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MSU

Elliptic flow in p-p collision and proton structure

D. d'Enterria, G.Kh. Eyyubova, V.L. Korotkikh, I. P. Lokhtin, A. M. Snigirev

Proposal of Paper

16 May 2008, CMS HI East meeting


·First measurements on CMS LHC are the minimum bias p-p collisions

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Motivation

·David suggested to calculate V2 in p-p by PYTHIA to see non-flow effects. It is useful to understand such kind effects in A-A collisions, using results of PYTHIA. ·There are the theoretical works (Drescher, Strikman, Frankfurt, 2004) where impact parameter dependence of parton-proton interaction is studied in the black disk regime (BDR ) . This regime will be increased at LHC and we would be able to observe it in p-p collisions ·We need the p-p minimum bias events or the di-jet events and to study V2 in p-p collisions. · Ratio V2/ can be calculated as a function of ratio 1/SdN/dy, where S is transverse overlap area of two disks. Nobody knows 1/SdN/dy for p-p collision. It may be enough to measure V2/ . ·There is a method (Lokhtin, Sarycheva, Snigirev, 2003) for calculation a coefficient of jet azimuthal anizotropy without reconstruction of the nuclear reaction plane between the azimuthal position of jet axis and the angles of particles not in the jet.


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Goals

·To find the parameters of parton spatial distribution in nucleon which is a black body disk in the frame of approximation of BDR ·To calculate an eccentricity of p-p collision in initial state as function of impact parameter between two disk centers ·To find the dependence V2/ as function of p-p multiplicity dN/dy, using an incomplete thermolization formula (Bhalerao - 2005, Drescher-2007) and calculate the azimuthal anisotropy in p-p ·To incorporate the calculated azimuthal anisotropy in the PYTHIA particle distribution at LHC energies and extract the inserted collective flow by cumulant or Li-Yang-zero methods from this sum


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Some excerpts of recent theoretical works

1. There are expectations that gluon density in the nucleon is comparable to this in nuclei . (see [1])

(1) H.J. Drescher et al, hep-ph arXiv:/0712.3209[hep-ph]


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Some excerpts of recent theoretical works

2. At LHC energies the soft parton interactions take place in the disk edges and the hard parton interactins happen in more central region. (see [2])

(2) L. Frankfurt et al, Phys.Rev. D69(2004)114010, [hep-ph/031123]


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Some excerpts of recent theoretical works

3. It is possible to find the impact parameter dependence of the generic inelastic probability Pin(b) from experimental data of p-p collision at high energies. (see [2])

(2) L. Frankfurt et al, Phys.Rev. D69(2004)114010, [hep-ph/031123]


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Some excerpts of recent theoretical works

4. We can also find the impact parameter dependence of the parton-disk probability Pblack(b) from model Frankfurt-StrikmanWeiss. (see [2])

Then we can make the inverse analysis and find nucleon thickness function t(b) as analog of nucleus thickness function TA(b) from Pin(b)
(2) L. Frankfurt et al, Phys.Rev. D69(2004)114010, [hep-ph/031123]


Eccentricity scaling and incomplete thermolization
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There is a simple model based on eccentricity scaling and incomplete thermolization (see [3]). Free parameters: V2 hydro and

V2





V2

h yd r o

0



1 1 K / K

(2)
0

where

1 0 dN c K S dy

s

(4)

Here are K0=0.7 (transport calculations) 0 -- parton-medium cross-section Cs=1/sqrt(3) -- velocity of sound S --- transverse overlap area of nuclei dN/dy ­ multiplicity at y=0 V2 hydro -- thermodynamic limit value
(3) H.J. Drescher et al, Phys.Rev. C76(2007)024905, [nucl-th/0704.3553]


Eccentricity scaling and incomplete thermolization
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Glauber initial conditions

(V2 /)

hydro

CGS initial conditions (ydro/) V2 h

S --- transverse overlap area of nuclei, dN/dy ­ multiplicity at y=0, V2

hydro

-- thermodynamic limit value

(3) H.J. Drescher et al, Phys.Rev. C76(2007)024905, [nucl-th/0704.3553]


Eccentricity scaling and incomplete thermolization
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hydro

Free parameters in [3] : (V2

/ ) and
V2

0

hydro

/

0 , mb

Glauber initial conditions CGS initial conditions

0.30 0.02 0.22 0.01

4.3 0.6 5.5 0.5

Our question: If some equilibrium state of medium is formed in A-A collision why it can not be produced in p-p collision ?
(3) H.J. Drescher et al, Phys.Rev. C76(2007)024905, [nucl-th/0704.3553]


Formalism of impact parameter dependence in p-p collision
____________________________________________________________________________

We write by analogy

2 in ( pp) d B(1 e

0 t1,

2

( B)

)

Here is 0 -- parton-medium cross-section. Probability of p-p generic inelastic interaction as function of impact parameter is
0 t1, ( B)

Pin ( B) P , 2 ( B) (1 e 1

2

)

where the proton-proton thickness function is equal to

B B t1, 2 ( B) dxdy t1 ( x , y) t2 ( x , y) 2 2
Here is t1 (b) ­ parton-proton thickness function, which determines parton-disk inelastic interaction .
0 t1 ( b )

Pb

la ck

(b) (1 e

)


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Formalism of impact parameter dependence in p-p collision

So, a density of binary parton-parton collisions is

B B n1, 2 ( x, y, B) 0 t1 ( x , y) t2 ( x , y) 2 2
And we can calculate the eccentricity of p-p

y2 x2 1, 2 ( B) 2 2 2 2 dxdy ( y x )n1, 2 ( x, y, B) y x
where B is an impact parameter of two black body disks.



dxdy ( y 2 x 2 )n1, 2 ( x, y, B)

Transverse overlap area of two disks is

S1, 2 ( B) x 2 y 2
The average number of binary parton-parton collisions is

n1, 2 ( B)



dxdy n1, 2 ( x, y, B) P , 2 ( B) 1


Formalism of impact parameter dependence in p-p collision
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By analogy with A-A collision a multiplicity of p-p is proportional to average number of binary parton-parton collisions
0 dN ch dN ch ( y 0, B) ( y 0) n1, 2 ( B) dBdy dy

or after integration over all region of rapidaty we have

dN ch 0 ( B) N ch ( pp) n1, 2 ( B) dB
or after integration over impact parameter
pp 0 dN ch dN ch ( y 0) ( y 0) d 2 B n1, 2 ( B) dy dy

We get dNppch (y=0)=5 from approximation to LHC energy and from this equation calculate normalization constant dN0ch (y=0)=X.XX .


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What can we do:

1. Let's fit the probability of parton-disk inelastic interaction Pblack(b) , which is received in [2], with the help of parton-proton thickness function t1(b)

Pb

la ck

(b) (1 e


0 t1 ( b )

)
,

For that aim the next formula is used

t1 ( x, y ) t (b) dz
0

e

0 (r R) /



1

r ( x 2 y 2 z 2 ) (b 2 z 2 )
Hear 0 is defined from condition: So, we have three free parameters: 0 , R and
Then we calculate n12 (B) , eccentricity (pp) of p-p and transverse overlap area S(pp), using our formalism



dx dy dz ( x, y) 1


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What can we do:

2. For estimations let's take a formulae (2) and (4) with parameters K0, Cs and (V2 hydro/) from work [3] and calculate V2 for p-p collision

V2 hyd ro 1 V2 ( B) ( B) 1 K ( B) / K K0 0 ( di sk ) dN h ( B )c s K 0 K ( B) S ( B) dBdy

0

We have to remind the values (pp) , s (pp) and dNch/dy(pp) depend on impact parameter B. So we need to calculate also dNch/dy(pp) in different impact parameter bins as

dN ch dN ch ( y 0, B) ( y 0) n1, 2 ( B) dBdy dy
where dN/dy(y=0) is taken from PYTHIA calculations at y=0 in p-p collisions


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What can we do:

3. Then we can calculate the collective azimuthal anisotropy in p-p collision

dN

N 0flo w flo w ( ) {1 2V2 ( pp) cos[2( RP )]} d 2
dN flo w dN su m dN PYT ( ) ( ) d d d

4. The sum of flow and background is
HIA

( )

5. Using Li-Yang method we extract V2{LYZ} and compare it with input V2(pp)


Preliminary Results


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Figures

From fig.12 [2] at x=10-5, Q2=10 GeV

2

Sharp edge

Degraded edge

Fig.1 Impact parameter dependence of parton-disk inelastic interaction after the fit with parameters: a) sharp edge R=0.876 fm, 0=1.336 fm2. b) degraded edge: R=0.556 fm, 0=0.851 fm2, ksi=0.231 fm.


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Table. Parameters of spatial density

Au-Au

p- p
Degrade edge

p- p
Sharp edge

R,fm
, fm

6.37
0.54

0.556
0.231

0.86
0.

0, mb

40.

8.5

13.4


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Figures

Au-Au

>0

Fig.2a. Density of the binary nucleon-nucleon collisions (standard parameters of Au-Au collisions, B=6 fm, R=6.357 fm)

p-p

<0
Sharp edge, R=0.86 fm, B=8fm
Degraded edge, R=0.556fm, ksi=0.231fm, B= 3fm

Fig.2. Density of the binary parton-parton collisions


dN ch ( y 0, B) Figures dBdy_______________________________________________________________
Sharp edge Degraded edge

Fig.3 Impact parameter dependence of dNch(y=0, B) in pp collisions at dN0ch (y=0)=~5.0


S1, 2 ( B), fm

2

Figures Degraded ____ Sharp edge _______________________________________edge ____________________

Fig.5 The transverse overlap area S and eccentricity of two disks in p-p collision


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Figures

Fig.5 The eccentricity of two disks in p-p collision for different ratio diffuseness to radius


Two disks 1 / S1, 2 ( B) dN / dy, fm 2 Figures _______________________________________________________________ Sharp edge Degraded edge

Fig.6. Dependence

1 dN ch ( y 0, B)B, fm 2 on impact parameter B S1, 2 dy


V2/eps

Two disks Sharp edge

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Figures

Degraded edge

Fig.7. V2/eps as a function of particle multiplicity over overlap area of two disks in p-p collisions


V2

Two disks Sharp edge

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Figures

Degraded edge

Fig.8. V2 as a function of centrality in p-p collisions at dNppch (y=0)=~5


Two disks

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Table. V2 in pp at B2·Rpp=1.2 fm



|V2/|

V2

Sharp edge Degrade edge

0.6

0.219

0.13

-0.075

0.216

-0.016

(V2/)

hydro

=0.22 0.01 [3]

[3] H.J. Drescher et al, Phys.Rev. C76(2007)024905, [nucl-th/0704.3553]


Conclusions
The main results:
1. The initial spacial eccentricity changes a sign with impact parameter for degraded edge depending on the parameter of ratio edge diffuseness to radius. So, the elliptic flow can be negative or it changes a sign with impact parameter. 2. The multiplicity density over transverse overlap area (centrality) 1/SdN/dy increases with B. It is larger for sharp edge than for degraded edge at large inpact parameter. 3. Absolute value of elliptic flow |v2| is measurable for case of sharp edge and very small for degraded edge of parton density. 4. The changing of elliptic flow sign following the eccentricity sign gives the important information on the proton structure.


Nearest plans
1. Calculate V2 / and V2 with other parameters of incomplete thermolization model, where Glauber initial conditions are used. 2. Repeat the study with minimum bias for generic inelastic probability Pin(b) from experimental data of p-p 3. Incorporate the flow with our model V2 in the PYTHIA azimuthal distribution of p-p collision and to extract V2 by Li-Yang method


Back up slides


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1. Formalism A-A collision

Let's remind the formulae of A-A collisions

2 pr ( A1 A2 ) d b (1 e
PA1 A2 (b) {1 e
T
A1 A2

0 ( NN ) A1 A2T

A1, A 2

(b )

)

Probability of A-A generic inelastic interaction as function of impact parameter is
0 A1 A2T
A A2 1

(b )

}
b , y) 2 )

where the nuclear-nuclear thickness function is equal to

b ( b) dxdy TA1 ( x , y )TA2 ( x 2 dxdy TA1 ( x , y )TA2 ( x b, y

The average number of binary nucleon-nucleon collisions is

N

A1 , A 2

( b)

0 ( NN ) A1 A2T
PA1 A2 ( b)

A1 A2

( b)


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2. Formalism A-A collision

A density of binary nucleon-nucleon collisions is [4]

N

BC

A1 , A 2

b b ( x , y, b) 0 TA1 ( x , y ) TA2 ( x , y ) 2 2

and a density of wounded nucleon-nucleon collisions is

N

WN

A1 , A 2

0 b ( x , y , b) 0 ( NN ) TA1 ( x , y ) (1 e 2 b TA2 ( x , y ) (1 e 2

( NN ) A2T

A2

b ( x , y) 2

) )

0

b ( NN ) A1T A1 ( x , y ) 2



(4) P.F.Kolb and U. Heinz, Quark Gluon Plasma 3, ed. World Scientific, Singapur, p.634, [nucl-th/0305084]


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Figures



n1, 2 ( B)

P , 2 ( B) 1
n1, 2 ( B)

Fig.4 The non-normalized average number of binary parton-parton collisions in the frame of BDR and the probability of disk-disk interaction