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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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