Particle Physics Phenomenology 4. Parton distributions and...
Transcript of Particle Physics Phenomenology 4. Parton distributions and...
Particle Physics Phenomenology4. Parton distributions and initial-state showers
Torbjorn Sjostrand
Department of Astronomy and Theoretical PhysicsLund University
Solvegatan 14A, SE-223 62 Lund, Sweden
Lund, 6 February 2018
Parton Distribution Functions
Hadrons are composite, with time-dependent structure:
fi (x ,Q2) = number density of partons iat momentum fraction x and probing scale Q2.
Linguistics (example):
F2(x ,Q2) =∑
i
e2i xfi (x ,Q2)
structure function parton distributions
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 2/70
PDF evolution – 1
Initial conditions at small Q20 unknown: nonperturbative.
Resolution dependence perturbative, by DGLAP:
DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi)
dfb(x ,Q2)
d(lnQ2)=∑
a
∫ 1
x
dz
zfa(y ,Q2)
αs
2πPa→bc
(z =
x
y
)
DGLAP already introduced for (final-state) showers:
dPa→bc =αs
2π
dQ2
Q2Pa→bc(z) dz
Same equation, but different context:
dPa→bc is probability for the individual parton to branch; while
dfb(x ,Q2) describes how the ensemble of partons evolveby the branchings of individual partons as above.
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 3/70
PDF evolution – 1
Initial conditions at small Q20 unknown: nonperturbative.
Resolution dependence perturbative, by DGLAP:
DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi)
dfb(x ,Q2)
d(lnQ2)=∑
a
∫ 1
x
dz
zfa(y ,Q2)
αs
2πPa→bc
(z =
x
y
)DGLAP already introduced for (final-state) showers:
dPa→bc =αs
2π
dQ2
Q2Pa→bc(z) dz
Same equation, but different context:
dPa→bc is probability for the individual parton to branch; while
dfb(x ,Q2) describes how the ensemble of partons evolveby the branchings of individual partons as above.
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 3/70
PDF evolution – 2
Note 1: Pa→bc(z) only same to leading order;at NLO different for FSR (timelike) and ISR (spacelike).
Note 2: In ISR more common to use Pb/a(z) = Pa→b(c)(z).
Note 3: Properly speaking gain+loss equations, e.g.
dq(x ,Q2)
d(lnQ2)= +(q at y > x branches to x)
−(q at x branches to y < x)
= +
∫ 1
xdy q(y ,Q2)
∫ 1
xdz
αs
2πPq/q(z) δ(x − yz)
−q(x ,Q2)
∫ 1
0dz
αs
2πPq/q(z)
(neglecting g → qq).
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 4/70
PDF evolution – 3
Singularity in Pq/q(z) ∝ 1/(1− z) for z → 1 must be addressed.
Not too bad: consider emissions with 1− ε ≤ z ≤ 1,
xdq(x ,Q2)
d(lnQ2)= +
∫ 1
1−ε
dz
zxq(
x
z,Q2)
αs
2πPq/q(z)
−xq(x ,Q2)
∫ 1
1−εdz
αs
2πPq/q(z)
=
∫ 1
1−εdz[xzq(
x
z,Q2)− xq(x ,Q2)
] αs
2πPq/q(z)
where [. . .] → 0 for z → 1 for smooth q(x ,Q2),so net effect of z ≈ 1 branchings on q(x ,Q2) is smooth and finite.
Put another way: infinitely many infinitely soft gluons are emitted,but they carry away a finite amount of momentum,since
∫ 10 (1− z) P(z) dz is finite.
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 5/70
PDF evolution – 4
Conventional approach is to use conservation of (valence) quarknumber: ∫ 1
0Pq/q(z) dz = 0
to replace
Pq/q(z) =4
3
1 + z2
1− z→ 4
3
1 + z2
(1− z)++ 2δ(1− z)
where 1/(1− z)+ prescription is defined by∫ 1
0dz
f (z)
(1− z)+=
∫ 1
0dz
f (z)− f (1)
(1− z)
for a function f (z) well-behaved in limit z → 1.Whole change to be associated with emissions “at” z = 1.
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 6/70
PDF evolution: moments – 1
(moments useful analytically, but outdated numerically)∫ 1
0xn dx
dfb(x ,Q2)
d(lnQ2)
=∑
a
∫ 1
0xn dx
∫ 1
0dy fa(y ,Q2)
∫ 1
0dz
αs
2πPb/a(z) δ(x − yz)
=
∫ 1
0yn dy fa(y ,Q2)
∫ 1
0zn dz
αs
2πPb/a(z)
so with
fa(n,Q2) =
∫ 1
0xn fa(x ,Q2) dx
Pb/a(n) =
∫ 1
0zn Pb/a(z) dz
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 7/70
PDF evolution: moments – 2
one obtains
dfb(n,Q2)
d(lnQ2)=∑
a
fa(n,Q2)αs
2πPb/a(n)
i.e. convolution replaced by multiplication,which gives simpler equation system to solve.Recover fa(x ,Q2) by inverse Mellin transform ⇒ numerical.
Warning: often (usually) moments are defined offset one step:
Pb/a(n) =
∫ 1
0zn−1 Pb/a(z) dz
etc., so be careful what people mean by “first moment” and“second moment”.
Always remember: need boundary conditions at Q2 = Q20 .
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 8/70
PDF examples – 1
Convenient “Durham” plotting interface, but by now out-of-date:http://hepdata.cedar.ac.uk/pdf/pdf3.html
Intended replacement by APFEL (more ambitious, less convenient):http://apfel.mi.infn.it/
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 9/70
PDF examples – 2
Peaking of PDF’s at small x and of QCD ME’s at low p⊥=⇒ most of the physics is at low transverse momenta . . .. . . but New Physics likely to show up at large masses/p⊥’s
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 10/70
PDF positivity issues – 1
At NLO PDFs are not physical objects and not required to bepositive definite everywhere. Neither are cross sections!(Recall negative δ function in NLO p⊥ spectrum in lecture 2.)
More generally, consider structure of perturbative expansionexpressed in toy language:
dσ
dx=∣∣f0(x) + αs f1(x) + α2
s f2(x) + · · ·∣∣2 > 0
but written order-by-order
dσ
dx= |f0(x)|2 + αs 2Re(f ∗0 (x)f1(x))
+ α2s
(|f1(x)|2 + 2Re(f ∗0 (x)f2(x))
)where interference terms may have either sign (and do!)
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 11/70
PDF positivity issues – 2
Problem most acute for gluon at small x for small Q;disappers at larger Q by evolution from larger x
Dangerous for LO MCs: attempts with MC-adapted PDFs• allow
∑i
∫ 10 xfi (x ,Q2) > 1 as “built-in K factor”
• use NLO-calculated pseudodata as target for tunes• force NLO PDFs to have “sensible” small-x behaviour
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 12/70
PDF sets
A few groups regularly producing updated sets:
CTEQ: (Coordinated Theoretical-Experimental Project onQCD) (Tung, Huston, . . . ), recently CT14
“MRSTW”: (first name initials, currently Thorne), recentlyMMHT16
NNPDF: (Neural Net PDF) (Forte et al.) recently NNPDF 3.1
HERAPDF: (Cooper-Sarkar, Radescu, . . . ): only HERA data,recently HERAPDF 2.0
Alekhin et al.: recently ABMP16
PDF4LHC15: combination of CT14, MMHT14, NNPDF 3.0
Nowadays LO is rare, NLO the standard,NNLO starting to take over.
More on PDFs later.
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 13/70
Initial-State Shower Basics
• Parton cascades in p are continuously born and recombined.• Structure at Q is resolved at a time t ∼ 1/Q before collision.• A hard scattering at Q2 probes fluctuations up to that scale.• A hard scattering inhibits full recombination of the cascade.
• Convenient reinterpretation:
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 14/70
Forwards vs. backwards evolution
Event generation could be addressed by forwards evolution:pick a complete partonic set at low Q0 and evolve,consider collisions at different Q2 and pick by σ of those.Inefficient:
1 have to evolve and check for all potential collisions,but 99.9. . . % inert
2 impossible (or at least very complicated) to steer theproduction, e.g. of a narrow resonance (Higgs)
Backwards evolution is viable and ∼equivalent alternative:start at hard interaction and trace what happened “before”
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 15/70
Backwards evolution master formula
Monte Carlo approach, based on conditional probability : recast
dfb(x ,Q2)
dt=∑
a
∫ 1
x
dz
zfa(x
′,Q2)αs
2πPa→bc(z)
with t = ln(Q2/Λ2) and z = x/x ′ to
dPb =dfbfb
= |dt|∑
a
∫dz
x ′fa(x′, t)
xfb(x , t)
αs
2πPa→bc(z)
then solve for decreasing t, i.e. backwards in time,starting at high Q2 and moving towards lower,with Sudakov form factor exp(−
∫dPb)
Webber: can be recast by noting that total change of PDF at x isdifference between gain by branchings from higher x and loss bybranchings to lower x .
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 16/70
The ladder
Ladder representation combines whole event:
DGLAP:Q2
max > Q21 > Q2
2 ∼ Q20
Q2max > Q2
3 > Q24 > Q2
5
One possible Monte Carloorder:
1 Hard scattering
2 Initial-state showerfrom center outwards
3 Final-state showers
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 17/70
Coherence in spacelike showers
with Q2 = −m2 = spacelike virtuality
kinematics only:Q2
3 > z1Q21 , Q2
5 > z3Q23 , . . .
i.e. Q2i need not even be ordered
coherence of leading collinear singularities:Q2
5 > Q23 > Q2
1 , i.e. Q2 orderedcoherence of leading soft singularities (more messy):E3θ4 > E1θ2, i.e. z1θ4 > θ2
z � 1: E1θ2 ≈ p2⊥2 ≈ Q2
3 , E3θ4 ≈ p2⊥4 ≈ Q2
5
i.e. reduces to Q2 ordering as abovez ≈ 1: θ4 > θ2, i.e. angular ordering of soft gluons
=⇒ reduced phase space
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 18/70
Evolution procedures
DGLAP: Dokshitzer–Gribov–Lipatov–Altarelli–Parisievolution towards larger Q2 and (implicitly) towards smaller xBFKL: Balitsky–Fadin–Kuraev–Lipatovevolution towards smaller x (with small, unordered Q2)CCFM: Ciafaloni–Catani–Fiorani–Marchesiniinterpolation of DGLAP and BFKLGLR: Gribov–Levin–Ryskin (⇒JIMWLK)nonlinear equation in dense-packing (saturation) region,where partons recombine, not only branch
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 19/70
Initial-State Shower Comparison – 1
Two(?) CCFM Generators:(SMALLX (Marchesini, Webber))
CASCADE (Jung, Salam)LDC (Gustafson, Lonnblad):reformulated initial/final rad.=⇒ eliminate non-Sudakov
Test 1) forward (= p direction) jet activity at HERA
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 20/70
Initial-State Shower Comparison – 2
2) Heavy flavour production
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but also explained by DGLAP with leading order pair creation+ flavour excitation (≈ unordered chains)+ gluon splitting (final-state radiation)
CCFM requires off-shell ME’s + unintegrated parton densities
F (x ,Q2) =
∫ Q2dk2⊥
k2⊥F(x , k2
⊥) + (suppressed with k2⊥ > Q2)
so not ready for prime time in pp
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 21/70
Initial-State Shower Comparison – 3
Mueller-Navelet: trigger on two jets and look for third in between.Figure: inclusive/exclusive 2-jet rate ⇒ rate of extra jets.
5
y|∆|0 1 2 3 4 5 6 7 8 9
incl
R
1
1.5
2
2.5
3
3.5
4
4.5
52010 data
PYTHIA6 Z2
PYTHIA8 4C
HERWIG++ UE-7000-EE-3
HEJ + ARIADNE
CASCADE
= 7 TeVsCMS, pp,
dijets > 35 GeV
Tp
|y| < 4.7
y|∆|0 1 2 3 4 5 6 7 8 9
MN
R1
1.5
2
2.5
3
3.5
4
4.5
52010 data
PYTHIA6 Z2
PYTHIA8 4C
HERWIG++ UE-7000-EE-3
HEJ + ARIADNE
CASCADE
= 7 TeVsCMS, pp,
dijets > 35 GeV
Tp
|y| < 4.7
Figure 1: Ratios of the inclusive to exclusive dijet cross sections as a function of the rapidityseparation |Dy| between the two jets, Rincl (left panel) and RMN (right panel), compared to thepredictions of the DGLAP-based MC generators PYTHIA6, PYTHIA8 and HERWIG++, as well asof CASCADE and HEJ+ARIADNE which incorporate elements of the BFKL approach. The shadedband indicates the size of the total systematic uncertainty of the data. Statistical uncertaintiesare smaller than the symbol sizes. Because of limitations in the CASCADE generator it was notpossible to obtain a reliable prediction for |Dy| > 8.
y|∆|0 1 2 3 4 5 6 7 8 9
MC
/ da
ta
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
y|∆|0 1 2 3 4 5 6 7 8 9
MC
/ da
ta
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5dataPYTHIA6 Z2PYTHIA8 4CHERWIG++
= 7 TeVsCMS, pp,
incldijets, R > 35 GeV
Tp
|y| < 4.7
y|∆|0 1 2 3 4 5 6 7 8 9
MC
/ da
ta
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
y|∆|0 1 2 3 4 5 6 7 8 9
MC
/ da
ta
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5dataPYTHIA6 Z2PYTHIA8 4CHERWIG++
= 7 TeVsCMS, pp,
MNdijets, R > 35 GeV
Tp
|y| < 4.7
Figure 2: Predictions for Rincl (left) and RMN (right) from DGLAP-based MC generators pre-sented as ratio to data corrected for detector effects. Both BFKL-motivated generators CASCADEand HEJ+ARIADNE (not shown) lead to a MC/data ratio well above unity. The shaded bandindicates the size of the total systematic uncertainty of the data while statistical uncertaintiesare shown as bars.
Left: |∆y | for any jet pairRight: only largest |∆y | in event
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 22/70
Initial- vs. final-state showers
Both controlled by same evolution equations
dPa→bc =αs
2π
dQ2
Q2Pa→bc(z) dz · (Sudakov)
but
Final-state showers:Q2 timelike (∼ m2)
decreasing E ,m2, θboth daughters m2 ≥ 0physics relatively simple⇒ “minor” variations:Q2, shower vs. dipole, . . .
Initial-state showers:Q2 spacelike (≈ −m2)
decreasing E , increasing Q2, θone daughter m2 ≥ 0, one m2 < 0physics more complicated⇒ more formalisms:DGLAP, BFKL, CCFM, GLR, . . .
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 23/70
PYTHIA (new) showers: objective
Originally PYTHIA showers used Q2 = ±m2 as evolution variable.Complete rewrite since ∼ 14 years.Incorporate several of the good points of the dipole(like ARIADNE) within the shower approach (⇒ hybrid)
± explore alternative p⊥ definitions
+ p⊥ ordering ⇒ coherence inherent
+ ME merging works as with Q2 = ±m2
(unique p2⊥ ↔ m2 mapping; same z)
+ g → qq natural
+ kinematics constructed after each branching(partons explicitly on-shell until they branch)
+ showers can be stopped and restarted at given p⊥ scale ⇒well suited for ME/PS matching (CKKW-L, etc.)
+ well suited for interleaved multiple interactions
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 24/70
PYTHIA showers: simple kinematics
Consider branching a → bc in lightcone coordinates p± = E ± pz
p+b = zp+
a
p+c = (1− z)p+
a
p− conservation
=⇒ m2a =
m2b + p2
⊥z
+m2
c + p2⊥
1− z
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 25/70
PYTHIA showers: general strategy – 1
1 Definep2⊥evol = z(1− z)Q2 = z(1− z)m2 for FSR
p2⊥evol = (1− z)Q2 = (1− z)(−m2) for ISR
2 Find list of radiators = partons that can radiate.Evolve them all downwards in p⊥evol from common p⊥max
dPa =dp2⊥evol
p2⊥evol
αs(p2⊥evol)
2πPa→bc(z) dz exp
(−∫ p2
⊥max
p2⊥evol
· · ·
)
dPb =dp2⊥evol
p2⊥evol
αs(p2⊥evol)
2π
x ′fa(x′,p2
⊥evol)
xfb(x ,p2⊥evol)
Pa→bc(z) dz exp (− · · · )
Pick the one with largest p⊥evol to undergo branching; alsopick associated z .
3 DeriveQ2 = p2
⊥evol/z(1− z) for FSR
Q2 = p2⊥evol/(1− z) for ISR
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 26/70
PYTHIA showers: general strategy – 2
4 Find recoiler = takes recoil when radiator is pushed off-shellusually nearest colour neighbour for FSRincoming parton on other side of event for ISR
5 Interpret z as energy fraction (not lightcone)in radiator+recoiler rest frame for FSR,in mother-of-radiator+recoiler rest frame for ISR,so that Lorentz invariant (2Ei/Ecm = 1−m2
jk/E 2cm)
and straightforward match to matrix elements
6 Do kinematics based on Q2 and z ,a) assuming yet unbranched partons on-shellb) shuffling energy–momentum from recoiler as required
7 Continue evolution of all radiators from recently picked p⊥evol.Iterate until no branching above p⊥min.⇒ One combined sequence p⊥max > p⊥1 > . . . > p⊥min.
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 27/70
PYTHIA showers: FSR detailed – 1
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 28/70
PYTHIA showers: FSR detailed – 2
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 29/70
PYTHIA showers: FSR p⊥ – 1
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 30/70
PYTHIA showers: FSR p⊥ – 2
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 31/70
PYTHIA showers: FSR check/tune
Checked/tuned against LEP data
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 32/70
PYTHIA showers: ISR detailed – 1
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 33/70
PYTHIA showers: ISR detailed – 2
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 34/70
PYTHIA showers: ISR p⊥
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 35/70
PYTHIA showers: ISR check/tune
Checked/tuned against Tevatron data, primarily p⊥ of Z0
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 36/70
PYTHIA showers: final comment
PYTHIA solution for combining ISR and FSR:
ISR does boost whole contained system, like normal showers
FSR uses dipoles, including dipoles stretched out to remnants,but remnant end of dipole does not radiate; that is in ISR
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 37/70
Combining FSR with ISR
Separate processing of ISR and FSR misses interference(∼ colour dipoles)
ISR+FSR add coherentlyin regions of colour flow
in “normal” shower byazimuthal anisotropies
automatic in dipole(by proper boosts)
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 38/70
Coherence tests – 1
old normal showers with/without ϕ reweighting:η3: pseudorapidity of third jetα: angle of third jet around second jet
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 39/70
Coherence tests – 2
current-day generators for psuedorapidity of third jet:
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 40/70
The role of radiation (Peter Skands)
ISR/FSR give important corrections to the event topologyat all scales, from hard to soft.
Corrections:
Scale machine energy accordingly or else E -p conservation
PDF evolution gives scaling violations
αs(Q2) smaller for high-scale processes
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 41/70
Resummation introduction
Assume perturbative series 1− x +O(x2).Ill-behaved if x is large ⇒ need to resum.
Example 1: non-ordered as in αem evolution:loop integrals
∫dQ2/Q2
Q1 and Q2 unordered
⇒ 1− x + x2 − x3 + · · · = 1
1 + x
Example 2: ordered as in parton showerQ1 > Q2 by kinematics (& coherence)∫
dQ21
Q21
∫ Q21 dQ2
2
Q22
=1
2
(∫dQ2
Q2
)2
.
⇒ 1− x +x2
2− x3
6+ · · · = e−x
Q22
Q21
Q21 Q2
2
Q22
Q21
Q21 Q2
2
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 42/70
Resummation – 1
Resummation = analytical method to add the effect of multiple(soft) gluon emission.Most used for ISR, e.g.
p⊥Z = −p⊥ = −n∑
i=1
p⊥i .
Start from single gluon emission
1
σ0
dσ
d2p⊥= δ(2)(p⊥) +
4
3
αs
2π
(· · ·p2⊥
)+
= δ(2)(p⊥) + αs ν(p⊥)
where + prescription gives virtual correction at p⊥ = 0so that net O(αs) contribution to σtot vanishes.
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 43/70
Resummation – 2
The n-gluon emission is assumed to factorize(at least for soft gluons, which dominate)
1
σ0
dσ
d2p⊥=
=
∫· · ·∫
1
σ0
dσ
d2p⊥1 . . .d2p⊥nd2p⊥1 . . .d2p⊥n δ(2)(p⊥ −
n∑i=1
p⊥i )
=
∫· · ·∫
αns
n!ν(p⊥1) . . . ν(p⊥n) d2p⊥1 . . .d2p⊥n δ(2)(p⊥ −
n∑i=1
p⊥i )
=1
n!
(n∏
i=1
αs ν(p⊥i ) d2p⊥i
)δ(2)(p⊥ −
n∑i=1
p⊥i )
where 1/n! comes from symmetrization of identical gluons.
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 44/70
Resummation – 3
Now technical trick: Fourier transform to impact parameter b∫d2p⊥ exp(ibp⊥)
(n∏
i=1
αs ν(p⊥i ) d2p⊥i
)δ(2)(p⊥ −
n∑i=1
p⊥i )
= exp(ibn∑
i=1
p⊥i )
(n∏
i=1
αs ν(p⊥i ) d2p⊥i
)
=n∏
i=1
αs ν(p⊥i ) exp(ibp⊥i ) d2p⊥i = (αs ν(b))n
Then sum over different number of emissions
σ(b) =
∫dσ
d2p⊥exp(ibp⊥) d2p⊥ = σ0
∞∑n=0
1
n!(αs ν(b))n
= σ0 exp(αs ν(b))
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 45/70
Resummation – 4
Inverse Fourier transform gives answer
1
σ0
dσ
d2p⊥=
1
(2π)2
∫d2b exp(−ibp⊥) exp(αs ν(b))
Now need to be specific about ν(p⊥) choice.Early ansatz gave
1
σ0
dσ
dp2⊥∼ 4
3
αs
π
1
p2⊥
exp
(−4
3
αs
2πln2 s
p2⊥
)which → 0 for p⊥ → 0.Gradually more fancy
running αs
large b corresponds to small p⊥, which is nonperturbative
longitudinal momentum constraints
higher orders
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 46/70
Deeply Inelastic Scattering (DIS) (Fred Olness)Key ideas Experimental observables PDF parametrizations Statistical aspects Practical applications
Neutral-current ep DIS: kinematics! s = (pe + pp)2 –total energy
! Q2 = −q2 = −(pe − p′e)2– momentum
transfer
! x = Q2/(2pp · q) – Bjorken scaling variable
! y = Q2/(xs) – inelasticity
! W 2 = Q2(1− x)/x – energy of thehadronic final state
d2σ(e±p)
dQ2dx=
2πα2
Q4xY+
!F2 −
y2
Y+FL ±
Y−Y+
xF3
",
with Y± ≡ 1± (1− y)2
The data is fitted either in the form of F2(x,Q2) or d2σ/(dQ2dx)Pavel Nadolsky (SMU) CTEQ school Lecture 1, 2017-07-21 28
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 47/70
The scaling of proton structure function (Fred Olness)
F2(x ,Q2) ≈∑q
e2q
(xq(x ,Q2) + xq(x ,Q2)
)
The Scaling of the Proton Structure Function
Data is (relatively) independent of energy
Scaling Violations observed at extreme x
values
Varies with energy
Varies with energy
10
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 48/70
NLO corrections (Ilkka Helenius)NLO corrections
I Now focus on the �⇤ + qi interaction:
9>>>>>>>=
>>>>>>>;
X
p
k
k0
q = k � k0
M
p0
p
Three types of O(↵s) process
I Virtual correctionsI Gluon radiationI Initial state gluon
I Detailed calculations not done here, see e.g. H. Paukkunen’s PhDThesis: http://arxiv.org/abs/0906.2529 and references therein
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 49/70
Virtual corrections (Ilkka Helenius)Virtual corrections
I LO contribution
= A0µ
I Virtual corrections
+ + = AV µ
I For the |M|2 we need
|A0 + AV |2 = |A0|
2 + 2Re(A0A⇤V ) + |AV |
2
but here |AV |2 is order ⇠ ↵
2s so need only the interference term.
I The virtual corrections contains divergences (UV and IR)!
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 50/70
Real corrections (Ilkka Helenius)Real corrections
As we are considering inclusive process there can be more particles in thefinal state
I Gluon emission
+ = ACµ
I Initial state gluons
+ = AGµ
I Also these diagrams contain singularities (IR and collinear)!I However, when combining all the O(↵s) processes together all but the
collinear divergences cancel!
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 51/70
Redefinition in NLO (Ilkka Helenius)F2 at order ↵s
After careful calculations one should end up with
1
xF2(x, Q
2) =X
q,q
e2q
1Z
x
d⇠
⇠q0(⇠)
�(1� z)�
↵s
2⇡
✓1
✏+ log
µ2
Q2
◆Pqq(z) +
↵s
2⇡Cq(z)
�
+2X
q
e2q
1Z
x
d⇠
⇠g0(⇠)
�
↵s
2⇡
✓1
✏+ log
µ2
Q2
◆Pqg(z) +
↵s
2⇡Cg(z)
�
where z = x/⇠, Pij(z) splitting functions and Ci(z) coefficient functionsTwo problem remains in F2 (and similarly for F1):
I It depends on arbitrary scale µ
(arising from dimensional regularization)I It is divergent: 1/✏ = 2/✏� �E + log(4⇡) and where ✏! 0
Solution:I Define scale dependent parton distribution functions and absorb the
collinear 1/✏ singularity to the redefinition
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 52/70
DGLAP equations (Ilkka Helenius)DGLAP equations
Defining convolution as
P ⌦ f =
1Z
x
d⇠
⇠P (x/⇠)f(⇠)
the full set of DGLAP equations can be written as:
@fqi(x, Q2)
@ log Q2=
↵s(Q2)
2⇡
⇥Pqiqj ⌦ fqj (Q
2) + Pqig ⌦ fg(Q2)⇤
@fg(x, Q2)
@ log Q2=
↵s(Q2)
2⇡
⇥Pgg ⌦ fg(Q
2) + Pgqj ⌦ fqj (Q2)⇤
Where the splitting functions are power series in ↵s:
Pfifj (z) = P(0)fifj
(z) +↵s
2⇡P
(1)fifj
(z) +⇣
↵s
2⇡
⌘2P
(2)fifj
(z) + . . .
Current state of the art is P(2)fifj
(z) but the expressions are rather messy
already for P(1)fifj
(z).
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 53/70
Where are PDFs measured? (Fred Olness)
Calculable from theoretical model
Pc = fPa ⊗ ac
Where do PDFs come from???? Universality!!!
Must extract from experiment
Note we can combine different experiments.
FACTORIZATION!!!
27
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 54/70
PDFs from DIS (Pavel Nadolsky)Key ideas Experimental observables PDF parametrizations Statistical aspects Practical applications
PDF combinations in DIS at the lowest order
! Neutral current ℓ±p:
F ℓ±p2 (x,Q2) =
4
9(u + u + c + c) +
1
9
!d + d + s + s + b + b
"
" PDFs are weighted by the fractional EM quark couplinge2
i = 4/9 or 1/9
" 4 times more sensitivity to u and c than to d, s, and b
" No sensitivity to the gluon at this order
! Neutral current (ℓ±N) DIS on isoscalar nuclei (N = (p + n)/2):
F ℓ±N2 (x,Q2) =
5
9
!u + u + d + d + smaller s, c, b contributions
"
! Charged current (νN) DIS :
F νN2 (x,Q2) = x
#
i=u,d,s...
(qi + qi)
xF νN3 (x,Q2) = x
#
i=u,d,s
(qi − qi)
Pavel Nadolsky (SMU) CTEQ school Lecture 1, 2017-07-21 30
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 55/70
PDFs from pp/pp (Pavel Nadolsky)Key ideas Experimental observables PDF parametrizations Statistical aspects Practical applications
Inclusive jet production, pp(−) → jet + X
50 100 150 200 250 300 350 400 450 500ET (GeV)
0
0.5
1
Subp
roce
ss fr
actio
n
p− p −−> jet +X√s = 1800 GeV CTEQ6M µ = ET /2 0< |η| <.5
CTEQ6MCTEQ5M
qg
gg
High-ET jets are mostlyproduced in qq scattering; yetmost of the PDF uncertaintyarises from qg and ggcontributions
Here typical x is of order2ET /
√s ! 0.1;
e.g., x ≈ 0.2 for ET = 200 GeV,√s = 1.8 TeV
At such x, u(x,Q) and d(x,Q)are known very well; uncertaintyarises mostly from g(x,Q)
Pavel Nadolsky (SMU) CTEQ school Lecture 1, 2017-07-21 41
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 56/70
Where do PDFs come from? (Pavel Nadolsky)Key ideas Experimental observables PDF parametrizations Statistical aspects Practical applications
Practical answer:from the Les Houches Accord PDF library (LHAPDF)
Almost all recent PDFs are included in the LHAPDF C++ libraryavailable at lhapdf.hepforge.org.
Thousands of PDF sets are provided and can be linked to yourcomputer code. Which one should you use?
Pavel Nadolsky (SMU) CTEQ school Lecture 1, 2017-07-21 16
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 57/70
How are PDFs obtained? (Pavel Nadolsky)Key ideas Experimental observables PDF parametrizations Statistical aspects Practical applications
The flow of the global analysis Data sets in the CT10analysis
Modern fits involve up to 40 experiments, 3000+ data points,and 100+ free parameters
Pavel Nadolsky (SMU) CTEQ school Lecture 1, 2017-07-21 19
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 58/70
Ansatz for PDFs (Pavel Nadolsky)Key ideas Experimental observables PDF parametrizations Statistical aspects Practical applications
Boundary conditions at Q0In practice, independent parametrizations fa/p(x,Q0) areintroduced for
! g, u, d, s, u, d, s (always)contribute > 97% of the proton’s energy Ep at Q0
" even in this case, the data are usually insufficient forconstraining all PDF parameters; some of them can be fixedby hand
" e.g., u = d = s in outdated fits
! c and or b (occasionally; in a model allowingnonperturbative “intrinsic heavy-quark production”)
! photons γ (in QCD+QED PDFs by CT, LUX, MRST, NNPDF...groups)
" a QCD+QED fit is more complicated than one might think: itmust account for violation of charge symmetry by EM effects,
up(x, Q) = dn(x, Q); dp(x, Q) = un(x, Q)Pavel Nadolsky (SMU) CTEQ school Lecture 1, 2017-07-21 21
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 59/70
PDFs for heavy flavours (Pavel Nadolsky)Key ideas Experimental observables PDF parametrizations Statistical aspects Practical applications
General-mass variable-flavor number scheme
! A series of factorization schemes with Nf active quark flavorsin αs(Q) and fa/p(x,Q)
" Nf is incremented sequentially at momentum scalesµNf
≈ mNf
! incorporates essential mc,b dependence near, and awayfrom, heavy-flavor thresholds
! implemented in all latest PDF fits except ABM
µ
µ4 ≈ mc µ5 ≈ mb
Nf = 4 Nf = 5Nf = 3 + −
Pavel Nadolsky (SMU) CTEQ school Lecture 1, 2017-07-21 23
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 60/70
Nontrivial symmetry breaking (Pavel Nadolsky)Key ideas Experimental observables PDF parametrizations Statistical aspects Practical applications
SU(2) and charge symmetry breaking
d(x) = u(x), q(x) = q(x)
May be caused by
! DGLAP evolution
! Fermi motion
! Electromagnetic effects
! Nonperturbative meson fluctuations
! Chiral symmetry breaking
! Instantons
! . . .
Pavel Nadolsky (SMU) CTEQ school Lecture 1, 2017-07-21 34Similar discussions whether s(x) = s(x)and if s(x ,Q0) = kd(x ,Q0) with k = 1, 1/2, 1/3?
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 61/70
Origin of PDF set differences (Pavel Nadolsky)
Experimental observables Theoretical cross sections PDF parametrizations
Origin of differences between PDF sets
1. Corrections of wrong or outdated assumptions
lead to significant differences between new (≈post-2014) andold (≈pre-2014) PDF sets
! inclusion of NNLO QCD, heavy-quark hard scatteringcontributions
! relaxation of ad hoc constraints on PDF parametrizations
! improved numerical approximations
Pavel Nadolsky (SMU) CTEQ school Lecture 2, 2017-07-22 6
Experimental observables Theoretical cross sections PDF parametrizations
Origin of differences between PDF sets
2. PDF uncertainty
a range of allowed PDF shapes for plausible input assumptions,partly reflected by the PDF error band
is associated with
! the choice of fitted experiments
! experimental errors propagated into PDF’s
! handling of inconsistencies between experiments
! choice of factorization scales, parametrizations for PDF’s,
higher-twist terms, nuclear effects,...
leads to non-negligible differences between the newest PDF sets
Pavel Nadolsky (SMU) CTEQ school Lecture 2, 2017-07-22 6
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 62/70
Requirement on PDF parametrizations (Pavel Nadolsky)Experimental observables Theoretical cross sections PDF parametrizations
3. Requirements for PDF parametrizationsA. A valid set of fa/p(x,Q) must satisfy QCD sum rules
Valence sum rule! 1
0[u(x,Q) − u(x,Q)] dx = 2
! 1
0
"
d(x,Q) − d(x,Q)#
dx = 1! 1
0[s(x,Q)− s(x,Q)] dx = 0
A proton has net quantum numbers of 2 u quarks + 1 d quark
Momentum sum rule
[proton] ≡$
a=g,q,q
! 1
0x fa/p(x,Q) dx = 1
momenta of all partons must add up to the proton’s momentum
Through this rule, normalization of g(x,Q) is tied to the firstmoments of quark PDFs
Pavel Nadolsky (SMU) CTEQ school Lecture 2, 2017-07-22 11
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 63/70
Approaches to PDF fitting – 1 (Pavel Nadolsky)Experimental observables Theoretical cross sections PDF parametrizations
Requirements for PDF parametrizationsPDF parametrizations for fa/p(x,Q) must be “flexible just enough”to reach agreement with the data, without violating QCDconstraints (sum rules, positivity, ...) or reproducing randomfluctuations
good fit
F2(x, Q2)
x
Traditional solution
“Theoretically motivated” functionswith a few parameters
fi/p(x,Q0) = a0xa1(1− x)a2
×F (x; a3, a4, ...)
! x→ 0: f ∝ xa1 – Regge-likebehavior
! x→ 1: f ∝ (1− x)a2 – quarkcounting rules
! F (a3, a4, ...) affects intermediatePavel Nadolsky (SMU) CTEQ school Lecture 2, 2017-07-22 16Historical approach, with increasing number of parametersas data have become more precise and revealed shortcomings
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 64/70
Approaches to PDF fitting – 2 (Pavel Nadolsky)Experimental observables Theoretical cross sections PDF parametrizations
Requirements for PDF parametrizationsPDF parametrizations for fa/p(x,Q) must be “flexible just enough”to reach agreement with the data, without violating QCDconstraints (sum rules, positivity, ...) or reproducing randomfluctuations
F2(x, Q2)
x
Radical solutionNeural Network PDF collaboration
! Generate N replicas of theexperimental data, randomlyscattered around the original datain accordance with the probabilitysuggested by the experimentalerrors
! Divide the replicas into a fittingsample and control sample
Pavel Nadolsky (SMU) CTEQ school Lecture 2, 2017-07-22 16
Warning: individual replicas can look quite weird;only average over 100–1000 members will givesensible answers and uncertainties
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 65/70
Statistical aspects (Pavel Nadolsky)Experimental observables Theoretical cross sections PDF parametrizations
4. Statistical aspectsJ. Pumplin et al., JHEP 0207, 012 (2002), and references therein; J. Collins & J. Pumplin, hep-ph/0105207
χ2 for one experiment is
χ2k =
Mk!
i=1
1
σ2i
"
Di − Ti({a}) −Rk!
n=1
rnβni
#2
+Rk!
n=1
r2n
Di and Ti are data and theory values at each point
σi =$
σ2stat + σ2
syst,uncor is the total statistical + systematical
uncorrelated error%
n βnirk are correlated systematic shifts
βni is the correlation matrix; is provided with the data ortheoretical cross sections before the fit%
n r2n is the penalty for deviations of rn from their expected
values, rn = 0
Pavel Nadolsky (SMU) CTEQ school Lecture 2, 2017-07-22 23
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 66/70
The Hessian method – 1 (Ilkka Helenius)The Hessian method 1
I The fits minimize the �2 defined as
�2 =
NX
i
Di � Ti({a})
�i
�2
where Di is a data point, �i its (statistical) uncertainty and Ti
corresponding theory pointI Correlated and normalization uncertainties need to treated differentlyI Expanding this in terms of parameters ai around minimum �
20 gives
�2⇡ �
20 +X
ij
Hij(ai�a0i )(aj�a
0j ), where Hij =
1
2
@2�
2
@ai@aj
����a=a0
where parameter set {a0} gives the �
20
I If the Hessian matrix H is not diagonal there are correlations betweendifferent parameters
correlations e.g. from momentum conservation and quark sum rules
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 67/70
The Hessian method – 2 (Pavel Nadolsky)Experimental observables Theoretical cross sections PDF parametrizations
Tolerance hypersphere in the PDF space
(a)�
Original parameter basis
(b)�
Orthonormal eigenvector basis
zk
Tdiagonalization and�
rescaling by�
the iterative method
ul
ai
2-dim (i,j) rendition of N-dim (22) PDF parameter space
Hessian eigenvector basis sets�
ajul
p(i)
s0s0
contours of constant 2global
ul: eigenvector in the l-direction
p(i): point of largest ai with tolerance T
s0: global minimump(i)
zl
A hyperellipse ∆χ2 ≤ T 2 in space of N physical PDF parameters{ai} is mapped onto a filled hypersphere of radius T in space ofN orthonormal PDF parameters {zi}
Pavel Nadolsky (SMU) CTEQ school Lecture 2, 2017-07-22 25
Gives central tune and ± tunes along parameter eigenvectors,e.g. 20 parameters for f (x ,Q0) gives 1 + 2× 20 = 41 sets
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 68/70
The shape of proton PDFs (Fred Olness)
Sample PDFs: The rich structure of the proton
Scaling violations are essential feature of PDFs
m2=10 GeV2m2=10,000 GeV2
26
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 69/70
The PDF momentum fractions (Fred Olness)
u-bar
strangestrange
d-bar
down
up
charm
gluon
bottom
Scale m
Momentum Fraction
PDF Momentum Fractions vs. scale m
SU(3)
Scaling violations are essential feature of PDFs
46
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 70/70