K meson in dense matter L. Tols 1 , R. Molina 2 , E. Oset 2 and A. - - PowerPoint PPT Presentation

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K meson in dense matter L. Tols 1 , R. Molina 2 , E. Oset 2 and A. - - PowerPoint PPT Presentation

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K self-energy from the s-wave K K N interaction K in dense matter Nuclear transparency in the A Introduction Results K meson in dense matter L. Tols 1 , R. Molina 2 , E. Oset 2 and A. Ramos 3 1 Theory

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

¯ K∗ meson in dense matter

  • L. Tolós1, R. Molina2, E. Oset2 and A. Ramos3

1Theory Group. KVI. University of Groningen,

Zernikelaan 25, 9747 AA Groningen, The Netherlands

2Instituto de Física Corpuscular (centro mixto CSIC-UV)

Institutos de Investigación de Paterna, Aptdo. 22085, 46071, Valencia, Spain

3Estructura i Constituents de la Materia. Facultat de Física. Universitat de

Barcelona,

  • Avda. Diagonal 647, 08028 Barcelona, Spain
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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

Outline

1

Introduction

2

¯ K∗ self-energy from the s-wave ¯ K∗N interaction

3

¯ K∗ → ¯ Kπ in dense matter

4

Results

5

Nuclear transparency in the γ A → K+ K∗− A′ reaction

6

Conclusions

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

Previous results on the interaction of vector mesons with nuclear matter Within the Nambu Jona Lasinio model there is no shift of the vector masses while the σ-mass decreases sharply with the

  • density. V. Bernard and U. G. Meissner, Nucl. Phys. A 489, 647 (1988).

Recent calculations show no shift of the ρ meson and a broadening of the vector meson in nuclear matter. Rapp, 97; Urban, 99;

  • D. Cabrera, E. Oset and M. J. Vicente Vacas, Nucl. Phys. A 705, 90 (2002):

△M(ρ0) ∼ 30 MeV, Γ(ρ0) ∼ 200 MeV. In the case of the φ meson, theoretical calculations also show no shift of the mass and a broadening of the meson.

  • D. Cabrera and
  • M. J. Vicente Vacas, Phys. Rev. C 67, 045203 (2003):

△M(ρ0) ∼ 8 MeV, Γ(ρ0) ∼ 30 MeV. The case of the ω meson had been more controvelsial [1,2,3,4,5].

1.

  • M. Post, S. Leupold and U. Mosel, Nucl. Phys. A 741, 81 (2004)

2.

  • M. Kaskulov, H. Nagahiro, S. Hirenzaki and E. Oset, Phys. Rev. C 75, 064616 (2007)

3.

  • D. Trnka et al. [CBELSA/TAPS Collaboration], Phys. Rev. Lett. 94, 192303 (2005)

4. Mariana Nanova, Talk given at the XIII International Conference on Hadron Spectroscopy, December 2009, Florida State University. 5.

  • M. Kotulla et al. [CBELSA/TAPS Collaboration], Phys. Rev. Lett. 100, 192302 (2008)
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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

Previous results on the interaction of vector mesons with nuclear matter In M. Kaskulov, E. Hernandez and E. Oset, Eur. Phys. J. A 31, 245 (2007) the authors predict no shift and a width of Γ(ρ0) ∼ 90 MeV at normal nuclear density. Experiments done by the CLASS Collaboration confirms this null shift for the ρ mass and a broadening of the ρ, ω, φ mesons at normal nuclear density. M. H. Wood et al. [CLAS Collaboration], Phys. Rev. C 78, 015201

(2008). C. Djalali et al., Nucl. Part. Phys. 35, 104035 (2008).

These theoretical predictions and the results of CLASS are in contradiction with the parametrization derived by Hatsuda and Lee: m = m0(1 − 0.16 ρ ρ0 ) and with the 20 % decrease in the ρ meson mass predicted by Brown and Rho. While the KEK team had earlier reported an attractive mass shift

  • f the ρ. Conclusions could depend on the way the background

is subtracted.

  • R. Muto et al. [KEK-PS-E325 Collaboration], Phys. Rev. Lett. 98, 042501 (2007)
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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

The VB → VB interaction LIII = − 1

4VµνVµν

L(3V)

III

= ig(∂µVν − ∂νVµ)VµVν L(c)

III = g2 2 VµVνVµVν − VνVµVµVν

Vµν, g

Vµν = ∂µVν − ∂νVµ − ig[Vµ, Vν] g = MV

2f

Vµ

    

ρ0 √ 2 + ω √ 2

ρ+ K∗+ ρ− − ρ0

√ 2 + ω √ 2

K∗0 K∗− ¯ K∗0 φ     

µ

LBBV = g 2 (¯ Bγµ[Vµ, B] + ¯ BγµBVµ)

1. Klingl,97; Palomar,02; M. Bando, T. Kugo, S. Uehara, K. Yamawaki, 1985, 88, 03 2.

  • E. Oset and A. Ramos, arXiv:0905.0973 (2009, Eur. Phys. J. A)
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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

The VB → VB interaction Figure: |T|2 for

different channels for I = 0, 1 and strangeness S = −1. Channel thresholds are indicated by vertical dotted

  • lines. 3Λ and 2Σ

appear: Λ(1783), Λ(1900), Λ(2158) and Σ(1830), Σ(1987). PDG: Λ(1800),

Λ(2000), Σ(1750), Σ(1940), Σ(2000).

100 200

I=0, S=-1 5 I=1, S=-1

10 20 1 2 3 10 20 0.5 1800 2000 2200

Ecm [MeV]

50 100 1800 2000 2200

Ecm [MeV]

0.5 10 20

100x|T|

2 [MeV

  • 2]

1 2 3 1 2 3

K

*N

ωΛ ρΣ φΛ K

*Ξ

K

*N

ρΛ ρΣ ωΣ K

*Ξ

φΣ

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

¯ K∗ self-energy from the s-wave ¯ K∗N interaction

Meson-baryon function loop in nuclear matter: Gρ(P) = G0(√s) + limΛ→∞δGρ

Λ(P),

δGρ

Λ(P)

≡ Gρ

Λ(P) − G0 Λ(√s)

= i2M

  • Λ

d4q (2π)4

  • Dρ

B(P − q) Dρ M(q) − D0 B(P − q) D0 M(q)

  • In-medium propagators:

Dρ

N(p)

= 2MN 2EN( p ) {

  • 1 − n(

p ) p0 − EN( p ) + iε + n( p ) p0 − EN( p ) − iε

  • ur(

p)¯ ur( p) + vr(− p)¯ vr(− p) p0 + EN( p ) − iε } = D0

N(p) + 2πi n(

p ) δ

  • p0 − EN(

p )

  • 2EN(

p ) Dρ

¯ K∗ (q)

=

  • (q0)2 − ω(

q)2 − Π¯

K∗ (q)

−1 = ∞ dω S¯

K∗(ω,

q ) q0 − ω + iε − SK∗(ω, q ) q0 + ω − iε

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

¯ K∗ self-energy from the s-wave ¯ K∗N interaction

Gρ¯

K∗N(P) = G0 ¯ K∗N(√s) +

  • d3q

(2π)3 MN EN( p )

  • −n(

p ) (P0 − EN( p ))2 − ω( q )2 + iε + (1 − n( p ))

  • −1/(2ω(

q )) P0 − EN( p ) − ω( q ) + iε + ∞ dω S¯

K∗ (ω,

q ) P0 − EN( p ) − ω + iε

  • p=

P−

  • q

Tρ(I)(P) = 1 1 − VI(√s) Gρ(I)(P) VI(√s) The in-medium ¯ K∗ self-energy is then obtained by integrating Tρ ¯

K∗N

  • ver the nucleon Fermi sea,

Π¯

K∗(q0,

q ) =

d3p (2π)3 n(

p )

  • Tρ(I=0)

¯ K∗N (P0,

P) + 3Tρ(I=1)

¯ K∗N (P0,

P)

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

¯ K∗ selfenergy coming from its decay into ¯ Kπ

¯ K∗ ¯ K∗ π ¯ K = ⇒ ¯ K∗ ¯ K∗ ¯ K π

Figure: The ¯ K propagator in the free space (l), and in the medium (r). In the free space, we have: −iΠ = −6g2

  • d4q

(2π)4 i q2 − m2

π

i (P − q)2 − m2

K + iǫǫ′ µqµǫνqν

For low momenta,

  • P

mK∗− ∼ 0, ǫµqµǫ′ νqν −

→ ǫ ′ · ǫ 1

3

q 2 δij Cutkosky rules − → ImΠ = g2

4π

ǫ · ǫ ′q3 1

P0 −

→ Γ = 42 MeV

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

¯ K∗ selfenergy coming from its decay into ¯ Kπ

In the medium, we have: 1 q2 − m2

π

→ 1 q2 − m2

π − Ππ(q0,

q) 1 (P − q)2 − m2

K

→ 1 (P − q)2 − m2

K − ΠK(P0 − q0,

P − q) We use their Lehmann representations, this is: −iΠ¯

K∗(P0,

P) = 2 g2 ǫ · ǫ ′

  • d4q

(2π)4 q 2 ∞ dω π (−2ω) ImDπ(ω, q) (q0)2 − ω2 + iǫ × ∞ dω′ π (−){ ImD¯

K(ω′,

P − q) P0 − q0 − ω′ + iη − ImDK(ω′, P − q) P0 − q0 + ω′ − iη }

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

¯ K∗ selfenergy coming from its decay into ¯ Kπ

The real part of the free ¯ K∗ selfenergy is subtracted to get its physical mass at ρ = 0 The last term is ∝ ImDK is small and as approximation can be cancelled with the term in the free space Therefore, Π¯

K∗(P0,

P) = 2g2ǫ · ǫ′{

  • d3q

(2π)3 q 2 1 π2 ∞ dω ImDπ(ω, q) ∞ dω′ ImD¯

K(ω′,

P − q) P0 − ω − ω′ + iη −Re

  • d3q

(2π)3

  • q 2

2ωπ(q) 1 2ωK(P − q) 1 P0 − ωπ(q) − ωK(P − q) + iǫ}

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

¯ K∗ selfenergy coming from its decay into ¯ Kπ

The ¯ K selfenergy includes s and p waves that take into account the processes ¯ KN → ¯ KN, ¯ KN → πΣ..., ¯ K → ΛN−1, ΣN−1 and Σ∗N−1 in the nuclear medium, Pauli-blocking, mean-field binding

  • n the nucleons and hyperons and self-consistency.

The π sefenergy incorporates a dominant p wave contribution that comes from ph and ∆h excitations, 2p2h pieces and NN, N∆ short-range correlations by means of a single Landau-Migdal parameter (g′).

1.

  • A. Ramos and E. Oset, Nucl. Phys. A671,481 (2000)

2.

  • E. Oset, P

. Fernandez, L. L. Salcedo and R. Brockmann, Phys. Rept. 188, 79(1990) 3.

  • Phys. Rept. 188, 79(1990),A. Ramos, E. Oset and L. L. Salcedo, Phys. Rev. C50,2314 (1994)

The ¯ K∗ selfenergy accounts for the processes ¯ K∗N → ¯ K∗N, ¯ K∗N → ρΣ..., ¯ K∗ → ¯ Kπ in the medium which allows new decay channels as ¯ K∗NN → Λ(1405)N and includes Pauli-blocking, mean-field binding on the nucleons and self-consistency.

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

¯ K∗ selfenergy coming from its decay into ¯ Kπ

We also include vertex correcions in the selfenergy of the pion

¯ K∗ π ¯ K +

˜ Π(p) q 2 = ⇒ ˜ Π(p)( q 2 + D(π)−1 (q) + 3 4 D(π)−2 (q)

  • q 2

) , (1)

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

Results

400 600 800 1000 1200 1400 q0 [MeV]

  • 120
  • 100
  • 80
  • 60
  • 40
  • 20

ImΠΚ∗−>Κ π/(2q0) (ρ0,q=0 MeV/c) [MeV] free dressing π (w/o v.c.) dressing π and K (w/o v.c.) dressing π (w/ v.c.) and K (w/o v.c.)

Figure: Imaginary part of the ¯ K∗ self-energy at ρ0 coming from the ¯ Kπ decay in dense matter.(i) calculation in free space, (ii) including the π self-energy, (iii) including the π and ¯ K self-energies, and (iv) including the π dressing with vertex corrections and the ¯ K self-energy.

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

Results

Figure: The ¯ K∗ self-energy showing the different contributions: (i) self-consistent calculation of the ¯ K∗N interaction, (ii) self-energy coming from ¯ K∗ → ¯ Kπ decay, (iii) combined self-energy from both previous sources.

  • 200
  • 150
  • 100
  • 50

50 Re Π/(2 q0) [MeV] q=0 MeV/c, ρ=ρ0 600 800 1000 1200 q0 [MeV]

  • 200
  • 150
  • 100
  • 50

Im Π/(2 q0) [MeV] K*N K*-> K π K*N + K*-> Kπ

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

Results

  • 200
  • 150
  • 100
  • 50

50 Re Π/(2 q0) [MeV] q=0 MeV/c q=300 MeV/c q=450 MeV/c ρ0 ρ=0.5ρ0 ρ=ρ0 ρ=1.5ρ0 q=0 MeV/c 600 800 1000 1200 q0 [MeV]

  • 200
  • 150
  • 100
  • 50

Im Π/(2 q0) [MeV] 600 800 1000 1200 q0 [MeV]

Figure: The ¯ K∗ self-energy as a function of the meson energy q0 for different momenta and densities.

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

Results

600 800 1000 1200 q0 [MeV] 1 2 3 4 5 6 7 8 9 SK*(q=0,q0) [GeV

  • 2]

ρ=0 ρ=0.5ρ0 ρ=ρ0 ρ=1.5ρ0

m − m∗(ρ = ρ0) = 50 MeV Γ0 = 50 MeV Γ(ρ = ρ0) = 280 MeV

Figure: The ¯ K∗ spectral function for different densities.

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

Transparency ratio Definition ˜ TA = σγA→K+ K∗− A′ AσγN→K+ K∗− N Describes the loss of flux of K∗−-mesons in the nuclei and is related to its width in the medium In the Eikonal Approximation, ˜ TA ∝ exp ∞ dlImΠK∗−(ρ( r ′)) | pK∗−|

  • ”Survivalprobability”
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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

Transparency ratio

12 6 C, 14 7 N, 23 11Na, 27 13Al, 28 14Si, 35 17Cl, 32 16S, 40 18Ar, 50 24Cr, 56 26Fe, 65 29Cu, 89 39Y, 110 48 Cd, 152 62 Sm, 207 82 Pb, 238 92 U

TA = ˜ TA ˜ T12C

0.2 0.4 0.6 0.8 1 50 100 150 200 250 TA A s1/2 = 3000 MeV s1/2 = 3500 MeV

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Introduction ¯ K∗ self-energy from the s-wave ¯ K∗N interaction ¯ K∗ → ¯ Kπ in dense matter Results Nuclear transparency in the γ A

Conclusions

We have studied the properties of ¯ K∗ mesons in symmetric nuclear matter within a self-consistent coupled-channel unitary approach using hidden-gauge local symmetry. The corresponding in-medium solution incorporates Pauli blocking effects and the ¯ K∗ meson self-energy self-consistently. We have found a small shift of the mass of the ¯ K∗ resonance in the medium. We found that at ρ = ρ0 the ¯ K∗ width is increased to about 250 MeV, or about six times larger than its free width. This spectacular increase is much bigger than the one evaluated for the ρ meson in matter. We have made estimation of the transparency ratios in the γA → K+ ¯ K∗A′ reaction and found substantial reduction from unity of that magnitude which should be easier to observe experimentally by the CLASS collaboration.