Corrigendum to “renormalization of the cottingham formula” nucl phys b 149 (1979) 90–100

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

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Nuclear Physics B ••• (••••) •••–•••

www.elsevier.com/locate/nuclphysb

Corrigendum

John C Collins

104 Davey Lab., Penn State University, University Park, PA 16802, USA

Received 19 December 2016; accepted 20 December 2016

Editor: Tommy Ohlsson

Abstract

AnerrorinCollins(1979)[1]iscorrected,concerningthevalueofthecoefficientofthegluonicoperator

intherenormalizationofthematrixelementintheCottinghamformula.Thechangedoesnotaffectthe con-clusionsofthepaperfortheapplicationoftheCottinghamformulatotheneutron–protonmassdifference, butitdoesaffectmoregeneralapplicationstoelectromagneticcorrectionstostrong-interactionphenomena Minorerrorsinintermediatestepsofthederivationsarealsocorrected

©2016ElsevierB.V.Allrightsreserved

In Ref.[1], the following corrections should be applied, in addition to those in Ref.[2]:

1 In the Appendix, Nfshould be replaced by 

i =u,d,s κ i2, where, in the notation of the paper,

κ i 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)

2 As a consequence, the same correction applies also to the value of K used in Eq (2.14) and

subsequent equations, where

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.

JID:NUPHB AID:13951 /ERR [m1+; v1.241; Prn:29/12/2016; 12:23] P.2 (1-2)

2 J.C Collins / Nuclear Physics B ••• (••••) •••–•••

K = (48π2)−1 

i =u,d,s

κ i2

should be used

3 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 G2counterterm in (2.4)

4 The left-hand side of (A.2)should be μ∂A/∂μ.

5 There should be an additional factor of Q2on the right-hand side of the un-numbered equa-tion after (A.2)

6 There should be an additional factor of 1/Q2on the right-hand side of (A.3)

7 Throughout, the function A depends on e2 and all the κis, as well as on the explicitly

in-dicated arguments g2and /μ This can be seen from the definition (2.4) and the explicit

formula (A.2)

8 Similarly Bi also depends on e2as well as the indicated arguments

9 On the left-hand side of (3.1), C() should be replaced by C().

The equations in the Appendix thus read:

 L = 2e2

g2

i =u,d,s κ i2

A(g2, e2, {κi }, /μ) F2

μ∂A/∂μ = −g2 

i =u,d,s

κ i2( 2-loop β)/(4e3)

= −e2g2 

i =u,d,s

μ∂A/∂μ = 3e2Q2C 1,G2(Q2= μ2)/( 16π2),

C 1,G2(Q2) = −g2 

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.