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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
C. Cio� degli Atti 2 SEATTLE, O tober 2009
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".
C. Cio� degli Atti 4 SEATTLE, O tober 2009
2 Nu leon Nu leon orrelations in nu lei:re ent theoreti al developments and experimental results
C. Cio� degli Atti 5 SEATTLE, O tober 2009
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
C. Cio� degli Atti 6 SEATTLE, O tober 2009
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.
C. Cio� degli Atti 7 SEATTLE, O tober 2009
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 )
C. Cio� degli Atti 8 SEATTLE, O tober 2009
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)
C. Cio� degli Atti 9 SEATTLE, O tober 2009
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
C. Cio� degli Atti 10 SEATTLE, O tober 2009
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.
C. Cio� degli Atti 11 SEATTLE, O tober 2009
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)
C. Cio� degli Atti 12 SEATTLE, O tober 2009
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)
C. Cio� degli Atti 13 SEATTLE, O tober 2009
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.
C. Cio� degli Atti 14 SEATTLE, O tober 2009
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
C. Cio� degli Atti 16 SEATTLE, O tober 2009
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
C. Cio� degli Atti 17 SEATTLE, O tober 2009
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.
C. Cio� degli Atti 18 SEATTLE, O tober 2009
2.5 Formation of old dense nu lear matter in the laboratory
How are neutron star properties a�e ted by SRC?
C. Cio� degli Atti 19 SEATTLE, O tober 2009
!"#$%&'($)*(+,-(+.&/-(0)*)(1)&-(&2)*34)1,5)3&"(&6+7*-("1&2%83"13&
'B4."1+,-(3&0-*&=)<$*-(&!$+*3&
y!)EF)FG%)#&.%)&-)'%"F.&')(FB.(H)/&(F)B##%IF%J)/&J%$()B(("/%)DK@6L)'%"F.&'(H)K6L)I.&F&'()
y!)!%M$%#N'M)FG%)'I2345)0'F%.B#N&'(H)&'%)#B')B(("/%)FO&)(%IB.BF%)P%./0)MB(%()
y!)30'#%)'I)0'F%.B#N&')0()$B.M%)#&/IB.%J)F&)''H)')MB()G%BF()FG%)I)MB()
y!)QG0()#&"$J)%R%#F)FG%)"II%.)$0/0F)&')/B(()&-)'%"F.&')B'J)B$$&O)FG%)'%"F.&'()0')FG%)(FB.)J%#BS))))&
C. Cio� degli Atti 20 SEATTLE, O tober 2009
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
C. Cio� degli Atti 23 SEATTLE, O tober 2009
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
C. Cio� degli Atti 24 SEATTLE, O tober 2009
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)
]
C. Cio� degli Atti 25 SEATTLE, O tober 2009
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)
C. Cio� degli Atti 26 SEATTLE, O tober 2009
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 .
C. Cio� degli Atti 27 SEATTLE, O tober 2009
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 . . .
C. Cio� degli Atti 28 SEATTLE, O tober 2009
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)
C. Cio� degli Atti 29 SEATTLE, O tober 2009
4. RESULTS of CALCULATIONS(σtot, σel and σqe hadroni ross se tions at HERA B, RHIC andLHC energies)
C. Cio� degli Atti 30 SEATTLE, O tober 2009
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
C. Cio� degli Atti 31 SEATTLE, O tober 2009
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.
C. Cio� degli Atti 32 SEATTLE, O tober 2009
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
C. Cio� degli Atti 34 SEATTLE, O tober 2009
•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.
C. Cio� degli Atti 35 SEATTLE, O tober 2009
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
C. Cio� degli Atti 36 SEATTLE, O tober 2009
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
C. Cio� degli Atti 37 SEATTLE, O tober 2009
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
C. Cio� degli Atti 38 SEATTLE, O tober 2009
• 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.
C. Cio� degli Atti 40 SEATTLE, O tober 2009