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Reaction-diusion approach in soft diraction

Rodion Kolevatov

(SPbSU and SUBATECH)

in collab oration with K.Boreskov (ITEP)

QFTHEP-2013, StPetersburg, 28 June

R. Kolevatov

RD approach in soft diraction


Power-like contributions to the amplitude

PDG t:

tot

pp (p ) Ї

=

18.3

s

0 095

.

+

60.1

s

-0.

34

±

32.8

s

-0.

55

Optical theorem:

tot =

1 2

s

Ael (q

= 0)

2

Mel (q

= 0)

Indication: High energy elastic scattering go es via quasiparticle (Reggeon) exchanges with p owerlike asymptotic in c.m.energy.

y

Leading contirbution Pomeron,

=

ln

s

MP



s





e

y , (

>

0

overall rapidity)

R. Kolevatov

RD approach in soft diraction


Elastic scattering shrinkage of diractive cone

B

d el dt

= 21 |M (s , t )|2

-

d d el dt ln dt t

=0

M (s
of

= ey , t)

exp[

y

- (R 2 + y )t ]

implies reasonable assumptions ab out the analytic prop erties

T (s , t

)

Caveat: The p ower-like b ehaviour violates unitarity b ound (

tot

C

ln

2

s

).

R. Kolevatov

RD approach in soft diraction


Impact parameter representation

Fourier transform:

f (Y , b

)=

(2 )2

1

d qe
2

-i qb

M (Y , q

).
2

el =

d 2q (2 )2
2

|M (Y , q)|2 =

d b |f (Y , b
2
2

)|2 . ),

tot (Y ) =

M (Y , q

= 0) =

d b f (Y , b

Denition inel (b)
Unitarity constraint: 0
Interpretation:

2

<

f (b) - |f (b) f (b ) < 2

|2
0

inel (b)

1

inel (b)

probability of inelastic interaction

R. Kolevatov

RD approach in soft diraction


Geometrical models

Unitarity limit: Black disk

f (b) limit: f (b

= 2(R - b) inel (b) = 0. ) = (R - b) inel (b) = (R - b), el = 1/2 tot
growing size

.

The data suggest: shrinkage of diracitve cone presence of dip



deviations from Gaussian prole shap e

The inelastic prole in the center is close to the upp er limit (e.g.

inel (b) = 0.

94 at



s

=

53 GeV)

R. Kolevatov

RD approach in soft diraction


Inelastic diraction a special case of inelastic event

y

=

1 2

ln

E +pz E -pz

Illustration: talk by Chris Quigg at Spaatind'2012 R. Kolevatov RD approach in soft diraction


Soft inelastic diraction
Single diraction

Double diraction

Central diraction

ygap = SD
10 mbn @ 7TeV

ln

s /M

X

2

rapidity gap

[ TOTEM preliminary @ Trento pA workshop 05/2013 ]

R. Kolevatov

RD approach in soft diraction


Low-

M

2

diraction

Multichannel approach (Go o d, Walker '60):

| |
1 , |2

p

= 1 |1 + 2 |2 ;

2 2 1 + 2 =

1

scattering eigenstates (amplitudes

iF (y , b
1

)

and

iF (y , b
2

)

)

tot =

2

db
2

2 [1

F (b
1

2 )+2

F (b
2

)];

el =

db
2

2 [1

F (b
1

2 )+2

F (b
2

)]2

SD =

db
2

[1 2 (F1 (b) -

F (b
2

))]

2

R. Kolevatov

RD approach in soft diraction


Low-

M

2

diraction

Multichannel approach (Go o d, Walker '60):

| |
1 , |2

p

= 1 |1 + 2 |2 ;

2 2 1 + 2 =

1

scattering eigenstates (amplitudes

iF (y , b
1

)

and

iF (y , b
2

)

)

Lessons from the example 1: Has a p eripheral nature 2: Black disc limit for the elastic amplitude implies (growing ring). This holds also for largedierent origin.

dir

ln :

s

M

2

diraction which however has a

R. Kolevatov

RD approach in soft diraction


Contributions to tot from inelastic cuts

Contributions to imaginary part (Cutkosky rules): Cut the diagram for the elastic scattering amplitude Put cut lines on the mass shell, integrate over the phase space Single ladder exchange uniform rapidity distribution 2

M

1

=

2

=

=

d
+

n -

Double ladder 2

=
elastic + LM SD

+
abs. corrections to2

+
T1
double

dN /dy

R. Kolevatov

RD approach in soft diraction


High-

M

2

diraction

Rapidity gaps splitting of the ladder: Single diraction disso ciation + abs. corrections

Double diraction disso ciation + abs. corrections

Motivates the eective theory of the Pomeron (Reggeon) exchanges and interactions Multy-

P exchantes, enhanced & loop graphs Tame the growth, restore s -channel unitarity Give inelastic contributions with rapidity gaps

Account of

all

enhanced graphs is an untrivial task

R. Kolevatov

RD approach in soft diraction


Systematic account of enhanced graphs RFT
The elastic amplitude

iT

A/(8 s )
n ,m

is factorized:

T G
mn

=

V

n Gnm Vm
):

pro cess indep endent, obtained within 2D+1 eld theory (only

P

- - 1 L = (y - y ) - ( b )( b ) + + Lint .
2 Minimal choice (classic): Innite of vertices

Lint =

i r3P

( + )

[ KMR, Ostap chenko, MP+ABK ]:

r

m n mn

Fine tuning of the vertices, some contributions neglected

R. Kolevatov

RD approach in soft diraction


Systematic account of enhanced graphs RFT
The elastic amplitude

iT

A/(8 s )
n ,m

is factorized:

T G
mn

=

V

n Gnm Vm
):

pro cess indep endent, obtained within 2D+1 eld theory (only

P

- - 1 L = (y - y ) - ( b )( b ) + + Lint .
2 Minimal choice (classic): Innite of vertices

Lint =

i r3P

( + )

[ KMR, Ostap chenko, MP+ABK ]:

r

m n mn

Almost minimal:

the reaction-diusion approach is applicable for numerical computation of all-lo op Green functions.
[ Grassb erger'78; Boreskov'01 ]

i r3P

( + ) +

2

2

R. Kolevatov

RD approach in soft diraction


The reaction-diusion (stochastic) approach.
Consider tons in a system the of classic plane parwith:

transverse

Diusion (chaotical movement) Splitting ( Death (

D

;


)

prob. p er unit time)

m

1

Fusion (



Annihilation (

d b p (b)) m2 d b pm2 (b
2 2

))

Parton numb er and p ositions are describ ed in terms of

probability densities N (y , BN ) (N = 0, 1, ...; BN {b , . . . , bN })
1

with normalization

pN (y

1 ) N!

N (y , BN )

d



BN ;
0

pN

=

1.

R. Kolevatov

RD approach in soft diraction


Inclusive distributions

S -parton inclusive distributions: fs (y d Zs fs (y
allowed.

; Zs ) =
N! (N -s )!

N

(

N -s

1

)!

d

BN N (y ; BN )

s i=
1

(zi - bi );

; Zs ) =

pN (y

) µs (y ).

factorial moments.

Example: Start with a single parton with only diusion and splitting

f

1 parton

1

(y , b) =

exp(

y

)

exp( 4



the bare Pomeron propagator in

b

Dy

-b2 /

4

Dy

)

.

-representation.

The set of evolution equations for fs (Zs ), (s = 1, . . .) coincides with the set of equations for the Green functions of the RFT.
R. Kolevatov RD approach in soft diraction


The amplitude.
To compute the RFT elastic amplitude: Hadron at

y

n

P

vertices



distribution of partons

=

0 evolution time:

MC evolution



set of

~ fs (0, Zs ) = Ns (Zs ) ~ ~ fs (y , Bs ) (fs (y , Bs ))
2

/s
for

/2

the projectile (target) With some narrow

T (Y , b


=
s=
1

) (-1)s s!

-

g (b), g (b)d b ~ M (Y , b) = A|T |A =
1



:

d

~ ~ ~ Zs d Zs fs (y ; Zs )fs (Y - y ; Zs )

s i =1

g (z

z i - ~i - b). if

T does not depend on the linkage point y (boost g (b)d b = pm2 (b)d b + p (b)d b
2 2 1 2

invariance)



2

,

equality of fusion and splitting vertices in the RFT.

R. Kolevatov

RD approach in soft diraction


Correspondence RFTStochastic model
We use the simplest form of

pm

2

(

with

a

some

g (b), pm2 (b) and p (b): b) = m (a - |b|); p (b) = (a - |b g (b) = (a - |b|);. small scale; a .
2 2

|);

RFT Rapidity

y
1

sto chastic mo del Evolution time

y

Slop e = (0) - Fusion vertex

Diusion co ecient

Splitting vertex

r3P

r3P


2

Quartic coupling

- m1 (m2 + 1 ) 2 1 (m2 + ) 2

D

Few things to note:
Bo ost invariance ( The 2

=

m

2

+

)



equality of fusion and splitting vertices.



2 vertex cannot b e set to zero (

m

2

, >

0).

R. Kolevatov

RD approach in soft diraction


Calculation method elastic amplitude

Convenient choice set the linkage p oint to target rapidity:

~ fs (y

= 0, Zs ) = Ns (Zs )/ s , Zs ) =
N

/2

for a given realization via MC evolution

fssample (Y
el Tsample

^ ^ (z1 - xi1 ) . . . (zs - xis ) ~^ ps (xi1 - b ^ , . . . , xis - b).

^ {^i1 ,..,^is }XN x x

=
s =1

(-1)s

-

1

µs s ~

i1
R. Kolevatov

RD approach in soft diraction


Calculation method the SD cut

For the SD cut substituting event-by-event Green functions gives

SD Tsample

=

2

el Tsample - Tsample el Tsample
with two distinctions:

Tsample

is computed the same way as

Not one, but two sets from the projectile side which are evolved indep endently until the combined into a single one

ygap

and then

ResumИ:

The elastic scattering amplitude and its SD cut are

computed within the same numerical framework.

R. Kolevatov

RD approach in soft diraction


Model parameters

Two-channel eikonal diraction

p n

P

vertices to incorp orate low-

M

2

Account the secondary Reggeons contribution to the lowest order Real part of the Pomeron exchange amplitude evaluated via Grib ovMigdal relation Neglect central diraction in calculation of SD cross sections (CD contribution is accounted twice in calculation of 2-side SD, the extra contribution should have b een subtracted).

R. Kolevatov

RD approach in soft diraction


Model parameters

r a

3

P xed from [Kaidalov'79]
regularization scale

+ bare Pomeron intercept Pomeron slope |p = 1 |1 + 2 |2 ; |1 |2 C1 ; |2 |2 C2 = 1 - P couplings to |1 and |2 : g1/2 = g0 (1 ± ) R1 , R2 size of the pP vertex (Gaussian)
1 Strategy: 1 Eikonal t to low-

C

1

.

M

2

2 All-lo op from [1]

tot , el , d el /dt SD 1.5mbn at s = t to tot , el , d el /dt

keeping 35

GeV /c

starting with parameter set

3 Calculation of diractive cross sections with parameters obtained at [2]

R. Kolevatov

RD approach in soft diraction


Calculation results

Total and elastic cross sections:

tot (



s

),

mbn

el (



s

),

mbn

d el dt (t )

, mbn GeV

-

2



s

, GeV


GeV
1

s

, GeV
-1

t
=

, GeV

2

= 0.19; = 0.236 GeV-2 ; C1 = 0.1, C2 = 1 - C1 = 0.9; R1 = 0.51 r3P = 0.087 GeV-1 [Kaidalov'79].

-

; R2

= 2.

8 GeV

; g1

46.7 GeV

-

1

; g2

=

11.7 GeV

-1

;

R. Kolevatov

RD approach in soft diraction


Single diraction

SD (



s ), mbn

Prole,



s

= 240 GeV/c

Prole,



s

= 13.5 TeV/c



s , GeV

b, fm

b, fm

R. Kolevatov

RD approach in soft diraction


M

2

(rapidity gap) dependence, preliminary

Single-diractive cross section as a function of

2 (linear behaviour corresponds to 1/MX -scaling of d /dM 2 ) min SD (ygap ), mbn

ygap

min

,



s

=

5

TeV

:

2d d 2 min dygap = -MX dMX

y

gap

min

R. Kolevatov

RD approach in soft diraction


Conclusions

Total, elastic and single diractive cross sections are computed in RFT within the same numerical framework to all orders in the numb er of lo ops; A satisfactory description on total and elastic cross sections is obtained within the all-lo op framework; The single diractive cross sections energy b ehaviour is compatible with logarithmic growth.

R. Kolevatov

RD approach in soft diraction


Backup scale and 4P vertex dependence

Fits with C1 = C2 = 0.5 and R1 = R2 . Much worse description of ddtel at larger t compared to ts with C1 = C2 and R1 = R2 (though still a nice t of slope B )

3 > 1 = 4 > 2 ; a1 = -2 = 0.018 fm; a3 = a4 = 0.036 a2 2 = 0.195; = 0.154 GeV ; R = 3.62 GeV-2 ; g0 = 4.7

fm. C1 = C2 GeV-1 ; r3P

= 0.5, = 0.55. = 0.087 GeV-1 [Kaidalov'79].

R. Kolevatov

RD approach in soft diraction


Backup scale and 4P vertex dependence

Inelastic and diractive proles

R. Kolevatov

RD approach in soft diraction


Backup secondary trajectories

pp SD:

pp: fpp fpp pp: fpp fpp

(b ) = (b ) =

(b ) = (b ) =

AP (b) + [ A+ (b) + AR+ + ReAR- [1 - AP (b) + [ A+ (b) - AR+ - ReAR- [1 -

A- (b)] [1 AP (b)] A- (b)] [1 AP (b)]

- -

AP (b AP (b

)] )]

Di fpp (b

Di ) = fpp (b ) Ponly 1 + |AR+ (b ) + AR- (b )|2 - 2 (AR+ (b ) + AR- (b ))

A ± (y , b

2 ) = ± ±

exp(± y ) b2 2 exp - 4( y + R 2 ) 2± y + 2R± ± ± 1 ± cos ± (0) ± = ±i - sin ± (0)

R. Kolevatov

RD approach in soft diraction