Brief review of the theory of the muonic hydrogen Lamb shift and the proton radius

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1 Brief review of the theory of the muonic hydrogen Lamb shift and the roton radius Antonio Pineda Gru de Física Teòrica, Universitat Autònoma de Barcelona, E-0893 Bellaterra, Barcelona, Sain Recently the muonic hydrogen lamb shift has been measured with unrecedented accuracy, allowing for a recise determination of the roton radius. This determination is 5 sigma away from the revious CODATA value obtained from (mainly) the hydrogen lamb shift and the electron-roton scattering. Within an effective field theory formalism, I will define the roton radius and briefly review some asects of the theoretical rediction for the muonic hydrogen lamb shift, studying both the ure QED-like comutation and the hadronic effects. Introduction The recent measurement of the muonic hydrogen Lamb shift, E2P 3{2 F 2 E2S {2 F, E ex meV and the associated determination of the electromagnetic roton radius []: () r fm. has led to a lot of controversy. The reason is that this number is 5 sigma away from the CODATA value, r fm [2]. If instead one uses this value in the theoretical exression of the muonic Hydrogen Lamb shift one gets the following discreancy: (2) E ex E th 0.3 mev between theory and exeriment. Two main otions are clearly at hand: either the theoretical determination is not correct (or not as recise as claimed), or revious determinations of the roton radius were incorrect (or not as recise as claimed). Here we would like to study the theoretical exression of the muonic Hydrogen Lamb shift within an effective field theory ersective. We do it artially, and only focus on some few asects, as a full analysis would reuire much more sace. In articular sin effects will not be considered. We believe that the use of effective field theories hels in organizing the comutation by roviding with ower counting rules to asses the imortance of the different contributions. This will be even more imortant once higher order effects are included. For the resent

2 XIV International Conference on Hadron Sectroscoy (hadron20), 3-7 June 20, Munich, Germany discussion a Om r α 5 recision is enough to visualize the discreancy. Higher order effects are way smaller than the discreancy found in E. (2). Moreover Om r α 5 is the only thing comletely known at resent. The dynamics of the muonic hydrogen is characterized by several scales: m Λ χ, m µ m π m r m µm m `m µ, m r α m e. By considering ratios between them the main exansion arameters are obtained: m π m µ m m 9, m r α m rα 2 m r m r α α 37. We use the effective field theory Potential Non-Relativistic QED (NRQED) [3]. Secially relevant for us is Ref. [4], which contains much more detailed information on the alication of NRQED to the muonic hydrogen, and we refer to it for details (see also [5 7]). NRQED rofits from the hierarchy m µ " m µ α " m µ α 2 and the Lagrangian reads (3) ż # L NRQED d 3 rd 3 RdtS : r, R, t ib 0 2 2m r + ż Vr,, σ, σ 2 ` er ER, t Sr, R, t d 3 r 4 F µνf µν, where S is the field reresenting the muonic hydrogen, R the center of mass coordinate and r the relative distance. V stands for the otential and admits an exansion in owers of {m µ : (4) Vr,, σ, σ 2 V 0 r ` V r m µ ` V2 r m 2 µ `. The otential is obtained through matching to the underlying theory. Since NRQED describes degrees of freedom with E m µ α 2, any other degree of freedom with larger energy is integrated out. This imlies treating the roton and muon in a non-relativistic fashion and integrating out ions. This is the ste of going from Heavy Baryon Effective Theory (HBET) to NRQCD. By integrating out the scale m µ α, NRQED is obtained and the otentials aear. Schematically the ath followed is the following: HBETm π {m µ Ñ NRQEDm µ α Ñ NRQED. This is also the recision resently reached in heavy uarkonium sectrum comutations. This made rovide with some cross checks between both systems. 2

3 XIV International Conference on Hadron Sectroscoy (hadron20), 3-7 June 20, Munich, Germany 2 Pure QED contributions We first focus on the ure QED contributions. We mostly follow Pachucki s work [8 0], as it mainly follows an strict order by order in α comutation, trying to accommodate their results in our formalism. See however [] (or [, 2]) for more comlete list of references. The static otential can be written in the following way in momentum sace (5) Ṽ 0 4πZ µ Z α V k k 2, (6) α e f f k α ` Π k 2, where Πk 2 is the vacuum olarization due to electrons and can be comuted order by order in α: Πk 2 απ k 2 ` α 2 Π 2 k 2 ` α 3 Π 3 k 2 `... (7) α V k α e f f k ` ÿ n,m 0 n`m eveną0 Z n µz m α n,m e f f k α e f f k ` δαk, δαk Oα 4. E, E, k 0, k Figure : Leading correction to the Coulomb otential due to the electron vacuum olarization. k and k 0 E E. The leading order contribution to the lamb shift comes from the one-loo vacuum olarization correction to the static otential (see Fig. ) E LO xn δv ny mev Om r α 3. The Om r α 4 contribution to the lamb shift comes from the two-loo static otential and from the iteration of the one-loo otential in uantum mechanics erturbation theory. The latter yields E 0.5 mev. The former is urely due to vacuum olarization corrections (see Fig. 2) and yields E.5079 mev. 3

4 XIV International Conference on Hadron Sectroscoy (hadron20), 3-7 June 20, Munich, Germany µ µ µ µ µ µ µ µ e e e e e Figure 2: The static otential at two loos. The three-loo static otential contribution due to the vacuum olarization (and the associated iterations from erturbation theory were comuted in Ref. [3] (see also [4] for an small correction). The result was uite small E mev Om r α 5. All revious contributions were due to the vacuum olarization. The only contribution from the static otential that is not due to the vacuum olarization at Om r α 5 comes from δα. It is a light-by-light (Wichmann-Kroll and Delbrück) contribution and very small [5] E» mev. It should be mentioned that the limit m e Ñ 0 of the static otential is known at three loos from QCD [6, 7], which could be used as a check. It is also reasonable to think that the result with finite m e could also be obtained from these results (albeit numerically) with a finite amount of work. There are no corrections due to the {m µ otential at Om r α 5. From the {m 2 µ otential (see [4] for its exression in NRQED) there are the tree level relativistic corrections, which give E mev Omα 4. The incororation of the one-loo vacuum olarization to the relativistic {m 2 µ tree-level otential gives the following result E mev Omα 5 [0]. In order to comlete the ure QED Omα 5 corrections one has to include the interaction with the ultrasoft hotons (see Fig. 3). They yield the result (taken from [8]) The m µ m ultrasoft effects contribute E mev Omα 5. E mev Omα 5 m µ m. In NRQED these results would not come from the interaction with the ultrasoft hotons only, as it would be factorization scale deendent, they also include effects due to the NRQED matching coefficients encoded in the {m 2 µ otentials. The rocedure is retty much the same the one used for ositronium in Ref. [5]. The details for muonic hydrogen will be worked out elsewhere. 4

5 XIV International Conference on Hadron Sectroscoy (hadron20), 3-7 June 20, Munich, Germany Figure 3: self-energy correction to the muonic hydrogen energy due to the interaction with ultrasoft hotons. At Omα 5 m2 µ m 2 one starts to have overla with hadronic effects, which we discuss in the next section. 3 Hadronic Contributions In the revious section we have considered the roton to be oint-like. We now incororate the finite-size effects due to the hadronic structure of the roton. These effects are encoded in the coefficient multilying the delta otential (note that the combination of NRQED matching coefficients that aears in the otential is always the same). (8) δv 2 had r m 2 Dd had δ 3 r Ñ E m 2 D had d π m rα n 3 δ l0 where (9) D had d c had 3 6παd had 2 ` πα 2 chad D. We define c 3 c oint like 3 ` c3 had, d 2 d oint like 2 ` d2 had c D c oint like D ` c had. D are the left-over of the matching coefficients of NRQED Lagrangian c had D (0) δl d2 m 2 F µν D 2 F µν ` e c D m 2 N : EN ` ` c3 m 2 N N : µ : µ,so that chad 3, d2 had, after subtraction of the oint-like contributions. We do in this way because traditionally the oint-like contributions are already included in the "ure" QED corrections described in the revious section 2. I more extended discussion can be found in Refs. [4, 6]. d had 2 encodes the hadronic vacuum olarization effect. Its contribution to the Lamb shift is tiny, E 0.0 mev, and not much subject to uncertainty as it can be determined with enough recision from disersion relations. 2 Note though, that for an strict effective theory oint of view, at scales of the order of m, it is not a good aroximation to consider the roton oint like. Therefore, in a way, we are introducing an "surious" contribution in the hadronic matching coefficients. 5

6 XIV International Conference on Hadron Sectroscoy (hadron20), 3-7 June 20, Munich, Germany More subject to discussion are the hadronic corrections associated to c3 had. They are usually slit into two terms (see the discussion in Refs. [4, 6]): c3 had c3,zemach had ` chad 3, 206 ol. We symbolically draw them in Figs. 4 and 5, and discuss them in the next subsections. A common feature of both of them is that they are ower-like chiral enhanced: m µ m π. This is very imortant, as it allows chiral erturbation theory to redict the leading order term without introducing any extra arameter. The resulting correction to the Lamb shift is of Om µ α 5 ˆ m2 µ ˆ mµ Λ 2 χ m π. l i γ m π G (2) E γ G E (0) Figure 4: Symbolic reresentation (lus ermutations) of the Zemach xr 3 y correction, E. (). l i l i l i γ m π γ Figure 5: Symbolic reresentation of E. (7). 3. Zemach correction, xr 3 y It is the one analogous to the Zemach correction defined in the hyerfine slitting [8]. It is also common to rewrite it in terms of a coefficient xr 3 y () c had 3,Zemach π 3 α2 m 2 m µ xr 3 y, xr 3 y fm 3 96 ż π d D k k 6 G0 E G2 E, 6

7 XIV International Conference on Hadron Sectroscoy (hadron20), 3-7 June 20, Munich, Germany where G n E is the electric Sachs form factor to order n in the chiral counting (G0 E ). We also use dimensional regularization (D 4 ` 2ɛ). This gets rid of ower-like divergences which are then automatically set to zero (no need for counterterms as one would with cutoff regularization). The final result is finite and it is ossible to obtain an analytic exression for the leading term in the chiral and large N c exansion (by including the article contribution). It reads [4] (2) ˆ c3,zemach had 2πα 2 m 4πF 0 ` 2 m π π g2 πn 2 m µ 8ÿ r 0 m π " 3 4 g2 A ` 8 mπ 2r 8ÿ C r ` g 2 πn r + mπ 2r H r, where ( M M 300 MeV) (3) C r r " * Γ 3{2 2r ` 2 B 6`2r Γr ` Γ 3{2 r 3 ` 2r B 4`2r, r ě 0, (4) B n ż 8 0 «t 2 n dt? ln t 2 t ` c ff t 2 H n n!2n!!γr 3{2s 22n!!Γr{2 ` ns. This exression roduces the following number for xr 3 y and the associated energy shift: (5) xr 3 y χpt fm 3.9 Pineda Ñ E 0.00 xr3 y 0.09 mev fm3 This number can be comared with some recent determinations of xr 3 y using disersion relations [9 22] xr 3 y ex fm 3 " 2.73 Friar Sick Bernauer Arrington * Ñ E In rincile the difference between these two determinations comes from different fit functions and data, which may give a first estimate of the associated uncertainty of the disersion relation analysis. We find uite reassuring that the difference with the chiral comutation is around 40 %, which could be easily accommodated with higher order corrections. Much more difficult to accommodate would be the value advocated in Ref. [23], xr 3 y 36.5, from a direct fit to the muonic hydrogen Lamb shift using the CODATA value for the roton value. This would reuire that higher order corrections in the chiral comutation to be a factor 5 larger than the leading order result. We believe this is at odds with chiral symmetry, even more so taking into account that one of the motivations of such roosal was the lack of exerimental data at low momentum, but it is recisely in this region where chiral erturbation theory should work better. 7

8 XIV International Conference on Hadron Sectroscoy (hadron20), 3-7 June 20, Munich, Germany 3.2 Polarizability correction The determination of the olarizability correction from exeriment is on more shaky grounds than for the Zemach correction, roducing the larger uncertainty in the theoretical exression for the Lamb shift. The reason is that disersion relations do not fix the result comletely. The final number used in [] was taken from the average in Ref. [24] E using the results from, [9] E mev, [25] E mev, and [26] E 0.06 mev. For a recent discussion see Ref. [27]. Here again chiral comutations may turn out to be crucial to asses the size of this correction. The reason, as before, is that the olarizability correction is ower-like chiral enhanced. Therefore, chiral erturbation theory can redict the leading order term with no new arameter. This is the attitude followed in Ref. [7], where a chiral comutation using disersion relations yielded c had 3,ol e4 m m µ ż d 4 k E 2π 4 k 4 E k 4 E ` 4m2 µk 2 0,E 3k 2 0,E ` k 2 S ik 0,E, k 2 E k2 S 2 ik 0,E, k 2 E ( where ż T µν i d 4 x e i x x, s TJ µ xj ν 0, sy, which has the following structure (ρ {m): T µν ˆ g µν ` µ ν (6) 2 i m ɛ µνρσ ρ s σ A ρ, 2 S ρ, 2 ` ˆ m 2 µ mρ ˆ 2 µ i m 3 ν mρ 2 ν S 2 ρ, 2 ɛ µνρσ ρ`m ρs σ s σ A2 ρ, 2 After introducing the chiral exressions for the structure factors from the diagrams in Fig. 6, one obtains (7) where (for D 4) (8) ˆ 2 ż m µ ga m π f π d c3,ol had e4 m 2 D ż k E 8 dw 2π D ` k ˆ 2 ` ` k 2 2 A E w 2, k 2 ` ` k 2 2 k 2 w 2 B E w 2, k 2 ( «ż B E 8π 0 A E 4π «3 2 ` a ` w 2 ` ż 2x dx a ` x 2 w 2 ` x xw 2 k π wd 5 w 2 ` 4 m2 µ m 2 π ff x dx a, ` x 2 w 2 ` x xw 2 k 2 ż 0 `k 2 2 ff x 2x 2 dx. ` x 2 w 2 ` x xw 2 k 2 2 3

9 XIV International Conference on Hadron Sectroscoy (hadron20), 3-7 June 20, Munich, Germany µ π ν µ ν π µ () π ν (2) µ ν π (3) (4) µ ν π (Seagull) Figure 6: Diagrams contributing to T ij. Crossed diagrams are not exlicitly shown but calculated. This gives the number (9) E χpt ions 0.08 mev. We consider a more through chiral study of this object, in articular including the article, comulsory. The introduction of the article roduced a large effect in the case of the Zemach correction, something similar may haen here. Whereas we can (and should, see Ref. [28] for a recent discussion) further analyze the error associated to the olarizability correction, we would like to emhasize that in order to exlain E. (2), the corrections to the leading order chiral comutation should be a factor 5 larger than the number obtained in E. (9). 3.3 Definition of the roton radius From the effective theory oint of view, the roton radius corresonds to an secific combination of the Wilson coefficients of the effective theory. Let us see how this relation aears. One first considers the following matrix element (20) x, s J µ, sy u F 2 γ µ ` if 2 2 σµν ν 2m We are interested in the low energy limit of the form factors (2) F i 2 F i ` 2 m 2 Fi `... 9 j u.

10 XIV International Conference on Hadron Sectroscoy (hadron20), 3-7 June 20, Munich, Germany or more recisely of the Sachs form factors (22) G E 2 F 2 ` 2 4m 2 F 2 2, G M 2 F 2 ` F 2 2. The roton radius is usually defined as the derivative of the Electric Sachs form factor at zero momentum: (23) r 2 ν 6 d d 2 G E, c 4 D ν m 2 (24) c D ` 2F 2 ` 8F ` 8m2 dg,e 2 d 2 ˇ ˇ ˇ2 0, This set of euations allows to visualize the relation between the roton radius and the matching coefficients of the effective theory [4]. They also highlight the roblem of defining the roton radius through the derivative of the Electric Sachs form factor at zero momentum, as this object is infrared divergent. Then, what is the definition of the roton radius used to obtain E. ()? It corresonds to subtract the oint-like effect of c D : r m 2 `cd ν c D,oint like ν where, in the MS scheme, c D,oint like ` α 4 π 3 ln m2 ν 2. 4 Conclusions We have briefly reviewed the theoretical determination of the sin-indeendent corrections to the Lamb shift that contribute at Om r α 5. We believe that it is imortant to have a model indeendent and efficient aroach to the roblem. Effective Field Theories are suitable for this task. The use of effective theories highlights that the roton radius is a matching coefficient of the effective theory and, in general, an scheme/scale deendent object. In rincile, this is not a roblem as far as one knows the definition one is using. The ure QED comutation aears to be solid, not to say extremely reliable. Yet it could be interesting to reanalyze some arts of the comutation from an effective theory ersective. We also remark that the correction coming from the three-loo static otential has only been comuted by one grou. On the other hand, the analogous QCD static otential has been obtained by two grous. We believe that with a reasonable amount of effort, such comutations may yield cross checks of the QED comutation for muonic hydrogen. 0

11 XIV International Conference on Hadron Sectroscoy (hadron20), 3-7 June 20, Munich, Germany For the hadronic corrections there are recise determinations of the effects associated to the Zemach correction, xr 3 y, using disersion relations. Chiral erturbation theory rovides with a highly non-trivial double check of the magnitude of this correction. The reason is that the chiral comutation is ower-like chiral enhanced. It actually linearly diverges in the chiral limit. Therefore, the leading order comutation in chiral erturbation theory is a ure rediction, with no free arameter. This rules out much larger values of xr 3 y than those obtained from exeriment, as such values would be in tension with the chiral erturbation theory rediction. The olarizability correction is the major source of uncertainty. The reason is that disersion relations alone are not able to fully determine this uantity, suffering from some ambiguity in the arameterization. Therefore, the chiral erturbation theory result may turn out to be crucial here to determine the size of this correction. Again the chiral comutation is ower-like chiral enhanced and linearly diverges in the chiral limit. Thus, the leading order comutation in chiral erturbation theory is a ure rediction, with no free arameter. At resent there is room for imrovement over the result obtained in Ref. [7] using chiral erturbation theory with disersion relations. In articular, it does not include the contribution due to the article, which, in the case of the Zemach term, turned out to be uite imortant. It will then be very imortant to comute it to really asses the size of this correction. Obviously any eventual determination from lattice of this uantity would be most welcome. Acknowledgments This work was artially suorted by the sanish grants FPA and FPA , and by the catalan grant SGR References [] R. Pohl et al., Nature 466 (200) 23. [2] P. J. Mohr, B. N. Taylor and D. B. Newell, Rev. Mod. Phys. 80, 633 (2008) [arxiv: [hysics.atom-h]]. [3] A. Pineda and J. Soto, Nucl. Phys. Proc. Sul. 64, 428 (998) [arxiv:he-h/970748]. [4] A. Pineda, Phys. Rev. C 7, (2005) [arxiv:he-h/04242]. [5] A. Pineda and J. Soto, Phys. Rev. D 59, (999) [arxiv:he-h/ ]. [6] A. Pineda, Phys. Rev. C 67, (2003) [arxiv:he-h/02020].

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