pp elastic scattering with infraredmodifications: an overview

Werner K. Sauter

Grupo de Fenomenologia de Partıculas de Altas Energias

IF–UFRGS

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.1

Summary• Brief introduction to the IPomeron;• The model: two gluon exchange;• Results;• Improvements;• Conclusions.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.2

Introduction• The IPomeron : Regge theory versus QCD

(BFKL): different predictions for the crosssections with small momentum transfer inhigh energy collisions.

• LN Model: QCD vacuum properties modifiedthe gluon propagator in long range (soft)interactions

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.3

Introduction• The IPomeron : Regge theory versus QCD

(BFKL): different predictions for the crosssections with small momentum transfer inhigh energy collisions.

• LN Model: QCD vacuum properties modifiedthe gluon propagator in long range (soft)interactions

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.3

The model: Introduction• Two gluon exchange: Low-Nussinov model

for the IPomeron;

• Modeling the hadron collisions asphoton-photon collisions;

• Gunion & Soper (after Cheng & Wu): multiplegauge boson exchange in s Channel;

• pp scattering with two gluon exchange:

App2 = 8isα2

s

∫

dkD(k2)D(

(∆− k)2)

[E1(∆) − E2(k,∆− k)]2.

(1)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.4

The model: Introduction• Two gluon exchange: Low-Nussinov model

for the IPomeron;• Modeling the hadron collisions as

photon-photon collisions;

• Gunion & Soper (after Cheng & Wu): multiplegauge boson exchange in s Channel;

• pp scattering with two gluon exchange:

App2 = 8isα2

s

∫

dkD(k2)D(

(∆− k)2)

[E1(∆) − E2(k,∆− k)]2.

(1)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.4

The model: Introduction• Two gluon exchange: Low-Nussinov model

for the IPomeron;• Modeling the hadron collisions as

photon-photon collisions;• Gunion & Soper (after Cheng & Wu): multiple

gauge boson exchange in s Channel;

• pp scattering with two gluon exchange:

App2 = 8isα2

s

∫

dkD(k2)D(

(∆− k)2)

[E1(∆) − E2(k,∆− k)]2.

(1)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.4

The model: Introduction• Two gluon exchange: Low-Nussinov model

for the IPomeron;• Modeling the hadron collisions as

photon-photon collisions;• Gunion & Soper (after Cheng & Wu): multiple

gauge boson exchange in s Channel;• pp scattering with two gluon exchange:

App2 = 8isα2

s

∫

dkD(k2)D(

(∆− k)2)

[E1(∆) − E2(k,∆− k)]2.

(1)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.4

The model: Introduction• The E ’s are form factors: E1 ⇒ proton’s Dirac

elastic form factor,

E1(t = −∆2) =4m2

p − 2, 79t

(4m2p − t)

(

1 − t0,71

)2. (2)

• E2 has a arbritary functional form but theinfrared divergences goes to zero when k’s→ 0

E2(ka,kb) = E1(k2a + k2

b − fka · kb), (3)

where f is a hadron depend parameter. First,Cudell & Ross used f = 7, later Cudell &Nguyen used f = 1 (used here).

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.5

The model: Introduction• The E ’s are form factors: E1 ⇒ proton’s Dirac

elastic form factor,

E1(t = −∆2) =4m2

p − 2, 79t

(4m2p − t)

(

1 − t0,71

)2. (3)

• E2 has a arbritary functional form but theinfrared divergences goes to zero when k’s→ 0

E2(ka,kb) = E1(k2a + k2

b − fka · kb), (3)

where f is a hadron depend parameter. First,Cudell & Ross used f = 7, later Cudell &Nguyen used f = 1 (used here).

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.5

The model: Introduction• Rewrite the amplitude as

App2 (s, t) = 8isα2

s (T1 − T2) , (4)

T1 =

∫ s

0

dk D(p+)D(p−)E21 (∆),

T2 =

∫ s

0

dk D(p+)D(p−)E2(p+,p−)[

2E21 (∆) + E2(p+,p−)

]

,

p± = ∆/2 ± k,

• T1: the gluons strucks the same quark insidethe proton and T2: different quarks inside theproton.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.6

The model: Introduction• Rewrite the amplitude as

App2 (s, t) = 8isα2

s (T1 − T2) , (4)

T1 =

∫ s

0

dk D(p+)D(p−)E21 (∆),

T2 =

∫ s

0

dk D(p+)D(p−)E2(p+,p−)[

2E21 (∆) + E2(p+,p−)

]

,

p± = ∆/2 ± k,

• T1: the gluons strucks the same quark insidethe proton and T2: different quarks inside theproton.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.6

The Model: Cross Sections• Cross sections by optical theorem:

σ0tot =

App2 (s, 0)

is, (4)

dσ0

dt=

|App2 (s, t)|216πs2

. (5)

• Note that the cross sections are independentof energy ⇒ Too simple approx.

• In low momentum transfer:dσ

dt= A eBt ⇒ A =

σ2tot

16π, B =

d

dt

(

lndσ

dt

)∣

∣

∣

∣

t=0

. (6)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.7

The Model: Cross Sections• Cross sections by optical theorem:

σ0tot =

App2 (s, 0)

is, (6)

dσ0

dt=

|App2 (s, t)|216πs2

. (6)

• Note that the cross sections are independentof energy ⇒ Too simple approx.

• In low momentum transfer:dσ

dt= A eBt ⇒ A =

σ2tot

16π, B =

d

dt

(

lndσ

dt

)∣

∣

∣

∣

t=0

. (5)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.7

The Model: Cross Sections• Cross sections by optical theorem:

σ0tot =

App2 (s, 0)

is, (5)

dσ0

dt=

|App2 (s, t)|216πs2

. (5)

• Note that the cross sections are independentof energy ⇒ Too simple approx.

• In low momentum transfer:dσ

dt= A eBt ⇒ A =

σ2tot

16π, B =

d

dt

(

lndσ

dt

)∣

∣

∣

∣

t=0

. (4)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.7

The model: Pros and cons• Problems:

• no energy dependence• B(t = 0) → ∞ for D(k2) = 1/k2

• Possible solution:• ad hoc Regge energy term.

σtot =

(

s

m2p

)0,08

σ0tot, (5)

dσ

dt=

(

s

m2p

)0,168dσ0

dt. (6)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.8

The model: Pros and cons• Problems:

• no energy dependence

• B(t = 0) → ∞ for D(k2) = 1/k2

• Possible solution:• ad hoc Regge energy term.

σtot =

(

s

m2p

)0,08

σ0tot, (6)

dσ

dt=

(

s

m2p

)0,168dσ0

dt. (6)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.8

The model: Pros and cons• Problems:

• no energy dependence• B(t = 0) → ∞ for D(k2) = 1/k2

• Possible solution:• ad hoc Regge energy term.

σtot =

(

s

m2p

)0,08

σ0tot, (5)

dσ

dt=

(

s

m2p

)0,168dσ0

dt. (5)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.8

The model: Pros and cons• Problems:

• no energy dependence• B(t = 0) → ∞ for D(k2) = 1/k2

• Possible solution:

• ad hoc Regge energy term.

σtot =

(

s

m2p

)0,08

σ0tot, (4)

dσ

dt=

(

s

m2p

)0,168dσ0

dt. (4)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.8

The model: Pros and cons• Problems:

• no energy dependence• B(t = 0) → ∞ for D(k2) = 1/k2

• Possible solution:• ad hoc Regge energy term.

σtot =

(

s

m2p

)0,08

σ0tot, (3)

dσ

dt=

(

s

m2p

)0,168dσ0

dt. (3)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.8

The model: Previous results• LN model (pp → pp at

√s = 53 GeV)

• Cudell & Ross ⇒ EDS’ solutions;• Halzen, Krein & Natale ⇒ Cornwall’s

solution;• UKQCD Coll. ⇒ Lattice solution;• Aguilar, Mihara & Natale ⇒ Comparison of

propagators.

• Other approaches: Jenkovszky et al.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.9

The model: Previous results• LN model (pp → pp at

√s = 53 GeV)

• Cudell & Ross ⇒ EDS’ solutions;

• Halzen, Krein & Natale ⇒ Cornwall’ssolution;

• UKQCD Coll. ⇒ Lattice solution;• Aguilar, Mihara & Natale ⇒ Comparison of

propagators.

• Other approaches: Jenkovszky et al.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.9

The model: Previous results• LN model (pp → pp at

√s = 53 GeV)

• Cudell & Ross ⇒ EDS’ solutions;• Halzen, Krein & Natale ⇒ Cornwall’s

solution;

• UKQCD Coll. ⇒ Lattice solution;• Aguilar, Mihara & Natale ⇒ Comparison of

propagators.

• Other approaches: Jenkovszky et al.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.9

The model: Previous results• LN model (pp → pp at

√s = 53 GeV)

• Cudell & Ross ⇒ EDS’ solutions;• Halzen, Krein & Natale ⇒ Cornwall’s

solution;• UKQCD Coll. ⇒ Lattice solution;

• Aguilar, Mihara & Natale ⇒ Comparison ofpropagators.

• Other approaches: Jenkovszky et al.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.9

The model: Previous results• LN model (pp → pp at

√s = 53 GeV)

• Cudell & Ross ⇒ EDS’ solutions;• Halzen, Krein & Natale ⇒ Cornwall’s

solution;• UKQCD Coll. ⇒ Lattice solution;• Aguilar, Mihara & Natale ⇒ Comparison of

propagators.

• Other approaches: Jenkovszky et al.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.9

The model: Our approach• Our approach: using IR modified gluon props

but not fixing parameters with LN formulae.

• Cornwall propagator:

D−1C (q2) =

[

q2 + m2(q2)]

b g2 ln

[

q2 + 4 m2(q2)

Λ2QCD

]

m2(q2) = m2g

ln(

q2+4 m2g

Λ2QCD

)

ln(

4 m2g

Λ2QCD

)

−12/11

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.10

The model: Our approach• Our approach: using IR modified gluon props

but not fixing parameters with LN formulae.• Cornwall propagator:

D−1C (q2) =

[

q2 + m2(q2)]

b g2 ln

[

q2 + 4 m2(q2)

Λ2QCD

]

m2(q2) = m2g

ln(

q2+4 m2g

Λ2QCD

)

ln(

4 m2g

Λ2QCD

)

−12/11

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.10

The model: Propagators• Cornwall modified propagator:

DC ′(q2) =1

q2 + m2(q2)

• Massive:

DM(q2) =1

q2 + m2g

• Häbel/Gribov propagator:

DHG(q2) =q2

q4 + b4

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.11

The model: Propagators• Cornwall modified propagator:

DC ′(q2) =1

q2 + m2(q2)

• Massive:

DM(q2) =1

q2 + m2g

• Häbel/Gribov propagator:

DHG(q2) =q2

q4 + b4

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.11

The model: Propagators• Cornwall modified propagator:

DC ′(q2) =1

q2 + m2(q2)

• Massive:

DM(q2) =1

q2 + m2g

• Häbel/Gribov propagator:

DHG(q2) =q2

q4 + b4

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.11

The model: Propagators• Gorbar/Natale propagator

DGN(q2) =1

q2 + m2GN(q2)

,

m2GN (q2) = µ2

gΘ(χµ2g − q2) +

µ2g

q2Θ(q2 − χµ2

g)

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.12

The model: Coupling Constant• αs fixed: αs = 0, 9 and αs = αs(0).

• αs running:

αs(k2) =

4π

β0 ln(

k2+4m2D

Λ2QCD

) ,

• The scale of the coupling is the momentum ofthe gluon.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.13

The model: Coupling Constant• αs fixed: αs = 0, 9 and αs = αs(0).• αs running:

αs(k2) =

4π

β0 ln(

k2+4m2D

Λ2QCD

) ,

• The scale of the coupling is the momentum ofthe gluon.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.13

The model: Coupling Constant• αs fixed: αs = 0, 9 and αs = αs(0).• αs running:

αs(k2) =

4π

β0 ln(

k2+4m2D

Λ2QCD

) ,

• The scale of the coupling is the momentum ofthe gluon.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.13

Results

0.0 0.2 0.4 0.6 0.8 1.0|t| (GeV2)

10-3

10-2

10-1

100

101

102

103

dσ/d

t (m

b/G

eV2 )

CornwallCornwall (var)

Massivo (mg=0.8 GeV2)

Habel/GribovGorbar/Natale

αs = 0.9

0.0 0.2 0.4 0.6 0.8 1.0|t| (GeV2)

αs = αs(0)

Differential cross section.√

s = 53 GeV. Fixed coupling.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.14

Results

0.0 0.1 0.2 0.3 0.4|t| (GeV2)

100

101

102

103

dσ/d

t (m

b/G

eV2 )

CornwallCornwall (var)

Massivo (mg=0.8 GeV2)

HabelGribovGorbar/Natale

αs = 0.9

0.0 0.1 0.2 0.3 0.4|t| (GeV2)

αs = αs(0)

Differential cross section.√

s = 1.8 TeV. Fixed coupling.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.15

Results

0.0 0.2 0.4 0.6 0.8 1.0|t| (GeV2)

10-3

10-2

10-1

100

101

102

103

dσ/d

t (m

b/G

eV2 )

CornwallCornwall (var)

Massivo (mg=0.8 GeV2)

HabelGribovGorbar/Natale

αs = 0.9

0.0 0.2 0.4 0.6 0.8 1.0|t| (GeV2)

αs = αs(0)

Differential elastic cross section.√

s = 53 GeV. Fixed coupling.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.16

Results

0.0 0.1 0.2 0.3 0.4|t| (GeV2)

10-1

100

101

102

103

dσ/d

t (m

b/G

eV2 )

CornwallCornwall (var)

Massivo (mg=0.8 GeV2)

HabelGribovGorbar/Natale

αs = 0.9

0.0 0.1 0.2 0.3 0.4|t| (GeV2)

αs = αs(0)

Differential elastic cross section.√

s = 1.8 TeV. Fixed coupling.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.17

Results

0.0 0.2 0.4 0.6 0.8 1.0|t| (GeV2)

10-3

10-2

10-1

100

101

102

103

dσ/d

t (m

b/G

eV2 )

CornwallCornwall (var.)Massivo (mg = 0.8 GeV)

HabelGribovGorbar/Natale

sqrt(s) = 53 GeV

0.0 0.1 0.2 0.3 0.4|t| (GeV2)

10-1

100

101

102

103sqrt(s)= 1.8 TeV

Differential elastic cross section.√

s = 1.8 TeV. Running coupling.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.18

Conclusions• Good (or poor) agreement with the data in

particular cases (with same parameters ofprevious analisis).

• The model with running coupling have a goodagreement with the data.

• Strong dependence of the running coupling.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.19

Conclusions• Good (or poor) agreement with the data in

particular cases (with same parameters ofprevious analisis).

• The model with running coupling have a goodagreement with the data.

• Strong dependence of the running coupling.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.19

Conclusions• Good (or poor) agreement with the data in

particular cases (with same parameters ofprevious analisis).

• The model with running coupling have a goodagreement with the data.

• Strong dependence of the running coupling.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.19

Improvements• Cudell & Nguyen: next leading order (but not

BFKL).

• BFKL with non perturbative propagators:Hancock & Ross, Vacca & Venturi, Nikolaev,Zakharov & Zoller, Forshaw & Harriman.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.20

Improvements• Cudell & Nguyen: next leading order (but not

BFKL).• BFKL with non perturbative propagators:

Hancock & Ross, Vacca & Venturi, Nikolaev,Zakharov & Zoller, Forshaw & Harriman.

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.20

References• Cudell, Ross Nucl. Phys B 359 (1991) 247

• Cudell, Nguyen Nucl. Phys. B 420 (1994) 669

• UKQCD Coll. Phys. Lett. B 369 (1996) 130

• Halzen et al. Phys. Rev. D 47 (1993) 295

• Aguilar et al. Phys. Rev. D 65 (2002) 054011, hep-ph/0208095

• NNZ JETP Lett. 60 (1994) 694, Phys. Lett. B 328 (1994) 486

• Jenkovszky et al. Sov. J. Nucl. Phys. 53 (1991) 329, Z. Phys. C 63 (1994) 131

• Ross J. Phys. G 15 (1989) 1175

• Hancock, Ross Nucl. Phys. B 383 (1992) 575, Nucl. Phys. B 394 (1993) 200

• Vacca, Venturi Il Nuovo Cim. A 108 (1995) 77

• Forshaw & Harriman Phys. Rev. D 46 (1992) 3778

Seminarios de Fenomenologia de Partıculas de Altas Energias – 2003/1 – p.21