1. K+ - Scattering from Nuclear Targets
Ed Hungerford
University of Houston
E. V. Hungerford
University of Houston
Panic 11/10-14/08

2. What does a Nucleon Look Like in the Nuclear medium?
The EMC effect shows that the nuclear medium modifies nuclear structure
1) Measured at high momentum transfer ( Q^{2} ≈ 2Mν )
2) This is an assumed consequence of Nuclear density
Early measurements of K+ total cross sections were enhanced,
----
Interpreted as :
-----
nuclear “swelling”
dibaryon structures
change in the effective mass
of mesonic coupling
The K+ was proposed as a
hadronic “replacement”
for the electron
E. V. Hungerford
University of Houston
Panic 11/10-14/08

3. K+ Mean Free path Useful K+ beams E. V. Hungerford Panic 11/10-14/08
University of Houston
Panic 11/10-14/08

4. K+ Amplitude and Cross Section
Im f
K+ p σT
Im(f) fm
σT mb
Re f
L=0,1
L=0
Re(f) fm
Because the Amplitudes are weak and relatively momentum independent the tρ approximation for construction of the optical potential is expected to be valid
K+ p/n Amplitudes
Soild s
Dashed p
E. V. Hungerford
University of Houston
Panic 11/10-14/08

5. Elementary K+-Nucleon Interaction
K+ Nucleon (particularly the K+- n) interaction is not well known
The spin dependence is weak
L=0 is dominated by K+- p,
L=1 dominated by K+- n
S wave dominates for momenta less than 800 MeV/c
Cross section relatively flat for momenta less than 800 MeV/c
Because amplitudes show no resonance behavior they should be easily modeled
Implications
Should be sensitive to the
Nuclear Volume
K - much more diffractive –
deeper minimum at lower Q2
E. V. Hungerford
University of Houston
Panic 11/10-14/08

6. Elastic and Total Cross Sections Additional data taken for Ca
An Early Example of
Elastic and Total Cross Sections
Additional data taken for Ca
12C(K+,K+)
800 MeV/c
PR 168(68)1466 x
Phase Shift
Increase by
25 %
Cross Section increase by
25 %
Ratio σc/(6σD)
Elastic
PR C25(82)2619
PRL 65(90)2110
Analysis
PR C45(92)2019
E. V. Hungerford
University of Houston
Panic 11/10-14/08

7. Initial K+ -- e Comparison
40Ca Target
Radius = 4 fm K+
550 MeV/c Incident Momentum
λK = 0.39 fm
Scattered at 80º
Q2 = (540 MeV/c)2
 = 156 MeV
x = q2/2M  = 0.91 e
1500 MeV/c Incident Momentum
λK = 0.20 fm
Scattered at 20º
Q2 = (515 MeV/c)2
 = 140 MeV
x = q2/2M  = 0.93
Obviously DIS can’t be explored with these kinematics, but they are appropriate for QHD
E. V. Hungerford
University of Houston
Panic 11/10-14/08

8. K+ Scattering Enhancements
Scattering from C shows an increase of 36% in the 2-body amplitude – in Ca it is about 64%
Can be explained by:
Increasing the elementary phase shifts
Decrease the effective mass of the exchanged mesons (effective mass varies with density
[1 – λρ/ρ0])
Other Density dependence
Calculations are not sensitive to common corrections (energy shifts, multiple scattering, etc)
Data are limited but there is a consistent discrepancy between data and theory.
E. V. Hungerford
University of Houston
Panic 11/10-14/08

9. K+ C Total Cross Section Ratio Compared to Calculations
Increase Vector Meson Mass PRL 60(88)2723
Scaled Swelling
PRC31(84)2184
Optical Models
PRC31(84)2184
Optical +MEX
PRC46(92)2462
E. V. Hungerford
University of Houston
Panic 11/10-14/08

10. A Second Series of Experiments
PR C49(94)2569
Ratio σ/(x σD)
Scaled Optical Model
E. V. Hungerford
University of Houston
Panic 11/10-14/08

11. Cross Section Ratios to [x D]
Re-Analysis of Ratios
Cross Section Ratios to [x D]
PR C(97)1304
PL B396(97)21
Super Ratio
Total
Reaction
ρSi >ρCa>ρC
E. V. Hungerford
University of Houston
Panic 11/10-14/08

12. Analysis σR less model dependent than σT
An optical potential is required to extract σR from the transmission experimental data
The results are not self-consistent
Need elastic data to get the optical potential
While a tρ approximation should be valid, it appears that potential is repulsive at low density but less so or attractive at high density
Energy dependence of K+-Li is similar to more complex nuclei
Super ratio removes Energy dependence
[σR(exp) /σR(cal)]/[σR(exp) /σR(cal)]
Meson Exchange is energy dependent
E. V. Hungerford
University of Houston
Panic 11/10-14/08

13. Elastic Scattering on C and Li at 635 and 715 MeV/c
C/Li Ratio
PL B382(96)29 NP A625(97)251
Elastic
PL B382(96)29
Li λ = 1.0
C λ = 1.2
DWIA
λ= 1.0, 1.3
PRL 55(85)592
PR C51(95)857
λ = 1.0
Analysis
PRL 55(85)592
PR C51(95)857
E. V. Hungerford
University of Houston
Panic 11/10-14/08

14. Analysis
Using a fit to σR and σT does not produce self consistent results, i.e. optical potentials constructed to fit σR and σT are not self consistent
Need an increase of ~15% of Im Vopt for Li plus an additional % of nuclear dependence
A density threshold is needed – use a linear density dependence for Im Vopt of the form
α = 1 + β(ρ – ρo)Θ(ρ – ρo)
ρo = 0.088; Β = 13.0
ρ the average nuclear density
Vopt → Re Vopt + Im Vopt [ α ]
ρo > ρ (Li) = 0.049
Self consistent fits possible
PR C(97)1304
PL B396(97)21
E. V. Hungerford
University of Houston
Panic 11/10-14/08

15. Inelastic Scattering Transition density < ρo
PL B382(96)29
ρ (C) = 0.104
ρ (T) = 0.063
DWIA
λ = 1.0
DWIA uses Transition Strength from π Scattering
DWIA
λ = 1.3
Transition density < ρo
Simple density dependence does not work
E. V. Hungerford
University of Houston
Panic 11/10-14/08

16. Quasi_free Scattering
On D, C, Ca, and Pb
PRL 71(93)2571
PR C51(95)669
Note Data Quality
and Resolution
Normalization ±11%
E. V. Hungerford
University of Houston
Panic 11/10-14/08

17. Number of Effective Nucleons
Aeff = N0 Aα
N0 ≈ 2 ; Α ≈ 0.6
Compare to
π+/π /0.56
e -- ≈1
A large σT increases Aeff but each cross section increases – Resulting Aeff insensitive
K+ reaches 55% of Central Nuclear Density in Pb
Verifies K+ Long mean free path
E. V. Hungerford
University of Houston
Panic 11/10-14/08

18. K+ Ca Quasi Free Compared to EM Theory
q = 500 MeV/c
Developed relativistic response in analogy to EM Scattering
EM response (Dirac V-A) expressed in Coulomb/Transverse – Coulomb quenched
4-vector current-current formulation for K+ uses scalar, vector, tensor forms and IA for nuclear response
Nuclear current obtained from the Walecka model with ρ, ω, σ exchange
Relativistic RPA applied to get the coherent nuclear response
EM quenching due to the polarization of Nucleon sea – relativistic RPS effect
K+ quenching sensitive to cancellation of kinematics and Relativistic RPA
Still need some renormalization so Aeff(exp) is used but shape is correct
PR C51(95)806
E. V. Hungerford
University of Houston
Panic 11/10-14/08

19. A Possible Round of New Experiments Previous Experimental Parameters
Momentum MeV/c
K+ flux -- 3 x 105 instantaneous; 0.5 x 105 avg
π/K – 1
Acceptance – 15%
Beam Dimensions – 8 x 3 cm2
Targets – Mostly Natural 2-4 g/cm2
ΔE – 3-4 MeV (FWHM)
Δθ – 2-4 Deg
PID especially for pions and protons
Normalizations carefully handled
Dominated by systematics – 15%
E. V. Hungerford
University of Houston
Panic 11/10-14/08

20. We need more data and more inspired theory
Conclusions
Reaction, total, elastic, inelastic, quasi-free all seems to have the same problem
While it appears that the issue is related to nuclear density, some simple, direct dependence does not seem to work
What the data seems to indicate is an increase in the Im Vopt with perhaps a more attractive real component
A. Gal NP A639(98)485c suggests that there are “missing” medium effects
Have we asked the right questions about the K+-nuclear interaction ?
We need more data and more inspired theory
E. V. Hungerford
University of Houston
Panic 11/10-14/08

21. The End
E. V. Hungerford
University of Houston
Panic 11/10-14/08