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Proton Polarisability Contribution to the Lamb Shift in Muonic Hydrogen

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Proton Polarisability Contribution to the Lamb Shift in Muonic Hydrogen Proton polarisability contribution to the Lamb shift in muonic hydrogen Mike Birse University of Manchester Work done in collaboration with Judith McGovern Eur. Phys. J. A 48 (2012) 120 Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 The Lamb shift in muonic hydrogen Much larger than in electronic hydrogen: DEL = E(2p1) − E(2s1) ' +0:2 eV 2 2 Dominated by vacuum polarisation Much more sensitive to proton structure, in particular, its charge radius th 2 DEL = 206:0668(25) − 5:2275(10)hrEi meV Results of many years of effort by Borie, Pachucki, Indelicato, Jentschura and others; collated in Antognini et al, Ann Phys 331 (2013) 127 Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 The Lamb shift in muonic hydrogen Much larger than in electronic hydrogen: DEL = E(2p1) − E(2s1) ' +0:2 eV 2 2 Dominated by vacuum polarisation Much more sensitive to proton structure, in particular, its charge radius th 2 DEL = 206:0668(25) − 5:2275(10)hrEi meV Results of many years of effort by Borie, Pachucki, Indelicato, Jentschura and others; collated in Antognini et al, Ann Phys 331 (2013) 127 Includes contribution from two-photon exchange DE2g = 33:2 ± 2:0 µeV Sensitive to polarisabilities of proton by virtual photons Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Two-photon exchange Integral over T µn(n;q2) – doubly-virtual Compton amplitude for proton Spin-averaged, forward scattering ! two independent tensor structures Common choice: qµqn 1 p · q p · q T µn = −gµn + T (n;Q2)+ pµ − qµ pn − qn T (n;Q2) q2 1 M2 q2 q2 2 multiplied by scalar functions of n = p · q=M and Q2 = −q2 Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Two-photon exchange Integral over T µn(n;q2) – doubly-virtual Compton amplitude for proton Spin-averaged, forward scattering ! two independent tensor structures Common choice: qµqn 1 p · q p · q T µn = −gµn + T (n;Q2)+ pµ − qµ pn − qn T (n;Q2) q2 1 M2 q2 q2 2 multiplied by scalar functions of n = p · q=M and Q2 = −q2 µn µn µn Amplitude contains elastic (Born) and inelastic pieces: T = TB + T • elastic: photons couple independently to proton (no excitation) • need to remove terms already accounted for in Lamb shift (iterated Coulomb, 2 leading dependence on hrEi) • inelastic: proton excited ! polarisation effects Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Doubly-virtual Compton scattering Elastic amplitude from Dirac nucleon with Dirac and Pauli form factors K. Pachucki, Phys. Rev. A 60 (1999) 3593 2 " 4 2 2 # 2 Q FD(Q ) + FP(Q ) B 2 e 2 2 T (n;Q ) = − FD(Q ) 1 M Q4 − 4M2n2 2 2 " 2 # B 2 4e MQ 2 2 Q 2 2 T (n;Q ) = FD(Q ) + FP(Q ) 2 Q4 − 4M2n2 4M2 Other choices have been used: nonpole terms only But depend on choice of tensor basis (energy-dependent tensors) cf Walker-Loud et al, Phys Rev Lett 108 (2012) 232301 Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Low-energy theorems V2CS not directly measurable, but constrained by LETs Two independent tensors of order q2: correspond to polarisabilities a (electric) and b (magnetic) determined from real Compton scattering 2 2 2 4 T 1(n;Q ) = 4pQ b + 4pn (a + b) + O(q ) 2 2 4 T 2(n;Q ) = 4pQ (a + b) + O(q ) Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Low-energy theorems V2CS not directly measurable, but constrained by LETs Two independent tensors of order q2: correspond to polarisabilities a (electric) and b (magnetic) determined from real Compton scattering 2 2 2 4 T 1(n;Q ) = 4pQ b + 4pn (a + b) + O(q ) 2 2 4 T 2(n;Q ) = 4pQ (a + b) + O(q ) Other choices for elastic amplitude ! LETs containing charge radius cf Hill and Paz, Phys Rev Lett 107 (2011) 160402 Just important to use consistent definitions of elastic and inelastic amplitudes Here: all results expressed using Pachucki’s choice Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Dispersion relations 2 2 Information on forward V CS away from q = 0 from structure functions F1;2(n;Q ) F1;2 well determined from electroproduction experiments eg at JLab Dispersion relation for T 2 converges since F2 ∼ 1=n at high energies Z ¥ F (n0;Q2) T (n;Q2) = − dn02 2 2 2 02 2 nth n − n Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Dispersion relations 2 2 Information on forward V CS away from q = 0 from structure functions F1;2(n;Q ) F1;2 well determined from electroproduction experiments eg at JLab Dispersion relation for T 2 converges since F2 ∼ 1=n at high energies Z ¥ F (n0;Q2) T (n;Q2) = − dn02 2 2 2 02 2 nth n − n But F1 ∼ n so need to use subtracted dispersion relation Z ¥ dn02 F (n0;Q2) T (n;Q2) = T (0;Q2) − n2 1 1 1 2 02 02 2 nth n n − n Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Dispersion relations 2 2 Information on forward V CS away from q = 0 from structure functions F1;2(n;Q ) F1;2 well determined from electroproduction experiments eg at JLab Dispersion relation for T 2 converges since F2 ∼ 1=n at high energies Z ¥ F (n0;Q2) T (n;Q2) = − dn02 2 2 2 02 2 nth n − n But F1 ∼ n so need to use subtracted dispersion relation Z ¥ dn02 F (n0;Q2) T (n;Q2) = T (0;Q2) − n2 1 1 1 2 02 02 2 nth n n − n 2 Problem: subtraction function T 1(0;Q ) not experimentally accessible 2 2 2 Satisfies LET: T 1(0;Q )=Q ! 4pb as Q ! 0 Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Subtraction term Define form factor 2 2 2 T 1(0;Q ) = 4pbQ Fb(Q ) 2 2 −4 Large Q : operator-product expansion, quark counting rules give Fb(Q ) ∝ Q Small Q2: use heavy-baryon chiral perturbation theory at 4th order plus leading effect of gND form factor • same as for real Compton scattering McGovern et al, Eur Phys J A 49 (2013) 12 • minor modifications for different kinematics • subtract elastic (Born) contribution calculated to this order • relevant LETs satisfied; consistent with value for b Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Form factor 1.0 7 2 . F b Q 1- F Q 2 6 b 0.8 H L 5 Q 2H L 0.6 2 - 4 0.4 GeV 3 2 0.2 1 0.0 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Q 2 GeV 2 Q 2 GeV 2 Extrapolate to higher QH 2 byL matching ChPT form onto dipole H L 1 F (Q2) ∼ b ( + Q2= M2)2 1 2 b 2 2 2 Match at Q = 0 ! Mb = 462 MeV; at Q ∼ mp ! Mb = 510 MeV Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Form factor 1.0 7 2 . F b Q 1- F Q 2 6 b 0.8 H L 5 Q 2H L 0.6 2 - 4 0.4 GeV 3 2 0.2 1 0.0 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Q 2 GeV 2 Q 2 GeV 2 Extrapolate to higher QH 2 byL matching ChPT form onto dipole H L 1 F (Q2) ∼ b ( + Q2= M2)2 1 2 b 2 2 2 Match at Q = 0 ! Mb = 462 MeV; at Q ∼ mp ! Mb = 510 MeV Mb = 485 ± 100 ± 40 ± 25 MeV • generous allowance for higher-order effects and uncertainties in input (shaded) −4 3 • b = (3:1 ± 0:5) × 10 fm Griesshammer et al, Prog Part Nucl Phys 67 (2012) 841 • matching uncertainty Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Energy shift . 2 0s 13 2 Z ¥ 2 2 ! 2 2g aEM f(0) 2 T 1(0;Q ) Q 4m DE = dQ × 41 + 1 − @ + 1 − 1A5 sub 4pm 0 Q2 2m2 Q2 • with dipole form, 90% comes from Q2 < 0:3 GeV2 • rather insensitive to value of Mb • main source of error: b = 3:1 ± 0:5 Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Energy shift . 2 0s 13 2 Z ¥ 2 2 ! 2 2g aEM f(0) 2 T 1(0;Q ) Q 4m DE = dQ × 41 + 1 − @ + 1 − 1A5 sub 4pm 0 Q2 2m2 Q2 • with dipole form, 90% comes from Q2 < 0:3 GeV2 • rather insensitive to value of Mb • main source of error: b = 3:1 ± 0:5 Result: 2g DEsub = −4:2 ± 1:0 µeV Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Energy shift . 2 0s 13 2 Z ¥ 2 2 ! 2 2g aEM f(0) 2 T 1(0;Q ) Q 4m DE = dQ × 41 + 1 − @ + 1 − 1A5 sub 4pm 0 Q2 2m2 Q2 • with dipole form, 90% comes from Q2 < 0:3 GeV2 • rather insensitive to value of Mb • main source of error: b = 3:1 ± 0:5 Result: 2g DEsub = −4:2 ± 1:0 µeV 2g Combined with inelastic (dispersive) contribution: DEinel = 12:7 ± 0:5 µeV Carlson and Vanderhaeghen, Phys Rev A 84 (2011) 020102 2g ! total polarisability contribution: DEpol = 8:5 ± 1:1 µeV Mike Birse Proton polarisability contribution to the Lamb shift Mainz, June 2014 Energy shift .
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