Some radiative corrections to the hyperfine splitting of true muonium: two-photon-exchange contributions
SSome radiative corrections to the hyperfine splitting of true muonium:Two-photon-exchange contributions
Yao Ji ∗ Institut f¨ur Theoretische Physik, Universit¨at Regensburg, Regensburg 93040, Germany
Henry Lamm † Department of Physics, University of Maryland, College Park, MD 20742 (Dated: August 12, 2018)We consider a set of radiative contributions from one-loop lepton vacuum polarization tothe hyperfine splitting of true muonium. Improving previous results, we obtain values for theelectron-loop coefficients and extract their leading dependence upon β = m µ /m e . The coeffi-cients are C µ , = 0 . β , C µ , = 0 . β , and C µ , = 0 . β .The mass-independent contribution from three one-loop vacuum polarization is C , VPX = (cid:0) − − π + 51840 ζ (3) (cid:1) . Contributions from τ up to O ( m µ α ) are calculated. True muonium is the yet unidentified ( µ ¯ µ ) bound statewith lifetimes on the order of ps [1]. QED dominates thecharacteristics of true muonium, while QCD and Elec-troweak effects appear at O ( m µ α ) [2] and O ( m µ α ) [3]respectively. The need to discover and study true muo-nium is motivated by the existing discrepancies in muonphysics [4–8]. Both new physics models [9–30] and sys-tematic errors in the experiments have been proposedto resolve these discrepancies. Other works have sug-gested a more subtle understanding of known physics isrequired [31–36]. True muonium can produce compet-itive constraints on all these solutions if the standardmodel predictions are known to the 100 MHz level, cor-responding to O ( m µ α ) [37]. Today, the Heavy Pho-ton Search (HPS) [38] experiment is searching for truemuonium [39], and DImeson Relativistic Atom Complex(DIRAC) [40] has discussed a search during an upgradedrun [41]. In both situations, the true muonium will berelativistic, necessitating consideration to the effect ofboost on wave functions and production rates [42–44].In this work, we will focus on improving the theoreticalprediction for the hyperfine splitting (hfs). To review, theexpression for the hfs corrections to true muonium fromQED can be written∆ E hfs = m µ α (cid:20) C + C απ + C α ln (cid:18) α (cid:19) + C (cid:16) απ (cid:17) + C α π ln (cid:18) α (cid:19) + C α π ln (cid:18) α (cid:19) + C (cid:16) απ (cid:17) + · · · (cid:21) , (1)where C ij indicate the coefficient of the term propor-tional to ( α ) i ln j (1 /α ) and for i = 0 , C ij include any dependence on mass scalesother than m µ (e.g m e , m π , m τ , ...). The coefficients of ∗ [email protected] † [email protected] single-flavor QED bound states, used in positronium, areknown up to O ( m e α ). Partial results have been com-puted for O ( m e α ) and are an active research area inlight of upcoming experiments (For an updated reviewof the coefficients see [45, 46]). The exchange m e → m µ translates these results to true muonium.True muonium receives further contributions that aretypically neglected in positronium. Most importantly,the existence of the lighter electron allows for largevacuum loop contributions. The relative smallness of m τ /m µ ≈
17 and m π /m µ ≈ . C µij , only a few terms areknown.The hadronic contribution to the annihilation chan-nel, C µ , hvp = − . R ( s ) [2]. The leptonic-loops in the two-photon annihilation channel coefficient C µ , γ = − . C µ , γ = − . Z -bosons has also beencomputed [3].Ref. [48] presented calculations for the correction tothe n = 1 , S -states from electron vacuum polarizationin the Coulomb line with an additional transverse pho-ton (VPCT), vacuum polarization in the Coulomb linewith an annihilation photon (VPCA), and the vacuumpolarization in a transverse photon (VPT). With C , hvp now known to a higher precision, the uncertainty from fi-nite precision in these contributions, 200 MHZ, is as largeas the unknown higher-order contributions and thereforemust be removed. These results would naively scale as O ( m µ α ), but the electron loops modify this scaling to O ( m µ m µ m e α ) ≈ O ( m µ α ). This large enhancement overnaive α scaling led [48] to assign this contribution to O ( m µ α ), finding it to be for the ground state˜ C µ ,e = απ m µ m e (cid:104) C µ , + C µ , + C µ , (cid:105) =0 .
353 + 0 .
265 + 0 .
201 = 0 . . (2) a r X i v : . [ phy s i c s . a t o m - ph ] A p r Together, these contributions predict ∆ E s hfs =42329429(16) had (200) ,e (700) miss MHz where thefirst uncertainty estimate is from hadronic experimentaluncertainties, the second from the finite precision of˜ C ,e computed in [48], and the final is from uncalculated O ( m µ α ) contributions.In this work, we will instead assign these contributionsto their correct α scaling and compute them to higherprecision. Along the way we will also correct errors in theliterature and in the case of the VPT derive an analyticexpression. This completely reduces the uncertainty fromfinite precision. We compute these contributions for thedimensionless variable β = m µ m e .Additionally, we use the scattering approximation tocompute the full radiative contribution at O ( m µ α n )from µ and τ loops in the two-photon exchangechannel where n = 6 , ,
8. We reproduce the µ -loop values of C , VPX = [49] and C , VPX = (cid:16) π − (cid:17) [50, 51] (note that E P sF = mα / C , VPX = (cid:0) − − π + 51840 ζ (3) (cid:1) term.For the τ − loops, numerical results are obtained. I. VPC-T AND VPC-A
The vacuum polarization insertion into the Coulombline with an additional transverse photon (VPCT) andwith an annihilation photon (VPCA) are related by theirrelative contribution to the hyperfine splitting (4/7 and3/7, respectively) and therefore we only need computeone. Following [48], we compute VPCT. The contribu-tion is given by∆ E n VPCT = 47 απ E F n (cid:20) ∆ ψ nS (0) ψ nS (0) (cid:21) = m µ α π C µ ,n VPCT , (3)where n is the energy level and we emphasize that thecoefficients have an n dependence, and E F is the Fermienergy. ∆ ψ nS (0) /ψ nS (0) is the correction to the wavefunction, given by∆ ψ nS (0) ψ nS (0) = 2 (cid:90) Ω d r ¯ G nS ( E nS ; 0 , r ) V U ( r ) ψ nS ( r ) . (4)where Ω indicates an integration over all space. TheUehling potential is given by V U ( r ) = − α πr (cid:90) d v v (1 − v / − v e − λr , (5)where λ = 2 m e / √ − v . The reduced Coulomb Green’sfunction, ¯ G nS ( E nS ; 0 , r ), can be expressed in closed formfor S states [52]. For the cases of n = 1 ,
2, these formulae are¯ G S ( E S ; 0 , r ) = αm r π e − z / z × (cid:2) z (ln z + γ ) + z − z − (cid:3) , (6)and¯ G S ( E S ; 0 , r ) = − αm r π e − z / z × (cid:20) z ( z − z + γ ) + z − z + 6 z + 4 (cid:21) , (7)where z n = 2 αm r r/n and γ is Euler’s constant. The in-tegrals over r can be done analytically and the remaining v integral for the 1 S state is given by C µ , = πβ (cid:90) d v v (3 − v )9 θ (2 + xθ ) × (cid:20) x + 3 x θ − xθ (2 + xθ ) log (cid:18) xθ xθ (cid:19)(cid:21) , (8)where θ = √ − v , and x = αβ = αm µ m e . An integral ex-pression in terms of only x and v can be found for 2 S aswell, but is omitted for length. Numerically integratingthese expressions, we find C µ , = 0 . β ,and C µ , = 0 . β . Multiplying by 3 / C µ , = 0 . β , and C µ , =0 . β . Multiplying these results by α/π , we findagreement with the values of [48] with improved preci-sion. We point out there is an error in their Eq. (18)as printed because when we recomputed this expressionnumerically, we find it equivalent to C µ , = 1 . β ,much larger than the correct value of 0 . β . Thecontributions to the hfs are∆ E = m µ α π (cid:104) C µ , + C µ , (cid:105) =103948 . , (9)∆ E = m µ α π (cid:104) C µ , + C µ , (cid:105) =89933 . . (10) II. VPT
The contribution of one-loop electron vacuum polar-ization in a transverse photon is given by the interactionbetween the magnetic field induced by the Uehling po-tential and the muons. The general expression for theseterms to a given nS state is∆ E n VPT =8 π m µ (cid:90) ∞ dr r (cid:18) − ∂∂r (cid:12)(cid:12) ψ nS ( r ) (cid:12)(cid:12) (cid:19) (cid:18) ∂∂r V U ( r ) (cid:19) . (11)In deriving their expressions, it is clear that [48] omittedthe nontrivial boundary term to write the expressions − ( (cid:126) ∇| ψ | ) · ( (cid:126) ∇ V U ) as | ψ | (cid:126) ∇ V U in their intermediate steps,while their final results are correct. The integrals can be reexpressed into our standard notation as∆ E n VPT = α π C µ ,n VPT , (12)where for n = 1 and x <
2, we have derived an analyticexpression: C µ , = πβ x (cid:26) x (24 + x ) + 3 16 − x + x √ − x (cid:20) π − (cid:18) x √ − x (cid:19)(cid:21) − π (cid:27) , (13)similarly for n = 2 and x <
4, we have: C µ , = 7 πβ x (16 − x ) / (cid:26) x (cid:112) − x (49152 − x − x + 11 x )+ 6 (cid:20) π − (cid:18) x √ − x (cid:19)(cid:21) (65536 − x + 1056 x − x + x ) − π (16 − x ) / (cid:27) . (14)For true muonium, these coefficients are C µ , =0 . β and C µ , = 0 . β , whichagree with Eq. (25) of Ref. [48] but are exact up to un-certainties in the physical constants. These correspondto ∆ E = 33876 . , (15)∆ E = 21 . . (16) III. USING THE SCATTERINGAPPROXIMATION
We wish to consider the heavy lepton contributionsfrom Fig. 1 to the 1 S state within the scattering approx-imation. This approximation multiplies | ψ (0) | by theloop integral with the on-mass-shell fermions. These re-sults are valid when the loop momenta are much largerthan the momenta of the bound state, O ( m µ α ). Inser-tions of vacuum polarization loops to the skeleton dia-gram effectively makes exchanged momenta O ( m j ) where j is the loop particles. The infrared-divergent skeleton di-agram of the two-photon exchange obtained in the scat-tering approximation which requires m (cid:48) (cid:96) (cid:29) αm (cid:96) . It isfound to have a simple form [50, 51, 53]:∆ E VPX = mα π (cid:20) (cid:90) ∞ d q f p ( q ) (cid:21) , (17)where f p ( q ) = 16 + 2 q + q − q (cid:112) q + 44 q (cid:112) q + 4 . (18)While this integral itself is infrared-divergent, all radia-tive insertions into the expression will render it conver-gent. µ − µ + µ − µ − µ + µ + (cid:96) + (cid:96) + µ − µ + µ − µ − µ + µ + (cid:96) + (cid:96) + (cid:96) + (cid:96) + µ − µ + µ − µ − µ + µ + (cid:96) + (cid:96) + (cid:96) + (cid:96) + µ − µ + µ − µ − µ + µ + (cid:96) + (cid:96) + (cid:96) + (cid:96) + (cid:96) + (cid:96) + µ − µ + µ − µ − µ + µ + (cid:96) + (cid:96) + (cid:96) + (cid:96) + (cid:96) + (cid:96) + FIG. 1. Radiative loop corrections from heavy leptonic loopsto the two photon exchange graph considered in this work.
The one-loop polarization insertion given in momen-tum space is απ I ,β ( q ) = απ (cid:90) d v v (cid:16) − v (cid:17) (1 − v ) q + β . (19)The contribution of n one-loop vacuum polarization in-sertions is obtained from Eq.(17) with (cid:0) απ q I ,β ( q ) (cid:1) n andthe appropriate combinatoric factor. The integral over v can be performed analytically, and the resulting q inte-gral is∆ E = mα π (cid:16) απ (cid:17) n − n (cid:90) ∞ d q f p ( q ) (12 − q β ) + 6( q β − (cid:113) q β +4 q β arctanh (cid:16)(cid:113) q βq β +4 (cid:17) q β n − = mα π (cid:16) απ (cid:17) n − C n , VPX , (20)where the factor of n reflects the combinatorics.In the case of β = 1, we can analytically solvethis equation for n = 2 , ,
4. Our results agreewith previous calculations for C , VPX = 5 / C , VPX = (cid:16) π − (cid:17) [50, 51]. These terms havealready been included in the hfs of true muonium.While experimental precision even for positronium hasnot reach reached this level, we found C , VPX = (cid:0) − − π + 51840 ζ (3) (cid:1) . The contributionfrom this term is 0 . . β <
1, we have been unable to find ananalytic expression, and have instead numerically inte-grated Eq. 20 for n = 2 , ,
4. The results for physi-cally relevant cases are found in Tab. I. The contribu-tion from τ -loop terms in true muonium is 2 . . . . β scaling for n = 2 , ,
4. The coefficients are increas-ingly well-approximated by C n , VPX ∝ β with increasing n . The C n , VPX ∝ β behavior at large n agrees withnaive scaling expectations inside loops, and can be usedto guide estimates of higher order corrections. IV. CONCLUSION
In conclusion, we have computed several critical cor-rections to the hyperfine splitting of true muonium nec-essary to reach 100 MHz precision. The recalculationof the VPCT, VPCA, and VPT diagrams has reducedthe error budget by 200 MHz by removing the errorfrom finite precision and in the case of VPT has yieldedanalytic results. We have further computed C n , VPX for n = 2 , , β that are physically rel-evant. The current theoretical prediction is ∆ E s hfs =42329435(16) had (700) miss MHz. In light of this work, thelargest uncertainty arises from uncalculated O ( m µ α )electron loops. ACKNOWLEDGMENTS
HL would like to express thanks to Michael Eides forpointing out errors in an earlier draft of this work. HLis supported by the National Science Foundation un-der Grant Nos. PHY-1068286 and PHY-1403891 andby the U.S. Department of Energy under Contract No.DE-FG02-93ER-40762. YJ acknowledges the DeutscheForschungsgemeinschaft for support under grant BR2021/7-1. [1] S. J. Brodsky and R. F. Lebed, Phys. Rev. Lett. ,213401 (2009), arXiv:0904.2225 [hep-ph].[2] H. Lamm, (2016), arXiv:1611.04258 [physics.atom-ph].[3] H. Lamm, Phys. Rev. D , 073008 (2015).[4] G. Bennett et al. (Muon G-2 Collaboration), Phys. Rev. D73 , 072003 (2006), arXiv:hep-ex/0602035 [hep-ex].[5] A. Antognini et al. , Science , 417 (2013).[6] R. Aaij et al. (LHCb collaboration), Phys. Rev. Lett. , 151601 (2014), arXiv:1406.6482 [hep-ex].[7] R. Aaij et al. (LHCb), Phys. Rev. Lett. , 111803 (2015), [Addendum: Phys. Rev.Lett.115,no.15,159901(2015)], arXiv:1506.08614 [hep-ex].[8] R. Pohl et al. (CREMA), Science , 669 (2016).[9] D. Tucker-Smith and I. Yavin, Phys. Rev.
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TABLE I. Coefficients from two-photon exchange from one-loop leptonic vacuum polarization for physical values of β ij where i is the valence lepton, and j is the lepton in the loop. β C β ij , VPX C β ij , VPX C β ij , VPX β eτ = 2 . × − . × − . × − . × − β eµ = 4 . × − . × − . × − . × − β µτ = 5 . × − . × − . × − . × − β (cid:96)(cid:96) = 1
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