http://dispatchesfromturtleisland.blogspot.com/2013/09/the-not-quite-boson-fermion-mass.html">2</a> <a href="http://dispatchesfromturtleisland.blogspot.com/2013/12/more-lc-p-papers.html">3</a>). <br /><br />It has a mild resemblance to the <a href="http://snarxivblog.blogspot.com/2013/08/t-h-w-z-weinberg-veltman.html">"Veltman condition"</a>, 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.<br /><br />However, the original form of the Veltman condition is specific to the unadorned standard model. Today, Ernest Ma - <a href="http://arxiv.org/abs/hep-ph/0612022">one of the few theorists to tackle the Koide formula</a> - 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; <i>skotos</i> means darkness, thus scotogenic, generated from the dark).<br /><br /><a href="http://arxiv.org/abs/1401.3284">The paper is here.</a> 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 <a href="http://arxiv.org/abs/1308.1242">here</a>.) <br /><br />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 <i>is</i> natural. Mitchellnoreply@blogger.com0tag:blogger.com,1999:blog-416737812331434120.post-70998395716400373282014-01-06T14:58:00.001-08:002014-01-06T14:58:31.164-08:00α-numerology from M-theoryThe fine-structure constant might be the most popular target of physics numerologists. <span class="st">α numerology has a long history, such as Eddington's efforts and Feynman's remark. It's a recurring topic in <a href="http://www.physicsforums.com/showthread.php?t=46055">this long thread</a> which might be the high point of Internet-era physics numerology. </span><br /><span class="st"><br /></span><span class="st">Today on vixra there is <a href="http://vixra.org/abs/1401.0037">an article</a> which speculates about how to obtain one of the numerological formulas for </span><span class="st">α, 4</span><span class="st">π<sup>3</sup>+</span><span class="st">π<sup>2</sup>+</span><span class="st">π</span>. 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 <i>l<sub>p</sub></i>-sized three-cycles on <i>G<sub>2</sub></i> holonomy manifolds". (<i>l<sub>p</sub></i> would be the Planck length.)<br /><br />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. <br /><br />I haven't really gone over Mulic's article (I note that he had a similar one <a href="http://arxiv.org/abs/hep-th/9908086">on arxiv</a> 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.<br /><br />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 <span class="s"></span><a href="http://vixra.org/author/a_garces_doz">Angel Garcés Doz</a>, 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. <br /><br />On the other side, consider this item of <a href="http://arxiv.org/abs/1005.3033">F-theory phenomenology</a> (via <a href="http://motls.blogspot.com/2010/05/fuzzy-f-theory.html">Lubos</a>). Here we have a genuine example of how a string-theory background geometry might determine a particular value of <span class="st">α: in this case, it's "</span>the number of fuzzy points" in "a non-commutative four-cycle" wrapped by a 7-brane. But the value of <span class="st">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. </span><br /><span class="st"><br /></span><span class="st">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 <a href="http://pseudomonad.blogspot.com/2010/05/m-theory-lesson-321.html">parametrization of the CKM matrix</a>, 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.</span>Mitchellnoreply@blogger.com10tag:blogger.com,1999:blog-416737812331434120.post-77599722999384023132013-12-05T22:06:00.000-08:002013-12-05T22:06:29.857-08:00MeVs and GeVs III 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 5<sup>3</sup>=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...<br /><br />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.<br /><br />It is tempting to suppose that some unknown physics has forced the Higgs to a critical value. In the <a href="http://snarxivblog.blogspot.com/2013/08/mevs-and-gevs.html">previous post</a> 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.<br /><br />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?<br /><br />It turns out that <a href="http://en.wikipedia.org/wiki/Extended_periodic_table#Feynmanium_and_elements_above_the_atomic_number_137">the problem</a> <a href="http://www.blogger.com/null">emerges </a>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.<br /><br />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. <br /><br />A search for "feynmanium higgs" turned up <a href="http://backreaction.blogspot.com/2012/01/molecei.html?showComment=1326807020020#c7019625395478148155">a blog comment by "Juan F."</a> 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 <a href="http://en.wikipedia.org/wiki/Magic_number_%28physics%29">"magic number" in nuclear physics</a>. Mitchellnoreply@blogger.com3