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ohwilleke
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I. Background
The magnetic moment of the muon, g, is predicted by the Standard Model, to be equal to 2 and a bit more, with the quantity that we look at being g-2. We have both experimental measurements and theoretical predictions that are close to each other to many significant digits, but have a roughly 3 sigma discrepancy. More precisely:
A. The Current Experimental Result
The most definitive measurement of the anomalous magnetic moment of the muon (g-2) was conducted by the Brookhaven National Laboratory E821 experiment which announced its results on January 8, 2004.
This measurement was
which differs somewhat from a precisely calculated theoretical value. In units of 10-11 and combining the errors in quadrature, the experimental result was:
E821 116 592 091 ± 63
Dividing the total value by the error gives an error of 540 parts per million.
B. The Current Theoretical Prediction
The current state of the art theoretical prediction from the Standard Model of particle physics regarding the value of muon g-2 in units of 10-11 requires considering contributions to the value from quantum electrodynamics (QED), electroweak contributions other the QED (EW), and a hadronic contribution involving quantum chromodynamics (QCD) that is conventionally broken into two components: HVP and HLbL.
QED 116 584 718.95 ± 0.08
EW 153.6 ± 1.0
HLbL 105 ± 26
HVP 6 850.6 ± 43
Total SM 116 591 828 ± 49
The main contribution to the overall absolute value of muon g-2 comes by far from QED, which is known to five loops (tenth order) and has a small, well-understood uncertainty. Sensitivity at the level of the electroweak (EW) contribution was reached by the E821 experiment.
The hadronic contribution dominates the uncertainty (0.43 ppm compared to 0.01 ppm for QED and EW grouped together). This contribution splits into two categories, hadronic vacuum polarization (HVP) and hadronic light-by-light (HLbL).
The HVP contribution dominates the correction, and can be calculated from e + e − → hadrons cross-section using dispersion relations. The HLbL contribution derives from model-dependent calculations.
Lattice QCD predictions of these two hadronic contributions are becoming competitive, and will be crucial in providing robust uncertainty estimates free from uncontrolled modeling assumptions. Lattice QCD predictions have well-understood, quantifiable uncertainty estimates. Model-based estimates lack controlled uncertainty estimates, and will always allow a loophole in comparisons with the SM.
In other words, unsurprisingly, almost all of the uncertainty in the theoretical prediction comes from the QCD part of the calculations. One part of the QCD calculation (HVP) has a precision of ± 0.6% (roughly the precision with which the strong force coupling constant is known), the other part of the QCD calculation (hadronic light by light) has a precision of only ± 25%.
The key point is that theoretically, in the Standard Model, muon g-2 is globally sensitive to the entire particle content of the Standard Model and all of its coupling constants. So, it is a probe
C. Experiment and Theoretical Prediction Compared
The experimental and theoretical values agree up to the first eight digits after the decimal point.
Disregarding the significant digits of the experimental value and theoretical prediction that agree, in units of 10-11 (muon g-2 is a dimensionless number) you are left with:
Experiment: 2 091 ± 63
Theory: 1 828 ± 49
Experiment-Theory= 263 where a one sigma gap is 83, so the discrepancy is so there is roughly a 3.2 sigma tension between experiment and theory at this time.
A direct calculation of the gap and the margin of error from a 2013 paper from the Snowmass on the Mississippi Charge Lepton paper at Equations 4.3-4.5 on page 35 sets forth a discrepancy between theory from a pair of 2011 papers and experiment from BNL E821 (2006) of 287 ± 80 *(10-11), although some of the difference between this number and the one I use above may be due to an updated theoretical calculation and theoretical margin of error. (At any rate they're very close for the purposes relevant to this post.)
A 2011 paper has suggested that most of the discrepancy was between theory and experiment that we no observe is probably due to errors in the theoretical calculation.
II. Hypothetical Situation
Suppose that both the experimental value and the theoretical prediction improve by a factor of 10 at some time in the medium term future and that the new results confirm each other at the one sigma level (about twice the improvement forecast with efforts currently underway in the Fermilab E989 experiment).
For example, suppose that the parallel numbers to the comparison above in on New Year's Eve 2024 is (in units of 10-11):
Experiment: 116 592 005
Theory: 116 591 998
leading to a difference between them of is 7*10-11 where a one sigma gap is 8 in the last digit.
This would be an agreement in the first ten digits after the decimal point.
Thus all tension between the experimental measurement and the theoretical measurement has disappeared in a measurement 10 times more precise than the current status quo.
III. Questions
This leads to several closely related questions.
My understanding is that the muon g-2 limitations on supersymmetry are particularly notable because unlike limitations from collider experiments, the muon g-2 limitations tend to cap the mass of the lightest supersymmetric particle, or at least to strongly favor lighter sparticle masses in global fits to experimental data of SUSY parameters. As a paper from 2013 noted:
Similarly, the theoretical "preference of the global fits for light super particle masses is driven by one measurement: the long standing 3 sigma excess in the muon anomalous magnetic moment." See here (2008) (usually favoring less than 1 TeV see also, e.g. this 2015 meta-analysis from ATLAS).
But, of course, as of 2017, no SUSY particles have been discovered, sparticles with a mass of the order 100 GeV have been largely excluded by the LHC, very little parameter space of less than 1 TeV is left from direct LHC search exclusions, and in my hypothetical, there is a much smaller discrepancy between experiment and theory leaving less room for new physics contributions.
1. Is my understanding that a tight fit between experimental muon g-2 and the Standard Model theoretical value imposes a cap on the size of new physics particles correct, or could their also be very high energy scale new particles that basically decouple from the muon g-2 calculation?
(I thought I recalled Jester stating at his blog that the existing muon g-2 measurement naively implied no new physics below the high tens of TeVs but can't find the reference with a word search, so if he said that he did so using key words that I'm not thinking to search with, or I may be confusing this with some other global constraint on BSM physics, such as this post which also considers the constraints from the Higgs boson mass measurement to get a naive 10 TeV floor.)
2. Assuming that the hypothetical above is true, what impact would this have on new physics, in general, and in particular on SUSY models?
3. Would a hypothetical muon g-2 result such as this one generically rule out SUSY models when taken together with existing (as of 2017 publications) LHC exclusions?
4. Wouldn't this hypothetical furthermore generically rule out models where new physics particles interact via Standard Model forces as ordinary Standard Model scale coupling constants and charges (as opposed to particles that only interact via new forces or have tiny non-integer strong force or electro-weak quantum numbers)? For example, what would this do to lepto-quark models?
5. What kinds of new physics models could escape the implications of a very tight fit between measured muon g-2 and the Standard Model theoretical prediction of that value?
6. If improved muon g-2 measurements and theoretical calculations can clarify globally what kind of new physics could be out there, doesn't that suggest that a decision on a next collider should be postponed at least until we have the Fermilab E989 experiment results, so we can gauge how likely a "nightmare scenario" would be with a new collider?
(In other words, does it make sense to build a new collider if muon g-2 data and existing LHC data strongly disfavor new physics in the energy scales above the peak LHC scale and up to the peak new collider energy scale?)
7. If we had a theory of quantum gravity, would it enter into the muon g-2 calculation? If so, would the order of magnitude of any quantum gravity effect be big enough to be material relative to the margin of error in the existing experimental measurements and theoretical calculations?
(Obviously, we do not have a theory of quantum gravity, but we know, from GR, the magnitude of gravitational forces generally and there is a considerable and not terribly contentious literature of the magnitude of a quantum gravity coupling constant in many of the more conventional efforts to construct a quantum gravity theory. Presumably, this would make to conceivably possible to get a back of napkin estimate of the order of magnitude of an quantum gravity adjustment even if the exact value of such an adjustment was model dependent and had a considerable degree of uncertainty.)
IV. Closing Observation
It seems naively to me like a tight fit between the experimental and theoretically calculated values for muon g-2 in the Standard Model has the potential to globally rule out in a not very model dependent way, almost all SUSY theories and a whole lot of other popular BSM theories leaving only a very, very few such theories in the running, with those theories carefully constructed around this constraint.
And, it also seems to me like this possibility is one that could very realistically happen in the medium term future.
But, this seems too "good" to be true, so I'm wondering if I'm missing something important in this analysis.
The magnetic moment of the muon, g, is predicted by the Standard Model, to be equal to 2 and a bit more, with the quantity that we look at being g-2. We have both experimental measurements and theoretical predictions that are close to each other to many significant digits, but have a roughly 3 sigma discrepancy. More precisely:
A. The Current Experimental Result
The most definitive measurement of the anomalous magnetic moment of the muon (g-2) was conducted by the Brookhaven National Laboratory E821 experiment which announced its results on January 8, 2004.
This measurement was
E821 116 592 091 ± 63
Dividing the total value by the error gives an error of 540 parts per million.
B. The Current Theoretical Prediction
The current state of the art theoretical prediction from the Standard Model of particle physics regarding the value of muon g-2 in units of 10-11 requires considering contributions to the value from quantum electrodynamics (QED), electroweak contributions other the QED (EW), and a hadronic contribution involving quantum chromodynamics (QCD) that is conventionally broken into two components: HVP and HLbL.
QED 116 584 718.95 ± 0.08
EW 153.6 ± 1.0
HLbL 105 ± 26
HVP 6 850.6 ± 43
Total SM 116 591 828 ± 49
The main contribution to the overall absolute value of muon g-2 comes by far from QED, which is known to five loops (tenth order) and has a small, well-understood uncertainty. Sensitivity at the level of the electroweak (EW) contribution was reached by the E821 experiment.
The hadronic contribution dominates the uncertainty (0.43 ppm compared to 0.01 ppm for QED and EW grouped together). This contribution splits into two categories, hadronic vacuum polarization (HVP) and hadronic light-by-light (HLbL).
The HVP contribution dominates the correction, and can be calculated from e + e − → hadrons cross-section using dispersion relations. The HLbL contribution derives from model-dependent calculations.
Lattice QCD predictions of these two hadronic contributions are becoming competitive, and will be crucial in providing robust uncertainty estimates free from uncontrolled modeling assumptions. Lattice QCD predictions have well-understood, quantifiable uncertainty estimates. Model-based estimates lack controlled uncertainty estimates, and will always allow a loophole in comparisons with the SM.
In other words, unsurprisingly, almost all of the uncertainty in the theoretical prediction comes from the QCD part of the calculations. One part of the QCD calculation (HVP) has a precision of ± 0.6% (roughly the precision with which the strong force coupling constant is known), the other part of the QCD calculation (hadronic light by light) has a precision of only ± 25%.
The key point is that theoretically, in the Standard Model, muon g-2 is globally sensitive to the entire particle content of the Standard Model and all of its coupling constants. So, it is a probe
C. Experiment and Theoretical Prediction Compared
The experimental and theoretical values agree up to the first eight digits after the decimal point.
Disregarding the significant digits of the experimental value and theoretical prediction that agree, in units of 10-11 (muon g-2 is a dimensionless number) you are left with:
Experiment: 2 091 ± 63
Theory: 1 828 ± 49
Experiment-Theory= 263 where a one sigma gap is 83, so the discrepancy is so there is roughly a 3.2 sigma tension between experiment and theory at this time.
A direct calculation of the gap and the margin of error from a 2013 paper from the Snowmass on the Mississippi Charge Lepton paper at Equations 4.3-4.5 on page 35 sets forth a discrepancy between theory from a pair of 2011 papers and experiment from BNL E821 (2006) of 287 ± 80 *(10-11), although some of the difference between this number and the one I use above may be due to an updated theoretical calculation and theoretical margin of error. (At any rate they're very close for the purposes relevant to this post.)
A 2011 paper has suggested that most of the discrepancy was between theory and experiment that we no observe is probably due to errors in the theoretical calculation.
II. Hypothetical Situation
Suppose that both the experimental value and the theoretical prediction improve by a factor of 10 at some time in the medium term future and that the new results confirm each other at the one sigma level (about twice the improvement forecast with efforts currently underway in the Fermilab E989 experiment).
For example, suppose that the parallel numbers to the comparison above in on New Year's Eve 2024 is (in units of 10-11):
Experiment: 116 592 005
Theory: 116 591 998
leading to a difference between them of is 7*10-11 where a one sigma gap is 8 in the last digit.
This would be an agreement in the first ten digits after the decimal point.
Thus all tension between the experimental measurement and the theoretical measurement has disappeared in a measurement 10 times more precise than the current status quo.
III. Questions
This leads to several closely related questions.
My understanding is that the muon g-2 limitations on supersymmetry are particularly notable because unlike limitations from collider experiments, the muon g-2 limitations tend to cap the mass of the lightest supersymmetric particle, or at least to strongly favor lighter sparticle masses in global fits to experimental data of SUSY parameters. As a paper from 2013 noted:
Similarly, the theoretical "preference of the global fits for light super particle masses is driven by one measurement: the long standing 3 sigma excess in the muon anomalous magnetic moment." See here (2008) (usually favoring less than 1 TeV see also, e.g. this 2015 meta-analysis from ATLAS).
But, of course, as of 2017, no SUSY particles have been discovered, sparticles with a mass of the order 100 GeV have been largely excluded by the LHC, very little parameter space of less than 1 TeV is left from direct LHC search exclusions, and in my hypothetical, there is a much smaller discrepancy between experiment and theory leaving less room for new physics contributions.
1. Is my understanding that a tight fit between experimental muon g-2 and the Standard Model theoretical value imposes a cap on the size of new physics particles correct, or could their also be very high energy scale new particles that basically decouple from the muon g-2 calculation?
(I thought I recalled Jester stating at his blog that the existing muon g-2 measurement naively implied no new physics below the high tens of TeVs but can't find the reference with a word search, so if he said that he did so using key words that I'm not thinking to search with, or I may be confusing this with some other global constraint on BSM physics, such as this post which also considers the constraints from the Higgs boson mass measurement to get a naive 10 TeV floor.)
2. Assuming that the hypothetical above is true, what impact would this have on new physics, in general, and in particular on SUSY models?
3. Would a hypothetical muon g-2 result such as this one generically rule out SUSY models when taken together with existing (as of 2017 publications) LHC exclusions?
4. Wouldn't this hypothetical furthermore generically rule out models where new physics particles interact via Standard Model forces as ordinary Standard Model scale coupling constants and charges (as opposed to particles that only interact via new forces or have tiny non-integer strong force or electro-weak quantum numbers)? For example, what would this do to lepto-quark models?
5. What kinds of new physics models could escape the implications of a very tight fit between measured muon g-2 and the Standard Model theoretical prediction of that value?
6. If improved muon g-2 measurements and theoretical calculations can clarify globally what kind of new physics could be out there, doesn't that suggest that a decision on a next collider should be postponed at least until we have the Fermilab E989 experiment results, so we can gauge how likely a "nightmare scenario" would be with a new collider?
(In other words, does it make sense to build a new collider if muon g-2 data and existing LHC data strongly disfavor new physics in the energy scales above the peak LHC scale and up to the peak new collider energy scale?)
7. If we had a theory of quantum gravity, would it enter into the muon g-2 calculation? If so, would the order of magnitude of any quantum gravity effect be big enough to be material relative to the margin of error in the existing experimental measurements and theoretical calculations?
(Obviously, we do not have a theory of quantum gravity, but we know, from GR, the magnitude of gravitational forces generally and there is a considerable and not terribly contentious literature of the magnitude of a quantum gravity coupling constant in many of the more conventional efforts to construct a quantum gravity theory. Presumably, this would make to conceivably possible to get a back of napkin estimate of the order of magnitude of an quantum gravity adjustment even if the exact value of such an adjustment was model dependent and had a considerable degree of uncertainty.)
IV. Closing Observation
It seems naively to me like a tight fit between the experimental and theoretically calculated values for muon g-2 in the Standard Model has the potential to globally rule out in a not very model dependent way, almost all SUSY theories and a whole lot of other popular BSM theories leaving only a very, very few such theories in the running, with those theories carefully constructed around this constraint.
And, it also seems to me like this possibility is one that could very realistically happen in the medium term future.
But, this seems too "good" to be true, so I'm wondering if I'm missing something important in this analysis.
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