Proton radius from electron scattering: the experimental side
Jan C. Bernauer
INT 18-2a - June 2018 100 years of protons! Proton is a composite system. It must have a size! How big is it?
What is ”stuff”?
The matter around us is described by non-perturbative quantum chromodynamics. npQCD is hard. Simplest QCD system to study: Protons
2 Proton is a composite system. It must have a size! How big is it?
What is ”stuff”?
The matter around us is described by non-perturbative quantum chromodynamics. npQCD is hard. Simplest QCD system to study: Protons
100 years of protons!
3 What is ”stuff”?
The matter around us is described by non-perturbative quantum chromodynamics. npQCD is hard. Simplest QCD system to study: Protons
100 years of protons! Proton is a composite system. It must have a size! How big is it?
4 Two transitions for two unknowns: Rydberg constant R 1S Lamb shift = radius∞ ⇒ Direct Lamb shift 2S 2P →
Motivation: ”Normal” Hydrogen Spectroscopy
8S 4S 3S 3D
2S 2P
R L1S EnS u ∞2 + 3 − n n
2 1S L1S = 8171.626(4) + 1.5645 rp MHz
5 Direct Lamb shift 2S 2P →
Motivation: ”Normal” Hydrogen Spectroscopy
8S 4S 3S 3D 2S 8S 2S 8D → → 2S 2P
R L1S EnS u ∞2 + 3 − n n Two transitions for two unknowns: Rydberg constant R ∞ 1S 2S 1S Lamb shift = radius → ⇒
2 1S L1S = 8171.626(4) + 1.5645 rp MHz
6 Motivation: ”Normal” Hydrogen Spectroscopy
8S 4S 3S 3D 2S 2P 2S → 2P
R L1S EnS u ∞2 + 3 − n n Two transitions for two unknowns: Rydberg constant R 1S Lamb shift = radius∞ ⇒ Direct Lamb shift 2S 2P →
2 1S L1S = 8171.626(4) + 1.5645 rp MHz
7 ”Normal” Hydrogen Spectroscopy Results
8 Elastic lepton-proton scattering
Method of choice: Lepton-proton scattering Point-like probe No strong force Lepton interaction ”straight-forward”
Measure cross sections and reconstruct form factors.
9 Cross section for elastic scattering
dσ dΩ 1 2 2 2 2 = εGE Q + τGM Q dσ ε (1 + τ) dΩ Mott h i with:
2 1 Q θ − τ = , ε = 1 + 2 (1 + τ) tan2 e 4m2 2 p Rosenbluth formula Electric and magnetic form factor encode the shape of the proton Fourier transform (almost) gives the spatial distribution, in the Breit frame
10 How to measure the proton radius
dG d (G /µ ) r2 6 2 E r2 6 ² M p E = ~ 2 M = ~ 2 − dQ 2 − dQ 2 Q =0 Q =0 D E D E
1 0.995
. 0.99 dip
. 0.985
std 0.98
/G 0.975 E dGE G 0.97 dQ2 → Q2=0 0.965
0.96 0 0.05 0.1 0.15 0.2 0.25 0.3 Q2 [(GeV/c)2]
11 History of unpolarized electron-proton scattering
100
10 2 )
/c 1
(GeV 0.1 / 2 Q 0.01
0.001 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 Year Andivahis Borkowski Janssens Rock Walker Bartel Bosted Litt Sill Berger Christy Price Simon Bernauer Goitein Qattan Stein
12 Mainz Microtron cw electron beam 10 µA polarized, 100 µA unpolarized MAMI A+B: 180-855 MeV MAMI C: 1.6 GeV
A1 3-spectrometer facility 28 msr acceptance angle resolution: 3 mrad 4 momentum res.: 10−
High-precision p(e,e’)p measurement at MAMI
13 A1 3-spectrometer facility 28 msr acceptance angle resolution: 3 mrad 4 momentum res.: 10−
High-precision p(e,e’)p measurement at MAMI
Mainz Microtron cw electron beam 10 µA polarized, 100 µA unpolarized MAMI A+B: 180-855 MeV MAMI C: 1.6 GeV
14 High-precision p(e,e’)p measurement at MAMI
Mainz Microtron cw electron beam 10 µA polarized, 100 µA unpolarized MAMI A+B: 180-855 MeV MAMI C: 1.6 GeV
A1 3-spectrometer facility 28 msr acceptance angle resolution: 3 mrad 4 momentum res.: 10−
15 Measured settings
A 1 B C
0.8 ] 2 )
/c 0.6 (GeV [ 2
Q 0.4
0.2
0 0 0.2 0.4 0.6 0.8 1
1422 settings JCB et al., Phys. Rev. Lett. 105 (2010) 242001, M. O. Distler, JCB, Th. Walcher, Phys. Lett. B 696, 343 (2011) JCB et al., Phys. Rev. C90 (2014) 015206
16 Cross sections
100 100
10
10
1 ] ] 1 µb µb [ [ 0.1 exp exp σ σ
0.1 180 0.01 MeV 315 585 MeV MeV 0.01 720 450 0.001 MeV MeV 855 MeV
0.001 0.0001 0◦ 20◦ 40◦ 60◦ 80◦ 100◦ 120◦ 140◦ 0◦ 20◦ 40◦ 60◦ 80◦ 100◦ 120◦ 140◦ Central angle of spectrometer Central angle of spectrometer
Polynomial Inv. poly. Friedrich-Walcher Poly. + dip Spline Double dipole Poly. dip Spline dip Extended G.K. × ×
17 Muonic Hydrogen Spectroscopy
Replace electron with muon 200 times heavier = 200 times smaller orbit ⇒ Probability to be ”inside” 2003 higher!
18 From the 2017 Review of Particle Physics Until the difference between the e p and µ p values is understood, it does not make sense to average the values together. For the present, we give both values. It is up to the workers in this field to solve this puzzle.
The proton radius puzzle
7.9 σ µp 2013 electron avg.
scatt. JLab
µp 2010 scatt. Mainz
H spectroscopy
0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 Proton charge radius R [fm] ch
19 The proton radius puzzle
From the 2017 Review of Particle Physics Until the difference between the e p and µ p values is understood, it does not make sense to average the values together. For the present, we give both values. It is up to the workers in this field to solve this puzzle.
20 seems solid ep experiments wrong? both scattering and H-spectroscopy wrong? Theory wrong? checked thoroughly ...but maybe framework is wrong? Everybody is right? New physics! ”Naive” dark matter models essentially excluded But can play cancelation games E.g.: Electrophobic force (Liu, Cloët, Miller arXiv:1805.01028) WE NEED MORE DATA
Solutions?
µp experiment wrong?
21 Theory wrong? checked thoroughly ...but maybe framework is wrong? Everybody is right? New physics! ”Naive” dark matter models essentially excluded But can play cancelation games E.g.: Electrophobic force (Liu, Cloët, Miller arXiv:1805.01028)
WE NEED MORE DATA
Solutions?
µp experiment wrong? seems solid ep experiments wrong? both scattering and H-spectroscopy wrong?
22 Everybody is right? New physics! ”Naive” dark matter models essentially excluded But can play cancelation games E.g.: Electrophobic force (Liu, Cloët, Miller arXiv:1805.01028)
WE NEED MORE DATA
Solutions?
µp experiment wrong? seems solid ep experiments wrong? both scattering and H-spectroscopy wrong? Theory wrong? checked thoroughly ...but maybe framework is wrong?
23 WE NEED MORE DATA
Solutions?
µp experiment wrong? seems solid ep experiments wrong? both scattering and H-spectroscopy wrong? Theory wrong? checked thoroughly ...but maybe framework is wrong? Everybody is right? New physics! ”Naive” dark matter models essentially excluded But can play cancelation games E.g.: Electrophobic force (Liu, Cloët, Miller arXiv:1805.01028)
24 Solutions?
µp experiment wrong? seems solid ep experiments wrong? both scattering and H-spectroscopy wrong? Theory wrong? checked thoroughly ...but maybe framework is wrong? Everybody is right? New physics! ”Naive” dark matter models essentially excluded But can play cancelation games E.g.: Electrophobic force (Liu, Cloët, Miller arXiv:1805.01028)
WE NEED MORE DATA
25 Deuteron (arxiv:1607.03165)
In CODATA, rd is correlated strongly to rp because it uses ”isotope shift” from 1s-2s in both systems. But: Can built independent value from deuteron 1s-2s and 2s-8s/8d/12d. This gives a difference to muonic deuterium! another puzzle. → Rydberg from electric Hydrogen and Deuterium in perfect agreement. The muonic deuterium R is in slight disagreement with the muonic hydrogen∞ R (<3 sigma) ∞
26 New hydrogen results: MPQ (A. Beyer et al., Science 358, 79 (2017))
27 New results: Paris (Fleurbaey et al., Phys. Rev. Lett. 120, 183001 (2018))
28 Timeline of proton radius results
0.92 0.9 0.88 0.86 [fm] 0.84 >
E 0.82 0.8 < r 0.78 0.76 0.74 Paz ’10 Sick ’03 Pohl ’10 Zhan ’11 Hand ’63 Wong ’94 Udem ’97 Simon ’80 Lorenz ’12 Mergell ’96 Akimov ’72 McCord ’91 Blunden ’05 Eschrich ’01 Melnikov ’00 Bernauer ’10 CODATA ’06 CODATA ’02 Belushkin ’06 Antognini ’13 Borkowski ’75 Rosenfelder ’00 Frerejacque ’66
29 Comments on some newer scattering results 2010: >0.870 Hill, Paz: old data, z expansion with disp. bounds Bounds on infinite exp. bounds for truncated exp.? → 2012: 0.840(10) Lorenz, Hammer, Meissner: Disp. relation fit. Same value but a lot more data. Probably model dominated. 2014: 0.84 Lorenz, Meissner: z expansion without bounds Fit did not converge. In real minimum, large radius is found. 2014: 0.8989(1) Gracyk/Juszczak: Bayesian estimation Interesting technique, unbelievable? small errors 2016: 0.84? Higinbotham: F-Test to select max. order Misunderstood F-test. Absence of proof = proof of absence. 6 2016: 0.84? Horbatsch/Hessels/Griffioen/Carlson/Maddox... Low-Q Low-Q fits with low order don’t work. 2018: XXX Yan/Higinbotham/... Small radius fraction finally does bias testing
30 Volume of Mainz data set
7
2 6 σ σ ∆ 5 P q 4
3
2
Relative statistical weight 1
0 All others Mainz
Mainz data will dominate any fit. Need similar data set to validate!
31 Extrapolation to Q2 = 0
Have to extrapolate form factor to Q2 = 0. Mainz lowest Q2 = 0.0033 (GeV/c)2. We use a 10th order polynomial to fit data up to 1 (GeV/c)2. This gets people scared.
Can we fit just a linear term?
32 dσ But: Need to measure A to 1%, so measure dΩ to 6 0.002 0.01 = 0.012%. Good luck. · ·
Can a linear fit work?
dσ 1 A Q2 + B Q4 + ... dΩ ∝ − · · (6) (30) O O (Q in units of GeV/c) |{z} |{z} We want to measure the radius (~√A) to within 0.5%, without knowingB. So:
B/A Q2 0.01 Q2 0.002 (GeV/c)2 · −→
33 Can a linear fit work?
dσ 1 A Q2 + B Q4 + ... dΩ ∝ − · · (6) (30) O O (Q in units of GeV/c) |{z} |{z} We want to measure the radius (~√A) to within 0.5%, without knowingB. So:
B/A Q2 0.01 Q2 0.002 (GeV/c)2 · −→
dσ But: Need to measure A to 1%, so measure dΩ to 6 0.002 0.01 = 0.012%. Good luck. · ·
34 Why do low Q2 then?
Test / fix normalization Similar arguments apply, but helpful when dataset contains also higher Q2. Test for new physics / ultra long range structure Signal can easily, but doesn’t have to be undetectable small and still change the radius!
Measure rM Low Q2 at = 1 means lowish Q2 at = 0
35 Three ways to get to lower Q2
θ Q2 = 4EE sin2 0 2 Smaller scattering angle PRad −→ Lower beam energy MESA −→ Initial State Radiation
36 Status Data taken successfully Analysis ongoing
JLAB: PRoton RADius
High resolution, large acceptance hybrid calorimeter Windowless target Simultaneous measure ep ep and Møller scattering → Q2 range: 2 10 4 to 6 10 2 (GeV/c)2 × − × −
37 JLAB: PRoton RADius
High resolution, large acceptance hybrid calorimeter StatusWindowless target SimultaneousData taken successfully measure ep ep and Møller scattering → QAnalysis2 range: ongoing 2 10 4 to 6 10 2 (GeV/c)2 × − × −
38 Slide stolen from Weizhi Xiong’s talk at CIPANP
Form Factor GE (Preliminary)
Proton Electric Form Factor GE
E 1.05 1.1 GeV data (PRad Preliminary) G • Proton electric form factor 2.2 GeV data (PRad Preliminary) G v.s. Q2, with 2.2 and 1.1 E GE, J. C. Bernauer et al. PRC 90 (2014) 015206, R = 0.8868 fm G , J. J. Kelly. PRC 70 (2004) 068202), R = 0.8630 fm GeV data (preliminary) E
1 GE, S. Venkat et al. PRC 83(2011)015203), R = 0.8779 fm • Systematic uncertainties Preliminary shown as colored error 0.95 bars
0.9 • Preliminary GE slope seems to favor smaller radius 0 0.01 0.02 0.03 0.04 0.05 Q2 (GeV2)
18
39 Slide stolen from Weizhi Xiong’s talk at CIPANP
Form Factor GE (Preliminary)
Proton Electric Form Factor GE
E 1.1 GeV data (PRad Preliminary) G • Proton electric form factor 2.2 GeV data (PRad Preliminary) 2 G v.s. Q , with 2.2 and 1.1 1.05 G , J. C. Bernauer et al. PRC 90 (2014) 015206, R = 0.8868 fm E E G , J. J. Kelly. PRC 70 (2004) 068202), R = 0.8630 fm GeV data (preliminary) E
GE, S. Venkat et al. PRC 83(2011)015203), R = 0.8779 fm
1 • Systematic uncertainties shown as colored error bars 0.95 Preliminary
• Preliminary GE slope 0.9 seems to favor smaller radius 2×10−4 10−3 2×10−3 10−2 2×10−2 Q2 (GeV2)
19
40 ISR method
2 QReconstruct
2 QVertex
Use initial state radiation to reduce effective beam energy Have to subtract FSR
41 Status Published: PLB 771:194-198 Radiative correction correct on the 1% level deep in the tail! Radius extraction not competitive in precision
ISR at MAMI
Simulation Data at 195 MeV + 7 Elastic Settings H(e, e′)n π and 10 0 Data at 495 MeV H(e, e′)p π Contributions Data at 330 MeV Background
106
5 MeV) 105 ·
4 ISR small E (0.1 mC 10 −→ 2 −→ 3 small Q Counts / 10
21 19 6 4 2 Extract F.F. from 102 22 20 18 7 5 3 1 17 15 13 11 9 radiative tail 16 14 12 10 8 3%
Or: test radiative tail 2% 1 − description 1% 0% Simulation / -1%
Data -2% Systematic uncertainty
-3% 100 150 200 250 300 350 400 450 500 Electron energy E ′ [MeV] See: arXiv:1612.06707
42 ISR at MAMI
Simulation Data at 195 MeV + 7 Elastic Settings H(e, e′)n π and 10 0 Data at 495 MeV H(e, e′)p π Contributions Data at 330 MeV Background
106
5 MeV) 105 ·
4 ISR small E (0.1 mC 10 −→ 2 −→ 3 small Q Counts / 10
21 19 6 4 2 Extract F.F. from 102 22 20 18 7 5 3 1 Status 17 15 13 11 9 radiative tail 16 14 12 10 8 Published: PLB 771:194-1983% Or: test radiative tail 2% 1 − descriptionRadiative correction correct1% on the 1% level deep in the tail! 0% Simulation / -1%
Radius extraction not competitiveData -2% in precisionSystematic uncertainty
-3% 100 150 200 250 300 350 400 450 500 Electron energy E ′ [MeV] See: arXiv:1612.06707
43 point-like no walls but less density
Electron- beam 49.5 mm 11.5 mm liquid hydrogen
10 µm Havar
Target cell Heat Ventilator exchanger
"Basel-Loop"
Background from target walls Scattering chamber Acceptance correction for extended target Eliminate with jet target ⇓
Rinse, repeat with D,3He,4He, ...
Target dominant source of uncertainty
For Mainz data, systematic errors dominate
44 point-like no walls but less density
Eliminate with jet target ⇓
Rinse, repeat with D,3He,4He, ...
Electron- beam 49.5 mm 11.5 mm liquid hydrogen
10 µm Havar Target dominant source of uncertainty
Target cell Heat Ventilator exchanger
For Mainz data, systematic errors dominate "Basel-Loop"
Background from target walls Scattering chamber Acceptance correction for extended target
45 Electron- beam 49.5 mm 11.5 mm liquid hydrogen
10 µm Havar Target dominant source of uncertainty
Target cell Heat Ventilator exchanger
For Mainz data, systematic errors dominate "Basel-Loop"
Background from target walls Scattering chamber Acceptance correction for extended target Eliminate with jet target point-like ⇓ no walls but less density Rinse, repeat with D,3He,4He, ...
46 Mainz future plans
Repeat ISR with new target Use new target also for classic approach
1 108 1 × MAGIX at E=50 MeV MAMI at E=180 MeV MAGIX at E=50 MeV MAMI at E=180 MeV 1 107 MAGIX at E=75 MeV MAMI at E=240 MeV MAGIX at E=75 MeV MAMI at E=240 MeV × MAGIX at E=100 MeV MAMI at E=300 MeV MAGIX at E=100 MeV MAMI at E=300 MeV 1 106 1 week × 0.1 100000 1 day ] 2 )
10000 /c 1 hour 0.01
1000 (GeV) [ 2 100 1 minute Q 0.001 10 Time to 0.1% statistical error [s] 1 1 second
0.1 0.0001 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ε ε
Took first data in April! Full MAMI experiment next year, MESA 2021.
47 The missing piece
rE [fm] ep µp Spectroscopy 0.8758 0.077 0.84087 0.00039 ± ± Scattering 0.8770 0.060 ???? ±
Measure radius with muon-proton scattering!
48 MUSE - Muon Scattering Experiment at PSI
World’s most powerful low-energy e/π/µ-beam: Direct comparison of ep and µp!
+ + + Beam of e /π /µ or e−/π−/µ− on liquid H2 target Species separated by ToF, charge by magnet Absolute cross sections for ep and µp Ratio to cancel systematics Charge reversal: test TPE Momenta 115-210 MeV/c Rosenbluth G ,G ⇒ E M 49 Experiment layout
Scattered Particle Beam-Line Scintillator (SPS) Monitor
Straw-Tube Tracker (STT) Secondary beam = track beam particles ⇒
Target Low flux (5 MHz)= large Chamber acceptance ⇒
3 GEM Detectors Mixed beam = PID in ⇒ Beam trigger Hodoscope Veto Scintillator pM1 Beam-Line
~ 100 cm
R. Gilman et al., arXiv:1303.2160 [nucl-ex]
50 Difference: Common uncertainties cancel! factor two more −→sensitivity
MUSE can verify 7σ effect with similar significance!
Predicted performance
Absolute radius extraction uncertainties similar to current exp’s.
51 Predicted performance
Absolute radius extraction uncertainties similar to current exp’s. Difference: Common uncertainties cancel! factor two more sensitivity−→
MUSE can verify 7σ effect with similar significance!
52 Summary
Proton radius puzzle persists since 2010 We need new data to resolve it A lot of data incoming in the next years, but pretty hard limit on achievable errors MUSE, with electron and muon scattering, will test existing radius value lepton universality two photon exchange / proton polarizability
The most exciting phrase to hear in science, the one that heralds new discoveries, is not “Eureka!” but “That’s funny . . . ” — Isaac Asimov
53 Taylor expansions and polynomial fits
It’s a common theme that a polynomial fit is related to a Taylor expansion around 0, sharing important traits mainly radius of convergence. ”We will fit ... a simple Taylor series expansion.” R.J. Hill and G. Paz, Phys. Rev. D 82, 113005 (2010) ”correct inclusion of the lowest singularity” I. Lorenz and U.G-Meißner, Phys. Lett. B 737, 57 (2014) ”Maclaurin fits”, D. W. Higinbotham et al., Phys.Rev. C93, 055207 (2016) ”We do not advocate using polynomial fits.... since convergence ... is not assured...” K. Griffioen et al., arxiv:1509.06676 This is wrong.
54 Traits of Taylor, Weierstrass, Fits
Taylor expansion
Is correct in all order (to truncated order) at x0. Converges on a radius up to the next pole. (k+1) f (ξc) k Error is R = (x ξc) (x x ) k k! − − 0 Weierstrass theorem Any function continuos over [a, b] can be approximated with a polynomial in that range. The convergence is uniform: > 0, poly., so that f (x) p(x) < , x [a, b] ∀ ∃ || − ||∞ ∈ Polynomial fit Minimizes L2-norm over the points: f (x) p(x) || − ||2 Will converge to the function, NOT to the Taylor expansion of the function
55 We have no choice
Taylor expansion
Is correct in all order (to truncated order) at x0. Converges on a radius unto the next pole. (k+1) f (ξc) k Error is R = (x ξc) (x a) k k! − − Weierstrass theorem Any function continuous over [a, b] can be approximated with a polynomial in that range. The convergence is uniform: > 0, poly. .so that f (x) p(x) < , x [a, b] ∀ ∃ || − ||∞ ∈ Polynomial fit Minimizes L2-norm over the points: f (x) p(x) || − ||2 Will converge to the function, NOT to the Taylor expansion of the function
56 What does that mean in reality?
Let’s fit perfect pseudo data Compare with Taylor expansion Input function: dipole, i.e. pole at Q2 = 0.71 (GeV/c)2 −
57 Conclusion: Taylor convergence radius has no consequence for polynomial fit. is not a reason to use conformal mapping. is not a reason to limit Q2 range.
Fit results
1200 10th order Taylor expansion around 0 1000 Polynomial fit ( 107) × 800
600
400
200
0 Relative difference to dipole
200 − 0 0.2 0.4 0.6 0.8 1 Q2 [(GeV/c)2]
Fit within 40 ppm over data range, better than expansion for Q2 > 0.15 (GeV/c)2
58 Fit results
1200 10th order Taylor expansion around 0 1000 Polynomial fit ( 107) × Conclusion: 800 Taylor convergence radius has600 no consequence for polynomial fit. is not400 a reason to use conformal mapping. is not a reason to limit Q2 range. 200
0 Relative difference to dipole
200 − 0 0.2 0.4 0.6 0.8 1 Q2 [(GeV/c)2]
Fit within 40 ppm over data range, better than expansion for Q2 > 0.15 (GeV/c)2
59 Problems of (unconstrained) conformal mapping
remaps flexibility: a lot of flexibility to small Q2: Gap is 2.2% of data range instead of 0.4% not enough at high Q2 harder to fit: many local minima
60 Failures to fit conformal mapping polynomials I. Lorenz and U.G. Meißner, ”Reduction of the proton radius discrepancy by 3 σ”, Phys. Lett. B 737, 57 (2014)
2000 1900 1800 2
χ 1700 1600 4 6 8 10 12 14 16 1.40 1.20
1.00 0.80 0.60 4 6 8 10 12 14 16
0.85
0.84 0.83
0.82 4 6 8 10 12 14 16
61 ”Knee” 9 = 6 ⇒ No stable plateau in rm= we would reject the model ⇒ At knee, re compatible with large radius!
Finding better minima via random search
1.2 1 0.8 2000 from Lorenz et al. [fm] 0.6
π± mass m
1950 r π± mass, phys. radii 0.4 π0 mass 1900 π0 mass, phys. radii 0.2 0 1850 2 4 6 8 10 12 14 16
1800 kmax 1.02
2 1750 1 χ 0.98 1700 0.96 0.94
[fm] 0.92 1650 e
r 0.9 0.88 1600 0.86 0.84 1550 0.82 2 4 6 8 10 12 14 16 1500 2 4 6 8 10 12 14 16 kmax π mass π0 mass kmax ±
62 Finding better minima via random search
1.2 1 0.8 2000 from Lorenz et al. [fm] 0.6
π± mass m
1950 r π± mass, phys. radii 0.4 π0 mass 1900 π0 mass, phys. radii 0.2 0 1850 ”Knee” 9 = 6 2 4 6 8 10 12 14 16 1800 ⇒ kmax No stable plateau in rm= we1.02 would reject the model
2 1750 ⇒ 1 χ At knee, re compatible with0 large.98 radius! 1700 0.96 0.94
[fm] 0.92 1650 e
r 0.9 0.88 1600 0.86 0.84 1550 0.82 2 4 6 8 10 12 14 16 1500 2 4 6 8 10 12 14 16 kmax π mass π0 mass kmax ±
63 Questions Why 0.02? What happens at 0.01? 0.03? What is the bias of this method?
Low-Q polynomial fits
Griffioen et al. ”Are Electron Scattering Data Consistent with a Small Proton Radius?”, arxiv:1509.06676 advocate a fit up to 0.02 (GeV/c)2. They find:
Linear fit: re = 0.835(3) fm.
Quadratic fit: re = 0.850(15) fm.
64 Low-Q polynomial fits
Griffioen et al. ”Are Electron Scattering Data Consistent with a Small Proton Radius?”, arxiv:1509.06676 advocate a fit up to 0.02 (GeV/c)2. They find:
Linear fit: re = 0.835(3) fm.
Quadratic fit: re = 0.850(15) fm.
Questions Why 0.02? What happens at 0.01? 0.03? What is the bias of this method?
65 Simulate experiment with pseudo data
Use two input parametrizations Our 10th order polynomial fit 10th order polynomial fit with radius forced to 0.841 fm. Generate 1000 pseudo data sets each Fit
Look at extracted re as function of cut off. Compare with known input radius
66 Results on pseudo data
Linear fit Quadratic fit 0.92 0.92
0.9 0.9
0.88 0.88
0.86 0.86
0.84 0.84 [fm] [fm] e e r r 0.82 0.82
0.8 0.8
0.78 0.78
0.76 0.76
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Q2 cut off [(GeV/c)2] Q2 cut off [(GeV/c)2]
67 Results on pseudo data
Linear fit Quadratic fit 0.92 0.92
0.9 0.9
0.88 0.88
0.86 0.86
0.84 0.84 [fm] [fm] e e r r 0.82 0.82
0.8 0.8
0.78 0.78
0.76 0.76
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Q2 cut off [(GeV/c)2] Q2 cut off [(GeV/c)2]
68 Results on pseudo data
Linear fit Quadratic fit 0.92 0.92
0.9 0.9
0.88 0.88
0.86 0.86
0.84 0.84 [fm] [fm] e e r r 0.82 0.82
0.8 0.8
0.78 0.78
0.76 0.76
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Q2 cut off [(GeV/c)2] Q2 cut off [(GeV/c)2]
69 Results on pseudo data
Linear fit Quadratic fit 0.92 0.92
0.9 0.9
0.88 0.88
0.86 0.86
0.84 0.84 [fm] [fm] e e r r 0.82 0.82
0.8 0.8
0.78 0.78
0.76 0.76
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Q2 cut off [(GeV/c)2] Q2 cut off [(GeV/c)2]
70 Results on pseudo data
Linear fit Quadratic fit 0.92 0.92
0.9 0.9
0.88 0.88
0.86 0.86
0.84 0.84 [fm] [fm] e e r r 0.82 0.82
0.8 0.8
0.78 0.78
0.76 0.76
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Q2 cut off [(GeV/c)2] Q2 cut off [(GeV/c)2]
71 Results on data
Linear fit Quadratic fit 0.92 0.92
0.9 0.9
0.88 0.88
0.86 0.86
0.84 0.84 [fm] [fm] e e r r 0.82 0.82
0.8 0.8
0.78 0.78
0.76 0.76
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Q2 cut off [(GeV/c)2] Q2 cut off [(GeV/c)2]
72 Dipole fits
Horbatsch and Hessels, ”Evaluation of the strength of electron-proton scattering data for determining the proton charge radius”, Phys. Rev. C 93 015204 compare conformal mapping fits (large radius) and dipole fits (small radius) with varying cut-off. We already know that dipole fit to the whole range has a large bias. But what about smaller range?
73 Results dipole fit to (pseudo) data
Dipole fit 0.92
0.9
0.88
[fm] 0.86 e r 0.84
0.82
0.8 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Q2 cut off [(GeV/c)2]
74 Fisher-Snedecor F follows a Fisher-Snedecor distribution if H0 is true Otherwise: Non-central Fisher-Snedecor distribution (best case)
F-test to determine fit order
Nested models! Compare two hypothesis: H0: The true model has order j H1: The true model has order j+k (or any order>j) χ2 χ2 N j F = H0 − H1 − 2 k χH1
75 F-test to determine fit order
Nested models! Compare two hypothesis: H0: The true model has order j H1: The true model has order j+k (or any order>j) χ2 χ2 N j F = H0 − H1 − 2 k χH1
Fisher-Snedecor F follows a Fisher-Snedecor distribution if H0 is true Otherwise: Non-central Fisher-Snedecor distribution (best case)
76 F-test
Can rule out H0 at a given CL if F > Fcrit Type I error: If H0 is falsely rejected, H0 is true. = F is Fisher-Snedecor distributed ⇒ = Can calculate how often F > Fcrit by random chance⇒
Can NOT rule out H1 at same CL if F < Fcrit Type II error: Have to assume H1 is correct! = F NOT Fisher-Snedecor distributed ⇒ = Small F can reject BOTH H0 and H1 This⇒ is what D. Higinbotham et al. do wrong James does it (semi) correct in explanation, but wrong in example
77 Here: We know that the form factor 0 for Q2 . Any finite polynomial goes to → → ∞ Neither H0 nor any H1 can be±∞ true Everything Should Be Made as Simple as Possible, But Not Simpler (Einstein, probably) Let’s pretend we are John Snow and know nothing.
Remarks
One can disprove H0 without assuming H1 to be right! Science: We can disprove a theory (because a prediction is off), we can not prove one. Other tests: similar story Akaike Information Criterion (AIC) tells you if data can disprove that a certain model is ”enough” This does not touch the problem of bias!
78 Remarks
One can disprove H0 without assuming H1 to be right! Science: We can disprove a theory (because a prediction is off), we can not prove one. Other tests: similar story Akaike Information Criterion (AIC) tells you if data can disprove that a certain model is ”enough” This does not touch the problem of bias! Here: We know that the form factor 0 for Q2 . Any finite polynomial goes to → → ∞ Neither H0 nor any H1 can be±∞ true Everything Should Be Made as Simple as Possible, But Not Simpler (Einstein, probably) Let’s pretend we are John Snow and know nothing.
79 F-test results for low-order polynomials
Linear fit Quadratic fit 0.92 0.92
0.9 0.9
0.88 0.88
0.86 0.86
0.84 0.84 [fm] [fm] e e r r 0.82 0.82
0.8 0.8
0.78 0.78
0.76 0.76
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Q2 cut off [(GeV/c)2] Q2 cut off [(GeV/c)2]
80 1 0.9 0.8 0.7 Orders < 9 ruled out by F-test0.6 and AIC 0.5 0.4
Order 10 ”optimal” accordingCL of rejection 0.3 AIC 0.2 0.1 This is 100% in accordance to1 our2 model3 4 5 selection6 7 8 9 10 2 Order of polynomial N.B: AIC disfavors any model50 with χ > 1620 even if no parameter 40 30 AIC
∆ 20
10
0 2 3 4 5 6 7 8 9 10 11 Order of polynomial
F-test /AIC results for full data range
0.9
0.88
0.86
0.84
0.82 [fm] e
r 0.8
0.78
0.76
0.74 Fit to data Fit to pseudo-data 0.72 0 2 4 6 8 10 12 Order of polynomial
81 Orders < 9 ruled out by F-test and AIC Order 10 ”optimal” according AIC This is 100% in accordance to our model selection N.B: AIC disfavors any model with χ2 > 1620 even if no parameter
F-test /AIC results for full data range
0.9 1 0.9 0.8 0.88 0.7 0.6 0.86 0.5 0.4
0.84 CL of rejection 0.3 0.2 0.1 0.82 1 2 3 4 5 6 7 8 9 10
[fm] Order of polynomial e
r 0.8 50
0.78 40
30 0.76 AIC
∆ 20 0.74 Fit to data 10 Fit to pseudo-data 0.72 0 0 2 4 6 8 10 12 2 3 4 5 6 7 8 9 10 11 Order of polynomial Order of polynomial
82 F-test /AIC results for full data range
0.9 1 0.9 0.8 0.88 0.7 Orders < 9 ruled out by F-test0.6 and AIC 0.86 0.5 0.4
0.84 Order 10 ”optimal” accordingCL of rejection 0.3 AIC 0.2 0.1 0.82 This is 100% in accordance to1 our2 model3 4 5 selection6 7 8 9 10
[fm] Order of polynomial e
r 0.8 2 N.B: AIC disfavors any model50 with χ > 1620 even if no 0.78 parameter 40 30 0.76 AIC
∆ 20 0.74 Fit to data 10 Fit to pseudo-data 0.72 0 0 2 4 6 8 10 12 2 3 4 5 6 7 8 9 10 11 Order of polynomial Order of polynomial
83 Conclusion
Fitting is hard Taylor and polynomial fits are unrelated Have to balance between bias and overfitting Cutting data set to small Q2 makes balance HARDER. Statistical tests do not tell you about bias. Statistical tests, done right, support our analysis. Test your method on pseudo data! = If you want to disprove large radius, show that you can⇒ replicate the large radius! The resolution of the puzzle can not be found in refitting of the data For more info: arXiv:1606.02159
84 2 Poly/Taylor possible Q0
0.3 4 1 2 4 3 2 0.2 2 150 1 3 03 6 0 2 4 5 4 14 3 6 5 0.1 025 7 6 78978 6 3 ) ) 5 91010109 78 16 8 2 0 2 0 7810 109 204 9 3261504321 Q Q 0 0 2 4 6 810 12304567083960011023456789 0 1357 9 0 12345678910 ( 9 21504321 ( 7 10 109 204
= 8 = 5 91010109 78 16 8 6 78978 6 3 3 6 5 0.1 025 7 2 4 5 4 − 14 − 3 03 6 0 2 150 1 2 0.2 4 3 2 4 1 − − 0.3 10 8 6 4 2 0 − 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 − − − − − − − (Q2) (Q2) < 0 < 0
85 PSI setup (CREMA)
86 Result Two semi-independent measurements Consistent results
Muonic Hydrogen Spectroscopy Results
87 Muonic Hydrogen Spectroscopy Results
Result Two semi-independent measurements Consistent results
88 8Be is special
Many images from arXiv:1707.09749 8Be is special: two narrow, highly energetic states which can decay to ground state via E/M
89 Decay modes of 8Be(18.15)
Hadronic, electromagnetic and through internal pair conversion
90 The Atomkin experiment
θ
ATOMKI PAIR SPECTROMETER
1.04 MeV proton beam on 7Li to 8Be(18.15) + γ. Followed by decay. Looked at e± pairs from internal conversion.
91 The beryllium anomaly
92 Why believe it?
This model has χ2/d.o.f . of 1.07, significance of 6.8σ Bump, not last bin effect Rises/falls when scanning through proton energies around resonance Excess only happens for symmetric-energy pairs Preliminary reports of same excess in 8Be(17.6) (same group)
93 Why not believe it?
Group has a history of finding peaks IIUC, the detector acceptance has a minimum at 140◦ DM boson interpretation is proto-phobic to evade NA48/2 limits Actually: p coupling below 8%. Z 0 is 7% n ± ∼
94 Best kinematics: highest production rate if X takes all electron energy. CS rise beats all with limited out-of-plane acceptance, symmetric angle optimal
We can measure it!
In DarkLight, production is via Bremsstrahlung, predominantly ISR off the electron. We can look at e Ta e TaX, followed by X e e+ − → − → − Irreducible background: e Ta e Taγ? e Tae+e − → − → − −
95 We can measure it!
In DarkLight, production is via Bremsstrahlung, predominantly ISR off the electron. + We can look at e−Ta e−TaX, followed by X e−e → ? → + Irreducible background: e−Ta e−Taγ e−Tae e− Best kinematics: → → highest production rate if X takes all electron energy. CS rise beats all with limited out-of-plane acceptance, symmetric angle optimal
96 Reach
4 10−
-preferred 5 2) 10− (gµ −
6 10− -1c-1c 45 MeV 150 µA @ 10 µm 2000 hh-1c 5th force 10 7 − MESA APEX
2 Mu3e HPS 8 VEPP-3 10−
LHCb 9 10−
10 10 − SeaQuest SHIP 11 10− 101 102 103 mA [MeV]
97