CIOFI A TTI - cs.infn.it · JLab (2006-2008); 12 Ge V Jlab The cor re la tion pizza in 12C R. Sub edi et al, Sci enc e 320, 1476 ... NN correlations leads to a mo di cation the n

C. CIOFI degli ATTINu leon-Nu leon Correlations and Gribov Inelasti Shadowing inNu lear CollisionsDIFFRACTION 2010International Workshop on Di�ra tion in High Energy Physi sOtranto, ItalySeptember 10-15, 2010

C. Cio� degli Atti 1 Otranto, September 10-15, 2010

OUTLINE1. Short Range Correlations (SRC) in nu lei: re ent theoreti al andexperimental advan es.2. Beyond the Glauber approximation: e�e ts of SRC and Gribovinelasti shadowing by the light one dipole approa h.3. Results of al ulations I:(The total hadron-Nu leus ross se tions at HERA B, RHIC andLHC energies).4. Results of al ulations II:(The number of inelasti ollisions in hadron-Nu leus and Nu leus-Nu leus s attering).5. Con lusionsCollaboration Perugia-UTFSM, ValparaisoBoris Kopeliovi h, Irina Potashnikova, Ivan S hmidt


1. SHORT RANGE CORRELATIONS IN NUCLEI: RECENTTHEORETICAL AND EXPERIMENTAL ADVANCES


THE STANDARD MODEL OF NUCLEI[− h2

2 m

∑i ∇

2i +

∑i<j v(i, j) +

∑i<j<k v(i, j, k) + . . .

]Ψo = Eo Ψ0

vij(xi, xj) =

18∑

n=1

v(n)(rij)O(n)ij rij ≡ |ri − rj|

O(n)ij =

[1 , σi · σj , Sij , (L · S)ij , ...

]⊗[1 , τ i · τ j

].

• short-range repulsion ( ommon to many systems)

• intermediate- to long-range tensor hara ter (unique to nu lei)

Very dif� ult many-body problem but re ent theoreti aldevelopments lead to a urate solutions with a full treatment of NN orrelations.


Short Range Correlations(SRC): the strong modi�- ation of the Mean Fieldtwo-nu leon density distri-bution in the region r ≤1.2 fm due the interpalybetween the ore repulsionand the tensor attra tion .

Di�erent many-body approa hes and intera tions, lead to the samestru ture of ρNN (r). No model dependen e.Urbana-Argonne Group Phys. Rev. Lett. 98 (2007) 132501.Perugia Group Phys. Rev. Lett. 100 (2008) 162503; Phys. Rev.Lett. 100 (2008) 122301.


THE NOVEL VIEW OF THE ATOMIC NUCLEUS

Nu lei onsist also of drops of old high density matter whosepredi ted per entage is ∼ 10− 20%.Can su h a per entage be measured? Yes


A(p,p'pN)X AGK BNL (2003); A(e,e'p)X, A(e,e'pn)X JLab(2006-2008); 12 GeV JlabThe orrelation pizza in 12C

R. Subedi et al,S ien e 320,1476 (2008)


2 BEYOND THE GLAUBER APPROXIMATION: SRC ANDGRIBOV INELASTIC SHADOWING BY THE LIGHT CONEDIPOLES APPROACH


2.1 SRC orrelationsThe h−A elasti amplitudeΓhA(b) = 1−

[1−

∫|Ψ0(r1 . . . rA)|2

A∏

j=1

ΓhNj (b− sj)d

3 rj

]

Evaluation of ΓhA(b)⇒ (3A−3)-fold integration. For omplex nu leiimpra ti able, unless MC integration⇓EXPANSION OF |Ψ0(r1 . . . rA)|2


The exa t expansion of |Ψ0|2 (Glauber, Foldy & Wale ka ):

|Ψ0(r1, ..., rA)|2 =

A∏

j=1

ρ(rj) +

A∑

i<j=1

∆(ri, rj)

A−2∏

k 6=(il)

ρ(rk) +

+∑

(i<j) 6=(k<l)

∆(ri, rj)∆(rk, rl)

A−4∏

m6=(i,j,k,l)

ρ1(rm) + . . .

∆(ri, rj) = ρ(2)(ri, rj) − ρ(1)(ri) ρ(1)(rj) ;

ρ(1)(r1) =

∫|Ψ0(r1, ..., rA)|2

A∏

i=2

dri ; ρ(2)(r1, r2) =

∫|Ψ0(r1, ..., rA)|2

A∏

i=3

dri∫

drj ρ(2)(ri, rj) = ρ(1)(ri) ;

∫drj∆(ri, rj) = 0

ρ(1)(r) ≡ ρ(r)


1. "GLAUBER APPROXIMATION"Single density approximation: usual approximation in Glauber-type al ulations

|Ψ(r1, ..., rA)|2 ≃A∏

j=1

ρ(rj)

2. BEYOND THE GLAUBER APPROXIMATIONAll terms of the expansion ontaining ALL possible produ ts ofUNLINKED two-body ontra tions ∆′s, e.g.∆(i, j)∆(k, l)∆(m,n) . . . (i, j) 6= (k, l) 6= (m,n) . . . are exa tlysummed up so that two nu leon orrelations are taken exa tlyinto a ount. The summation yields :


∫|Ψ0(1, 2, . . . , A)|2

A∏

i=1

[1− Γ(b− bi)]

A∏

j=1

dj =

=

[1−

∫ρ(1)Γ(b− 1)d1

]A A2∑

m=0

[12

∫∆(12)Γ(b− 1)Γ(b− 2)d1d2(1−

∫ρ(1)Γ(b− 1)d1

)2

]m

Linked produ ts of ∆′s represents higher order orrelations, e.g.

∆(i, j)∆(j, k) represents 3-nu leon orrelations and will be oupledto the produ t of three Γ(b− i), and so on.

In the opti al (A >> 1) limit one obtains:


Glauber approximation

σtot = 2 Re ∫ d2b{

1− e−12σ

totNNTh

A(b)}

Beyond Glauberσtot = 2 Re ∫ d2b

{1− e−

12σ

totNN Th

A(b)}

ThA(b)⇒ Th

A(b) = ThA(b)−∆Th

A(b)

∆ThA(b) =

=1

σtotNN

∫d2s1 d2s2 Γ(s1)Γ(s2)

∫ ∞

−∞d z1d z2 A2 ∆(b− s1, z1; b− s2, z2)


Glauber in 1971 made an estimate of the e�e ts of orrelations onthe thi kness fun tion. His formula :

ThA(b)⇒ Th

A(b) = ThA(b)−∆TGl

A (b)

∆TGlA (b) =

(2πAf (0)

k

)lc

∫ +∞

−∞ρ2(b, z)d z

lc " orrelation length"Basi approximation:Range of NN for e aRange of orrelations lc

<< 1 (1)


R. J. Glauber, High Energy Collision Theory, 1971"Various types of orrelations in positions and spin may exist be-tween nu leons of an a tual nu leus . . . If the system being onsid-ered is spatially uniform an idea of the magnitude and nature of thee�e ts due to pair orrelations may be obtained by assuming thatthe range of NN for e a is smaller than the range of orrelations lcand the nu lear radius R

lc≫ a and R≫ aBe ause R is not vastly larger than a, and the orrelation length lcis not too di�erent in magnitude from the for e range, the approx-imations that follow from these onditions should only be used forrough estimates".


2.2 Gribov orre tions by the light one dipole approa h(Zamolod hikov, Kopeliovi h, Lapidus, JEPT Lett. 33 (1981) 612)Key ingredients:the universal dipole nu leon ross se tionσqq(rT , s) = σ0(s)

[1− exp

(−

r2T

R20(s)

)]

the light one wave fun tion of the proje tileq-2q model: |ΨN (r1, r2, r3)|2 = 2π R2

pexp

(−

2r2T

R2p

)

σpAtot = 2

∫d2b[1− 〈e−

12σqq(rT ,s) T

qqA (b,rT ,α)〉

] (2)

〈. . .〉 =

∫ 1

0dα

∫d2rT . . .


GL plus IS BY LIGHT CONE DIPOLES(Kopeliovi h, Potashnikova, S hmidt, Phys. Rev. C73 (2006)034901 ):

TqqA (b, rT , α) = 2

σqq(rT )

∫d2s Re ΓqqN (s, rT , α) TA(b− s)

GL plus SRC plus IS BY LIGHT CONE DIPOLES(Alvioli, CdA, Kopeliovi h, Potashnikova, S hmidt, Phys. Rev. C81(2010) 025204):∆T

qqA (b, rT , α) =

1σqq(rT )

∫d2s1d

2s2∆⊥A(s1, s2)×ReΓqq,N (b−s1, rT , α)ReΓqq,N (b−s2, rT , α)


3. RESULTS of CALCULATIONS-I(σtot, σel, σqe, σsd, σdd hadroni ross se tions atHERA B, RHIC and LHC energies )


The in lusion of NN orrelations leads to a modi� ation of thenu lear thi kness fun tion

ThA(b)⇒ Th

A(b)−∆ThA(b)

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

0.0

0.2

0.4

0.6

0.8

Th A(b

), |

Th A(b

)| [fm

-2]

b [fm]

208Pb

ThA(b)

- ThA(b)

12C

Nu lear thi kness fun tion ThA(b) andthe orre tion due to the NN orrela-tions ∆Th

A(b) al ulated at HERA en-ergies for 12C and 208P respe tively.


The total neutron−Nucleus ross se tion at high energies:(M. Alvioli, C.d.A, et al Phys. Rev. C78(R),031601(2008) )320

340

360120

125

130

135

140

420440460

10 1002900300031003200

10 100

12C

nA tot [mb]

G

G + IS

4He

plab [Gev/c]

16O

208Pb

G

G+ SRC

G+ SRC+ IS

•No free parameters!!• Full SRC.•Gribov inelasti shadowing atlowest order.•Main result: SRC in rease theopa ity, Gribov IS de reases it,the two effe ts being of aboutthe same order in this energyrange.•What about higher order Gri-bov orre tions?.


CALCULATION of σtot, σel, σqe, σsd, σdd. . .(Alvioli, CdA, Kopeliovi h, Potashnikova, S hmidt, Phys. Rev. C81(2010) 025204)

Full SRC and Gribov inelasti shadowing to all orders by the light one dipole approa h. RHIC12C GL + SRC q-2q + SRC 3q + SRC

σtot 413.71 425.73 391.12 406.90 394.96 410.20

σel 112.13 119.68 97.94 109.16 100.34 111.29

σqe 30.14 28.14 26.13 26.72

208Pb GL + SRC q-2q SRC 3q + SRC

σtot 3297.56 3337.57 3155.29 3228.11 3208.92 3262.58

σel 1368.36 1398.08 1246.73 1314.04 1293.75 1343.76

σqe 80.42 74.36 71.99 73.92


4. RESULTS OF (PRELIMINARY) CALCULATIONS II :THE EFFECTS OF SRC AND IS ON THE NUMBER OFINELASTIC COLLISIONS IN h− A AND A− A SCATTERING

(CdA, Mezzetti, Kopeliovi h, Potashnikova, S hmidt, To appear)


4.1 The number of inelasti ollisions in hA s atteringIn order to know the absolute value of a hard nu lear ross se -tion, the measured fra tion of the total number of inelasti eventsNhA

hard/NhAin is normalized as follows

RhardA/N =

σhAin NhA

hard

AσhNin NhN

hard

=1

Ncoll

NhAhard

NhNhard

, Ncoll = AσhN

in

σhAinThe number of hard ollisions at a given impa t parameter shouldbe de�ned as follows

RhardA/N (b) =

NhAhard(b)

ncoll(b) NhNhard

ncoll(b) =σhN

in TA(b)

Pin(b)and Pin(b) is the probability for an inelasti intera tion to o ur atimpa t parameter b; it is affe ted by both SRC and IS through ThA(b):


The following quantities have been onsidered

nGlcoll(b) =

σhNin TA(b)

1− e−σhNin Th

A(b)

nGl+SRCcoll (b) =

σhNin TA(b)

1− e−σhNin Th

A(b)

nGl+SRC+IScoll (b) =

σhNin TA(b)

Pin(b)where

Pin(b) =dσtot

d2b−

dσel

d2b−

dσdiff

d2b−

dσqel

d2b−

dσqsd

d2b,with

1

2

dσtot

d2b= 1− e

12IA(b)

⟨e−

12σdipT

hA(b)⟩


p−208 PbGLAUBER

σNNin [mb] σNA

tot [mb] σNAel [mb] σNA

qel [mb] σNAin [mb] NcollRHIC 42.1 3297.6 1368.4 66.0 1863.2 4.70LHC 68.3 3850.6 1664.8 121.0 2064.8 6.88GLAUBER+SRC

σNNin [mb] σNA


qel [mb] σNAin [mb] NcollRHIC 42.1 3337.6 1398.1 58.5 1881.0 4.65LHC 68.3 3885.8 1690.5 112.6 2082.7 6.82GLAUBER+SRC+GRIBOV IS (q− 2q)

σNNin [mb] σNA


qel [mb] σNAin [mb] NcollRHIC 42.1 3228.1 1314.0 71.99 1842.1 4.75LHC 68.3 3833.3 1655.7 113.4 2064.2 6.88Gribov IS in rease Ncoll, SRC de reases it to the Glauber value thee�e ts are of the order of few per ent in agreement with the resultsof deuteron-Gold al ulations (Kopeliovi h, Phys. Rev. C68 (2003)025204)


4.2 The number of inelasti ollisions in A− A s atteringCollision of two heavy nu lei A and B with nu leon numbers AAand AB respe tively.

5.2.1 Glauber approximation :

ThAB(~b) =

∫d2bA TA(~bA) Th

B(~b−~bA)

=2

σNNtot

AAAB

∫d2bAd2bBρA

1 (~bA)ReΓNN (~b−~bA +~bB)ρB1 (~bB)

5.2.2 Beyond the Glauber approximation :

ThB(~b−~bA)→ Th

B(~b−~bA) = ThB(~b−~bA)−∆Th

B(~b−~bA)


The �nal Nu leus-Nu leus thi kness fun tion is thusTh

AB(~b) = ThAB(~b)−∆Th

AB(~b)

the orrelation ontribution being∆Th

AB(~b) =1

σNNtot

AA A2B

∫d2 bAρA(~bA)

×

∫d2bB1d

2bB2∆⊥B(~bB1,~bB2)Γ

NN (~b−~bA +~bB1)ΓNN (~b−~bA +~bB2) +

+{A←→ B}


0 2 4 6 8 10 12 14 16 18 20

0.000

0.005

0.010

RHIC

A-2 Τ

h ΑΒ(b

) [fm

-2]

b [fm]

Th

AB(b)

∆Th

AB(b)

A-2 TOTAL

208Pb-208Pb~

Large e�e ts on the thi kness fun tion but what about NABcoll ?


The number of ollisions at impa t parameter b isnAB

coll(b) =σhN

in TAB(b)

PABin (b)

• σhNin TAB(b) is a�e ted neither by SRC nor by IS.

• how to al ulate PABin (b)? The usual formula

PABin (b) = 1− exp[−σNN

in ThAB(b)]misses many terms of the Glauber theory. However the probabil-ity of no intera tion exp[−σNN

in ThAB(b)] is expe ted to be very smallwith or without the missed terms, the Gribov IS orre tions andthe e�e ts of NN orrelations. Ex ept the very peripheral olli-sions PAB

in (b) ≃ 1 and it seems therefore that nABcoll is pra ti allynot a�e ted by NN orrelations and IS .


4. CONCLUSIONS


• Advan ed solutions of the nu lear many-body problem lead to nu- lear wave fun tions exhibiting a ri h orrelation stru ture whi hhas experimentally been observed: the Nu leus is neither a Fermigas nor a system of independent parti les but rather a self-boundsaturated liquid whi h nowadays an theoreti ally be des ribedwith high degree of on�den e.

• A reliable theoreti al approa h has been developed to treat si-multaneously NN orrelations and Gribov inelasti shadowing inhigh energy nu lear ollisions.• In the onsidered pro esses the e�e ts from NN orrelations arenot ex eptionally large, but still of the same order of other ef-fe ts that are ommonly being onsidered in high energy nu learintera tions.


Documents

CIOFI A TTI - cs.infn.it · JLab (2006-2008); 12 Ge V Jlab The cor re la tion pizza in 12C R. Sub edi et al, Sci enc e 320, 1476 ... NN correlations leads to a mo di cation the n