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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.
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 3/39
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.
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 3/39
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 4/39
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 5/39
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 6/39
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 7/39
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 8/39
PDF examples – 1
Convenient plotting interface:http://durpdg.dur.ac.uk/hepdata/pdf.html
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 9/39
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 10/39
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
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 11/39
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
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 12/39
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 13/39
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 14/39
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 15/39
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 16/39
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 17/39
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
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 18/39
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 19/39
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 20/39
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 21/39
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
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 22/39
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 23/39
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 24/39
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 25/39
PYTHIA showers: FSR detailed – 1
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 26/39
PYTHIA showers: FSR detailed – 2
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 27/39
PYTHIA showers: FSR p⊥ – 1
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 28/39
PYTHIA showers: FSR p⊥ – 2
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 29/39
PYTHIA showers: FSR check/tune
Checked/tuned against LEP data
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 30/39
PYTHIA showers: ISR detailed – 1
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 31/39
PYTHIA showers: ISR detailed – 2
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 32/39
PYTHIA showers: ISR p⊥
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 33/39
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 34/39
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 35/39
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 36/39
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 37/39
Coherence tests – 2
current-day generators for psuedorapidity of third jet:
Torbjorn Sjostrand PPP 4: Parton distributions and initial-state showers slide 38/39
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 39/39