Embed Size (px)
Citation preview
Michelangelo ManganoTH Division, CERN
Introduction to the physics of hard probes in hadron collisions:
lecture II
Jet production
12 3
4 13 2
412 4
3gg→gg
qq→gg_
qg→qg
qq’→qq’
qq→qq_ _
gg→qq_
• Inclusive production of jets is the largest component of high-Q phenomena in hadronic collisions
• QCD predictions are known up to NLO accuracy
• Intrinsic theoretical uncertainty "at NLO# is approximately 10%
• Uncertainty due to knowledge of parton densities varies from 5-10% "at low transverse momentum, pT to 100% "at very high pT, corresponding to high-x gluons#
• Jet are used as probes of the quark structure "possible substructure implies departures from point-like behaviour of cross-section#, or as probes of new particles "peaks in the invariant mass distribution of jet pairs#
Phase space and cross-section for LO jet production
d[PS] =d3p1
(2p)22p01
d3p2
(2p)22p02(2p)4 d4(Pin−Pout) dx1 dx2
"a# d(Ein−Eout)d(Pzin−Pz
out)dx1 dx2 =1
2E2beam
"b# d pz
p0 = dy ≡ dh
d[PS] =1
4pSpT d pT dh1 dh2
d3sd pT dh1dh2
=pT
4pS Âi, j
fi(x1) f j(x2)12s Â
kl|M(i j→ kl)|2
The measurement of pT and rapidities for a dijet final state uniquely determines the parton momenta x1 and x2. Knowledge of the partonic cross-section allows therefore the determination of partonic densities f"x#
Some more kinematics
x1,2 =pT
Ebeamcosh y∗ e±yb
Prove as an exercise that
wherey∗ =
h1−h2
2 , yb =h1 +h2
2
We can therefore reach large values of x either by selecting large invariant mass events:
or by selecting low-mass events, but with large boosts "yb large# in either positive of negative directions. In this case, we probe large-x with events where possible new physics is absent, thus setting consistent constraints on the behaviour of the cross-section in the high-mass region, which could hide new phenomena.
pT
Ebeamcosh y∗ ≡ √t→ 1
Example, at the Tevatron
0<η<0.5
0.5<η<11<η<1.51.5<η<2
2<η<2.5
DO jet data, and PDF fits
CDF data, using fits from high-η
region
0<η<0.9
Small-angle jet production, a useful approximation for the determination of the matrix elements and of the cross-section
At small scattering angle, t = (p1− p3)2 ∼ (1− cosq)→ 0and the 1/t2propagators associated with t-channel gluon exchange dominate the matrix elements for all processes. In this limit it is easy to evaluate the matrix elements. For example:
p p’
q q’k ∼ (la)i j (la)kl (2pµ)
1t(2qµ) =
2st
(la)i j (la)kl
where we used the fact that, for k=p-p’<<p "small angle scattering#,
u(p′)gµu(p) ∼ u(p)gµu(p) = 2pµ
Using our colour algebra results, we then get: Âcol,spin
|M|2 =1
N2c
N2c −14
4s2
t2
Noting that the result must be symmetric under s↔u exchange, and setting Nc=3, we finally obtain: Â
col,spin|M|2 =
49
s2 +u2
t2
which turns out to be the exact result!
Quark-gluon and gluon-gluon scatteringWe repeat the exercise in the more complex case of qg scattering, assuming the dominance of the t-channel gluon-exchange diagram:
a,p
j,q’i,q
b,p’
c,k ∼ f abc lci j 2pµ
1t
2qµ = 2st
f abc lci j
Using the colour algebra results, and enforcing the s↔u symmetry, we get: Â
col,spin|M|2 =
s2 +u2
t2
Âcol,spin
|M|2 =s2 +u2
t2 − 49
s2 +u2
uswhich di%ers by only 20% from the exact result even in the large-angle region, at 90o
In a similar way we obtain for gg scattering "using the t↔u symmetry#: Â
col,spin|M(gg→ gg)|2 =
92
(s2
t2 +s2
u2
)compared to the exact result Â
col,spin|M(gg→ gg)|2 =
92
(3− ut
s2 −ust2 −
stu2
)with a 20% di%erence at 90o
Note that in the leading 1/t approximation we get the following result:sgg : sqg : sqq =
94 : 1 : 4
9and thereforeds jet =
∫dx1 dx2 Â
i jfi(x1) f j(x2)dsi j =
∫dx1 dx2 Â
i jF(x1)F(x2)dsgg
where we defined the `e%ective parton density’ F"x#:
F(x) = g(x)+49 Â
i[qi(x)+ qi(x)]
As a result jet data cannot be used to extract separately gluon and quark densities. On the other hand, assuming an accurate knowledge of the quark densities "say from HERA#, jet data can help in the determination of the gluon density
Exercise: prove that the 1/t2 behaviour of the cross-section implies Rutherford’s scattering law.Event
rates in PbPb:
Final-state evolution
• Typical final state resulting from a jet includes many particles• How can a calculation of a 2->2 cross-section have anything to
do with reality? What are the quantities which can be properly described by such calculation?
• Hadronization takes place at time scales much larger than the hard 2->2 scattering "1/ΛQCD>>1/pT#. What happens between the hard process and hadronization can be described by perturbative QCD. However hadronization itself is a phenomenon of catastrophic intensity, which could totally disrupt the structure of the final state, by forcing any pair of colour-connected partons which are separated by more than 1/ΛQCD to bind together.
• Fortunately, QCD is kind enough to ensure that the perturbative evolution prepares a partonic state which will be left almost unperturbed by hadronization. We’ll now discuss how.
Soft gluon emission
p×k = p0 k0 "1-cosq#Þ singularities for collinear "cosqfi1) or soft "k0fi0) emission
Collinear emission does not alter the global structure of the final state, since its preserves its “pencil-like-ness”. Soft emission at large angle, however, could spoil the structure, and leads to strong interferences between emissions from di%erent legs. So soft emission needs to be studied in more detail.In the soft "k0fi0) limit the amplitude simpifies and factorizes as follows:
Aso f t = glai j
(p · ep · k
− pep · k
)ABorn
Factorization: it is the expression of the independence of long-wavelength "soft# emission on the nature of the hard "short-distance# process.
Similar, but more structured, result in the case of more complex colour configurations:
Aso f t = g (la lb)i j
[QeQk− pe
pk
]+ g (lb la)i j
[pepk−Qe
Qk
]The two terms correspond to the two possible ways colour can flow in these diagrams:
The interference between the two colour structures is suppressed by 1/Nc2
As a result, the emission of a soft gluon can be described, to the leading order in 1/Nc
2, as the incoherent sum of the emission from the two colour currents
Âa,b,i, j
|(lalb)i j|2 = Âa,b
tr(lalblbla) =
N2−12 CF = O(N3)
Âa,b,i, j
(lalb)i j[(lbla)i j]∗ = Âa,b
tr(lalblalb) =N2−1
2 (CF −CA
2 )︸ ︷︷ ︸− 1
2N
= O(N)
Angular ordering in soft-gluon emission
dsg = Â |Aso f t|2 d3k(2p)32k0 Â |A0|2 −2pµpn
(pk)(pk)g2 Âeµe∗n
d3k(2p)32k0
= ds0asCF
pdk0
k0df2p
1− cosqi j
(1− cosqik)(1− cosq jk)d cosq
1− cosqi j
(1− cosqik)(1− cosq jk)=
12
[cosq jk− cosqi j
(1− cosqik)(1− cosq jk)+
11− cosqik
]+
12[i↔ j]≡W(i) +W( j)
You can easily prove that:
W(i)→ f inite i f k ‖ j (cosq jk → 1)W( j)→ f inite i f k ‖ i (cosqik → 1)where:
The probabilistic interpretation of W"i# and W"j# is a priori spoiled by their non-positivity. However, you can prove that after azimuthal averaging:
∫ df2p
W(i) =1
1− cosqikif qik < qi j , 0 otherwise∫ df
2pW( j) =
11− cosq jk
if q jk < qi j , 0 otherwise
j2
2
j1
j2
Q(j-j1)
Q(j-j2)
2
= +
Further branchings will obey angular ordering relative to the new angles. As a
result emission angles get smaller and smaller, squeezing the jet
Total colour charge of the system is equal to the quark colour charge. Treating the system as the
incoherent superposition of N gluons would lead to artificial growth of gluon multiplicity. Angular
ordering enforces coherence, and leads to the proper evolution with energy of particle
multiplicities.
The structure of the perturbative evolution therefore leads naturally to the clustering in phase-space of colour-singlet parton pairs "”preconfinement”#. Long-range correlations are strongly suppressed. Hadronization will only act locally, on low-mass colour-singlet clusters.
HadronizationHadronizationAt the end of the perturbative evolution, the final state consists ofquarks and gluons, forming, as a result of angular-ordering, low-mass clusters of colour-singlet pairs:
p
p p
pNp
p p
p pp p
p p
p p
p p
p p
N p
Thanks to the cluster pre-confinement, hadronization is local and independent of the nature of the primary hard process, as well as of the details of how hadronization acts on di%erent clusters. Among other things, one therefore expects: N(pions) = C N(gluons),
C=constant~2
Heavy quark production
(p1−Q)2−m2 =−2p1 · Q =− s2(1−bcosq)
2p1 · Q≥ s2(1−b) =
s2(1+b)
(1−b2) =2m2
1+b≥ m2
⇒ Total HQ cross-sections are finite, and calculable in pQCD
Singularity structure of t-channel propagator:
"for pedagogical introduction, see: hep-ph/9711337#
Matrix elements:
Phase-space:
Total cross-section:
Di%erential distributions:
Very strong rapidity correlation: ∆y<1
Total cross-sectionsI-V
I: di
%ere
nt in
put
para
met
ers
Evolution of a heavy quark
mQ≠0
Aso f t = glai j
(p · ep · k
− pep · k
)ABorn same as for mQ=0. However now:
pk = p0 k0 "1 - βcosθ #, where β is the quark velocity, β<1.
Therefore radiation in the very forward region "collinear radiation# is suppressed. A massive quark therefore looses less energy during the evolution, and its “quenching profile” will be di%erent than the one of a light quark or gluon.
Some interesting questions
d
D
Trigger jet, J1
Quenched jet, J2
* Plot dN/dφ vs d/D. Possible?
* Particle pT in J2 is degraded; this means that more particles must share the same amount of momentum -> increased multiplicity. Can it be measured?
* What do we learn about the QGP form the study of quenching? What is the quantitative relation between the parameters of the QGP Eq of State and the fragmentation properties of J2?
* What are the PDG-like parameters that we can extract from these measurements?
