# Top, Higgs, Higgs VEV relation from conformal symmetry?

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1. Sep 1, 2014

### mitchell porter

http://arxiv.org/abs/1409.0492
Is the Standard Model saved asymptotically by conformal symmetry?
A.Gorsky, A.Mironov, A.Morozov, T.N.Tomaras
(Submitted on 1 Sep 2014)
It is pointed out that the top-quark and Higgs masses and the Higgs VEV satisfy with great accuracy the relations 4m_H^2=2m_T^2=v^2, which are very special and reminiscent of analogous ones at Argyres - Douglas points with enhanced conformal symmetry. Furthermore, the RG evolution of the corresponding Higgs self-interaction and Yukawa couplings \lambda(0)=1/8 and y(0)=1 leads to the free-field stable point \lambda(M_Pl)= \dot \lambda(M_Pl)=0 in the pure scalar sector at the Planck scale, also suggesting enhanced conformal symmetry. Thus, it is conceivable that the Standard Model is the low-energy limit of a distinct special theory with (super?) conformal symmetry at the Planck scale. In the context of such a "scenario" one may further speculate that the Higgs particle is the Goldstone boson of (partly) spontaneously broken conformal symmetry. This would simultaneously resolve the hierarchy and Landau pole problems in the scalar sector and would provide a nearly flat potential with two almost degenerate minima at the electroweak and Planck scales.

2. Sep 2, 2014

### torsten

Thanks mitchell for the interesting paper. It seems one need a non-perturbative explanation of the relations (5).

3. Sep 2, 2014

### ohwilleke

I think it is more likely that the observed relationship is really an approximation of the relationships

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.

To be concrete about it: v^2 is about 60,628.23(1) GeV^2/c^4. Half of this number is is about 30,314.

To three significant digits, the square of the (rest pole) mass of the top quark is about 30,000(250), the square of the (rest pole) mass of the b quark is about 17.5(2), the square of the mass of the tau lepton is about 3.154(2), the square of the (rest pole) mass of the charm quark is about 1.7(2), and so on. Thus, the margin of error in the top quark measurement exceeds the sum of square of all of the other fermion rest masses at the pole mass value. This formula is a slightly better fit than the current measured value of the top quark mass alone.

The proposed value of the top quark from the September 1, 2014 preprint formulation is 174.11 GeV which is a bit on the high side relative to current experimental measurements.

On the boson side, photons and gluons have no rest mass. The Higgs boson mass squared is about 15,700(125). The Z boson mass squared is about 8315.17(2) and the W boson mass squared is about 6,460(2). The sum of these three values is about 30,475(126), which is a closer fit to v^2/2 than 2*H^2 as proposed, although both are within the margin of error.

The proposed value of the Higgs boson from September 1, 2014 preprint formulation is a Higgs boson mass of 125.11 GeV which is on the light side.

Less restrictively, sum((Fi(^2)+sum((Bj)^2)=v^2, which is off at the squares of currently experimental values by about 0.14%, is a significantly better fit than either the fermion fit or the boson fit separately under either formula, although perhaps spuriously given the margins of error in the top quark and Higgs boson masses (which overwhelm all of the other uncertainties). In other words, the sum of the appropriate Yukawas and Higgs field couplings of the bosons is unitary.

The fact that this is probably unitary, in my view, is 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. The former is an unexpected statement about the entire fundamental particle rest mass creation mechanism of the Standard Model, and this conjecture, if true, dramatically limits the possibility that the set of fundamental particle mass constants in the Standard Model is incomplete and reduces the number of independent experimentally measured parameters of the Standard Model by one.

More precisely, it suggests that Standard Model fundamental particles derive their masses entirely from the Standard Model Higg mechanism, while any non-Standard Model fundamental particles heavy enough not to be detected so far under the assumptions of LHC particle exclusion models, must arise from a completely different mechanism (e.g. a different scalar boson and its field that do not couple to Standard Model particles, e.g. in a dark sector).

If the fermion mass squared sum side and boson mass squared sum side are in fact exactly equal, the number of independent experimentally measured parameters of the Standard Model falls by two, and the largeness of the top quark mass and Higgs boson mass seems to flow naturally from the fact that these are residual quantities that must total up to the V^2/2 in each case. It also suggests that the hierarchy problem might be due to a fermion/boson symmetry different from the one present in SUSY theories without having to have superpartners to balance the scales.

The latter is more of a side observation about the side effects of Nature's particular choice of the largest fermion mass and largest boson mass out of the fifteen particle mass constants in the Standard Model.

Current experimental data favor a scenario in which sum((Bj)^2)-sum((Fi(^2)=425. In other words, the boson side is about 0.65% over and the fermion side is about 0.65% under equality. But, a result with a sufficiently heavy top and a sufficiently light Higgs to even the scales is well within experimental bounds.

Last edited: Sep 2, 2014
4. Sep 2, 2014

Wow.

5. Sep 2, 2014

### ohwilleke

"It is pointed out that the top-quark and Higgs masses and the Higgs VEV satisfy with great accuracy the relations 4m_H^2=2m_T^2=v^2". According to the paper this "great accuracy", which is called "miraculous accuracy" in the body text, involved a three sigma discrepancy in the top quark mass to Higgs boson mass ratio. Personally, I have a higher expectation for my miracles, although the Higgs boson mass that they use is 125.66 and the current experimental best fit is more like 125.35, which improves their fit a bit.

6. Sep 19, 2014

### mitchell porter

In comment #3, ohwilleke refers to the unexplained (and little-noted) sum rule of Lopez Castro & Pestieau. His idea that this provides the real context of the relation touted by Gorsky et al is plausible to me. In fact, we can go further and include the very approximate relation mH^2 = mW^2 + mZ^2 due to S. Vik of Wilfrid Laurier University, to produce a refinement of the Lopez-Castro-Pestieau sum rule in which the Higgs vev squared is broken down as follows: half of it is top mass squared, a quarter of it is Higgs mass squared, and the other quarter of it is W mass squared plus Z mass squared. (For the skeptical reader, I would remark that sum rules connecting squares of masses are already known in physics, such as the Gell-Mann-Okubo formula.)

Could this extended set of relations fit into the paradigm of Gorsky et al (the paper which started this thread)? I don't know, but I can express at least one doubt about their paradigm: the quantities they want to explain include the top quark mass, but the top quark gets its mass from a Yukawa coupling to the Higgs. Gorsky et al talk about what happens at Argyres-Douglas fixed points, but the fermion there gets its mass in quite a different way, I think.

I also want to mention another recent paper, "Holomorphy in the Standard Model Effective Field Theory", which might actually be worth far more attention than Gorsky et al, given that it involves deep mathematical study of a canonical standard-model extension (all the "dimension 6" operators that can be added). Numerous mysterious patterns turn up, including mH, mW, mZ mass relations. If I try to sum it up... it suggests that the standard model is the low-energy limit of an unknown special class of theory, and that the specific values of standard model parameters tell us about properties of the deeper theory, in a way more subtle than the usual running of parameters in a grand unified theory.

7. Jun 18, 2016

### nikkkom

How well does this hold up two years later? Are there better top mass measurements?

8. Jun 22, 2016

### ohwilleke

The relationships still holds, but the uncertainty in the mass of the top quark continues to dwarf any other consideration in attempting to verify the relationships. The reality experimentally is that reducing the uncertainty in the top quark mass measurement is going to be much harder than reducing the uncertainty in the Higgs boson measurement, which could easily improve by a factor of ten or so in the next few years, something that might take a decade or two to do with the top quark measurement. So it is easier to make statements about the boson side of the relationship than the quark side of the relationship.

The particle data group estimate for the Higgs boson mass is currently 125.09 +/- 0.24 GeV. http://pdglive.lbl.gov/Particle.action?node=S126 and I am not aware of any recent experimental results from the LHC (or elsewhere) that are not included in that value that pull the value significantly up or down from the value. Thus, the Higgs boson mass squared is now roughly 15647(120) GeV^2. This is moving in the right direction to support the relationship compared to values two years ago. Equating the sum of the square of the fundamental boson masses to half of the square of the Higgs vev would imply a Higgs boson mass of about 124.66 GeV, which is a little less than two sigma lighter than the current best estimate of the Higgs boson mass.

If the the sum of the square of the boson masses equals the sum of the square of the fermion masses the implied top quark mass is 174.03 GeV if pole masses of the quarks are used, and 174.05 GeV if MS masses at typical scales are used. Using half of the Higgs vev as a measure implies a top quark mass of 174.04 GeV. This is less than two sigma heavier than the current global average value from the particle data group.

The expected value of the top mass from the formula that the sum of the square of each of the fundamental particle masses equals the square of the Higgs vacuum expectation value (a relaxed formula that does not require that the sum of fermion masses squared equals the sum of boson masses squared), given the state of the art Higgs boson mass measurement (and using a global fit value of 80.376 GeV for the W boson rather than the PDG value of 80.385 +/- 0.015 GeV) is 173.73 GeV (because a somewhat high value of the Higgs boson mass permits a somewhat lower value for the top quark mass).

Some recent measurements of the top quark mass are in line with these relatively high top quark masses implied by this relationship. Others are significantly lower. But, all of the individual measurements have rather large error bars.

As of March 9, 2016, the latest lepton decay based measurement of the top quark mass from CMS at the LHC was 173.8 -1.7+1.8 GeV. This tends to pull up the LHC and world average measurement of this key parameter of the Standard Model. On March 22, 2016, another new CMS measurement is here: http://arxiv.org/abs/1603.06536 with an independent methodology found that ""A top quark mass of 173.68 +/- 0.20 (stat) +1.58 -0.97 (syst) GeV is measured."

As of December 2015, the best available estimate of the mass of the top quark from the Large Hadron Collider (LHC) combining data from both the CMS and ATLAS experiments was 172.38 +/- 0.66 GeV. The final Tevatron mass measurement for the top quark was 174.34 +/- 0.64 GeV. This brings the error weighted world average mass measurement of the top quark to about 173.35 GeV, which is consistent with both the LHC measurement and the Tevatron measurement at the 1.5 sigma level. The previous top quark mass estimate from ATLAS (as of April of 2015) was 172.99 +/- 0.91 GeV. The latest combined LHC measurement excluding that ATLAS estimate was 173.34 +/- 0.76 GeV. Thus, the LHC mass measurement is trending down.

There have been a couple new top quark mass measurement papers announced at the LHC this summer, but I have not had an opportunity to review and analyze them due to a busy work schedule. There was a CMS paper in May http://arxiv.org/pdf/1605.02890.pdf and an ATLAS paper in June http://arxiv.org/abs/1606.02179 and I'm sure that there were one or two others as well since March. A quick glance at these papers suggests that recent top quark mass measurements are trending towards the low end of the range, although there is quite a bit of tension between individual measurements by different means and by different experiments.

Last edited: Jun 22, 2016
9. Jun 22, 2016

### kodama

if true that "Thus, it is conceivable that the Standard Model is the low-energy limit of a distinct special theory with (super?) conformal symmetry at the Planck scale."

are there any theories of QG that respect "(super?) conformal symmetry at the Planck scale"

10. Jun 27, 2016

### mitchell porter

Conformal gravity. It emerges in the twistor string, one version of which (see section 5 here) has 3 colors and 6 quark flavors like the standard model, and $\mathcal{N}=2$ supersymmetry like the superconformal field theories considered in this paper.

11. Jun 27, 2016

### kodama

how well accepted is it?

12. Jul 2, 2016

### mitchell porter

Conformal gravity by itself is not unitary when quantized. However, it contains general relativity as a subsector, and unlike GR it is renormalizable. So it has been studied (especially by the new twistor theorists) as a pathway to quantum GR.

Also, the ideas in this paper do not demand the use of conformal gravity. You could just have a conformal field theory coupled to ordinary gravity.

13. Jul 2, 2016

### kodama

any relation to asymptotically safe gravity

14. Jul 12, 2016

### mitchell porter

Ohta and Percacci seem to say that conformal gravity is asymptotically safe! They say (end of section 7.1) the fixed point is similar to that claimed for general relativity... I still don't know whether to believe GR is asymptotically safe, but the AS program does keep advancing.

15. Nov 5, 2016

### mitchell porter

The only paper I've seen, that explains the first of these relations (mt = √2 mH):

https://arxiv.org/abs/1401.3311
Strong dynamics behind the formation of the 125 GeV Higgs boson
M.A. Zubkov
(Submitted on 14 Jan 2014 (v1), last revised 20 Mar 2014 (this version, v4))
We consider the scenario, in which the new strong dynamics is responsible for the formation of the 125 GeV Higgs boson. The Higgs boson appears as composed of all known quarks and leptons of the Standard Model. The description of the mentioned strong dynamics is given using the ζ - regularization. It allows to construct the effective theory without ultraviolet divergences, in which the 1/Nc expansion works naturally. It is shown, that in the leading order of the 1/Nc expansion the mass of the composite h - boson is given by Mh=mt/√2≈125 GeV, where mt is the top - quark mass.

Zubkov's starting point is his equation II.4. There is an unspecified BSM strong force which induces four-fermion interactions with a form factor. In II.7, he specifies the form factor for interactions involving the top quark... From there - and I haven't worked through this - in III.2, he arrives at a relation between mH and mt. In that formula, Nc = 3, the number of colors in QCD, and Ntotal = 24, the number of SM fermion species, if each color of a quark is counted separately.

It was a toss-up whether to report this paper here or in the thread on Kahana and Kahana's 1993 prediction of top and Higgs masses. There are some superficial similarities, e.g. the use of a four-fermion interaction that involves all the SM fermions. On the other hand, Kahana and Kahana have no new physics until something like the usual GUT scale, whereas Zubkov says something new should show up around 5 TeV. In the end I posted here, because although Gorsky et al write about the behavior of beta functions in the high UV, the Argyres-Douglas theory that is their inspiration is typically an infrared phenomenon, and Zubkov is in effect offering an IR explanation.

16. Feb 5, 2017

### mitchell porter

More Russians... From 2012, "Condensate Mechanism of Conformal Symmetry Breaking and the Higgs Boson" (2013 talk), and today by the same authors, "Radiative breaking of conformal symmetry in the Standard Model". Their idea is that, instead of having an explicit quadratic mass term in the Higgs potential, it can be induced by the QCD top quark condensate. They also posit a feedback loop between top condensate and Higgs condensate, whereby the scale of the latter is set by the constituent quark mass scale.

I also recommend the 2002 review "Strong Dynamics and Electroweak Symmetry Breaking", which covers technicolor and the NJL model.

17. Feb 6, 2017

### arivero

Each time I read the comments about relationships in the electroweak set (W,Z,Higgs,Top) -or v instead of Top-, I am remembered of old de Vries relationship here in PF in November 2004, and where Higgs and Top happen to be the imaginary mass solutions, or negative mass square, of the equation for W and Z.

Namely, Hans had postulated $$M^2_s= \frac 12 m^2 (- s(s+1) + \sqrt { s^2 (s+1)^2 + 4 s (s+1)) })$$
which in the limit of high "spin" approaches to $m^2$. You can notice that the terms in the postulate are simple combination of the two casimirs of a generic representation $(m,s)$ of Poincare Group, $C_1 \equiv m^2$ and $C_2 \equiv -m^2 s (s+1)$ but I never got into their logic. Still, you can use them to rewrite the equation as$$M^2_s M^2_s - M^2_s C_2 + C_1 C_2=0$$

Well, anyway, the point was that when the mass of Z, 91.1874 GeV, was assigned to the s=1 solution, then automagically the mass of the W did appear in as $M^2_{s=1/2,+}=(80.372 GeV)^2$ and it made a nice "mnemonic" for Weinberg angle. At that time, in 2006, we noticed that the negative solutions were $$M^2_{s=1,-}=-(176.15 GeV)^2$$ and $$M^2_{s=1/2,-}=-(122.38 GeV)^2$$ but the "prediction" for the top was not so nice as the one of the W, and the other mass was irrelevant, there was no particle near there.

Nowadays, we could then look at the equations, say that $M_W^2 - M_H^2 = m^2 s (s+1), \; s=1/2$ and $M_Z^2 - M_t^2 = m^2 s (s+1), \; s=1$ and then wipe out $m^2$ to claim a relationship for the quotient of differences
$${M_W^2 - M_H^2 \over M_Z^2 - M_t^2}= \frac 38$$
and of course also the same for the quotient of products. It would be amazing if some of these surface in the literature at the end :-)

Last edited: Feb 6, 2017
18. Feb 6, 2017

### mitchell porter

If I tried to make a theory for that, I might study its compatibility with the model in the Kahana thread, since that model includes weak interactions. I'd also think about whether we are somehow dealing with superfields, in order to explain those wrong spins. (This would require the top superpartner to be spin 1 rather than spin 0, as if the top were a gaugino.) And I would try to understand Hans de Vries's physical concept which led to the formula.

However, it's late in the day here, so I will just perform a little trick of my own (and hope I am not making a stupid mistake out of tiredness). Suppose we evaluate your final formula using the approximations $M_t \sim 2M_Z$, $M_H \sim \sqrt 2 M_Z$, $M_W = M_Z cos \theta_W$. Then we get that $cos^2 \theta_W \sim 7/8$. Meanwhile, in reality, $sin^2 \theta_W \sim 2/9$, and thus $cos^2 \theta_W \sim 7/9$.

19. Mar 15, 2017

### mitchell porter

Karananas and Shaposhnikov, "Gauge coupling unification without leptoquarks". Their basic idea is to start with a GUT like SU(5) or SO(10), and then to solve all its problems by imposing constraints.

The concrete example they give is an SU(5) gauge theory, with three generations of fermions in the 5* and 10 representations, and scalars in the 5 and 24 representations. The way this normally works, the 24 breaks SU(5) to the SM gauge group, and then the 5 breaks the SM gauge group to SU(3) x U(1)em. But the "leptoquark" (X and Y) gauge bosons in SU(5), though now superheavy, still exist and can cause proton decay; and the 5 needs to be split into a light doublet (the SM Higgs) and a heavy triplet.

The theory that K & S use is still just this same SU(5) gauge theory; but they also impose constraints, so that the quantum states are built on a highly restricted space of classical field configurations. Eqn 9 and 11 constrain the 24 scalar and the X and Y bosons; eqn 12 constrains the 5. There is also a fine-tuning of parameters (eqn 13), but that part is not unusual.

The artificiality of their procedure may be seen in figure 1: the couplings unify at a GUT scale, but then just "pass through" each other, and diverge at still higher scales. They have two ideas for how to fix this. One is for higher-dimensional operators (eqn 14) to shift the running of the couplings, so that they unify at the Planck scale. The other, if I understand it correctly, is for the couplings to remain unified in a scale-invariant phase above the GUT scale - see figure 2. This option is developed in section 4.

Quantum constraints can be tricky, and I would be completely unsurprised to discover that the procedures in this paper are significantly altered or even invalidated, at the quantum level. Even if they are valid, one would like to motivate them somehow. Eqn 9 is likened to something from a sigma model, eqn 11 to "coset constructions". Perhaps it could all come from something like Peter West's nonlinear E11 symmetry.

20. May 24, 2017 at 7:32 AM

### mitchell porter

From "A QCD Lagrangian including renormalizable NJL terms" by A. Cabo: "A sort of Top condensate model might be the effective action for massless QCD. In it, a Top quark condensate... could play the role of the Higgs field... In a first step, the six quarks could get their masses thanks to a flavour symmetry breaking determined by the quark and gluon condensates. Afterwards, the electron, muon and tau leptons, would receive their intermediate masses thanks to radiative corrections mediated by the mid strength electromagnetic interactions with quarks. Finally, the only weak interacting character of the three neutrinos with all the particles, could determine their even smaller mass values."

In an NJL framework, particle mass shows up e.g. as a quantity in a propagator, and the value of the mass is whatever allows the propagator to be well-defined. This can produce some nice numerology, but since we are now in an era when the Higgs and its couplings can be measured, one has to have something in the model (some bound state) that behaves like the Higgs. Cabo seems to envisage, that the Higgs is really toponium, an excitation of a top quark condensate, and that the SM yukawas will be effective couplings of the fermions with that condensate, perhaps mediated by loop effects and interactions with other condensates.

Koide's yukawaon models involve many condensates (many vevs), and he enforces desired relations between the vevs through a superpotential (look for references to SUSY vacuum conditions). I have no idea how far one could go towards engineering a desired set of mass relations, by combining NJL and superpotential techniques, but it would be interesting to know.