High-Order QED Calculations in Physics of Positronium

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1 current status and future perspectives of High-Order QED Calculations in Physics of Positronium Alexander A. Penin II. Institut für Theoretische Physik, Universität Hamburg, Germany and Institute for Nuclear Research, Moscow, Russia ETH, Zürich, May 2003

2 Contents Current status of the high-order perturbative analysis of the positronium spectrum in QED. Past and present days of QED theory of weakly bound systems. Problems and future perspectives of the high-order calculations.

3 Positronium spectroscopy Theoretical analysis is not plagued by strong-interaction uncertainties, thanks to m e M had. Expansion is α is applicable up to very high orders. Unique laboratory for testing the QED theory of weakly bound systems. The analysis is complicated by recoil and annihilation contributions. Not sensitive to new physics at short distance. Laboratory for testing new physics at large distance. Apparent discrepancy of the high order QED predictions and high precision experimental measurements of orthopositronium decay width and hyperfine splitting.

4 Orthopositronium decay 3 S 1 orthopositronium (o-ps) ground state three-photon width Γ th o = 2(π2 9)α 6 m e 9π α 3 π [ 3 2 ln2 α + [ 1 + α ( α ) [ 2 π 2 π (10) + π 3 ( (3) 229 ) ln2 lnα + D o π 2 ] lnα (26) ]}, O(α) Caswell, Lepage, Sapirstein 77; O(α 2 ) Adkins, Fell, Sapirstein 00, 02; O(α 3 ln 2 α) Karshenboim 93; O(α 3 lnα) Kniehl, Penin 00; Melnikov, Yelkhovsky 00; Hill, Lepage 00

5 Orthopositronium lifetime problem Γ th o = (11) µs 1, Γ exp o = (14) µs 1, Ann Arbor 89 (gas) Γ exp o = (16) µs 1, Ann Arbor 90 (vacuum) Γ exp o = ± (stat.) ± (syst.) µs 1, Tokyo 00 (SiO 2 powder) Ann Arbor: 8 σ (gas) and 5 σ (vacuum) deviation from theory. Tokyo: perfect agreement with theory, larger reported uncertainty. the five-photon decay channel included (Adkins, Brown 83; Lepage at al. 83)

6 Parapositronium decay 1 S 0 parapositronium (p-ps) ground state two-photon width Γ th p = α5 m e 2 α 3 π { 1 + α ( π 2 π [ 3 2 ln2 α + ) 4 5 ( ) π ln2 ( α 2 [ + 2π π) 2 lnα (33) ] lnα + D ]} p π 2, O(α) Harris, Brown 57; O(α 2 ) Czarnecki, Melnikov, Yelkhovsky 99; Adkins,McGovern, Fell, Sapirstein 03; O(α 3 ln 2 α) Karshenboim 93; O(α 3 lnα) Kniehl, Penin 00; Melnikov, Yelkhovsky 00 Γ th p = (2) µs 1, Theory vs Experiment Γ exp p = (1.7) µs 1, Ann Arbor 94 the four-photon decay channel included (Lepage at al. 83; Adkins, Pfahl 99)

7 Positronium hyperfine splitting Hyperfine splitting (HFS) between o-ps and p-ps ground state energies ν = E ( 1 3 S 1 ) E ( 1 1 S 0 ) ν th = 7m eα 4 12 ( π2 { 1 α ( 32 π ) ln ζ(3) ) 7 ln2 ] + α3 π ( [ α 2 + π) 5 14 π2 lnα [ 3 ( 62 2 ln2 α ln π2 ) lnα + D ]} hfs π 2 O(α) Karplus, Klein 52; O(α 2 ), many contributors, in particular: one-photon annihilation - Adkins, Fell, Mitrikov 97; Hoang, Labelle, Zebarjad 97; radiative recoil - Pachucki, Karshenboim 98; pure recoil - Pachucki 97 (num.calc.); Czarnecki, Melnikov, Yelkhovsky 99 (an.calc.); O(α 3 ln 2 α) Karshenboim 93; O(α 3 lnα) Hill 00; Melnikov, Yelkhovsky 00; Kniehl, Penin 00

8 Positronium HFS problem ν th = GHz ν exp = (16)GHz, Mills, Bearman 83 ν exp = (74)GHz, Ritter et.al 84 HFS is the most precisely measured quantity in positronium spectroscopy (absolute precision). Theoretical value exceeds the experimantal data by 2.8 σ and 3.9 σ.

9 Numerical convergence of QED series ( Γ th o = Γ LO o α α 2 lnα α α 3 ln 2 α 1.76α 3 lnα + D o π 3 α3 ( Γ th p = Γ LO p α 2.00α 2 lnα α α 3 ln 2 α α 3 lnα + D ) p π 3 α3 ν = ν LO ( α 0.36α 2 lnα 0.67α α 3 ln 2 α α 3 lnα + D hfs π 3 α3 ) ) HFS in muonium (Nio, Kinoshita 94, 97; Eides, Shelyuto 95, Pachucki 96) D hfs π 3 = 5.17

10 Theoretical vs experimental uncertainties Theory Experiment Γ o µs µs 1 vacuum µs 1 SiO 2 powder Γ p µs µs 1 ν 400 khz 740 khz Theoretical uncertainties correspond to D i /π 3 = ±5 To cover the discrepancy one needs D o π 3 = 3000 to fit Ann Arbor 90 data D hfs π 3 = 33 to fit Ritter et.al 84 data

11 Possible solutions Breakdown of QED Signature of new physics at long distance The uncertainty estimate in some of the current experiments is too optimistic Warning! Evolution of g 2 problem: from... fits supersimmetry like a glove (2000) to obviously this is all work in progress (2002)

12 Historical Overview s Quantum mechanics, time-independent perturbation theory, Dirac s radiation theory s Early days of Quantum Field Theory, noncovariant perturbation theory 1949 Feynman s covariant perturbation theory 1951 Bethe-Salpiter equation Schwinger:...there is a moral here for us. The artificial separation of high and low frequencies, which are handled in different ways, must be avoided 1986 Beginning of the nonrelativistic effective theory era (Caswell, Lepage) Late 1990s Dimensional regularization and method of threshold expansion

13 Nonrelativistic effective theory Characteristic modes in the threshold problem (electron velocity v α) hard soft potential ultrasoft energy and momentum scale like m e energy and momentum scale like vm e energy scales like v 2 m e, momentum scales like vm e energy and momentum scale like v 2 m e Dynamical degrees of freedom in potential nonrelativistic QED (pnrqed) (Pineda, Soto 98): potential electron-positron pair ultrasoft photons

14 Perturbative calculations in pnrqed Schrödinger equation for e + e pair propagator (H E)G(r,r,E) = δ(r r ), G(r,r,E) Re(E i ) and Im(E i ) give the energy and width of i-th state ψ i (r)ψ i(r ) E E i E i E Corrections to Coulomb solution Effective Hamiltonian H = H C + δh. Multiplole interaction of the bound state to ultrasoft photons. Dimensional regularization and threshold expansion (Beneke, Smirnov 98; applications to QED/QCD bound state problem Pineda, Soto; Czarnecki, Melnikov, Yelkhovsky; Kniehl, Penin, Smirnov, Steinhauser,)

15 Perspectives What do we need to get D o? Effective Hamiltonian at O(αv 2 ), corresponding correction to the Green function. Leading retardation effects due to dipole interaction to ultrasoft photons. Three-loop hard renormalization of the on-shell on-threshold decay amplitude: the diagram looks terrifying but structure of the infrared divergences is known and numerical evaluation is tractable the diagram is relevant for HFS and heavy quarkonium production/annihilation, to be computed first

16 Summary Experimental uncertainty exceeds theoretical (QED) one by two orders of magnitude for decay rates and by factor 2 for HFS. The calculation of O(α 3 ) non-logarithmic terms would be one of the most complicated perturbative calculations in quantum field theory though conceptually the problem is clear, all the necessary tools are at hand and a number of partial results has been obtained. New measurements of essentially better accuracy are mandatory to unambiguously establish or remove the apparent discrepancy between experimental data and QED and to inspire the theorists for O(α 3 ) feat.