Proton-proton scattering
from low to LHC-energies
Finn Ravndal, Dept of Physics, UiO
• Coulomb scattering
• Low-energy strong interactions
• Regge poles and the pomeron
• The Froissart bound
• Feynman parton model
• Recent ideas and conclusion
HEP-coll, 13/11 - 2009
Coulomb scattering
in CM-frame:
e
2
Q
2
2
Q = 4p sin θ/2
e
ressed by the free Green’s function
dσ
2
Differential cross-section:
= |f (Q)|
1
dΩ
G0 (E) =
E − H0 + iε
m
V (Q)
In first Born approximation f (Q) =
2π
approximation
(1)
! b
V
with
=
"p
|
V
!
p p
(Q)
| p# =
=
Z
3
−iQ·r
d r V (r) e
g on potential of finite range R
in CM-frame where proton momentum
V(r)
and reduced mass m = M/2.
p = mv
e2
V (r) =
4πr
Coulomb potential
with Feynman diagram
e
= V (Q) =
+
Q
e
+
e2
= 2
Q
+ !!!
α
giving scattering amplitude f (Q) =
2
4E sin (θ/2)
=
+
where E = p2 /2m and
α = e2 /4π = 1/137
Figure 6: Coulomb propagator as an infinite sum.
or Kummer function M (a, b; x)[23]. For the repulsive Coulomb potential VC = α/r the
Full Coulomb propagator
=
+
=
+
+
+ !!!
results in non-perturbative scattering amplitude
Figure 6: Coulomb propagator as an infinite sum.
α
−iη ln sin2 (θ/2)
= M (a, b; x)[23].2For the repulsive
e Coulomb potential V = α/r the
or f
Kummer
function
C (θ)
in-state solution with 4E
outgoing
spherical
waves in the future is
sin
(θ/2)
C
1
ψp(+) (r) = e− 2 πη Γ(1 + iη)M (−iη, 1; ipr − ip · r) eip·r
(20)
η
=
α/v
where
parameter
gives effective strength
The corresponding out-state has incoming spherical waves in the distant past and is given
by the wavefunction
of the Coulomb interaction.
1
ψp(−) (r) = e− 2 πη Γ(1 − iη)M (iη, 1; −ipr − ip · r) eip·r
(21)
Coulomb cross-section is unmodified:
dσ
α
=
2
dΩ
4E sin (θ/2)
!
"2
Probability to find two protons at zero separation:
2πη
|ψ(0)| = 2πη
e
−1
2
Becomes exponentially small when η > 1
i.e. when p < 10 MeV. Thus Coulomb interaction
dominates for energies E < 1 MeV.
E − p2 /2MV
+ef
iε f
= Cδ(r)
Low-energy strong interactions
mentary vertex
4π
a
M
ere coupling constant C =
in first Born
Effective potential, valid
atMeV
low energies:
E < 100
roximation.
Vef f = Cδ(r)
her order Born corrections can now be obtaine
4π
ctive
field
theory
order
Born
correction
from
Feynman
diagram
where
coupling
constant
C = M a in first Born
described
by effective
Langrangian
approximation.
1 ∗ 2
C ∗ 2
∗
L = iψ ψ̇ +
ψ ∇ ψ − (ψ ψ)
2M corrections 2can now be obta
Higher order Born
4
dσ
effective
field
theory
2 theor
ng standard
field-theoretic
perturbation
Cross-section to lowest order
=a
-relativistic propagator
1
∗
∗
2
dΩ C
∗
L = iψ ψ̇ +
ψ ∇ ψ
m− (ψ ψ)
2M a =
2
where scattering length
C
1
2
2π
0 (E, p) =
using G
standard
field-theoretic
perturbation the
2
3 p2 − k 2 + i"
I0 (p) (2π)
=
(2π)3 p2 − k 2 + i"
Higher order
order
corrections
Higher
order diagram
with
bubbles
Higher
diagram
with
n nbubbles
...
+ ...... +
+ ....
...
with value C I (p) with bubble integral
2
0
I0 (p) with bubble
integral
gives
similarly C n+1 I0n (p). Total Z
scattering
amplitude:
3
2M
d
k
Z
ˆ I0 (p) = 2
˜
given
3
2M T (p)integral
d3 k by bubble
= C 1 + CI0 + (CI0 ) +
(CI03) p
+2· ·−
· k2 +
(2π)
0 (p) =
(2π)3 p2 − k 2 + i"
1
C
2
=
= n
n+1
− CI0
1/C − I0 (p)
C
I01(p)
bubbles
gives
similarly
. Total
scattering
Fullwith
scattering
amplitude:
Higher
order
diagram
with n
diagram
n
i"
amplitude
bubbles
Bubble integral
d = 3 dimensions. ˜
ˆ I0 (p) is divergent in
2
3
TRegularize
T (p)
= Cwith
1 +cut-off
CI0 Λ+≈(CI
)
+
(CI
)
+ ···
0
1/R 0giving
„
«
M
1i πp
C
Λ
+
I
(p)
=
−
0
=2π2
=
1 − CI0
1/C −2I0 (p)
which implies that bare coupling C = C(Λ).
...
...
ove cut-off Λ by1 introducing
renormalized co
1
M
Λ
ularize
with
cut-off
Λ
≈+ 1/Rregularization:
giving
= cut-off
Bubble
integral
with
C
C
2π
ant
rmalization
«
Renormalization„
2
R
with value CR =
4π
a
M
which
amplitude
i
M gives scattering
πp
+M
IΛ0 (p)
=1
− 4π2 1Λ
Λ
1 renormalized
ve cut-off
by
introducing
coupling
Remove cut-off
renormalized
co
T (p) 2π
= Λ by introducing
2
=
+
M 1/a + ip
2
ant Remove
C
C
2π
R by introducing
constant
cut-off
renormalized
and is now
unitary.ΛDifferential
cross-section
ch implies that
bare
coupling
C = C(Λ).
1
1
M
Λ
coupling constant
1
MΛ
4π
= dσ + a21
gives
amp
value CR C=R M dΩaC =which
2
+ scattering
2π
1+
(ap) =
2
CR
2
C
2π
Renormalized
4π coupling CR goes to
5 zero as Λ → ∞, no scattering
scattering
amplitude
value C
4π
R = M a which gives
4π
1
aTheory
which
gives
value
CR
= limit.
trivial. scattering amp
onwith
δ-function
potential
in this
is then
M
T (p) =
Instead of cut-off regularization,
dimensional
4π
1 1/a
M
M
+
ip
4πwith
1 aand
T (p)
= introduced by Kaplan,
T was
Thus
=
C
R
regularization
Savage
T (p)
M 1/a
+ ip=
4π
Wise (1998) by the name PDS givingM 1/a + ip
is now unitary. “Differential
cross-section
Z
”
s nowand
unitary.
Differential
differential
cross-section
becomes
µ "
ddcross-section
k
2M
I
(p)
=
0
and is now unitary.
Differential cross-section
2
(2π)d p2 − k 2 + i#
2
a
dσ
=
a22
=(ap)
dΩ
1
+
2
dΩ pole1 at
+d(ap)
where d = 3 − # after subtracting
= 2, i.e.
dσ
dΩ
2
´
a
M`
+ ip
== − 4π µ dσ
1 + (ap)2
For proton-proton
scattering with E << 100 MeV
must include Coulomb
repulsion.
Blatt & Weisskopf: Theoretical Nuclear Interactions
E = 3 MeV
Full scattering amplitude:
f (θ) = fC (θ) + fS
h Coulomb-dressed strong interaction bubble J0 (p):
Coulomb-modified
strongnow
interactions:
Strong interactions
modified by Coulomb effects
=
+
+
+ !!!
fS =
+
+
+
!!!
such Feynman diagrams again form geometric series
ng scattering
amplitude
Again form
geometric
series strong interaction bubble J0 (p):
with
Coulomb-dressed
C0
=bubble J0 (p) is amplitude for protons
TSCf
(p)
S=
Coulomb-dressed
1= − C0 J0 (p)
+
+
+ !!!
Cη2
to move from zero separation back to zero separation,
ves where
Coulomb-modified
length
Coulomb-dressed
bubble
now is
! scattering
i.e. J0 (p) =AllGsuch
(E;
r
=
0,
r
=
0)
or
C
Feynman diagrams again form geometric series
1 giving
4π scattering
1
Z
3 −amplitude
=
αM
H(η)
2πη(k)
d
k
1
a
M
T
(p)
C
SC
J0 (p) = M
C20 − k 2 + i#
3 e2πη(k) 2− 1 p
(2π)
TSC (p) = Cη
1 − C0 J0 (p)
ere standard function
Can be done
Givesanalytically
Coulomb-modified
1 (!): scattering length
H(η) = ψ(iη) +
2
»− log(iη)√
–
4π
4π
Finn Ravndal, University of Oslo
with
constant
CE =Kong
0.5772and
. . ..Ravndal (1998)
Can Euler
be done
analytically
and gives
Coulomb-modified
scattering
length:
Gives
Coulomb-modified
scattering
length
»
√
1
1
µ π
3
= a δ-function
− αM ln potential
+ 1 − CE
Scattering
on
aC
a(µ)
αM
2
–
when divergence 1/# is absorbed in strong interaction
Renormalization
Blatt and Jackson (1950)
coupling
from quantum mechanics
in aand
potential
model.
Proton-neutron
scattering
the
deuteron
D
4π
1
C0 (µ) =
M 1/a(µ) − µ
Proton-proton
and Coulomb effects
Proton-protonscattering
fusion:
Expect a(µ) ≈ apn ≈ −20 fm for µ ≈ Λ ≈ mπ .
+
Solar
fusion
p + pscattering
→ D + lengths
e + νfirst
e
Relation
between
derived by
Blatt and Jackson (1950) in potential model:
Conclusion
1
1
»
1
–
Regge poles and the pomeron
E > 1 GeV
s = (p1 + p2 )2
= 2mplab
2
t = (p1 − p3 )
= −Q2
2
2
= −4p sin (θ/2)
Differential x-section:
Total x-section:
dσ
2
= π|T (s, t)|
dt
σT = 4πImT (s, 0)
Including Coulomb interaction
2α 2
F (t)eiφ(t) + T (s, t)
T (s, t) =⇒
t
1
with proton form factor F (t) =
(1 − t/0.71)2
and calculable phase φ(t)
ISR:
√
s = 24 GeV
p) =
1
E − p2 /2M + iε
ertex Strong
amplitude
=
correction from Feynman diagram
T (s, t) = βR (t)sαR (t)−1 + βP (t)sαP (t)−1
Regge trajectory:
αR (t) = 0.5 +
!
αR t
4
Pomeron trajectory:
αP (t) = 1.0 +
!
αP t
√
σT (s → ∞) = 4πImβP (0) + O(1/ s)
Total x-section should approach a constant value
at high energies!
Pomeranchuk theorem (1958):
σT (AB) = σT (ĀB)
as s → ∞
Isaak Pomeranchuk (1913 - 1966)
Impact parameter representation from partial wave
expansion:
∞
!
" 2iδ (s)
#
1
(2" + 1)P! (cos θ) e ! − 1
f (s, θ) =
2ip
!=0
Eikonal approximation: Replace sum over partial waves
with integral over impact parameter b:
1:
2:
! + 1/2 −→ pb
" ∞
"
∞
!
−→
d! = p
!=0
0
∞
db
0
3:
P! (cos θ) −→ J0 [(2" + 1) sin θ/2]
4:
2δ! (s) → E(s, b)
! 2 "
#
d b
T (s, Q) = i
1 − eiE(s,b) eiQ·b
2π
=⇒
Eikonal expansion:
1
i
T (s, Q) = E(s, Q) + E ⊗ E − E ⊗ E ⊗ E + . . .
2
6
=
E
E⊗E
Hamer & Ravndal, 1970:
E(s, Q =
√
−t) = βR (t)sαR (t)−1 + βP (t)sαP (t)−1
Total, asymptotic cross-section becomes
"
!
C
σT (s → ∞) = 4πC 1 − !
8αP log s
with C = Im βP (0)
Serphukov, 1970:
plab = 60 GeV
√
s = 10 GeV
LHC, 2010:
√
s = 14 000 GeV
ISR (1971 - 1984):
pp
pp
Total x-section increases at higher energies !!
eron exchange, and they go nicely through the E710 Tevatron
Landshoff
&
Donnachie(1984)
TeV only the soft-pomeron term 21.7s0.0808 survives, givin
!
.
Non-standard pomeron: αP (t) = 1.08 + αP
t
σ
(mb)
80
p̄p : 21.70s0.0808 + 98.39s−0.4525
pp : 21.70s0.0808 + 56.08s−0.4525
70
Landshoff:
0811.0260
60
SPS+TeV
50
40
30
10
Violates Froissart bound!
100 √
s (GeV)
1000
Froissart bound
Radius of proton:
R ! 1fm = 10
−13
2
cm
Classical x-section: σT = πR = 30 mb
Simplest inelastic process when one pion produced in
overlap region where fraction
√
energy s is available:
−bmπ
e
Thus
bmax
Total x-section
where
e
of total
s ≥ mπ
s
1
log 2
=
2mπ
mπ
σT ≤
2
π/4mπ
smaller value.
√
−bmπ
πb2max
! 15 mb
(Heisenberg)
π
2 s
=
log
4m2π
m2π
Fits need much
Froissart saturation:
s
σT ∝ log
m2π
2
M. Block: 0705.3037
Feynman parton model
f (x)|x→0
1
∝ α
x
Feynman, 1970:
Assume completely absorptive scattering amplitude
T (s, t) = iA(s, t)
so that scattering operator in impact-parameter
representation
S(s, b) = 1 − A(s, b)
Incoming parton wave function
| Ψ! =
∞
!
Cn (x1 , x2 , · · · , xn )| P, n!
n=0
Only wee partons contribute
2
|C
|
c
c c
0
2
|Cn | =
···
n! x1 x2
xn
where expect c << 1.
Normalization !Ψ | Ψ" = 1 gives now:
" #
$n
1
1
dx
2
1 = |C0 |
c
= |C0 |2 sc
n!
x
1/s
n=0
∞
!
Each parton with energy xs scatters with amplitude
!
S(xs,
b) so that scattered state becomes
"
S(s, b) = !Ψ | S | Ψ" = exp − c
!
Self-consistency:
=⇒
%&
dx #
$
1 − S(xs,
b)
1/s x
1
! b) = S(s, b)
S(s,
1
S(s, b) =
1 + F (b)sc
where unknown function F(b) must decrease faster
than any power - from unitarity.
−b/a
F
(b)
=
e
1:
with a ! 1 fm
Transition amplitude:
A(s, b) = 1 − S(s, b) =
1
eb/a−c log s + 1
FD-distribution!
bmax = ac log s and total x-section σT !
2
2
σT = π(ac) log s
Froissart!
2
πbmax
2: Better fit to differential x-section with
−b2 /a2
F (b) = e
which now gives total x-section
2
σT = πca log s
h1
h1
h1
h1
h
QCD:
h2
h2
h2
Recent ideas and? conclusion
2
(a)
(b)
h1
h1
h1 h1
h1
h1
h1h1
Pomeron:
h2
h2
(b)
h2
(b)
h2
h1
h1
(c)
+ ....
h2
h2
h1
h1
h1
+ ... +
h2
h2
h1
(c)
h1
h2
h2
h2
(d)
h1
+
+ ....
Figure 3: Hadron-hadron scattering: (a) what happens?, (
(c) two gluon exchange, (d) exchange of a reggeised gluon
vacuum gluon field
BFKL
h2
(d)
h2
h2
(e)
pomeron is a single Regge pole, consists of two Regge poles
4
proceedings
AdS/QCD:
4
printed on April 22, 2008
proceedings
printed on April 22, 2008
a strongly coupled plasma of deconfined gauge fields. In this case, one
may expect that
features coupled
of the AdS/CFT
be relea strongly
plasma of correspondence
deconfined gaugemay
fields.
In this case, one
4-dim
QCDtesting
vant. Hence this
problem
appears
to giveofa the
stimulating
ground for may be relemay
expect
that Minkowski:
features
AdS/CFT
correspondence
the Gauge/Gravity
and its
physical
relevance
for QCDtesting
and ground for
vant.correspondence
Hence this problem
appears
to give
a stimulating
particle physics.the Gauge/Gravity correspondence and its physical relevance for QCD and
Our aim inparticle
these lectures
physics. is to provide one possible introduction to
those aspects of the
string is
theory
and its
Ourconstruction
aim in theseoflectures
to provide
oneapplications,
possible introduction to
those aspects
of the construction
theory
its applications,
mainly the AdS/CFT
correspondence,
which could of
be string
of interest
for and
the stumainly
thephenomenology.
AdS/CFT correspondence,
which could
be “strongof interest for the students in QCD and
QGP
The presentation
is thus
dents in
QCD
andreasons
QGP phenomenology.
The presentation
is thus “stronginteraction oriented”,
with
both
that it uses as much
as possible the
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with
that it uses
as muchasas possible the
particle language,
and thatoriented”,
the speaker
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is more
a particle physicist
thanlanguage,
a string theorist.
respect
he isappropriately
deeply grate- considered as
particle physicist
thancollaborators,
a string theorist.
In this
respect
he is deeply grateful to his stringatheorists
friends and
in first
place
Romuald
to his
string subtle
theorists
collaborators,
place Romuald
Janik, for theirful
help
in many
andfriends
often and
technical
aspects in
of first
string
Janik, for
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in many subtle
often
technical
aspects of string
theory. In this respect,
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to5-dim
takeand
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Anti-de-Sitter:
String theory
In this
bridge between theory.
“particles
and respect,
strings”.it is quite stimulating to take part in casting a new
bridge between “particles and strings”.
2. The Veneziano Formula and Dual Resonance Models
2. The Veneziano Formula and Dual Resonance Models
s{
q^2
{
{
q^2
Reggeon:
Pomeron:
s{
Open
string:
J=1
Veneziano
Amplitude
string: J
Shapiro-VirasoroClosed
Amplitude
=2
LHC (CMS):
TOTEM
180
Cosmic ray
data
" = 2.2 (best fit)
+-1 !
" = 1.0
160
140
!tot (mb)
120
100
20
0
10
102
FIGURE 1. Experimental
103
LHC
!pp
UA4
UA5
40
!pp
ISR
60
TEVATRON
80
104
105
s (GeV)
pp
p̄p
σtot
and σtot
with the prediction of [5].
Cosmic rays: hep-ph/0011167