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
Odense, 24/11 - 2009
Coulomb scattering
in CM-frame:
e
2
Q
2
2
Q = 4p sin θ/2
e
e2
V (r) =
4πr
Coulomb potential
e
=
V (Q)
=
+
Q
+
e
giving= scattering amplitude
+
e2
= 2
Q
+ !!!
α
f (Q) =
2
4E sin (θ/2)
dσ
2
Figure 6:
Coulomb propagator as an infinite sum.= |f (Q)|
Differential
cross-section:
dΩ
or Kummer function M (a, b; x)[23]. For the repulsive Coulomb potential VC = α/r the
n-state solution with outgoing spherical waves in the future is
1
ψ (+) (r) = e− 2 πη Γ(1 + iη)M (−iη, 1; ipr − ip · r) eip·r
(20)
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 /2M + iε
Low-energy
interactions
Vefstrong
f = Cδ(r)
mentary vertex
Effective
potential,
valid
at
low
energies:
4π
E
<
100
MeV
here coupling constant C = a in first Born
M
proximation.
Vef f = Cδ(r)
4πnow be obtain
gherBorn
order
Born
corrections
can
order
correction
from
Feynman
diagram
where coupling constant C = a in first Born
described by effective Langrangian M
ective
field theory
approximation.
1 corrections
C now
Higher order
can
∗ Born
∗
2
∗
2be obta
L = iψ4 ψ̇ +
ψ ∇ ψ − (ψ ψ)
2M
2
effective field theory
ing standard field-theoretic
1 ∗ perturbation
C ∗ 2theor
∗
2
L = iψ ψ̇ +
ψ ∇ ψ − (ψ ψ)
2M
2
n-relativistic propagator
using standard field-theoretic perturbation the
I0 (p) =
(2π)3 p2 − k 2 + i"
1
1
Higher order
corrections
Higher
order diagram with n bubbles
=
MΛ
+
C
2π 2
CR
with value
CR
+ ......
+ =
4π ...
a
M
which+gives
scatt
....
with value C 2 I0 (p) with4π
bubble 1integral
T (p) =
2
I0 (p) with bubble
integral
gives
similarly C n+1 I0n (p). Total
amplitude:
Z scattering
3
Z
2M
d
k
ˆ
˜
3
2
3
2M
d
k
I
(p)
=
where bubble Tintegral
(p) = C 1 0+ CI0 + (CI0 ) + (CI
0 ) 2+ · · · 2
3
0 (p) =
(2π) p − k +
(2π)3 p2 − k 2 + i"
1
C
M 1/a + ip
i"
and is now =unitary.
= Differential cross-s
1 − CI
1/C − I (p)
diagram with n bubbles
Higher
0
0
order diagram
with
n bubbles
2
Bubble integral I0 (p) is divergent in d = 3 dimensions.
a
dσ
Differential
x-section:
Regularize with cut-off Λ ≈ 1/R giving
...
=
„
«
2
dΩ
1
+
(ap)
i
M
Λ + πp
I (p) = −
0
2π 2
2
...
Renormalized
coupling
C
goes
to
which implies that bare coupling
R C = C(Λ). zero as Λ
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)√
–
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
propagator
1
E − p2 /2M + iε
Strong interaction:
E, p) =
vertex
=
rn correction from Feynman diagram
Regge amplitude:
Total
cross-section:
4
Regge trajectory:
T (s, t) = βR (t)sαR (t)−1
αR (0)−1
σT = 4π Im βR (0)s
αR (t) = 0.5 +
!
αR t
Pomeranchuk theorem (1958):
σT (AB) = σT (ĀB)
as s → ∞
Isaak Pomeranchuk (1913 - 1966)
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!
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
σT
R
P
P ⊗P
√
−→ const
s
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
xα(0)
1
Pomeron: α(0) = 1 −→ wee partons: f (x) ∝
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!
Expanding, black disk!
bmax = ac log s
σT = π(ac)2 log2 s
and
σT ! πb2max
Froissart!
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
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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