Gribov-Regge approach and EPOS

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Klaus Werner

To cite this version:

Gribov-Regge approach and EPOS

Klaus Werner

SUBATECH, University of Nantes - IN2P3/CNRS

- IMT Atlantique, Nantes, France

Contents

1

Introduction

8

1.1

EPOS and Gribov-Regge approach

. . . 9

1.2

Secondary interactions: An example

. . . 15

1.3

Radial flow visible in particle distributions

. . . 30

1.4

Ridges & flow harmonics

. . . 33

1.5

Flow harmonics, identified particles

. . . 42

2

T-matrix properties and Pomerons

45 2.1

Poles, analytical continuation

. . . 46

2.3

Cut diagrams

. . . 57

2.4

Cuts in the s-plane

. . . 69

2.5

Dispersion relations

. . . 75

2.6

Watson-Sommerfeld transform

. . . 77

2.7

The signature

. . . 85

2.8

Reggeons and Pomerons

. . . 90

3

Pomerons and pQCD parton ladders

101 3.1

Parton evolution in ep scattering

. . . 102

3.3

Soft domain

. . . 112

3.4

Semihard Pomeron

. . . 117

3.5

Gribov-Regge approach

. . . 119

3.6

Factorization approach

. . . 120

3.7

Models

. . . 122

4

Multiple Pomeron exchange in EPOS

134 4.1

The Pomeron in EPOS

. . . 135

4.2

Hadronic vertex

. . . 144

4.3

Multiple scattering

. . . 148

4.5

Configurations via Markov chains

. . . 163

4.6

Metropolis algorithm

. . . 169

4.7

Glauber and Gribov Regge

. . . 176

4.8

Gribov-Regge and Factorization

. . . 187

5

The Pomeron structure in EPOS

198 5.1

The parton ladder T-matrix

. . . 202

5.2

Parton saturation

. . . 204

5.3

The parton ladder cross section

. . . 207

5.4

Parton generation

. . . 221

5.6

From partons to strings

. . . 241

5.7

Hadron production

. . . 256

6

Collectivity in EPOS

263 6.1

Core-corona procedure

. . . 264

6.2

Hydrodynamic evolution

. . . 271

6.3

Flow in small systems

. . . 285

6.4

Statistical particle production

. . . 300

7

New trends on the foundations

of hydrodynamics

345

7.1

Introduction

. . . 346

7.2

Covariant derivative

. . . 349

7.3

Bjorken hydrodynamics

. . . 354

7.4

Truncated conformal Bjorken hydrodyn.

. . . 357

7.5

Divergent series, Borel transforms,

trans-series

. . . 365

—————————————————————

1

Introduction

1.1

EPOS and Gribov-Regge approach

EPOS is an event generator to treat consistently



e+e- annihilation

(test string fragmentation)



ep scattering

(test parton evolution)



pp, pA, AA collisions

at high energies

Basic structure of EPOS

(for modelling pp, pA, AA)



Primary interactions

Multiple scattering, instantaneously, in parallel

(Parton Based Gribov-Regge Theory)

–

in pA and AA: multiple NN scattering

–

but also in pp : Multiple parton scattering

(or for each NN scattering in pA, AA)



Secondary interactions

formation of “matter” which expands

Some history of Gribov-Regge Theory

(the heart of EPOS)



1960-1970: Gribov-Regge Theory of multiple scattering.

pp = multiple exchange of “Pomerons”

(with amplitudes based on Regge poles)



1980-1990: pQCD processes

added into GRT scheme (Capella)



1990: M.Braun, V.A.Abramovskii, G.G.Leptoukh:

problem with energy conservation



2001: H.J.Drescher, M.Hladik, S.Ostapchenko, T. Pierog,

and K. Werner, Phys. Rept. 350, p93:

Marriage pQCD + GRT,

with energy sharing (NEXUS)

x

+2 2

x

−

x

1 − +

x

1 ☛ ✡ ✟ ✠

∑

_x

_i±

+

x

±_remn

=

1

Multiple scatterings

(in parallel !!)

in pp, pA, or AA

Single scattering

=

hard elementary

scattering

including

IS + FS

radiation



~ 2003 NEXUS split into



QGSJET (S. Ostapchenko)

– Triple Pomeron contributions and more, to all orders



EPOS (T. Pierog, KW)

– Saturation scale, secondary interactions

– two versions, EPOSLHC and EPOS3, going to be “fused”, with a rigorous (selfconsistent) treatment of new key features

(HF, saturation & factorization)

=> new public version (β version exists since few days ...)

The current (2019) EPOS team

Tanguy Pierog, Karlsruhe, Germany

Yuriy Karpenko, Nantes, France

Benjamin Guiot, Valparaiso, Chile

Gabriel Sophys, Nantes, France

Maria Stefaniak, Nantes & Warsaw, Poland

Mahbobeh Jafarpour, Nantes, France

1.2

Secondary interactions: An example

In the following:

An example of a EPOS simulation

of expanding matter in pp scattering

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 50 100 150 200 250 300 350

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 10 20 30 40 50 60

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 5 10 15 20

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 9

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 1 2 3 4

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2

pp @ 7TeV EPOS 3.119

x [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y [fm] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

1.3

Radial flow visible in particle distributions

Particle spectra affected by radial flow

10-2 10-1 1 10 102 0 1 2 3 pt dn/dptdy _ π K- p Λ hydrodynamics (solid) string decay (dotted)

pPb at 5TeV

CMS, arXiv:1307.3442

Strong variation of shape with multiplicity

for kaon and even more for proton pt spectra

Λ/K

s

versus pT (

high compared to low multiplicity)

in pPb

(left)

similar to PbPb

(right)

ALICE (2013) arXiv:1307.6796 ALICE (2013) arXiv:1307.5530

Phys. Rev. Lett. 111, 222301 (2013)

1.4

Ridges & flow harmonics

Anisotropic radial flow

visible in dihadron-correlations

R

=

1

N

trigg

dn

d∆φ∆η

Anisotropic flow due to initial

azimuthal anisotropies

Initial “elliptical” matter

distribution:

Preferred expansion

along φ

=

0

and φ

=

π

η

s

-invariance

same form at any η

s

η

s

=

1₂

ln

t_t+₋z_z

Particle

distribution:

Preferred

directions

φ

=

0 and φ

=

π

∝ 1

+

2v

2

cos

(

2φ

)

0

0.05

0.1

0.15

0.2

-1

0

1

2

3

4

φ

f(

φ

) = dn / d

φ

Dihadrons:

Initial

“triangular”

matter distribution:

Preferred

expansion

along φ

=

0, φ

=

2₃

π,

and φ

=

4₃

π

η

s

-invariance

φ

Particle

distribution:

Preferred

directions

φ

=

0, φ

=

2 3

π,

and φ

=

4 3

π

∝ 1

+

2v

3

cos

(

3φ

)

0

0.05

0.1

0.15

0.2

-1

0

1

2

3

4

φ

f(

φ

) = dn / d

φ

Dihadrons:

preferred ∆φ

=

0, and ∆φ

=

2 3

π, and ∆φ

=

4 3

π

In general, superposition of several eccentricities ε

n

,

ε

n

e

inψ PP n

_{= −}

R

dxdy r

2

e

inφ

e

(

x, y

)

R

dxdy r

2

_e

₍

_{x, y}

₎

Particle distribution characterized by harmonic flow

coef-ficients

v

n

e

inψ EP n

=

Z

dφ e

inφ

f

(

φ

)

At φ

=

0:

Here, v

2

and v

3

non-zero

The

ridge

∝ 1

+

2v

2

cos

(

2φ

) +

2v

3

cos

(

3φ

)

(extended in η)

Awayside

peak

may

originate

from jets, not the

ridge (for large

∆η)

0

0.1

0.2

-1

0

1

2

3

4

φ

f(

φ

) = dn / d

φ

CMS: Ridges

(in dihadron correlation functions)

also seen in pp

(left)

and pPb

(right)

CMS (2010) arXiv:1009.4122

JHEP 1009:091,2010 CMS (2012) arXiv:1210.5482_{Phys. Lett. B 718 (2013) 795}

Ridges also realized in simulations in pPb

(and even pp)

Central - peripheral

(to remove jets)

Phys. Lett. B 726 (2013) 164-177

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1 0 1 2 3 4 0.72 0.74 0.76 0.78 2.0-4.0 GeV/c trigg T p 1.0-2.0 GeV/c assoc T p η ∆ φ ∆ R EPOS3.074 ALICE

1.5

Flow harmonics, identified particles

Flow shifts

parti-cles to higher p

t

Effect

increases

with mass

Also true for v

2

vs p

t

p

t

v

₂

ALICE: v2 versus pT: mass splitting

(π, K, p)

in pPb

(left)

similar to PbPb

(right)

ALICE (2013)arXiv:1307.3237 _{ALICE (2012) F. Noferini QM2012}

So : “Flow-like phenomena” are also seen in pp

and pA, therefore:

Heavy ion approach

= primary (multiple) scattering

+ subsequent fluid evolution

becomes interesting for pp and pA

—————————————————————

2

T-matrix properties and Pomerons

—————————————————————

primary interactions

providing initial conditions

for secondary interactions

2.1

Poles, analytical continuation

Even functions f

(

x

)

of a real variable x

may need to be considered in the complex plane,

to understand their properties.

Example

_f

₍

_x

_{) =}

_∑

∞ n=0

a

n

x

n

=

∞

∑

n=0

 x

2i



n

.

The radius of convergence is

ρ

=

lim

n→∞

|

a

n

|

−1/n

Which is obvious, since f

considered as function of a

complex variable z, is (within

the radius of convergence)

equal to

˜f

(

z

) =

1

1

₋

z/

(

2i

)

having a pole at z

=

2i,

Im z

Re z

1

1

Blue area: convergence

Consider again

f

(

x

) =

∞

∑

n=0

a

n

x

n

=

∞

∑

n=0

 x

2i



n

,

defined within the radius of

convergence,

whereas

˜f

(

z

) =

1

1

₋

z/

(

2i

)

is defined in C

_\{

2i

_}

Im z

Re z

1

1

Blue: convergence of f

1

1

Re z

Im z

Pink: convergence of ˜f

Going from f to ˜f is (a trivial case) of a



analytic continuation

of a holomorphic function into

a new region



where an infinite series representation in terms of which

it is initially defined becomes divergent.

Holomomorphic function:



complex differentiable in a neighbourhood,



infinitely differentiable

Even without an analytical formula for the Taylor series,

there is a systematic approach to extend the region:

Proceed via a sequence of

discs

D

k

, with centers z

k

,

with

f

k

(

z

) =

∞

∑

n=0

a

n

(

z

−

z

k

)

n

,

being convergent in D

k

,

and f

k

=

f

k−1

in the

over-lap region.

Re z

Im z

Identity theorem:

For given functions f and g, holomorphic on a domain D,



if f

=

g on some S

⊆

D then f

=

g on D.

Consequence:

2.2

Branch cuts

An example: The logarithm.

The exponential function defines a mapping M

M :

C

→

C

w

_→

z

=

exp

(

w

)

Consider w

=

x

+

iy, with x

fixed and y going from

−

π

to π.

(Trajectory γ going from

w

1

=

x

−

iπ to w

2

=

x

+

iπ)

x = Re w

w

w

2

1

y = Im w

The mapped trajectory γ

′

=

M

(

γ

)

is given as

z

=

exp

(

w

) =

exp

(

x

)

exp

(

iy

)

=> A circle with start and end point z

1

=

z

2

= −

e

x

x = Re w

w

w

2

1

y = Im w

z =

₁

z

₂

Im z

Re z

Doing the inverse mapping

M

−1

: z

_→

w

=

log

(

z

)

,

we get for z

1

=

z

2

two different values w

1

and w

2

!!

One has to define log in C

−

R

_≤0

_(branch).

The negative real axis is called branch cut.

Im z

Re z

z

₂

z

₁

x = Re w

w

w

2

1

y = Im w

The discontinuity at z

_{= −}

e

x

:

2.3

Cut diagrams

The scattering operator ˆ

S is defined via

|

ψ

(

t

= +

∞

_{i =}

_S

ˆ

_|

_ψ

(

t

_{= −}

∞

_i

Unitarity relation ˆ

S

†

S

ˆ

=

1 gives (

considering a discrete Hilbert space

)

1

_{= h}

i

_|

S

ˆ

†

S

ˆ

_|

i

_i

=

∑

f

h

i

_|

S

ˆ

†

_|

f

_{i h}

f

_|

S

ˆ

_|

i

_i

=

_∑

f

h

f

_|

S

ˆ

_|

i

_i

∗

_h

f

_|

S

ˆ

_|

i

_i

Expressed in terms of the S-matrix:

1

=

_∑

f

S

∗_{f i}

S

f i

Using

S

f i

=

δ

f i

+

i

(

2π

)

4

δ

(

p

f

−

p

i

)

T

f i

dividing by i

(

2π

)

4

_δ

₍

₀

₎

_:

1

i

(

T

ii

−

T

∗ ii

) =

∑

f

(

2π

)

4

δ

(

p

f

−

p

i

)





T

_{f i}





2

✬

✫

✩

✪

Be φ le current of incoming particles hitting a target of surface A containing N particles. The transsition rate τ is

τ=φAσN

A =φ σ N, The cross section is _σ= τ

Nφ = τ V φ ρ = W TV φ ρ≡ W TV w. The transition probability W= |S_{f i}|2 _is



(2π)4_δ4_(p f−pi)

2

|Tf i|2=TV(2π)4δ4(pf−pi)|Tf i|2. The cross section is then _σ= 1

w|Tf i|

2_(2π)4_δ4_(p f−pi).

with w=2E1v12E2. We need a covariant form of f = E1v1E2. In the lab frame, we have f2_{= |~p}

1|2m22 = (E12−m21)m22, which gives the invariant form f = q (p1p2)2−m21m22. With 2p1p2=s−m2₁−m22, we get 2 f = q (s−m2₁−m2₂)2_−4m2 1m22, and thus w=4 f =2 r  s− (m1+m2)2   s− (m1−m2)2  →2s for s→∞

Using

1

i

(

T

ii

−

T

∗

ii

) =

2ImT

ii

,

we get the optical theorem

2ImT

ii

=

∑

f

(

2π

)

4

δ

(

p

f

−

p

i

)



Assume:



T

ii

is Lorentz invariant

→

use s, t



T

ii

(

s, t

)

is an analytic function of s, with s considered

as a complex variable

(Hermitean analyticity)



T

ii

(

s, t

)

is real on some part of the real axis

Using the Schwarz reflection principle, T

ii

(

s, t

)

first defined

for Ims

_≥

0 can be continued in a unique fashion

via T

ii

(

s

∗

, t

) =

T

ii

(

s, t

)

∗

.

So:

1

i

(

T

ii

(

s, t

) −

T

ii

(

s, t

)

∗

_{) =}

1

i

(

T

ii

(

s, t

) −

T

ii

(

s

∗

_{, t}

₎₎

Def:

We have finally

1

i

disc T

= (

2π

)

4

_δ

₍

_p

f

−

p

i

)

∑

f





T

_{f i}





2

=

2s σ

_tot

Interpretation:

1_i

disc T can be seen as a so-called “cut

di-agram”, with modified Feynman rules, the “intermediate

particles” are on mass shell.

Modified Feynman rules :



Draw a dashed line from top to bottom



Use “normal” Feynman rules to the left



Use the complex conjugate expressions to the right



For lines crossing the cut: Replace propagators by mass

Cutting a diagram representing elastic scattering

corresponds to inelastic scattering

2

=

Cutting diagrams is useful in case of substructures:

=

Precisely the multiple scattering structure

in EPOS

= +

+

+ + ...

Cut diagram

= sum of products of cut/uncut subdiagrams

What are the blocks, called Pomerons?

=



Pomeron = parton ladders

2.4

Cuts in the s-plane

The inelastic cross section,

1

2s

(

2π

)

4

_δ

₍

_p

f

−

p

i

)

∑

f





T

_{f i}





2

will be

nonvanishing beyond some

s threshold

allowing

massive particles in the final state, and because of

1

i

disc T

= (

2π

)

4

_δ

₍

_p

f

−

p

i

)

∑

f





T

_{f i}





2

we have a

non-vanishing discontinuity

.

Non-vanishing discontinuity beyond some threshold means:

There is a branch cut

along the positive Re s axis

Re s

Im s

On the other hand, analyticity allows to continue T

(

s, t

)

from the “physical region”

s

>

_{0, t}

<

_{0, u}

<

₀

into the “non-physical region”:

s

<

_{0, t}

<

_{0, u}

>

₀

being the “physical region” for the u-channel reaction

1

+

¯4

_→

¯2

+

3

We have also a branch cut in u in the u-channel, and with

s

=

_∑

m

2_i

₋

t

₋

u

we get in addition a branch

cut for negative Re s.

Im s

Useful (later) for example for doing integrals.

We may have

Z

∞ −∞

...

=

Z

C

...

Im s

Re s

C

which may be

transformed into

Z

C′

...

Im s

C’

Re s

and we get

Z

∞ −∞

f

(

s

)

ds

=

Z

∞ s_thr

disc f

(

s

)

ds

2.5

Dispersion relations

One may write (Cauchy)

T

(

s, t

) =

1

2πi

Z

c

T

(

s

′

, t

)

s

′

₋

_s

ds

′

where on may choose C

as in the figure

C

Re s

Im s

So we find (assuming the semi-circles do not contribute)

T

(

s, t

) =

1

2πi

Z

∞ s_thr+

disc T

(

s

′

, t

)

s

′

₋

_s

ds

′

+

1

2πi

Z

_−s− thr −∞

disc T

(

s

′

, t

)

s

′

₋

_s

ds

′

This is called “dispersion relation”.

Will be useful later.

2.6

Watson-Sommerfeld transform

Consider a reaction (all particles having mass m)

1

+

2

_→

3

+

4

in the x

₋

z plane.

In the CMS, we have (P

_{= |~}

p

_|

)

P

1

=

P

2

=

P

3

=

P

4

(

E, p

x

, p

z

) =



















(

√

P

2

₊

_m

2

_{, 0, P}

₎

₍

₁

₎

(

√

P

2

₊

_m

2

_{, 0,}

₋

_P

₎

₍

₂

₎

(

√

P

2

₊

_m

2

_{, P sin θ, P cos θ}

₎

₍

₃

₎

(

√

P

2

₊

_m

2

_,

₋

_{P sin θ,}

₋

_{P cos θ}

_{) (}

₄

₎

which gives

s

=

4 P

2

+

m

2

t

=

0

₋

P

2

sin

2

θ

₋

P

2

(

1

₋

cos θ

)

2

_{= −}

2P

2

(

1

₋

cos θ

)

Defining

z

=

cos θ

we have

t

=

2

(

s

4

−

m

2

₎₍

_z

−

1

)

u

=

2

(

s

4

−

m

2

)(−

z

₋

1

)

We now consider a T matrix element as a function T

(

z, s

)

One may always expand (partial wave expansion)

T

(

s, t

) =

∞

∑

j=0

(

2j

+

1

_{)T (}

j, s

)

P

j

(

z

)

with

T (

j, s

) =

1

2

Z

₁ −1

dz T

(

s, t

)

P

j

(

z

)

due to the orthogonality property of Legendre

polynomi-als

2n

+

1

2

Z

1

We analytically continuate



to the unphysical region of the 1

+

2

→

3

+

4 process:

T

1+2→3+4

(

s

<

0,

t

>

0

)



corresponding to the physical region

of the 1

+

¯3

_→

¯2

+

4 (t-channel) process:

T

_{1+¯3→¯2+4}

(

t

>

_0,

_s

<

₀

₎



and after exchanging s and t :

T

_{1+¯3→¯2+4}

(

s

>

_0,

_t

<

₀

_{) ≡}

_T

₍

_{s, t}

₎

_.

With this new definition of T, the partial wave expansion

reads

T

(

s, t

) =

∞

∑

j=0

(

2j

+

1

)T (

j, t

)

P

j

(

z

)

with

T (

j, t

) =

1

2

Z

1 −1

dz T

(

s, t

)

P

j

(

z

)

with

z

=

1

+

2s

t

since we also exchanged s and t in z

=

1

+

2t

s

(neglecting

The Watson-Sommerfeld transform amounts to write the

partial wave expansion as

T

(

s, t

) =

1

2i

Z

C

dj

1

sin πj

(

2j

+

1

)T (

j, t

)

P

j

(

z

)

with a contour

integra-tion in the complex j

plane

C

Re j

Im j

Opening the contour to

integrate along the

imag-inary axis (Imj

_{= −}

1₂

)=>

one picks up poles of

T

at j

=

α

n

(

t

)

, residue β

′_n

(

t

)

Re j

Im j

C’

T

(

s, t

) =

1

2i

Z

C′

dj

2j

+

1

sin πj

T (

j, t

)

P

j

(

z

)

+

_∑

π

2α

n

(

t

) +

1

sin πα

n

(

t

)

β

′_n

(

t

)

|

{z

}

βn(t)

P

αn(t)

(

z

)

2.7

The signature

Problem



The partial wave amplitudes have contributions which

alternate in sign, so there is a factor

(−

1

)

j

=

exp

(

iπj

)

which diverges on the imaginary axis.

Solution

We use the dispersion relation

T

(

z, t

) =

1

2πi

Z

∞ z0

disc T

(

z

′

, t

)

z

′

₋

_z

dz

′

+

1

2πi

Z

_−z0 −∞

disc T

(

z

′

, t

)

z

′

₋

_z

dz

′

put into

_{T (}

j, t

) =

1 2

R

1 −1

T

(

z, t

)

P

j

(

z

)

dz :

T (

j, t

) =

1

2πi

Z

∞ z0

disc T

(

z

′

, t

)

1

2

Z

1 −1

P

j

(

z

)

z

′

₋

_z

dzdz

′

+

1

2πi

Z

₋z0 −∞

disc T

(

z

′

_{, t}

₎

1

2

Z

1 −1

P

j

(

z

)

z

′

₋

_z

dzdz

′

We use “Neumanns formula”

1

2

Z

1 −1

P

j

(

z

)

z

′

₋

_z

dz

=

Q

j

(

z

′

)

with Legendre functions Q

j

of the second kind, and get

T (

j, t

) =

1

2πi

Z

∞ z0

disc T

(

z

′

, t

)

Q

j

(

z

′

)

dz

′

+

1

2πi

Z

∞ z0

disc T

₍₋

z

′

, t

)

Q

j

(−

z

′

)

dz

′

We use the symmetry property Q

j

(−

z

) = (−

1

)

j+1

Q

j

(

z

)

,

and D

±_j

=

1 2πi

R

∞

z0

disc T

(±

z

and we get

T (

j, t

) =

n

D

+_j

_{+ (−}

1

)

j+1

D

−_j

o

=

_∑

η∈{−1,1}

1

+

η

₍₋

1

)

j

2

n

D

+_j

_{+ (−}

1

)

j+1

D

−_j

o

≡

∑

η∈{−1,1}

1

+

η

₍₋

1

)

j

2

T

η

₍

_{j, t}

₎

with a so-called signature η.

The Watson-Sommerfeld transform then reads

T

(

s, t

) =

_∑

η

1

2i

Z

C′

dj

1+η(−1)j 2

1

sin πj

(

2j

+

1

)T

η

₍

_{j, t}

₎

_P

j

(

z

)

+

_∑

η

∑

nη

1

+

η

₍₋

1

)

αnη(t)

2

β

nη

(

t

)

P

α_nη(t)

(

z

)

The poles are called



even signature Regge poles (η

= +

1) or



odd signature Regge poles (η

= −

1)

2.8

Reggeons and Pomerons

Asymptotically

, for s

≫

t, i.e. z

=

1

+

2s

t

≫

1, the

Legen-dre polynomial

P

j

(

z

) =

j

∑

k=0



j

k

 

j

+

k

k

  z

₋

1

2



k

is dominated by its leading term,

P

j

(

z

) →



2j

j

  z

₋

1

2



j

so we have

P

j

(

z

) =

P

j

(

1

+

2s

t

) →

Γ

₍

_2j

₊

₁

₎

Γ

2

₍

_j

₊

₁

₎

 s

t



j

.

Contour integration

(along j

_{= −}

1₂

+

iy

)

, for s

_≫

t

vanishes, because of factor

(

s/t

)

−1/2

from asymptotic form of P

j

(

z

)

Regge pole part

(absorbing the Γ functions and t

−j

into β),

for s

_≫

t

∑

η

∑

nη

1

+

η

₍₋

1

)

αnη(t)

2

β

nη

(

t

)

s

α_nη(t)

With α

(

t

)

being the rightmost (leading) pole,

in the limit s

_≫

t :

T

(

s, t

) =

1

+

ηe

−

iπα

(

t

)

2

β

(

t

)

s

α(

t

)

This is the Regge pole expression for the T-matrix,

viewed as Reggeon exchange amplitude

Regge trajectories

With t

_≪

s (Regge limit), we expect

α

(

t

) =

α

(

0

) +

α

′

t ,

i.e. linear Regge trajectories

Considering

t-channel

processes with t

>

_0:

α

(

t

)

=

ˆ

L

(

M

2

)

Can be checked:

Consider known hadrons

Chew, Frautschi, 1961

=> α

(

0

) ≈

0.5

The Pomeron

pp cross section via Reggeon exchange:

σ

tot

=

1_s

ImT

(

t

=

0

)

, so

σ

tot

∝ s

α(0)−1

with α

(

0

) =

0.5 :

σ

tot

∝ s

−0.5

Pomeranchuk (1956) :



Any exchange of charged hadrons leads asymptotically

to vanishing cross section

Solution :



Exchange of Pomeron, having vacuuum quantum

num-bers, α

(

0

) >

1. Like α

(

0

) =

1.08 seems to work.

Donnatchie-Landhoff fit

(1991):

Two contributions: Pomeron, α

(

0

) =

1.08

and Reggeon, α

(

0

_{) ≈}

0.5

α

(

0

) =

1.08 seemed to work, but

σ

tot

∝ s

0.08

violated the Froissart bound (1963), saying that unitarity

requires

σ

tot

<

const

× (

log s

)

2

Solution: Gribov (1960s): Multiple Pomeron exchange, to

recover unitarity

—————————————————————

3

Pomerons and pQCD parton ladders

3.1

Parton evolution in ep scattering

A fast moving proton

t

z

proton

cloud

of gluons

emits

successively

partons

(mainly

gluons),

quasi-real

(large gamma

fac-tors)

... which can be probed by a virtual photon

(emitted from an electron)

t

z

proton

photon

k

p

qbar

q

color

dipole

photon

splits

into q-qbar

→

Color dipole

p and k are

pro-ton and phopro-ton

momentum

What precisely the photon “sees” depends on two

kine-matic variables,

the virtuality

Q

2

_{= −}

k

2

and the Bjorken variable

x

=

Q

2

2pk

which probes partons with momentum fraction x.

It determines also the approximation scheme to compute

the parton cloud.

s

−λ

Q = x

ln Q

2

ln 1/x

soft

BFKL

saturation

DGLAP

DGLAP:

sum-ming to all

or-ders of α

s

ln Q

2

BFKL:

sum-ming

to

all

orders of α

s

ln

1_x

Linear

equations

BFKL (Balitsky, Fadin, Kuraev, and Lipatov):

∂ϕ

(

x, q

)

∂ ln

1_x

α

s

N

c

π

2

Z

d

2

k K

(

q

, k

)

ϕ

(

x, k

)

with xg

(

x, Q

2

) =

Z

Q2 0

d

2

k

k

2

ϕ

(

x, k

)

,

DGLAP (Dokshitzer, Gribov, Lipatov,

Altarelli and Parisi):

∂g

(

x, Q

2

)

∂ ln q

2

=

Z

₁ x

dz

z

α

s

2π

P

(

z

)

g

(

x

z

, Q

2

₎

Very large ln 1/x : Saturation domain

t

z

proton

Non-linear

ef-fects

Gluon

from

one cascade is

absorbed

by

another one

3.2

pp scattering (linear domain)

proton

proton

Different way of plotting the same reaction

nucleon

nucleon

t=0 time log(x /x ) 0.5 + −

Corresponding cut diagram

referred to as

“cut parton ladder”

Corresponding elastic diagram

3.3

Soft domain

Very small ln Q

2

: No perturbative treatment!

=> use general features of the T-matrix. Let α

(

s

)

be the

rightmost Regge pole of the partial wave amplitude :

t

_→

∞:

T

(

s, t

)

∝ s

α(t)

with the Regge trajectory

α

(

t

) =

α

(

0

) +

α

′

t

Im j

Re j

Perturbative:

Parton ladder

T-matrix computed

(DGLAP)

Soft:

Soft Pomeron

gluon fields

T-matrix parametrized

Formulas (

see Phys.Rept. 350 (2001) 93-289

):

T

soft

(

ˆs, t

) =

8πs

0

i γ

Pom2 −parton

 ˆs

s

0



α_soft(0)

×

exp

(

λ

soft

t

)

,

with

λ

_soft

=

2R

2_Pom−parton

+

α

′soft

ln

ˆs

s

0

.

Cut soft Pomeron (Schwarz reflection principle):

1

i

disc T

soft

(

ˆs, t

)

=

1

i

[

T

soft

(

ˆs

+

i0, t

)

−

T

soft

(

ˆs

−

i0, t

)]

Interaction cross section,

σ

soft

(

ˆs

) =

1

2ˆs

2Im T

soft

(

ˆs, 0

)

,

=

8πγ

2_part

 ˆs

s

0



α_soft(0)−1

,

using the optical theorem (with t

=

0),

3.4

Semihard Pomeron

soft

soft

parton

ladder

Space-time picture of semihard Pomeron

soft preevolution

cascade

parton

3.5

Gribov-Regge approach

T-matrix approach

allowing multiple

(pQCD-)Pomeron

exchange

cross section via

cutting rules

+

+ +

3.6

Factorization approach

Hypothesis:

The inclusive

cross section factorizes:

dσ

_inclpp

dp

2 t

=

f

_⊗

dσ

partons

dp

2 t

⊗

f

with f from DIS

F µ2 Q₀2 Q₀2 p_t2 ... ...

Inclusive pp cross section :

dσ

_inclpp

dp

2 t

=

∑

mn

Z

dxdx

′

dp

2_t

dσ

kl→mn Born

dp

2 t

(

xx

′

s, p

2_t

)

×

f

k

(

x, Q

20

, µ

2F

)

f

l

(

x

′

, Q

20

, µ

2F

)

with

µ

2_F

=

µ

2_F

(

p

2_t

)

3.7

Models

For pp scattering

model Gribov Dipole Facto used authors Regge risation for CR

QGSJETII X

I

X Ostapchenko

EPOSLHC X

I

X Pierog, Werner

EPOS3 X

I

Werner, Pierog

DIPSY

I

X Lönnblad, Bierlich

IP-Glasma

I

X Tribedy, Schenke

SIBYLL

I

X X Engel, Riehn

DPMJETIII

I

X X Engel, Fedynitch

PYTHIA

I

X Sjostrand, Skands

HERWIG

I

X Marchesini, Webber Gieseke

with secondary interactions

model primary secondary scatterings interactions

EPOS Gribov Regge viscous hadronic hydrodynamical cascade expansion of QGP

IP-Glasma Dipole model “

Ig

DIPSY Dipole model flow-like effects

AMPT Minijets partonic cascade “ from Pythia

Plus many hydro models with assumed initial conditions

Ig

Cascade means:

The Gribov-Regge approach as implemented in EPOS will

de discusses in very much detail later ...

Nonlinear effects in QGSJET

Pomeron-Pomeron coupling

+

+

+ ...



Summing of all orders



No energy conservation

DIPSY

(from Christian Bierlich)

Initial nucleon: Three dipoles

LL BFKL in b-space + corrections: A dipole (~x,~_{y) can emit a gluon at position}~_z

with probability (P) per unit rapidity (Y) dP dY = ¯α 2πd 2_~z (~x− ~y)2 (~x−~z)2_{(~z− ~y)}2

Multiple scattering:

Multiple color exchange

between dipoles i and j with

probabilities

α

2_s

4

"

log

(~

x

i

− ~

y

j

)

2

_(~

_y

i

− ~

x

j

)

2

(~

x

i

− ~

x

j

)

2

(~

y

i

− ~

y

j

)

2

!#

2

-> kinky strings



Two “leading“ strings



Additional strings

from loops



No Remnants

Many strings:

Lund strings may overlap

=> color ropes

IP-Glasma

(from Prithwish Tribedy)

IP-Sat dipole model

(r

_⊥

=

dipole size):

dσ

d

2

_b

=

2

[

1

−

exp

(

−

F

(

r

⊥

, x, b

)]

, F ∝ r

2

⊥

α

s

(µ

2

)

xg

(

x, µ

2

)

T

(

b

)

T

(

b

)

: Gaussian profile, µ

2

=

4/r

2_⊥

+

µ

2₀

, xg : DGLAP evolution

Saturation scale

Q

s

defined via

F



r

_⊥

, x

=

2

Q

2 s

, b



=

1

2

IP-Glasma

: Color charge squared for projectile A and target B :

g

2

µ

2_A

=

∑

_nucleons

g

2

µ

2_i

, with g

2

µ

2_i

∝ Q

2

Multiple Scattering:

Color charge density

ρ

A/B

generated from Gaussian

dis-tribution with variance g

2

µ

2_A

(contains DGLAP, saturation)

Current

J

ν

=

δ

ν±

ρ

A/B

(

x

∓

,

x

⊥

)

Field from

[

D

µ

,

F

µν

] =

J

ν

Numerical (lattice) solution,

fields can be expressed in terms

of initial ones:

A

i

=

A

i_A

+

A

i_B

,

A

η

=

ig₂

[

A

i_A

, A

i_B

]

Initial configuration JIMWLK evolution Single gluon emission A (classical field)

Multiple scattering:

Nonlinearity in terms of A:

Infinite number of g

+

g

→

g

processes

Multiple scattering in factorization approaches

First step:

Compute (based on factorization)

σ

_inclpp

=

Z

_dσ

pp incl

dp

2_t

dp

2 t

Second step:

Multiple scattering introduced via eikonal

formula for the probability of n scatterings:

Ei

(

b, n

) =



σ

_inclpp

(

s

)

T

(

s, b

)



n

n!

exp

−

σ

pp incl

(

s

)

T

(

s, b

)

with

_Z

d

2

b

∑

n

n

_×

Ei

(

b, n

) =

σ

_inclpp

(

s

)

Multiple scattering in

SIBYLL

From F. Riehn

Multiple

scattering

via

eikonal model with soft and

hard component



No Remnants



Main scattering

=> qq-q strings



Further scatterings

=> strings

between

gluon pairs

nucleon nucleon

Saturation scale

from

α

s

N

c

Q

2

×

1

N

_c2

−

1

xG

πR

2

=

1

Multiple

scattering

in Pythia

arXiv:1101.2599

Color

reconnections

—————————————————————

4

Multiple Pomeron exchange in EPOS

—————————————————————

Parton based Gribov-Regge theory

By H.J. Drescher, M. Hladik, S. Ostapchenko, T. Pierog, K. Werner. hep-ph/0007198. Published in Phys.Rept. 350 (2001) 93-289.

EPOS3

4.1

The Pomeron in EPOS

First overview, precise formulas later

T-matrix = sum of 5 terms

The val-val Pomeron

T

val−val

(

s, t

)

Parton ladder starting

from valence quarks

(emission starts long before

the actual interaction)

T-matrix computed

The soft Pomeron

T

soft

(

s, t

)

contains only soft partons

parametrized as Regge-pole

T

(

s, t

)

∝

_s

α(0)+α′(t)

non-perturbative gluon

_fields

T-matrix parametrized

Important! Our model should work down to few GeV

The sea-sea Pomeron

T

sea−sea

(

s, t

)

parton ladder starting

from sea quarks

Sea quarks emitted

from a soft Pomeron

(soft pre-evolution)

soft

soft

parton

ladder

Parton ladder computed

(DGLAP)

Soft part parametrized

(Regge pole)

The sea-val Pomeron

T

sea−val

(

s, t

)

parton ladder starting

from sea quarks (proj side)

and from valence quarks

(target side)

Parton ladder computed

(DGLAP)

Soft part parametrized

(Regge pole)

And for symmetry reasons we have of course

a sea-val Pomeron

Details later ... but one important remark:

Let p be the momentum

en-tering the soft part and k the

momentum of the first

emit-ted perturbative parton with

k

t

≪

k

+

≪

p

+

, p

−

≈

0

soft

k

p

Contribution to the T-matrix from the k-loop:

Z