Limitations on detecting Higgs couplings in transitions of muonic atoms Rajmund Krivec1
arXiv:1606.08257v4 [hep-ph] 1 Dec 2016
1
Department of Theoretical Physics, J. Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia∗
In light of the known Higgs mass and the current presumed range of the quark-lepton Higgs coupling, a recent work has proposed to extract limits on Higgs-nucleon coupling from experiments on heavy atoms as the Higgs term is enhanced by AZ. The approach utilizes the large value of the electron coupling modifier, and requires sufficiently precise electron wave functions. We instead revisit the old idea of extracting limits on muon-nucleon Higgs coupling from muonic transitions, e.g. the Lamb shift in light muonic atoms, where the coupling is enhanced by about 2013 AZ 3 , primarily due to large muon wave function at origin, and locally precise two- and three-body methods exist. As the muon coupling modifier is close to 1, an experimental precision well below 1 ppm is required to cover the upper range of possible Higgs couplings. Precision of theoretical calculations of the non-Higgs contributions to the Lamb shift is limited by uncertainties of the nuclear finite-size and polarization terms. The current focus of muon physics on the extraction of charge radii drives improved calculations of the polarization terms but not the finite-size terms. Unless isotope shifts could be employed, further improvement is limited by dependence on nuclear structure parameters, either nuclear potential parameters or assumed nuclear charge distributions in nonperturbative theories. PACS numbers: 14.80.Bn, 14.60.Ef, 36.10.Ee, 31.15.ac, 31.15.xj
I.
INTRODUCTION
There has been a proposal to derive limits on Higgs nucleon-lepton coupling constant from valence electron transitions in heavy atoms [1–3]. The approach is based on the enhancement of the coupling due to a large atomic number A; the stability of systems with large A allowing the experiment to reach precision of the order of 1 Hz in atomic clock transitions; and the relatively small other relevant corrections like the weak force, despite the fact that the latter may mask the Higgs contribution. A disadvantage of using electronic states of heavy atoms is the approximate nature of the factorization of the screened, relativistic electron wave function at the origin whose value determines the transition matrix element. To eliminate uncertainties in theoretical corrections depending on total charge Z, Refs. [1, 2] propose to test the deviations from linearity in the King’s plots [4] in isotope shift measurements instead of measuring transition frequencies directly. This employs the fact that change in A affects the transitions via nuclear recoil, electron correlations and nuclear charge radius independently of the transition measured. Another disadvantage is the reliance on the existence of new physics via the relatively large current value (of the order of 103 ) of the coupling modifier κe for the electron-Higgs coupling constant ye relative to its Standard Model (SM) value, ye = κe yeSM . We examine the alternative approach of testing the Higgs couplings in muonic atoms or ions, preferably in a two- or three-body atomic system exhibiting a transition where the Higgs contributions to the states are different. With the Higgs mass now known, we start by revisit-
∗ Electronic
address:
[email protected]
ing the earlier estimates [5] of the Higgs contribution to the muonic 4 He 2S − 2P Lamb shift. Muonic orbits being smaller than electronic orbits by the large ratio of the respective reduced masses mµA and meA results in a large coupling enhancement of (mµA /meA )3 for shortrange potentials. There are several disadvantages of this proposal. First, the 2S state lifetime is of the order of 1 µs [6, 7] as a result of the finite muon lifetime of 2.2 µs, the 2S − 1S two photon decay time of 8 µs and the collisional quenching rate in gas. This places a lower bound of the order of 1 MHz on the magnitude of measurable effects. Second, while the sum of QED corrections (1813.02 meV for (µ4 He)+ [5]) is already rather well known, the small muon orbit results in a large finite nuclear size correction (hundreds of meV in helium) and the nuclear polarization correction (a few meV). Both have large uncertainties due to the dependence on theoretical nuclear structure parameters multiplied by large coefficients which also scale unfavorably as Zm3µA . Third, muonic experiments currently concentrate on extracting the finite-size corrections themselves to resolve the charge radius puzzle in light nuclei. The Lamb shift measurements are used to extract the charge radii with the help of the polarization terms calculated theoretically and the Zemach radii extracted from the hyperfine splitting (HFS) measurements of the ground states. Consequently theoretical nuclear calculations concentrate only on the uncertainty of the polarization terms. The current experiments, e.g. by the CREMA collaboration [8], determine the charge radii to several digits but the finite-size terms themselves still carry large absolute errors due to large coefficients. The Higgs term extraction would require much higher experimental precision and more precise theoretical calculations of non-Higgs terms including the finite-size corrections now extracted from experiment.
2 Light muonic systems from µH to eµ4 He are two- and three-body systems. Locally precise nonrelativistic wave functions can be calculated even for the three-body problem, e.g. by the Correlation-function Hyperspherical Harmonic Method (CFHHM) [9]. For relativistic or nonperturbative calculations it is still possible to remain within the two-body problem by studying systems up to (µ4 He)+ as losing an electron does not change the muon states much. In heavy muonic systems, the small muonic orbits experience weak electron screening as opposed to the heavily screened valence electron orbits but also are dominated by nuclear structure effects and, moreover, the perturbation theory in (αZ), where α is the fine structure constant, well developed for light muonic atoms up to the order (αZ)6 , may make systematic study of perturbation terms much more difficult [10] The aim of the paper is to calculate the orders of magnitude of the parameters involved according to highenergy experimental status, in particular the coupling modifiers, identify candidate muonic transitions for Higgs term extraction and suggest ways to reduce the uncertainty of the bottlenecks in theoretical calculations. II.
BOUNDS ON HIGGS CONTRIBUTION IN MUONIC SYSTEMS
Higgs exchange between a nucleus and a bound electron or muon results in a potential of the Yukawa type, e−mH r , VH (r) = −gHµ,A r
For the muon, κµ = 0.2+1.2 −0.2 in Table 15 of Ref. [14] so we do not get the advantage of a weak experimental upper bound, resulting practically in the SM coupling value, yµ . 1.4 × 207 × yeSM ≈ 0.6 × 10−3 ,
using the upper bound of κµ . The nuclear coupling is approximately proportional to the atomic number A, yA = (A − Z)yn + Zyp ≈ A yN ,
yn ≈ 7.7yu + 9.4yd + 0.75ys , yp ≈ 11yu + 6.5yd + 0.75ys .
yµ yA . 4π
(2)
mF y F = κF , v = 246 GeV. v
(3)
For the electron, ye = κe × 2.1 × 10−6 .
(4)
Ref. [1] uses ye at the upper bound set by the LHC data on H → e+ e− [11–13], where κe < 611 [13], ye < 611 × 2.1 × 10−6 ≈ 1.3 × 10−3 .
(5)
This κe value corresponds to the lowest new physics scale, 5.8 TeV [13], but the bounds are likely to improve, lowering the ye value and reducing the feasibility of the proposal [1, 2].
(9)
and in muonic 4 He: y4 ≈ 4 × {10−3 , 0.2, 3}.
(10)
The corresponding bounds on gHµ,A are linear in A: gHµ,1 . {0.5 × 10−7 , 1 × 10−5 , 0.1 × 10−3 }
The SM fermion-Higgs coupling constants are proportional to the fermion (F ) mass according to the assumed hierarchy leading to fermion masses, as well as to the coupling modifiers based on experimental upper bounds on couplings which allow for the existence of new physics:
(8)
The weakest bounds on individual quark couplings are yq . 0.3 [20–22], where yq is one of yu , yd , ys , or yc , resulting in yN . 3 due to suppression of light quarks. LHC and electroweak data give a medium bound yq . 1.6 × 10−2 [21, 23, 24], resulting in yN . 0.2. Indirect bounds may be even lower, yq . 5 × 10−3 [1, 25] resulting in yN . 10−3 . These results translate to the following range of current upper bounds on the nuclear coupling yA in muonic hydrogen
where gHµ,A is proportional to the muon and nuclear coupling constants, gHµ,A =
(7)
where yn ≈ yp ≈ yN are the neutron and proton coupling constants which are linear combinations of the quark and gluon coupling constants. In more detail [1, 15–18] and neglecting the cg term [19],
y1 ≈ {10−3 , 0.2, 3} (1)
(6)
(11)
for µH and gHµ,4 . {2 × 10−7 , 4 × 10−5 , 0.6 × 10−3 }
(12)
for (µ4 He)+ /eµ4 He. The Higgs contribution to a transition energy and the corresponding required experimental precision at the above range of gHµ,A upper bounds can be estimated perturbatively due to the large Higgs mass, unlike Ref. [5] where calculations were performed for relatively small mH from 0.15 MeV to 750 MeV. The Higgs contribution in states with orbital angular momentum l > 0 is negligible and transitions of interest involve at least one S state. In muonic systems a sufficient level of accuracy is provided by the unmodified states of a muon bound to the nucleus, as the small muon orbit is is weakly screened. For instance, the ground state energy of muonic ion (µ4 He)+ scaled from the hydrogen value is E0 ≈ 4 × 201 × 13.6 eV ≈ 10.9 keV,
(13)
3 which is very close to the neutral eµ4 He atom ground state energy of 10.956 keV [9]. For the extensively measured Lamb shift at the muon principal quantum number n = 2, the Higgs effect on the transition energy ∆E2P −2S in the leading order (1/mH )2 comes from the 2S state matrix element 2 1 (14) δ(∆E2P −2S ) ≈ gHµ,A R20 (0) 2 , mH
where R20 is the n = 2, l = 0 radial wave function Rnl of the muon, 3 R20 (0) 2 = 4 Z , (15) 2aµ
and aµ is the reduced Bohr radius of the muon. We get 2013 -fold enhancement of the Higgs term relative to electronic transitions at the same n through |R20 (0)|2 by way of the smaller muon reduced Bohr radius. The Higgs term scales as ≈ AZ 3 resulting in total enhancement ≈ 2013 × AZ 3 . Most of it is used up to offset the finite muon lifetime by placing the Higgs effect above the MHz region. The transition energy scales as ≈ Z 3 , so the relative experimental accuracy (in ppm) is about 4 times less demanding in (µ4 He)+ /eµ4 He than in µH. TABLE I: Experimental requirements for extracting the Higgs contribution δ(∆E2P −2S ) to the 2S − 2P Lamb shift of the (µ4 He)+ ion for the Higgs mass of 125 GeV and the highest value from Ref. [5] for comparison. The assumed range of bounds on the nucleon-Higgs coupling yA is the same as in Ref. [1]. δν are the frequencies corresponding to δ(∆E2P −2S ). The last column gives the required precision η as the ratio of the Higgs contribution and the transition energy, η = |δ(∆E2P −2S )/∆E2P −2S |, where ∆E2P −2S ≈ 1664 meV, in ppm. Rows 4 and 5 correspond to the precision defined as the discrepancy between theory and experiment in Ref. [5]: row 4 shows the extrapolated, out of range gHµ,4 value that would be needed for observation at that η, and row 5 is from Ref. [5]; the experimental precision in Ref. [5] was 200 ppm. mH (GeV)
gHµ,4
δ(∆E2P −2S ) (meV)
δν (Hz)
η (ppm)
2 × 10−7 2 × 10−8 5 × 103 10−5 4 × 10−5 4 × 10−6 1 × 106 3 × 10−3 −3 −4 0.6 × 10 0.6 × 10 2 × 107 < 0.1 125 36 4 1.0 × 1012 3 × 103 0.75 1.3 × 10−3 4 1.0 × 1012 3 × 103 125
Tables I and II give experimental accuracy requirements for extracting the Higgs term form the 2S − 2P Lamb shift for (µ4 He)+ or eµ4 He and µH, respectively. The first three rows of Table I give the current requirements; for comparison we list the much more optimistic values assumed when the idea was first proposed [5]. Errors due to uncertainties of the fundamental constants are of the order of 10−5 meV [26].
TABLE II: As in Table I, but for µH; ∆E2P −2S ≈ 202 meV. mH (GeV) 125
gHµ,1
δ(∆E2P −2S ) (meV)
δν (Hz)
Precision (ppm)
0.5 × 10−7 0.7 × 10−9 2 × 102 3 × 10−6 1 × 10−5 1 × 10−7 3 × 104 10−3 −3 0.1 × 10 1 × 10−6 3 × 105 < 0.01
Tables I and II show that the δν requirements to measure the Lamb shift reach above the 2S state lifetime limit only for (µ4 He)+ /eµ4 He (or heavier) systems, and only for the upper range of gHµ,4 . In µH, the transition itself lies above the limit but the Higgs effect does not. The Higgs term extraction may thus avoid the more difficult measurements in muonic hydrogen [27–29] but would have to deal with larger nuclear structure effects in helium systems. The advantage of using the Lamb shift, apart from the non-cancellation of the Higgs shifts of the transition states and the well developed theory, is its small energy allowing relatively reasonable experimental precision requirements η. In contrast, in normal (electronic) heavy atoms of the proposal [1–3] the coupling is enhanced relative to single nucleon-electron coupling by up to two orders of magnitude through yA by way of large A, and through the electron wave function squared at origin, |ψe (0)|2 , by way of large Z (electron screening in heavy atoms precludes enhancement by Z 3 that would be expected from an unperturbed electron wave function, resulting in the enhancement factor Z instead [1]). Together with the current coupling modifier, the total enhancement is ≈ 611 × AZ. The electronic and muonic couplings for the same A, n are related by gHe,A ≈ 2.2 gHµ,A . The finite size corrections scale with (meA /mµA )3 ≈ 0.1 × 10−6 and the Higgs term as gHe,A (meA /mµA )3 ≈ 0.27 × 10−6 . The helium ion 2S − 2P Lamb shift is 14 GHz [30] and the Higgs term appears at the 5 Hz (or 1 ppb) level at the upper limit on coupling (the third row of Table I).
III.
OVERVIEW OF LIGHT MUONIC ATOM SPECTROSCOPY
The 2S1/2 − 2P3/2 and 2S1/2 − 2P1/2 lines involved in the Lamb shift extraction [31, 32] in (µ4 He)+ were measured long ago [33, 34] to lie at about 1528 meV and 1381 meV, respectively. At the time of the initial proposal [5] based on the assumed rather light Higgs in the sub-GeV range, the discrepancy between theory and experiment was 4 meV, most of it originating in the theoretical finite-size correction (−288.9 ± 4.1 meV) and nuclear polarization (3.1 ± 0.6 meV) while the experimental uncertainties in the Lamb shift were an order of magnitude smaller, ±0.3 meV (±0.5 meV) or 200 ppm (330 ppm), respectively. The finite-size effect was calculated using the electron
4 scattering 4 He radius known at the time of 1.674 ± 0.012 fm [35]. A decade ago improved theoretical calculations of corrections to the 2S1/2 − 2P1/2 Lamb shift [26] reduced the uncertainty of the non-nuclear corrections to 10−3 meV (240 Hz, 0.6 ppm) for (µ4 He)+ . The uncertainty of the finite-size correction (−295.848 ± 2.8 meV) was still large (relative error 1 × 10−2 ), double the error of the charge radius itself [26], while the uncertainty of the nuclear polarization term of the two-photon exchange correction remained unchanged at 0.6 meV. The uncertainty of the 4 He charge radius was 1.676(8) fm (5 × 10−3 relative error) [36, 37]. The charge radius problem has resurfaced a few years ago in the form similar to the proton charge radius puzzle [38, 39], which is the 0.04 fm (5%) difference between the measured proton radii from electron-proton scattering [40] and from muonic hydrogen spectroscopy. The 4 He charge radius from electron scattering is known to 2 × 10−3 relative accuracy (1.681 ± 0.004 fm) [41, 42]. The very recent µD results [43] confirm the existence of the charge radius problem in the deuteron. Interestingly, discrepancies in µH and µD scale with the third power of the muon reduced mass, not contradicting a proposed explanation involving new physics beyond the Standard Model. Currently the precision of Lamb shift experiments is geared towards improving the charge radii via the extraction of the finite-size corrections using theoretical nuclear polarization term as input [8, 44]. These measurements are complemented by measurements of the electronic 1S − 2S transition in 4 He+ which serve to test the QED part [30]. Recent laser spectroscopy measurements of the Lamb shift in muonic He ions (µ3 He)+ and (µ4 He)+ have aimed at “moderate” precision of 50 ppm [6, 41] allowing the charge radius extraction to relative accuracy 1 × 10−3 [6]. The (µ3 He)+ and (µ4 He)+ Lamb shifts were measured in 2013 and 2014 to 40 ppm by the CREMA collaboration [7, 45] allowing determination of alpha particle and helion charge radii to about 3 × 10−4 [7]; this is the current precision guideline for theoretical calculations. The experimental precision of the hyperfine splitting of the ground state of the neutral muonic helium eµ4 He, for the purpose of extracting the Zemach radius from nuclear polarization or vice versa, 4465.004(29) MHz (1.8 × 10−3 meV), was 6.5 ppm [46], while the CREMA collaboration is aiming at 1 ppm [7]. These experiments are also used to check some terms in the Lamb shift: a contribution from the nuclear polarization term cancels the third Zemach moment in µ4 He+ [47] like it was observed earlier for µD [48, 49]. The current theoretical results for the Lamb shift and the hyperfine splitting are summarized in Ref. [50] up to (µ4 He)+ and in Ref. [51] for µH. Except for the nuclear structure corrections, most corrections expressed in meV are reported to four or five digits after decimal point which indicates the required precision for Higgs term extraction is reachable in those terms.
The largest term, the vacuum polarization (VP), contains the following uncertainties [50, 52–56]. The Uehling term in µH calculated perturbatively from relativistic wave functions is modified by finite-size effects by about 0.0079 meV and 0.0082 meV for the two proton radii involved in the proton radius problem, 0.842 fm and 0.875 fm, respectively. For (µ4 He)+ , the finite-size effect in VP is −0.3297 hrα2 i meV fm−2 which implies a similar uncertainty of 0.0016 meV for the current 5 × 10−3 4 He radius uncertainty. Neglecting finite nuclear size in muon-electron VP causes moderately increasing shifts of up to 0.0001 meV for (µ4 He)+ . The uncertainties of the “light-by-light” corrections reach 0.0006 meV for (µ4 He)+ , while the sixth-order VP theoretical uncertainties reach 0.003 meV. The largest uncertainty within the VP corrections is exhibited by the hadronic VP, reaching an estimated 5% uncertainty in the 0.225 meV value for (µ4 He)+ , or estimated 0.012 meV [50]. The relativistic recoil corrections amount to about 0.001 meV [52–56]. The largest uncertainties by far lie in the nuclear structure terms, specifically in the finite-size term. The onephoton exchange term proportional to the charge radius squared hrp2 i in µH has uncertainty 0.064 meV for the spectroscopic charge radius 0.875 fm and 0.010 meV for the Lamb shift radius of 0.842 fm; however, the corresponding values -3.978 meV and -3.6855 meV differ by 8%, or 0.3 meV. The situation in (µ4 He)+ is worse due to the scaled contribution, amounting to 1.4 − 2.8 meV uncertainty depending on which radius is taken [26, 50]. The Lamb shift is conventionally parametrized in terms of charge distribution moments as A + Bhr2 i + C(hr2 i)3/2 where r is the charge radius. For (µ4 He)+ , B = −106.344 hrα2 i meV fm−2 [50], and consists of six contributions, the largest being the leading term ba = −
2 2αZ αZmµA 3 2αZ 1 =− Rn0 (0) 3 n 3 4
(16)
in Eq. (5) of Ref. [10], amounting to −105.319 hrα2 i meV fm−2 . (The total includes the finite-size VP correction −0.3297 hrα2 i meV fm−2 quoted above.) The coefficients are typically calculated to a precision slightly exceeding the current radius precision. In 4 He+ , the nonleading five contributions already amount to about a percent of the total. However, some depend on the assumed, e.g. exponential, charge distribution via terms e.g. ba (αZ)2 hln(αZmµA r)i (Ref. [50], Appendix B, and Ref. [10]). Eq. (16) is written for nonrelativistic perturbation theory, and the relativistic corrections start at (αZ)6 as verified using perturbation theory based on both the Schr¨odinger and the Dirac wave functions in Ref. [10]. Radius-independent corrections for µH are summarized in Table 1 of Ref. [51]. Recent ab-initio theoretical calculations of the inelastic term in the two-photon exchange correction in light muonic atoms using state of the art nuclear potentials (AV18 and χEFT) and the hyperspherical harmonic EIHH method [47, 57, 58] have significantly reduced the
5 extracted charge radius uncertainty. The nuclear problem was solved separately and the polarization terms evaluated in second-order perturbation theory where the perturbation was the residual Coulomb potential in the point nucleon approximation. A 5 × 10−2 accuracy is required for determining the 3 He and 4 He charge radii squared to 3 × 10−4 accuracy [7, 41, 47, 57], ensuring the same absolute errors in both terms. The new value of the nuclear polarization term −2.47(14) meV [47] has 6 × 10−2 accuracy (absolute error is misquoted as 0.015 meV in Ref. [50]) compared with the old [5] value of 3.1 ± 0.6 meV (2 × 10−1 accuracy). The potentials AV18 and χEFT are tuned to the 3 He binding energy but they give different charge radii. In this context and at this level of accuracy, uncertainty of the polarization term may plausibly be further reduced using the 4 He charge radius to constrain the nuclear potential models [47]. Nonperturbative calculations have been performed for muonic hydrogen, however they did not employ microscopic nuclear models. A calculation solving the Dirac equation to 500 neV accuracy [59] describes recoil only via the reduced mass of the muon thus leaving further recoil corrections to perturbation theory. The charge distributions used were again parametrized in terms of moments, i.e. hrp2 i, to express results in the conventional form. Also, a number of other corrections were not calculated [50] (two- and three-loop VP, muon self-energy, muon and hadron VP, and nuclear polarization). The terms calculated differ from the perturbation theory to the order of 0.03 meV. Another calculation [60] used the proton dipole form factor, Gaussian, uniform, Fermi and experimentally fitted charge distributions seems better converged, listing the subset of contributions calculated to better than 0.001 meV accuracy, but the differences between charge distributions cause, for example, an uncertainty of the order of 0.004 meV in the Coulomb and vacuum polarization terms. Current results [50, 59, 60] for the parametrized µH Lamb shift are compared in Table III.
TABLE III: Coefficients of the µH Lamb shift parametrization A + Bhrp2 i + C(hrp2 i)3/2 for perturbative [50] and nonperturbative calculations. Higher terms from Ref. [60] are not quoted. Ref. A (meV)
B (meV fm−2 ) C (meV fm−3/2 )
[50] 206.0579(60) -5.22713 [59] 206.0604 -5.2794 [60] 206.0465137 -5.226988356
0.0365(18) 0.0546 0.03530609322
The proton charge problem notwithstanding, individual terms, with some mutual cancellations, still differ to the order of 0.05 meV, the method of Ref. [60] being in better agreement with the perturbative method at this stage. The corresponding uncertainties in (µ4 He)+ may scale up appreciably.
IV.
EXTRACTING THE HIGGS TERM
For the Lamb shift in light muonic systems, the muonic helium represents a tradeoff between experimental accuracy demands and the size of the nuclear structure corrections. The bottleneck is the theoretical finite-size term together with other small contributions assembled in the Bhr2 i component, where non-leading contributions to B increase with Z, and are at the percent level, similar to the accuracy of hr2 i, already for helium. The uncertainty of the theoretical polarization term is currently three orders of magnitude larger than the requirements in Table I. The uncertainty of the finite-size term is an additional order of magnitude larger due to the fact that for Higgs term extraction, this term cannot be improved using the polarization term as input or by constraining nuclear models using a charge radius as input [47]. The estimated accuracy of the theoretical charge radius based on the accuracy of the polarization term [47, 57, 58] would be about 6 × 10−2 as limited by the differences between nuclear models. The coefficient B is more precise (Ref. [50], Table 14). Other terms like the electron-muon HFS which for the ground state of neutral muonic He is 1.8 × 10−3 meV [9] need to be taken into account. Increasing precision appreciably to exceed the experimental precision 3 × 10−4 [7] may be difficult in perturbation theory even up to order (αZ)6 [10] as a charge-distribution model-independent calculation is prohibitively difficult [50], and may require higher order corrections. Current nonperturbative methods using assumed charge distributions are similarly limited (Table III). Thus even improved current methods may only work in the case cancellations of finite-size terms among different measurable transitions could be employed.
Both the finite-size correction and the Higgs term scale as n−3 , so they subtract proportionally in transitions like 1S − 2S. In microscopic methods like in Ref. [47] the proportionality would not be exact as the nuclear structure terms contain explicit integration over nucleonic states and excited muonic states, while the Higgs term only samples the muon wave functions at the origin. A number of transitions, e.g. various 2S − 2P muon transitions analogously to the electronic states [61] where η remained roughly constant, if measurable in view of the great effort needed for the precise measurement of the Lamb shift [44], would help reduce the uncertainty. While in light muonic systems only low-lying states satisfy requirements of Table I, for heavy muonic systems the nuclear structure effect uncertainty increases rapidly and only transitions with large n may prove useful provided perturbation theory and the separation of charge distribution moments are tractable at all [27], but at the expense of increasing electronic screening [62].
6 A.
Isotope shifts
In normal heavy atoms of proposal [1–3] isotope shifts can be used to extract the Higgs term by looking for the departure from the linearity of the King’s plots [2, 4] for a pair of transitions. This relies on the fact that the A − A′ isotope shift for a transition depends on hr2 iA − hr2 iA′ via a coefficient essentially independent of A, A′ . For example, the relative difference of electron reduced masses for A = 100, A′ = 101 is 5 × 10−8 and the relative difference of finite-size coefficients is 10−22 ; the latter value is 10−15 even if electron is replaced by the muon. By measuring the isotopic shift of another transition we can then eliminate the terms proportional to hr2 iA − hr2 iA′ . The validity across several isotope pairs A, A′ of the resulting linear relation between the two isotope shifts can then be studied. This requires at least two transitions measured for three different A. In light muonic systems none of the above conditions are met. The leading contribution ba , Eq. (16), to the Lamb shift coefficient B differs by 3 percent for A = 3, A′ = 4 due to the difference of the muon reduced masses mµA , and its higher order terms also depend on A via mµA . This dependence is negligible [50] for the charge radius determination. The relative isotope shift of the (µ4 He)+ Higgs contribution with respect to (µ3 He)+ is (4/3)(mµ4 /mµ3 )3 − 1 ≈ 0.37 and the relative shift in the leading finite-size term is (hrα2 i/hr32 i)(mµ4 /mµ3 )3 − 1 ≈ −0.25. It would be interesting to see if the difference m3µ4 hrα2 i − m3µ3 hr32 i exhibited a correlation such that it were less model-dependent than the charge radii thus providing some help with the the Higgs term extraction. Due to the A-dependence of B, in light muonic systems three transitions are required to eliminate the finite-size corrections from isotope shifts. The latter is nevertheless desirable, even more so in order to eliminate any influence of the charge radius problems. If only two transitions and two isotopes are available, for estimating the orders of magnitude, we might try to write the isotopic shift of transition i, in the notation where [a]AA′ = aA − aA′ (and leaving out the discussion of the C term as well as the weak interaction term [1, 2]), as [∆Ei ]AA′ = [Ai ]AA′ +Bi′ (A)[R(A)]AA′ +c[AHi ]AA′ (17) where R(A) =
mµA mµ0
3
2 hrA i
(18)
contains the leading dependence on A via mµA (Eq. (16)), Bi′ (A) depends on A via mµA only in higher orders [50], and 2 1 yµ yN c = gHµ,1 = , Hi = Rn0 (0) 2 . (19) 4π mH Then using another transition j,
[∆Ei ]AA′ − βij [∆Ej ]AA′ − [Ai ]AA′ − βij [Aj ]AA′ c= [AHi ]AA′ − βij [AHj ]AA′ (20)
where βij = Bi′ (A)/Bj′ (A) and we neglected the term [Bi′ (A)]AA′ R(A′ ) where the difference [Bi′ (A)]AA′ is of higher order in (αZ). The Lamb shift (i = 1) related measured transitions are the (µ4 He)+ transitions 2S1/2 − 2P3/2 and 2S1/2 − 2P1/2 and the six (µ3 He)+ transitions between the HFS-split 2S and 2P states planned in Ref. [6, 41]; the transitions measured in µH F =1 F =2 F =0 F =1 are 2S1/2 − 2P3/2 and 2S1/2 − 2P3/2 [7, 31, 32]. The Lamb shifts are obtained by combining these measurements with the theoretical 2P3/2 − 2P1/2 fine splittings and the HFS for nonzero-spin nuclei, or for electron-muon for high precision [9], and are therefore suitable as one transition in the pair. The second transition (i = 2) is for example the 1S − 2S transition or the appropriate sum using the centroid energies [63, 64]. In perturbation theory the denominator of Eq. (20) is of the order (αZ)6 . In nonperturbative calculations [50, 59, 60] it has state and charge distribution dependence. To detect c ≈ 10−4 (Eq. (11)), there need to be precise cancellations in the numerator of Eq. (20) and the neglected term should be smaller still. The A-dependence of Bi′ (A) includes finite-size corrections to the 2P states of the order (αZ)6 [10, 50]; the hr2 i term for the 2S1/2 − 2P1/2 transition in hydrogen is shifted by 0.00134 meV [50]; the 2P1/2 binding energy of (µ4 He)+ is reduced by 0.0148 meV for the Gaussian charge distribution of radius 1.676 fm and 0.0001 meV less for the radius 1.681 fm. A detailed summary is given in [51]. For the Lamb shift and the 1S − 2S transition, contributions to [Bi′ (A)]43 up to the order (αZ)6 cancel except for ba (αZ)2 hln(αZmµA r)i (cf. Appendix B of Ref. [50]); using the fact that the VP contribution to B for (µ4 He)+ is below 1 percent (Ref. [50], Table 14), the contribution to the isotopic shift of the neglected term relative to the retained term is about [Bi′ (A)]43 R(3) Bi′ (A)[R(A)]43 (αZ)2 [hln(αZmµ4 r)i − hln(αZmµ3 r)i]R(3) ≈ [R(A)]43 ≈ 5 × 10−4 ,
(21)
where the reduced masses appear again. As [∆E1 ]43 in Eq. (17) is a few meV, this approximation might be justifiable in view of Table I. Let us note in passing that the isotope shifts of the charge radius squared between 3 He and 4 He for several electronic transitions [63] agree to only about 0.01 fm2 among transitions involving P states and differ to the order of 0.04 fm2 from transitions involving only S states. For heavy muonic systems, the Higgs term scales faster than the finite-size term by the factor A/Z but the perturbation theory starts breaking down; let us nevertheless consider some orders of magnitude in isotope shifts. They may be estimated by neglecting electronic screening (which may be a few percent of the transition energy [5] but muonic states in contrast to electronic states still scale quite well from light muonic atoms). In muonic
7 208
Pb the possible upper range of coupling scaled from Table I satisfies gHµ,208 . 0.029. The 2S1/2 − 2P1/2 transition is about 1215 keV [65], and the Higgs term is 0.2 eV, or 0.2 ppm. The Higgs term becomes very small with higher states; for the 3D1/2 − 2P1/2 transition at 2642 keV it is about 0.04 eV, and for the 4F5/2 − 3D3/2 transition at 972 keV it is about 0.5 meV. These effects appear above 1011 Hz, well above the muon decay limit, as the muon states recover the Z 2 enhancement missing [1] in the screened electronic states, resulting in roughly 2013 Z 2 /611 coupling enhancement relative to the corresponding (screened) electronic states. In the assumed muonic analogs of the Yb+ (Z = 70, A = 168, . . ., 176) E2 and E3 transitions dealt with in Ref. [2] involving a 6S state the Higgs term would be of the order of 4.6 meV, or in the range of 1012 Hz. The relative isotopic shift in the finite-size coefficient B of hr2 i for |A − A′ | = 1 near A = 200 is of the order of 10−17 , so the relative shift of the finite size term is still dominated by the relative shift of hr2 i, like in the valence electrons, indicating that the King’s plot approach using A-independent coefficients is feasible in principle but not in the conventional perturbation theory. B.
Other transitions
Combined electronic-muonic approach would sacrifice some enhancement for a larger set of transition data. An example is the threefold reduction in uncertainty of the deuteron polarizability correction against the direct µD measurements, by using the deuteron charge radius extracted from the electronic isotope shift of the 1S − 2S transitions in hydrogen and deuterium and the proton charge radius from muonic hydrogen [7, 66, 67]. Electronic states screen the nuclear charge for the muon, the effect being more pronounced in higher angular momentum states where both the finite-size and Higgs terms are smaller. Electron screening in muonic ions has been treated long ago by Hartree-Fock-type techniques and found to be non-negligible for n ≥ 8 states in heavy elements [62] (the situation in lighter elements was less clear due to unknown rates of electron state refilling during the muon cascade [5]). For example, if it were possible to calculate the QED corrections and measure the muon Lamb shift in the neutral three-body system eµ4 He, subtraction from (µ4 He)+ would suppress the contributions of the 2S states more than those of the 2P states. The order of magnitude of the equivalent
[1] Cedric Delaunay, Roee Ozeri, Gilad Perez and Yotam Soreq, arXiv:1601.05087v1 [hep-ph] (2016). [2] Cedric Delaunay and Yotam Soreq, arXiv:1602.04838v1 [hep-ph] (2016). [3] Claudia Frugiuele, Elina Fuchs, Gilad Perez and Matthias Schlaffer, arXiv:1602.04822v1 [hep-ph] (2016).
relative Z change ∆Z/Z can be estimated as 10−3 from the effect on the muon 1S1/2 − 2P3/2 transition in oxygen [62], so the relative remainder ≈ (∆Z/Z)4 for the nuclear finite size terms Bhr2 i would be smaller than (∆Z/Z)3 of the Higgs term. So far only the electronic Lamb shift has been studied in the neutral eµ4 He atom [68] assuming that the electronic shifts could be measured by methods similar to those used for measuring the hyperfine splitting of the ground state of eµ4 He [46].
V.
CONCLUSION
Current upper bounds of the Higgs muon-nucleon coupling and the finite muon lifetime constrain the possibility of Higgs term extraction from the Lamb shift to muonic helium and heavier systems. The Z 4 scaling of the nuclear structure corrections amplifying the uncertainties in charge radii, extensive experimental work on light muonic systems, as well as more difficult control of the number of ejected electrons during the muon cascade in heavy systems, favor helium. For extraction from the muonic helium Lamb shift, experimental precision should be at least 0.1 ppm, possibly requiring taking into account small effects [69]. The charge radius puzzle which has recently spread to the deuteron [43] may provide experimental incentive. Theoretical precision would need to reach 10−4 meV, a task limited by the uncertainties in the nuclear structure contributions due to differences between effective nuclear potentials or between assumed charge distributions, depending on the method used. The state of the art 0.14 meV uncertainty of the (µ4 He)+ nuclear polarization term [47, 57, 58] used to extract charge radii results in the 1.4−2.8 meV uncertainty of the finitesize corrections [26, 47]. Isotope shifts may offer a way to eliminate the nuclear structure corrections to sufficiently high order.
Acknowledgments
I thank S. Fajfer for suggesting to look for atomic Higgs effects in precisely solvable systems, J. F. Kamenik for discussions and Nir Barnea for comments on the state of the art ab-initio nuclear polarization and charge radius calculations.
[4] [5] [6] [7] [8]
W.H. King, J. Opt. Soc. Am. 53, 638 (1963). E. Borie and G.A. Rinker, Rev. Mod. Phys. 54, 67 (1982). T. Nebel et al., Hyperfine Int. 212, 195 (2012). R. Pohl et al., arXiv:1609.03440v1 (2016). A. Antognini, arXiv:1512.01765v2 [physics.atom-ph] (2016).
8 [9] R. Krivec and V.B. Mandelzweig, Phys. Rev. A 56, 3614 (1997), Phys. Rev. A 57, 4976 (1998). [10] J.L. Friar, Ann. Phys. 122, 151 (1979). [11] G. Aad et al. (ATLAS), Phys. Lett. 68, B738 (2014). [12] V. Khachatryan et al. (CMS Collaboration), Phys. Lett. B 744, 184 (2015). [13] W. Altmannshofer, J. Brod and M. Schmaltz, JHEP 05, 125 (2015). [14] http://cds.cern.ch/record/2052552/files/ ATLAS-CONF-2015-044.pdf, last accessed 2016/06/10. [15] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Phys. Lett. B 78, 443 (1978). [16] G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 180, 747 (2009). [17] P. Junnarkar and A. Walker-Loud, Phys. Rev. D 87, 114510 (2013). [18] G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 185, 960 (2014). [19] ATLAS and CMS (2015), ATLAS-CONF-2015-044. [20] V. Khachatryan et al. (CMS Collaboration), Eur. Phys. C 75, 212 (2015). [21] G. Perez, Y. Soreq, E. Stamou and K. Tobioka, Phys. Rev. D 92, 033016 (2015). [22] Y. Zhou, arXiv:1505.06369 (2015). [23] C. Delaunay, T. Golling, G. Perez and Y. Soreq, Phys. Rev. D 89, 033014 (2014). [24] A.L. Kagan, G. Perez, F. Petriello, Y. Soreq, S. Stoynev and J. Zupan, Phys. Rev. Lett. 114, 101802 (2015). [25] G. Perez, Y. Soreq, E. Stamou and K. Tobioka, Phys. Rev. D 93, 013001 (2016). [26] A.P. Martynenko, Phys. Rev. A 76, 012505 (2007). [27] W.G. Bauer and H. Salecker, Foundations of Physics 13, 15 (1983). [28] G. Carboni et al., Proceedings of the 1st Course of the International School of Exotic Atoms, Erice, April 2430, 1977, G. Fiorentini and G. Torelli, Editors, p. 117; F. Kottmann, ibid. p. 129 (Servizio Documentazione dei Laboratori Nazionali di Frascati, Frascati, December 1977). [29] O. Egan, S. Dhawan, V.W. Hughes, D.C. Lu, F.G. Mariam, P.A. Souder, J. Vetter, G. Zu Putlitz, P.A. Thompson and A.B. Denison, Phys. Rev. A 23, 1152 (1981). [30] M. Herrmann et al., Phys. Rev. A 79, 052505 (2009). [31] C.E. Carlson, V. Nazaryan and K. Griffioen, Phys. Rev. A 83, 042509 (2011). [32] C.E. Carlson, Progress in Particle and Nuclear Physics 82, 59 (2015). [33] G. Carboni et al., Nucl. Phys. A 278, 38 (1977). [34] G. Carboni et al., Phys. Lett. B 73, 229 (1978). [35] I. Sick, J.S. McCarthy and R.R. Whitney, Phys. Lett. B 64, 33 (1976). [36] J.L. Friar, Lect. Notes Phys. 627, 59 (2003). [37] I. Sick, Phys. Lett. B 116, 212 1982.
[38] R. Pohl, R. Gilman, G.A. Miller and K. Pachucki, Ann. Rev. Nucl. Part. Sci. 63, 175 (2013); arXiv:1301.0905. [39] T.P. Gorringe and D.W. Hertzog, Progress in Particle and Nuclear Physics 84, 73 (2015). [40] J.C. Bernauer et al., Phys. Rev. Letters 105, 242001 (2010). [41] A. Antognini et al., Can. J. Phys. 89, 47 (2011). [42] I. Sick, Phys. Rev. C 77, 041302 (2008). [43] R. Pohl et al. (CREMA collaboration), Science 353, 669 (2016). [44] R. Pohl for the CREMA collaboration, Journal of the Physical Society of Japan 85, 091003 (2016). [45] A. Antognini et al., arXiv:1509.03235v2 (2015); Proc. 21st International Conference on Few-Body Problems in Physics, Chicago, USA, May 18-22, 2015, C. Elster, D.R. Phillips and C.D. Roberts (eds.). [46] C.J. Gardner et al., Phys. Rev. Lett. 48, 1168 (1982). [47] C. Ji, N. Nevo Dinur, S. Bacca and N. Barnea, Phys. Rev. Letters 111, 143402 (2013). [48] J.L. Friar, arXiv:1306.3269. [49] K. Pachucki, Phys. Rev. Lett. 106, 193007 (2011). [50] E. Borie, Ann. Phys. 327, 733 (2012); arXiv:1103.1772v7 (2014). [51] R. Pohl, A. Antognini, F. Nez, F.D. Amaro, F. Biraben et al., Nature (and Supplementary Material) 466, 213 (2010). [52] S.G. Karshenboim, V.G. Ivanov, E.Yu. Korzinin and V. A. Shelyuto, Phys. Rev. A 81, 060501 (2010). [53] S.G. Karshenboim, V.G. Ivanov and E.Yu. Korzinin, Phys. Rev. A 85, 032509 (2012). [54] U.D. Jentschura, Phys. Rev. A 84, 012505 (2011). [55] E.N. Elekina, A.A. Krutov and A.P. Martynenko, Phys. Part. Nucl. 8, 331 (2011). [56] A.A. Krutov et al., JETP Lett. 120, 73 (2015). [57] Chen Ji, O.J. Hernandez, N. Nevo Dinur, S. Bacca and N. Barnea, arXiv:1509.01430v1 (2016). [58] O.J. Hernandez, N. Nevo Dinur, Chen Ji, S. Bacca and N. Barnea, arXiv:1604.06496v1 (2016). [59] J.D. Carroll, A.W. Thomas, J. Rafelski and G.A. Miller, Phys. Rev. A 84, 012506 (2011); arXiv:1104.2971v3. [60] P. Indelicato, Phys. Rev. A 87, 022501 (2013). [61] A. Czarnecki, U.D. Jentschura and K. Pachucki, Phys. Rev. Lett. 95, 180404 (2005). [62] P. Vogel, Phys. Rev. A 7, 63 (1973). [63] K. Pachucki and V.A. Yerokhin, arXiv:1503.07727v2 (2015). [64] V.A. Yerokhin, Phy. Rev. A 83, 012507 (2011). [65] K.W. Ford and G.A. Rinker, Phys. Rev. C, 1206 (1973). [66] C.G. Parthey et al., Phys. Rev. Lett. 104, 233001 (2010). [67] U.D. Jentschura, Phys. Rev. A 83, 042505 (2011). [68] S.G. Karshenboim, V.G. Ivanov and M.Ya. Amusia, Phys. Rev. A 91, 032510 (2015). [69] P. Amaro et al., Phys. Rev. A 92, 022514 (2015).
