The muonic hydrogen Lamb shift and the proton radius
TThe muonic hydrogen Lamb shift and the proton radius
Clara Peset ∗ Grup de F´ısica Te`orica and IFAE, Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Barcelona
Abstract
We obtain a model independent expression for the muonic hydrogen Lamb shift up to O ( m µ α , m µ α m µ m ρ ). The hadronice ff ects are controlled by the chiral theory, which allows for their model independent determination. We give theircomplete expression including the pion and Delta particles. Out of this analysis and the experimental measurement ofthe muonic hydrogen Lamb shift we determine the electromagnetic proton radius: r p = σ variance with respect to the CODATA value. The parametric control of the uncertainties allows us to obtain amodel independent determination of the error, which is dominated by hadronic e ff ects. Keywords:
Chiral Lagrangians, Bound states, Heavy quark e ff ective theory, Specific calculations
1. Introduction
The recent measurement of the muonic hydrogen ( µ p )Lamb shift, E (2 P / ) − E (2 S / ) [1, 2], ∆ E exp L = . r p = . σ away from previous determinations of this quantitycoming from hydrogen and electron-proton ( ep ) scatter-ing [3, 4].In order to asses the significance of this discrepancy itis of fundamental importance to perform the computa-tion of this quantity in a model independent way. Inthis respect, the use of e ff ective field theories (EFTs)is specially useful. They help organizing the computa-tion by providing with power counting rules that assesthe importance of the di ff erent contributions. This be-comes increasingly necessary as higher order e ff ects areincluded. Even more important, these power count-ing rules allow to parametrically control the size of thehigher order non-computed terms and, thus, give an es-timate of the error.The EFT approach is specially convenient in the caseof bound states where there are di ff erent, well separatedscales, namely, the hard scale or reduced mass ( m r ), the ∗ Speaker
Email address: [email protected] (Clara Peset) soft scale or typical momentum ( m r v ∼ m r α ) and theultrasoft scale or typical binding energy ( m r v ∼ m r α ).In the case of µ p we need to deal with several scales: m p ∼ m ρ , m µ ∼ m π ∼ m r ≡ m µ m p m p + m µ , m r α ∼ m e . from which we obtain the main expansion parametersby considering ratios of them m π m p ∼ m µ m p ≈ m e m r ∼ m r α m r ∼ m r α m r α ∼ α ≈ . (2)These, together with the counting rules given by theEFT provide the necessary tools to perform the full anal-ysis of the Lamb shift in µ p up to leading-log O ( m µ α )terms and leading O ( m µ α m µ m ρ ) hadronic e ff ects.In our approach we combine the use of Heavy BaryonE ff ective Theory (HBET) [5, 6], Non-Relativistic QED(NRQED) [7] and, specially, potential NRQED (pN-RQED) [8–10]. Partial results following this approachcan be found in [11–13] (see [14] for a review). In Ref.[15] we computed the n = O ( m µ α , m µ α m µ m ρ ). A more detailed account of thehadronic part can be found in [16]. These proceedingsare based on the work carried out in Refs. [15, 16].
2. Lamb shift and extraction of the proton radius
All contributions to the Lamb shift up to the order of ourinterest are summarized in Table 1. Together they sum
Preprint submitted to Nuc. Phys. (Proc. Suppl.) August 22, 2018 a r X i v : . [ h e p - ph ] N ov p ∆ E L = . − . r p fm + . , (3)which compared to Eq. (1), gives a value of the protonradius r p = . O ( m r α ), plus the leading logs at O ( m r α )which allow us to estimate the error of this number. Themain contribution is the electron vacuum polarization at O ( m r α ). The remaining amount corresponds to higherorder e ff ects such as higher loops, relativistic correc-tions, ultrasoft photons or perturbation theory e ff ects.A more detailed description (with comprehensive refer-ences) of this contribution can be found in [15].The lower part of Table 1 summarizes all the hadroniccontributions to the Lamb shift up to the order of ourinterest, which we explain here in more detail. All thehadronic contributions are encoded in the 1 / m poten-tial in pNRQED: D had d ≡ − c had3 − πα d had2 + πα r p m p , (4) δ V (2)had ( r ) ≡ m p D had d δ ( r ) → ∆ E = − D had d m p ( m r α ) n π δ l . (5)Entries 11 and 12 correspond to the r p -dependent termin Eq. (3) (i.e. the Wilson coe ffi cient c had D ), and the 13thentry allows us to estimate the uncertainty of this num-ber. The last term of Eq. (3) comes from the two lastentries of Table 1. The 14th entry of the table corre-sponds to the hadronic vacuum polarization (encoded inthe matching coe ffi cient d had2 ), which can be determinedfrom dispersion relations (DR) [17] with a small errorfor our purposes, and this is the number we quote here.The last entry of Table 1 corresponds to the two photonexchange (TPE) and deserves more care since it gen-erates most of the uncertainty in the Lamb shift. Thiscontribution is encoded in the Wilson coe ffi cient c had3 ,which is unique from an EFT point of view, althoughit is customary to split it into Born and polarizabilitypieces so that c had3 = c Born3 + c pol3 . We have computedboth of them separately, in the pure chiral limit and alsoincluding the contribution due to the ∆ (1232), whichcould give the largest subleading contribution not onlyfor being the closest resonance to the proton, but alsobecause both of them are degenerate in the large- N c limit [18]. When going from HBET to NRQED, weintegrate out the pions and the Delta and we can write c had3 ∼ α m µ m π F ( m π / ∆ ) + O (cid:18) α m µ m ρ (cid:19) , where no counterterms are needed to compute the leading order of the contribu-tion, as it is argued in Refs. [12, 15].1 O ( m r α ) V (0)VP . O ( m r α ) V (0)VP . O ( m r α ) V (0)VP . O ( m r α ) V (0)VP . O ( m r α ) V (0)LbL − . O ( m r α × m µ m p ) V (2) + V (3) . O ( m r α ) V (2)soft / ultrasoft − . O ( m r α ) V (2)VP . O ( m µ α × ln( m µ m e )) V (2) ; c ( µ ) D − . O ( m µ α × ln α ) V (2) VP ; c ( µ ) D − . O ( m r α × m r r p ) V (2) ; c ( p ) D ; r p − . r p fm O ( m r α × m r r p ) V (2)VP ; c ( p ) D ; r p − . r p fm O ( m r α ln α × m r r p ) V (2) ; c ( p ) D ; r p − . r p fm O ( m r α × m r m ρ ) V (2)VP had ; d had2 . O ( m r α × m r m ρ m µ m π ) V (2) ; c had3 ; (cid:104) r (cid:105) . Table 1:
The di ff erent contributions to the µ p Lamb shift in meV units. The Born contribution at leading order in the NR expan-sion (which guarantees that only the low energy modescontribute to the integral) reads c pl i , Born = πα ) M p m l i (cid:90) d D − q (2 π ) D − q G (0) E G (2) E ( − q ) , (6)where G (0) E = G (2) E ( q ) together with an analyticexpression for c pl i , Born can be found in [16]. This co-e ffi cient can also be related with (one of) the Zemachmoments: c pl i , Born = π α M p m l i (cid:104) r (cid:105) (2) , (7) (cid:104) r (cid:105) (2) = π (cid:90) ∞ dQQ (cid:32) G E ( − Q ) − + Q (cid:104) r (cid:105) (cid:33) . (8)The Zemach moments can be determined in a similarway as the moments of the charge distribution of theproton. We have studied some and compared them totheir values obtained applying DR techniques. A set ofthese results is summarized in Table 2 (a more completediscussion on this can be found in [16]).One would expect the chiral prediction to give the domi-nant contribution of (cid:104) r n (cid:105) for n ≥ n =
2. Nevertheless we observe large di ff erences2 r (cid:105) (cid:104) r (cid:105) (cid:104) r (cid:105) (cid:104) r (cid:105) (2) π . .
619 20 .
92 0 . π & ∆ . .
522 20 .
22 0 . . .
775 7 .
006 2 . . .
209 19 .
69 2 . . . . . − − − . Table 2:
Values of (cid:104) r n (cid:105) in fermi units. The first two rows give theprediction from the EFT at LO and LO + NLO. The third row in thestandard dipole fit. The last two rows are di ff erent DR analyses. (bigger than the errors) with di ff erent determinations fit-ting experimental data to di ff erent functions [19–21]. Inthis respect, the chiral result could help shaping the ap-propriate fit function and thus, resolving the di ff erencesbetween the fitted results as well as assessing their un-certainties. This di ff erence in the fit functions has animpact on the determination of the proton radius, as canbe clearly seen in Ref. [26] v.s. Refs. [4, 27] for directfits to the ep scattering data, where the determinationdi ff ers in about 3- σ . In any case, the reason for suchlarge discrepancies should be further investigated. Notethat for all n ≥
3, the chiral expressions give the lead-ing (non-analytic) dependence in the light quark mass aswell as in 1 / N c . This is a valuable information for even-tual lattice simulations of these quantities where one cantune these parameters.We can extract the contribution of the Born term to theenergy shift from Eq. (5), and this is what we quotein the last two entries of Table 3. The first two en-tries correspond to two di ff erent DR-analyses. Notethat in the HBET computation the addition of the Deltahas a good convergence. On the other hand, our resultis much smaller than the standard ones obtained fromDR. Whether this discrepancy is due to relativistic cor-rections or to a need for refining the fitting procedureshould be further investigated.The polarizability contribution is computed through thediagrams represented in Fig. 1 for the pure chiral caseand in Fig. 2 both for the tree level Delta exchange (topdiagram) and for the one-loop Delta contribution. Thesediagrams are summed up in the polarizability tensor: T µν pol = (cid:32) − g µν + q µ q ν q (cid:33) S ( ρ, q ) + M p (cid:32) p µ − M p ρ q q µ (cid:33) (cid:32) p ν − M p ρ q q ν (cid:33) S ( ρ, q ) . (9)The polarizability energy shift cannot be fully obtainedfrom DR and thus, needs some subtractions. This fact µpµpµp µp µppµ (3) (4) (5)(6) (7) (8) µp µp (1) (2) Figure 1:
Diagrams corresponding to the pure chiral contribution(only pions) of the TPE. µ eV [22] [23] [12]( π ) [16]( π & ∆ ) ∆ E Born . .
0) 24 . .
6) 10 . .
1) 8 . . Table 3:
Predictions for the Born contribution to the n = Lambshift. The first two entries correspond to DR analyses. The last twoentries are the predictions of HBET: at LO and at LO + NLO. makes our model independent computation even morerelevant. The Lamb shift obtained in HBET is: ∆ E pol = c pl µ , pol M p π (cid:18) m r α (cid:19) = . π ) − . ∆ ) + . π ∆ ) = . . µ eV (10)( µ eV) [22] [23] [24] ∆ E pol . .
4) 15 . . µ eV) B χ PT[25]( π ) HBET[13]( π ) [15]( π & ∆ ) ∆ E pol . + . − . ) 18 . .
3) 26 . . Table 4:
Predictions for the polarizability contribution to the n = Lamb shift. The first 3 entries use DR for the inelastic term anddi ff erent modeling functions for the subtraction term. In Table 4, we compare our HBET results to others ob-tained by a combination of DR for the inelastic term anddi ff erent modelling functions for the subtraction term,and also to the result obtained using B χ PT. This lastone is carried out within a theory that treats the baryonrelativistically. The result incorporates some subleadinge ff ects, which are sometimes used to give an estimate ofhigher order e ff ects in HB χ PT, but it also assumes thata theory with only baryons and pions is appropriate atthe proton mass scale, which should be taken with duecaution. Still, it would be desirable to have a deeper3heoretical understanding of this di ff erence, which maysignal that relativistic corrections are important for thepolarizability correction. In any case, the B χ PT com-putation di ff ers from our chiral result by around 50%which we consider reasonable. µp µpµpµp µp µppµ (3) (4) (5)(6) (7) (8) µp µp (1) (2) Figure 2:
Diagrams corresponding to the Delta tree level and loopcontribution (pions & Deltas) of the TPE.
For the total TPE energy shift we obtain: ∆ E TPE = ∆ E Born + ∆ E pol = . π ) + . π & ∆ ) = . . µ eV , (11)which, however is in good agreement with the total re-sult [28] used for the determination of the proton radiusin [2]. This result is a pure prediction of the EFT, and itis also the most precise result that can be obtained in amodel independent way since O ( m µ α m µ Λ QCD ) e ff ects arenot controlled by the chiral theory and would requirenew counterterms.
3. Conclusions
We have computed in a completely model independentway the Lamb shift for n = . σ away from theCODATA value and has much larger uncertainties. Wehave computed the pure chiral contribution to the TPE,and also the contribution due to the ∆ (1232). This com-putation of the TPE gives a similar result to the one ob-tained by the combination of DR plus the use of di ff er-ent models. However, the partial computations (Bornand polarizability) di ff er from the partial results ob-tained in these frameworks, fact that should be furtherunderstood. Acknowledgements
The author thanks Antonio Pineda for his collaborationin the development of this work. This work was sup-ported by the Spanish grant FPA2011-25948 and theCatalan grant SGR2009-00894.
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