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C. CIOFI degli ATTIDIFFRACTION IN NUCLEI:EFFECTS OF NUCLEON-NUCLEON CORRELATIONS

INT WORKSHOP:Physi s at a High Energy Ele tron Ion ColliderO tober 19-23 2009

C. Cio� degli Atti 1 SEATTLE, O tober 2009

LIST OF CONTENS1. Introdu tion: R. J. Glauber and NN orrelations2. NN orrelations: re ent theoreti al developments and experimen-tal investigations3. Di�ra tion on nu lei and NN orrelations4. Results of al ulations (total nA ross se tions at high energies,

σtot, σel and σqe hadroni ross se tions at HERA B, RHIC and LHCenergies)5. Con lusions


1 Introdu tion: R. J. Glauber and NN orrelations


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 land the nu lear radius Rl ≫ a and R ≫ aBe ause R is not vastly larger than a, and the orrelation length lis 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 Nu leon Nu leon orrelations in nu lei:re ent theoreti al developments and experimental results


THE STANDARD MODEL OF NUCLEI[− h2

2m

∑i ∇

2i +

∑i<j v(i, j) +

∑i<j<k v(i, j, k) + . . .

]Ψo = EoΨ0

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


THE MEAN FIELD APPROXIMATION[− h2

2m

∑i ∇

2i +

∑i<j vij

]Ψo = EoΨo

⇓[− h2

2m

∑i ∇

2i +

∑i V (ri)

]Φo = ǫoΦoIndependent parti le motion → Shell Model

Mean �eld or Shell Model: nu leons o upy all available states belowthe Fermi level; all states above the Fermi level are empty. Greatsu ess (magi numbers, magneti moments, low levels, et )but an-not a ount for the short-range (high momentum behavior) of nu- leon motion. Re ent theoreti al developments lead to realisti so-lutions of the nu lear many-body problem with a full treatment ofNN orrelations.


THE "EXACT" MANY-BODY SOLUTIONRe ent developments towards the exa t solution ofHΨo = EnΨo , with vij =

∑n v(n)(rij) O

(n)ijThe same operatorial dependen e appearing in vij is ast onto thetrial fun tion Ψo:

Ψo = FΦowhere Φo is the mean-�eld wave fun tion andF = S

∏

i<j

fij = S∏

i<j

∑

n

f (n)(rij) O(n)ijis a orrelation operator.Variational monte arlo (Urbana Group)Cluster expansion te hniques ( Here: Alvioli, Cda, Morita, Phys.Rev. C72 (2005) 054310; Phys. Rev. Lett. 100 (2008) 162503 )


The orrelation fun tions fij: Central, Spin-Isospin, Tensor . . .

0 1 2 3 4 5

-0.04-0.020.000.020.040.060.080.10 f(r)

4 - 5 - S 6 - S

16O - AV8'

r [fm]

1 - fc (/10) 2 - 3 -

SRC → two nu leons at separation shorter than average separation(≃ 1.7 fm)


THE NOVEL VIEW OF THE ATOMIC NUCLEUS

Nu lei onsist also of drops of old high density matter whosepredi ted per entage is ∼ 10%.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)

SCIENCE 320, 1476 (2008)R. SUBEDI et al Probing Cold Dense Nu lear MatterThe protons and neutrons in a nu leus an form strongly orrelatednu leon pairs. S attering experiments, in whi h a proton is kno kedout of the nu leus with high-momentum transfer and high missingmomentum, show that in arbon-12 the neutron-proton pairs arenearly 20 times as prevalent as proton-proton pairs and, by infer-en e, neutron-neutron pairs. This di�eren e between the types ofpairs is due to the nature of the strong for e and has impli ationsfor understanding old dense nu lear systems su h as neutron stars.


The orrelation pizza in 12C

R. Subedi et al,S ien e 320,1476 (2008)Dominant role of the tensor for e (S=1 T=0 states)


High momentum omponents. The role of the ore and the tensorfor e0 1 2 3 4

10-7

10-6

10-5

10-4

10-3

10-2

10-1

VMC

Mean Field Central Central

+ Tensor

n(k)

[fm

3 ]

k [fm-1]

16O

Tensor for es a ting in T=0 S=1 states produ e the largest amount of highmomentum omponents. Mean �eld distributions drop to zero at

k ≥ 1.5− 2 fm−1 (Adapted from Alvioli, CdA, Morita,Phys. Rev.C72(2005)054310)


Role of orrelations better seen in the2-body momentum distributions0 1 2 3 4 5

10-1

100

101

102

103

104

105

106n(

k rel,K

CM=0

) [fm

6 ]

krel [fm-1]

12C Total p - p p - n deuteron

0 1 2 3 4 5100101102103104105106107

40Ca

krel [fm-1]

Total p - p p - n deuteron

M. Alvioli, CdA, H.Morita, Phys. Rev. Lett, 100, 162503, (2008)The tensor deuteron-like ( T=0, S=1)dominan e at

krel ≥ 1.5− 2 fm−1o urs both in few-nu leon systems and omplexnu lei.


RELEVANCE OF SRC IN DIFFERENT PROCESSES


2.1. Study of ore and tensor orrelationsQuark, gluon or pion des riptions of NN intera tion?

Adapted from: W. Weise, Nu l. Phys. A 805(2008)145


2.2 Transition from hadron to quark gluon des riptions of nu lei

Nu leon radius < r2 >1/2≃ 0.8fm−1 ⇒ Nu leon overlap.

2.3 Medium indu ed modi� ations of hadron properties ??

EMC e�e t and SRC

CdA, L. Kaptari, L. Frankfurt, M. Strikman, Phys. Rev. C 76(2007) 055206


2.4 High energy s attering pro essesThe little bang in the laboratory

Signals depends upon the way hadrons propagated in the medium.SRC do a�e t hadron propagation.


2.5 Formation of old dense nu lear matter in the laboratory

How are neutron star properties a�e ted by SRC?


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3 EFFECTS OF SRC ON DIFFRACTION ON NUCLEI


3.1 Glauber formalismKey assumption: the hadron-nu leus partial elasti amplitude at impa tparameter b has the eikonal formΓpA(b; {lj, zj}) = 1−

A∏

k=1

[1− ΓpN(b− lk)

]

{~lj, zj} - transverse and longitudinal oordinates of the target nu leon j;

iΓpN - elasti s attering amplitude on a nu leon normalized asσpNtot = 2

∫d2b ReΓpN(b)One has to al ulate the matrix element of the amplitude with the nu learwave fun tion,Ψ0(1, . . . , A) i.e. (Ψ0 = |0 >)

GA(b) = 〈 0|ΓpA(b; {lj, zj})|0 〉 = 1 − 〈0|A∏

i=1

GpN(b− li)|0 〉

GpN(b− li) = 1− ΓpN(b− li)All nu lear effe ts are ontained in |Ψ0(1, . . . , A)|2.C. Cio� degli Atti 22 SEATTLE, O tober 2009

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∏

k 6=(il)

ρ(rk) +

+∑

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

∆(ri, rj)∆(rk, rl)∏

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


1. NO SRC or GLAUBERSingle parti le approximation: usual approximation in Glauber-type al ulations

|Ψ(r1, ..., rA)|2 ≃

A∏

j=1

ρ(rj)

2. SRC or GLAUBER plus SRCAll terms of the expansion ontaining all possible number of un-linked two-body ontra tions are summed up and so that twonu leon orrelations are taken exa tly into a ount


NO SRC (Glauber)

σtot = 2 Re ∫ d2b{1− e−

12σ

totNNTh

A(b)}

σel =

∫d2b∣∣∣1− e−

12σ

totNNTh

A(b)∣∣∣2

σqe =

∫d2b

{e−

12σ

totNN Th

A(b) − e−σtotNNThA(b)

}

ThA(b) =

2

σNNtot

∫d2s Re Γ(s)TA(b− l) (1)

TA(b− s) =

∫ ∞

−∞dz Aρ1(b− l, z) , (2)The tilded pro�le has Γ(b, b1) repla ed by [2 Γ(b, b1) − Γ2(b, b1)

]


Glauber plus SRC

σel =

∫d2b

∣∣∣∣∣1− e−1

2σtotNN

(ThA(b)−∆Th

A(b))∣∣∣∣∣

2

σqe =

=

∫d2b

{e−1

2 σtotNN

[ThA(b)−∆Th

A(b)]

− e−σtotNN

[ThA(b)−∆Th

A(b)]}

σtot = 2Re ∫ d2b

{1− e

−12σ

totNN

(ThA(b)−∆Th

A(b))}

∆ThA(b) =

=1

σtotNN

∫d2s1 d

2s2 Γ(s1)Γ(s2)

∫ ∞

−∞d z1d z2A

2∆(b− s1, z1; b− s2, z2)


3.2 Gribov orre tions (inelasti shadowing)

3.2.1 Lowest order(Karmanov-Kondratyuk)∆σIStot ∝

∫d2b ΓIS00 (b)

ΓIS00 (b) = −(2π)A2∫

d2σ

d2qT dM2X

∣∣∣qT=0

dM2Xe

−σtotnN

2T (bn)

|F (qL,bn)|2 .


3.2.2 All orders by light one dipoles(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

(−

r2TR20(s)

)]

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

pexp

(−2r2TR2p

)

3q model: |ΨN (r1, r2, r3)|2 = 3(π R2

p)2 δ (r1 + r2 + r3) exp(−r21+r

22+r

23

R2p

)

σpAtot = 2

∫d2b[1− 〈e−

12σqq(rT ,s) T

qqA (b,rT ,α)〉

] (3)

〈. . .〉 =

∫ 1

0dα

∫d2rT . . .


NO SRC(Kopeliovi h, Potashnikova, S hmidt, Phys. Rev. C73 (2006)034901):

TqqA (b, rT , α) =

2σqq(rT )

∫d2l Re ΓqqN (l, rT , α) TA(b− l)

SRC(Alvioli, CdA, Kopeliovi h, Potashnikova, S hmidt, to appear):

∆TA(b, rT , α) =1

σqq(rT )

∫d2l1 d

2l2 ∆⊥A(l1, l2)× Re Γqq(b− l1, rT , α) Re Γqq(b− l2, rT , α)


4. RESULTS of CALCULATIONS(σtot, σel and σqe hadroni ross se tions at HERA B, RHIC andLHC 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) and the orre tion due to the NN orrelations

∆ThA(b) al ulated at HERA energies for

12C and 208P respe tively.


The total neutron−Nucleus ross se tion at high energies:M. Alvioli, C.d.A, I. Mar hino, H. Morita, V. Palli, Phys. Rev. C78(R),031601(2008)Glauber + Inelasti shadowing(Di�ra tive ex itation of the proje tile)V. N. Gribov, Sov. JETP 29 (1969) 483;NN

AA

fNN N

A A

NX XNfN f N N

A A

NX f XNf f XX

(Glauber) (Inelasti Shadowing)total neutron-Nu leus ross se tion

σtot = 4 πk Im

[FG00(0)

]+ ∆σin = σG + ∆σinC. Cio� degli Atti 33 SEATTLE, O tober 2009

320

340

360120

125

130

135

140

420440460

10 1002900300031003200

10 100

12CnA tot [mb]

G

G + IS

4He

plab [Gev/c]

16O

208Pb

G

G+ SRC

G+ SRC+ IS


•No free parameters!!

• Full SRC: entral (O1 = 1),spin (O2(ij) = σi · σj), isospin(O3(ij) = τ i · τ j), spin-isospin(O4(ij) = (σi · σj) (τ i · τ j)),tensor (O5(ij) = Sij), tensor-isospin (O6(ij) = Sij (τ i · τ j)), orrelations.

•Gribov inelasti shadowing atlowest order.


CALCULATION of σtot, σel, σqe, σsd, σdd. . .• Full SRC: entral (O1 = 1), spin (O2(ij) = σi ·σj), isospin (O3(ij) =

τ i·τ j), spin-isospin (O4(ij) = (σi·σj) (τ i·τ j)), tensor (O5(ij) = Sij),tensor-isospin (O6(ij) = Sij (τ i · τ j)), orrelations.•Gribov inelasti shadowing at all orders by the dipole approa h.(Kopeliovi h et al)

M. Alvioli, CdA, B. Kopeliovi h, I. Potashnikova, I. S hmidt,to be submitted for publi ation


HERA B

12C GL + SRC % q-2q + SRC 3q + SRCσtot 353.71 364.11 +3% 344.16 349.37σel 86.90 92.96 +7% 82.39 85.42σqe 22.85 19.62 -14% 21.12 22.05

208Pb GL + SRC % q-2q + SRC 3q + SRCσtot 3052.11 3117.62 +1% 2955.57 3018.21

σel 1243.00 1274.60 +3% 1147.01 1199.14

σqe 62.55 54.11 -13% 61.01 64.39RHIC12C GL + SRC % q-2q + SRC 3q + SRC

σtot 413.71 425.73 +3% 391.12 406.90 394.96 410.20

σel 112.13 119.68 +7% 97.94 109.16 100.34 111.29

σqe 30.14 28.14 -7% 26.13 26.72

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

σtot 3297.56 3337.57 +1% 3155.29 3228.11 3208.92 3262.58

σel 1368.36 1398.08 +2% 1246.73 1314.04 1293.75 1343.76

σqe 80.42 74.36 -8% 71.99 73.92


LHC

12C GL + SRC % q-2q + SRC 3q + SRCσtot 598.79 613.68 +3% 578.42 591.05 579.84 592.12σel 198.11 208.59 +5% 183.19 194.84 184.28 195.65σqe 38.11 34.00 -11% 45.03 45.22

208Pb GL + SRC % q-2q + SRC 3q + SRCσtot 3850.63 3885.77 +1% 3811.74 3833.26 3823.32 3839.00

σel 1664.76 1690.48 +2% 1631.95 1655.70 1642.63 1660.67

σqe 87.28 80.54 -8% 113.37 113.88


5. CONCLUSIONS


• Advan ed solutions of the nu lear many-body problem lead tonu lear wave fun tions exhibiting a ri h orrelation stru ture.• Su h a orrelation stru ture has been on�rmed by experimentaldata from BNL and, parti ularly from JLab, and will be system-ati ally investigated at the 12 GeV upgraded Jlab and, hopefully,at hadron fa ilities like GSI and JPARC.• A theoreti al approa h has been developed to reliably treat thee�e ts of NN orrelations and inelasti shadowing in di�ra tion onnu lei. The results of al ulations show that both e�e ts, oftena ting in opposite dire tions, have to be onsidered simultane-ously, being almost of the same order.


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