# proton-proton scattering from low to lhc- ?· proton-proton scattering from low to lhc-energies...

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Proton-proton scattering from low to LHC-energies

Coulomb scattering Low-energy strong interactions Regge poles and the pomeron The Froissart bound Feynman parton model Recent ideas and conclusion

Finn Ravndal, Dept of Physics, UiO

HEP-coll, 13/11 - 2009

Coulomb scattering

e

e

Q Q2 = 4p2 sin2 /2

in CM-frame:

Differential cross-section:

In first Born approximation

in CM-frame where proton momentum

d

d= |f(Q)|2

f(Q) =m

2V (Q)

with

becomes infinite Born series

T = V + V G0V + V G0V G0V +

when expressed by the free Greens function

G0(E) =1

E H0 + i

First Born approximation

T (1)pp

= p |bV |p =Z

d3r V (r) eiQr

Scattering on potential of finite range R

V(r)

rR

at low energies p ! 1/R:

dd

= a20

where S-wave scattering length

a0 =M

4

Zd3r V (r)

in this approximation.

3

V (Q) =

and reduced mass m = M/2.

p = mv

Coulomb potential

with Feynman diagram

= + + + ! ! !

+=

Figure 6: Coulomb propagator as an infinite sum.

or Kummer function M(a, b;x)[23]. For the repulsive Coulomb potential VC = /r thein-state solution with outgoing spherical waves in the future is

(+)p (r) = e 1

2(1 + i)M(i, 1; ipr ip r) eipr (20)

The corresponding out-state has incoming spherical waves in the distant past and is givenby the wavefunction

()p (r) = e 1

2(1 i)M(i, 1;ipr ip r) eipr (21)

where = M/2p is the parameter which also appeared in the earlier perturbative calcu-lation. The probability to find the two protons at zero separation is thus

C2 |()p (0)|2 = e(1 + i)(1 i) =2

e2 1(22)

which is the well-known Sommerfeld factor[23][24]. When the relative velocity betweenthe particles goes to zero, it becomes exponentially small. At higher velocities < 1 andthe Coulomb repulsion is perturbative. We then recover to lowest order the result 1 obtained from the Feynman diagram Fig. 4 in the previous section.

With these Coulomb eigenstates we can now find a more useful expression for theGreens functions (18). Since the scattering states form a complete set in the repulsivecase we consider here, we can instead write it as in (17). Taking the matrix element incoordinate space, we then have for the retarded function

r |G(+)C |r = M

d3q

(2)3(+)q (r)

(+)q (r)

p2 q2 + i%(23)

In the next section we will see that this propagator gives the main part of the non-perturbative Coulomb corrections of the strong scattering amplitude.

9

giving scattering amplitude

and = e2/4 = 1/137where

V (r) =e2

4r

Q

e

e

=

e2

Q2V (Q) =

f(Q) =

4E sin2(/2)

E = p2/2m

= + + + ! ! !

+=

Figure 6: Coulomb propagator as an infinite sum.

or Kummer function M(a, b;x)[23]. For the repulsive Coulomb potential VC = /r thein-state solution with outgoing spherical waves in the future is

(+)p (r) = e 1

2(1 + i)M(i, 1; ipr ip r) eipr (20)

The corresponding out-state has incoming spherical waves in the distant past and is givenby the wavefunction

()p (r) = e 1

2(1 i)M(i, 1;ipr ip r) eipr (21)

where = M/2p is the parameter which also appeared in the earlier perturbative calcu-lation. The probability to find the two protons at zero separation is thus

C2 |()p (0)|2 = e(1 + i)(1 i) =2

e2 1(22)

which is the well-known Sommerfeld factor[23][24]. When the relative velocity betweenthe particles goes to zero, it becomes exponentially small. At higher velocities < 1 andthe Coulomb repulsion is perturbative. We then recover to lowest order the result 1 obtained from the Feynman diagram Fig. 4 in the previous section.

With these Coulomb eigenstates we can now find a more useful expression for theGreens functions (18). Since the scattering states form a complete set in the repulsivecase we consider here, we can instead write it as in (17). Taking the matrix element incoordinate space, we then have for the retarded function

r |G(+)C |r = M

d3q

(2)3(+)q (r)

(+)q (r)

p2 q2 + i%(23)

In the next section we will see that this propagator gives the main part of the non-perturbative Coulomb corrections of the strong scattering amplitude.

9

Full Coulomb propagator

results in non-perturbative scattering amplitude

where parameter = /v gives effective strength

of the Coulomb interaction.

fC() =

4E sin2(/2)ei ln sin

2(/2)

Probability to find two protons at zero separation:

|(0)|2 =2

e2 1

Becomes exponentially small when > 1

i.e. when p < 10 MeV.

Coulomb cross-section is unmodified:

d

d=

(

4E sin2(/2)

)2

Thus Coulomb interaction

dominates for energies E < 1 MeV.

Low-energy strong interactions

Effective potential, valid at low energies:

Veff = C(r)

where coupling constant C = 4M

a in first Born

approximation.

Higher order Born corrections can now be obtained ineffective field theory

L = i + 12M

2 C2

()2

using standard field-theoretic perturbation theory withnon-relativistic propagator

G0(E,p) =1

E p2/2M + i

and elementary vertex

Second order Born correction from Feynman diagram

4

Effective potential, valid at low energies:

Veff = C(r)

where coupling constant C = 4M

a in first Born

approximation.

Higher order Born corrections can now be obtained ineffective field theory

L = i + 12M

2 C2

()2

using standard field-theoretic perturbation theory withnon-relativistic propagator

G0(E,p) =1

E p2/2M + i

and elementary vertex

Second order Born correction from Feynman diagram

4

Cross-section to lowest order

Effective potential, valid at low energies:

Veff = C(r)

where coupling constant C = 4M

a in first Born

approximation.

Higher order Born corrections can now be obtained ineffective field theory

L = i + 12M

2 C2

()2

using standard field-theoretic perturbation theory withnon-relativistic propagator

G0(E,p) =1

E p2/2M + i

and elementary vertex

Second order Born correction from Feynman diagram

4

described by effective Langrangian

d

d= a

2

where scattering length a =m

2C

E < 100 MeV

Higher order corrections

with value C2I0(p) with bubble integral

I0(p) =

Zd3k

(2)32M

p2 k2 + i"

Higher order diagram with n bubbles

...

gives similarly Cn+1In0 (p). Total scattering amplitude:

T (p) = C1 + CI0 + (CI0)

2 + (CI0)3 +

=C

1 CI0=

11/C I0(p)

Bubble integral I0(p) is divergent in d = 3 dimensions.Regularize with cut-off 1/R giving

I0(p) = M22

+

i2p

which implies that bare coupling C = C().

5

with value C2I0(p) with bubble integral

I0(p) =

Zd3k

(2)32M

p2 k2 + i"

Higher order diagram with n bubbles

...

gives similarly Cn+1In0 (p). Total scattering amplitude:

T (p) = C1 + CI0 + (CI0)

2 + (CI0)3 +

=C

1 CI0=

11/C I0(p)

Bubble integral I0(p) is divergent in d = 3 dimensions.Regularize with cut-off 1/R giving

I0(p) = M22

+

i2p

which implies that bare coupling C = C().

5

+ ...... + + ....

given by bubble integral

with value C2I0(p) with bubble integral

I0(p) =

Zd3k

(2)32M

p2 k2 + i"

Higher order diagram with n bubbles

...

gives similarly Cn+1In0 (p). Total scattering amplitude:

T (p) = C1 + CI0 + (CI0)

2 + (CI0)3 +

=C

1 CI0=

11/C I0(p)

Bubble integral I0(p) is divergent in d = 3 dimensions.Regularize with cut-off 1/R giving

I0(p) = M22

+

i2p

which implies that bare coupling C = C().

5

with value C2I0(p) with bubble integral

I0(p) =

Zd3k

(2)32M

p2 k2 + i"

Higher order diagram with n bubbles

...

gives similarly Cn+1In0 (p). Total scattering amplitude:

T (p) = C1 + CI0 + (CI0)

2 + (CI0)3 +

=C

1 CI0=

11/C I0(p)

Bubble integral I0(p) is divergent in d = 3 dimensions.Regularize with cut-off 1/R giving

I0(p) = M22

+

i2p

which implies that bare coupling C = C().

5

Full scattering amplitude:

T

Bubble integral with cut-off regularization:

with value C2I0(p) with bubble integral

I0(p) =

Zd3k

(2)32M

p2 k2 + i"

Higher order diagram with n bubbles

...

gives similarly Cn+1In0 (p). Total scattering amplitude:

T (p) = C1 + CI0 + (CI0)

2 + (CI0)3 +

=C

1 CI0=

11/C I0(p)

Bubble integral I0(p) is divergent in d = 3 dimensions.Regularize with cut-off 1/R giving

I0(p) = M22

+

i2p

which implies that bare coupling C = C().

5

Remove cut-off by introducing renormalizedcoupling constant

Renormalization

Remove cut-off by introducing renormalized couplingconstant

1CR

=1C

+M22

with value CR = 4M a which gives scattering amplitude

T (p) =4

M

1

1/a + ip

and is now unitary. Differential cross-section

d

d=

a2

1 + (ap)2

Renormalized coupling CR goes to zero as , no scattering

on -function potentia

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