New string formulae for Higgs mass and cosmological constant

[mitchell porter]

TL;DR Summary

based on UV-IR duality; similarities to Veltman condition

https://arxiv.org/abs/2106.04622
Calculating the Higgs Mass in String Theory
Steven Abel, Keith R. Dienes
[Submitted on 8 Jun 2021]

In this paper, we establish a fully string-theoretic framework for calculating one-loop Higgs masses directly from first principles in perturbative closed string theories. Our framework makes no assumptions other than worldsheet modular invariance and is therefore applicable to all closed strings, regardless of the specific string construction utilized. This framework can also be employed even when spacetime supersymmetry is broken (and even when this breaking occurs at the Planck scale), and can be utilized for all scalar Higgs fields, regardless of the particular gauge symmetries they break. This therefore includes the Higgs field responsible for electroweak symmetry breaking in the Standard Model.

Notably, using our framework, we demonstrate that a gravitational modular anomaly generically relates the Higgs mass to the one-loop cosmological constant, thereby yielding a string-theoretic connection between the two fundamental quantities which are known to suffer from hierarchy problems in the absence of spacetime supersymmetry. We also discuss a number of crucial issues involving the use and interpretation of regulators in UV/IR-mixed theories such as string theory, and the manner in which one can extract an EFT description from such theories. Finally, we analyze the running of the Higgs mass within such an EFT description, and uncover the existence of a "dual IR" region which emerges at high energies as the consequence of an intriguing scale-inversion duality symmetry. We also identify a generic stringy effective potential for the Higgs fields in such theories. Our results can therefore serve as the launching point for a rigorous investigation of gauge hierarchy problems in string theory.

There are at least two properties of the Higgs mass one might hope to explain with such a fundamental calculation: its criticality, and its participation in the Veltman-like sum rule discovered by Lopez-Castro and Pestieau. Veltman's condition does rate a mention (after 6.11), and the running of the Higgs mass is discussed (figure 3, page 37), but at first glance the formulas look too complicated. On the other hand, the role of the supertrace is very promising - the supertrace is ideal for creating Veltman-like relations - and we are told directly (end of page 54) that "in string theory... the supertraces... are evaluated over the entire spectrum of string states and not merely the light states within the EFT". The LC&P sum rule deviates from the Veltman condition; perhaps this is a clue for how to embed it in string theory.

 

[ohwilleke]

Summary:: based on UV-IR duality; similarities to Veltman condition

https://arxiv.org/abs/2106.04622
Calculating the Higgs Mass in String Theory
Steven Abel, Keith R. Dienes
[Submitted on 8 Jun 2021]There are at least two properties of the Higgs mass one might hope to explain with such a fundamental calculation: its criticality, and its participation in the Veltman-like sum rule discovered by Lopez-Castro and Pestieau. Veltman's condition does rate a mention (after 6.11), and the running of the Higgs mass is discussed (figure 3, page 37), but at first glance the formulas look too complicated. On the other hand, the role of the supertrace is very promising - the supertrace is ideal for creating Veltman-like relations - and we are told directly (end of page 54) that "in string theory... the supertraces... are evaluated over the entire spectrum of string states and not merely the light states within the EFT". The LC&P sum rule deviates from the Veltman condition; perhaps this is a clue for how to embed it in string theory.

Criticality

Criticality really needs to be tweaked to adjust the beta function of the Higgs mass to reflect a massless spin-2 graviton in addition to usual SM beta function.

The difference between metastable and stable is quite modest and it would be a reasonable conjecture that expanding the beta function to account for quantum gravity could easily bring the Higgs boson mass to exactly the line between them.

It shouldn't be overwhelmingly difficult, in principle, to determine the beta function for the Higgs boson mass (and for that matter all of the SM beta functions) in this situation, which does not rely on any experimentally measured SM or GR constants and can be calculated (in theory, at least) exactly using plain vanilla assumptions about the graviton.

The argument in the notable predictions of the Higgs boson mass before the value was known was derived from an asymptotic safety gravity theory by Mikhail Shaposhnikova and Christof Wetterichb (2010) employed basically this line of reasoning and should perhaps be revisited.

Lopez-Castro and Pestieau

LC&P (i.e. that the sum of the square of the fundamental particle rest masses in the SM is equal to the square of the Higgs vev) is still consistent with the data at under 2 sigma.

The predominant sources of uncertainty in the LC&P relationship comparison to the data are the top quark and Higgs boson masses, with the top quark mass playing the larger part. All of the other uncertainties are negligible. The best fit is for the PDG values of both the top quark mass and the Higgs boson mass to be on the low side.

If you use the best fit values of fundamental particle masses from PDG, you can back out the Higgs boson mass, but it will be ruled out experimentally because the top quark mass uncertainty (which is actually quite low in relative terms compared to the other quark mass estimates, but is high in absolute terms because it is the largest SM fundamental particle mass) needs to pick up some of the slack for a good global fit.

Also, a more strict version of the LC&P relationship is not true. The sum of the square of the fundamental boson rest masses in the SM is slightly, but statistically significantly, more than half of the square of the Higgs vev, while the sum of the square of the fundamental fermion rest masses in the SM is slightly, but statistically significantly, less than the half of the square of the Higgs vev. So, there is not an exact symmetry between the fundamental fermion sector and the fundamental boson sector in the SM by this measure.

One approach would be to argue that at first order/tree level that the sum of the squares of the fundamental boson rest masses should be equal to half the square of the Higgs boson mass, and that the same should be true of the sum of the squares of the fundamental fermion rest masses. Then, you could look for a second (plus) order/loop level adjustment that shifts mass from the fermions to the bosons.

In the SM, the electromagnetic force coupling constant gets larger at higher energy scales, while the weak force and strong force get smaller at higher energy scales. All of the SM fundamental boson masses have beta functions as well. The Higgs vev is a function of the W mass divided by the weak force coupling constant. It could be that evaluating the LP&C relationship at other theoretically plausible energy scales (or some sort of average of all of energy scales) rather than pole masses could make it fit.

OTHER POSSIBILITIES

Observation of Global Electroweak Fits

Prior to the detection of the Higgs boson mass, its mass was estimated with global electroweak fits. Those predicted a Higgs boson mass significantly lower than was ultimately measured (at ca. 84 GeV with big error bars). If one understands what was wrong with the global electroweak fit theory one might be enlightened regarding why it has the mass that it does.

2W+Z = 2H

There is an argument advanced in the literature that sum of the Standard Model gauge boson rest masses (i.e. the sum of the masses of the W+, W- and Z boson masses) is equal to twice of the Higgs boson rest mass, at tree level with adjustments by some higher loop factor. See here, and here, and here. (Using this sum without adjustment produced a value about 4.4 sigma too high and so is strongly disfavored without loop level adjustments.)

Another Higgs mass property

The Higgs boson mass is very close to the mass which maximizes the branching fraction of the diphoton decay channel. This could have a deeper meaning rather than being a coincidence.

Entropy In Higgs Boson Mass Decays

The entropy of the Higgs boson decay probabilities distribution in the Standard Model (SM) is maximized for a Higgs mass value that is less than one standard deviation away from the current experimental measurement. (Source). There is a theoretical argument that this too is not just a coincidence.

 

[mitchell porter]

This paper is cited in Quanta Magazine's recent article on UV-IR mixing.

I'll also mention an old thread on supertrace sum rules - in particular see @fzero's final comment, for how the coefficients could be different than expected.

 

[mitchell porter]

On the Running of Gauge Couplings in String Theory (Abel, Dienes, Nutricati)

is a follow-up paper about this refined approach to the field-theory limit of a string theory model. In field theory, one is used to Feynman diagrams being integrals, often divergent, over the possible momenta of virtual particles. But one-loop string diagrams have an extra symmetry, modular invariance - it's a kind of arbitrariness in how you place coordinates along a closed string - which interchanges high-energy and low-energy states, implying a different way of organizing the Feynman integral. (At some point I would like to study the difference between their modular-invariant approach to vacuum energy, and the classic stringy approach as described by Witten on pages 42-43 here.)

Back in #1, I talked about the Lopez-Castro-Pestieau sum rule for the particle masses, which at present is just numerology, but which has an unusual resemblance to a "Veltman condition" known from supersymmetry. In #1, I speculated that these Abel-Dienes supertrace formulas might be able to produce the LC&P sum rule. This paper again offers tantalizing hope, with comments on pages 31-32 about how a Higgs mass correction of -1/12, which in field theory is just renormalized away, here must undergo "modular completion" to a function which does make a difference. Fans of the zeta function will recognize the controversial identity "1+2+3+ ... = -1/12" at work here, something employed in string theory to sum the zero point energies of all the modes of vibration of a closed string.

The authors say two more papers are coming (references 19 and 20).

 

[malawi_glenn]

The Higgs boson mass is very close to the mass which maximizes the branching fraction of the diphoton decay channel. This could have a deeper meaning rather than being a coincidence.

And near the maximum of BR(H->gg) ;)

 

[ohwilleke]

Fans of the zeta function will recognize the controversial identity "1+2+3+ ... = -1/12" at work here

I recognize that it is out there but I'm not a fan.

 

[mitchell porter]

The authors say two more papers are coming

One of the promised papers has arrived:

A New Non-Renormalization Theorem from UV/IR Mixing (Abel, Dienes, Nutricati)

On the surface, this is just technical progress in understanding the running of couplings in string theory. In a Kaluza-Klein field theory, when you get to high enough energies that the extra compact dimensions become relevant, you expect that couplings run according to a power law (see page 24). Here the authors argue that in string theory, you instead get a fixed point in the UV. Hopefully this refined understanding will eventually help in calculating the empirical properties of string vacua, and making the theory more testable.

At the mathematical level, there may be something going on that is of interest to number theorists. This paper is about the interplay between the "modular invariance" of closed strings (basically, the freedom to twist around coordinates on the string) and what happens when extra dimensions go from being small to large. This reminds me of a paper in which the Riemann hypothesis is encoded into the interplay of Kaluza-Klein modes and wrapped branes. That the Riemann hypothesis can be encoded in string theory is not that surprising - it is a statement of pure math and there are many ways to encode it into quantum dynamics - but there is some technical overlap between these papers (e.g. both mention "Eisenstein" series).

Another consideration here is "misaligned supersymmetry". This is a 30-year-old observation due to the second author (Keith Dienes), that even in non-supersymmetric string vacua, excited bosonic and fermionic states still cancel out in a certain way. If there is an excess of bosonic states at one level, there will be an overcompensating excess of fermionic states at the next level, and it goes back and forth indefinitely, with larger and larger excesses (e.g. see figure 2, page 9 in Dienes's 1994 paper). These oscillations remind me of the Mertens function that is intimately related to the Riemann zeta function.

Yet another recurring concept is that of a "trace formula". The trace of a square matrix is the sum of the entries along its main diagonal. This is equal to the sum of its eigenvalues, so any sum of all eigenvalues is potentially a trace formula. Trace formulas and zeta functions are often related. Here, there are "supertrace" formulas summing over all string states, which as noted above, resemble the theoretical "Veltman condition" and the empirical Lopez-Castro & Pestieau sum rule.

Obviously it would be stunning if relations like the LC&P sum rule could be shown to be a generic consequence of the supertrace constraints obeyed even by non-supersymmetric string vacua. At this point, I think no one knows enough to say yes or no to that proposition. But it's something to look forward to.

P.S. As a final comment: all these results only pertain to closed strings (which are the ones for which modular invariance is meaningful). What about open strings? Lubos has some old comments on how open strings fit into this picture.