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

NBI, Copenhagen, 4 October 2011

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 2/39

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 plotting interface:http://durpdg.dur.ac.uk/hepdata/pdf.html


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

At NLO PDFs are not physical objects and not required positivedefinite everywhere (and neither are cross sections):

Dangerous for LO MCs: recently introduce new MC-adapted PDFs• allow

∑i

∫ 10 xfi (x ,Q2) > 1 as “built-in K factor”

• use NLO-calculated pseudodata as target for tunes


PDF sets

Current usage of LO PDFs (needed for LO MCs):• conventional: CTEQ 5L, CTEQ 6L, CTEQ 6L1, MSTW 2008 LO• MC-adapted: MRST LO* and LO**; CT09 MC1, MC2 and MCS

Current usage of NLO PDFs:

MSTW 08

CT 10

NNPDF 2.5

HERAPDF 1.5

. . .

also some NNLO sets available

More in talk by Alberto Guffanti


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–Ryskinnonlinear 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


Initial-State Shower Comparison – 2

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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


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 ∼ 7 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


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