The Proton Radius Puzzle- Why we all should care Gerald A. Miller, University of Washington Feb. 2014 Pohl et al Nature 466, 213 (8 July 2010)
4 % Difference muon H rp =0.84184 (67) fm Small 2 2 dGE(Q ) rp 6 ⌘ dQ2 electron H rp =0.8768 (69)fm Large Q2=0 Pohl, Gilman, Miller, Pachucki electron-p scattering rp =0.875 (10)fm (ARNPS63, 2013) PRad at JLab- lower Q2 C Carlson PPNP 4 2 2 2 3 10 Q 5 10 GeV 82,59(2015) ⇥ ⇥ 3 2 2
1 The Proton Radius Puzzle- Why we all should care Gerald A. Miller, University of Washington Feb. 2014 Pohl et al Nature 466, 213 (8 July 2010)
4 % Difference muon H rp =0.84184 (67) fm Small 2 2 dGE(Q ) rp 6 ⌘ dQ2 electron H rp =0.8768 (69)fm Large Q2=0 Pohl, Gilman, Miller, Pachucki electron-p scattering rp =0.875 (10)fm (ARNPS63, 2013) PRad at JLab- lower Q2 C Carlson PPNP 4 2 2 2 3 10 Q 5 10 GeV 82,59(2015) ⇥ ⇥ 3 2 2
1 4 % in radius: why care?
• Can’t be calculated to that accuracy?
Is the muon-proton interaction the same as the electron-proton interaction? Something violation of universality happening connections with muon g-2? here ??? connections with LHCb ? Kaons ? Kohl talk Outline - a) review history experiments b) List & explain possible resolutions 2 university of melbourne cssm, university of adelaide TheMuonic Experiment hydrogen experiment and rp Muonic Hydrogen
2P1/2 The Lamb shift is the splitting of the degenerate 2S1/2 and 2P1/2 eigenstates The Lamb shift is the splitting ∆ 2S−2P ELamb of the degenerate 2S1/2 and 2P1/2 eigenstates,Dominant due to in vacuum μH polar- vacuum polarization 205 of 206 meV ization
2S1/2 Dominant in eH electron self-energy ∞ 2 Zα α d(q ) −meqr 4 2 VVP(r)=− 2 e 1 − 2 1+ 2 2 r 3π 4 q q q Proton radius in! " # $ = rp ? Lamb shift J. Carroll — Proton Radius Puzzle — Slide 8 pt 2 2 E = V V = ⇡↵ (0) ( 6G0 (0)) h S| C C | Si 3 | S | E • Muon/electron mass ratio 205! 8 million times larger for muon
3 1 RESEARCH ARTICLES
Figure 3 shows the two measured mp res- The uncertainties result from quadratically This rE value is compatible with our pre- onances. Details of the data analysis are given adding the statistical and systematic uncertain- vious mp result (6), but 1.7 times more precise, in (12). The laser frequency was changed every ties of ns and nt. and is now independent of the theoretical pre- few hours, and we accumulated data for up to The charge radius. The theory (14, 16–22) diction of the 2S-HFS. Although an order of 13 hours per laser frequency. The laser frequen- relating the Lamb shift to rE yields (13): magnitude more precise, the mp-derived proton cy was calibrated [supplement in (6)] by using radius is at 7s variance with the CODATA-2010 th 2 well-known water absorption lines. The reso- DE 206:0336(15 − 5:2275(10 r DETPE (7)valueofr =0.8775(51)fmbasedonHspec- L ¼ Þ Þ E þ E nance positions corrected for laser intensity ef- 7 troscopy and electron-proton scattering. fects using the line shape model (12)are ð Þ Magnetic and Zemach radii. The theoretical stat sys where E is in meV and rE is the root mean prediction (17, 18, 27–29)ofthe2S-HFSis(13) ns 54611:16(1:00) (30) GHz 2 ¼ ð Þ square (RMS) charge radius given in fm and 2 3 2 th pol stat sys defined as rE = ∫d rr rE(r) with rE being the DE 22:9763(15 − 0:1621(10)rZ DE nt 49881:35(57) (30) GHz 3 HFS ¼ Þ þ HFS ¼ ð Þ normalized proton charge distribution. The first 9 where “stat” and “sys” indicate statistical and sys- term on the right side of Eq. 7 accounts for ð Þ tematic uncertainties, giving total experimental un- radiative, relativistic, and recoil effects. Fine and where E is in meV and rZ is in fm. The first term is certainties of 1.05 and 0.65 GHz, respectively. hyperfine corrections are absent here as a con- the Fermi energy arising from the interaction Although extracted from the same data, the fre- sequence of Eqs. 4. The other terms arise from between the muon and the proton magnetic mo- quency value of the triplet resonance, nt,isslightly the proton structure. The leading finite size effect ments, corrected for radiative and recoil con- 2 more accurate than in (6)owingtoseveralimprove- −5.2275(10)rE meV is approximately given by tributions, and includes a small dependence of 2 ments in the data analysis. The fitted line widths Eq. 1 with corrections given in (13, 17, 18). −0.0022rE meV = −0.0016 meV on the charge are 20.0(3.6) and 15.9(2.4) GHz, respectively, com- Two-photon exchange (TPE) effects, including the radius (13). patible with the expected 19.0 GHz resulting from proton polarizability, are covered by the term The leading proton structure term depends the laser bandwidth (1.75 GHz at full width at half DETPE =0.0332(20)meV(19, 24–26). Issues on rZ,definedas maximum) and the Doppler broadening (1 GHz) related with TPE are discussed in (12, 13). th exp of the 18.6-GHz natural line width. The comparison of DEL (Eq. 7) with DEL 3 3 ′ ′ ′ rZ d r d r r r (r)r (r − r ) 10 The systematic uncertainty of each measure- (Eq. 5) yields ¼ ∫ ∫ E M ð Þ ment is 300 MHz, given by the frequency cal- exp th ibration uncertainty arising from pulse-to-pulse rE 0:84087(26) (29) fm with rM being the normalized proton mag- fluctuations in the laser and from broadening ¼ 0:84087(39) fm 8 netic moment distribution. The HFS polariz- effects occurring in the Raman process. Other ¼ ð Þ systematic corrections we have considered are Muonic Hydrogen Experiment the Zeeman shift in the 5-T field (<60 MHz), AC and DC Stark shifts (<1 MHz), Doppler Randolf shift (<1 MHz), pressure shift (<2 MHz), and black-body radiation shift (<<1 MHz). All these Pohl typically important atomic spectroscopy system- atics are small because of the small size of mp. The Lamb shift and the hyperfine splitting. From these two transition measurements, we can independently deduce both the Lamb shift
(DEL = DE2P1/2−2S1/2) and the 2S-HFS splitting (DEHFS) by the linear combinations (13)
1 3 hns hnt DEL 8:8123 2 meV 4 þ 4 ¼ þ ð Þ hns − hnt DEHFS − 3:2480 2 meV 4 ¼ ð Þ ð Þ Finite size effects are included in DEL and Aldo DEHFS. The numerical terms include the cal- culated values of the 2P fine structure, the 2P3/2 hyperfine splitting, and the mixing of the 2P Antognini states (14–18). The finite proton size effects on the 2P fine and hyperfine structure are smaller than 1 × 10−4 meV because of the small overlap between the 2P wave functions and the nu- cleus. Thus, their uncertainties arising from the proton structure are negligible. By using From 2013 the measured transition frequencies ns and nt in Eqs. 4, we obtain (1 meV corresponds to 241.79893 GHz) Science paper Fig. 1. (A)Formationofmpinhighlyexcitedstatesandsubsequentcascadewithemissionof“prompt” DEexp 202:3706(23) meV 5 L ¼ ð Þ Ka, b, g.(B)Laserexcitationofthe2S-2Ptransitionwith subsequent decay to the ground state with Ka emission. (C)2Sand2Penergylevels.Themeasuredtransitionsns and nt are indicated together with DEexp 22:8089(51) meV 6 4 HFS ¼ ð Þ the Lamb shift, 2S-HFS, and 2P-fine and hyperfine splitting. EMBARGOED UNTIL 2PM U.S. EASTERN TIME ON THE THURSDAY BEFORE THIS DATE: 418 25 JANUARY 2013 VOL 339 SCIENCE www.sciencemag.org The proton radius puzzle In a picture
The proton rms charge radius measured with electrons: 0.8770 ± 0.0045 fm muons: 0.8409 ± 0.0004 fm
7.9 σ Large µp 2013 electron avg.
Small scatt. JLab
µp 2010 scatt. Mainz
2.5 % 0.01% H spectroscopy
0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 R. Pohl et al.,Nature466,213(2010). Proton charge radius R [fm] A. Antognini et al., Science 339, 417 (2013). 5 ch Randolf Pohl Mainz, 2nd June 2014 11 Deuteron is smaller too 2 2 rd(fm ) ���� CODATA
���� H-D isotope shift mu-D PRL 104,23301 ���� Science 353,669
����
Smaller Small Large rp(fm) ���� ���� ���� ���� ���� ���� Deuteron is smaller too 2 2 rd(fm ) ���� CODATA
���� H-D isotope shift mu-D PRL 104,23301 ���� Science 353,669 Deuteron radius puzzle ����
Smaller Small Large rp(fm) ���� ���� ���� ���� ���� ���� 3He charge radii
3He ion PRELIMINARY This work Sick 2001
Amroun 1994
Ottermann 1985 Retzlaff 1984 Collard 1965 4He charge radii Dunn 1983 1.82 1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2 helion charge radius [fm]
PRELIMINARY This work 4He ion Sick 2008 Randolf Pohl PhiPsi17 , 28 June 2017 20
wrong µ4He Carboni 1977
Ottermann 1985
1.66 1.665 1.67 1.675 1.68 1.685 alpha charge radius [fm] A. Antognini slides nuclear structure Barnea et al
Randolf Pohl PhiPsi17 , 28 June 2017 18 So far
• effect in the proton • effect in the deuteron, implies effect on neutron • no effect in 3He (large error bars) • no effect in 4He (small error bars) • any explanation must account for the above • how would you react to this?
8/25 RESEARCH | RESEARCH ARTICLE which are suppressed by using methods spe- ancies in the value of the fine structure constant ment. Here we remove this necessity and a cifically developed for this measurement and de- a extracted from various precision spectroscopy source of potential inaccuracies by a suitable tailed below. experiments in helium (20, 21). The root of the line shape model to compensate for the line matter is that natural line shapes of atomic reso- shape distortions. Quantum interference nances may experience deviations from a perfect Line shape distortions caused by quantum inter- Lorentzian when off-resonant transitions are Two driven oscillators ference from distant neighboring atomic reso- present. One common way of dealing with these Within the framework of perturbation theory, → nances have recently come into the focus of the effects has been to perform sophisticated nu- the induced dipole moment D w of an atom → precision spectroscopy community (18). To the merical simulations to correct the experimental driven by a laser field E at frequencyð Þ w is given best of our knowledge, this effect has been con- results (18, 20, 22–26). These simulations re- by the Kramers-Heisenberg formula (27–29). For sidered in the analysis of only one of the previous quire a highly accurate characterization of the two resonances at w0 and w0 + D with identical Hexperimentsandwasfoundtobeunimportant experimental geometry if the line center needs damping constants G,theresultingdipolemo- for that particularBeyer experimental scheme et (19). Theal toScience be determined with high accuracy358,79 relative to ment electron is given by H - effect was found to be responsible for discrep- the line width, as is the case in this measure- → → D D w º 0 ð Þ w0 w iG=2 ð À→ Þþ small radius D 1 2 þ w0 D w iG=2 ð Þ 2S-4Pð þ transitionÀ Þþ It is analogous to two coherently driven resonat- → → ing classical dipoles D0 and D1.Inthequantum Downloaded from takedescription, 2017 each ofresult these dipoles is-no constructed puzzle through an absorbing and an emitting dipole, useconnecting old eH- the initial puzzle state ( i )withthefinal as before state ( f )viatheexcitedstates(j i e ; e′ ) (see Fig. j i j i j i 2A). With the atomic dipole matrix elements djk onewith morej; k∈i; e; e′; f , thenew contributing eH dipole needed mo- Smallest radius → → → → → http://science.sciencemag.org/ ments are given by D0º E d ie d ef and D1º → → → ð Á →Þ agrees w. D E d ie′ d e′f .Theinduceddipolego halfwayD w generates a → ðfieldÁ º Þ→r D w →r= →r 3 at positionð Þ →r whose powerð spectrum ð ÞPÞ Âw; →r jis proportionalj to the → square modulus ofðD wÞ. It consists of two real valued Lorentzians andð Þ a non-Lorentzian cross term. The latter depends not only on the relative → → orientation of D0 and D1 but also on the direction of the emitted radiation relative to the orienta- tion of the dipoles. Because the orientation of the dipoles is itself a function of the laser polariza-
→ on October 22, 2017 tion, i.e., the orientation of E,theobservedcross term will effectively depend on the orientation of Averagethe laser polarization over relative to detectiontwo direc- levels tion. If the detection is not pointlike, as is the case Asymmetricin our fit measurement, function, eliminates which is designed line shifts for an as-large-as-possible collection efficiency, the from quantumexact detection interference geometry of neighboring will enter in the atomic ob- resonances. served cross term. The relative strength of the cross term tends to decrease with increasing 2707 coupleddetection solid angle, withdiff the crosseqs term com-solved pletely disappearing for detection of all radiation dependemitted, i.e., inon a 4p solid exp. angle. set up For a sufficiently large separation of the two resonances (G/D << 1), the second resonance at w0 + D can be treated as a perturbation to the resonance at w and the full line shape P w; →r 0 ð Þ can be expanded around the resonance at w0 (28)
→ C P w; r ≈ 2 2 a w w0 ð Þ w w0 G=2 þ ð À Þ ð À Þ þð Þ Fig. 1. Rydberg constant R and proton RMS charge radius rp. Values of rp derived from this 1 b w w work (green diamond) and spectroscopy of mp(mp; pink bar and violet square) agree.We find a discrepancy 0 ð 2À Þ 2 3 þ w w0 G=2 ð Þ of 3.3 and 3.7 combined standard deviations with respect to the H spectroscopy world data (12)(bluebar ð À Þ þð Þ and blue triangle) and the CODATA 2014 global adjustment of fundamental constants (3) (gray hexagon), respectively. The H world data consist of 15 individual measurements (black circles, optical measure- The first term represents the Lorentzian line ments; black squares, microwave measurements). In addition to H data, the CODATA adjustment shape with amplitude C of the isolated, unper- includes deuterium data (nine measurements) and elastic electron scattering data. An almost identical turbed resonance at w0, whereas the other two plot arises when showing R instead of rp because of the strong correlation of these two parameters. terms denote perturbations caused by the pres- 1 This is indicated by the R axis shown at the bottom. ence of the second resonance. The second term, 1
Beyer et al., Science 358, 79–85 (2017) 6 October 2017 2 of 7 RESEARCH | RESEARCH ARTICLE which are suppressed by using methods spe- ancies in the value of the fine structure constant ment. Here we remove this necessity and a cifically developed for this measurement and de- a extracted from various precision spectroscopy source of potential inaccuracies by a suitable tailed below. experiments in helium (20, 21). The root of the line shape model to compensate for the line matter is that natural line shapes of atomic reso- shape distortions. Quantum interference nances may experience deviations from a perfect Line shape distortions caused by quantum inter- Lorentzian when off-resonant transitions are Two driven oscillators ference from distant neighboring atomic reso- present. One common way of dealing with these Within the framework of perturbation theory, → nances have recently come into the focus of the effects has been to perform sophisticated nu- the induced dipole moment D w of an atom → precision spectroscopy community (18). To the merical simulations to correct the experimental driven by a laser field E at frequencyð Þ w is given best of our knowledge, this effect has been con- results (18, 20, 22–26). These simulations re- by the Kramers-Heisenberg formula (27–29). For sidered in the analysis of only one of the previous quire a highly accurate characterization of the two resonances at w0 and w0 + D with identical Hexperimentsandwasfoundtobeunimportant experimental geometry if the line center needs damping constants G,theresultingdipolemo- for that particularBeyer experimental scheme et (19). Theal toScience be determined with high accuracy358,79 relative to ment electron is given by H - effect was found to be responsible for discrep- the line width, as is the case in this measure- → → D D w º 0 ð Þ w0 w iG=2 ð À→ Þþ small radius D 1 2 þ w0 D w iG=2 ð Þ 2S-4Pð þ transitionÀ Þþ It is analogous to two coherently driven resonat- → → ing classical dipoles D0 and D1.Inthequantum Downloaded from takedescription, 2017 each ofresult these dipoles is-no constructed puzzle through an absorbing and an emitting dipole, useconnecting old eH- the initial puzzle state ( i )withthefinal as before state ( f )viatheexcitedstates(j i e ; e′ ) (see Fig. j i j i j i 2A). With the atomic dipole matrix elements djk onewith morej; k∈i; e; e′; f , thenew contributing eH dipole needed mo- Smallest radius → → → → → http://science.sciencemag.org/ ments are given by D0º E d ie d ef and D1º → → → ð Á →Þ agrees w. D E d ie′ d e′f .Theinduceddipolego halfwayD w generates a → ðfieldÁ º Þ→r D w →r= →r 3 at positionð Þ →r whose powerð spectrum ð ÞPÞ Âw; →r jis proportionalj to the → square modulus ofðD wÞ. It consists of two real valued Lorentzians andð Þ a non-Lorentzian cross term. The latter depends not only on the relative → → orientation of D0 and D1 but also on the direction of the emitted radiation relative to the orienta- tion of the dipoles. Because the orientation of the dipoles is itself a function of the laser polariza-
→ on October 22, 2017 tion, i.e., the orientation of E,theobservedcross term will effectively depend on the orientation of Averagethe laser polarization over relative to detectiontwo direc- levels tion. If the detection is not pointlike, as is the case Asymmetricin our fit measurement, function, eliminates which is designed line shifts for an as-large-as-possible collection efficiency, the from quantumexact detection interference geometry of neighboring will enter in the atomic ob- resonances. served cross term. The relative strength of the cross term tends to decrease with increasing 2707 coupleddetection solid angle, withdiff the crosseqs term com-solved pletely disappearing for detection of all radiation dependemitted, i.e., inon a 4p solid exp. angle. set up For a sufficiently large separation of the two resonances (G/D << 1), the second resonance at w0 + D can be treated as a perturbation to the resonance at w and the full line shape P w; →r 0 ð Þ can be expanded around the resonance at w0 (28)
→ C P w; r ≈ 2 2 a w w0 ð Þ w w0 G=2 þ ð À Þ ð À Þ þð Þ Fig. 1. Rydberg constant R and proton RMS charge radius rp. Values of rp derived from this 1 b w w work (green diamond) and spectroscopy of mp(mp; pink bar and violet square) agree.We find a discrepancy 0 ð 2À Þ 2 3 þ w w0 G=2 ð Þ of 3.3 and 3.7 combined standard deviations with respect to the H spectroscopy world data (12)(bluebar ð À Þ þð Þ and blue triangle) and the CODATA 2014 global adjustment of fundamental constants (3) (gray hexagon), respectively. The H world data consist of 15 individual measurements (black circles, optical measure- The first term represents the Lorentzian line ments; black squares, microwave measurements). In addition to H data, the CODATA adjustment shape with amplitude C of the isolated, unper- includes deuterium data (nine measurements) and elastic electron scattering data. An almost identical turbed resonance at w0, whereas the other two plot arises when showing R instead of rp because of the strong correlation of these two parameters. terms denote perturbations caused by the pres- 1 This is indicated by the R axis shown at the bottom. ence of the second resonance. The second term, 1
Beyer et al., Science 358, 79–85 (2017) 6 October 2017 2 of 7 26 Jan., 2018 1801.08816 Paris 1S-3S hydrogen Phys. Rev. Lett. 120, 183001 4(2018)
TABLE I. Optimal velocity distribution parameters (Eq. 1), and frequency before and after light shift correction, for each data set. Only the last four digits of the 1S − 3S(F =1) frequency are given in the table, ν =2922742936xxx.xkHz.
Data set σ [km/s] v0 [km/s] ν [kHz] νLS [kHz] 2013 1.526(27) 0.75(28) 724.8(1.5) 718.9(1.9) LP1 1.515(52) 1.23(55) 734.2(3.9) 723.8(4.9) LP2 1.495(32) 1.33(31) 730.3(2.4) 718.2(4.3) HP 1.521(85) 0.87(78) 713.2(7.1) 706.9(12.4)
been used: the voltage of the photodiode recording the transmitted UV power (for the 2013 recordings) and the square root of the two-photon absorption signal height Is there still a proton radius puzzle? (for 2016-2017). As the signal height depends on pres- FIG. 3. Proton charge radius values from H spectroscopy, sure, the correction coefficient was determined separatelyIs electron-hydrogenwith 1σ errorbars. The spectroscopy pink bar is the value accurate from muonic enough? hy- for each pressure value. This coefficient is defined as the drogen spectroscopy [6]. The CODATA-2014 H-spectroscopy slope of a linear extrapolation of the frequency with re- average [3] (light blue bar and hexagon) includes RF mea- spect to the chosen parameter. The light-shift-corrected surements (blue triangles) as well as combinations of optical transitions with the 1S − 2S frequency (blue circles). Green average frequencies are given in the last column of Ta- squares are obtained from optical transitions measured in ble I. Garching since 2014 [8, 10], and the red diamond is the present Collisions between atoms can also induce frequency work. shifts, depending linearly on the pressure. Unfortunately, the pressure gauge was replaced between the two record- ing sessions, so that we cannot merge the two sessions for take into account the frequency shift resulting from the the collisionnal shift analysis. To determine this pressure cross-damping effect, following our theoretical estimation shift for the 2013 data set, measurements were carried of this shift [20]. out several times during that recording session, for two All the frequency measurements were done with re- or three pressure values in the same day, with no applied spect to the 100 MHz reference signal from LNE-SYRTE. magnetic field. At the time, the velocity distribution was This reference was obtained from a hydrogen maser, measured for only one pressure value and our velocity dis- whose frequency was continuously measured by the LNE- tribution model could allow for pressure dependence of SYRTE atomic fountains realizing the frequency of the SI second to a few 10−16 [21, 22]. Using a simple linear the parameter v0, so that we did not know which param- −16 eters should be used to analyse the other pressure points frequency drift of the order of 10 per day to model the [19]. The analysis of the 2016 data gave us insight on H-maser behavior over each period, we estimate the aver- this question. In fact, the velocity distribution does not age fractional shift of the reference signal with respect to − × −15 − × −15 seem to depend significantly on pressure, at least within the SI to be 205(2) 10 in 2013, and 357(2) 10 experimental uncertainties (see Table I). To check this in 2016-2017. This yields an absolute correction to the 1S − 3S transition frequency of −599(6) Hz for the 2013 assumption, we have fitted a number of signals using the − various optimal distributions. The resuting change in measurement and 1043(6) Hz for the 2016-2017 mea- the center frequency was at most of about 3 kHz. Hence, surement. when analyzing the 2013 recordings, we use the same ve- The centroid value of the transition is calculated by locity distribution for all pressure values. The slope of adding a hyperfine correction of +341949.077(3) kHz de- a linear fit gives the collisional shift coefficient. To take rived from experimental values of the 1S and 2S hyper- into account the uncertainty on the velocity distribution fine splittings [23]. Eventually, we obtain for the two determination, an uncertainty of 3 kHz divided by the recording sessions, maximal pressure difference is added in quadrature to 2013 ν1S−3S =2922743278671.6(2.8) kHz, (2) that of the obtained coefficient. This entails a correction 2017 ν − =2922743278671.0(4.9) kHz. (3) of +3.6(2.0) kHz on the frequency quoted in the last col- 1S 3S umn of Table I. For the 2016 session, since the velocity We estimate a correlation coefficient of 0.186. The distribution was determined for each pressure value, we weighted average of our two measurements is then simply extrapolate the light-shift corrected frequencies of ν1S−3S =2922743278671.5(2.6) kHz. Combining this the three data sets to zero pressure. result with the 1S − 2S transition frequency [1], one At this point, we add a correction of +0.6(2) kHz to can derive values of the Rydberg constant, R∞ = Possible resolutions
• Electron H spectroscopy not so accurate • Strong interaction effect in two photon exchange diagram • Muon interacts differently than electron!- new scalar boson
The last two resolutions could be halved and not be in conflict with data
Shift from history to our11 efforts to explain The search begins • Pohl et al table 32 terms! • Most important -HFS- measured Jan ’13 muon electron
A new effect on mu-H energy shift must vary at least as fast as lepton mass to the fourth power, if short-ranged
12/25 The search begins • Pohl et al table 32 terms! • Most important -HFS- measured Jan ’13 muon electron
A new effect on mu-H energy shift must vary at least as fast as lepton mass to the fourth power, if short-ranged
12/25 Suppose radii extracted in earlier H experiments is correct, some new muon effect is responsible What energy difference corresponds to 4% in radius? 2 3 Measured =206.2949(32)= 206.0573(45)-5.2262 rp +0.0347 rp meV computed
Explain puzzle with radius as in earlier H atom measures: increase 206.0573 meV by 0.31 meV-attractive effect on 2S state
Can go half way and not disagree with data
13 Two photon exchange
2 3 Measured =206.2949(32)= 206.0573(45)-5.2262 rp +0.0347 rp meV computed
2
′ ℓ − k of momentum q = p − p.as: ℓ ℓ µ ′ µ Miller2 2 PLB2 µ 2012 Γ (p ,p)=γN F1(−q )+F1(−q )F (−q )Oa,b,c (2) ′ µ ′ µ (p + p ) ′ (p · γN − M) (p · γN − M) O = [Λ+(p ) + Λ+(p)] energy shift proportional a 2M M M µ 2 2 2 ′2 2 2 µ 4 Ob =((p − M )/M +(p − M )/M )γN to lepton mass P P ′ µ ′ µ (p · γN − M) (p · γN − M) µ Oc = Λ+(p )γN + γN Λ+(p), M M FIG. 1: Direct two-photon exchange graph corresponding to where three possible forms are displayed. OtherRe terms part of the of hitherto virtual neglected term. Compton The dashed line denotes scattering the the vertex function needed to satisfy the WT identity do lepton; the solid line, the nucleon; the wavy lines photons; and the ellipse the off-shell nucleon. not contribute significantly to the Lamb shift and are not Im part is measured 2 shown explicitly. The proton Dirac form factor, F1(−q ) 2 is empirically well represented as a dipole F1(−q )=(1− q2/Λ2)−2, (Λ = 840 MeV) for the values of −q2 ≡ Q2 > 0 use dispersion relations of up to about 1 GeV2 needed here. F (−q2) is an off- interference between one on-shell and one off-shell part shell form factor, and Λ+(p)=(p · γNbut+ M)/ (2unknownM) is an of the vertex subtraction function. The change infunction the invariant am- is needed µ operator that projects on the on-mass-shell proton state. plitude, MOff , due to using Eq. (2) along with Oa ,tobe We use Oa unless otherwise stated. evaluated between fermion spinors, is given in the rest We take the off-shell form factor F (−q2) to vanish at frame by ) q2 = 0. This means that theCan charge of theaccount off-shell proton2 for 0.31 meV,2 no conflict with e-HIm T1(⌫,Q will be the same as the chargeT of1 a( freeq proton,0,Q and is)=T1(0,Q )+( )q0P d⌫ 4 4 2 2 2 ⌫(⌫ q ) demanded by current conservation as expressed through e d k F1 (−k )F (−k ) 14 0 MOff = ··· (4) the Ward-Takahashi identity [24, 25]. We assume 2M 2 " (2π)4 (k2 + iϵ)2 2 2 µ ν ν µ −λq /b ×(γN (2p + k) + γN (2p + k) ) F (−q2)= . (3) 2 2 1+ξ (l · γ − k · γ + m) (l · γ + k · γ + m) R (1 − q /Λ ) × γ γ + γ γ , # µ k2 − 2l · k + iϵ ν ν k2 +2l · k + iϵ µ$ This purely phenomenological form! is simple and clearly not unique. The parameter b is expected to be of the order of the pion mass, because these longest range com- where the lepton momentum is l =(m, 0, 0, 0), the vir- ponents of the nucleon are least bound and more suscep- tual photon momentum is k and the nucleon momentum tible to the external perturbations putting the nucleon p =(M,0, 0, 0). The intermediate proton propagator off its mass shell. At large values of |q2|, F has the same is cancelled by the off-mass-shell terms of Eq. (2). This fall-off as F1, if ξ =0.WetakeΛ = Λ here. graph can be thought of as involving a contact interaction We briefly discuss the expected influence of using and the amplitude in Eq. (4) as a new proton polariza- Eq. (2). The ratio, R,ofoff-shell! effects to on-shell ef- 2 tion correction corresponding to a subtraction term in the (p·γN −M) q 2 2 fects, R ∼ M λ b2 , (|q | ≪ Λ ) is constrained by dispersion relation for the two-photon exchange diagram a variety of nuclear phenomena such as the EMC effect that is not constrained by the cross section data [34]. (10-15%), uncertainties in quasi-elastic electron-nuclear The resulting virtual-photon-proton Compton scattering scattering [26], and deviations from the Coulomb sum µ ν amplitude, containing the operator γN γN corresponds to rule [27]. For a nucleon experiencing a 50 MeV central the T2 term of conventional notation [35], [36]. Eq. (4) 2 2 potential, (p · γN − M)/M ∼ 0.05, so λq /b is of or- is gauge-invariant; not changed by adding a term of the der 2. The nucleon wave functions of light-front quark- form kµ kν /k4 to the photon propagator. models [33] contain a propagator depending on M 2. Thus the effect of nucleon virtuality is proportional to Evaluation proceeds in a standard way by taking the the derivative of the propagator with respect to M,orof sum over Dirac indices, performing the integral over k0 the order of the wave function divided by difference be- by contour rotation, k0 →−ik0, and integrating over the tween quark kinetic energy and M. This is about three angular variables. The matrix element M is well approx- times the average momentum of a quark (∼ 200 MeV/c) imated by a constant in momentum space, for momenta divided by the nucleon radius or roughly M/2. Thus typical of a muonic atom, and the corresponding poten- 2 2 r r R ∼ (p · γN − M)2/M , and the natural value of λq /b tial V = iM has the form V ( )=V0δ( ) in coordinate is of order 2. space. This is the “scattering approximation” [3]. Then 2 The lowest order term in which the nucleon is suffi- the relevant matrix elements have the form V0 |Ψ2S(0)| , ciently off-shell in a muonic atom for this correction to where Ψ2S is the muonic hydrogen wave function of the produce a significant effect is the two-photon exchange state relevant to the experiment of Pohl et al. We use 2 3 diagram of Fig. 1 and its crossed partner, including an |Ψ2S(0)| =(αmr) /(8π), with the lepton-proton re- 2 Almost unknown T¯1(0,Q ) Miller PLB 2012
subt 2 2 dQ2 2 E ↵ m S(0) Q2 Floop(Q ) / ··· Soft proton number of R 2 Infinite Floop(Q ) give 0.31 meV OPE satisfy all constraints Ch, PT Q2 Recast in EFT- parameters seem natural So far size of this term cannot be determined from theory-experiment is needed lattice QCD may disprove above sentence
15 Nuclear dependence of short-ranged mu-p effects 7 The first example we consider is the model of Ref. [4]. To make a prediction for the nucleus, one needs to know the contribution of the neutrons. Using the single-quark dominance idea of Sect II gives a specific result obtained by considering the factors of the square of the quark charge that would appear in the calcuation. A proton contains two up quarks and one• downEnergy quark, with shift a resulting is proportional quark charge squared to square factor of 2(2 of/ 3)2e2 + 1(1/3)2e2 = e2 . For a neutron one would have 1(2muon/3)2e2 + wave 2(1/3) 2functione2 =2/3e2. Thusat the the origin neutron contribution would be 2/3 that of the proton. As result, if one considers the Lamb shift in the electron-deuteron atom, the e↵ect would 5/3 as large as for a proton. Such an e↵ect would contradict the existing good agreement between theory and experiment [11]. GAM More generally, suppose theSuppose contribution you of a protonhave toeffect the Lamb that shift gives is E (0.3meV) energy to resolve the proton radius 1501.01036 • p puzzle) and that of the neutronshifts is E nEp. Then (on for proton) a nucleus with En A(onnucleons neutron) and Z protons, we find m 3 m 3 1+ µ 2 1+ µ mp 3 RA mp 3 EA = mµ Z (ZEp + NEn) 1 ( 2 ) mµ Z (ZEp + NEn), (23) 1+ O aµ ⇡ 1+ Amp ! ✓ ◆ Amp ! where aµ is the muon Bohr radius (>100 times larger than nuclearSize radius, ofR Anucleus). The meaning of Eq. (23) is that the contributions of such contact interactions increase very rapidly with atomic number. Nuclear shift 4 In particular, the prediction of Ref. [4](with En =2/3Ep) for He is a Lamb shift that is (1.27) 8 (2)5/3 10 meV, a huge number. The expressionSquare Eq. (23)of wave applies tofun all modelsCounting in which the contribution to the Lamb shift⇡ enters as a delta function (or of very short range) in the lepton-nucleon coordinate, including [9, 10]. In Ref. [9], which My model: ~0.3 meV (1+0.3)(8)(2) =-6.3 meV about 6 st. dev 4 concerns polarizability corrections, the neutron contribution, En could vanish, so that the contribution for He would 4 be 20 Ep=6 meV. In Ref. [10], which concerns a gravitational e↵ect, En = Ep, so the prediction for He would be 16 off RIP any short range idea 40Ep=12 meV. These various predictions will be tested in an upcoming experiment [12]. It is also worth mentioning the MUon proton Scattering Experiment (MUSE) [13], a simultaneous measurement of µ+ p and e+ p scattering and also a simultaneous measurement of µ p and e p scattering that is particularly sensitive to the presence of contact interactions [9] .
VII. SUMMARY
The work presented here supports the idea of the existence of a non-perturbative lepton-pair content of the proton. Such components are not forbidden by any symmetry principle and therefore must appear. However, the presented calculations show that such a content is not a candidate to explain the proton radius puzzle. This is because computation of the necessary loop e↵ects is cannot yield an e↵ective Hamiltonian of the strength and form of Eq. (2). 2 The 1/ml behavior of that equation in Ref. [4] is necessary to obtain the needed magnitude of the separate electron and muon Lamb shifts. Instead, loop calculations are expected to lead to a a dependence of 1/m2,withm the constituent quark mass. This means that the e↵ect of Ref. [4] is expected to be entirely negligible. This is in accord with the estimate: (↵/⇡) times the polarizability correction that is obtained from evaluating the relevant Feynman diagrams. More generally: it can be said that the proton does have Fock -space components containing lepton pairs. However, the simplest and most reliable method of treating such pairs is to compute gauge-invariant sets of Feynman diagrams using QED perturbation theory.
VIII. ACKNOWLEDGEMENTS
This material is based upon work supported by the U.S. Department of Energy O ce of Science, O ce of Basic En- ergy Sciences program under Award Number DE-FG02-97ER-41014. I thank S. Paul, U. Jentschura, S. Karshenboim and M. Eides for useful discussions.
[1] R. Pohl, A. Antognini, F. Nez, F. D. Amaro, F. Biraben, J. M. R. Cardoso, D. S. Covita, A. Dax, S. Dhawan, L. M. P. Fernandes, A. Giesen, T. Graf, T. W. H¨ansch, P. Indelicato, L. Julien, C.-Y. Kao, P. Knowles, E.-O. Le Bigot, Y. W. Liu, J. A. M. Lopes, L. Ludhova, C. M. B. Monteiro, F. Mulhauser, T. Nebel, P. Rabinowitz, J. M. F. dos Santos, L. A. Schaller, K. Schuhmann, C. Schwob, D. Taqqu, J. F. C. A. Veloso, and F. Kottmann, Nature (London) 466,213(2010). [2] A. Antognini, F. Nez, K. Schuhmann, F. D. Amaro, F. Biraben, J. M. R. Cardoso, D. S. Covita, A. Dax, S. Dhawan, M. Diepold, L. M. P. Fernandes, A. Giesen, A. L. Gouvea, T. Graf, T. W. H¨ansch, P. Indelicato, L. Julien, C.-Y. Kao, P. Knowles, F. Kottmann, E.-O. Le Bigot, Y.-W. Liu, J. A. M. Lopes, L. Ludhova, C. M. B. Monteiro, F. Mulhauser, T. 2
′ ℓ − k of momentum q = p − p.as: ℓ ℓ µ ′ µ 2 2 2 µ Γ (p ,p)=γN F1(−q )+F1(−q )F (−q )Oa,b,c (2) ′ µ ′ (p + p ) ′ (p · γN − M) (p · γN − M) energy shift proportional Oµ = [Λ (p ) + Λ (p)] a 2M + M M + ′ Oµ =((p2 − M 2)/M 2 +(p 2 − M 2)/M 2)γµ 4 b N to lepton mass P P ′ µ ′ µ (p · γN − M) (p · γN − M) µ Oc = Λ+(p )γN + γN Λ+(p), ExplainM puzzleM withFIG. radius 1: Direct two-photon as exchange in graphH correspondingatom to increase 206.0573 meV where three possible forms are displayed. Other terms of the hitherto neglected term. The dashed line denotes the the vertex function needed to satisfy the WT identity do lepton; the solid line, the nucleon; the wavy lines photons; and the ellipse the off-shell nucleon. not contributeby significantly0.31 tomeV-attractive the Lamb shift and are not effect on 2S state, reproduce Lamb shift in 2 shown explicitly. The proton Dirac form factor, F1(−q ) 2 is empirically well representedR as a dipole F1(−q )=(1− 2 2 −2 2 2 proton, deuterium q /Λ ) , (Λ = 840 MeV) for thefractional values of −q ≡ Q > 0 of up to about 1 GeV0.072 needed here. F (−q2) is an off- interference between one on-shell and one off-shell part shell form factor, and Λ+(p)=(p · γN + M)/(2M) is an of the vertex function. The change in the invariant am- µ operator that projects on the on-mass-shellchange proton state. plitude, MOff , due to using Eq. (2) along with Oa ,tobe We use Oa unless otherwise0.06 stated. evaluated between fermion spinors, is given in the rest We take the off-shell form factor F (−q2) to vanish at frame by q2 = 0. This means that the charge of the off-shell proton 10-20% effect will be the same as the charge of a free proton, and is 0.05 4 4 2 2 2 demanded by current conservation as expressed through e d k F1 (−k )F (−k ) MOff = (4) the Ward-Takahashi identity [24, 25]. We assume 2M 2 " (2π)4 (k2 +iniϵ)2 scattering 2 2 µ ν ν µ −λq /b ×(γN (2p + k) + γN (2p + k) ) F (−q20.04)= . (3) 2 2 1+ξ (l · γ − k · γ + m) (l · γ + k · γ + m) (1 − q /Λ ) × γ γ + γ γ , 200 MeV/c # µ k2 − 2l · k + iϵ ν ν k2 +2l · k + iϵ µ$ This purely phenomenological0.03 form! is simple and clearly not unique. The parameter b is expected to be of the 100 MeV/c order of the pion mass, because these longest range com- where the lepton momentum is l =(m, 0, 0, 0), the vir- ponents of the nucleon are least bound and more suscep- tual photon momentum is k and the nucleon momentum tible to the external0.02 perturbations putting the nucleon p =(M,0, 0, 0). The intermediate proton propagator off its mass shell. At large values of |q2|, F has the same is cancelled by the off-mass-shell terms of Eq. (2). This fall-off as F1, if ξ =0.Wetake0.01 Λ = Λ here. graph can be thought of as involving a contact interaction We briefly discuss the expected influence of using and the amplitude in Eq. (4) as a new proton polariza- Eq. (2). The ratio, R,ofoff-shell! effects to on-shell ef- 2 tion correction corresponding to a subtraction term in the (p·γN −M) q 2 2 fects, R ∼ λ 2 , (|q | ≪ Λ ) is constrained by dispersion relation for the two-photon exchange diagram M b qcm a variety of nuclear phenomena such0.5 as the1.0 EMC1.5 effect 2.0that2.5 is not constrained3.0 by the crossRadians section data [34]. (10-15%), uncertainties in quasi-elastic electron-nuclear The resulting virtual-photon-proton Compton scattering scattering [26], and deviations from the Coulomb sum amplitude, containing the operator γµ γν corresponds to N N rule [27]. For a nucleon experiencing a 50 MeV central the T2 term of conventional notation [35], [36]. Eq. (4) 2 2 potential, (p · γN − M)/M ∼ 0.05, so λq /b is of or- is gauge-invariant; not changed by adding a term of the der 2. The nucleon wave functions of light-front quark- form kµ kν /k4 to the photon propagator. models [33] contain a propagator dependingfails on M 2. in 4He Thus the effect of nucleon virtuality is proportional to Evaluation proceeds in a standard way by taking the the derivative of the propagatorµA with respect3 to M,orof psum over Dirac indices,n performing the integral over k0 4 the order of the wave E functionL dividedZ by difference(Z be-ELby contour+ N rotation, EkL0 →−)ik0, and 4 integrating standard over the deviations o↵ He tween quark kinetic energy and M/. This is about three angular variables. The matrix element M is well approx- times the average momentum of a quark (∼ 200 MeV/c) imated by a constant in momentum space, for momenta or 1 st dev 17 divided by the nucleon radius or roughly M/2. Thus typical of a muonic atom, and the corresponding poten- 2 2 r r R ∼ (p · γN − M)2/M , and the natural value of λq /b tial V = iM has the form V ( )=V0δ( ) in coordinate is of order 2. space. This is the “scattering approximation” [3]. Then 2 The lowest order term in which the nucleon is suffi- the relevant matrix elements have the form V0 |Ψ2S(0)| , ciently off-shell in a muonic atom for this correction to where Ψ2S is the muonic hydrogen wave function of the produce a significant effect is the two-photon exchange state relevant to the experiment of Pohl et al. We use 2 3 diagram of Fig. 1 and its crossed partner, including an |Ψ2S(0)| =(αmr) /(8π), with the lepton-proton re- + + e /e and µ /µ scattering on proton So what? MUSE expt using RF time measurements. Magnet polarities can be reversed to allow the channel to transport either positive or negative polarity particles. http://www.physics.rutgers.edu/~rgilman/elasticmup/ B. Detector Overview PSI proposal R-12-01.1 • constrains two photon effect, which still survives at significant level • if large radius correct and no two photon all leptons see large radius
FIG. 3. A Geant4 simulation showing part of the MUSE experimental system. Here one sees the beam going through the GEM chambers and the scattering chamber, along with the spectrometer wire chambers and scintillator hodoscopes. The beam SciFi’s, quartz Cherenkov, and beam monitor scintillators are missing from this view. • if small radius correct and no two photon all The ⇡M1 channel features a momentum dispersed ( 7 cm/%) intermediate focal point (IFP) ⇡ leptons see small radius and a small beam spot ( x,y < 1 cm) at the scattering target. The base line design for the MUSE beam detectors has a collimator and a scintillating fiber detector (SciFi) at the intermediate focus. Some of the detectors in the target region are shown in Fig. 3. After the• channel and immediatelywill not see a new light particle, but all leptons before the target there are a SciFi detector, a quartz Cherenkov detector, and a setsee of GEM large radius chambers. A high precision beam line monitor scintillator hodoscope is downstream of the target. 18 The IFP collimator serves to cut the ⇡M1 channel acceptance to reduce the beam flux to manageable levels. The IFP SciFi measures the RF time, for use in determining particle type, and measures beam particle position, to determine the particle momentum and thus the beam momentum spectrum.
The target SciFi measures the RF time, for use in determining particle type. The quartz
14 + + e /e and µ /µ scattering on proton So what? MUSE expt using RF time measurements. Magnet polarities can be reversed to allow the channel to transport either positive or negative polarity particles. http://www.physics.rutgers.edu/~rgilman/elasticmup/ B. Detector Overview PSI proposal R-12-01.1 • constrains two photon effect, which still survives at significant level • if large radius correct and no two photon all leptons see large radius
FIG. 3. A Geant4 simulation showing part of the MUSE experimental system. Here one sees the beam going through the GEM chambers and the scattering chamber, along with the spectrometer wire chambers and scintillator hodoscopes. The beam SciFi’s, quartz Cherenkov, and beam monitor scintillators are missing from this view. • if small radius correct and no two photon all The ⇡M1 channel features a momentum dispersed ( 7 cm/%) intermediate focal point (IFP) ⇡ leptons see small radius and a small beam spot ( x,y < 1 cm) at the scattering target. The base line design for the MUSE beam detectors has a collimator and a scintillating fiber detector (SciFi) at the intermediate focus. Some of the detectors in the target region are shown in Fig. 3. After the• channel and immediatelywill not see a new light particle, but all leptons before the target there are a SciFi detector, a quartz Cherenkov detector, and a setsee of GEM large radius chambers. A high precision beam line monitor scintillator hodoscope is downstream of the target. 18 The IFP collimator serves to cut the ⇡M1 channelN acceptanceex tot reduce- them beamu fluxo ton interacts differently than electron manageable levels. The IFP SciFi measures the RF time, for use in determining particle type, and measures beam particle position, to determine the particle momentum and thus thein beamtroduce new scalar boson momentum spectrum.
The target SciFi measures the RF time, for use in determining particle type. The quartz
14 20 Sep 2004 16:46 AR AR228-NS54-05.tex AR228-NS54-05.Sgm LaTeX2e(2002/01/18) P1: JRX
20 Sep 2004 16:46 AR AR228-NS54-05.tex AR228-NS54-05.Sgm LaTeX2e(2002/01/18) P1: JRX 116 DAVIER ! MARCIANO
1. INTRODUCTION
One of the great successes of the116 DiracDA equationVIER ! MARCIANO (1) was its prediction that the magnetic dipole moment, µ,ofaspin s 1/2 particle such as the electron (or ⃗ |⃗|= muon) is given by 1. INTRODUCTION e µl gl s,One ofl thee great,µ..., successes of the Dirac equation1. (1) was its prediction that the ⃗ = 2ml ⃗ magnetic= dipole moment, µ,ofaspin s 1/2 particle such as the electron (or muon) is given by ⃗ |⃗|= with gyromagnetic ratio gl 2, a value already implied by early atomic spec- = e troscopy. Later it was realized that a relativistic quantumµl fieldgl theorys, suchl e as,µ..., 1. quantum electrodynamics (QED) can give rise via quantum⃗ fluctuations= 2ml ⃗ to a shift= with gyromagnetic ratio g 2, a value already implied by early atomic spec- in gl , l troscopy. Later it was realized= that a relativistic quantum field theory such as quantumgl 2 electrodynamics (QED) can give rise via quantum fluctuations to a shift al − , 2. ≡ in g2l , gl 2 called the magnetic anomaly. In a now classic QED calculation,a Schwingerl − , (2) 2. ≡ 2 found the leading (one-loop) effect (Figure 1), called the magnetic anomaly. In a now classic QED calculation, Schwinger (2) α found the leading (one-loop) effect (Figure 1), al 0.00116 = 2π ≃ α al 0.00116 e2 = 2π ≃ α 1/137.036. e2 3. ≡ 4π ≃ α 1/137.036. 3. ≡ 4π ≃ This agreed beautifully with experiment (3), thereby providing strong confidence This agreed beautifully with experiment (3), thereby providing strong confidence in the validity of perturbative QED. Today,in the we validity continue of perturbative the tradition QED. of Today, testing we QED continue the tradition of testing QED and its SU(3)C SU(2)L U(1)Y standard-modeland its SU(3) (SM)C SU(2) extensionL U(1)Y (whichstandard-model includes (SM) extension (which includes × × strong and electroweak×exp interactions)× by measuring aexp for the electron and muon strong and electroweak interactions) by measuring al for the electron and muon l ever more precisely and comparingSM these measurements with aSM expectations, ever more precisely and comparing these measurements with al expectations, l calculated to much higher order in perturbationcalculated to theory. muchmuon higher Such order comparisons in perturbation anomalous test theory. Such comparisons test moment by University of Washington on 01/27/13. For personal use only. by University of Washington on 01/27/13. For personal use only. 3.6 st. dev anomaly now
Annu. Rev. Nucl. Part. Sci. 2004.54:115-140. Downloaded from www.annualreviews.org fix add heavy photon Annu. Rev. Nucl. Part. Sci. 2004.54:115-140. Downloaded from www.annualreviews.org interacts preferentially with + muon Figure 1 The first-order QED correction to g-2 of the Muon data is g-2 - BNL exp’t, muon. H Figure 1 The first-order Hertzog- FermiLab now... QED correction to g-2 of the muon. Maybe dark Postulate new scalar matter, boson! energy particles look for violation show up in of 4 momentum conservation in muon elastic ep scattering physics! e p Dark Light interest µ p 19 New scalar bosons assumes puzzle exists • give μ-p Lamb shift • almost no hyperfine in μ proton • small effect for D, almost no effect 4He • consistent with g-2 of μ and electron • avoid many other constraints be found be found • • week ending PRL 117, 101801 (2016) PHYSICAL REVIEW LETTERS 2 SEPTEMBER 2016
Electrophobic Scalar Boson and Muonic Puzzles
Yu-Sheng Liu,* David McKeen,† and Gerald A. Miller‡ Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA (Received 24 May 2016; published 2 September 2016) A new scalar boson which couples to the muon and proton can simultaneously solve the proton radius puzzle and the muon anomalous magnetic moment20 discrepancy. Using a variety of measurements, we constrain the mass of this scalar and its couplings to the electron, muon, neutron, and proton. Making no assumptions about the underlying model, these constraints and the requirement that it solve both problems limit the mass of the scalar to between about 100 keVand 100 MeV. We identify two unexplored regions in the coupling constant-mass plane. Potential future experiments and their implications for theories with mass-weighted lepton couplings are discussed.
DOI: 10.1103/PhysRevLett.117.101801
Recent measurements of the proton charge radius using the lepton l μ;e muon’s magnetic moment due to the Lamb shift in muonic hydrogen are troublingly dis- one-loop ϕ ðexchange¼ Þ is given by [12] crepant with values extracted from hydrogen spectroscopy and electron-proton scattering. The value from muonic 2 1 2 αϵl 1 − z 1 z hydrogen is 0.84087(39) fm [1,2] while the CODATA Δal dz ð 2 Þ ð þ Þ 2 : 1 ¼ 2π 0 1 − z m =m z ð Þ average of data from hydrogen spectroscopy and e-p Z ð Þ þð ϕ lÞ scattering yields 0.8751(61) fm [3]; these differ at more Scalar exchange between fermions f and f leads to a than 5σ. Although the discrepancy may arise from subtle 1 2 Yukawa potential, V r −ϵ ϵ αe−mϕr=r. In atomic lepton-nucleon nonperturbative effects within the standard ð Þ¼ f1 f2 model or experimental uncertainties [4,5], it could also be a systems, this leads to an additional contribution to the signal of new physics involving a violation of lepton Lamb shift in the 2S-2P transition. For an atom of A and universality. Z this shift is given by [13] The muon anomalous magnetic moment provides lN α another potential signal of new physics. The BNL [6] δEL − ϵl Zϵp A − Z ϵn f alNmϕ ; 2 measurement differs from the standard model prediction ¼ 2alN ½ þð Þ ð Þ ð Þ exp th by at least 3 standard deviations, Δaμ aμ − aμ −11 ¼ ¼ where f x x2= 1 x 4 [9,14], with a Zαm −1 287 80 × 10 [7,8]. ð Þ¼ ð þ Þ lN ¼ð lNÞ Að newÞ scalar boson ϕ that couples to the muon and the Bohr radius and mlN the reduced mass of the lepton- proton could explain both the proton radius and g − 2 nucleus system. Throughout this Letter we set ð Þμ puzzles [9]. We investigate the couplings of this boson to −11 μH standard model fermions f, which appear as terms in the Δaμ 287 80 × 10 ; δEL −0.307 56 meV ¯ ¼ ð Þ ¼ ð Þ Lagrangian, L ⊃ eϵfϕff, where ϵf gf=e and e is the 3 electric charge of the proton. Other authors¼ have pursued ð Þ this idea, but made further assumptions relating the μH within 2 standard deviations. This value of δEL is equal couplings to different species; e.g., in Ref. [9] ϵp is taken equal to ϵ and in Ref. [10],mass-weightedcouplingsare to the energy shift caused by using the different values of μ the proton radius [1 3,15]. Using Eq. (3) allows us to assumed. References [9] and [10] both neglect ϵ . We make – n determine both ϵ and ϵ as functions of m . The no a priori assumptions regarding signs or magnitudes of p μ ϕ the coupling constants. The Lamb shift in muonic hydrogen unshaded regions in Figs. 1 and 3 show the values of ϵp and ϵμ, as functions of the scalar’s mass, which lead to fixes ϵμ and ϵp to have the same sign which we take to be the values of a and δEμH in Eq. (3). positive. ϵe and ϵn are allowed to have either sign. Δ μ L We focus on the scalar boson possibility because scalar We study several observables sensitive to the couplings exchange produces no hyperfine interaction, in accord with of the scalar to neutrons ϵn and protons ϵp to obtain new observation [1,2]. The emission of possible new vector bounds on mϕ. particles becomes copious at high energies, and in the (i) Low energy scattering of neutrons on 208Pb has been absence of an ultraviolet completion, is ruled out [11]. used to constrain light force carriers coupled to nucleons Scalar boson exchange can account for both the proton [16], assuming a coupling of a scalar to nucleons of gN. radius puzzle and the g − 2 discrepancy [9]. The shift of Using the replacement ð Þμ
0031-9007=16=117(10)=101801(6) 101801-1 © 2016 American Physical Society New scalar bosons assumes puzzle exists give μ-p Lamb shift • X
• almost no hyperfine in μ proton X
• consistent with g-2 of μ and electron X
Electrophobic Scalar Boson and Muonic Puzzles
Yu-Sheng Liu,* David McKeen,† and Gerald A. Miller‡ Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA (Received 24 May 2016; published 2 September 2016) A new scalar boson which couples to the muon and proton can simultaneously solve the proton radius puzzle and the muon anomalous magnetic moment20 discrepancy. Using a variety of measurements, we constrain the mass of this scalar and its couplings to the electron, muon, neutron, and proton. Making no assumptions about the underlying model, these constraints and the requirement that it solve both problems limit the mass of the scalar to between about 100 keVand 100 MeV. We identify two unexplored regions in the coupling constant-mass plane. Potential future experiments and their implications for theories with mass-weighted lepton couplings are discussed.
DOI: 10.1103/PhysRevLett.117.101801
Recent measurements of the proton charge radius using the lepton l μ;e muon’s magnetic moment due to the Lamb shift in muonic hydrogen are troublingly dis- one-loop ϕ ðexchange¼ Þ is given by [12] crepant with values extracted from hydrogen spectroscopy and electron-proton scattering. The value from muonic 2 1 2 αϵl 1 − z 1 z hydrogen is 0.84087(39) fm [1,2] while the CODATA Δal dz ð 2 Þ ð þ Þ 2 : 1 ¼ 2π 0 1 − z m =m z ð Þ average of data from hydrogen spectroscopy and e-p Z ð Þ þð ϕ lÞ scattering yields 0.8751(61) fm [3]; these differ at more Scalar exchange between fermions f and f leads to a than 5σ. Although the discrepancy may arise from subtle 1 2 Yukawa potential, V r −ϵ ϵ αe−mϕr=r. In atomic lepton-nucleon nonperturbative effects within the standard ð Þ¼ f1 f2 model or experimental uncertainties [4,5], it could also be a systems, this leads to an additional contribution to the signal of new physics involving a violation of lepton Lamb shift in the 2S-2P transition. For an atom of A and universality. Z this shift is given by [13] The muon anomalous magnetic moment provides lN α another potential signal of new physics. The BNL [6] δEL − ϵl Zϵp A − Z ϵn f alNmϕ ; 2 measurement differs from the standard model prediction ¼ 2alN ½ þð Þ ð Þ ð Þ exp th by at least 3 standard deviations, Δaμ aμ − aμ −11 ¼ ¼ where f x x2= 1 x 4 [9,14], with a Zαm −1 287 80 × 10 [7,8]. ð Þ¼ ð þ Þ lN ¼ð lNÞ Að newÞ scalar boson ϕ that couples to the muon and the Bohr radius and mlN the reduced mass of the lepton- proton could explain both the proton radius and g − 2 nucleus system. Throughout this Letter we set ð Þμ puzzles [9]. We investigate the couplings of this boson to −11 μH standard model fermions f, which appear as terms in the Δaμ 287 80 × 10 ; δEL −0.307 56 meV ¯ ¼ ð Þ ¼ ð Þ Lagrangian, L ⊃ eϵfϕff, where ϵf gf=e and e is the 3 electric charge of the proton. Other authors¼ have pursued ð Þ this idea, but made further assumptions relating the μH within 2 standard deviations. This value of δEL is equal couplings to different species; e.g., in Ref. [9] ϵp is taken equal to ϵ and in Ref. [10],mass-weightedcouplingsare to the energy shift caused by using the different values of μ the proton radius [1 3,15]. Using Eq. (3) allows us to assumed. References [9] and [10] both neglect ϵ . We make – n determine both ϵ and ϵ as functions of m . The no a priori assumptions regarding signs or magnitudes of p μ ϕ the coupling constants. The Lamb shift in muonic hydrogen unshaded regions in Figs. 1 and 3 show the values of ϵp and ϵμ, as functions of the scalar’s mass, which lead to fixes ϵμ and ϵp to have the same sign which we take to be the values of a and δEμH in Eq. (3). positive. ϵe and ϵn are allowed to have either sign. Δ μ L We focus on the scalar boson possibility because scalar We study several observables sensitive to the couplings exchange produces no hyperfine interaction, in accord with of the scalar to neutrons ϵn and protons ϵp to obtain new observation [1,2]. The emission of possible new vector bounds on mϕ. particles becomes copious at high energies, and in the (i) Low energy scattering of neutrons on 208Pb has been absence of an ultraviolet completion, is ruled out [11]. used to constrain light force carriers coupled to nucleons Scalar boson exchange can account for both the proton [16], assuming a coupling of a scalar to nucleons of gN. radius puzzle and the g − 2 discrepancy [9]. The shift of Using the replacement ð Þμ
0031-9007=16=117(10)=101801(6) 101801-1 © 2016 American Physical Society New scalar bosons assumes puzzle exists give μ-p Lamb shift • X
• almost no hyperfine in μ proton X
• consistent with g-2 of μ and electron X
Electrophobic Scalar Boson and Muonic Puzzles
Yu-Sheng Liu,* David McKeen,† and Gerald A. Miller‡ Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA (Received 24 May 2016; published 2 September 2016) A new scalar boson which couples to the muon and proton can simultaneously solve the proton radius puzzle and the muon anomalous magnetic moment20 discrepancy. Using a variety of measurements, we constrain the mass of this scalar and its couplings to the electron, muon, neutron, and proton. Making no assumptions about the underlying model, these constraints and the requirement that it solve both problems limit the mass of the scalar to between about 100 keVand 100 MeV. We identify two unexplored regions in the coupling constant-mass plane. Potential future experiments and their implications for theories with mass-weighted lepton couplings are discussed.
DOI: 10.1103/PhysRevLett.117.101801
Recent measurements of the proton charge radius using the lepton l μ;e muon’s magnetic moment due to the Lamb shift in muonic hydrogen are troublingly dis- one-loop ϕ ðexchange¼ Þ is given by [12] crepant with values extracted from hydrogen spectroscopy and electron-proton scattering. The value from muonic 2 1 2 αϵl 1 − z 1 z hydrogen is 0.84087(39) fm [1,2] while the CODATA Δal dz ð 2 Þ ð þ Þ 2 : 1 ¼ 2π 0 1 − z m =m z ð Þ average of data from hydrogen spectroscopy and e-p Z ð Þ þð ϕ lÞ scattering yields 0.8751(61) fm [3]; these differ at more Scalar exchange between fermions f and f leads to a than 5σ. Although the discrepancy may arise from subtle 1 2 Yukawa potential, V r −ϵ ϵ αe−mϕr=r. In atomic lepton-nucleon nonperturbative effects within the standard ð Þ¼ f1 f2 model or experimental uncertainties [4,5], it could also be a systems, this leads to an additional contribution to the signal of new physics involving a violation of lepton Lamb shift in the 2S-2P transition. For an atom of A and universality. Z this shift is given by [13] The muon anomalous magnetic moment provides lN α another potential signal of new physics. The BNL [6] δEL − ϵl Zϵp A − Z ϵn f alNmϕ ; 2 measurement differs from the standard model prediction ¼ 2alN ½ þð Þ ð Þ ð Þ exp th by at least 3 standard deviations, Δaμ aμ − aμ −11 ¼ ¼ where f x x2= 1 x 4 [9,14], with a Zαm −1 287 80 × 10 [7,8]. ð Þ¼ ð þ Þ lN ¼ð lNÞ Að newÞ scalar boson ϕ that couples to the muon and the Bohr radius and mlN the reduced mass of the lepton- proton could explain both the proton radius and g − 2 nucleus system. Throughout this Letter we set ð Þμ puzzles [9]. We investigate the couplings of this boson to −11 μH standard model fermions f, which appear as terms in the Δaμ 287 80 × 10 ; δEL −0.307 56 meV ¯ ¼ ð Þ ¼ ð Þ Lagrangian, L ⊃ eϵfϕff, where ϵf gf=e and e is the 3 electric charge of the proton. Other authors¼ have pursued ð Þ this idea, but made further assumptions relating the μH within 2 standard deviations. This value of δEL is equal couplings to different species; e.g., in Ref. [9] ϵp is taken equal to ϵ and in Ref. [10],mass-weightedcouplingsare to the energy shift caused by using the different values of μ the proton radius [1 3,15]. Using Eq. (3) allows us to assumed. References [9] and [10] both neglect ϵ . We make – n determine both ϵ and ϵ as functions of m . The no a priori assumptions regarding signs or magnitudes of p μ ϕ the coupling constants. The Lamb shift in muonic hydrogen unshaded regions in Figs. 1 and 3 show the values of ϵp and ϵμ, as functions of the scalar’s mass, which lead to fixes ϵμ and ϵp to have the same sign which we take to be the values of a and δEμH in Eq. (3). positive. ϵe and ϵn are allowed to have either sign. Δ μ L We focus on the scalar boson possibility because scalar We study several observables sensitive to the couplings exchange produces no hyperfine interaction, in accord with of the scalar to neutrons ϵn and protons ϵp to obtain new observation [1,2]. The emission of possible new vector bounds on mϕ. particles becomes copious at high energies, and in the (i) Low energy scattering of neutrons on 208Pb has been absence of an ultraviolet completion, is ruled out [11]. used to constrain light force carriers coupled to nucleons Scalar boson exchange can account for both the proton [16], assuming a coupling of a scalar to nucleons of gN. radius puzzle and the g − 2 discrepancy [9]. The shift of Using the replacement ð Þμ
0031-9007=16=117(10)=101801(6) 101801-1 © 2016 American Physical Society Our approach any signs
m r V (r)= ✏ ✏ ↵ e . ✏ ✏ f1 f2 r e µ Bohr radius | | ⌧
2 `N ↵ (a`Nm ) EL = ✏`[Z✏p +(A Z)✏n] 4 2a`N (1 + (a`Nm )) g-2
11 12 87 a = 287(80) 10 , a =1.5 10 From Rb µ ⇥ e ⇥
For light mass- evades 4He data 1
10-1 Unshaded allowed by
p muon g-2 and muon-p ϵ
-2 μH 10 Δaμ+δEL Lamb shift δB in NM Two anomalies 3He-3H 10-3 have same scale 10-1 1 10 102 mϕ (MeV) Unshaded allowed by 10-2 muon g-2 μ ϵ
10-3
Δaμ
10-1 1 10 102 22 mϕ (MeV) 1
10-1 Unshaded allowed by
p muon g-2 and muon-p ϵ
-2 μH 10 Δaμ+δEL Lamb shift δB in NM Two anomalies 3He-3H 10-3 have same scale 10-1 1 10 102 mϕ (MeV) Unshaded allowed by 10-2 muon g-2
Using new eH experiment μ ϵ ✏µ✏p is reduced by 10-3 a factor of 3:
Δaμ barely visible in loglog plot
10-1 1 10 102 22 mϕ (MeV) Nuclear Physics constraints ✏n/✏p 208 If ✏n = ✏p scalar is ruled out by n- Pb scattering If ✏n has opposite sign as ✏p parameter space widens Other constraints: NN scattering, nuclear matter & 3He 3 Hbinding 3 2 muonic D, He n-208Pb
NN
1 μD δEL
μ4He+ δEL p ϵ / 0 n ϵ
-1
-2 10-1 1 10 102
m (MeV) 23 ϕ Beam dump experiments Beam dumps absorb beam of charged particles e+27Al to dissipate energy 27 hope e+ Al makes penetrating new particles e+ LIU, MCKEEN, and MILLER PHYSICAL REVIEW D 95, 036010 (2017) processes. In Sec. III, the cross sections for the 2 to 3 and 2 to 2 processes are calculated in the lab frame without any approximation. In Sec. IV, we introduce the WW approxi- e- mation. In Sec. V, we derive and compare the cross sections with and without approximations. In Sec. VI, we compare the number of new particles produced in beam dump experiments with and without approximations. In Sec. VII, we assume that this new scalar boson is observed FIG. 1. Lowest orderPHYSICAL 2 to REVIEW 3 production D 95, 036010 process: (2017) and measured in beam dump experiment, determine the e pValidityA Pi of→ thee p0 Weizsäcker-WilliamsA Pf ϕ k . A, γ, approximatione, and ϕ stand and the analysis mass and coupling constant, and compare the results with forð theÞþ targetð Þ atom,ð photon,Þþ ð electron,Þþ ð andÞ the new scalar boson. and without approximations. A discussion is presented in of beam dump experiments: Production of a new scalar boson Sec. VIII. * † ‡ Yu-Sheng2 Liu,2 David McKeen, and2 Gerald A. Miller s~ − p0 k − me −2p0 · k m Department¼ ofð Physics,þ Þ University¼ of Washington,þ Seattle,ϕ Washington 98195-1560,Better USA analysis if discovery (Received2 152 OctoberPHYSICAL 2016; REVIEW published2 D 96, 14016004 February (2017) 2017) II. DYNAMICS—ANEWSCALARBOSON u~ − p − k − me 2p · k mϕ ? ¼ ð ValidityÞ of the Weizsäcker-Williams¼ þ approximation and the analysis AS AN EXAMPLE Beam dump experiments have2 been used to search2 for new particles with null results interpreted in terms t2 − pof0 − beamp dump2p experiments:0 · p 2me Production of an axion, a dark photon, of limits on masses¼ ðmϕ andÞ coupling¼ constantsþ ϵ. However these limits have been obtained by using For simplicity, we assume that the new scalar boson ϕ approximations [including the Weizsäcker-Williamsor a new (WW) axial-vector approximation] boson or Monte-Carlo simulations. q Pi − Pf only couples to electron by a Yukawa interaction, i.e. the We display methods,¼ using a new scalar bosonYu-Sheng as an Liuexample,* and Gerald to obtain A. Miller the† cross section and the resulting 2 24 particle productiont q numbersDepartment without of Physics, using approximations University of Washington, or Monte-Carlo Seattle, Washington5 simulations. 98195-1560, We USA show that the scalar boson does not couple to other standard model (Received 15 May 2017; published 10 July 2017) approximations¼ cannot be used to obtain accurate values of cross sections.ð Þ The corresponding exclusion ? fermions other than electron. The Lagrangian in the mostly- plots differ by substantialBeam dump amounts experiments when have seen been on used a linear to search scale. for new In particles, the eventϕ, with of a null discovery, results interpreted we generate in plus metric is pseudodatawhich satisfy (assumingterms given of limits values on masses of mmϕ andand couplingϵ) in the constants currentlyϵ. However allowed these regions limits have of been parameter obtained by space. using The approximations [including theϕ Weizsäcker-Williams (WW) approximation] or Monte Carlo simulations. use of approximationsWe displayto analyze methods the to pseudodata obtain the cross for section the future and the experiments resulting particle is shown production to rateslead without to considerable using errors in determiningapproximations the parameters. on the phase Furthermore, space integral2 or a Monte new Carlo region simulations. of parameter In our previous space work can we examined be explored 1 1 s~ t2 u~ t mϕ: 6 2 2 2 without using one ofthethe case common ofþ the newþ scalar approximations,þ boson¼ production;m in≫ thism paper. Our we explore methodð allÞ canpossible be new used spin-0 as a and consistency spin-1 L ⊃− ϕ − mϕϕ eϵϕψψ¯ 1 particles. We show that the approximations cannotϕ be usede to obtain accurate values of cross sections. The 2 ð∂ Þ 2 þ ð Þ check for Monte-Carlocorresponding simulations. exclusion plots differ by substantial amounts when seen on a linear scale. Furthermore, a new region (mϕ < 2me) of parameter space can be explored without using one of the common DOI: 10.1103/PhysRevD.95.036010 For definiteness,approximations, we assumemϕ ≫ me. We the derive atom new expressions is a scalar for the boson three-photon decays of dark photon and where ϵ g=e, e is the electric charge, and ψ is the electron four-photon decays of new axial-vector bosons. As a result, the production cross section and exclusion ¼ (its spin is not consequential here) so that the Feynman rule field. Once the scalar boson is produced, it will decay to region of different low mass (mϕ < 2me) bosons are very different. Moreover, our method can be used as a photons pairs through the electron loop, for the photon-atomconsistency vertex check for isMonte Carlo simulations. I. INTRODUCTIONDOI: 10.1103/PhysRevD.96.016004 work in a beam dump experiment, it needs the incoming beam energy to be much greater than the mass of the new Beam dump experiments have been2 aimed at searching 2 ieF q Pi Pf ieF q particle,Pμ m , and7 electron mass m . 3 m3 m2 for new particles, such as dark photonsð Þð andþ axionsÞμ (see,≡ e.g.ð Þ ϕ ð Þ e 2 α ϕ ϕ I. INTRODUCTION Thelike previous a plane wave work and can[2] beused approximated the following by a real three Γϕ→γγ ϵ 2 2 f 2 ; [1] and2 references therein) that decay to lepton and/or photon. The use of the WW approximation in bremsstrah- 4π m 4m In our2 previous work [1], as an example, we usedapproximations: the ¼ e e photonð Þ pairs.where ElectronF q beamis dumps the form in particular factor which have accountslung processes for the was developed in Refs. [8,9] and applied to beam dump experiment E137 [2] and the production of a ð Þ (1)beam WW dump approximation; experiments in Refs. [4,10]. The WW approxi- received a largenuclear amountnew form scalar of theoretical boson factor to demonstrate[15] attentionand our the technique in recent atomic for analysis form factor [16]. mation simplifies evaluation of the integral over phase of beam dump experiments. In this paper, we further(2) a further simplification of the phase space integral, years [2,3]. TheHere, typical we setup only of include an electron the beam elastic dump form factorspace since and approximates the the 2 particle to 3 particle (2 to 3) where me is the electron mass and f τ include all possible new spin-0 and spin-1 particles, which see Eq. (31); experiment is to dump an electron beam into a target, in cross section in terms of a 2 particle to 2 particle (2 to 2) 1 1 −1 2 2 ð Þ¼ contributionwe denote ofϕ; they the are inelastic pseudoscalar, one vector, is much and axial-vector smaller and can1 be 2 1 1 − sin pτ . If mϕ > 2me, the scalar boson (3) crossmϕ ≫ section.me. For the WW approximation to work in a 4τ τ which the electrons arebosons. stopped (For a discussion of proton j þð Þð Þ j neglected in computing the cross section. The amplitudebeam dump experiment, of it needs the incoming beam energy can also decay to electron pairs, beam dumps, which isBeam beyond dump the experiments scope of have this been work, aimed see, at searchingThe combination of the first two approximations has been to be much greater than the mass of the new particle, m , ffiffiffi the processfor new particles, in Fig. such1 is as dark photons and axions (see, ϕ e.g. [4,5]). The new particles produced by the bremsstrah- denotedand[8] electronthe improved mass m . WW (IWW) approximation. The e.g., [3] and references therein) that decay to lepton pairs “ e ” lung-like process passand/or through photons. a Electron shield beamregion dumps and in decay. particular havename improvedThe previous WW work might[4] used be the somewhat following three misleading 2 3=2 approximations: 2 α 4me These new particlesreceived can a be largeF q detected amount2 of theoreticalby their attention decay in recentsince the procedure reduces the computational time but not − ϵ m 1 : 3 2→3 2 1. WW approximation; Γϕ→eþe ϕ − 2 M years e[4,5]g. The typicalu¯ setup of an electron beam dumpto improve accuracy). In this paper, we will focus on products, electron and/or photonð 2 pairs,Þ p0;s measured0 by the 2. a further simplification of the phase space integral ¼ 2 mϕ ð Þ experiment isq to dump an electron beam into a target, in detector downstream¼ of the decay region. Previous work examining[see the Eq. validity(28)]; of WW and IWW approximations which the electrons are stopped. The new particles pro- 3. m m . simplified the necessaryduced phase by the space bremsstrahlunglikep integralk m by process using passp the throughk(The a m validityϕ ≫ of WWe approximation is also discussed in − − e − 0 Thee combination of the first two approximations has been shield× regionP andð decay.Þþ These new particlesð þ canÞþother be processes,P up;s e.g. [11]). The third approximation used to Weizsäcker-Williams (WW) approximation [6,7] which, denoted [8] the improved WW (IWW) approximation. (The detected by their decay− products,u~ electronþ and/or− photons~ also known as method of virtual quanta, is a semiclassical simplifyname the“improved calculation WW of” might amplitude, be somewhat however, misleading is not in our pairs, measured by the detector downstream of the decay A. 2 to 3 production since the procedure reduces the computational time but approximation. Theregion. idea is Previous that the earlier electromagnetic work simplified field the necessaryscope because it8 is merely a special case by cutting off our does not improve accuracy.) In this paper, we will focus on generated by a fastphase moving space integral charged by particleusing the Weizsäcker-Williamsis nearly results when mð Þ 2m . Nevertheless, we should point out The leading production process is the bremsstrahlung- examining theϕ validitye of WW and IWW approximations (WW) approximation [6,7] which, also known as the ≲ transverse which is like a plane wave and can be approxi- that withoutfor the production using the of third axions, approximation dark photons and we new can axial- use beam like radiation of the scalar from the electron, shown in method of virtual quanta, is a semiclassical approximation. where up;s is the electron spinor; s and s0 aredump equalvector experiments tobosons.1 The. to third explore approximation a larger used parameter to simplify the space. mated by real photon.The The idea use is that of the the WWelectromagnetic approximation field generated in by a Fig. 1, calculationÆ of amplitude, however, is not in our scope bremsstrahlungAfter processesfast-moving averaging was charged developed and particle summing in is Refs. nearly[8,9] over transverseand initial which and isAs final an example, spins, we use the beam dump experiment E137 1 because it is merely a special case by cutting off our results applied to beamwe havedump experiments in Refs. [2,10]. The [12] and the production of a new scalar boson, which we when mϕ 2me. Nevertheless, we should point out that * denote ϕ. Interest≲ in a new scalar boson arose recently WW approximation [email protected] evaluation of the integral without using the third approximation we can use beam e p A P → e p0 A P ϕ k 4 † i f over phase space [email protected] approximates the 2 particle to 3 becausedump such experiments particle to explore which a couples larger parameter to standard space. model ð Þþ ð Þ ð Þþ ð Þþ ð ÞðÞ 1 F q2 2 particle (2 to 3)M cross23~ 2 section in terms ofM a2 2→ particle3 2 e4 tog2 2 ð fermionsÞ A2→3 can solve9 the proton radius puzzle and muonic particle (2 to 2) cross section.2 For the WWapproximation to q4anomalous magnetic moment discrepancy simultaneously where e, A, and ϕ stand for electron, target atom, and the j j ¼ s s j j ¼ ð Þ 2470-0010 =2017=96(1) =016004(13)0 [13,14] 016004-1. However, the techniques © 2017 American we introduce Physical Society can be used new scalar boson, respectively. We define the following X X *[email protected] for the production and possible detection of other particles. quantities using the mostly-plus metric †[email protected] The outline of this paper is as follows. In Sec. II,we ‡[email protected] calculate the squared amplitude for 2 to 3 and 2 to 2
036010-2 2470-0010=2017=95(3)=036010(11) 036010-1 © 2017 American Physical Society Constraining Beam dump experiments do not see +
Now
⌘
η scalar boson could survive the constraints if its coupling to the Γ )/
? ? u quark were zero. However, investigating the allowed regions
10-8 of the parameter spaces, we find that this is not the case. The η→πγγ
u ( Γ exclusion of coupling to u quark is equivalent to excluding observed the existence of the . 10-12 u e We can further use the ⌘ ⇡ through u quark loop
1 ! d μ channel to constrain ✏u and ✏p. The results are shown in Figs. 3 1 10 4 and 1. The allowed ✏u and ✏p are both around 10 , as well mϕ (MeV) as ✏µ.
Figure 11. Decay rate of ⌘ ⇡0 ⇡0 (shaded region is ! ⇤ ! Allowed mass range fromallowed to200 decay). KeV The decay to width 3 MeV with a superscript f indicates VII. COEXISTENCE WITH THE OLD AND THE NEW the process decaying through f fermion loop. The solid blue, dashed green, dotted red, and dotted dashed yellow lines are ⌘ through ! To allow the new physics, the electron-proton scattering u quark, d quark, muon, and electron loops. The horizontal gray line (CODATA value), and the new experiment to coexist, we is the observed decay width. The vertical lines indicate where the assume that the actual proton radius lies within 3 of both decay rate of ⌘ ⇡ through u quark loop channel is greater than ! observed value plus 3 . new and old experiments. Such a value is between 0.8568 to 0.8620 fm. First, following our previous work, we obtain the constraint of ✏p, ✏n, ✏µ, ✏u, and ✏d, and the results are shown Using Eq. (22), the decay rate is found to be in Figs. 12–16, respectively. Second, we repeat the analysis in Secs. V and VI, and obtain the decay rate in di erent 0 f 2m channels in Fig. 17, ✏e exclusion plot in Fig. 18, and ⌘ ⇡ 2 2 4 4 ⌘ ! = ✏f ✏u ✏d ↵ Q f decay rate in di erent channels in Fig. 19. Finally, we obtain ( ) 12⇡3m2 f the allowed m is from 198 keV to 2.47 MeV and ✏u, ✏p, and 2 3 smax 2 4 2 ✏µ are all around 10 . Note that ✏n is completely ruled out 13 s m⌘ 4mf ds 13 I p˜ p˜ , with the new result and this means that the scalar coupling to 13 2 2 2 2 3 0 s m m s m π 13 ⌘ 13 ⌘ ! the neutron is zero, i.e. ✏u = 2✏d. ( ) f 2m5 At first glance, one might think that the e ect of new physics 2 2 4 4 ⌘ = ✏ ✏u ✏d ↵ Q is approximately halved, therefore the constraint should be less f ( ) f ⇡3 2 24 mf strict. This intuition is partially correct. Comparing with the 2 smax 5 2 2 results in Sec. VI, we see that the lower bound of m is smaller 13 s / 4mf ds 13 I p˜. (33) (as expected) while the upper bound also becomes smaller. The 13 2 2 2 2 0 s13m⌘ m s13m⌘ ! problem of the upper bound of m is that the e ect of ✏ should π ( ) n be included carefully. To correctly analyze it, examine Eq. (34). We used the fact that there are two photons in the final state The ⌘ ⇡0 decay rate through u quark loop seems to be (S = 2), smin = 0, and p˜ = ps 2. We can further apply the ! 2 2 13 3 13/ proportional to ✏u ✏u ✏d . However there is ,total in the narrow width approximation denominator and it( is dominated ) by through u quark loop decay channel (Fig. 7). Therefore! the ⌘ ⇡ decay 2 2 4 4 ! 2 = ✏f ✏u ✏d ↵ Q f rate through u quark loop is proportional to ✏u ✏d (this ( ) 0 ( ) 2 4 2 2 factor comes from the ⌘⇡ vertex). In the previous scenario, f m 4m 2 f ✏d can be positive (see Fig. 5), so ✏u ✏d can be smaller I p˜ m2 . (34) ⇥ 2 2 2 2 ✏2 ( )✏ 24⇡ mf m⌘ ,total m ! s13= 2 than u in the first scenario; in this section, d stays negative m⌘ 2 in the allowed m range (see Fig. 16), so ✏u ✏d is bigger 2 ( ) The process ⌘ ⇡0 is observed and the value is than ✏u. This e ect surpasses the smaller ✏u (✏p) value in the ! second scenario. The root cause of this strange behavior can 0 be traced back to ✏n can be non-zero in the first scenario, but ⌘ ⇡ 4 ( ! ) = 2.56 0.22 10 . (35) strictly zero in the second scenario. ⌘,total ( ± ) ⇥ The decay rate of a virtual to two photons cannot be too big to spoil the observed value. We define that the scalar boson is VIII. CONCLUSION excluded if the decay rate is greater than the observed value 0 4 plus 3 , i.e. ⌘ ⇡ ⌘,total > 3.22 10 . Our previous work limits the existence of the new scalar In Fig. 11,( we! show)/ a virtual decay⇥ to two photons boson to the mass range of about 160 keV to 60 MeV. Here through di erent fermion loops. We can read from the plot that we reanalyze the beam dump experiments and find that the Summary
• If the proton radius puzzle exists : new scalar boson of mass from 300 KeV to 3 MeV may exist- narrow target • Direct detection is needed. Does 4% matter?
26 Summary
• If the proton radius puzzle exists : new scalar boson of mass from 300 KeV to 3 MeV may exist- narrow target • Direct detection is needed. YesY Does 4% matter?
26 Spares follow electron Exclusion plot
10-2 APEX HPS
B.D. DarkLight 10-4 A Δae VEPP-3 E141 e+e-
| MESA e
ϵ H | Orsay δEL Izaguirre10-6 et al B PLB740 E137 61 scalar 10-8 10-1 1 10 102
mϕ (MeV) allowed mass range 28 More constrains Coupling to electron ✏e Hydrogen atom Lamb shift H EL < 45 kHz + e e resonant (?) scattering not seen electron anomalous magnetic moment
10-2 BABAR A1 2014
10-4 A Δae E141 e+e-
| H e δEL
ϵ Orsay | 10-6 BD BD is beam dump B E137 discussed next
10-8 scalar Electrophobic
10-1 1 10 102
mϕ (MeV) 29 Science 12 Aug 2016: Vol. 353, Issue 6300, pp. 669-673 Deuteron is smaller too
Electron (D-H) isotope shift (2S-1S) 2 photon spectroscopy PRL 104, 233001 r2 r2 =3.82007(65) d p µ D Lamb shift r =2.12562(78) fm Science 353 (2016) 669 d CODATA (2010)rd =2.1424(26) fm - mainly electron scattering Use r =0.84087 in r2 r2 =3.82007(65) gives r =2.12769 fm p d p d µD and Electron (D-H) isotope shift are consistent redo eD scattering? !
If NO proton radius puzzle, there still is a missing Lamb shift OR: CODATA deuteron radius is too large 2.1424 vs 2.12769 remeasure deuteron ? Nuclear Physics constraints
✏n/✏p 208 Low energy scattering of neutrons on Pb using -nucleon coupling gN . • 2 gN A Z 2 Z 2 ✏ + ✏ ✏ cancellation evades previous limits e ! A n A p n NN charge-independence breaking scattering length • a =(a + a )/2 a , measured: 5.64(60) fm, theory: 5.6(5) pp nn np Scalar boson exchange: a 1 V uu¯ dr 1.6 fm (2 S.D.) / 0 np Change in binding energy/A infiniteR nuclear matter: less than 1 MeV • binding energy B(3He) B(3H)= 763.76 keV due to Coulomb (693 keV) • + strong force charge symmetry breaking (68 keV) exchange < 30 keV
31 Lepton-universality violating one boson exchange • Tucker-Smith & Yavin PRD83, 101702 new particle scalar or vector coupling • Brax & Burrage scalar particles PRD 83, 035020 &’14 • Batell, McKeen & Pospelov PRL 107, 011803 new gauge boson kinetically mixing with Fμν plus scalar for muon mag. mom. 1401.6154 W decays enhanced • Carlson Rislow PRD 86, 035013 fine tune scalar pseudoscalar or polar and axial vector couplings • Barger et al PRL106,153001 - new particles ruled out but assumes universal coupling Kaon decays provide constraints • 32 Looking for newc scalars is not new Low mass Higgs searches 19 16 p(1.88 MeV) + F ↵ + O⇤(6.05) 16 16 ! O⇤(6.05) O(GS)+ ! Kohler et al PRL 33, 1628 (1974)
Freedman et al. PRL 52, 240 (1984)
p +3 H 4 He(20.1) 4He(GS)+ ! ! No Scalars found, but assumed coupling constants were much larger than what we will use
33 2 Almost unknown T 1(0,Q ) Miller PLB 2012
2 2 subt ↵ 2 1 2 h(Q ) 2 Soft proton E = S(0) dQ 2 T 1(0,Q ) m 0 Q 2Z 2 2m 2 M 2 lim h(Q ) , chiral PT : T 1(0,Q )= Q + Q2 ⇠ Q2 ↵ ··· !1 Logarithmic divergence ! Typo bleow T (0,Q2) M Q2F (Q2) Cuts o↵integral 1 ! ↵ loop Birse & McGovern assumedipole : Esubt =0.004 meV very small 2 n 2 Q 1 Miller Floop(Q )= ,n 2,N Nn +3 M 2 (1 + aQ2)N n+3 ✓ 0 ◆ Infinite parameter set gets needed 0.31meV, NO constraint on neutron Choose parameters so shift in proton mass <0.5 MeV (current uncertainty) Recast in EFT- parameters seem natural Physics Letters B 718 (2013) 1078–1082
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Physics Letters B www.elsevier.com/locate/physletbTwo photon exchange Proton polarizability contribution: Muonic hydrogen Lamb shift and elastic 2 3 scattering Measured =206.2949(32)= 206.0573(45)-5.2262 rp +0.0347 rp meV
Gerald A. Miller computed Department of Physics, Univ. of Washington, Seattle,Explain WA 98195-3560, United puzzle States with radius as in H atom increase 206.0573 meV article info abstractby 0.31 meV-attractive effect on 2S state2 needed Article history: The uncertainty in the contribution to the Lamb shift in muonic hydrogen, !Esubt arising from proton Received 2 October 2012 ′ ℓ − k of momentum q = p − p.as:polarizability effects in the two-photon exchange diagramℓ at large virtual photon momenta is shownℓ large Received in revised form 27 October 2012 enough to account for the proton radius puzzle. This is because !Esubt is determined by an integrand that Accepted 6 November 2012′ µ µ µ 2 falls very2 slowly2 with very large virtual photon momenta. We evaluate the necessary integral using a set Available online 8Γ November(p ,p)= 2012γN F1(−q )+F1(−q )F (−q )Oa,b,c (2) of chosen form factors and also a dimensional regularization procedure which makes explicit the need for Editor: W. Haxton ′ µ ′ µ (p + p ) ′ (p · alowenergyconstant.Theconsequencesofourtwo-photonexchangeinteractionforlow-energyelasticγN − M) (p · γN − M) Oa = [Λ+(p ) + Λ+(p)] energy shift proportional 2M lepton–protonMOur scattering ideaM are evaluated and could be observable in a planned low energy lepton–proton ′ Oµ =((p2 − M 2)/M 2 +(pscattering2 − M 2) experiment/M 2)γµ planned to run at PSI. 4 b N © 2012 Elsevier B.V. All rights reserved. to lepton mass P P ′ ′ (p · γ − M) (p · γ − M) Oµ = Λ (p )γµ N + N γµ Λ (p), c + N M M N + 1. Introduction FIG. 1: Direct two-photon exchange graph corresponding to where three possible forms are displayed. Other terms of the hitherto neglected term. The dashed line denotes the lepton; the solid line, the nucleon; the wavy lines photons; The protonthe radius vertex puzzle function is one of needed the most to satisfy perplexing the physics WT identity do µ⌫ and the ellipse the off-shell nucleon. issues of recentnot times. contribute The extremely significantly precise to extraction theT Lamb of the shift= pro- and are not 2 ton radius [1]shownfrom the explicitly. measured The energy proton difference Dirac between form factor, the F1(−q ) 2P F 2 and 2S F 1 states of muonic hydrogen disagrees with that 2 3/=2 is1/= empirically2 well represented as a dipole F1(−q )=(1− µ⌫ µ µ 2 2 −2 2 Fig.2 1. The box diagram for the (α5m4) corrections. The graph in which the pho- extracted fromq electronic/Λ ) , hydrogen.(Λ = 840 The MeV) extracted for the value values of the of − pro-q ≡ Q > 0 O = tons(g cross is also included. )T1 +(P )(P )T2 ton radius isof smaller up to than about the 1 CODATA GeV2 needed[2] value here. (basedF mainly(−q2) is an off- interference between one on-shell and one off-shell part on electronic H) by about 4% or 5.0 standard deviations. This im- of the vertex··· function. The change in the invariant ··· am- ··· shell form factor, and Λ+(p)=(p · γN + M)/(2M)tions is an entering in the two-photon exchange term, see Fig. 1,can plies [1] that either the Rydberg constant hasDispersion to be shifted by relation:plitude,ImM [,T due1] to usingW Eq.1 (2)measured along with Oµ,tobe operator that projects on the on-mass-shell protonaccount state. for the 0.31 meV.Off This idea is worthy of consideration be-a 4.9 standard deviations or that present QED calculations for hy- We use Oa unless otherwise stated. cause the computedevaluated effect between is proportional fermion/ spinors, to the lepton is given mass in to the rest drogen are insufficient. The Rydberg constantLarge is extremely virtual2 well photon energy ⌫, W ⌫ integral over energy diverges We take the off-shell form factor F (−q ) to vanishthe atfourthframe power, by and so is capable of being relevant1 for muonic measured and the2 QED calculations seem to be very extensive and q = 0. This means that the charge of the off-shell protonatoms, but irrelevant for electronic atoms. ⇠2 highly accurate, so the muonic H finding is a significant puzzle for ¯ will be the same as the chargeSubtraction of a free proton, and function is needed: T1(0,Q ) zero energy the entire physics community. subt demanded by current conservation as expressedF 2 through2. !E and its evaluatione4 d4k F 2(−k2)F (−k2) 35 Pohl et al. show that the energy difference between the 2P 3/=2 1 the Ward-Takahashi identityHill [24, 25]. & We assumePaz- big uncertaintyMOff = 2 4 in 2 dispersion2 (4)approach and 2S F 1 states, !E,isgivenby 2M " (2π) (k + iϵ) 1/=2 The basic idea is that the two-photon exchange term depends 2 2 µ ν ν µ −λq /b ×(γN (2p + k) + γN (2p + k) ) µν 2 2 2 3 on the forward virtual Compton scattering amplitude T (ν, q ) !E 209.9779(49)! 5.2262Fr(−q 0)=.0347r meV, . (1) (3) 2 µ = − p + (1 p− q2/Λ2)1+ξ where q is the square(l · γ − ofk the· γ + fourm) momentum,(l · qγ +ofk · theγ + vir-m) tual photon and× γνµ is its2 time componentγν (+ν γνq p2/M)withp as γµ , where rp is given in units of fm. Using this equation and the exper- # k − 2l · k + iϵ ≡ ·k +2l · k + iϵ $ imentally! measuredThis purely value ! phenomenologicalE 206.2949 meV, form one! can is simple see that and clearlythe proton momentum and M as its mass. One uses symmetries = µν 2 the differencenot between unique. the Pohl The and parameter CODATA valuesb is expected of the proton to be ofto thedecompose T (ν, q ),intoalinearcombinationoftwoterms, 2 2 order of the pion mass, because these longest rangeT com-1,2(ν, q ).TheimaginarypartsofT1,2(ν, q ) are related to struc- radius would be removed by! an increase of the first term on the where the lepton momentum is l =(m, 0, 0, 0), the vir- rhs of Eq. (1) byponents 0.31 meV of the3. nucleon1 10 10 areMeV. least bound and more suscep-ture functions F1,2 measured in electron– or muon–proton scat- = × − tual photon momentum is k and the nucleon momentum This protontible radius to puzzle the external has been perturbations attacked from many putting differ- the nucleontering, so that T1,2 can be expressed in terms of F1.2 through p =(M,0, 0, 0). The intermediate proton2 propagator2 ent directionso[3–21]ff its massThepresent shell. At communication large values ofis intended|q2|, F has to thedispersion same relations. However (for fixed values of q ), F1(ν, q ) falls off toois slowly cancelled for large by valuesthe off of-mass-shellν for the dispersion terms of relation Eq. (2). This investigate thefall-o hypothesisff as F that, if ξ the=0.Wetake proton polarizabilityΛ = Λ contribu-here. 1 to converge.graph Hence, can one be makesthought a subtractionof as involving at ν a contact0, requiring interaction We briefly discuss the expected influence of using 2 = ! that an additionaland the function amplitude of q in(the Eq. subtraction (4) as a function) new proton be in- polariza- Eq. (2). The ratio, R,ofoff-shell effects to on-shell ef- 2 E-mail address: [email protected]. 2 troduced. Becausetion correctionq < 0forlepton–nucleonscattering,oneoften corresponding to a subtraction term in the (p·γN −M) q 2 2 fects, R ∼ λ 2 , (|q | ≪ Λ ) is constrained by dispersion relation for the two-photon exchange diagram 0370-2693/$ – see front matter © 2012M Elsevierb B.V. All rights reserved. http://dx.doi.org/10.1016/j.physletb.2012.11.016a variety of nuclear phenomena such as the EMC effect that is not constrained by the cross section data [34]. (10-15%), uncertainties in quasi-elastic electron-nuclear The resulting virtual-photon-proton Compton scattering scattering [26], and deviations from the Coulomb sum µ ν amplitude, containing the operator γN γN corresponds to rule [27]. For a nucleon experiencing a 50 MeV central the T2 term of conventional notation [35], [36]. Eq. (4) 2 2 potential, (p · γN − M)/M ∼ 0.05, so λq /b is of or- is gauge-invariant; not changed by adding a term of the der 2. The nucleon wave functions of light-front quark- form kµ kν /k4 to the photon propagator. models [33] contain a propagator depending on M 2. Thus the effect of nucleon virtuality is proportional to Evaluation proceeds in a standard way by taking the the derivative of the propagator with respect to M,orof sum over Dirac indices, performing the integral over k0 the order of the wave function divided by difference be- by contour rotation, k0 →−ik0, and integrating over the tween quark kinetic energy and M. This is about three angular variables. The matrix element M is well approx- times the average momentum of a quark (∼ 200 MeV/c) imated by a constant in momentum space, for momenta divided by the nucleon radius or roughly M/2. Thus typical of a muonic atom, and the corresponding poten- 2 2 r r R ∼ (p · γN − M)2/M , and the natural value of λq /b tial V = iM has the form V ( )=V0δ( ) in coordinate is of order 2. space. This is the “scattering approximation” [3]. Then 2 The lowest order term in which the nucleon is suffi- the relevant matrix elements have the form V0 |Ψ2S(0)| , ciently off-shell in a muonic atom for this correction to where Ψ2S is the muonic hydrogen wave function of the produce a significant effect is the two-photon exchange state relevant to the experiment of Pohl et al. We use 2 3 diagram of Fig. 1 and its crossed partner, including an |Ψ2S(0)| =(αmr) /(8π), with the lepton-proton re- 16O(6.05, 0+) 16 O(GS, 0+)+c , No single photon decay ! From electron g-2 + 2 ⌧(A⇤ A+e e ) 3 g ee¯ m 2 5/2 ⌧(A! A+ =3.3 10 e2 1 ( 6MeV ) ⇤! ⇥ + 10 ⌧(A⇤ A + e e ) : lifetime is 10 s Decay length! (m) nuclear emission of
scalar boson
m (MeV) Several new electron spectroscopy experiments
• Independent measurement of Rydberg constant. This would change only extracted rp nothing else • 2S-6S UK, 2S-4P Germany,1S-3S France • 2S-2P classic, Canada • Highly charged single electron ions NIST 2S-4P has reported preliminary results- small radius not yet published 37 Yes it really is GE
• Non-relativistic reduction of one-photon exchange leads to the spin independent 2 2 interaction being GE (Q )/Q • All recoil effects properly accounted for:Breit- Pauli Hamiltonian computed for non-zero lepton and proton momentum uate terms up to fourth-order in chiral perturbation theory to find The function h(t) is monotonically falling, approaching 1/⌅t for small values of t, and 2 BM ⇥ Q ⇥ 1 falling as 3/(4t) large2 valuesM 2 of t. The subtraction4 functionM 2 T (0, Q2) is not available T1 (0, Q ) Q 1 2 + (Q ) Q 1 2 , (6) ⇧ ⇤ M O ⌅ ⌅ 2 ⇥ 1 + 2 Q from experimental measurements, except at the real photon point Q 2M=2 0. It comes from ⇥ ⇥ the excitation of the proton, and can be described, at small values of Q2, in terms of the with M⇥ = 460 50 MeV. They also use the most recent evaluation of ⇥M, based on a fit electric ( E) and± magnetic (⇥M) polarizabilities. For small values of Q and ⌅ = 0 one to real Compton scattering [29] that finds 2 Q2 sees [23] lim⌅2,Q2 0 T1(0, Q )= ⇥M, where is the fine structure constant. Using this ⇤ 2 4 3 2 simple linear Q -dependence⇥ in Eq.=( (2)3.1 shows0.5) that10 thefm integral, over T1(0, Q ) converges (7) M ± ⇥ at the lower limit, but diverges logarithmically at the upper limit. Thus obtaining a non- where only statistical and Baldin Sum Rule errors are included. Their result is a negligible infinite result depends on including an arbitrary form factor that cuts off the integrand Esubt = 4.1µ eV [20]. The form Eq. (6) achieves the correct 1/Q2 asymptotic behavior for large values of Q2 or some other renormalization procedure. 2 of T1(0, Q ) but the coefficient¯ ⇥M2/ is not the same as obtained from the operator prod- We note that limQ2 ⇥ T1(0, Q ) can be obtained from the operator production expan- uct expansion. The coefficient⇤ of Eq. (6) is about twice the asymptotic limit obtained by sion [26, 27]. Using Eq. (2.18) of Ref. [26], neglecting the term proportional to light quark Collins [26]. 2 1 2 masses, and accounting for different conventions yields T¯ (0, Q ) 2.1 fm /Q . This 1 ⇥ Previous2 authors [17, 20] noted the sensitivity of the integrand of Eq. (5)2 to large values 1/Q behavior removes the putative logarithmic divergence of T¯1(0, Q ), but this func- of Q2. Our aim here is to more fully explore the uncertainty in the subtraction term tion is far from determined. 2 that arises from theArbitrary logarithmic divergence. functions We shall use a form of Floop(Q ) that is We follow the previous literature by including a form factor defined as Floop. Then consistent with the constraint on the Q4 term found Birse & McGovern [20]. This is done ⇥ by postulating a term that beginsT (0, atQ order2)=Q6MinQ Eq.2F (4),(Q such2) . as (4) 1 loop n Q2 1 Using Eqs. (2,3,4) one( finds2)= the energy shift to be + Floop Q 2 2 N , n 2, N n 3, (8) ⇤ M0 ⌅ (1 + aQ ) ⇤ ⇤ 2 2 ⇥ 2 subt ⇤ (0) ⇥M 2 1 2 2 4m 2 where E M0=, a are parameters to2dQ be determined(1 2Q /(4 bym requiring)) 1 + that2 the1 computed+ 1 Floop contribu-(Q ). mT 1(0 ,Q0 ) ⇤ or faster ⌦ , QM ⇥ ⌅ 4 2 tion to the Lamb shift reproduce⇠ the desiredQ 0.31 meV. With Eq. (8)! the low Q behavior ⇧ ⌥ ⌃ (5) ¯ 2 6 4 2 of T1(0, Q ) is of order Q or greater and it falls as 1/Q or greater1 for large values of Q . subt 2 2 n 2 6 4 SoThe far issue as we here know,E is the there arbitrary3 are↵ nom nature constraintsS(0) of the on functionB the(N,n coefficientF)loop, (Q of) the. Pachucki2Q term [24] or the used 1/Q the ⇡ ↵ ⌘ M0 a term.dipole However, form, 1/ weQ shall4, often determine used to the characterize subtraction the term’s proton contribution electromagnetic to the form Lamb factors. shift ⇥ 3 parameters: n, N, a (M0 = M ) asBut a general the subtraction functionChoose function of parametersn, N. We should note not thatsuch be⇥ M computedthatis anomalously shift fromin proton the small proton duemass formto a< cancellation factors, be- betweencause virtual pion component cloud andelectromagnetic intermediate scattering includes terms uncertainty a [30] term , so in thatwhich of 0.5 one the canMeV photon use a isvalue absorbed ten times and largeremitted than from appears the same in Eq. quark (7) [28]. to set Carlson the overall and Vanderhaeghen scale of the subtraction [17] evaluated term. a Thus loop we di- replaceagram using the term a specific⇥ of Eq. model (4) by and a foundgeneral a form form of factor the same1/Q dimensions2 log Q2, leading⇥: ⇥ to a⇥ larger. M ⇥ M ⌅ contributionThe use of to Eq. the (8) subtraction in Eq. (5) allows term than one previous to state the authors. expression Birse & for McGovern the energy [20] shift eval- in
5 4 2 Almost unknown T 1(0,Q ) Miller PLB 2012
2 2 subt ↵ 2 1 2 h(Q ) 2 Soft proton E = S(0) dQ 2 T 1(0,Q ) m 0 Q 2Z 2 2m 2 M 2 lim h(Q ) , chiral PT : T 1(0,Q )= Q + Q2 ⇠ Q2 ↵ ··· !1 Logarithmic divergence ! Typo bleow T (0,Q2) M Q2F (Q2) Cuts o↵integral 1 ! ↵ loop Birse & McGovern assumedipole : Esubt =0.004 meV very small Q2 n 1 Miller F (Q2)= ,n 2,N N +3 loop M 2 (1 + aQ2)N ✓ 0 ◆ Infinite parameter set gets needed 0.31meV, NO constraint on neutron Choose parameters so shift in proton mass <0.5 MeV (current uncertainty) Recast in EFT- parameters seem natural Form factors Unknown asymptotic region Floop 1.0 Miller 0.8
0.6 CPT 0.4
0.2 Birse McGovern Q2 GeV2 0.0 0.2 0.4 0.6 0.8 1.0 If recast into effective field theory strength seems natural I M Decay modes 2 2 3/2 + g e m 4me ( e e )= 4⇡ 2 1 m2 ,m > 2me ! ⇣ ⌘ c⌧(m)
m (MeV) 2 3 2 ↵ m ( )=g e 3 2 ! 144⇡ me c⌧( ) + c⌧(m) c⌧(e e ) ?
m (MeV) ?
1
1 doi: 10.1038/nature09250 SUPPLEMENTARY INFORMATION Pohl et al. Table of calculations
? ** # "0" * % ( .*5L7 *"9 )"$M 10" / M 10" / M 10" / M 5 !) #" 0 "0" 1% 5I 6 64944<8 6 )"0*** " * D " " E 5L7I 9 445;> 7 )"0*** " 0 1% 9 64946=6 64946=6 Lamb 8 !) K0 "0" 1% 9I 58 594=5 594<> 594=5 9 %0 * * *" * * 02 0*" 5I 6I 9 4594> 4594> 45954 ; !) ( ""K0 "0" 1% 55 44496> < %0 ** *" * * 55I 56 444667 shift: ( "" 02 0*" D " " E = -( ""K0 1% D0I " " E 444<; 444<;5 > 3* (2K 00 9I 59I 5; 444547 444547 54 *&( 0*&( "0" 0 ** ; 444579 4M44579 444579 4M44459 vacuum D1* 0 "0 .G " *&E 55 ) **" ( "0" 0 * * 5I 6 444944 4M4454 444; 4M445 4449 * (" 02 0*" 6DE8 56 0" 0 * (" **" ( 5
-0" 5@ 00 . *K ** (" 2 (*$ * $ 2 *" " ( I (" " " "0" " M 3" $00 (" 2" 0 " $ * " < -0" ** )"$M "0" * % ( .*6 *"9 6; 6 " *& 0" * " ** 95><89 95><8 95><5 6I 6; 6 ) **" " * 0" H*" * " "" 446<9 446=6 446<7 ; 6 5I 6 -0" 6@ 00 "0" *K ** 22 * " * * " " 0M 7 www.nature.com/nature 3 doi: 10.1038/nature09250 SUPPLEMENTARY INFORMATION Pohl et al. Table of calculations ? ** # "0" * % ( .*5L7 *"9 )"$M 10" / M 10" / M 10" / M 5 !) #" 0 "0" 1% 5I 6 64944<8 6 )"0*** " * D " " E 5L7I 9 445;> 7 )"0*** " 0 1% 9 64946=6 64946=6 Lamb 8 !) K0 "0" 1% 9I 58 594=5 594<> 594=5 9 %0 * * *" * * 02 0*" 5I 6I 9 4594> 4594> 45954 ; !) ( ""K0 "0" 1% 55 44496> < %0 ** *" * * 55I 56 444667 shift: ( "" 02 0*" D " " E = -( ""K0 1% D0I " " E 444<; 444<;5 > 3* (2K 00 9I 59I 5; 444547 444547 54 *&( 0*&( "0" 0 ** ; 444579 4M44579 444579 4M44459 vacuum D1* 0 "0 .G " *&E 55 ) **" ( "0" 0 * * 5I 6 444944 4M4454 444; 4M445 4449 * (" 02 0*" 6DE8 56 0" 0 * (" **" ( 5 -0" 5@ 00 . *K ** (" 2 (*$ * $ 2 *" " ( I (" " " "0" " M 3" $00 (" 2" 0 " $ * " < -0" replaced by 6 9 ** )"$M "0" * % ( .* *" " *& 0" * " ** 6; 95><89 6 95><8 95><5 ) **" " * 0" H*" * " "" 6I 6; 446<9 6 446=6 446<7 experiment ; 6 5I 6 -0" 6@ 00 "0" *K ** 22 * " * * " " 0M 7 www.nature.com/nature 3 doi: 10.1038/nature09250 SUPPLEMENTARY INFORMATION Pohl et al. Table of calculations ? ** # "0" * % ( .*5L7 *"9 )"$M 10" / M 10" / M 10" / M 5 !) #" 0 "0" 1% 5I 6 64944<8 6 )"0*** " * D " " E 5L7I 9 445;> 7 )"0*** " 0 1% 9 64946=6 64946=6 Lamb 8 !) K0 "0" 1% 9I 58 594=5 594<> 594=5 9 %0 * * *" * * 02 0*" 5I 6I 9 4594> 4594> 45954 ; !) ( ""K0 "0" 1% 55 44496> < %0 ** *" * * 55I 56 444667 shift: ( "" 02 0*" D " " E = -( ""K0 1% D0I " " E 444<; 444<;5 > 3* (2K 00 9I 59I 5; 444547 444547 54 *&( 0*&( "0" 0 ** ; 444579 4M44579 444579 4M44459 vacuum D1* 0 "0 .G " *&E 55 ) **" ( "0" 0 * * 5I 6 444944 4M4454 444; 4M445 4449 * (" 02 0*" 6DE8 56 0" 0 * (" **" ( 5 -0" 5@ 00 . *K ** (" 2 (*$ * $ 2 *" " ( I (" " " "0" " M 3" $00 (" 2" 0 " $ * " < -0" replaced by 6 9 ** )"$M "0" * % ( .* *" " *& 0" * " ** 6; 95><89 6 95><8 95><5 ) **" " * 0" H*" * " "" 6I 6; 446<9 6 446=6 446<7 experiment ; 6 5I 6 -0" 6@ 00 "0" *K ** 22 * " * * " " 0M 7 www.nature.com/nature 3 I. INTRODUCTION The proton radius puzzle is one of the most perplexing physics issues of recent times. The extremely precise extraction of the proton radius [1] from the measured energy dif- F=2 F=1 ference between the 2P3/2 and 2S1/2 states of muonic hydrogen disagrees with that ex- tracted from electronicwhere hydrogen.nmin is The chosen extracted to be value one order of the higher proton than radius the is power smaller determined than by chiral per- the CODATA [2] valueturbation (based theory. mainly on The electronic free parameters H) by aboutan, cn 4%, M orn could 5.0 standard then be devi- varied to reproduce the ations. This implies [1]desired that either 0.31 meV the Rydberg shift in energy. constant This has means to be shifted that the by application 4.9 standard of chiral perturbation deviations or that presenttheory QED to any calculations finite order for does hydrogen not prevent are insufficient. the choice of The a subtraction Rydberg function that gives constant is extremelythe well necessary measured shift and in the energy. QED calculations seem to be very extensive and highly accurate, soThe the muonic above paragraphs H finding is show a significant that the current puzzle forprocedure the entire used physics to estimate the size of the community. subtraction term is rather arbitrary. This arises because the chiral EFT is being applied F=2 F=1 Pohl et al. show thatto the the virtual-photon energy difference nucleon between scattering the 2P3/2 amplitude.and 2S1/2 Anotherstates, effectiveDE is field theory tech- given by nique would be to develop an procedure to determine the short-distancee lepton-nucleon amplitude implied by the subtraction term. This is the direction we pursue now. DE = 209.9779(49) 5.2262r2 + 0.0347r3 meV, (1) p p where r is given ine units of fm. Using this equation, one can see that the difference p III. EFFECTIVE FIELD THEORY FOR THE µp INTERACTION between the Pohl and CODATA values of the proton radius would be removed by an increase of the first termThe on previous the rhs of considerations Eq. (1) by 0.31 show meV=3.1 the sensitivity10 10 MeV. to assumptions regarding the behav- ⇥ 2 2 This proton radiusior puzzle of T1(0, hasQ been) for attackedlarge values fromQ manyabout different which little directions or nothing [3]-[21] is known. This results The present communicationform the is logarithmic intended to divergence investigate in the the hypothesis integral of that Eq. the (5) proton for the po- case Floop = 1, and is a larizability contributions,symptom that that enter an in inefficient the two-photon technique exchange has been term, used see [27]. Fig. A more1, can efficient way to pro- Caswell Lepage ’86 account for the 0.31 meV.ceed would This idea be us is worthyEFT to use an of of effective considerationµp interaction field theory because (EFT) the computed for the lepton-proton ef- interaction. fect is proportionalIn to EFT the lepton logarithmic• massCompute divergences to the fourth Feynman identified power, diagram, and through so is dimensional capableremove of log being regularization are renor- relevant for muonicmalized atoms, but away irrelevant by includingdivergence for electronic a lepton-proton using atoms. dimensional contact interaction regularization in the Lagrangian. We may handle• include the divergence counter using term standard in Lagrangian dimensional regularization (DR) tech- niques by evaluating the scattering amplitude of Fig. 1. The term of interest is obtained by includingq only T (0, Q2) of Eq. (3)q with F = 1. We evaluate the integral in d = 4 e 1 loop dimensions and obtain the result: relevant data is the 0.31 meV needed to account for the proton radius puzzle. Thus we 3 b 2 µ2 5 DR = i a2m M + log + g + log 4p u u U U , (13) FIG. 1: Thewrite box the diagram resulting for the scattering(a5m2 4 amplitude) corrections. as The graph in2 which theE photons crossf isi f i O M 2 a e m 6 also included. b ⇥ ⇤ where lower caseDR spinors= i a2m representM (l + 5/4 leptons) u u ofU massU m, and upper case proton(14) of mass M, M2 a f i f i q is momentum transferred to the proton, and g is Euler’s constant, 0.577216 . E ··· where l is determinedThe resultChoose by fittingEq. (13) to2 corresponds the Lambto get shift. to 0.31 The an infiniteµ dependence meV contribution shift of the to counter the Lamb term shift in the is chosen solimit that that the resulte goes is to independent zero. In EFT of oneµ. Eq. removes (14) corresponds the divergent to piece using by the addingMS a contact scheme because the term log(4p) g is absorbed into l. interaction to the Lagrangian E that removes the divergence, replacing it by an unknown The correspondingfinite part. contribution The finite part to the is obtained Lamb shift by fitting is given to by a relevant piece of data. Here the only b DEDR = a2m M f2(0)(l + 5/4). (15) a 6 Setting DEDR to 0.31 meV in the above equation requires that l = 769 which seems like a large number. However bM is extraordinarily small. The natural units of polarizability are bM 4p/L3 , [28] where L 4p f ,(f is the pion decay constant). Then Eq. (14) a ⇠ c c ⌘ p p becomes DR 2 4p 2 = i 3.84 a m 3 u f uiU f Ui. (16) M Lc The coefficient 3.84 is of natural size. Thus standard EFT techniques result in an effective lepton-proton interaction of natural size that is proportional to the lepton mass. The present results, Eq. (11) and Eq. (15) represent an assumption that there is a lepton- proton interaction of standard-model origin, caused by the high-momentum behavior of the virtual scattering amplitude, that is sufficiently large to account for the proton radius puzzle. Fortunately, our hypothesis can be tested in an upcoming low-energy µ± p, e± p scattering experiment [22] planned to occur at PSI. IV. LEPTON PROTON SCATTERING AT LOW ENERGIES Our aim is to provide a prediction for the PSI experiment. It is well-known that two- photon exchange effects in electron-proton scattering are small at low energies. Our con- tact interaction is proportional to the lepton mass, so it could provide a measurable effect for muon-proton scattering but be ignorable for electron-proton scattering. We shall in- vestigate the two consequences of using form factors (FF) and EFT. 7 contact interaction to the Lagrangian that removes the divergence, replacing it by an unknown finite part. The finite part is obtained by fitting to a relevant piece of data. Here the only relevant data is the 0.31 meV needed to account for the proton radius puzzle. The low energy term contributes DR(LET)=iC(µ), (14) M2 where C(µ) is chosen such that the sum of the terms of Eq. (13) and Eq. (14), DR, is ⌘ M2 finite and independent of the value of µ. Thus we write the resulting scattering amplitude as b DR = i a2m M (l + 5/4) u u U U (15) M2 a f i f i where l is determined by fitting to the Lamb shift. Eq. (15) corresponds to using the MS scheme because the term log(4p) g is absorbed into l. E The corresponding contributionsubt to the Lamb shift2 is givenM by2 E (DR)=↵ m S(0)( +5/4) b DEDR = a2m M f2(0)(l + 5/4↵). (16) Esubt(DR)=0a .31 meV = 769 Setting DEDR to 0.31 meV in the above equation requires that!l = 769 which seems like a large number. However, bM is extraordinarily small due to a4 cancellation3 between M (magnetic polarizability) = 3.1 10 fm very small paramagnetic effects of an intermediate D and diamagnetic effects⇥ of the pion cloud [30]. 3 Natural units MbM/↵ 4⇡/3(4⇡f⇡) Butler & Savage 092 The natural units of polarizability are ⇠4p/L , [31] where Lc 4p fp,(fp is the a ⇠ c ⌘ pion decay constant). Then Eq. (15) becomes DR 2 4p 2 = i 3.95 a m 3 u f uiU f Ui. (17) M Lc The coefficient 3.95 is of natural size.3.95 Thus =natural standard EFT techniques result in an effective lepton-proton interaction of natural size that is proportional to the lepton mass. The present results, Eq. (11) and Eq. (16) represent an assumption that there is a lepton- proton interaction of standard-model origin, caused by the high-momentum behavior of the virtual scattering amplitude, that is sufficiently large to account for the proton radius puzzle. Fortunately, our hypothesis can be tested in an upcoming low-energy µ± p, e± p scattering experiment [23] planned to occur at PSI. 7