Corrigendum to “Renormalization of the Cottingham formula” [Nucl Phys B 149 (1979) 90–100] JID NUPHB AID 13951 /ERR [m1+; v1 241; Prn 29/12/2016; 12 23] P 1 (1 2) Available online at www sciencedirect[.]
JID:NUPHB AID:13951 /ERR [m1+; v1.241; Prn:29/12/2016; 12:23] P.1 (1-2) Available online at www.sciencedirect.com ScienceDirect 1 2 Nuclear Physics B ••• (••••) •••–••• www.elsevier.com/locate/nuclphysb Corrigendum 10 Corrigendum to “Renormalization of the Cottingham formula” [Nucl Phys B 149 (1979) 90–100] 11 10 11 John C Collins 12 13 12 13 14 104 Davey Lab., Penn State University, University Park, PA 16802, USA 15 Received 19 December 2016; accepted 20 December 2016 16 14 15 16 17 17 Editor: Tommy Ohlsson 18 18 19 19 20 20 21 21 Abstract 22 23 24 25 26 27 22 An error in Collins (1979) [1] is corrected, concerning the value of the coefficient of the gluonic operator in the renormalization of the matrix element in the Cottingham formula The change does not affect the conclusions of the paper for the application of the Cottingham formula to the neutron–proton mass difference, but it does affect more general applications to electromagnetic corrections to strong-interaction phenomena Minor errors in intermediate steps of the derivations are also corrected © 2016 Elsevier B.V All rights reserved 23 24 25 26 27 28 28 29 29 30 30 31 31 32 In Ref [1], the following corrections should be applied, in addition to those in Ref [2]: 33 34 35 36 37 38 39 40 i=u,d,s κi , In the Appendix, Nf should be replaced by where, in the notation of the paper, κi is the electric charge of the quark of flavor i in units of the positron charge, i.e., κu = 2/3, κd = κs = −1/3 This correction applies to the second line of the Appendix, and to Eqs (A.1)–(A.3) As a consequence, the same correction applies also to the value of K used in Eq (2.14) and subsequent equations, where 41 44 45 46 47 33 34 35 36 37 38 39 40 41 42 43 32 42 DOI of original article: http://dx.doi.org/10.1016/0550-3213(79)90158-5 E-mail address: jcc8@psu.edu http://dx.doi.org/10.1016/j.nuclphysb.2016.12.017 0550-3213/© 2016 Elsevier B.V All rights reserved 43 44 45 46 47 JID:NUPHB AID:13951 /ERR [m1+; v1.241; Prn:29/12/2016; 12:23] P.2 (1-2) J.C Collins / Nuclear Physics B ••• (••••) •••–••• 2 10 11 12 13 14 15 16 17 18 19 20 21 22 27 L = 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 2e2 A(g , e2 , {κi }, /μ) κ i=u,d,s i κi2 (2-loop β)/(4e3 ) μ∂A/∂μ = −g i=u,d,s g2 = −e g 2 κi2 /(2 π ), μ∂A/∂μ = 3e2 Q2 C1,G2 (Q2 = μ2 )/(16π ), κi2 /(48π Q2 ) C1,G2 (Q2 ) = −g 10 11 12 13 14 15 16 17 Fμν , (A.1) 18 19 20 21 22 (A.2) 23 24 i=u,d,s 28 29 The equations in the Appendix thus read: 24 26 should be used In (A.1) it is perhaps worth emphasizing that L refers to a two-loop counterterm in QED, and that this equation expresses it in terms of the coefficient A of the G2 counterterm in (2.4) The left-hand side of (A.2) should be μ∂A/∂μ There should be an additional factor of Q2 on the right-hand side of the un-numbered equation after (A.2) There should be an additional factor of 1/Q2 on the right-hand side of (A.3) Throughout, the function A depends on e2 and all the κi s, as well as on the explicitly indicated arguments g and /μ This can be seen from the definition (2.4) and the explicit formula (A.2) Similarly Bi also depends on e2 as well as the indicated arguments On the left-hand side of (3.1), C() should be replaced by C() 23 25 κi2 i=u,d,s K = (48π )−1 25 26 (A.3) i=u,d,s None of these corrections affect the conclusions of the paper about the neutron–proton mass difference However, the change in the coefficient of the gluonic operator, in (2.14), (A.3), etc., affects more general applications to electromagnetic corrections to strong interaction phenomena, as in Ref [3] Acknowledgements I thank Richard Hill and Gil Paz for pointing out the important error about the coefficient of the gluonic operator, as reported in Ref [3] This work was supported in part by the U.S Department of Energy under Grant No DE-SC0013699 References [1] J.C Collins, Renormalization of the Cottingham formula, Nucl Phys B 149 (1979) 90–100, http://dx.doi.org/10 1016/0550-3213(79)90158-5, Nucl Phys B 153 (1979) 546 (Erratum) [2] J.C Collins, Erratum to “Renormalization of the Cottingham formula”, Nucl Phys B 153 (1979) 546, http:// dx.doi.org/10.1016/0550-3213(79)90616-3 [3] R.J Hill, G Paz, Nucleon spin-averaged forward virtual Compton tensor at large Q2 , arXiv:1611.09917 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 ... formula, Nucl Phys B 149 (1979) 90–100, http://dx.doi.org/10 1016/0550-3213(79)90158-5, Nucl Phys B 153 (1979) 546 (Erratum) [2] J.C Collins, Erratum to “Renormalization of the Cottingham formula”, Nucl. .. in terms of the coefficient A of the G2 counterterm in (2.4) The left-hand side of (A.2) should be μ∂A/∂μ There should be an additional factor of Q2 on the right-hand side of the un-numbered equation... (A.3) i=u,d,s None of these corrections affect the conclusions of the paper about the neutron–proton mass difference However, the change in the coefficient of the gluonic operator, in (2.14), (A.3),