part1 - Desy

Transcription

part1 - Desy

From HERA to the LHC
H. Jung (DESY)
Hannes Jung, From HERA to the LHC, Graduiertenkolleg, Bullay 2007
Page 1
H. Jung (DESY)
or
Why HERA physics is important
for discoveries at LHC !
“…The mechanic, who wishes to do his work well, must first sharpen his tools …”
—Chapter15, “The Analects” attributed to Confucius, translated by James Legge.
(from X. Zu talk at DIS05)
Page 2
Many thanks to all conveners and authors !
hepph/0601012
hepph/0601013
Available on request
from CERN/DESY libs
e
ag
>6
p
0
5
s
Next working group week:
29. Oct. - 2. Nov. 2007, DESY
next workshop
May 2008 CERN
Page 3
H. Jung (DESY)
HERA and the structure of the proton
QCD is challenging
from inclusive x-section measurements to detailed investigations of
QCD
measurements of hadronic final states:
➔ lead to a detailed understanding of QCD
➔ jets, heavy flavours
next lecture:
➔ implications and applications for LHC
➔ PDFs, multiparton interactions, etc
lectures based on lecture series:
“QCD & collider physics” H.Jung,J. Bartels University HH, 2005 -2007
Contributions to “HERA and the LHC” workshops: www,desy.de/~heralhc
Page 4
HERA collider and experiments
E
pe = 27:6 GeV
s = 319 GeV
Ep = 920 GeV
! E = 54 TeV
¯xed target
Page 5
What is HERA doing in Hamburg ?
electron proton collider HERA √s = 320 GeV HERA: QCD
structure of the proton
E
pe = 27:6 GeV
s = 319 GeV
Ep = 920 GeV
! E = 54 TeV
¯xed target
Physics Program:
structure functions, parton
density functions
jets
heavy quarks
diffraction in QCD
high energy behavior of QCD
precision machine for QCD, like
LEP was for electroweak...
running until mid 2007
Page 6
How was HERA performing?
Page 7
Why HERA and LHC ?
proton proton collider LHC
√s = 14 TeV LHC: Higgs, SUSY etc., but mostly QCD...
Page 8
A typical ep event at HERA
Page 9
Kinematics
d¾ep
2
°¤p
2
=
F
(y;
Q
)¾
(W;
Q
)
°=e
2
dydQ
with
Page 10
Picture of the Proton
Flavor sum rules for proton:
Z 1
dx uV (x) = 2
0
Z
Z
Z
1
dx dV (x) = 1
0
}
p=(uud)
dx x q(x) » 0:1 [0:9 + 0:95 + 0:85 + 0:7 + 0:35 + 0:15 + 0:1 + 0:05] = 0:1¢4:05 = 0:405
dx x¹
q (x) » 0:1 [0:42 + 0:2 + 0:06 + 0:03 + 0:01] = 0:1 ¢ 0:72 = 0:072
Page 11
Picture of the Proton
Flavor sum rules for proton:
Z 1
dx uV (x) = 2
0
Z
1
dx dV (x) = 1
0
}
p=(uud)
Momentum sum of quarks:
XZ
q
1
0
dx x[q(x) + q¹(x)] » 0:5
Where are the other 50 % of the
proton's momentum ?
Page 12
Naive picture of the proton: F2
From Halzen & Martin: Quarks & Leptons, p201
Page 13
Structure functions from HERA
Proton structure
function does not
depend on Q2 for
large x
F2 scales ...
Quarks are pointlike
constituents of
proton
BUT things change
at smaller x..... and
smaller Q2
Page 14
What about the
scaling violations ?
15
F2(x,Q2): DGLAP evolution equation
QPM: F2 is independent of Q2
Q2 dependence of structure function:
DokshitzerGribovLipatovAltarelliParisi
➔
Probability to find
parton at small x
increases with Q2
F2 = ➔ Test of theory: Q2 evolution of F2(x,Q2) !!!!!
Page 16
DGLAP, collinear factorization and Pqq
2
jM Ej
=
=
·
¸
¢ 4 ¡t^ s^ 2^
uQ
32¼
¡ +
3 s^
t^
s^t^
· 2
¸
2
¡ 2
¢ 4 ¡1 Q (1 + z )
2
32¼ eq ®® s
+ ¢¢¢
3 t^
z(1 ¡ z)
2
¡
e2q ®® s
2
Q2
z=
s^ + Q2
integral over kt generates log, BUT what is the lower limit
d¾
2 ®s 1
= ¾0 eq
[Pqq (z) + ¢ ¢ ¢ ]
2
2
dk?
2¼ k?
4 1 + z2
Pqq (z) =
3 1¡z
¾
QC DC
=
2 ®s
¾0 eq
2¼
·
Pqq (z) log
µ
2
Q (1 ¡ z)
Â2 z
¶
4¼2 ®
¾0 =
s^
+ ¢¢¢
¸
Page 17
Collinear factorization
2
qi (x ;¹ ) =
qi0 (x)
Z
®s
+
2¼
q0 (x)
1
x
·
µ ¶
µ 2¶
µ ¶¸
d» 0
x
¹
x
qi (»)Pqq
log
+ Cq
+ :::
»
»
Â2
»
bare distributions
are not measurable (like the bare charges .... )
collinear singularities are absorbed into this bare distributions at a
factorization scale ¹2 À Â2 , defining renormalized distributions
F2
=
x
X
·
µ ¶
µ 2¶
¸
Z
dx2
®s
x
Q
2
eq q0 (x) +
q0 (x) Pqq
log
+ Cq (z; ::)
2
x2
2¼
x2
Â
Z
· µ
¶
µ ¶
µ 2¶
¸
dx2
x
®s
x
Q
2
q(x2 ; ¹ ) ± 1 ¡
+
Pqq
log
+C
2
x2
x2
2¼
x2
¹
now F2 becomes:
F2 = x
X
e2q
separating or factorizing the long distance contributions to structure
functions is a fundamental property of the theory
factorization provides a description for dealing with the logarithmic
singularities, there is arbitrariness in how the finite (non-logarithmic) parts
are treated.
Be aware that factorization is just an approximation to the full story
Page 18
Collinear factorization: DGLAP
introduce new scale ¹ À Â and include soft, non-perturbative physics
into renormalised partonZdensity:
·
µ ¶
µ ¶¸
µ 2¶
1
2
2
qi (x ;¹ ) =
2
®s
+
2¼
qi0 (x)
x
d» 0
qi (»)Pqq
»
x
»
0
+ g (»)Pqg
x
»
log
¹
Â2
DokshitzerGribovLipatovAltarelliParisi equation (take derivative of the above eq):
V.V. Gribov and L.N. Lipatov Sov. J. Nucl. Phys. 438 and 675 (1972) 15, L.N. Lipatov Sov. J. Nucl. Phys 94 (1975) 20,
G. Altarelli and G. Parisi Nucl.Phys.B 298 (1977) 126, Y.L. Dokshitser Sov. Phys. JETP 641 (1977) 46
2
dqi (x ;¹ )
®s
=
d log ¹2
2¼
Z
1
x
·
µ ¶
µ ¶¸
d»
x
x
qi (»; ¹2 )Pqq
+ g(»; ¹2 )Pqg
»
»
»
BUT there are also gluons....
2
dg(x ;¹ )
®s
=
d log ¹2
2¼
Z
1
x
#
µ
¶
µ
¶
d» X
x
x
2
2
qi (»; ¹ )Pgq
+ g(»; ¹ )Pgg
»
»
»
"
i
Page 19
What the hell
is factorization ?
20
Collinear factorization
(V h)
F2
(x ;Q2 )
=
X Z
i=f;f¹;G
0
1
(V i)
d»C2
Ã
2
x Q
; 2;
» ¹
!
see handbook of pQCD, chapter IV, B
¹2f
2
2
2
;
®
(¹
)

f
(»;
¹
;
¹
)
s
i=h
f
2
¹
Factorization Theorem in DIS (Collins, Soper, Sterman, (1989) in Pert. QCD, ed. A.H. Mueller, Wold
Scientific, Singapore, p1.)
generalization of the parton model result
(V i)
hard-scattering function C2
is infrared finite and calculable in pQCD,
depending only on vector boson V, parton i, and renormalization and
factorization scales. It is independent of the identity of hadron h.
2
2
pdf fi=h(»; ¹f ; ¹ ) contains all the infrared sensitivity of cross section,
and is specific to hadron h, and depends on factorization scale. It is
universal and independent of hard scattering process.
Generalization: applies to any DIS cross section defined by a sum over
hadronic final states .... but be careful what it really means....
explicit factorization theorems exist for:
diffractive DIS (... see above....)
Drell Yan (in hadron hadron collisions)
single particle inclusive cross sections (fragmentation functions)
Page 21
➔
Factorization is an approximation !!!
Page 22
Factorization proofs and all that ...
About factorization proofs (Wu-Ki Tung, pQCD and the parton structure of the nucleon, 2001, In *Shifman, M.
(ed.): At the frontier of particle physics, vol. 2* 887-971)
d¾
=
dy
XZ
a;b
1
xA
d»A
Z
1
xB
³³ m ´p ´
d^
¾
(¹)
ab
a
b
d»B fA
(»A ; ¹)fB
(»B ; ¹)
+O
dy
P
The problem with Drell-Yan:
initial state interactions...
factorization here does not
hold graph-by-graph but
only for all ....
Page 23
Factorization is violated ...
arXiv:0705.2141v1 [hepph]
We come to that point later ....
Page 24
Solving the
DGLAP equations
25
Solving DGLAP equations ...
Different methods to solve integro-differential equations
brute-force (BF) method (M. Miyama, S. Kumano CPC 94 (1996) 185)
df (x)
f (x)m+1 ¡ f (x)m
=
dx
¢xm
Z
f (x)dx =
X
f (x)m ¢xm
Laguerre method (S. Kumano J.T. Londergan CPC 69 (1992) 373, and L. Schoeffel Nucl.Instrum.Meth.A423:439-445,1999)
Mellin transforms (M. Glueck, E. Reya, A. Vogt Z. Phys. C48 (1990) 471)
QCDNUM: calculation in a grid in x,Q2 space (M. Botje Eur.Phys.J. C14 (2000) 285-297)
CTEQ evolution program in x,Q2 space: http://www.phys.psu.edu/~cteq/
QCDFIT program in x,Q2 space (C. Pascaud, F. Zomer, LAL preprint LAL/94-02, H1-09/94-404,H1-09/94-376)
MC method using Markov chains (S. Jadach, M. Skrzypek hep-ph/0504205)
Monte Carlo method from iterative procedure
brute-force method and MC method are best suited for
detailed studies of branching processes !!!
Page 26
We have to deal with divergencies....
collinear divergencies factored into renormalized parton distributions
what about soft divergencies ?
treated with “plus” prescription
1
1
!
1¡z
1 ¡z+
Z
with
0
1
f (z)
dz
=
(1 ¡ z)+
soft divergency treated with Sudakov form factor:
· Z
¢(t) = exp ¡
t0
resulting in
@
t
@t
and
µ
f
¢
¶
t
1
=
¢
Z
dt
t0
0
Z
zmax
®s ~
dz P (z)
2¼
0
1
f (z) ¡ f (1)
dz
(1 ¡ z)
¸
dz ®s ~
P (z)f (x =z; t)
z 2¼
f (x ;t) = ¢(t)f (x ;t0 ) +
zmax
Z
Z
t
t0
0
dt ¢(t)
t0 ¢(t0 )
Z
zmax
dz ®s ~
P (z)f (x =z; t0)
z 2¼
Page 27
Sudakov form factor: all loop resum...
g ! gg
®
¹s
®
¹s
~
Splitting Fct P (z) = 1 ¡ z + z + :::
Sudakov form factor .... all loop resummation
µ Z
¶
Z
0
dq ®s ~
¢s = exp ¡ dz
P (z)
0
q 2¼
µ Z
¶1
µ Z
¶2
Z
Z
dq ®s ~
1
dq ®s ~
¢s = 1 + ¡ dz
P (z) +
¡ dz
P (z) :::
q 2¼
2!
q 2¼
"
P~ (z) 1 ¡
Z Z
dq ®s ~
dz
P (z) +
q 2¼
#
µ Z Z
¶2
1
dq ®s ~
¡
dz
P (z) + :::+
2!
q 2¼
Page 28
and again DGLAP evolution ....
@
t f (x ;t) =
@t
differential form:
Z
differential form using f =¢s with
µ Z
¢s (t) = exp ¡
zmax
dz
x
@ f (x ;t)
t
=
@t ¢s (t)
Z
Z
t
t0
³x ´
dz ®s
P+ (z) f
;t
z 2¼
z
¶
®s dt ~
P (z)
0
2¼ t
0
dz ®s P~ (z) ³ x ´
f
;t
z 2¼ ¢s (t)
z
integral form
f (x ;t) = f (x ;t0 )¢s (t) +
Z
dz
z
Z
³x ´
dt0 ¢s (t) ~
0
¢
P
(z)f
;
t
t0 ¢s (t0 )
z
no – branching probability from t0 to t
Page 29
and again DGLAP evolution ....
@
t f (x ;t) =
@t
differential form:
Z
differential form using f =¢s with
µ Z
¢s (t) = exp ¡
zmax
dz
x
@ f (x ;t)
t
=
@t ¢s (t)
Z
Z
t
t0
³x ´
dz ®s
P+ (z) f
;t
z 2¼
z
We need only:
¶
®s dt ~
P (z)
0
2¼ t
0
1
P (z) !
1¡z
dz ®s P~ (z) ³ x ´
f
;t
z 2¼ ¢s (t)
z
integral form
f (x ;t) = f (x ;t0 )¢s (t) +
Z
dz
z
Z
³x ´
dt0 ¢s (t) ~
0
¢
P
(z)f
;
t
t0 ¢s (t0 )
z
no – branching probability from
to
Page 30
Solving integral equations
Integral equation of Fredholm type:
Á(x) = f (x) + ¸
solve it by iteration (Neumann series):
Á0 (x)
Á1 (x)
Á2 (x)
Án (x)
=
=
=
=
f (x)
f (x) + ¸
f (x) + ¸
n
X
Z
Z
Z
b
K(x ;y)Á(y)dy
a
b
K(x ;y)f (y)dy
a
b
2
K(x ;y1 )f (y1 )dy1 + ¸
a
Z
a
b
Z
b
K(x ;y1 )K(y1 ; y2 )f (y2 )dy2 dy1
a
¸i ui (x)
i=0
u0 (x)
=
u1 (x)
=
f (x)
Z b
K(x ;y)f (y)dy
a
un (x)
=
Z
a
b
Z
a
b
K(x ;y1 )K(y1 ; y2 ) ¢ ¢ ¢ K(yn¡1; yn )f (yn )dy2 ¢ ¢ ¢ dyn
with the solution:
Á(x) = lim qn (x) = lim
n!1
n!1
n
X
¸i ui (x)
i=0
Page 31
DGLAP re-sums leading logs...
f (x ;t) = f (x ;t0 )¢s (t) +
Z
Z
dz
z
³x ´
dt0 ¢s (t) ~
0
¢
P
(z)f
;
t
t0 ¢s (t0 )
z
solve integral equation via iteration:
f0 (x ;t)
f1 (x ;t)
=
=
f (x ;t0 )¢(t)
f (x ;t0 )¢(t) +
Z
from t' to t w/o branching
Z
t
0
t0
dt ¢(t)
t0 ¢(t0 )
branching at t'
from t0 to t' w/o branching
dz ~
P (z)f (x =z; t0 )¢(t0 )
z
Page 32
DGLAP re-sums leading logs...
f (x ;t) = f (x ;t0 )¢s (t) +
Z
dz
z
Z
³x ´
dt0 ¢s (t) ~
0
¢
P
(z)f
;
t
t0 ¢s (t0 )
z
solve integral equation via iteration:
f0 (x ;t)
f1 (x ;t)
=
=
=
f2 (x ;t)
=
f (x ;t)
=
f (x ;t0 )¢(t)
Z
from t' to t w/o branching
t
0
Z
branching at t'
from t0 to t' w/o branching
dt ¢(t)
dz ~
0
P
(z)f
(x
=z;
t
)¢(t
)
0
0
0
z
t0 t ¢(t )
t
f (x ;t0 )¢(t) + log A ¢(t)f (x =z; t0 )
t0
t
f (x ;t0 )¢(t) + log A ¢(t)f (x =z; t0 ) +
t0
1
2 t
log
A A ¢(t)f (x =z; t0 )
2
t0
µ ¶
X 1
t
lim fn (x ;t) = lim
log n
An ¢(t)f (x =z; t0 )
n!1
n!1
n!
t0
n
f (x ;t0 )¢(t) +
log t
DGLAP resums to all orders !!!!!!!!!!!!!!!!
Page 33
DGLAP evolution equation... again...
for fixed x and Q2 chains with different branchings contribute
iterative procedure, spacelike parton showering
f (x ;t) =
1
X
fk (xk ; tk ) + f0 (x ;t0 )¢s (t)
k=1
Page 34
F2(x,Q2): DGLAP evolution equation
QPM: F2 is independent of Q2
Q2 dependence of structure function:
DokshitzerGribovLipatovAltarelliParisi
➔
Probability to find
parton at small x
increases with Q2
F2 = ➔ Test of theory: Q2 evolution of F2(x,Q2) !!!!!
Page 35
Is our theory
working at all ?
36
Q2 dependence of F2(x,Q2)
➔
F2 is rising with Q2 at
small x
Scaling violations !!!!!
➔ Well described by
theory ...
➔
Page 37
x dependence of F2(x,Q2)
new level of precision reached:
~1%
DGLAP fits data well even at
low Q2
strong rise towards small x
F2 ~ xq(x,Q2)
➔ probability to find parton at
small x increases
➔ How can rising F2 be
understood ?
➔ Does rise continue forever ?
➔ What limits F2 ?
Page 38
What is happening at small x ?
For x ! 0 only gluon splitting function matters:
Pgg = 6
Pgg » 6
µ
1¡z
z
+
+ z(1 ¡ z)
z
1¡z
1
for z ! 0
z
¶
=6
µ
1
1
¡ 2 + z(1 ¡ z) +
z
1¡z
¶
evolution equation is then:
f (x ;t) = f (x ;t0 )¢s (t) +
Z
dz
z
at small z: ¢s (t) ! 1
3®s
x g(x ;t) =
¼
when
f (x ;t0 )
Z
t
t0
d log t
0
Z
Z
1
x
³x ´
dt0 ¢s (t) ®s
0
¢
P
(z)f
;
t
t0 ¢s (t0 ) 2¼
z
d»
»g(»; t0 )
»
with
t = ¹2
is neglected (compared to evolved piece .... )
Page 39
Estimates at small x: DLL
3®s
x g(x ;t) =
¼
Z
t
0
d log t
t0
Z
1
x
d»
»g(»; t0 )
»
with
t = ¹2
use constant starting distribution at small t:
x g1 (x ;t)
=
x g2 (x ;t)
=
..
.
x gn (x ;t)
=
x g(x ;t)
=
x g(x ;t)
»
3®s
t
1
log
log C
¼
t0
x
µ
¶2
3®s 1
t 1
1
log
log
C
¼ 2
t0 2
x
µ
¶n
1 1 3®s
t
1
log
log
C
n! n!
¼
t0
x
¶n
X µ 1 ¶2 µ 3®s
t
1
log
log
C
n!
¼
t
x
0
n
µ r
¶
3®s
t
1
double leading log C exp 2
log
log
¼
t0
x
approximation (DLL)
Page 40
Results from DLL approximation
DLL arise from taking small x limit of x g(x ;t)
splitting fct:
log 1=x from small x limit of
splitting fct
log t=t0 from t integration...
gives evolution length....
®s
softer for running
DLL gives rapid increase of gluon density
from a flat starting distribution
gluons are coupled to F2... strong rise of
»
µ r
¶
3®s
t
1
C exp 2
log
log
¼
t0
x
Q2 ~2 F2 at small x:
consequences: ● rise continues forever ???
● what happens when too high gluon density ?
●
●
Page 41
Parton Distribution Functions
at small x sea quarks dominate
gluon density is very large
Page 42
PDF uncertainty: improvements
Using jets together with F2 (at large Q )
2
from C. Gwenlan, A. CooperSarkar, C. TargettAdams
in HERA – LHC workshop proceedings hepph/0601012
quark and gluon uncertainties
high statistics from HERA II is important (assumed 700 pb1)
Error on LHC jet xsection reduced !!!
Page 43
Average of HERA data
From M. CooperSakar, C. Gwenlan and S. Glazov
Average H1&ZEUS data sets
●
Combined PDF fit to H1& ZEUS ●
Much reduced uncertainties ....
Model independent analysis of data desirable
Activities started to get HERA – PDF !!!!!
Page 44
HERA measurements: FL
The gluon distribution
R. Thorne, hepph/0511351
From J. Feltesse, C. Gwenlan, S. Glazov, M. Klein, S. Moch
FL / ®s x g(x ;Q2 )
Precision measurement of FL at
●
●
Ep = 460 GeV and 5-10 pb-1
cleanest for gluon
crucial test of QCD at higher
orders and consistency of theory
➔
Is measured at HERA !! Hannes Jung, From HERA to the LHC, Graduiertenkolleg, Bullay 2007
Page 45
Structure Functions at HERA
Structure Functions at
HERA are well described
by
NLO DGLAP
Page 46
by
NLO DGLAP
is that all we can learn ?
Page 47
by
NLO DGLAP
HERA and QCD is more
and much richer !!!
Page 48
Conclusions
HERA physics is very rich:
from inclusive x-section measurements to detailed investigations of QCD
measurements of hadronic final states:
jets, heavy flavors
➔ lead to a detailed understanding of QCD
➔ what about saturation, diffraction multi-parton interactions ?
next lecture
➔ HERA implications for LHC
➔ PDFs, small x, multiple interactions, diffraction
Understanding of QCD at
high energies is still challenging !
Page 49

part1 - Desy

Transcription

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