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Erratum to “Reweighting NNPDFs: The W lepton asymmetry” [Nucl. Phys. B 849 (1) (2011) 112–143]
Available online at www.sciencedirect.com
Nuclear Physics B 854 (2012) 926–927 www.elsevier.com/locate/nuclphysb
Erratum
Erratum to “Reweighting NNPDFs: The W lepton asymmetry” [Nucl. Phys. B 849 (1) (2011) 112–143] NNPDF Collaboration Richard D. Ball , Valerio Bertone b , Francesco Cerutti c , Luigi Del Debbio a , Stefano Forte d , Alberto Guffanti b , José I. Latorre c , Juan Rojo d , Maria Ubiali e a,∗
a Tait Institute, University of Edinburgh, JCMB, KB, Mayfield Rd, Edinburgh EH9 3JZ, Scotland, United Kingdom b Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3,
D-79104 Freiburg i. B., Germany c Departament d’Estructura i Constituents de la Matèria, Universitat de Barcelona, Diagonal 647,
E-08028 Barcelona, Spain d Dipartimento di Fisica, Università di Milano and INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy e Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, D-52056 Aachen, Germany
Received 9 September 2011; accepted 12 September 2011 Available online 19 September 2011
Since writing the paper [R.D. Ball et al., NNPDF Collaboration, Reweighting NNPDFs: The W lepton asymmetry, Nucl. Phys. B 849 (1) (2011)112–143], we realised that the uniform measure on the space of data involves changes in χ ≡ χ 2 , and not χ 2 , and that this actually makes a small difference to the final result, since unlike y, χ depends on f . The revised result for the weights is equivalent to the old result when the number of added data, n, is large, but differs when n is small: in particular now when n = 1, the weights become purely Gaussian, as they must. Thus in Eqs. (3), (8), (9), (20), (21) of the paper, the n/2 − 1 in the exponents should be (n − 1)/2. Likewise in Eq. (37) the factor 12 Ndat − 1 should be 12 (Ndat − 1). In Eqs. (20), (21), (22), (24), (25), (33) dχ 2 should be replaced everywhere by dχ . Eqs. (20), (21) then also have an extra factor of 2 on the right-hand side. In Eqs. (3), (5), (6), (20), (22)–(29), (33)–(36) χ 2 should be replaced by χ in the arguments of the probabilities.
DOI of original article: 10.1016/j.nuclphysb.2011.03.017. * Corresponding author.
E-mail address: [email protected] (R.D. Ball). 0550-3213/$ – see front matter Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2011.09.011
NNPDF Collaboration / Nuclear Physics B 854 (2012) 926–927
927
To the sentence immediately before Eq. (3) on p. 115 a footnote should be added: “Note that this is the probability density of χ , not the probability density of χ 2 : the reason for this is explained in Section 3.1.” At the top of p. 119 below Eq. (19) the following sentence should be added: “The volume element d n y is independent of f : without a specific prediction, all data are assumed equally likely.” Then below Eq. (24) the text should read “since both the volume factor d n−1 Ω and the interval dχ are independent of the choice of replica, and may thus be taken out of the sum: this follows directly from the assumption that the measure d n y in Eq. (19) is independent of f .” The first sentence of the third paragraph on p. 121 should read “The probability density P(f |χK) is defined as the probability density for f given that χ lies in the finite interval [χ, χ + dχ], in the limit dχ → 0.” Similarly the last two complete sentences before Eq. (33) should read “Thus Vn is a thin shell of thickness dχ , and hence its total volume is proportional to An dχ . The limit Vn → 0 is then taken by letting dχ → 0.” We also realised that Eq. (34) was incorrect: it should read P(α|χfk K) dα P(χ|fk K) dχ = P(χ|αfk K) dχ P(α|fk K) dα. The likelihood P(χ|αfk K) may be evaluated using the usual formula Eq. (20), and by noting that the effect of α is to rescale χ 2 → χ 2 /α. The result Eq. (35) is unchanged. None of the numerical results are altered significantly by these changes. Acknowledgements We would like to thank J.C. Collins and J. Pumplin for remarks which prompted this revision.
Nuclear Physics B 854 (2012) 926–927 www.elsevier.com/locate/nuclphysb
Erratum
Erratum to “Reweighting NNPDFs: The W lepton asymmetry” [Nucl. Phys. B 849 (1) (2011) 112–143] NNPDF Collaboration Richard D. Ball , Valerio Bertone b , Francesco Cerutti c , Luigi Del Debbio a , Stefano Forte d , Alberto Guffanti b , José I. Latorre c , Juan Rojo d , Maria Ubiali e a,∗
a Tait Institute, University of Edinburgh, JCMB, KB, Mayfield Rd, Edinburgh EH9 3JZ, Scotland, United Kingdom b Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3,
D-79104 Freiburg i. B., Germany c Departament d’Estructura i Constituents de la Matèria, Universitat de Barcelona, Diagonal 647,
E-08028 Barcelona, Spain d Dipartimento di Fisica, Università di Milano and INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy e Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, D-52056 Aachen, Germany
Received 9 September 2011; accepted 12 September 2011 Available online 19 September 2011
Since writing the paper [R.D. Ball et al., NNPDF Collaboration, Reweighting NNPDFs: The W lepton asymmetry, Nucl. Phys. B 849 (1) (2011)112–143], we realised that the uniform measure on the space of data involves changes in χ ≡ χ 2 , and not χ 2 , and that this actually makes a small difference to the final result, since unlike y, χ depends on f . The revised result for the weights is equivalent to the old result when the number of added data, n, is large, but differs when n is small: in particular now when n = 1, the weights become purely Gaussian, as they must. Thus in Eqs. (3), (8), (9), (20), (21) of the paper, the n/2 − 1 in the exponents should be (n − 1)/2. Likewise in Eq. (37) the factor 12 Ndat − 1 should be 12 (Ndat − 1). In Eqs. (20), (21), (22), (24), (25), (33) dχ 2 should be replaced everywhere by dχ . Eqs. (20), (21) then also have an extra factor of 2 on the right-hand side. In Eqs. (3), (5), (6), (20), (22)–(29), (33)–(36) χ 2 should be replaced by χ in the arguments of the probabilities.
DOI of original article: 10.1016/j.nuclphysb.2011.03.017. * Corresponding author.
E-mail address: [email protected] (R.D. Ball). 0550-3213/$ – see front matter Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2011.09.011
NNPDF Collaboration / Nuclear Physics B 854 (2012) 926–927
927
To the sentence immediately before Eq. (3) on p. 115 a footnote should be added: “Note that this is the probability density of χ , not the probability density of χ 2 : the reason for this is explained in Section 3.1.” At the top of p. 119 below Eq. (19) the following sentence should be added: “The volume element d n y is independent of f : without a specific prediction, all data are assumed equally likely.” Then below Eq. (24) the text should read “since both the volume factor d n−1 Ω and the interval dχ are independent of the choice of replica, and may thus be taken out of the sum: this follows directly from the assumption that the measure d n y in Eq. (19) is independent of f .” The first sentence of the third paragraph on p. 121 should read “The probability density P(f |χK) is defined as the probability density for f given that χ lies in the finite interval [χ, χ + dχ], in the limit dχ → 0.” Similarly the last two complete sentences before Eq. (33) should read “Thus Vn is a thin shell of thickness dχ , and hence its total volume is proportional to An dχ . The limit Vn → 0 is then taken by letting dχ → 0.” We also realised that Eq. (34) was incorrect: it should read P(α|χfk K) dα P(χ|fk K) dχ = P(χ|αfk K) dχ P(α|fk K) dα. The likelihood P(χ|αfk K) may be evaluated using the usual formula Eq. (20), and by noting that the effect of α is to rescale χ 2 → χ 2 /α. The result Eq. (35) is unchanged. None of the numerical results are altered significantly by these changes. Acknowledgements We would like to thank J.C. Collins and J. Pumplin for remarks which prompted this revision.