The long-standing difference between the experimental measurement and the standard model prediction for the muon’s anomalous magnetic moment, aμ = (gμ − 2)/2, can be due to new particles flowing in loop contributions: such

discrepancy might thus signal the presence of new physics at the TeV scale. The vast majority of models explaining the muon discrepancy in terms of new physics (NP) predict sizable effects in ae = (ge−2)/2, too. We discuss the experimental prospects to reach sub-ppb precision on ae and test the NP origin of the muon anomaly in its electron counterpart

Terranova, F.

Tino, G. M.

English

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DOI 10.1393/ncc/i2015-11869-6Colloquia: LaThuile14IL NUOVO CIMENTO Vol. 37 C, N. 6 Novembre-Dicembre 2014Experimental prospects to observe the g − 2 muon anomalyin the electron sectorF. Terranova(1)(∗) and G. M. Tino(2)(1) Dipartimento di Fisica, Universita` di Milano-Bicocca and INFNSezione di Milano-Bicocca - Milano, Italy(2) Dipartimento di Fisica e Astronomia and LENS, Universita` di FirenzeINFN, Sezione di Firenze, Sesto Fiorentino (FI), Italyricevuto il 31 Luglio 2014Summary. — The long-standing difference between the experimental measurementand the standard model prediction for the muon’s anomalous magnetic moment,aμ = (gμ − 2)/2, can be due to new particles flowing in loop contributions: suchdiscrepancy might thus signal the presence of new physics at the TeV scale. The vastmajority of models explaining the muon discrepancy in terms of new physics (NP)predict sizable effects in ae = (ge−2)/2, too. We discuss the experimental prospectsto reach sub-ppb precision on ae and test the NP origin of the muon anomaly in itselectron counterpart.PACS 6.20.Jr – Determination of fundamental constants.PACS 13.40.Em – Electric and magnetic moments.1. – IntroductionThe magnetic moment of the muon, μμ = e2mμ (1 + aμ)σ, is a key Standard Model(SM) observable and one of the most precisely measured quantities in physics. It is alsoan important ingredient to electroweak precision tests [1]. Since 1960, the experimentalprecision in aμ has been improved by more than five orders of magnitude and furtherimprovements are expected soon.The experimental value of the anomalous contribution aμ from the latest and mostprecise E821 experiment at BNL [2] differs from the Standard Model prediction by morethan three standard deviations. This discrepancy could be due to theory or experimentalnuisance. It could be related to a statistical fluctuation, an overlooked systematic effect(∗) Speakerc© Societa` Italiana di Fisica 287288 F. TERRANOVA and G. M. TINOin aexpμ or to a genuine new physics effect. In this case, loop corrections from new particlesbeyond the SM could produce a differenceΔaμ = aexpμ − aSMμ = 2.90 (90)× 10−9,(1)explaining the discrepancy observed at BNL.The experimental value will soon be cross-checked by the E989 experiment at Fermilaband the planned g−2/EDM experiment at J-PARC in about five years. A NP explanationof the muon puzzle remains a viable option and should the new experiments confirm thefindings of E821, independent searches of these NP effects by direct production of newparticles at colliders or by the exploitation of other high-precision observables will playa key role.The vast majority of the models explaining the muon puzzle in term of new physics,predicts sizable effects in the electron sector, too. This is due to the fact that the size ofloop contributions generally scales as the squared ratio of the lepton masses. Hence, NPeffects are:– Enhanced by (mτ/mμ)2 in the tau counterpart of aμ. This enhancement hasnot been exploited yet due to the experimental challenges related with the tinyτ lifetime.– Suppressed by (me/mμ)2 in the electron counterpart of aμ. In spite of the superiorexperimental accuracy of ae compared with aμ, NP effects become invisible in theelectron sector due to (me/mμ)2  2× 10−5.This natural suppression factor (“naive scaling”- NS) can be sidestepped in specificmodels due to either non-universality of the couplings or of the NP mass spectrum [3].It has been noted in 2012 [3] that a combination of naive scaling violation and thehigher accuracy in ae could make NP effects visible to experiments in the electron sector.In fact [4], improvements in the electron sector measurements that can be achievedemploying atom interferometry are so significant that they can attain the level of precisionneeded to test NP effects even in the naive scaling hypothesis (0.06 ppb in ae). Theseresults will likely be achieved on a timescale comparable to next generation g − 2 muonexperiments at Fermilab and JPARC and, hence, will play a major role in clarifying theorigin of the muon anomaly.2. – The measurement of ae and the fine structure constantThe best measurements of ae have been obtained from single electrons in Penningtraps [5]. In the 70’s research groups at the University of Mainz and the University ofWashington (UW) developed methods to measure the electron magnetic moment using alarge number of electrons stored in a Penning trap. Finally, they were able to confine anddetect a single electron in the trap reaching an uncertainty of 4 ppt in g/2 (3.5 ppb in ae)in 1987. This result led the CODATA fits [6] for nearly 20 years. A major breakthroughoccurred in 2006 thanks to the development of cylindrical Penning traps. This device setsup boundary conditions that produce a controllable radiation field within the trap cavity.Spontaneous emission is strongly reduced at the same time as corresponding shifts of theelectron’s oscillation frequencies are avoided.The most recent Harvard measurement of ae with cylindrical Penning trap achieveda 0.24 ppb accuracy, i.e. a factor of four larger than the precision needed to seeEXPERIMENTAL PROSPECTS TO OBSERVE THE g − 2 MUON ANOMALY ETC. 289the new physics responsible for the muon g − 2 in the occurrence of naive scaling.This measurement (aexpe ) would already be able to constrain specific models that breakNS and enhance the new physics contributions in ae with respect to aμ. However, therole of ae as a probe for NF becomes quite marginal once we remove the correlationbetween aexpe , which is commonly used to extract α, and its theory expectation aSMe thatstrongly depends on α (aSMe is α/2π at leading order). If we use a fully independentdetermination of α based on cold atom techniques, the accuracy on the theory predictionfor ae (aSMe ) grows up to 0.66 ppb.This considerations strengthened the standard approach of exploiting the electrong− 2 for the determination of α and resorting to the muon (or tau) g− 2 to seek for NPeffects in loop contributions.3. – A new role for the electron g − 2In the next few years the role of ae will likely change, moving from the key measure-ment for the determination of α to a powerful observable to seek for physics beyond thestandard model [4] (as it is the case now for aμ and several classes of rare decays). Thisis mostly due to three reasons that are briefly recalled in the following.– Experimental determination of ae. Cylindrical Penning traps have not reached theirlimiting systematics, yet and additional improvements can be envisaged. The origi-nal 2006 Harvard measurement [7] was mostly dominated by cavity shift modeling.In cylindrical Penning traps the interaction of the trapped electron and the cavitymodes shifts the cyclotron frequency and the shift has to be properly modeled toextract ae. This shift cannot be avoided if we want to reduce spontaneous emissionof synchrotron radiation. Cavity shift is a source of systematics that is intrinsicto the technology of cylindrical Penning traps. It can, however, be reduced below0.08 ppb for ae (< 0.1 ppt for g/2).– Independent determination of α. The outstanding precision reached in the mea-surement of the Rydberg constant offers a different venue to determine α. Thisnew approach (see below) neither depends on the measurement of ae nor on thehigh-order perturbative calculation of QED corrections.– Theory uncertainty on ae. The uncertainty in the theoretical determination of aSMeis appropriate to check the aμ anomaly at NS level. The uncertainty is mostly dueto the numerical approximations employed for the evaluation of four and five loopQED contributions (0.06 ppb). It is worth mentioning that, unlike aμ, the hadronicterm uncertainty (0.02 ppb) is negligible if we aim at observing new physics in theNS scenario. Even now the overall theory uncertainty is within the error budgetfor NS (0.06 ppb) and further improvements are possible [3].4. – An independent determination of αHaving a ppb measurement of α independent of ae is a realistic goal thanks to theimproved measurement of the Rydberg constant. The optical comb generators allowedthe measurement of the narrow (1.3Hz) 1S-2S two-photon resonant line of the hydrogenwith a relative precision of 3.4 × 10−13 [8]. It corresponded to an improvement of twoorders of magnitude with respect to previous measurements. The new data on hydrogenspectroscopy resulted in a measurement of the Rydberg constant with a precision better290 F. TERRANOVA and G. M. TINOthan 0.01 ppb (7× 10−12 [6]). Since R∞ = meα2c/2h, the outstanding precision on R∞connects α to the evaluation of the quotient h/me. For a given atom X,α2 =2R∞cmXmehmX= 2R∞cmXmumumehmX.(2)Equation (2) opened many opportunities for an independent determination of α basedon cold atom interferometers, which are particularly well suited to determine the h/MXratio. The latter is also interesting to metrologists for the redefinition of the kilogram. Onthe other hand, using eq. (2) to extract α pays the penalty of two new systematic sources:the systematics due to the knowledge of the mass (MX/mu) of the atom employedto measure h/MX and the uncertainty on the electron mass expressed in atomic massunit(1).5. – The nuisance parameters MX/mu and A(me)Atom interferometers (see below) exploit mostly hydrogen-like atoms, i.e. atoms withonly one electron in partially filled shells. Alkali metals thus play a special role amongpossible atom candidates. For alkali metals, the atomic masses of the isotopes relevant forthe determination of α have been measured with high precision employing orthogonallycompensated Penning traps. The most recent results [9] from Washington Univ. andFlorida State University significantly improve the AME2012 evaluation for several atomicmasses, including 23Na, 39,41K, 85,87Rb and 133Cs. The most plausible candidates foratom interferometers are measured with a typical uncertainty of 0.1 ppb. This accuracy isstill too low for new physics tests in the NS framework. Fortunately, reaching precisionswell below 0.05 ppb for candidates as Rb, Cs or non-alkali atoms like Sr is possibleimproving the experiments that are currently operating [10] (double Penning traps withmultiply-charged ions) and we do not expect the uncertainty on MX/mu to dominatethe knowledge of α in the years to come. 4He is worth a special mention since its atomicmass is known with outstanding accuracy (0.016 ppb). 4He is not hydrogen-like but itsmetastable state is often employed in atom interferometers, making 4He∗ a speciallysuitable candidate for an α-oriented measurement of h/M4He.Until 2014, the most important nuisance parameter of eq. (2) was the limited knowl-edge of the electron mass expressed in atomic mass unit. According to CODATA2010 [6],the relative atomic mass of the electron Ar(me) ≡ me/mu isAr(me) = 5.4857990946(22)× 10−4 (0.4 ppb).(3)This estimate is not the outcome of a direct measurement of electrons in Penning trapscompared with the reference atom 12C because direct measurements do not reach theO(1 ppb) precision. The estimate of eq. (3) is obtained from ae in bound-state QED. Fora bound electron in hydrogen-like systems of nuclear charge Z, the electron anomalousmagnetic moment gb is different with respect to the free-particle value. At tree level,gb = gBreitb ≡ 2/3(1 + 2√1− Z2α2). The measurement of abe ≡ (gb − gBreitb )/gBreitbdone in Penning traps is not competitive with ae (1.9 ppm versus 0.24 ppb) but can beused to evaluate Ar(me). The most relevant drawback is that this determination heavily(1) Here, mu is the mass of 1/12 the mass of12C.EXPERIMENTAL PROSPECTS TO OBSERVE THE g − 2 MUON ANOMALY ETC. 291relies on theory calculations in bound-QED. However, calculations have been checkedagainst several atoms and these checks strengthen our confidence on theory modeling.Until 2014, bound QED provided Ar(me) with an accuracy of 0.4 ppb: this uncertaintypropagates to me/mRb bringing the error on this ratio to 0.44 ppb. A much improvedaccuracy (a factor of 13 better) has been obtained in 2014: details and implications forNP searches can be found in [11] and [4], respectively.6. – Atom interferometersAtom interferometers were first demonstrated in early 90’s and in the last 20 yearsadvances in atom interferometry led to the development of several innovative techniquesfor fundamental physics experiments and for applications (see [12] for an overview). Inparticular, atom interferometers are currently employed for precision measurements ofgravity acceleration, rotations, for the determination of the Newtonian constant G, fortesting general relativity and quantum gravity models, and for possible applications ingeophysics.The above-mentioned quotient h/MX results from the measurement of the recoilvelocity vr of an atom X when it absorbs a photon of frequency ν (vr = hν/M). Themeasurement is performed by combining a Ramsey-Borde´ atom interferometer with theBloch oscillations technique. Bloch oscillations can be observed in atomic systems whenthe atoms are shed with two counterpropagating laser beams whose frequency difference isswept linearly. This setup produce oscillations that resemble the oscillations experiencedby an electron inside a solid in the presence of an electric field (“Bloch oscillations”).During the frequency sweep, the internal state of the atom is unchanged, the atom isaccelerated and its velocity increases by 2vr per oscillation. The Doppler shift due tothis velocity variation is compensated by the frequency sweep itself. This technique isextremely effective in terms of photon momentum transfer and it has soon become thereference method for the measurement of h/MX .The most relevant atom candidates and the perspective for improvement of atominterferometers for the measurement of h/MX are discussed in [4] and briefly summarizedin the following.Cesium: The exploitation of eq. (2) to link α with the quotient h/MX has been intro-duced by the group of S. Chu at Stanford University and implemented using 133Cs [13].The value of α was measured with a precision σα/α = 7.4 × 10−9. The most recentexperiments achieved a relative uncertainty for α of ∼ 2 ppb [14], mainly limited by thestatistical error. Implementation of the Bloch oscillation technique in this scheme isexpected to reduce the overall uncertainty well below ppb [15].Rubidium: Measurement based on rubidium and performed in Paris currently providethe best measurement of h/MX . These experiments [16] exploit atom interferometersbased on a combination of a Ramsey-Borde´ interferometer with Bloch oscillations. Themost precise value obtained so far by atom interferometers is for 87Rb [16] and it corre-sponds to h/MRb = 4.5913592729(57)×10−9 m2s−1 (1.24 ppb in h/M , i.e. 0.62 ppb in α).The precision in [16] is mostly limited by laser beams alignment, wave front curvatureand Gouy phase effects and the current upgrades are aimed to a precision in α of about0.1 ppb.Strontium: Sr-based atom interferometers with long-lived Bloch oscillations have al-ready been developed and they are currently employed for gravity measurements at smallspatial scales [17,18]. These interferometers can also provide a measurement of the h/MSrratio with systematics and potential upgrades comparable to Rb [19].292 F. TERRANOVA and G. M. TINOHelium: Helium is not hydrogen-like but can be successfully used for atom interfer-ometry thanks to its long-lived metastable states. An experiment on He was started inAmsterdam using 4He in a 1D-lattice to perform Bloch oscillations and velocity mea-surement with an atom interferometer [20]. In principle, helium can achieve an accuracycomparable (or better than) Rb and, as noted above, the helium mass is known withoutstanding precision (relative uncertainty of 0.015 ppb).7. – ConclusionsFor many decades, the electron g− 2 has been considered an ideal tool for QED testsand for metrology, with very limited sensitivity to new physics in loop corrections. Theremarkable progress in the technology of the cylindrical Penning traps, in the determi-nation of the Rydberg constant and in the knowledge of the electron mass opens newvenues in this field. In particular, the accuracy that can be reached in aexpe and α is suchthat NP effects that might cause the g− 2 muon discrepancy will be evident even in theoccurrence of naive scaling.∗ ∗ ∗We wish to express our gratitude to K. Blaum, P. Clade´, E. Myers, H. Mu¨ller, P. Par-adisi, M. Passera and W. Vassen for many useful information and fruitful discussions.REFERENCES[1] Jegerlehner F. and Nyffeler A., Phys. Rep., 477 (2009) 1.[2] Bennett G. W. et al., Phys. Rev. D, 73 (2006) 072003.[3] Giudice G. F., Paradisi P. and Passera M., JHEP, 11 (2012) 113.[4] Terranova F. and Tino G. M., Phys. Rev. A, 89 (2014) 052118.[5] See, e.g., Hanneke D., Fogwell Hoogerheide S. and Gabrielse G., Phys. Rev. A, 83(2011) 052122 and references therein.[6] Mohr P. J., Taylor B. N. and Newell D. B. (CODATA 2010), Rev. Mod. Phys., 84(2012) 1527.[7] Odom B. C., Hanneke D., D’Urso B. and Gabrielse G., Phys. Rev. Lett., 97 (2006)030801.[8] Udem T., Huber A., Gross B., Reichert J., Prevedelli M., Weitz M. and HanschT. W., Phys. Rev. Lett., 79 (1997) 2646.[9] Myers E. G., Int. J. Mass Spectrom., 349 (2013) 107.[10] Myers E. G., FSU University, private communication.[11] Sturm S. et al., Nature, 506 (2014) 467.[12] Tino G. M. and Kasevich M. A. (Editors), Atom Interferometry, in Proceedings of theInternational School of Physics “Enrico Fermi”, Course CLXXXVIII (Societa Italiana diFisica and IOS Press) 2014.[13] Wicht A., Hensley J. M., Sarajlic E. and Chu S., Phys. Scr., 102 (2002) 82.[14] Lan S. Y., Kuan P.-C., Estey B., English D., Brown J. M., Hohensee M. A. andMuller H., Science, 339 (2013) 554.[15] Mu¨ller H., UC Berkeley, private communication.[16] Bouchendira R. et al., Phys. Rev. Lett., 106 (2013) 080801.[17] Ferrari G., Poli N., Sorrentino F. and Tino G. M., Phys. Rev. Lett., 97 (2006)060402.[18] Poli N., Wang F.-Y., Tarallo M. G., Alberti A., Prevedelli M. and Tino G. M.,Phys. Rev. Lett., 106 (2011) 038501.[19] Tino G. M., Testing gravity with atom interferometry in [12].[20] Vassen W., VU University, Amsterdam, private communication.

Experimental prospects to observe the g − 2 muon anomaly in the electron sector

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Experimental prospects to observe the g − 2 muon anomaly in the electron sector

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