Resummation at small x

Transcription

1 Resummation at small Anna Stasto Penn State University & RIKEN BNL & INP Cracow 09/13/0, INT Workshop, Seattle

2 Outline Limitations of the collinear framework and the fied order (DGLAP) calculations. Signals at HERA. High energy limit, problems with convergence. Resummation at small. Results for the gluon Green s function and the splitting function. Including quarks: matri formulation. Results for the matri of the splitting functions.

3 Collinear framework for DIS

4 Collinear framework for DIS Hard scattering coefficient. On-shell matri element H(Q/µ, /z, α s )= i α i s H i

5 Collinear framework for DIS Hard scattering coefficient. On-shell matri element H(Q/µ, /z, α s )= αs i H i i Integrated parton distribution: f(, µ)

6 Collinear framework for DIS Hard scattering coefficient. On-shell matri element H(Q/µ, /z, α s )= i α i s H i Factorization for structure function: F T,L (, Q )= j 1 Integrated parton distribution: f(, µ) dz z f j/h(z,µ)h j T,L (/z, Q/µ, α s(µ)) + O(Λ/Q)

7 Collinear framework for DIS Hard scattering coefficient. On-shell matri element H(Q/µ, /z, α s )= i α i s H i Factorization for structure function: 1 Integrated parton distribution: f(, µ) F T,L (, Q )= dz j z f j/h(z,µ)h j T,L (/z, Q/µ, α s(µ)) + O(Λ/Q) Renormalization group equations: µ d dµ f j/h(, µ) = k 1 dz z P jk(z,α s (µ))f k/h (/z, µ)

8 Collinear framework for DIS Hard scattering coefficient. On-shell matri element H(Q/µ, /z, α s )= i α i s H i Factorization for structure function: 1 Integrated parton distribution: F T,L (, Q )= dz j z f j/h(z,µ)h j T,L (/z, Q/µ, α s(µ)) + O(Λ/Q) Renormalization group equations: µ d dµ f j/h(, µ) = k Epansion for anomalous dimensions (splitting functions) 1 dz f(, µ) z P jk(z,α s (µ))f k/h (/z, µ) P jk (z,α s (µ)) = i (α s (µ)) i P (i) jk (z)

9 Large corrections to fied order at small 1 F L = C L,ns q ns + e ( C L,q q s +C L,g g ) Moch, Vermaseren, Vogt Singlet-quark and gluon coefficient of the longitudinal structure function up to NNLO order

10 Large corrections to fied order at small 1 F L = C L,ns q ns + e ( C L,q q s +C L,g g ) Moch, Vermaseren, Vogt Singlet-quark and gluon coefficient of the longitudinal structure function up to NNLO order MSTW PDFs Large gluon uncertainties from fits to the data. Problems already at NLO level.

11 Problems at low

12 Problems at low Uncertainties of the gluon distribution translate into the observable FL.

13 Problems at low Uncertainties of the gluon distribution translate into the observable FL. NLO,NNLO predictions allow for the negative structure function.

14 Problems at low Uncertainties of the gluon distribution translate into the observable FL. NLO,NNLO predictions allow for the negative structure function. Note that the problem remains even for larger values of Q, it is though pushed towards lower values of.

15 Compatibility of data with DGLAP NNPDF.0 dataset J.Rojo, S.Forte ] [ GeV T / p / M Q NMC-pd NMC SLAC BCDMS HERAI-AV CHORUS FLH8 NTVDMN ZEUS-H DYE886 CDFWASY CDFZRAP D0ZRAP CDFRKT = 0.5 = 1.0 = 1.5 = 3.0 = 6.0 diag Cut out the region of small and small Q and Fit with cuts monitor the quality of the fit. < < < < < 6.0 Fit without cuts < < 3.0 > 6.0 Figure : Left plot: the kinematical coverage of the data used in the NNPDF.0 analysis with the different regions in used to probe deviations from DGLAP. Right plot: the diagonal χ diag computed in the different kinematic slices in,whereχ diag has been computed using both the reference NNPDF.0 fit without kinematical cuts (yellow line) and the NNPDF.0 with the maimum =1.5 cut(redline). ( ' )$*+,-. ' ' /01 & '3* ' $3* $ "#$%"#&'(#$)(#* "#$%"#&$/ )(# $+$,-. "#$%"#&$/ )(# $+$0-. 1# ( ' )&*+,-./ Q =3.5 %*4 GeV ' 3 ' 01 $*# "#$%"#&'(#$)(#* $*' $ %*5 %*# "#$%"#&$/ )(# $+$,-. "#$%"#&$/ )(# $+$0-. 1#

16 Compatibility of data with DGLAP NNPDF.0 dataset J.Rojo, S.Forte ] [ GeV T / p / M Q NMC-pd NMC SLAC BCDMS HERAI-AV CHORUS FLH8 NTVDMN ZEUS-H DYE886 CDFWASY CDFZRAP D0ZRAP CDFRKT = 0.5 = 1.0 = 1.5 = 3.0 = 6.0 diag vary Cut out the region of small and small Q and Fit with cuts monitor the quality of the fit. < 1.0 Q λ 1.0 < < < < 6.0 Fit without cuts < < 3.0 > 6.0 Figure : Left plot: the kinematical coverage of the data used in the NNPDF.0 analysis with the different regions in used to probe deviations from DGLAP. Right plot: the diagonal χ diag computed in the different kinematic slices in,whereχ diag has been computed using both the reference NNPDF.0 fit without kinematical cuts (yellow line) and the NNPDF.0 with the maimum =1.5 cut(redline). ( ' )$*+,-. ' ' /01 & '3* ' $3* $ "#$%"#&'(#$)(#* "#$%"#&$/ )(# $+$,-. "#$%"#&$/ )(# $+$0-. 1# ( ' )&*+,-./ Q =3.5 %*4 GeV ' 3 ' 01 $*# "#$%"#&'(#$)(#* $*' $ %*5 %*# "#$%"#&$/ )(# $+$,-. "#$%"#&$/ )(# $+$0-. 1#

17 Compatibility of data with DGLAP NNPDF.0 dataset J.Rojo, S.Forte ] [ GeV T / p / M Q NMC-pd NMC SLAC BCDMS HERAI-AV CHORUS FLH8 NTVDMN ZEUS-H DYE886 CDFWASY CDFZRAP D0ZRAP CDFRKT = 0.5 = 1.0 = 1.5 = 3.0 = 6.0 diag vary Cut out the region of small and small Q and Fit with cuts monitor the quality of the fit. < 1.0 Q λ 1.0 < < < < 6.0 Fit without cuts (This prescription 1.5 < < 3.0 mimics the behavior of the > 6.0 saturation scale.) Figure : Left plot: the kinematical coverage of the data used in the NNPDF.0 analysis with the different regions in used to probe deviations from DGLAP. Right plot: the diagonal χ diag computed in the different kinematic slices in,whereχ diag has been computed using both the reference NNPDF.0 fit without kinematical cuts (yellow line) and the NNPDF.0 with the maimum =1.5 cut(redline). ( ' )$*+,-. ' ' /01 & '3* ' $3* $ "#$%"#&'(#$)(#* "#$%"#&$/ )(# $+$,-. "#$%"#&$/ )(# $+$0-. 1# ( ' )&*+,-./ Q =3.5 %*4 GeV ' 3 ' 01 $*# "#$%"#&'(#$)(#* $*' $ %*5 %*# "#$%"#&$/ )(# $+$,-. "#$%"#&$/ )(# $+$0-. 1#

18 6 NMC-pd NMC SLAC BCDMS HERAI-AV 5 CHORUS FLH8 NTVDMN ZEUS-H 6 DYE886 4 CDFWASY CDFZRAP D0ZRAP CDFRKT 5 3 = 0.5 = 1.0 = 1.5 = 3.0 A = 6.0 cut PDF.0 dataset -5 ] [ GeV T / p / M Q NNPDF.0 dataset NMC-pd NMC SLAC BCDMS HERAI-AV CHORUS FLH8 NTVDMN ZEUS-H DYE886 CDFWASY CDFZRAP D0ZRAP CDFRKT = 0.5 = 1.0 = 1.5 = 3.0 = Compatibility of data with DGLAP ent kinematic slices in,whereχ diag has been computed 1 using both the reference NNPDF.0 fit ut kinematical cuts (yellow line) and the NNPDF.0 with the maimum 0.5 =1.5 cut(redline). diag 3.5 < < < < < < < 6.0 Fit without cuts & Figure : Left plot: the kinematical coverage of the data used in the NNPDF.0 analysis with the different regions in used to probe deviations from DGLAP. Right plot: the diagonal χ "#$%"#&$/ '3* )(# $+$,-. "#$%"#&$/ )(# $+$,-. diag computed in the $*' "#$%"#&$/ different kinematic slices in,whereχ )(# $+$0-. "#$%"#&$/ diag has been computed using )(# $+$0-. both the reference NNPDF.0 fit 1# $ 1# ' without kinematical cuts (yellow line) and the NNPDF.0 with the maimum =1.5 cut(redline). ( ' 1# $ "# ' "& "' "# "& $% $% $% $% $% %*5 $3* Q = 15 GeV Q =3.5 %*4 GeV $ %*# )$*+,-. ' ' /01 & '3* diag < 1.0 "#$%"#&'(#$)(#* %*# "#$%"#&$/ )(# $+$,-. "#$%"#&$/ %*' )(# $+$ < < 1.5 ( ' )&*+,-./ ' 3 ' 01 $*# "#$%"#&'(#$)(#* $*' Fit with cuts Fit without cuts Cut out the region of small and small Q and Fit with cuts monitor the quality of the fit. Q 3.0 < A < 6.0 λ cut e : Left plot: the kinematical coverage of the data used in the NNPDF.0 analysis with the different 1 s in used to probe deviations from DGLAP. Right plot: the diagonal χ diag computed in the ( ' )$*+,-. ' ' /01 $3* $ %3* "#$%"#&'(#$)(#* ( ' )&*+,-./ ' 3 ' 01 $*# "#$%"#&'(#$)(#* %*5 %*4 1.5 vary e 3: Left: The proton structure function F (, Q = 15 GeV ) at small-, computed from PDFs ed from the NNPDF.0 fits with different values of. Right the same but at a lower Q = "' $% > 6.0 (This prescription 1.5 < < 3.0 mimics the behavior of the > 6.0 saturation scale.) "#$%"#&$/ )(# $+$,-. "#$%"#&$/ )(# $+$0-. 1# J.Rojo, S.Forte

19 6 NMC-pd NMC SLAC BCDMS HERAI-AV 5 CHORUS FLH8 NTVDMN ZEUS-H 6 DYE886 4 CDFWASY CDFZRAP D0ZRAP CDFRKT 5 3 = 0.5 = 1.0 = 1.5 = 3.0 A = 6.0 cut PDF.0 dataset -5 ] [ GeV T / p / M Q NNPDF.0 dataset NMC-pd NMC SLAC BCDMS HERAI-AV CHORUS FLH8 NTVDMN ZEUS-H DYE886 CDFWASY CDFZRAP D0ZRAP CDFRKT = 0.5 = 1.0 = 1.5 = 3.0 = Compatibility of data with DGLAP ent kinematic slices in,whereχ diag has been computed 1 using both the reference NNPDF.0 fit ut kinematical cuts (yellow line) and the NNPDF.0 with the maimum 0.5 =1.5 cut(redline). diag 3.5 < < < < < < < 6.0 Fit without cuts & Figure : Left plot: the kinematical coverage of the data used in the NNPDF.0 analysis with the different regions in used to probe deviations from DGLAP. Right plot: the diagonal χ "#$%"#&$/ '3* )(# $+$,-. "#$%"#&$/ )(# $+$,-. diag computed in the $*' "#$%"#&$/ different kinematic slices in,whereχ )(# $+$0-. "#$%"#&$/ diag has been computed using )(# $+$0-. both the reference NNPDF.0 fit 1# $ 1# ' without kinematical cuts (yellow line) and the NNPDF.0 with the maimum =1.5 cut(redline). found. ( ' 1# $ "# ' "& "' "# "& $% $% $% $% $% %*5 $3* Q = 15 GeV Q =3.5 %*4 GeV $ %*# )$*+,-. ' ' /01 & '3* diag < 1.0 "#$%"#&'(#$)(#* %*# "#$%"#&$/ )(# $+$,-. "#$%"#&$/ %*' )(# $+$ < < 1.5 ( ' )&*+,-./ ' 3 ' 01 $*# "#$%"#&'(#$)(#* $*' Fit with cuts Fit without cuts Cut out the region of small and small Q and Fit with cuts monitor the quality of the fit. Q 3.0 < A < 6.0 λ cut e : Left plot: the kinematical coverage of the data used in the NNPDF.0 analysis with the different 1 s in used to probe deviations from DGLAP. Right plot: the diagonal χ diag computed in the ( ' )$*+,-. ' ' /01 $3* $ %3* "#$%"#&'(#$)(#* ( ' )&*+,-./ ' 3 ' 01 $*# "#$%"#&'(#$)(#* %*5 %*4 1.5 vary e 3: Left: The proton structure function F (, Q = 15 GeV ) at small-, computed from PDFs ed from the NNPDF.0 fits with different values of. Right the same but at a lower Q = "' $% > 6.0 (This prescription 1.5 < < 3.0 mimics the behavior of the > 6.0 saturation scale.) At higher Q no deviations "#$%"#&$/ )(# $+$,-. "#$%"#&$/ )(# $+$0-. 1# J.Rojo, S.Forte

20 6 NMC-pd NMC SLAC BCDMS HERAI-AV 5 CHORUS FLH8 NTVDMN ZEUS-H 6 DYE886 4 CDFWASY CDFZRAP D0ZRAP CDFRKT 5 3 = 0.5 = 1.0 = 1.5 = 3.0 A = 6.0 cut PDF.0 dataset -5 ] [ GeV T / p / M Q NNPDF.0 dataset NMC-pd NMC SLAC BCDMS HERAI-AV CHORUS FLH8 NTVDMN ZEUS-H DYE886 CDFWASY CDFZRAP D0ZRAP CDFRKT = 0.5 = 1.0 = 1.5 = 3.0 = Compatibility of data with DGLAP ent kinematic slices in,whereχ diag has been computed 1 using both the reference NNPDF.0 fit ut kinematical cuts (yellow line) and the NNPDF.0 with the maimum 0.5 =1.5 cut(redline). diag 3.5 < < < < < < < 6.0 Fit without cuts & Figure : Left plot: the kinematical coverage of the data used in the NNPDF.0 analysis with the different regions in used to probe deviations from DGLAP. Right plot: the diagonal χ "#$%"#&$/ '3* )(# $+$,-. "#$%"#&$/ )(# $+$,-. diag computed in the $*' "#$%"#&$/ different kinematic slices in,whereχ )(# $+$0-. "#$%"#&$/ diag has been computed using )(# $+$0-. both the reference NNPDF.0 fit 1# $ 1# ' without kinematical cuts (yellow line) and the NNPDF.0 with the maimum =1.5 cut(redline). found. ( ' 1# $ "# ' "& "' "# "& $% $% $% $% $% %*5 $3* Q = 15 GeV Q =3.5 %*4 GeV $ %*# )$*+,-. ' ' /01 & '3* diag < 1.0 "#$%"#&'(#$)(#* %*# "#$%"#&$/ )(# $+$,-. "#$%"#&$/ %*' )(# $+$ < < 1.5 ( ' )&*+,-./ ' 3 ' 01 $*# "#$%"#&'(#$)(#* $*' Fit with cuts Fit without cuts Cut out the region of small and small Q and Fit with cuts monitor the quality of the fit. Q 3.0 < A < 6.0 λ cut e : Left plot: the kinematical coverage of the data used in the NNPDF.0 analysis with the different 1 s in used to probe deviations from DGLAP. Right plot: the diagonal χ diag computed in the ( ' )$*+,-. ' ' /01 $3* $ %3* "#$%"#&'(#$)(#* ( ' )&*+,-./ ' 3 ' 01 $*# "#$%"#&'(#$)(#* %*5 %*4 1.5 vary e 3: Left: The proton structure function F (, Q = 15 GeV ) at small-, computed from PDFs ed from the NNPDF.0 fits with different values of. Right the same but at a lower Q = "' $% > 6.0 (This prescription 1.5 < < 3.0 mimics the behavior of the > 6.0 saturation scale.) At higher Q no deviations "#$%"#&$/ )(# $+$,-. "#$%"#&$/ )(# $+$0-. 1# J.Rojo, S.Forte

21 6 NMC-pd NMC SLAC BCDMS HERAI-AV 5 CHORUS FLH8 NTVDMN ZEUS-H 6 DYE886 4 CDFWASY CDFZRAP D0ZRAP CDFRKT 5 3 = 0.5 = 1.0 = 1.5 = 3.0 A = 6.0 cut PDF.0 dataset -5 ] [ GeV T / p / M Q NNPDF.0 dataset NMC-pd NMC SLAC BCDMS HERAI-AV CHORUS FLH8 NTVDMN ZEUS-H DYE886 CDFWASY CDFZRAP D0ZRAP CDFRKT = 0.5 = 1.0 = 1.5 = 3.0 = Compatibility of data with DGLAP ent kinematic slices in,whereχ diag has been computed 1 using both the reference NNPDF.0 fit ut kinematical cuts (yellow line) and the NNPDF.0 with the maimum 0.5 =1.5 cut(redline). diag 3.5 < < < < < < < 6.0 Fit without cuts & Figure : Left plot: the kinematical coverage of the data used in the NNPDF.0 analysis with the different regions in used to probe deviations from DGLAP. Right plot: the diagonal χ "#$%"#&$/ '3* )(# $+$,-. "#$%"#&$/ )(# $+$,-. diag computed in the $*' "#$%"#&$/ different kinematic slices in,whereχ )(# $+$0-. "#$%"#&$/ diag has been computed using )(# $+$0-. both the reference NNPDF.0 fit 1# $ 1# ' without kinematical cuts (yellow line) and the NNPDF.0 with the maimum =1.5 cut(redline). found. ( ' 1# $ "# ' "& "' "# "& $% $% $% $% $% %*5 $3* Q = 15 GeV Q =3.5 %*4 GeV $ %*# )$*+,-. ' ' /01 & '3* diag < 1.0 "#$%"#&'(#$)(#* %*# "#$%"#&$/ )(# $+$,-. "#$%"#&$/ %*' )(# $+$ < < 1.5 ( ' )&*+,-./ ' 3 ' 01 $*# "#$%"#&'(#$)(#* $*' Fit with cuts Fit without cuts Cut out the region of small and small Q and Fit with cuts monitor the quality of the fit. Q 3.0 < A < 6.0 λ cut e : Left plot: the kinematical coverage of the data used in the NNPDF.0 analysis with the different 1 s in used to probe deviations from DGLAP. Right plot: the diagonal χ diag computed in the ( ' )$*+,-. ' ' /01 $3* $ %3* "#$%"#&'(#$)(#* ( ' )&*+,-./ ' 3 ' 01 $*# "#$%"#&'(#$)(#* e 3: Left: The proton structure function F (, Q = 15 GeV ) at small-, computed from PDFs ed from the NNPDF.0 fits with different values of. Right the same but at a lower Q = %*5 %*4 1.5 vary "' $% > 6.0 (This prescription 1.5 < < 3.0 mimics the behavior of the > 6.0 saturation scale.) At higher Q no deviations At lower Q the prediction "#$%"#&$/ )(# $+$,-. J.Rojo, S.Forte "#$%"#&$/ )(# $+$0-. obtained from the backward 1# evolution is below the data. NLO DGLAP evolves too fast as compared with the data.

22 What can we do? Troublesome region: small and small Q at HERA. Resummation of subleading effects in the splitting and coefficient functions. Saturation of parton density? Other higher twist effects? Something else?

23 High energy limit s, 0 Energy much larger than any other scale in the process At small there are potentially large logs: P gg () α n S lnn 1 (1/), P qg () α n S lnn (1/) and C L,g () α n S lnn (1/).

24 High energy limit s, 0 Energy much larger than any other scale in the process At small there are potentially large logs: P gg () α n S lnn 1 (1/), P qg () α n S lnn (1/) and C L,g () α n S lnn (1/). At high energy, or small we can have: α S ln 1/ 1

25 High energy limit s, 0 Energy much larger than any other scale in the process At small there are potentially large logs: P gg () α n S lnn 1 (1/), P qg () α n S lnn (1/) and C L,g () α n S lnn (1/). At high energy, or small we can have: α S ln 1/ 1 Need to resum them as well to all orders: (α S ln 1/) n

26 High energy limit s, 0 Energy much larger than any other scale in the process At small there are potentially large logs: P gg () α n S lnn 1 (1/), P qg () α n S lnn (1/) and C L,g () α n S lnn (1/). At high energy, or small we can have: α S ln 1/ 1 Need to resum them as well to all orders: (α S ln 1/) n Any fied order here would not be sufficient, potentially very large corrections.

27 Many soft gluon emissions in small limit Cascade of the n soft gluons

28 Many soft gluon emissions in small limit Cascade of the n soft gluons p + k + p + 1 p + p + 3 p + 4 p + n scattering

29 Many soft gluon emissions in small limit Cascade of the n soft gluons p + Strong ordering (in longitudinal momenta) k + p + 1 p + p + 3 p + 4 p + n scattering

30 Many soft gluon emissions in small limit Cascade of the n soft gluons p + Strong ordering (in longitudinal momenta) p + p + 1 p+ p+ n k + k + p + 1 p + p + 3 p + 4 p + n scattering

31 Many soft gluon emissions in small limit Cascade of the n soft gluons p + p + 1 p + p + 3 p + 4 Strong ordering (in longitudinal momenta) p + p + 1 p+ p+ n k + Note: transverse momenta are not ordered k + p + n scattering

32 Many soft gluon emissions in small limit Cascade of the n soft gluons p + p + 1 p + p + 3 p + 4 Strong ordering (in longitudinal momenta) p + p + 1 p+ p+ n k + Note: transverse momenta are not ordered α s N c π p + k + + dp1 p + 1 k + = p + = α sn c ln 1 π Large logarithm k + p + n scattering

33 Many soft gluon emissions in small limit Cascade of the n soft gluons p + k + p + 1 p + p + 3 p + 4 p + n scattering Strong ordering (in longitudinal momenta) p + p + 1 p+ p+ n k + Note: transverse momenta are not ordered α s N c π p + k + + dp1 p + 1 = α sn c ln 1 π Large logarithm Nested logarithmic integrals k + = p +

34 Many soft gluon emissions in small limit Cascade of the n soft gluons p + k + p + 1 p + p + 3 p + 4 p + n scattering Strong ordering (in longitudinal momenta) p + p + 1 p+ p+ n k + Note: transverse momenta are not ordered α s N c π p + k + + dp1 p + 1 = α sn c ln 1 π Large logarithm Nested logarithmic integrals αs N c π ln 1 k + = p + n

35 Many soft gluon emissions in small limit Cascade of the n soft gluons p + k + p + 1 p + p + 3 p + 4 p + n scattering Strong ordering (in longitudinal momenta) p + p + 1 p+ p+ n k + Note: transverse momenta are not ordered α s N c π p + Nested logarithmic integrals Resummation of the gluon emissions performed by the equation k + + dp1 p + 1 αs N c π k + = p + = α sn c ln 1 π Large logarithm ln 1 n

36 Many soft gluon emissions in small limit Cascade of the n soft gluons p + Strong ordering (in longitudinal momenta) p + p + 1 p+ p+ n k + Note: transverse momenta are not ordered α s N c π p + Nested logarithmic integrals Resummation of the gluon emissions performed by the equation df g (, k T ) d ln 1/ k + p + 1 p + p + 3 p + 4 p + n scattering = α sn c π k + + dp1 p + 1 αs N c π k + = p + = α sn c ln 1 π Large logarithm ln 1 d k T K(k T,k T ) f g (, k T ) n

37 Many soft gluon emissions in small limit Cascade of the n soft gluons p + Nested logarithmic integrals Resummation of the gluon emissions performed by the equation df g (, k T ) d ln 1/ k + p + 1 p + p + 3 p + 4 p + n scattering = α sn c π integral over transverse momenta Strong ordering (in longitudinal momenta) p + p + 1 p+ p+ n k + Note: transverse momenta are not ordered α s N c π p + k + + dp1 p + 1 αs N c π k + = p + = α sn c ln 1 π Large logarithm ln 1 d k T K(k T,k T ) f g (, k T ) n

38 Many soft gluon emissions in small limit Cascade of the n soft gluons p + Strong ordering (in longitudinal momenta) p + p + 1 p+ p+ n k + Note: transverse momenta are not ordered α s N c π p + Nested logarithmic integrals Resummation of the gluon emissions performed by the equation df g (, k T ) d ln 1/ k + p + 1 p + p + 3 p + 4 p + n scattering = α sn c π k + + dp1 p + 1 αs N c π k + = p + = α sn c ln 1 π Large logarithm ln 1 d k T K(k T,k T ) f g (, k T ) n integral over transverse momenta kernel describing branching of gluons

39 Many soft gluon emissions in small limit Cascade of the n soft gluons p + I.Balitsky, V.Fadin, E.Kuraev,L.Lipatov Nested logarithmic integrals Resummation of the gluon emissions performed by the equation df g (, k T ) d ln 1/ k + p + 1 p + p + 3 p + 4 p + n scattering = α sn c π integral over transverse momenta Strong ordering (in longitudinal momenta) p + p + 1 p+ p+ n k + Note: transverse momenta are not ordered α s N c π p + k + kernel describing branching of gluons + dp1 p + 1 αs N c π k + = p + = α sn c ln 1 π Large logarithm ln 1 d k T K(k T,k T ) f g (, k T ) n gluon density

40 Evolution equation in longitudinal momenta

41 Evolution equation in longitudinal momenta df g (, k T ) d ln 1/ = α sn c π d k T K(k T,k T ) f g (, k T )

42 Evolution equation in longitudinal momenta df g (, k T ) d ln 1/ Solution: = α sn c π d k T K(k T,k T ) f g (, k T )

43 Evolution equation in longitudinal momenta df g (, k T ) d ln 1/ Solution: = α sn c π f g (, k T ) ω P d k T K(k T,k T ) f g (, k T )

44 Evolution equation in longitudinal momenta df g (, k T ) d ln 1/ Solution: = α sn c π f g (, k T ) ω P d k T K(k T,k T ) f g (, k T ) ω P = j 1= α sn c π 4 ln Leading eponent(spin)

45 Evolution equation in longitudinal momenta df g (, k T ) d ln 1/ Solution: = α sn c π f g (, k T ) ω P d k T K(k T,k T ) f g (, k T ) ω P = j 1= α sn c π 4 ln Leading eponent(spin) Rise too strong for the data σ DIS γ p s ω P

46 Evolution equation in longitudinal momenta df g (, k T ) d ln 1/ Solution: = α sn c π f g (, k T ) ω P d k T K(k T,k T ) f g (, k T ) ω P = j 1= α sn c π 4 ln Leading eponent(spin) Rise too strong for the data Take higher order corrections. V.Fadin,L.Lipatov, G.Camici,M.Ciafaloni σ DIS γ p s ω P α s K 0 + α sk

47 Evolution equation in longitudinal momenta df g (, k T ) d ln 1/ Solution: = α sn c π f g (, k T ) ω P d k T K(k T,k T ) f g (, k T ) ω P = j 1= α sn c π 4 ln Leading eponent(spin) ω P Rise too strong for the data σ DIS γ p s ω P LL Take higher order corrections. V.Fadin,L.Lipatov, G.Camici,M.Ciafaloni α s K 0 + α sk NLL α s

48 Evolution equation in longitudinal momenta df g (, k T ) d ln 1/ Solution: = α sn c π f g (, k T ) ω P d k T K(k T,k T ) f g (, k T ) ω P = j 1= α sn c π 4 ln Leading eponent(spin) ω P Rise too strong for the data σ DIS γ p s ω P LL Take higher order corrections. V.Fadin,L.Lipatov, G.Camici,M.Ciafaloni α s K 0 + α sk ω P ᾱ s 4 ln (1 6.5ᾱ s ) NLL α s

49 Evolution equation in longitudinal momenta df g (, k T ) d ln 1/ Solution: = α sn c π f g (, k T ) ω P d k T K(k T,k T ) f g (, k T ) ω P = j 1= α sn c π 4 ln Leading eponent(spin) ω P Rise too strong for the data Take higher order corrections. V.Fadin,L.Lipatov, G.Camici,M.Ciafaloni σ DIS γ p s ω P α s K 0 + α sk ω P ᾱ s 4 ln (1 6.5ᾱ s ) LL NLL relevant values of α s α s

50 Rise too strong for the data Evolution equation in longitudinal momenta Solution: Take higher order corrections. V.Fadin,L.Lipatov, G.Camici,M.Ciafaloni df g (, k T ) d ln 1/ = α sn c π Very large net-to-leading correction f g (, k T ) ω P σ DIS γ p s ω P α s K 0 + α sk d k T K(k T,k T ) f g (, k T ) Problems with convergence. ω P = j 1= α sn c π 4 ln Leading eponent(spin) ω P ω P ᾱ s 4 ln (1 6.5ᾱ s ) LL NLL relevant values of α s α s

51 LL vs NLL BFKL solution for the gluon Green s function BFKL at NLL k 0 G(Y; k0 + ", k 0 ) 1 LL NLL # s (q ), s 0 =kk 0 NLL # s (q ), s 0 =k NLL # s (k ), s 0 =kk 0 NLL # s (k ), s 0 =k Scale of the coupling. Energy scale: Y = ln s/s 0 Differences large even though formally at NNLL. k 0 = 0 GeV Y Very large correction at NLL. Need higher orders: resummation at small.

52 Why NLL is so large in BFKL? Strong coupling constant is not a naturally small parameter in the Regge limit: s t, Λ QCD but α s (µ ), µ = s Regge limit is inherently nonperturbative. Compare DGLAP (collinear approach): Q Λ and α s (Q ) 1 No momentum sum rule, since the evolution is local in. In DGLAP: momentum sum rule satisfied at each order due to the initial assumption of the collinearity of the partons and the nonlocality of the evolution in. Approimations in the phase space (multi-regge kinematics, quasi multi-regge kinematics, etc..) cannot be recovered by the (fied number of) the higher orders of epansion in the coupling constant.

53 Resummation M. Ciafaloni, D. Colferai, G. Salam, A.Stasto ;G. Altarelli, R. Ball, S.Forte; R. Thorne ln 1/ Problem with two αs N c π ln 1 n large parameters αs N c π ln Q Q 0 n ln Q/Q 0 Mellin variables: γ ln k T ω ln 1/ dk k γ Kernel eigenvalue Anomalous dimension χ(γ) = γ(ω) = k K(k,k ) dzp (z)z ω k

54 Resummation M. Ciafaloni, D. Colferai, G. Salam, A.Stasto ;G. Altarelli, R. Ball, S.Forte; R. Thorne ln 1/ Problem with two αs N c π ln 1 n energy large parameters αs N c π ln Q Q 0 n scale (related to transverse momentum) ln Q/Q 0 Mellin variables: γ ln k T ω ln 1/ dk k γ Kernel eigenvalue Anomalous dimension χ(γ) = γ(ω) = k K(k,k ) dzp (z)z ω k

55 General setup Kinematical constraint. DGLAP splitting function at LO and NLO. NLL BFKL with suitable subtraction of terms included above. Momentum sum rule. Running coupling. Calculations done in momentum space, even though we use Mellin space as a guidance.

56 LL BFKL+LO DGLAP Representation of the kernel K = n=0 ˆα n+1 s K n ˆα s α s π Mellin variables: γ ln k T ω ln 1/ K 0 (γ, ω) = C A ω χω 0 (γ) + [γ gg 0 (ω) C A ω BFKL DGLAP χ ω 0 (γ) = ψ(1) ψ(γ) ψ(1 γ + ω) ] χω c (γ) LL BFKL with kinematical constraint (shift of poles, energy scale choice)

57 LL BFKL+LO DGLAP Representation of the kernel K = n=0 ˆα n+1 s K n ˆα s α s π Mellin variables: γ ln k T ω ln 1/ K 0 (γ, ω) = C A ω χω 0 (γ) + [γ gg 0 (ω) C A ω BFKL DGLAP χ ω 0 (γ) = ψ(1) ψ(γ) ψ(1 γ + ω) ] χω c (γ) LL BFKL with kinematical constraint (shift of poles, energy scale choice) Collinear kernel for the DGLAP part χ ω c (γ) = 1 γ γ + ω LO DGLAP anomalous dimension γ gg 0 (ω)

58 NLL BFKL+NLO DGLAP ˆαK 0 (γ, ω) + ˆα K 1 (γ, ω) K 1 (γ, ω) = (C A) ω BFKL χ ω 1 (γ) + γ gg 1 (ω) χω c (γ) DGLAP χ ω 1 (γ) and γ gg 1 (ω) Contain the subtractions to match to the NLL and NLO DGLAP. Subtract the double and triple poles already contained in the K 0 (γ, ω)

59 Gluon Green s function Solution to the BFKL equation (gluon Green s function) Single channel: gluons only. k 0 G(Y; k0 + ", k 0 ) 1 (a) LL scheme A scheme B Large suppression as compared to LL. Two schemes, small differences. k 0 = 0 GeV Y

60 Gluon Green s function uncertainties (a) Renormalization scale variation 1/ < µ < k = 4.5 GeV LL k G(Y; k+", k#") 1 $k = 0.50 GeV $k = 0.74 GeV $k = 1.00 GeV (frozen)$k = 0.74 GeV NLL B Change of the infrared cutoff k =(0.5, 1.0) Different schemes give similar results Y Uncertainties are under control

61 Splitting function Deconvolution of the integral equation. Calculate the integrated density: g(, Q ) = Solve numerically for the splitting function: Q dk T G (s 0=k T ) (; k T, k 0T ) dg(, Q ) d log Q = dz z P eff(z, Q ) g( z, Q ) At large values of Q the results should be independent of the regularization of the coupling and the choice of k 0. Factorization in of the non-perturbative and perturbative contributions. Q

62 Splitting function Gluon-gluon splitting function has logarithmic enhancements at small P gg () = n=1 a n αs n ln n b n αs n ln n n= LO NLO NNLO LL First small logarithmic term which belongs to NLL hierarchy recovered at NNLO 1.54ᾱ 3 s ln 1/ NLL Resummation at small is inevitable. P gg () 0.3 s (Q ) = / < µ /Q <

63 Splitting function Gluon-gluon splitting function has logarithmic enhancements at small P gg () = n=1 a n αs n ln n b n αs n ln n n= LO NLO NNLO LL First small logarithmic term which belongs to NLL hierarchy recovered at NNLO 1.54ᾱ 3 s ln 1/ NLL Resummation at small is inevitable. P gg () 0.3 s (Q ) = / < µ /Q < Moch, Vermaseren, Vogt

64 Resummed splitting function Full P gg (z) splittingfn 1 Q = 4.5 GeV " s (Q ) = 0.15 LL (fied " s ) LL ( " s (q )) NLL B LO DGLAP epansion (1999) NLL B (003) LO DGLAP z P(z) z P gg (z) Q = 4.5 GeV # " s (Q ) = z 1/4 < µ /Q < z Dependence on the renormalization scale Small growth delayed to much smaller values of (beyond HERA) 3 Interesting feature: a dip seen at around Is this universal feature? The same feature seen in other schemes of resummation (Altarelli,Ball,Forte; Thorne). Need to understand the origin of the dip.

65 Where the dip comes from? epansion (1999) NLL B (003) LO DGLAP NNLO DGLAP z P gg (z) Q = 4.5 GeV # " s (Q ) = z 1/4 < µ /Q < ᾱ 3 s ln 1/ Initial decrease seems to be consistent with the small NNLO term.

66 Understanding the structure of the splitting function Perturbative terms in the splitting function LL NLL NNLL... s " " s 0 n f " 3 s 0 4 s const. 5 s 0 ln 1/... ln 3 1/ ln 1/

67 Understanding the structure of the splitting function Perturbative terms in the splitting function Reorganise perturbative series LL NLL NNLL... LL NLL NNLL... " s = 0.05 s s 0 " n f " " s s 0 " n f " " P gg () 3 s 4 s 5 s s 4 s 5 s const. ln 1/ 0 0 const. ln 1/ At moderately small, first terms with -dependence are 1... ln 3 1/... ln 1/ ln 3 1/ ln 1/ 1.54 ᾱ 3 s ln ᾱ4 s ln3 1 Minimum when α s ln 1 ln 1 1 αs

68 Understanding the structure of the splitting function Perturbative terms in the splitting function LL NLL NNLL... s " " s 0 n f " 3 s 0 4 s const. 5 s 0 ln 1/... ln 3 1/ ln 1/

69 Understanding the structure of the splitting function Perturbative terms in the splitting function Reorganise perturbative series LL NLL NNLL... LL NLL NNLL... " s = 0.05 s s 0 " n f " " s s 0 " n f " " P gg () 3 s 4 s 5 s s const. 4 s ln 1/ 5 s 0 0 const. ln 1/ At moderately small, first terms with -dependence are 1.54 ᾱ 3 s ln ᾱ4 s ln ln 3 1/... ln 1/ ln 3 1/ ln 1/ Minimum when α s ln 1 ln 1 1 αs

70 Understanding the structure of the splitting function Perturbative terms in the splitting function LL NLL NNLL... s " " s 0 n f " 3 s 0 4 s const. 5 s 0 ln 1/... ln 3 1/ ln 1/

71 Understanding the structure of the splitting function Reorganise perturbative series Perturbative terms in the splitting function LL NLL NNLL... " s = 0.05 s s 0 " n f " " P gg () 3 s s const. 5 s 0 ln 1/... ln 3 1/ ln 1/

72 Understanding the structure of the splitting function Reorganise perturbative series Perturbative terms in the splitting function LL NLL NNLL... " s = 0.05 s s 0 " n f " " P gg () 3 s s const. 5 s 0 ln 1/... ln 3 1/ ln 1/ 1.54ᾱ 3 s ln 1/ 0.401ᾱ 4 s ln 3 1/ +

73 Understanding the structure of the splitting function Reorganise perturbative series Perturbative terms in the splitting function LL NLL NNLL... " s = 0.05 s s 0 " n f " " P gg () 3 s s const. 5 s 0 ln 1/... ln 3 1/ ln 1/ 1.54ᾱ 3 s ln 1/ 0.401ᾱ 4 s ln 3 1/ + There is a minimum when α s ln 1 αs 1 ln 1 1

74 Understanding the structure of the splitting function Reorganise perturbative series Perturbative terms in the splitting function LL NLL NNLL... " s = 0.05 s s 0 " n f " " P gg () 3 s s const. 5 s 0 ln 1/... ln 3 1/ ln 1/ 1.54ᾱ 3 s ln 1/ 0.401ᾱ 4 s ln 3 1/ + In general: dip comes from the interplay between NNLO and the resummation. There is a minimum when α s ln 1 αs 1 ln 1 1

75 Matri approach Resummation demonstrated to give stable results for the gluon channel only. For the complete description need to include quarks. Matri approach: consistent with the collinear matri factorization of the parton densities in the singlet evolution. Enable to calculate the anomalous dimensions matri, which can be directly compared with the standard DGLAP matri. Incorporating NLL BFKL + NLO DGLAP.

76 Frozen coupling features BFKL Pomeron resummed intercept in 0.5 the matri approach 0.4 s NL-NLO, nf = 4 NL-NLO, nf = 0 NL-LO, nf = 0 1-channel B Results similar to previous single - channel approach. " s

77 Splitting functions gg channel results similar to the older (003) calculations gq channel close to gg characteristic dip at present (in both channels) 3 still P qq () NL-NLO NL-NLO + NLO s =0., n f =4 0.5 < µ < P qg () onset of rise at 4 scale dependence grows with decreasing, but not larger than at plain NLO qg,qq splitting functions: larger scale uncertainty, but closer to NLO dip structure in qg,qq channels is much milder P gq () qq gq qg scheme B (nf=0) gg P gg ()

78 Summary and outlook I Formalism that includes DGLAP NLO and BFKL NLL and the higher order terms. Stability of the results demonstrated for scale changes and model changes. Characteristic feature: small growth delayed, dip of the splitting function Results on gluon Green s function and the splitting functions from the matri model. To do: Coefficient functions. Fit to the data. Etraction of the resummed pdfs: predictions for EIC, LHeC, applications for LHC (esp. forward processes). Etension to include the NNLO DGLAP.

79 Summary and outlook II What EIC can bring to clarify this picture? Increased luminosity: precision measurement of structure function. Longitudinal structure function? Running e/p and e/a can help disentangle resummation and/or saturation effects. Resummation at small in case of polarized scattering. Uneplored so far, but could be promising approach.

80 Backup

81 Gluon Green s function Resummation identical to the single channel case in gg part. qg channel suppressed by factor of the coupling. Quarks are generated radiatively therefore growth in Y follows gg channel. Small difference between two resummation schemes:nl-nlo and NL-NLO. + k 0 Gig (Y, k, k 0 ) [k= 1. k 0 ] NL-NLO NL-NLO + scheme B i=g i=q " s = < µ < Y = ln s/(k k 0 )

82 Matri kernel constraints General form of the matri kernel: K(α s, γ, ω) Single poles in Single poles in γ n,m,p=0 pk (m) n ˆα n+1 γ m 1 ω p 1, ˆα α s π, natural in DGLAP K = 1 γ K(0) (α s, ω) + K (1) (α s, ω) + γ K () (α s, ω) + O ( γ ) ω, natural in BFKL K = 1 ω 0 K(α s, γ) + 1 K(α s, γ) + ω K(α s, γ) + O ( ω ) K qq, K qg have no poles in ω at any order

83 Matri kernel constraints General form of the matri kernel: K(α s, γ, ω) Single poles in Single poles in γ ω K qq, K qg have no poles in n,m,p=0 pk (m) n ˆα n+1 γ m 1 ω p 1, ˆα α s π, natural in DGLAP K = 1 γ K(0) (α s, ω) + K (1) (α s, ω) + γ K () (α s, ω) + O ( γ ), natural in BFKL K = 1 ω 0 K(α s, γ) + 1 K(α s, γ) + ω K(α s, γ) + O ( ω ) ω at any order γ ω Higher order singularities in present at NLL (and beyond) as well as higher order singularities in present at NLO (and beyond) can be generated through the subleading dependences in both variables.

84 Kernel Kernel vs anomalous dimensions K(α s, γ, ω) n,m,p=0 pk (m) n ˆα n+1 γ m 1 ω p 1, ˆα α s π Anomalous dimensions Γ(ω) n=0 ˆα n+1 Γ n (ω)

85 Kernel Kernel vs anomalous dimensions K(α s, γ, ω) n,m,p=0 pk (m) n ˆα n+1 γ m 1 ω p 1, ˆα α s π Anomalous dimensions Γ(ω) n=0 ˆα n+1 Γ n (ω) From the kernel one can derive anomalous dimensions: Γ 0 = K (0) 0 Γ 1 = K (0) 1 + K (1) 0 K (0) 0 Γ = K (0) + K (1) 1 K(0) 0 + K (1) 0 K(0) 1 + K () 0 ( K (0) 0 ) + ( K (1) ) 0 K (0) 0 These recursive equations can be used to constrain the collinear singularities of the kernels by the known anomalous dimensions (same procedure used in the previous single channel approach).

86 Momentum sum rule α s Q [GeV] NL-NLO NL-NLO + j Γ jq(1) j Γ jg(1) j Γ jq(1) j Γ jg(1) Momentum sum rule satisfied to very good accuracy. Residual Q dependence (higher twist, non-perturbative regularization?)

87 Instabilities of the splitting function P gg () LL NLL LO DGLAP " s (Q ) = 0.15 (, ) NNNLO LO, NLO NNLO Splitting function: LO DGLAP, L,NLL BFKL e+05 1e+06 Plot from S. Forte talk, RBRC workshop, BNL

88 Collinear-anticollinear symmetry Sequence of splittings : collinear or anti-collinear collinear splitting d... Γ dc Γ cb Γ ba... c b { a { d < c < b < a k d k c k b k a anti-collinear splitting... Γ cd Γ bc Γ ab... =... (Γ T ) dc (Γ T ) cb (Γ T ) ba.... { d > c > b > a k d k c k b k a

89 Collinear-anticollinear symmetry Sequence of splittings : collinear or anti-collinear collinear splitting d... Γ dc Γ cb Γ ba... c b { a { d < c < b < a k d k c k b k a anti-collinear splitting... Γ cd Γ bc Γ ab... =... (Γ T ) dc (Γ T ) cb (Γ T ) ba.... { d > c > b > a k d k c k b k a Naively one would take: K(γ, ω) = K T (1 + γ ω)

90 Collinear-anticollinear symmetry Sequence of splittings : collinear or anti-collinear collinear splitting d... Γ dc Γ cb Γ ba... c b { a { d < c < b < a k d k c k b k a anti-collinear splitting... Γ cd Γ bc Γ ab... =... (Γ T ) dc (Γ T ) cb (Γ T ) ba.... { d > c > b > a k d k c k b k a Naively one would take: K(γ, ω) = K T (1 + γ ω) Need to take care of the proper arrangement of color and ω factors when putting collinear and anti-collinear splittings together. 1

91 Collinear-anticollinear symmetry Sequence of splittings : collinear or anti-collinear collinear splitting d... Γ dc Γ cb Γ ba... c b { a { d < c < b < a k d k c k b k a anti-collinear splitting... Γ cd Γ bc Γ ab... =... (Γ T ) dc (Γ T ) cb (Γ T ) ba.... { d > c > b > a k d k c k b k a Naively one would take: K(γ, ω) = K T (1 + γ ω) Need to take care of the proper arrangement of color and ω factors when putting collinear and anti-collinear splittings together. For each echanged particle the appropriate color factor should be taken only once. The same problem with 1/ω factor for the echanged gluon. 1

92 Very fast hadron(nucleus) p + k + slow parton Lifetime on the lightcone + (p) 1 p p+ m T Gribov s spacetime picture: fast parton eists for a long time, slow parton is a short fluctuation. LC momentum: p + = 1 (E + p z ) LC energy: p = 1 (E p z )

93 Very fast hadron(nucleus) p + k + slow parton Lifetime on the lightcone + (p) 1 p p+ m T Gribov s spacetime picture: fast parton eists for a long time, slow parton is a short fluctuation. LC momentum: p + = 1 (E + p z ) LC energy: p = 1 (E p z ) p + k + p +

94 Very fast hadron(nucleus) p + k + slow parton Lifetime on the lightcone + (p) 1 p p+ m T Gribov s spacetime picture: fast parton eists for a long time, slow parton is a short fluctuation. LC momentum: p + = 1 (E + p z ) LC energy: p = 1 (E p z ) p + k + p + Slow partons can only see the total charge of the fast partons.

95 p + p + k + ρ k + k +

96 p + p + k + ρ charge k + k +

97 p + p + k + ρ charge k + k + p + k + p + 1 Radiation of gluons: Bremsstrahlung One gluon emission p + p + 1 k+ Separation of scales ρ k + Renormalized charge

98 p + p + k + ρ charge k + k + p + k + p + 1 Radiation of gluons: Bremsstrahlung One gluon emission p + p + 1 k+ Separation of scales ρ k + Renormalized charge The effect of the additional gluon emission is to renormalize the effective color charge.

99 p + p + k + ρ charge k + k + p + k + p + 1 Radiation of gluons: Bremsstrahlung One gluon emission p + p + 1 k+ Separation of scales ρ k + Renormalized charge The effect of the additional gluon emission is to renormalize the effective color charge. (This framework is the basis of Color Glass Condensate) L.McLerran,R.Venugopalan; J.Jalilian- Marian,E.Iancu,A.Kovner,H.Weigert,A. Leonidvov

100 Running coupling α s (q T ) BFKL part k T emitted gluon q T α s (k >) k > = ma(k T, k T ) DGLAP part k T

101 Results on splitting function resummed (NLL B ) LO DGLAP A " s + A 31 " s 3 ln 1/ P gg () Q = 4.5 GeV " s (Q ) = The onset of small rise delayed to < Characteristic dip at around = Universal feature of the resummed approaches.

102 Results on splitting function

103 General form of the matri kernel K 0 (γ, ω) = Γ qq,0 (ω)χ ω c (γ) Γ gq,0 (ω)χ ω c (γ) Γ qg,0 (ω)χ ω c (γ) + qg (ω)χ ω ht (γ) Γ gg,0 (ω)χ ω c (γ) + C A [ χ ω ω 0 (γ) χ ω c (γ) ] ( )

104 General form of the matri kernel K 0 (γ, ω) = Γ qq,0 (ω)χ ω c (γ) Γ gq,0 (ω)χ ω c (γ) Γ qg,0 (ω)χ ω c (γ) + qg (ω)χ ω ht (γ) Γ gg,0 (ω)χ ω c (γ) + C A [ χ ω ω 0 (γ) χ ω c (γ) ] ( ) Etra term in qg channel: + qg (ω)χ ω ht (γ) [ Allows to set the pole part of the NLO anomalous dimension: ω Γ qg,1 (K (1) 0 ) qg for ω 0.

105 General form of the matri kernel K 0 (γ, ω) = Γ qq,0 (ω)χ ω c (γ) Γ gq,0 (ω)χ ω c (γ) Γ qg,0 (ω)χ ω c (γ) + qg (ω)χ ω ht (γ) Γ gg,0 (ω)χ ω c (γ) + C A [ χ ω ω 0 (γ) χ ω c (γ) ] ( ) Etra term in qg channel: + qg (ω)χ ω ht (γ) [ Allows to set the pole part of the NLO anomalous dimension: ω Γ qg,1 (K (1) 0 ) qg for ω 0. The rest of NLO DGLAP and NL BFKL terms added in K 1 (γ, ω)

106 Similarity transformation Introduce matri S such that: K(1 + ω γ, ω) = S(ω)K T (γ, ω)s 1 (ω). defined to guarantee the appropriate color and 1/ω factors when collinear and anticollinear splitting are combined. Use the freedom in defining S = to obtain K ab (γ, ω) = K ab (1 + ω γ, ω)