This document discusses parton distribution functions (PDFs) relevant for a future 100 TeV hadron collider. It summarizes that such a collider would probe PDFs at smaller x values than currently possible, down to x=10-9. It would also probe higher mass scales, up to masses of 10 TeV. Current PDF determinations have little constraint for x<10-4 or masses above 1 TeV. The increased energy would require considering effects like top quark PDFs, electroweak corrections, and high-energy resummation not important at lower energies. Polarized collisions could also provide insights into PDFs.

1. Parton Distributions
at a 100 TeV Hadron Collider
Juan Rojo
CERN, PH Division, TH Unit
Future Circular Colliders Kick-off Meeting
Geneva, 14.02.2014
Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014

2. Exploratory first study on the role of Parton Distributions at a Future Circular
Collider with CoM of 100 TeV
Concentrate on the basic results, avoid technical details (choice of PDF set etc),
and explore the new qualitative features of PDFs at the FCC
Outline of this talk:
Kinematic coverage from LHC 14 TeV to FCC 100 TeV
PDF luminosities from 14 to 100 TeV
New qualitative features of PDF in 100 TeV collisions: top-quark PDFs, LO
small-x PDFs, Electroweak corrections, high-energy BFKL resummation
Physics case for polarized collisions at the FCC
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Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014

3. Kinematic Coverage and
PDF Luminosities:
From LHC14 to FCC100
Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014

4. At Born level, for the production of a particle at rapidity y with mass M, the PDFs are evaluated at
Going from Ecm=14 TeV to Ecm=100 TeV, for a given M and y knowledge of PDFs at smaller x is
required:
x1,20( S=100 TeV ) = 0.14 x1,20( S=14 TeV )
In addition, for fixed M, the increased range in y allows to prove even further smaller x
Conversely, fixing x1,20 and y, the increased center of mass energy implies that higher masses will
be probed:
M( S=100 TeV ) = 7.14 M( S=14 TeV )
But recall that hadronic cross-sections fall as 1/M2 so this is not how BSM mass reach scales!
Some consequences:
For Higgs and SM probes (tt, W, Z), PDFs will be proved (at least) one order of magnitude
smaller x
For possible new high-mass particles of mass M, large-x PDFs (required at 14 TeV) turn into
medium-x PDFs (at 100 TeV), not only leading to larger cross-sections but with less uncertainties
For small masses M, a completely new kinematical range opens that has never been explored
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Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014

5. Kinematics of a 100 TeV FCC
Plot by J. Rojo, Dec 2013
105
FCC 100 TeV
MX ( GeV )
104
LHC 14 TeV
3
10
102
10
y=-8
1 -10
10
5
Juan Rojo
-9
10
y=-4
-8
10
10-7
y=0
-6
10
-5
10
x
y=4
10-4
-3
10
y=8
10-2
10-1
FCC Kick-off Meeting, Geneva, 14/02/2014

6. Kinematics of a 100 TeV FCC
Plot by J. Rojo, Dec 2013
105
20 TeV Z’
FCC 100 TeV
MX ( GeV )
104
2 TeV squarks
LHC 14 TeV
3
10
Higgs, top
102
W,Z
DY, low-pt jets
10
y=-8
1 -10
10
6
Juan Rojo
-9
10
y=-4
-8
10
10-7
y=0
-6
10
-5
10
x
y=4
10-4
-3
10
y=8
10-2
10-1
FCC Kick-off Meeting, Geneva, 14/02/2014

7. NNPDF2.3 dataset
Kinematics of a 100 TeV FCC
Plot by J. Rojo, Dec 2013
105
107
NMC-pd
NMC
SLAC
BCDMS
6
5
10
LHC 14 TeV
103
HERAI-AV
CHORUS
FLH108
Q2 / M2 / p2 [ GeV2 ]
T
MX ( GeV )
10
FCC 100 TeV
104
ZEUS-H2
ZEUSF2C
H1F2C
104
102
NTVDMN
DYE605
DYE886
CDFWASY
3
10
CDFZRAP
D0ZRAP
CDFR2KT
D0R2CON
102
10
ATLAS-WZ-36pb
CMS-WEASY-840PB
LHCB-WZ-36pb
ATLAS-JETS-10
y=-8
1 -10
10
-9
10
y=-4
-8
10
10-7
y=0
-6
10
-5
10
x
y=4
10-4
-3
10
y=8
10-2
10-1
10
1 -5
10
10-4
-3
10
x
10-2
10-1
Compare kinematical coverage to that of the NNPDF2.3 set:
Current PDF determinations have essentially no constraints for x < 10-4
The region of masses for M>1 TeV also unconstrained: rely on DGLAP evolution extrapolation
Poor constraints on high-x PDFs, relevant for high masses M
For M > 10 GeV, constraints from HERA only available for x > 10-3
Global PDF fits rely on QCD evolution, but EW effects will be required in multi-TeV region
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Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014
1

8. Compare the ratio of PDF luminosities between 100 TeV and 14 TeV in different channels as a
function of the final state mass
100 TeV vs 14 TeV PDF Luminosities, NNPDF2.3 NNLO
PDF luminosity ratio 100 TeV / 14 TeV
108
107
For final state masses M < 1
TeV moderate increase in PDF
luminosity, between a factor 10
and 100
106
105
104
For M > 1 TeV, much steeper
increase (since 14 TeV lumis
damped by large-x PDFs), up
to a factor 108 for M = 10 TeV
103
102
10
1
8
gg lumi ratio
102
Juan Rojo
10
3
MX [ GeV ]
104
Qualitative similar behavior
for other PDF luminosities
FCC Kick-off Meeting, Geneva, 14/02/2014

9. New qualitative features of
PDFs in 100 TeV collisions
Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014

10. At FCC100, the production of 100 GeV particles might require knowledge of PDFs down to x=1e-6, 1e-7,
depending on acceptance
For other processes, both perturbative (low mass Drell-Yan, Quarkonium production, low pt photons
and jets) and non-perturbative (Underlying Event, Minimum Bias) we need PDFs down to x=1e-8, 1e-9
Compare various PDFs at very small x out of the box using LHAPDF5.9.0: small-x PDFs frozen between
1e-6 and 1e-8 depending on the PDF set
Small-x NNLO PDFs for FCC studies
NNPDF2.3
CT10
cteq6
MSTW08
2
103
102
10-9
10
NNPDF2.3
CT10
cteq6
MSTW08
104
x u ( x, Q2)
104
x g ( x, Q )
Small-x NNLO PDFs for FCC studies
103
102
10-8
10-7
Juan Rojo
10-6
x
10-5
10-4
10-3
10-9
10-8
10-7
10-6
x
10-5
10-4
FCC Kick-off Meeting, Geneva, 14/02/2014
10-3

11. This is not a fundamental problem: is easy enough to provide a extrapolation down to very small-x ...
... but this extrapolation must be driven by theory + methodology, since no data available
The NNPDF2.3LO set can be extrapolated down to 1e-9 using the neural network extrapolation: huge
PDF uncertainties, but central values turn out to have a reasonable behavior
NNPDF2.3LO PDF set is already available as standalone code in Pythia8, where there is an ongoing
tuning effort. Torbjorn has been using it to provide UE predictions for FCC finding reasonable results
xg(x,Q), NNPDF23_lo_as_0119_qed.LHgrid members
500
Members
Generated by APFEL2.0.0: V.Bertone, S.Carrazza, J.Rojo (arXiv:1310.1394)
Central value
Std. deviation
400
68% c.l.
Q = 1.41 GeV
300
200
100
0
-9
10
11
Juan Rojo
-8
10
10-7
-6
10
-5
10
x
10-4
-3
10
10-2
10-1
1
FCC Kick-off Meeting, Geneva, 14/02/2014

12. In the Variable-Flavor-Number (VFN) scheme, heavy quark PDFs for charm and bottom are generated
radiatively from the gluon PDF
For very high-mass production at a 100 TeV FCC, the top quark can be considered as approximately
massless, and thus we need to generate a top quark PDF t(x,Q) just as we have a charm and bottom PDF
Note that missing the top PDF also feeds back into all other PDFs due to the different DGLAP evolution
Ratio to NNPDF2.1 GM-VFN, x=10-2
Ratio to NNPDF2.1 GM-VFN, x=10-2
1.15
missing charm
1.1
NNPDF2.1
1.2
NNPDF2.1 FFN NF=3
1.1
NNPDF2.1 FFN NF=4
x (x, Q 2)
xg (x, Q2)
1.05
1
1
0.9
0.95
NNPDF2.1
NNPDF2.1 FFN NF=3
0.9
missing charm
0.8
NNPDF2.1 FFN NF=4
0.85
10
12
Juan Rojo
102
Q 2 [ GeV 2 ]
3
10
104
0.7
10
102
Q 2 [ GeV 2 ]
3
10
FCC Kick-off Meeting, Geneva, 14/02/2014
104

13. Currently, only the NNPDF Collaboration provides sets with top quark PDF included (as well as sets in
the other VFN schemes, NF=3, NF=4 and NF=5)
At 10 TeV, the top quark PDF t(x,Q) is only a factor 2 smaller that all other quark PDFs (charm and
bottom are very close to light quark PDFs): should be included in theoretical predictions
2
x PDF ( x, Q = 10 TeV )
NNPDF2.3 NNLO NF = 6
102
Gluon PDF
Up quark PDF
10
Top PDF
1
10-1 -5
10
13
Gluon
Up
Charm
Bottom
Top
Juan Rojo
10-4
-3
10
x
10-2
10-1
FCC Kick-off Meeting, Geneva, 14/02/2014

14. As mentioned in the case of top quark, when Q >> mt it makes sense computationally to resum
large collinear-like logarithms into a top PDF
With a similar motivation, it might be efficient to resum multiple initial state emissions of W and Z
bosons into the corresponding parton distributions, FW(x,Q), FZ(x,Q)
The analogous of DGLAP evolution equations in QCD can be derived in the electroweak sector of
the Standard Model, but the resulting equations are very different (Ciafaloni and Comelli,
2002,2005)
The purely electroweak evolution equations must be supplemented with the QED evolution
equations and the determination of the photon PDF
(x,Q2) from experimental data
(NNPDF2.3QED)
An important difference between QED and EW evolution is that in the later case W and Z boson
PDFs are always generated dynamically, while there is an intrinsic photon PDF in the proton
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Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014

15. The analogous of DGLAP evolution equations in QCD can be derived in the electroweak sector of
the Standard Model, but the resulting equations are very different (Ciafaloni and Comelli, 2002,2005)
Evolution equation for the structure function of W bosons
No numerical implementation of EW evolution equations exist. Very different flavor/coupling
structure as compared to QCD evolution equations
In addition, EW must be combined with pure QED evolution, and then combined with QCD into
a complete set of Standard Model PDF evolution equations
15
Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014

16. Once QED corrections to PDFs are taken into account, photon-initiated contributions are relevant
at the LHC, could be even more the case at the FCC
Many examples, like photon-initiated corrections substantially modify the lepton pt in high-mass
Drell-Yan
They also induce substantial uncertainties for high-mass WW production, up to 100%
More FCC-specific studies required
WW production @ LHC 100 TeV, 68% CL
high-mass DY
σ(WW) [QCD+QED] / σ(WW) [QCD]
2.5
2
WW at FCC100
1.5
1
NNPDF2.3 QED, qq
0.5
(QCD+QED)/QCD
MRST04 QED, qq
NNPDF2.3 QED, qq + γ γ
MRST04 QED, qq + γ γ
0
0
16
Boughezal, Li, Petriello 2013
Juan Rojo
1000
2000
3000
4000
5000
Mcut ( GeV )
WW
6000
7000
8000
Snowmass QCD report
FCC Kick-off Meeting, Geneva, 14/02/2014

17. The kinematical coverage of the FCC reaches the small-x region where the usual DGLAP PDF
evolution might need to be completed with BFKL high-energy (small-x) resummation
Combining DGLAP and BFKL stabilizes the perturbative expansion of DGLAP evolution at small-x
gluon-gluon DGLAP kernel
The theory for DIS known (Altarelli, Ball, Forte 08, Ciafaloni, Colferai, Salam, Stasto 07), more
theoretical progress is required for computation of relevant hadronic cross sections at the FCC
17
Juan Rojo
DGLAP+BFKL
DGLAP NNLO
FCC Kick-off Meeting, Geneva, 14/02/2014

18. Crucial milestone for the next years is to produce a global PDF fit which combines NNLO DGLAP
evolution with NLL BFKL resummation
This would be a pre-requisite to study the relevance of high-energy resummation at the FCC,
specially for benchmark processes like Higgs, W, Z and top production
Ratio g(x,Q)/gNLO(x,Q)
High-energy resummation affects mostly the gluon PDF, and then feeds to all the quark PDFs
18
Juan Rojo
DGLAP NNLO
DGLAP+BFKL
x=10-4
FCC Kick-off Meeting, Geneva, 14/02/2014

19. The Physics Case for
Polarized Collisions at the FCC
Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014

20. What are the BSM opportunities for the polarized option of a 100 TeV machine?
Polarized beams could provide a useful tool for BSM characterization, where differences in polarized
asymmetries provide a handle on SM vs BSM discrimination
NNPDF2.3, Q2 = 104 GeV2
0.8
0.6
xd
xs
0.5
2
x ∆ q ( x, Q )
0.5
x∆g
x∆u
x∆d
x∆s
0.7
xu
0.6
2
0.8
xg
0.7
x q ( x, Q )
NNPDFpol1.1, Q2 = 104 GeV2
0.4
0.3
0.2
0.4
0.3
0.2
0.1
0.1
0
0
-0.1
-0.1
10-2
10-1
x
10-2
10-1
x
Different qualitative behaviors of PDFs wrt unpolarized PDFs
Polarized PDFs smaller than unpolarized due to positivity condition
and suppressed at small-x due to different DGLAP splitting functions
Only asymmetries which involve high-x polarized PDFs (final states with large invariant mass)
will be large enough to be accessible experimentally
20
Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014

21. Fuks, Proudom, Rojo, Schienbein,
in preparation
Measurement of sign of AL will discriminate between different BSM scenarios (different production modes)
21
A more precise measurement of AL leads to enhanced discrimination, ie, qq vs gg production mechanisms
Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014

22. Parton Distributions play a central role at the LHC program, will continue to do so
at a Future Circular Collider at 100 TeV
Many new qualitative features of PDFs need to be studied: top PDF, PDFs at very
large-x, PDFs with electroweak corrections, PDFs with high-energy resummation
It will be useful to provide specific PDF sets for FCC studies, with suitable
extrapolations at very small and very large-x as well as to a very large Q2 (and
include top PDF)
A polarized mode of the FCC might provide a unique way of characterizing BSM
dynamics uncovered in unpolarized collisions
22
Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014

23. High-mass production at FCC will probe PDFs at very high-x, we want to make sure that available PDF
sets properly extrapolate in this region
Michelangelo reported some weeks ago that using CTEQ at very large x returns zero
I have tried using LHAPDF5.9.0 with CT10, MSTW08 and NNPDF2.3. Consider only central PDF sets
(ignore for the time being PDF uncertainties), for Q = 10 TeV
Large-x PDFs seems to behave properly at very large-x, though MSTW08 extrapolation has a peculiar
behavior, at least using the LHAPDF library
Note substantial differences for d(x,Q) at high-x
Large-x NNLO PDFs for FCC studies
Large-x NNLO PDFs for FCC studies
NNPDF2.3
10-1
10-1
CT10
-2
CT10
-2
10
MSTW08
-3
10
MSTW08
-3
10
x d ( x, Q2)
10
x g ( x, Q2)
NNPDF2.3
10-4
10-4
10-5
10-5
10-6
10-7
10-7
10-8
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
x
23
10-6
10-8
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
x
Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014

24. At 10 TeV, the top quark PDF t(x,Q) is only a factor 2/3 smaller that all other quark PDFs (charm and
bottom are very close to light quark PDFs): should be included in theoretical predictions
The missing contribution from t(x,Q) also feeds back to the gluon PDF: up to 4% differences due to the
missing top quark PDF
NNPDF2.3 NNLO, x = 10-2
1.06
1.04
F
Ratio ( N = 5 ) / ( N = 6 )
1.05
Gluon
Up
Down
NF=6 vs NF=5
1.03
F
1.02
1.01
1
0.99
0.98
102
24
Juan Rojo
3
Q10GeV )
(
104
FCC Kick-off Meeting, Geneva, 14/02/2014

25. 100 TeV vs 14 TeV PDF Luminosities, NNPDF2.3 NNLO
PDF luminosity ratio 100 TeV / 14 TeV
108
107
106
GG
QG
QQbar
QQ
105
104
103
102
10
1
102
10
3
MX [ GeV ]
104
Gluon-initiated processes grow faster than quark initiated from 14 to 100 TeV
Very large PDF uncertainties affect QQbar luminosities since large-x antiquarks poorly
known
Using these results, it is possible to estimate how the cross-sections for different processes
will increase when going from 14 TeV to 100 TeV
25
Juan Rojo
FCC Kick-off Meeting, Geneva, 14/02/2014