Index: trunk/papers/bbz/bbz_fonll.bib
===================================================================
--- trunk/papers/bbz/bbz_fonll.bib (revision 48)
+++ trunk/papers/bbz/bbz_fonll.bib (revision 49)
@@ -1,1015 +1,1041 @@
@article{Collins:1978wz,
author = "Collins, John C. and Wilczek, Frank and Zee, A.",
title = "{Low-Energy Manifestations of Heavy Particles:
Application to the Neutral Current}",
journal = "Phys. Rev.",
volume = "D18",
year = "1978",
pages = "242",
doi = "10.1103/PhysRevD.18.242",
reportNumber = "COO-2220-127",
SLACcitation = "%%CITATION = PHRVA,D18,242;%%"
}
@article{Han:2014nja,
author = "Han, Tao and Sayre, Joshua and Westhoff, Susanne",
title = "{Top-Quark Initiated Processes at High-Energy Hadron
Colliders}",
journal = "JHEP",
volume = "04",
year = "2015",
pages = "145",
doi = "10.1007/JHEP04(2015)145",
eprint = "1411.2588",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "PITT-PACC-1405",
SLACcitation = "%%CITATION = ARXIV:1411.2588;%%"
}
@article{Dittmaier:2011ti,
author = "Dittmaier, S. and others",
title = "{Handbook of LHC Higgs Cross Sections: 1. Inclusive
Observables}",
collaboration = "LHC Higgs Cross Section Working Group",
doi = "10.5170/CERN-2011-002",
year = "2011",
eprint = "1101.0593",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CERN-2011-002",
SLACcitation = "%%CITATION = ARXIV:1101.0593;%%"
}
@article{Anastasiou:xxx,
author = "Anastasiou, C. and others",
title = "{Handbook of LHC Higgs Cross Sections: 4. Deciphering the nature of the Higgs sector}",
collaboration = "LHC Higgs Cross Section Working Group",
year = "2016",
eprint = "{\it in preparation}"
}
@article{Martin:1997ns,
author = "Martin, Stephen P.",
title = "{A Supersymmetry primer}",
year = "1997",
doi = "10.1142/9789812839657_0001, 10.1142/9789814307505_0001",
note = "[Adv. Ser. Direct. High Energy Phys.18,1(1998)]",
eprint = "hep-ph/9709356",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
SLACcitation = "%%CITATION = HEP-PH/9709356;%%"
}
@article{Brodsky:1980pb,
author = "Brodsky, S. J. and Hoyer, P. and Peterson, C. and Sakai,
N.",
title = "{The Intrinsic Charm of the Proton}",
journal = "Phys. Lett.",
volume = "B93",
year = "1980",
pages = "451-455",
doi = "10.1016/0370-2693(80)90364-0",
reportNumber = "NORDITA-80-18",
SLACcitation = "%%CITATION = PHLTA,B93,451;%%"
}
@book{Martin:1997ne,
author = "Martin, B. R. and Shaw, Graham",
title = "{Particle physics}",
journal = "Chichester, UK: Wiley (1997) 366 p",
year = "1997",
SLACcitation = "%%CITATION = INSPIRE-458129;%%"
}
% MSbar mass
@article{Kuhn:2007vp,
author = "Kuhn, Johann H. and Steinhauser, Matthias and Sturm,
Christian",
title = "{Heavy Quark Masses from Sum Rules in Four-Loop
Approximation}",
journal = "Nucl. Phys.",
volume = "B778",
year = "2007",
pages = "192-215",
doi = "10.1016/j.nuclphysb.2007.04.036",
eprint = "hep-ph/0702103",
archivePrefix = "arXiv",
primaryClass = "HEP-PH",
reportNumber = "SFB-CPP-07-03, TTP07-02",
SLACcitation = "%%CITATION = HEP-PH/0702103;%%"
}
% LHAPDF
@article{Buckley:2014ana,
author = "Buckley, Andy and Ferrando, James and Lloyd, Stephen and
Nordström, Karl and Page, Ben and Rüfenacht, Martin and
Schönherr, Marek and Watt, Graeme",
title = "{LHAPDF6: parton density access in the LHC precision
era}",
journal = "Eur. Phys. J.",
volume = "C75",
year = "2015",
number = "3",
pages = "132",
doi = "10.1140/epjc/s10052-015-3318-8",
eprint = "1412.7420",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "GLAS-PPE-2014-05, MCNET-14-29, IPPP-14-111, DCPT-14-222",
SLACcitation = "%%CITATION = ARXIV:1412.7420;%%"
}
% bbH calculations
% LO
@inproceedings{Spira:1998wh,
author = "Spira, Michael",
title = "{Higgs boson production and decay at the Tevatron}",
booktitle = "{Physics at Run II: Workshop on Supersymmetry / Higgs:
Summary Meeting Batavia, Illinois, November 19-21, 1998}",
url = "http://alice.cern.ch/format/showfull?sysnb=0292769",
year = "1998",
eprint = "hep-ph/9810289",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "DESY-98-159",
SLACcitation = "%%CITATION = HEP-PH/9810289;%%"
}
@article{Bonvini:2015pxa,
author = "Bonvini, Marco and Papanastasiou, Andrew S. and Tackmann,
Frank J.",
title = "{Resummation and matching of b-quark mass effects in $
b\overline{b}H $ production}",
journal = "JHEP",
volume = "11",
year = "2015",
pages = "196",
doi = "10.1007/JHEP11(2015)196",
eprint = "1508.03288",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "OUTP-15-16P, DESY-15-137",
SLACcitation = "%%CITATION = ARXIV:1508.03288;%%"
}
@article{Butterworth:2015oua,
author = "Butterworth, Jon and others",
title = "{PDF4LHC recommendations for LHC Run II}",
journal = "J. Phys.",
volume = "G43",
year = "2016",
pages = "023001",
doi = "10.1088/0954-3899/43/2/023001",
eprint = "1510.03865",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "OUTP-15-17P, SMU-HEP-15-12, TIF-UNIMI-2015-14,
LCTS-2015-27, CERN-PH-TH-2015-249",
SLACcitation = "%%CITATION = ARXIV:1510.03865;%%"
}
@article{Lim:2016wjo,
author = "Lim, Matthew and Maltoni, Fabio and Ridolfi, Giovanni and
Ubiali, Maria",
title = "{Anatomy of double heavy-quark initiated processes}",
year = "2016",
eprint = "1605.09411",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CAVENDISH-HEP-16-07, CP3-16-24",
SLACcitation = "%%CITATION = ARXIV:1605.09411;%%"
}
@article{Harland-Lang:2014zoa,
author = "Harland-Lang, L. A. and Martin, A. D. and Motylinski, P.
and Thorne, R. S.",
title = "{Parton distributions in the LHC era: MMHT 2014 PDFs}",
journal = "Eur. Phys. J.",
volume = "C75",
year = "2015",
number = "5",
pages = "204",
doi = "10.1140/epjc/s10052-015-3397-6",
eprint = "1412.3989",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "LCTS-2014-47, IPPP-14-97, DCPT-14-194",
SLACcitation = "%%CITATION = ARXIV:1412.3989;%%"
}
@article{Dulat:2015mca,
author = "Dulat, Sayipjamal and Hou, Tie-Jiun and Gao, Jun and
Guzzi, Marco and Huston, Joey and Nadolsky, Pavel and
Pumplin, Jon and Schmidt, Carl and Stump, Daniel and Yuan,
C. P.",
title = "{New parton distribution functions from a global analysis
of quantum chromodynamics}",
journal = "Phys. Rev.",
volume = "D93",
year = "2016",
number = "3",
pages = "033006",
doi = "10.1103/PhysRevD.93.033006",
eprint = "1506.07443",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
SLACcitation = "%%CITATION = ARXIV:1506.07443;%%"
}
@article{Gao:2013bia,
author = "Gao, Jun and Nadolsky, Pavel",
title = "{A meta-analysis of parton distribution functions}",
journal = "JHEP",
volume = "07",
year = "2014",
pages = "035",
doi = "10.1007/JHEP07(2014)035",
eprint = "1401.0013",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
SLACcitation = "%%CITATION = ARXIV:1401.0013;%%"
}
@article{Carrazza:2015aoa,
author = "Carrazza, Stefano and Forte, Stefano and Kassabov, Zahari
and Latorre, Jose Ignacio and Rojo, Juan",
title = "{An Unbiased Hessian Representation for Monte Carlo
PDFs}",
journal = "Eur. Phys. J.",
volume = "C75",
year = "2015",
number = "8",
pages = "369",
doi = "10.1140/epjc/s10052-015-3590-7",
eprint = "1505.06736",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "TIF-UNIMI-2015-1, OUTP-15-04P",
SLACcitation = "%%CITATION = ARXIV:1505.06736;%%"
}
@article{Watt:2012tq,
author = "Watt, G. and Thorne, R. S.",
title = "{Study of Monte Carlo approach to experimental
uncertainty propagation with MSTW 2008 PDFs}",
journal = "JHEP",
volume = "08",
year = "2012",
pages = "052",
doi = "10.1007/JHEP08(2012)052",
eprint = "1205.4024",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CERN-PH-TH-2012-132, LCTS-2012-11",
SLACcitation = "%%CITATION = ARXIV:1205.4024;%%"
}
@article{Bonvini:2016fgf,
author = "Bonvini, Marco and Papanastasiou, Andrew S. and Tackmann,
Frank J.",
title = "{Matched predictions for the $b\bar{b}H$ cross section at
the 13 TeV LHC}",
year = "2016",
eprint = "1605.01733",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "OUTP-16-08P, CAVENDISH-HEP-16-06, DESY-16-076",
SLACcitation = "%%CITATION = ARXIV:1605.01733;%%"
}
@article{Kunszt:1984ri,
author = "Kunszt, Z.",
title = "{Associated Production of Heavy Higgs Boson with Top
Quarks}",
journal = "Nucl. Phys.",
volume = "B247",
year = "1984",
pages = "339",
doi = "10.1016/0550-3213(84)90553-4",
reportNumber = "BUTP-84/10-BERN",
SLACcitation = "%%CITATION = NUPHA,B247,339;%%"
}
@article{Dicus:1988cx,
author = "Dicus, Duane A. and Willenbrock, Scott",
title = "{Higgs Boson Production from Heavy Quark Fusion}",
journal = "Phys. Rev.",
volume = "D39",
year = "1989",
pages = "751",
doi = "10.1103/PhysRevD.39.751",
reportNumber = "MAD/PH/440",
SLACcitation = "%%CITATION = PHRVA,D39,751;%%"
}
@article{Barnett:1987jw,
author = "Barnett, R. Michael and Haber, Howard E. and Soper,
Davison E.",
title = "{Ultraheavy Particle Production from Heavy Partons at
Hadron Colliders}",
journal = "Nucl. Phys.",
volume = "B306",
year = "1988",
pages = "697",
doi = "10.1016/0550-3213(88)90440-3",
reportNumber = "OITS-365",
SLACcitation = "%%CITATION = NUPHA,B306,697;%%"
}
% NLO 4FS
@article{Dicus:1998hs,
author = "Dicus, D. and Stelzer, T. and Sullivan, Z. and
Willenbrock, S.",
title = "{Higgs boson production in association with bottom quarks
at next-to-leading order}",
journal = "Phys. Rev.",
volume = "D59",
year = "1999",
pages = "094016",
doi = "10.1103/PhysRevD.59.094016",
eprint = "hep-ph/9811492",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "DOE-ER-40757-121, UTEXAS-HEP-98-23, ILL-TH-98-5,
ANL-HEP-PR-98-138, EFI-98-57",
SLACcitation = "%%CITATION = HEP-PH/9811492;%%"
}
% NLO 5FS
@article{Dittmaier:2003ej,
author = "Dittmaier, Stefan and Kramer, 1, Michael and Spira,
Michael",
title = "{Higgs radiation off bottom quarks at the Tevatron and
the CERN LHC}",
journal = "Phys. Rev.",
volume = "D70",
year = "2004",
pages = "074010",
doi = "10.1103/PhysRevD.70.074010",
eprint = "hep-ph/0309204",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "EDINBURGH-2003-13, MPP-2003-49, PSI-PR-03-14",
SLACcitation = "%%CITATION = HEP-PH/0309204;%%"
}
@article{Dittmaier:2006cz,
author = "Dittmaier, Stefan and Kramer, 1, Michael and Muck,
Alexander and Schluter, Tobias",
title = "{MSSM Higgs-boson production in bottom-quark fusion:
Electroweak radiative corrections}",
journal = "JHEP",
volume = "03",
year = "2007",
pages = "114",
doi = "10.1088/1126-6708/2007/03/114",
eprint = "hep-ph/0611353",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "MPP-2006-149, PITHA-06-11",
SLACcitation = "%%CITATION = HEP-PH/0611353;%%"
}
% NNLO 5FS
@article{Harlander:2003ai,
author = "Harlander, Robert V. and Kilgore, William B.",
title = "{Higgs boson production in bottom quark fusion at
next-to-next-to leading order}",
journal = "Phys. Rev.",
volume = "D68",
year = "2003",
pages = "013001",
doi = "10.1103/PhysRevD.68.013001",
eprint = "hep-ph/0304035",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "BNL-HET-03-4, CERN-TH-2003-067",
SLACcitation = "%%CITATION = HEP-PH/0304035;%%"
}
@article{Carena:2002es,
author = "Carena, Marcela and Haber, Howard E.",
title = "{Higgs boson theory and phenomenology}",
journal = "Prog. Part. Nucl. Phys.",
volume = "50",
year = "2003",
pages = "63-152",
doi = "10.1016/S0146-6410(02)00177-1",
eprint = "hep-ph/0208209",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "FERMILAB-PUB-02-114-T, SCIPP-02-07",
SLACcitation = "%%CITATION = HEP-PH/0208209;%%"
}
@article{Dawson:2003kb,
author = "Dawson, S. and Jackson, C. B. and Reina, L. and
Wackeroth, D.",
title = "{Exclusive Higgs boson production with bottom quarks at
hadron colliders}",
journal = "Phys. Rev.",
volume = "D69",
year = "2004",
pages = "074027",
doi = "10.1103/PhysRevD.69.074027",
eprint = "hep-ph/0311067",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "BNL-HET-03-17, FSU-HEP-2003-1015, UB-HET-03-05",
SLACcitation = "%%CITATION = HEP-PH/0311067;%%"
}
@article{Reina:2001sf,
author = "Reina, L. and Dawson, S.",
title = "{Next-to-leading order results for t anti-t h production
at the Tevatron}",
journal = "Phys. Rev. Lett.",
volume = "87",
year = "2001",
pages = "201804",
doi = "10.1103/PhysRevLett.87.201804",
eprint = "hep-ph/0107101",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "FSU-HEP-2001-0601, BNL-HET-01-20",
SLACcitation = "%%CITATION = HEP-PH/0107101;%%"
}
@article{Reina:2001bc,
author = "Reina, L. and Dawson, S. and Wackeroth, D.",
title = "{QCD corrections to associated t anti-t h production at
the Tevatron}",
journal = "Phys. Rev.",
volume = "D65",
year = "2002",
pages = "053017",
doi = "10.1103/PhysRevD.65.053017",
eprint = "hep-ph/0109066",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "FSU-HEP-2001-0602, BNL-HET-01-19, UR-1639",
SLACcitation = "%%CITATION = HEP-PH/0109066;%%"
}
@article{Plehn:2002vy,
author = "Plehn, Tilman",
title = "{Charged Higgs boson production in bottom gluon fusion}",
journal = "Phys. Rev.",
volume = "D67",
year = "2003",
pages = "014018",
doi = "10.1103/PhysRevD.67.014018",
eprint = "hep-ph/0206121",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "MADPH-02-1275",
SLACcitation = "%%CITATION = HEP-PH/0206121;%%"
}
@article{Dawson:2002tg,
author = "Dawson, S. and Orr, L. H. and Reina, L. and Wackeroth,
D.",
title = "{Associated top quark Higgs boson production at the LHC}",
journal = "Phys. Rev.",
volume = "D67",
year = "2003",
pages = "071503",
doi = "10.1103/PhysRevD.67.071503",
eprint = "hep-ph/0211438",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "BNL-HET-02-27A, FSU-HEP-2002-1115, UB-HET-02-09",
SLACcitation = "%%CITATION = HEP-PH/0211438;%%"
}
@article{Beenakker:2002nc,
author = "Beenakker, W. and Dittmaier, S. and Kramer, M. and
Plumper, B. and Spira, M. and Zerwas, P. M.",
title = "{NLO QCD corrections to t anti-t H production in hadron
collisions}",
journal = "Nucl. Phys.",
volume = "B653",
year = "2003",
pages = "151-203",
doi = "10.1016/S0550-3213(03)00044-0",
eprint = "hep-ph/0211352",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "DESY-02-177, EDINBURGH-2002-18, MPI-PHT-2002-70,
PSI-PR-02-22",
SLACcitation = "%%CITATION = HEP-PH/0211352;%%"
}
@article{Maltoni:2003pn,
author = "Maltoni, F. and Sullivan, Z. and Willenbrock, S.",
title = "{Higgs-boson production via bottom-quark fusion}",
journal = "Phys. Rev.",
volume = "D67",
year = "2003",
pages = "093005",
doi = "10.1103/PhysRevD.67.093005",
eprint = "hep-ph/0301033",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "ILL-TH-02-12, RM3-TH-02-21, FERMILAB-PUB-02-341-T",
SLACcitation = "%%CITATION = HEP-PH/0301033;%%"
}
@inproceedings{Rainwater:2002hm,
author = "Rainwater, David L. and Spira, Michael and Zeppenfeld,
Dieter",
title = "{Higgs boson production at hadron colliders: Signal and
background processes}",
booktitle = "{Physics at TeV colliders. Proceedings, Euro Summer
School, Les Houches, France, May 21-June 1, 2001}",
url = "http://lss.fnal.gov/cgi-bin/find_paper.pl?conf-02-412",
year = "2002",
eprint = "hep-ph/0203187",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "FERMILAB-CONF-02-412, MAD-PH-02-1260",
SLACcitation = "%%CITATION = HEP-PH/0203187;%%"
}
% FONLL
@article{Cacciari:1998it,
author = "Cacciari, Matteo and Greco, Mario and Nason, Paolo",
title = "{The P(T) spectrum in heavy flavor hadroproduction}",
journal = "JHEP",
volume = "05",
year = "1998",
pages = "007",
doi = "10.1088/1126-6708/1998/05/007",
eprint = "hep-ph/9803400",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CERN-TH-98-77, LPTHE-ORSAY-98-11, IFUM-613-FT,
LNF-98-008-P",
SLACcitation = "%%CITATION = HEP-PH/9803400;%%"
}
@article{Forte:2010ta,
author = "Forte, Stefano and Laenen, Eric and Nason, Paolo and
Rojo, Juan",
title = "{Heavy quarks in deep-inelastic scattering}",
journal = "Nucl. Phys.",
volume = "B834",
year = "2010",
pages = "116-162",
doi = "10.1016/j.nuclphysb.2010.03.014",
eprint = "1001.2312",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "IFUM-949-FT, NIKHEF-2010-001, ITP-UU-10-03, ITFA-2010-01",
SLACcitation = "%%CITATION = ARXIV:1001.2312;%%"
}
@article{Forte:2015hba,
author = "Forte, Stefano and Napoletano, Davide and Ubiali, Maria",
title = "{Higgs production in bottom-quark fusion in a matched
scheme}",
journal = "Phys. Lett.",
volume = "B751",
year = "2015",
pages = "331-337",
doi = "10.1016/j.physletb.2015.10.051",
eprint = "1508.01529",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "TIF-UNIMI-2015-12, CAVENDISH-HEP-15-06",
SLACcitation = "%%CITATION = ARXIV:1508.01529;%%"
}
+@article{Forte:2016sja,
+ author = "Forte, Stefano and Napoletano, Davide and Ubiali, Maria",
+ title = "{Higgs production in bottom-quark fusion: matching beyond
+ leading order}",
+ journal = "Phys. Lett.",
+ volume = "B763",
+ year = "2016",
+ pages = "190-196",
+ doi = "10.1016/j.physletb.2016.10.040",
+ eprint = "1607.00389",
+ archivePrefix = "arXiv",
+ primaryClass = "hep-ph",
+ reportNumber = "TIF-UNIMI-2016-6, IPPP-16-61, CAVENDISH-HEP-16-11",
+ SLACcitation = "%%CITATION = ARXIV:1607.00389;%%"
+}
% Other schemes
@article{Olness:1987ep,
author = "Olness, Fredrick I. and Tung, Wu-Ki",
title = "{When Is a Heavy Quark Not a Parton? Charged Higgs
Production and Heavy Quark Mass Effects in the QCD Based
Parton Model}",
booktitle = "{In *Lake Louise 1988, Proceedings, Quantum
chromodynamics* 515-525 and Fermilab Batavia -
FERMILAB-Pub-87-221 (87,rec.Jan.88) 20 p.}",
journal = "Nucl. Phys.",
volume = "B308",
year = "1988",
pages = "813",
doi = "10.1016/0550-3213(88)90129-0",
reportNumber = "IIT-TH-87-17, FERMILAB-PUB-87-221-T",
SLACcitation = "%%CITATION = NUPHA,B308,813;%%"
}
@article{Aivazis:1993pi,
author = "Aivazis, M. A. G. and Collins, John C. and Olness,
Fredrick I. and Tung, Wu-Ki",
title = "{Leptoproduction of heavy quarks. 2. A Unified QCD
formulation of charged and neutral current processes from
fixed target to collider energies}",
journal = "Phys. Rev.",
volume = "D50",
year = "1994",
pages = "3102-3118",
doi = "10.1103/PhysRevD.50.3102",
eprint = "hep-ph/9312319",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "SMU-HEP-93-17, MSUHEP-93-17, PSU-TH-138",
SLACcitation = "%%CITATION = HEP-PH/9312319;%%"
}
@article{Thorne:1997ga,
author = "Thorne, R. S. and Roberts, R. G.",
title = "{An Ordered analysis of heavy flavor production in deep
inelastic scattering}",
journal = "Phys. Rev.",
volume = "D57",
year = "1998",
pages = "6871-6898",
doi = "10.1103/PhysRevD.57.6871",
eprint = "hep-ph/9709442",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "RAL-TR-97-049",
SLACcitation = "%%CITATION = HEP-PH/9709442;%%"
}
@article{Kramer:2000hn,
author = "Kramer, 1, Michael and Olness, Fredrick I. and Soper,
Davison E.",
title = "{Treatment of heavy quarks in deeply inelastic
scattering}",
journal = "Phys. Rev.",
volume = "D62",
year = "2000",
pages = "096007",
doi = "10.1103/PhysRevD.62.096007",
eprint = "hep-ph/0003035",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "EDINBURGH-2000-02",
SLACcitation = "%%CITATION = HEP-PH/0003035;%%"
}
@article{Buza:1995ie,
author = "Buza, M. and Matiounine, Y. and Smith, J. and Migneron,
R. and van Neerven, W. L.",
title = "{Heavy quark coefficient functions at asymptotic values
Q**2 >> m**2}",
journal = "Nucl. Phys.",
volume = "B472",
year = "1996",
pages = "611-658",
doi = "10.1016/0550-3213(96)00228-3",
eprint = "hep-ph/9601302",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "NIKHEF-95-070, ITP-SB-95-59, INLO-PUB-22-95",
SLACcitation = "%%CITATION = HEP-PH/9601302;%%"
}
@article{Buza:1996wv,
author = "Buza, M. and Matiounine, Y. and Smith, J. and van
Neerven, W. L.",
title = "{Charm electroproduction viewed in the variable flavor
number scheme versus fixed order perturbation theory}",
journal = "Eur. Phys. J.",
volume = "C1",
year = "1998",
pages = "301-320",
doi = "10.1007/BF01245820",
eprint = "hep-ph/9612398",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "NIKHEF-96-027, ITP-SB-96-66, DESY-96-258, INLO-PUB-22-96",
SLACcitation = "%%CITATION = HEP-PH/9612398;%%"
}
@article{Chuvakin:1999nx,
author = "Chuvakin, A. and Smith, J. and van Neerven, W. L.",
title = "{Comparison between variable flavor number schemes for
charm quark electroproduction}",
journal = "Phys. Rev.",
volume = "D61",
year = "2000",
pages = "096004",
doi = "10.1103/PhysRevD.61.096004",
eprint = "hep-ph/9910250",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "YITP-SB-99-15, INLO-PUB-12-99",
SLACcitation = "%%CITATION = HEP-PH/9910250;%%"
}
@article{Campbell:2002zm,
author = "Campbell, John M. and Ellis, R. Keith and Maltoni, F. and
Willenbrock, S.",
title = "{Higgs-Boson production in association with a single
bottom quark}",
journal = "Phys. Rev.",
volume = "D67",
year = "2003",
pages = "095002",
doi = "10.1103/PhysRevD.67.095002",
eprint = "hep-ph/0204093",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "ANL-HEP-PR-02-027, FERMILAB-PUB-02-062-T, ILL-TH-02-3",
SLACcitation = "%%CITATION = HEP-PH/0204093;%%"
}
@article{Boos:2003yi,
author = "Boos, Eduard and Plehn, Tilman",
title = "{Higgs boson production induced by bottom quarks}",
journal = "Phys. Rev.",
volume = "D69",
year = "2004",
pages = "094005",
doi = "10.1103/PhysRevD.69.094005",
eprint = "hep-ph/0304034",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CERN-TH-2003-068",
SLACcitation = "%%CITATION = HEP-PH/0304034;%%"
}
@inproceedings{Buttar:2006zd,
author = "Buttar, C. and others",
title = "{Sect. 24 in: Les houches physics at TeV colliders 2005, standard
model and Higgs working group: Summary report}",
booktitle = "{Physics at TeV colliders. Proceedings, Workshop, Les
Houches, France, May 2-20, 2005}",
year = "2006",
eprint = "hep-ph/0604120",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
SLACcitation = "%%CITATION = HEP-PH/0604120;%%"
}
@article{Cascioli:2011va,
author = "Cascioli, Fabio and Maierhofer, Philipp and Pozzorini,
Stefano",
title = "{Scattering Amplitudes with Open Loops}",
journal = "Phys. Rev. Lett.",
volume = "108",
year = "2012",
pages = "111601",
doi = "10.1103/PhysRevLett.108.111601",
eprint = "1111.5206",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "ZU-TH-23-11, LPN11-66",
SLACcitation = "%%CITATION = ARXIV:1111.5206;%%"
}
@article{Kilgore:2002sk,
author = "Kilgore, William B.",
title = "{Higgs boson production at hadron colliders}",
booktitle = "{High energy physics. Proceedings, 31st International
Conference, ICHEP 2002, Amsterdam, Netherlands, July
25-31, 2002}",
year = "2002",
pages = "282-284",
doi = "10.1016/S0920-5632(03)90546-9",
note = "[Nucl. Phys. Proc. Suppl.117,282(2003)]",
eprint = "hep-ph/0208143",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "BNL-HET-02-18",
SLACcitation = "%%CITATION = HEP-PH/0208143;%%"
}
@article{Gleisberg:2008ta,
author = "Gleisberg, T. and Hoeche, Stefan. and Krauss, F. and
Schonherr, M. and Schumann, S. and Siegert, F. and Winter,
J.",
title = "{Event generation with SHERPA 1.1}",
journal = "JHEP",
volume = "02",
year = "2009",
pages = "007",
doi = "10.1088/1126-6708/2009/02/007",
eprint = "0811.4622",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "FERMILAB-PUB-08-477-T, SLAC-PUB-13420, ZU-TH-17-08,
DCPT-08-138, IPPP-08-69, EDINBURGH-2008-30, MCNET-08-14",
SLACcitation = "%%CITATION = ARXIV:0811.4622;%%"
}
+@article{Krauss:2017wmx,
+ author = "Krauss, Frank and Napoletano, Davide",
+ title = "{Towards a fully massive five-flavour scheme}",
+ year = "2017",
+ eprint = "1712.06832",
+ archivePrefix = "arXiv",
+ primaryClass = "hep-ph",
+ SLACcitation = "%%CITATION = ARXIV:1712.06832;%%"
+}
+
+
%Santander
@article{Harlander:2011aa,
author = "Harlander, Robert and Kramer, Michael and Schumacher,
Markus",
title = "{Bottom-quark associated Higgs-boson production:
reconciling the four- and five-flavour scheme approach}",
year = "2011",
eprint = "1112.3478",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CERN-PH-TH-2011-134, FR-PHENO-2011-009, TTK-11-17,
WUB-11-04",
SLACcitation = "%%CITATION = ARXIV:1112.3478;%%"
}
@Unpublished{LHhq,
author = "Rojo, J. and others",
title ="{Chapter 22 in: J.~R.~Andersen et al., The SM and NLO multileg working group: Summary report}",
year = "2010",
note = "arXiv:1003.1241",
primaryClass = "hep-ph",
SLACcitation = "%%CITATION = 1003.1241;%%"
}
% choice of scale
@article{Maltoni:2012pa,
author = "Maltoni, Fabio and Ridolfi, Giovanni and Ubiali, Maria",
title = "{b-initiated processes at the LHC: a reappraisal}",
journal = "JHEP",
volume = "07",
year = "2012",
pages = "022",
doi = "10.1007/JHEP04(2013)095, 10.1007/JHEP07(2012)022",
note = "[Erratum: JHEP04,095(2013)]",
eprint = "1203.6393",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CP3-12-15, TTK-12-11",
SLACcitation = "%%CITATION = ARXIV:1203.6393;%%"
}
@article{Ubiali:2014cva,
author = "Ubiali, Maria",
title = "{Are bottom PDFs needed at the LHC?}",
booktitle = "{Proceedings, 22nd International Workshop on
Deep-Inelastic Scattering and Related Subjects (DIS
2014)}",
journal = "PoS",
volume = "DIS2014",
year = "2014",
pages = "037",
SLACcitation = "%%CITATION = POSCI,DIS2014,037;%%"
}
% NNPDF
@article{Ball:2014uwa,
author = "Ball, Richard D. and others",
title = "{Parton distributions for the LHC Run II}",
collaboration = "NNPDF",
journal = "JHEP",
volume = "04",
year = "2015",
pages = "040",
doi = "10.1007/JHEP04(2015)040",
eprint = "1410.8849",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "EDINBURGH-2014-15, IFUM-1034-FT, CERN-PH-TH-2013-253,
OUTP-14-11P, CAVENDISH-HEP-14-11",
SLACcitation = "%%CITATION = ARXIV:1410.8849;%%"
}
% EXPERIMENTS
@article{Aad:2015zhl,
author = "Aad, Georges and others",
title = "{Combined Measurement of the Higgs Boson Mass in $pp$
Collisions at $\sqrt{s}=7$ and 8 TeV with the ATLAS and
CMS Experiments}",
collaboration = "ATLAS, CMS",
journal = "Phys. Rev. Lett.",
volume = "114",
year = "2015",
pages = "191803",
doi = "10.1103/PhysRevLett.114.191803",
eprint = "1503.07589",
archivePrefix = "arXiv",
primaryClass = "hep-ex",
reportNumber = "ATLAS-HIGG-2014-14, CMS-HIG-14-042, CERN-PH-EP-2015-075",
SLACcitation = "%%CITATION = ARXIV:1503.07589;%%"
}
@article{Wiesemann:2014ioa,
author = "Wiesemann, M. and Frederix, R. and Frixione, S. and
Hirschi, V. and Maltoni, F. and Torrielli, P.",
title = "{Higgs production in association with bottom quarks}",
journal = "JHEP",
volume = "02",
year = "2015",
pages = "132",
doi = "10.1007/JHEP02(2015)132",
eprint = "1409.5301",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CERN-PH-TH-2014-182, CP3-14-64, LPN14-114, MCNET-14-20,
ZU-TH-33-14",
SLACcitation = "%%CITATION = ARXIV:1409.5301;%%"
}
% Mitov et al
@article{Bertone:2017djs,
author = "Bertone, Valerio and Glazov, Alexandre and Mitov,
Alexander and Papanastasiou, Andrew and Ubiali, Maria",
title = "{Heavy-flavor parton distributions without heavy-flavor
matching prescriptions}",
year = "2017",
eprint = "1711.03355",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CAVENDISH-HEP-17-12, DAMTP-2017-40, DESY-17-168,
NIKHEF-2017-052, NSF-ITP-17-139",
SLACcitation = "%%CITATION = ARXIV:1711.03355;%%"
}
% Z production NNLO 5F
@article{Hamberg:1990np,
author = "Hamberg, R. and van Neerven, W. L. and Matsuura, T.",
title = "{A complete calculation of the order $\alpha-s^{2}$
correction to the Drell-Yan $K$ factor}",
journal = "Nucl. Phys.",
volume = "B359",
year = "1991",
pages = "343-405",
doi = "10.1016/S0550-3213(02)00814-3,
10.1016/0550-3213(91)90064-5",
note = "[Erratum: Nucl. Phys.B644,403(2002)]",
reportNumber = "DESY-90-129",
SLACcitation = "%%CITATION = NUPHA,B359,343;%%"
}
@article{Rijken:1995gi,
author = "Rijken, P. J. and van Neerven, W. L.",
title = "{Heavy flavor contributions to the Drell-Yan
cross-section}",
journal = "Phys. Rev.",
volume = "D52",
year = "1995",
pages = "149-161",
doi = "10.1103/PhysRevD.52.149",
eprint = "hep-ph/9501373",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "INLO-PUB-1-95",
SLACcitation = "%%CITATION = HEP-PH/9501373;%%"
}
@article{Stelzer:1997ns,
author = "Stelzer, T. and Sullivan, Z. and Willenbrock, S.",
title = "{Single top quark production via $W$ - gluon fusion at
next-to-leading order}",
journal = "Phys. Rev.",
volume = "D56",
year = "1997",
pages = "5919-5927",
doi = "10.1103/PhysRevD.56.5919",
eprint = "hep-ph/9705398",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
SLACcitation = "%%CITATION = HEP-PH/9705398;%%"
}
@article{Maltoni:2005wd,
author = "Maltoni, Fabio and McElmurry, Thomas and Willenbrock,
Scott",
title = "{Inclusive production of a Higgs or $Z$ boson in
association with heavy quarks}",
journal = "Phys. Rev.",
volume = "D72",
year = "2005",
pages = "074024",
doi = "10.1103/PhysRevD.72.074024",
eprint = "hep-ph/0505014",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CERN-PH-TH-2005-076",
SLACcitation = "%%CITATION = HEP-PH/0505014;%%"
}
% 4F calculations
@article{Campbell:2000bg,
author = "Campbell, John M. and Ellis, R. Keith",
title = "{Radiative corrections to Z b anti-b production}",
journal = "Phys. Rev.",
volume = "D62",
year = "2000",
pages = "114012",
doi = "10.1103/PhysRevD.62.114012",
eprint = "hep-ph/0006304",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "FERMILAB-PUB-00-145-T",
SLACcitation = "%%CITATION = HEP-PH/0006304;%%"
}
@article{FebresCordero:2008ci,
author = "Febres Cordero, F. and Reina, L. and Wackeroth, D.",
title = "{NLO QCD corrections to $Z b \bar{b}$ production with
massive bottom quarks at the Fermilab Tevatron}",
journal = "Phys. Rev.",
volume = "D78",
year = "2008",
pages = "074014",
doi = "10.1103/PhysRevD.78.074014",
eprint = "0806.0808",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "FSU-HEP-2008-0531",
SLACcitation = "%%CITATION = ARXIV:0806.0808;%%"
}
@article{Cordero:2009kv,
author = "Febres Cordero, Fernando and Reina, L. and Wackeroth, D.",
title = "{W- and Z-boson production with a massive bottom-quark
pair at the Large Hadron Collider}",
journal = "Phys. Rev.",
volume = "D80",
year = "2009",
pages = "034015",
doi = "10.1103/PhysRevD.80.034015",
eprint = "0906.1923",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "FSU-HEP-2009-0314, UCLA-09-TEP-49",
SLACcitation = "%%CITATION = ARXIV:0906.1923;%%"
}
@article{Frederix:2011qg,
author = "Frederix, Rikkert and Frixione, Stefano and Hirschi,
Valentin and Maltoni, Fabio and Pittau, Roberto and
Torrielli, Paolo",
title = "{W and $Z/\gamma*$ boson production in association with a
bottom-antibottom pair}",
journal = "JHEP",
volume = "09",
year = "2011",
pages = "061",
doi = "10.1007/JHEP09(2011)061",
eprint = "1106.6019",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CERN-PH-TH-2011-147, CP3-11-20, NSF-KITP-11-114,
ZH-TH-13-11",
SLACcitation = "%%CITATION = ARXIV:1106.6019;%%"
}
\ No newline at end of file
Index: trunk/papers/bbz/bbz_fonll.tex
===================================================================
--- trunk/papers/bbz/bbz_fonll.tex (revision 48)
+++ trunk/papers/bbz/bbz_fonll.tex (revision 49)
@@ -1,802 +1,561 @@
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\pdfoutput=1
\usepackage{graphicx}
\usepackage{epsfig,cite}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{dsfont}
\usepackage{multirow}
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\usepackage{subfigure,amstext,alltt,setspace}
\usepackage{amsbsy}
\usepackage{comment}
\usepackage{fullpage}
\usepackage{array}
\usepackage{booktabs,multirow,tabularx}
\usepackage{hyperref}
\usepackage{slashed}
\usepackage{url}
\interfootnotelinepenalty=10000
\textwidth=17.0cm \textheight=22.0cm
\topmargin 0cm \oddsidemargin 0cm
\setlength{\unitlength}{1mm}
\newcommand{\vfs}{{\abbrev VFS}}
\newcommand{\ffs}{{\abbrev FFS}}
\newcommand{\code}{\tt}
\newcommand{\abbrev}{\small}
\newcommand{\ep}{\epsilon}
\newcommand{\vep}{\varepsilon}
\newcommand{\api}{\frac{\alpha_s}{\pi}}
\newcommand{\apib}{\frac{\alpha_s^\bare}{\pi}}
\newcommand{\eqn}[1]{Eq.\,(\ref{#1})}
\newcommand{\fig}[1]{Fig.\,\ref{#1}}
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\newcommand{\refs}[1]{Refs.\,\cite{#1}}
\newcommand{\dd}{{\rm d}}
\newcommand{\ddoverdd}[1]{\frac{\dd}{\dd #1}}
\newcommand{\doverd}[1]{\frac{\partial}{\partial #1}}
\newcommand{\order}[1]{{\cal O}(#1)}
\newcommand{\bld}[1]{\boldmath{$#1$}}
\newcommand{\bsym}{\boldsymbol}
\renewcommand{\Re}{{\rm Re}}
\renewcommand{\Im}{{\rm Im}}
\newcommand{\cf}{C_{\rm F}}
\newcommand{\ca}{C_{\rm A}}
\newcommand{\tr}{T}
\newcommand{\lht}{l_{\higgs t}}
\newcommand{\Lx}{\left(}
\newcommand{\Rx}{\right)}
\newcommand{\LB}{\left[}
\newcommand{\RB}{\right]}
\newcommand{\Li}[1]{{\mathop{\rm Li}_{#1}\nolimits}}
\newcommand{\Di}[1]{{\cal D}_{#1}}
\newcommand{\lmut}{l_{\mu t}}
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\newcommand{\nnlo}{{\abbrev NNLO}}
\newcommand{\gfermi}{G_{\rm F}}
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\newcommand{\ptb}{p_{{\rm T}b}}
\newcommand{\muR}{\mu_R}
\newcommand{\muF}{\mu_F}
%----------------------------------------------------------------------
\newcommand{\sprod}[2]{#1\!\cdot\!#2}
\newcommand{\matel}[1]{\langle #1\rangle}
\newcommand{\msbar}{\overline{\mbox{\small MS}}}
\newcommand{\mmsbar}{\overline{\mbox{\scriptsize MS}}}
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\newcommand{\phiggs}{A}
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\newcommand{\mahiggs}{M_A}
\newcommand{\dglap}{{\abbrev DGLAP}}
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\allowdisplaybreaks[1]
\begin{document}
\begin{flushright}
TIF-UNIMI-2016-6\\
IPPP/16/61 \\
DAMTP-2018-xx
\end{flushright}
\vspace*{.2cm}
\begin{center}
{\Large \bf{$Z$ boson production in bottom-quark fusion:\\
study of $b$-mass effects beyond leading order}}
\end{center}
\vspace*{.7cm}
\begin{center}
Stefano Forte$^{1}$, Davide Napoletano$^2$ and Maria Ubiali$^{3}$
\vspace*{.2cm}
\noindent
{\it
$^1$ Tif Lab, Dipartimento di Fisica, Universit\`a di Milano and\\
INFN, Sezione di Milano,
Via Celoria 16, I-20133 Milano, Italy\\
$^2$ IPhT, CEA Saclay, CNRS UMR 3681,\\ F-91191, Gif-Sur-Yvette, France\\
$^3$ DAMTP, University of Cambridge,\\ Wilberforce Road, Cambridge, CB3 0WA, UK\\}
\vspace*{3cm}
%\begin{center}
{\bf Abstract}
\end{center}
\noindent
We compute the total cross-section for $Z$ boson production in
bottom-quark fusion using the so-called FONLL method for the matching
of a scheme in which the $b$-quark is
treated as a massless parton to that in which it is treated as a
massive final-state particle. The next-to-next-to-leading-log five-flavor
scheme result is combined with the next-to-leading-order $\order{\alpha_s^3}$
four-flavor scheme computation. Results are compared to those previously
obtained in the case of bottom-initiated Higgs production.
Differences with respect to a recent prescription based on a massless
-calculation with increased bottom-qaurk thresholds are discussed.
+calculation with increased bottom-quark thresholds are discussed.
\pagebreak
%\tableofcontents
The production of a $Z$ boson is one of the main standard candles at the LHC,
-its cross section being both measured and computed tp a very high degree of precision.
+its cross section being both measured and computed to a very high degree of precision.
[...]\\
$Z$-boson production at the leading order in perturbation theory involves quark-antiquark
fusion. Contrarily to the case of the Higgs boson production, the $Z$-boson coupling
to light quarks is not
suppressed by the small Yukawa coupling, thus the contribution of heavy flavor
fusion to the total $Z$ production cross section is of ${\cal O}(10\%)$. In particular
bottom-initiated contribution accounts only to ${\cal O}(4\%)$ of the total cross section.
However, in the high-precision regime required at LHC Run-II, it is important
to compute such contribution to the highest possible accuracy.
[...]\\
Like any process involving bottom quarks at the matrix-element level,
the bottom-initiated $Z$-boson production may be computed using two different
-factorization schemes, often dubbed four- and five-flavour schemes for short. In the
-four-flavour scheme (4FS), the bottom quark is treated as a massive
+factorization schemes, often dubbed four- and five-flavor schemes for short. In the
+four-flavor scheme (4FS), the bottom quark is treated as a massive
object, which is not endowed with a parton distribution (PDF), and it
decouples from QCD perturbative evolution. The latter is performed only
including the four lightest flavors and the gluon in the DGLAP
equations, and likewise it decouples from the running of $\alpha_s$ so
that $n_f=4$ in the computation of the QCD $\beta$ function.
-In the five-flavour scheme (5FS), instead, the bottom quark is treated on
+In the five-flavor scheme (5FS), instead, the bottom quark is treated on
the same footing as other quark flavors, there is a $b$ PDF, and
$n_f=5$ in both the DGLAP and renormalization-group equations.
-
-Each scheme presents advantages and disadvantages depending on the
-scale of the calculation: for high enough scales, mass effects become negligible,
-collinear logarithms related to $b$-quark radiation are large and must be
-resummed, thus the 5FS is always more accurate. On the
-other hand, very close to the production threshold mass effects are
-important while collinear logs are not large, and the 4FS is more accurate.
+%
+As it has been analysed in many recent papers
+~\cite{Maltoni:2012pa,Forte:2015hba,Lim:2016wjo,Forte:2016sja,Bertone:2017djs,Krauss:2017wmx},
+each scheme presents advantages and disadvantages depending on the
+scale of the calculation.
+%for high enough scales, mass effects become negligible,
+%collinear logarithms related to $b$-quark radiation are large and must be
+%resummed, thus the 5FS is always more accurate. On the
+%other hand, very close to the production threshold mass effects are
+%important while collinear logs are not large, and the 4FS is more accurate.
%In principle, a computation performed at high
%enough perturbative order in the 4FS will reproduce
%the 5FS result, while this is not the case for a
%5FS computation, in which $b$-mass effects are never
%included.
%In practice, however, for Higgs production in bottom fusion
%the leading-order production diagram, which is $\order{\alpha_s^0}$ (parton
%model) in the 5FS, is $\order{\alpha_s^2}$ in the
%4FS, so one must go to very high order indeed in the
%5FS computation in order to reproduce 4FS results.
In the 5FS, the $Z$-production cross section has been known up to
NNLO for almost three decades~\cite{Hamberg:1990np} and
-the heavy-quark initiated contribution has been studied in
+the heavy-quark initiated contribution has been analysed in
several works~\cite{Rijken:1995gi,Stelzer:1997ns,Maltoni:2005wd}.
%
The $Zb\bar{b}$ production cross section was originally computed
(neglecting the $b$-quark mass) in Ref.~\cite{Campbell:2000bg}
for exclusive 2-jet final states.
The effect of a non-zero $b$-quark mass was
considered in later works~\cite{FebresCordero:2008ci,Cordero:2009kv}
where the total cross section was also given. More recently,
in Ref.~\cite{Frederix:2011qg} leptonic decays of the $Z$ boson have taken into account,
together with the full correlation of the final state leptons and
the parton shower and hadronisation effects.
The best available 4FS and 5FS computations of the total cross section for
-$b$-initiated $Z$-boson were compared in ~\cite{Maltoni:2012pa,Lim:2016wjo}.
+$b$-initiated $Z$-boson were compared in ~\cite{Lim:2016wjo,Bertone:2017djs}.
For factorization and renormalization scales set to $m_Z$ the 5FS prediction
-at NNLO exceeds the 4FS NLO preduicruib by almost 30\%, while the difference is
+at NNLO exceeds the 4FS NLO prediction by almost 30\%, while the difference is
reduced at lower values of the scales.
It was also found that the characteristic
scale for this process is necessarily higher than the $b$
production threshold, but lower than the $Z$ mass
itself~\cite{Maltoni:2012pa,Lim:2016wjo}, being about $m_Z/3$. At that
-scale the 5FS prediction is lower than the 4FS one by about 10\%, which
-is due to the size of the resummed logarithms in the 5FS computation.
+scale the 5FS prediction is lower than the 4FS one by about 10\%. The difference
+mostly accounts for the resummed collinear logarithms in the 5FS computation, which
+are found to be dominant for this process.
In previous works~\cite{Forte:2015hba,Forte:2016sja} we have implemented
the so-called FONLL matched scheme, first proposed in
Ref.~\cite{Cacciari:1998it} for $b$ production and extended in
Ref.~\cite{Forte:2010ta} to deep-inelastic scattering, to the bottom-fusion
initiated Higgs production.
With the FONLL method, one can combine the 4FS and 5FS computations performed
at any given perturbative accuracy, retaining the accuracy of both,
i.e. in such a way that from the point of view of any of the two
computations that enter the combined results the terms which are added
-are subleading.
+are sub-leading.
In Ref.~\cite{Forte:2015hba}, this method was used to combine the
next-to-next-to-leading order 5FS result with the
leading-order 4FS computation --- this particular
combination was called FONLL-A, corresponding to the lowest order at
which the 4FS and 5FS results have a non-vanishing
overlap. In Ref.~\cite{Forte:2016sja} an extra perturbative order to the 4FS result
is included in comparison to
FONLL-A, thereby constructing the FONLL-B matched result (according to
the nomenclature introduced in Ref.~\cite{Forte:2015hba}).
This amounts to combining both the
4FS and 5FS computations at the highest order available for both.
The basic idea of the FONLL method was explained in great detail in
Refs.~\cite{Forte:2015hba,Forte:2016sja} and there is no need to
replicate the discussion. On the other hand, in the previous
works, the heavy flavor matching point for the $b$ PDF $\mu_b$ was
always taken to be equal to the mass of the bottom quark $m_b$.
As it is outlined in Ref.~\cite{Bertone:2017djs}, while decoupling implies that
the matching point $\mu_b$ has to be of the order of the mass $m_b$, this
does not mean that $\mu_b$ has to be set equal to $m_b$. A wide range
of $\mu_b = k\cdot m_b$ was explored in Ref.~\cite{Bertone:2017djs} and
it was argued that, increasing $\mu_b$ up to ten times the mass of the
bottom PDF, the difference between the 4FS and 5FS calculation was
substantially reduced. To compare our findings to the one of the
above-mentioned reference, it is useful to re-write the key equations
-of the FONLL formalism by distinguishing the physical mass of the $b$ PDF
-from the matching point $\mu_b$.
-
-In the FONLL method the two
-computations which are combined are performed in different
-renormalization and factorization schemes, thus $\alpha_s$ and PDFs in the 4FS
-are re-expressed in terms of their 5FS counterparts, so that
-one single $\alpha_s$ and set of PDFs is used everywhere.
-The 4FS cross section can be written as
-\begin{equation}
- \label{massive:1}
- \sigma^{(4)}=\int_{\tau_H}^{1} \frac{dx}{x}\int_{\frac{\tau_H}{x}}^{1} \frac{dy}{y^2}\sum_{ij=q,g}f_{i}^{(5)}(x,Q^2)f_j^{(5)}\left(\frac{\tau_H}{x y},Q^2\right)B_{ij}\left(y,L,\alpha_s^{(5)}(Q^2),\frac{Q^2}{m_b^2}\right),
-\end{equation}
-where $f_{i}^{(5)}$ and $\alpha_s^{(5)}$ are 5FS PDFs
-and $\alpha_s$, $L=\log(Q^2/\mu_b^2)$ and the coefficients
-\begin{equation}
- \label{massive:exp}
- B_{ij}\left(y,L,\alpha_s^{(5)}(Q^2),\frac{Q^2}{m_b^2}\right)=\sum_{p=2}^N\left(\alpha_s(Q^2)\right)^pB_{ij}^{(p)}\left(y,L,\frac{Q^2}{m^2_b}\right)
-\end{equation}
-are such that if $f_{i}^{(5)}$ and $\alpha_s^{(5)}$ are re-expressed in terms of
-$f_{i}^{(4)}$ and $\alpha_s^{(4)}$, then the expression of
-$\sigma^{(4)}$ in the 4FS is recovered.
-
-
-The expressions relating the 4FS and 5FS
-PDFs up to $\order{\alpha^2_s}$ are given in
-Ref.~\cite{Buza:1996wv}. {\bf Anything to say about $\mu$ vs m here?}
+of the FONLL formalism by distinguishing the physical mass of the $b$ quark $m_b$
+from the matching point $\mu_b$. All relevant formulae are spelled out in
+Appendix A.
%They turn out to be trivial at $\order{\alpha_s}$,
%so in our case it is only the redefinition of $\alpha_s$ (due to
%changing $n_f$ by one unit) which has an effect. Explicitly,
%the non-vanishing $B_{ij}^{(k)}$ coefficients are
%at $\order{\alpha_s^2}$
%\begin{align}
% B_{gg}^{(2)}\left(y,L,\frac{Q^2}{m^2_b}\right) & = \hat{\sigma}_{gg}^{(2)}\left(y,L,\frac{Q^2}{m_b^2}\right) \\
% B_{q\bar{q}}^{(2)}\left(y,L,\frac{Q^2}{m^2_b}\right) & = \hat{\sigma}_{q\bar{q}}^{(2)}\left(y,L,\frac{Q^2}{m_b^2}\right)
%\end{align}
%while at $\order{\alpha_s^3}$ the redefinition of $\alpha_s$ contributes:
%\begin{align}
% B_{gg}^{(3)}\left(y,L,\frac{Q^2}{m^2_b},\frac{\mu_R^2}{m_b^2},\frac{\mu_F^2}{m_b^2}\right) & = \hat{\sigma}_{gg}^{(3)}\left(y,L,\frac{Q^2}{m_b^2}\right) - \frac{2 T_R}{3\pi} \ln{\frac{\mu_R^2}{\mu_F^2}}\hat{\sigma}_{gg}^{(2)}\left(y,L,\frac{Q^2}{m_b^2}\right)\\
% B_{q\bar{q}}^{(3)}\left(y,L,\frac{Q^2}{m^2_b},\frac{\mu_R^2}{m_b^2},\frac{\mu_F^2}{m_b^2}\right) & = \hat{\sigma}_{q\bar{q}}^{(3)}\left(y,L,\frac{Q^2}{m_b^2}\right)- \frac{2 T_R}{3\pi} \ln{\frac{\mu_R^2}{m_b^2}}\hat{\sigma}_{q\bar{q}}^{(2)}\left(y,L,\frac{Q^2}{m_b^2}\right) \\
% B_{gq}^{(3)}\left(y,L,\frac{Q^2}{m^2_b}\right) & = \hat{\sigma}_{gq}^{(3)}\left(y,L,\frac{Q^2}{m_b^2}\right) \\
% B_{qg}^{(3)}\left(y,L,\frac{Q^2}{m^2_b}\right) & = \hat{\sigma}_{qg}^{(3)}\left(y,L,\frac{Q^2}{m_b^2}\right).
%\end{align}
%% %
%% \begin{figure}[!htb]
%% \begin{center}
%% \includegraphics[width=0.9\textwidth,angle=0]{diagrams.pdf}
%% \caption{\label{fig:bbh} Contributions to the 5FS
%% computation which are subtracted and
%% get replaced by massive 4FS
%% contributions. The diagrams circled with a dashed line become
%% massive in FONLL-A, while those circled with a solid
%% pink line are those that must be additionally subtracted in the
%% FONLL-B scheme.}
%% \end{center}
%% \end{figure}
-Next, we take all logarithms and
-constant terms in the 4FS NLO
-cross section and drop all terms suppressed by powers of
-$m_b/\mu$, namely by computing
-\begin{equation}
- \label{eq:massless_lim}
- \sigma^{(4),(0)}\left(\alpha_s(Q^2),L\right)=\int_{\tau_H}^{1} \frac{dx}{x}\int_{\frac{\tau_H}{x}}^{1} \frac{dy}{y^2}\sum_{ij=q,g}f_{i}(x,Q^2)f_j\left(\frac{\tau_H}{x y},Q^2\right)B_{ij}^{(0)}\left(y,L,\alpha_s(Q^2)\right),
-\end{equation}
-with
-\begin{equation}
- B_{ij}^{(0)}\left(y,L,\alpha_s(Q^2)\right) = \sum_{p=2}^N\left(\alpha_s(Q^2)\right)^pB_{ij}^{(0),(p)}\left(y,L\right),
-\end{equation}
-where the coefficients $B_{ij}^{(0),(p)}$ satisfy
-\begin{equation}
- \lim_{m_b\rightarrow 0}\left[B_{ij}^{(p)}\left(y, \frac{Q^2}{m^2_b}\right)-B_{ij}^{(0),(p)}\left(y,\frac{Q^2}{m^2_b}\right)\right]=0.
-\end{equation}
-
-As already mentioned, all $B_{ij}^{(0),(p)}$ terms in
-Eq.~(\ref{eq:massless_lim})
-may be equivalently viewed as contributions to the 5FS computation.
-As in the case of Higgs production,
-given that no simple closed-form expression of the massive coefficients
-$B_{ij}^{(p)}$ is available, it turns
-out to be more convenient to extract the $B_{ij}^{(0),(p)}$ from the
-5FS result. This is
-simply done by expressing the 5FS $b$ PDF in terms of
-the 4FS light quark and
-gluon PDFs up to $\order{\alpha_s}$ using the matching coefficients from
-Ref.~\cite{Buza:1996wv} and then re-expressing the result in
-terms of the 5FS quark and gluon PDF, and 5FS $\alpha_s$.
-Note that terms that in the matching coefficients contain logarithms of
-the hard scale $Q$ over the matching point $\mu_b$ and $m_b$ is not involved.
-
-The result has the structure {\bf [MU: continue from here!!!!!!!!!!!]}
-\begin{align}
- \label{eq:btildesh}
- f_b^{(5)}(x,Q^2) = \alpha^{(5)}_s(Q^2) \int_{x}^1\frac{dz}{z} & \left\{ \mathcal{A}_{gb}^{(1)}\left(z,L\right)f^{(5)}_g\left(\frac{x}{z},Q^2\right)\vphantom{\frac{\alpha_s^{(5)}(Q^2)}{2\pi}} \right. \nonumber \\
- +& \left. \alpha_s(Q^2)\left[\mathcal{A}_{gb}^{(2)}\left(z,L\right)f^{(5)}_g\left(\frac{x}{z},Q^2\right) + \mathcal{A}_{\Sigma b}^{(2)}\left(z,L\right)f^{(5)}_{\Sigma}\left(\frac{x}{z},Q^2\right)\right]\right\}.
-\end{align}
-where $f^{(5)}_{b}$, $f^{(5)}_{\Sigma}$ and $f^{(5)}_{g}$ are
-respectively the 5FS $b$ quark, singlet, and
-gluon PDFs, and
-\begin{align}
- \label{eq:btilde}
-\mathcal{A}_{gb}^{(1)}&=a_{gb}^{(1,1)}(z)\, L \nonumber,\\
-\mathcal{A}_{gb}^{(2)}&=a_{gb}^{(2,2)}(z)L^2 + a_{gb}^{(2,1)}(z) L + a_{gb}^{(2,0)}(z),\\
-\mathcal{A}_{\Sigma b}^{(2)}&= a_{\Sigma b}^{(2,2)}(z)L^2 + a_{\Sigma b}^{(2,1)}(z) L + a_{\Sigma b}^{(2,0)}(z)\nonumber
-\end{align}
-Note that, as well known, to $\order{\alpha_s^2}$ the expression of the 5FS
-$f^{(5)}_{b}$ in terms of the light quarks and gluon
-receives constant (i.e. non-logarithmic)
-contributions $a_{g b}^{(2,0)}(z)$ and $a_{\Sigma b}^{(2,0)}(z)$, and
-thus it is discontinuous at threshold $Q^2=m_b^2$ in the massless
-scheme, as a consequence of it being continuous in the fully massive
-calculation. The explicit expressions of the coefficients
-Eq.~(\ref{eq:btilde}) are given in Appendix A for completeness.
-
-We can now collect all contributions to $ \sigma^{(4),(0)}$.
-The $\order{\alpha_s^2}$ terms, already given in
-Ref.~\cite{Forte:2015hba}, are
-\begin{align}
- B_{gg}^{(0)(2)} (y,L) & = y\int_y^1\frac{dz}{z}\left[2\mathcal{A}_{gb}^{(1)}\left(z,L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right) + 4\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)\hat{\sigma}_{gb}^{(1)}(z)\right] + \hat{\sigma}_{gg}^{(2)}(y), \\
- B_{q\bar{q}}^{(0)(2)} (y,L) &= \hat{\sigma}_{q\bar{q}}^{(2)}(y);
-\end{align}
-while the new contributions to $\order{\alpha_s^3}$ are
-\begin{align}\label{eq:subtrexp}
- B_{gg}^{(0)(3)} (y,L) & = y\int_y^1\frac{dz}{z}\left[4\mathcal{A}_{gb}^{(2)}\left(z,L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right) + 2\mathcal{A}_{gb}^{(1)}\left(z,L\right)\mathcal{A}_{gb}^{(2)}\left(\frac{y}{z},L\right)\hat{\sigma}_{b\bar{b}}^{(1)}(z) \right.\nonumber \\
- & \left.\phantom{asdfdy\int_y^1\frac{dz}{z}4\mathcal{A}_{gb}^{(2)}\left(z,L\right)}+ 4\mathcal{A}_{gb}^{(2)}\left(\frac{y}{z},L\right)\hat{\sigma}_{gb}^{(1)}(z) + 4\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)\hat{\sigma}_{gb}^{(2)}(z)\right], \\
- B_{gq}^{(0)(3)} (y,L) & = y\int_y^1\frac{dz}{z}\left[2\mathcal{A}_{\Sigma b}^{(2)}\left(z,L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right) + 2\mathcal{A}_{\Sigma b}^{(2)}\left(\frac{y}{z},L\right)\hat{\sigma}_{gb}^{(1)}(z) \right.\nonumber \\
- & \left.\phantom{asdfdy\int_y^1\frac{dz}{z} 4 \mathcal{A}_{gb}^{(2)}\left(z,L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)}+ 2\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)\hat{\sigma}_{qb}^{(2)}(z)\right],
-\end{align}
-which completes our result.
-Note that in Eq.~(\ref{eq:subtrexp}) $\hat{\sigma}_{ij}^{(p)}(x)$ denotes
-the partonic cross-section in the 5FS, as indicated by the fact that
-it only depends on the momentum fraction and does not have any dependence
-on $m_b$ (unlike the
-4FS partonic cross sections
-$\hat{\sigma}_{ij}^{(p)}\left(x,L,\frac{Q^2}{m_b^2}\right)$ of
-Eq.~(\ref{massive:3})).
+%Next, we take all logarithms and
+%constant terms in the 4FS NLO
+%cross section and drop all terms suppressed by powers of
+%$m_b/\mu$, namely by computing
+%\begin{equation}
+% \label{eq:massless_lim}
+% \sigma^{(4),(0)}\left(\alpha_s(Q^2),L\right)=\int_{\tau_H}^{1} \frac{dx}{x}\int_{\frac{\tau_H}{x}}^{1} \frac{dy}
+%{y^2}\sum_{ij=q,g}f_{i}(x,Q^2)f_j\left(\frac{\tau_H}{x y},Q^2\right)B_{ij}^{(0)}\left(y,L,\alpha_s(Q^2)\right),
+%\end{equation}
+%with
+%\begin{equation}
+% B_{ij}^{(0)}\left(y,L,\alpha_s(Q^2)\right) = \sum_{p=2}^N\left(\alpha_s(Q^2)\right)^pB_{ij}^{(0),(p)}\left(y,L\right),
+%\end{equation}
+%where the coefficients $B_{ij}^{(0),(p)}$ satisfy
+%\begin{equation}
+% \lim_{m_b\rightarrow 0}\left[B_{ij}^{(p)}\left(y, \frac{Q^2}{m^2_b}\right)-B_{ij}^{(0),(p)}\left(y,\frac{Q^2}{m^2_b}\right)\right]=0.
+%\end{equation}
+
+%As already mentioned, all $B_{ij}^{(0),(p)}$ terms in
+%Eq.~(\ref{eq:massless_lim})
+%may be equivalently viewed as contributions to the 5FS computation.
+%As in the case of Higgs production,
+%given that no simple closed-form expression of the massive coefficients
+%$B_{ij}^{(p)}$ is available, it turns
+%out to be more convenient to extract the $B_{ij}^{(0),(p)}$ from the
+%5FS result. This is
+%simply done by expressing the 5FS $b$ PDF in terms of
+%the 4FS light quark and
+%gluon PDFs up to $\order{\alpha_s}$ using the matching coefficients from
+%Ref.~\cite{Buza:1996wv} and then re-expressing the result in
+%terms of the 5FS quark and gluon PDF, and 5FS $\alpha_s$.
+%Note that terms that in the matching coefficients contain logarithms of
+%the hard scale $Q$ over the matching point $\mu_b$ and $m_b$ is not involved.
+
+% The result has the structure
+% \begin{align}
+% \label{eq:btildesh}
+% f_b^{(5)}(x,Q^2) = \alpha^{(5)}_s(Q^2) \int_{x}^1\frac{dz}{z} & \left\{ \mathcal{A}_{gb}^{(1)}\left(z,L\right)f^{(5)}_g\left(\frac{x}{z},Q^2\right)\vphantom{\frac{\alpha_s^{(5)}(Q^2)}{2\pi}} \right. \nonumber \\
+% +& \left. \alpha_s(Q^2)\left[\mathcal{A}_{gb}^{(2)}\left(z,L\right)f^{(5)}_g\left(\frac{x}{z},Q^2\right) + \mathcal{A}_{\Sigma b}^{(2)}\left(z,L\right)f^{(5)}_{\Sigma}\left(\frac{x}{z},Q^2\right)\right]\right\}.
+% \end{align}
+% where $f^{(5)}_{b}$, $f^{(5)}_{\Sigma}$ and $f^{(5)}_{g}$ are
+% respectively the 5FS $b$ quark, singlet, and
+% gluon PDFs, and
+% \begin{align}
+% \label{eq:btilde}
+% \mathcal{A}_{gb}^{(1)}&=a_{gb}^{(1,1)}(z)\, L \nonumber,\\
+% \mathcal{A}_{gb}^{(2)}&=a_{gb}^{(2,2)}(z)L^2 + a_{gb}^{(2,1)}(z) L + a_{gb}^{(2,0)}(z),\\
+% \mathcal{A}_{\Sigma b}^{(2)}&= a_{\Sigma b}^{(2,2)}(z)L^2 + a_{\Sigma b}^{(2,1)}(z) L + a_{\Sigma b}^{(2,0)}(z)\nonumber
+% \end{align}
+% Note that, as well known, to $\order{\alpha_s^2}$ the expression of the 5FS
+% $f^{(5)}_{b}$ in terms of the light quarks and gluon
+% receives constant (i.e. non-logarithmic)
+% contributions $a_{g b}^{(2,0)}(z)$ and $a_{\Sigma b}^{(2,0)}(z)$, and
+% thus it is discontinuous at threshold $Q^2=m_b^2$ in the massless
+% scheme, as a consequence of it being continuous in the fully massive
+% calculation. The explicit expressions of the coefficients
+% Eq.~(\ref{eq:btilde}) are given in Appendix A for completeness.
+%
+% We can now collect all contributions to $ \sigma^{(4),(0)}$.
+% The $\order{\alpha_s^2}$ terms, already given in
+% Ref.~\cite{Forte:2015hba}, are
+% \begin{align}
+% B_{gg}^{(0)(2)} (y,L) & = y\int_y^1\frac{dz}{z}\left[2\mathcal{A}_{gb}^{(1)}\left(z,L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right) + 4\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)\hat{\sigma}_{gb}^{(1)}(z)\right] + \hat{\sigma}_{gg}^{(2)}(y), \\
+% B_{q\bar{q}}^{(0)(2)} (y,L) &= \hat{\sigma}_{q\bar{q}}^{(2)}(y);
+% \end{align}
+% while the new contributions to $\order{\alpha_s^3}$ are
+% \begin{align}\label{eq:subtrexp}
+% B_{gg}^{(0)(3)} (y,L) & = y\int_y^1\frac{dz}{z}\left[4\mathcal{A}_{gb}^{(2)}\left(z,L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right) + 2\mathcal{A}_{gb}^{(1)}\left(z,L\right)\mathcal{A}_{gb}^{(2)}\left(\frac{y}{z},L\right)\hat{\sigma}_{b\bar{b}}^{(1)}(z) \right.\nonumber \\
+% & \left.\phantom{asdfdy\int_y^1\frac{dz}{z}4\mathcal{A}_{gb}^{(2)}\left(z,L\right)}+ 4\mathcal{A}_{gb}^{(2)}\left(\frac{y}{z},L\right)\hat{\sigma}_{gb}^{(1)}(z) + 4\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)\hat{\sigma}_{gb}^{(2)}(z)\right], \\
+% B_{gq}^{(0)(3)} (y,L) & = y\int_y^1\frac{dz}{z}\left[2\mathcal{A}_{\Sigma b}^{(2)}\left(z,L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right) + 2\mathcal{A}_{\Sigma b}^{(2)}\left(\frac{y}{z},L\right)\hat{\sigma}_{gb}^{(1)}(z) \right.\nonumber \\
+% & \left.\phantom{asdfdy\int_y^1\frac{dz}{z} 4 \mathcal{A}_{gb}^{(2)}\left(z,L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)}+ 2\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)\hat{\sigma}_{qb}^{(2)}(z)\right],
+% \end{align}
+% which completes our result.
+% Note that in Eq.~(\ref{eq:subtrexp}) $\hat{\sigma}_{ij}^{(p)}(x)$ denotes
+% the partonic cross-section in the 5FS, as indicated by the fact that
+% it only depends on the momentum fraction and does not have any dependence
+% on $m_b$ (unlike the
+% 4FS partonic cross sections
+% $\hat{\sigma}_{ij}^{(p)}\left(x,L,\frac{Q^2}{m_b^2}\right)$ of
+% Eq.~(\ref{massive:3})).
%
+We have implemented our final FONLL-B expression by combining,
+4FS predictions up to NLO obtained using MC@NLO~\cite{Wiesemann:2014ioa},
+using a model in which the coupling of the $Z$ boson to the light quarks are turned off and
+a private 5FS computations up to NNLL obtained using a private code, cross-checked at LO and NLO against MC@NLO.
+Finally we have implementation of the subtraction term.
+
\begin{figure}
\begin{center}
\includegraphics[width=0.8\textwidth,angle=0]{muR_var.pdf}
\includegraphics[width=0.8\textwidth,angle=0]{muR_mh_var.pdf}
\caption{\label{fig:muR_var} Comparison of the FONLL
matched result and its 4FS and 5FS components,
Eq.~(\ref{FONLL}). Results are shown as a function of the
renormalization scale, with the factorization scale fixed at a high value
$\mu_F=m_H$ (top) or a low value $\mu_F=\frac{(m_H+2m_b)}{4}$ (bottom).}
\end{center}
\end{figure}
%
-We have implemented our final FONLL-B expression by combining,
-according to Eq.~(\ref{FONLL})
-4FS predictions up to NLO
-obtained using MC@NLO~\cite{Wiesemann:2014ioa}, 5FS computations up
-to NNLL obtained using the {\tt bbh@nnlo}
-code~\cite{Harlander:2003ai}, and our own implementation of the
-subtraction term Eq.~(\ref{eq:massless_lim}).
+
%
\begin{figure}
\begin{center}
\includegraphics[width=0.8\textwidth,angle=0]{muF_var.pdf}
\includegraphics[width=0.8\textwidth,angle=0]{muF_mh_var.pdf}
\caption{\label{fig:muF_var} Same as Fig.~\ref{fig:muR_var}, but now
with the factorization scale varied with the renormalization scale
kept fixed at a high value
$\mu_R=m_H$ (top) or a low value $\mu_R=\frac{(m_H+2m_b)}{4}$ (bottom) .}
\end{center}
\end{figure}
%
In Figs.~\ref{fig:muR_var}-\ref{fig:muF_var} we compare the 4FS, 5FS
and matched FONLL results: specifically we show both the LO and NLO
4FS predictions, and the FONLL-A and FONLL-B matched results in which
they are respectively combined with the NNLL 5FS result, also
shown. The 4FS results shown are those which enter the FONLL
combination, namely, the form Eq.~(\ref{massive:1}) of the 4FS result
is used, in which this is expressed in terms of 5FS PDFs and $\alpha_s$.
All results are computed using a PDF set presented and
discussed in Ref.~\cite{Bonvini:2016fgf}. This PDF set is based on the
PDF4LHC15 combined
sets~\cite{Butterworth:2015oua,Ball:2014uwa,Harland-Lang:2014zoa,Dulat:2015mca,Carrazza:2015aoa,Gao:2013bia,Watt:2012tq},
with which it is taken to coincide below the $b$ mass, but from which
it is then evolved up in the $5FS$ from $Q=m_b$, with the results
below and above threshold matched exactly as in
Eq.~(\ref{eq:btildesh}). This is not quite the
same as the original PDF4LHC15 combination, which is obtained by
combining sets which adopt different values of $m_b$, and also
incorporate
-subleading
+sub-leading
differences in the way the 4FS and 5FS are matched at threshold: it
thus has the advantage of being fully consistent. We use pole-mass
expressions and take a $b$ pole-mass value
$m_b=4.58$~GeV; the strong coupling is $\alpha_s(m_Z)=0.118$.
From Fig.~\ref{fig:muR_var} we see that the strong renormalization scale
dependence of the LO 4FS result is reduced at NLO, and also,
that at NLO the big gap between the 4FS and 5FS results gets compensated for
by the inclusion of higher order terms in the 4FS.
This, together with the fact that the 5FS shows very little
scale dependence, and that differences are significantly smaller for smaller values of
$\mu_F$, strongly suggests that the bulk of the difference
between the 4FS and the 5FS is due to large logs of
$\mu_F^2/m_b^2$ which are resummed into the PDF in the latter case.
This is in agreement with the conclusion of Ref.~\cite{Lim:2016wjo},
in which it was shown that resummation increases
the cross section in most cases by up to 30\% at the LHC, leading to a better precision.
On the other hand, the 4FS predictions at NLO also displays a consistent
perturbative behaviour only when evaluated at a suitably low scale.
The massive corrections which the 4FS result contains turn out to be
much smaller, though not entirely negligible. Indeed, whereas the FONLL-A result
essentially coincides with the 5FS, the FONLL-B, which only differs
from it
because of the inclusion of massive terms at one extra perturbative
order, departs somewhat from it\footnote{In Ref.~\cite{Forte:2015hba} the FONLL-A result
, while also close to the 5FS result, did not coincide exactly with it
for generic scales, because their respective scale dependences,
though slight, had different shapes. This difference in
shape was due to the fact that, unlike here, a fully consistent PDF
set was not used: rather, the PDFs were taken from a public set, with
a value of $m_b$ which differed from that used in the computation of
the matrix element, thereby leading to a mismatch in the scale dependence.}.
The factorization scheme dependence
shown in Fig.~\ref{fig:muF_var} is very mild in all scheme when
$\mu_R$ is high, but for low $\mu_R$, where the perturbative expansion
of the 4FS result is more reliable, both the 5FS and the FONLL-A results
show a contained scale dependence, comparable in size to the mass effects,
which is reduced in the FONLL-B result.
These results suggest that the main difference
between the FONLL-A and the FONLL-B schemes is the inclusion of
a higher order contribution from the 4FS computation which reduces the
the scale dependence of the FONLL-A result; because the latter is
essentially the same as that of the 5FS computation this contribution
is likely to be a constant, i.e., mass-independent.
In a recent paper \cite{Bertone:2017djs} a new approach to construct
heavy-flavor PDFs was advocated, namely, a standard Zero Mass (ZM)-VFNS
but with heavy flavor matching point taken at a larger scale than the
conventional value $m_b$.
%
\begin{figure}
\begin{center}
\includegraphics[width=0.8\textwidth,angle=0]{m_muF_var.pdf}
\includegraphics[width=0.8\textwidth,angle=0]{m_muR_var.pdf}
\caption{\label{fig:m_mu_var}
with the factorization (top) or renormalization (bottom) scales
varied with the renormalization (top) or factorization (bottom) scale
kept fixed at $\mu=\frac{(m_Z+2m_b)}{3}$.}
\end{center}
\end{figure}
%
We conclude that the FONLL-B result is the most reliable, and a low
choice of renormalization and factorization scheme seems to lead to a
more reliable perturbative expansion, but all in all mass corrections
are very moderate, so the usage of the 5FS result at all scales would be
-adequate in most cases. This rather disfavours phenomenological
+adequate in most cases. This rather disfavors phenomenological
combinations such as the so-called Santander
matching~\cite{Harlander:2011aa} in which the 4FS and 5FS results are
combined through an interpolation that gives each of them comparable
weight.
Matched results for this process were recently obtained in
Refs.~\cite{Bonvini:2015pxa,Bonvini:2016fgf} using an
effective field theory approach,
and a somewhat different counting of perturbative orders. A benchmarking
of our results with those of these references has been performed in the
context of the Higgs cross section working group, and it will be
presented there~\cite{Anastasiou:xxx}. The benchmarking shows
agreement between the matched calculations when results at the same
perturbative orders are included.
In summary, we have presented a matched computation of Higgs
production in association with bottom quarks including known results
to the highest available accuracy, namely, NLO in a four-flavor scheme
in which $b$ quark mass effects are fully accounted for, and NNLL in
a five-flavor scheme in which the $b$ quark is treated as a massless
parton with collinear logs resummed to all orders.
We find that mass corrections are very small while collinear logs are
substantial, so that in practice the fully matched result is very
close to the 5FS one. The fully matched result receives a small
correction from mass effects and it is very stable upon renormalization
and factorization scheme variation, suggesting that it is adequate for
precision phenomenology at the LHC.
\bigskip
A public implementation of our NNLL+NLO FONLL-B matched computation
will be made available from:
\begin{center}
\url{http://bbhfonll.hepforge.org/}
\end{center}
\section*{Acknowledgements}
We thank Fabio Maltoni, Giovanni Ridolfi and Paolo Nason for illuminating discussions.
We thank Marius Wiesemann for his help in comparing our results
to those obtained with MG5, and Marco Bonvini, Andrew Papanastasiou
and Frank Tackmann for
discussions on the their approach and on the PDFs of
Ref.~\cite{Bonvini:2016fgf}.
SF and DN are supported by the European Commission through the
HiggsTools Initial Training Network PITN-GA2012-316704.
%%%%%%%%
\begin{appendix}
-\section{Appendix A}
+\section{FONLL expressions with $\mu_b$ different from $m_b$}
\numberwithin{equation}{section}
\setcounter{equation}{0}
-We give for completeness the expressions of the coefficients
-Eq.~(\ref{eq:btilde}). These were computed in Ref.~\cite{Buza:1996wv}.
-There are a few differences compared to what is presented there.
-Firstly we separate contributions from $b$ and $\bar{b}$. Secondly
-our expansion is done in powers of $\alpha_s$ rather than in powers
-of $\frac{\alpha_s}{4 \pi}$. Lastly we have re-expressed the gluon
-and singlet PDFs in the 4FS in terms of those computed in the 5FS.
-\begin{align}
-%(B.1)
-\mathcal{A}^{(2)}_{\Sigma b}(z,L) & =
-%
-\frac{1}{32 \pi^2 }C_FT_f\Biggl\{
-%
-\Biggl[-8(1+z)\ln z-\frac{16}{3z}-4
-%
-+ 4 z +\frac{16}{3}z^2\Biggr] L^2
-%
-\nonumber \\ &
-%
--\Biggl[8(1+z)\ln^2z-\Biggl(8+40z+\frac{64}{3}z^2\Biggr)\ln z
-%
--\frac{160}{9z}
-%
-+16-48z+\frac{448}{9}z^2\Biggr] L
-%
-\nonumber \\ &
-%
-+ (1+z)\Biggl[32{\rm S}_{1,2}(1-z)+16\ln z{\rm Li}_2(1-z)
-%
--16\zeta(2)\ln z
-%
--\frac{4}{3}\ln^3z\Biggr]
-%
-\nonumber \\ &
-%
-+\Biggl(\frac{32}{3z}+8-8z-\frac{32}{3}z^2\Biggr) {\rm Li}_2(1-z)
-%
-+ \Biggl( -\frac{32}{3 z}-8+8z+\frac{32}{3} z^2\Biggr)\zeta(2)
-%
-\nonumber \\ &
-%
-+\Biggl(2+10z+\frac{16}{3}z^2\Biggr) \ln^2z
-%
--\Biggl(\frac{56}{3}+\frac{88}{3}z
-%
-+\frac{448}{9}z^2\Biggr)\ln z
-%
--\frac{448}{27z} - \frac{4}{3}
-%
--\frac{124}{3}z+\frac{1600}{27}z^2 \Biggr\} \,,
-%
-\end{align}
-%
-\begin{align}
-%(B.2)
-\mathcal{A}_{gb}^{(1)} (z,L) & = \frac{T_f}{2\pi} \Biggl[ ( z^2 + (1 - z)^2) L\Biggr] \, ,
-\end{align}
-and
-
-\begin{align}
-%(B.3)
-\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\mathcal{A}_{bg}^{(2)}(z,L) & =
-%
-\frac{1}{32 \pi^2 }\Biggl\{\Biggl\{C_FT_f[ (8 -16 z+16 z^2)\ln(1-z)
-%
--(4 -8 z+ 16 z^2)\ln z -(2 - 8 z)]
-%
-\nonumber \\ &
-%
-+C_AT_f\Biggl[-(8 - 16 z + 16 z^2)\ln(1-z)
-%
--(8 + 32 z)\ln z
-%
- -\frac{16}{3z} -4 - 32 z+\frac{124}{3}z^2\Biggr]
-%
-\nonumber \\ &
-%
-+ T_f^2 \Biggl[ - \frac{16}{3} ( z^2 + (1 - z)^2) \Biggr]
-%
-+ T_f\Biggl[\frac{2}{3}(z^2 + (1 - z)^2)\Biggr]
-%
-\Biggr\} L^2
-%
-\nonumber \\ &
-%
--\Biggl\{C_FT_f \Biggl[( 8 - 16 z + 16z^2)[2\ln z\ln(1-z)
-%
--\ln^2(1-z)+2\zeta(2)]
-%
-\nonumber \\ &
-%
--(4 - 8 z +16 z^2)\ln^2z-32z(1-z)\ln(1-z)
-%
--(12 - 16 z + 32 z^2)\ln z - 56+116z -80z^2 \Biggr]
-%
-\nonumber \\ &
-%
-+ C_AT_f\Biggl[(16 +32 z +32 z^2)[{\rm Li}_2(-z) + \ln z\ln(1+z) ]
-%
-+(8 - 16 z + 16 z^2)\ln^2(1-z)
-%
-\nonumber \\ &
-%
-+(8 + 16 z)\ln^2z
-%
-+32z\zeta(2)+32z(1-z)\ln(1-z)
-%
--\Biggl(8+64z+\frac{352}{3}z^2\Biggr)\ln z
-%
-\nonumber \\ &
-%
--\frac{160}{9z}+16-200z+\frac{1744}{9}z^2 \Biggr]\Biggl\}L
-%
-\nonumber \\ &
-%
-+ C_FT_f \Biggl\{(1-2z+2z^2) [8\zeta(3)
-%
-+\frac{4}{3}\ln^3(1-z)
-%
--8\ln(1-z){\rm Li}_2(1-z)
-%
-+8\zeta(2)\ln z
-%
-%
-\nonumber \\ &
-%
--4\ln z\ln^2(1-z)
-%
-+\frac{2}{3}\ln^3z
-%
--8\ln z{\rm Li}_2(1-z)
-%
-+8{\rm Li}_3(1-z)
-%
--24{\rm S}_{1,2}(1-z)]
-%
-\nonumber \\ &
-%
-+z^2\Biggl[-16\zeta(2)\ln z+\frac{4}{3}\ln^3z
-%
-+16\ln z{\rm Li}_2(1-z)+32{\rm S}_{1,2}(1-z)\Biggr]
-%
-\nonumber \\ &
-%
--(4+96z-64z^2){\rm Li}_2(1-z)
-%
--(4-48z+40z^2)\zeta(2)
-%
-\nonumber \\ &
-%
--(8+48z-24z^2)\ln z\ln(1-z)
-%
-+(4+8z-12z^2)\ln^2(1-z)
-%
-\nonumber \\ &
-%
--(1+12z-20z^2)\ln^2z-(52z-48z^2)\ln(1-z)
-%
-\nonumber \\ &
-%
--(16+18z+48z^2)\ln z
-%
-+26-82z+80z^2\Biggr\}
-%
-\nonumber \\ &
-%
-+C_AT_f\Biggl\{(1-2z+2z^2) [
-%
--\frac{4}{3} \ln^3(1-z)
-%
-\nonumber \\ &
-%
-+8\ln(1-z){\rm Li}_2(1-z)-8{\rm Li}_3(1-z)]
-%
-+(1+2z+2z^2)
-%
-\nonumber \\ &
-%
-\times [-8\zeta(2)\ln(1+z)
-%
--16\ln(1+z){\rm Li}_2(-z)
-%
--8\ln z\ln^2(1+z)
-%
-\nonumber \\ &
-%
-+4\ln^2z\ln(1+z) + 8\ln z{\rm Li}_2(-z)-8{\rm Li}_3(-z)
-%
--16{\rm S}_{1,2}(-z)]
-%
-\nonumber \\ &
-%
-+(16+64z)[2{\rm S}_{1,2}(1-z)
-%
-+\ln z{\rm Li}_2(1-z)]
-%
--\Biggl(\frac{4}{3} + \frac{8}{3} z\Biggr)\ln^3z
-%
-\nonumber \\ &
-%
-+(8-32z+16z^2)\zeta(3)-(16+64z)\zeta(2)\ln z+(16+16z^2)
-%
-\nonumber \\ &
-%
-\times [ {\rm Li}_2(-z) + \ln z\ln(1+z) ]
-%
-+\Biggl(\frac{32}{3z}+12+64z-\frac{272}{3}z^2\Biggr)
-%
-{\rm Li}_2(1-z)
-%
-\nonumber \\ &
-%
--\Biggl( 12 + 48 z - \frac{260}{3} z^2+\frac{32}{3 z}\Biggr)\zeta(2)
-%
--4z^2\ln z\ln(1-z)
-%
-\nonumber \\ &
-%
--(2+8z-10z^2)\ln^2(1-z)+\Biggl(2+8z+\frac{46}{3}z^2\Biggr)\ln^2z
-%
-\nonumber \\ &
-%
-+(4+16z-16z^2)\ln(1-z)
-%
--\Biggl(\frac{56}{3}+\frac{172}{3}z+\frac{1600}{9}z^2\Biggr)\ln z
-%
-\nonumber \\ &
-%
--\frac{448}{27z}-\frac{4}{3}-\frac{628}{3}z
-%
-+\frac{6352}{27}z^2\Biggr\}\Biggr\} \, .
-%
-\end{align}
-
+We give for completeness the FONLL expressions by using $m_b$ different from $\mu_b$.
+{\bf[Add relevant formulae, see below the equations I consider relevant, check that what I am writing
+is correct, both from theory point of view and same as the implementation (MU)]}
+\begin{equation}
+ \label{massive:1}
+ \sigma^{(4)}=\int_{\tau_H}^{1} \frac{dx}{x}\int_{\frac{\tau_H}{x}}^{1} \frac{dy}{y^2}\sum_{ij=q,g}f_{i}^{(5)}(x,Q^2)f_j^{(5)}\left(\frac{\tau_H}{x y},Q^2\right)B_{ij}\left(y,L,\alpha_s^{(5)}(Q^2),\frac{Q^2}{m_b^2}\right),
+\end{equation}
+where $L=\log(Q^2/\mu_b^2)$ - i.e. in the change of scheme $m_b$ becomes $\mu_b$. Also in the 5FS the b PDF
+depends on $\mu_b$ and is completely unrelated to $m_b$. Instead in the 4FS the only b-related scale is $m_b$.
+Thus in the massless limit of the 4FS one take all logarithms and constant terms in the 4FS NLO cross section
+and drops the terms suppressed by powers of $m_b/Q$. The logs left in the massless limits are $\log(Q^2/m_b^2)$.
+If one instead truncates the $b$-PDF, one gets $L=\log(Q^2/\mu_b^2)$.
\end{appendix}
%%%%%%%%%%%%%%%%
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\bibliographystyle{UTPstyle}
\bibliography{bbz_fonll}
%\input{bbH_FONLL.bbl}
\end{document}