Particle Physics Phenomenology 4. Parton distributions and ...home.thep.lu.se/~torbjorn/ppp2018/lec4.pdf · Particle Physics Phenomenology 4. Parton distributions and initial-state

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.


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.


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).


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.


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.


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


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 .


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/


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


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!)


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


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.


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:


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”


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 .


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


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


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


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



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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



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


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, . . .


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


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


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


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.


PYTHIA showers: FSR detailed – 1


PYTHIA showers: FSR detailed – 2


PYTHIA showers: FSR p⊥ – 1


PYTHIA showers: FSR p⊥ – 2


PYTHIA showers: FSR check/tune

Checked/tuned against LEP data


PYTHIA showers: ISR detailed – 1


PYTHIA showers: ISR detailed – 2


PYTHIA showers: ISR p⊥


PYTHIA showers: ISR check/tune

Checked/tuned against Tevatron data, primarily p⊥ of Z0


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


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)


Coherence tests – 1

old normal showers with/without ϕ reweighting:η3: pseudorapidity of third jetα: angle of third jet around second jet


Coherence tests – 2

current-day generators for psuedorapidity of third jet:


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


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


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.


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.


Resummation – 3

Now technical trick: Fourier transform to impact parameter b∫d2p⊥ exp(ibp⊥)

(n∏

i=1


)δ(2)(p⊥ −

n∑i=1

p⊥i )

= exp(ibn∑

i=1

p⊥i )

(n∏

i=1


)

=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))


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


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


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


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


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)!


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!


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


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).


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


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


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

qq

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)



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?



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



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


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 + −



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?


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


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



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



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


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


Warning: individual replicas can look quite weird;only average over 100–1000 members will givesensible answers and uncertainties


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



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


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}


Gives central tune and ± tunes along parameter eigenvectors,e.g. 20 parameters for f (x ,Q0) gives 1 + 2× 20 = 41 sets


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


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


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