Exotic and non-exotic meson spectroscopy, chiral symmetry ...crunch.ikp.physik.tu-darmstadt.de/erice/2015/sec/talks/tuesday/hilge… · Model and Strategy:::so far no comprehensive

Exotic and non-exotic meson spectroscopy,chiral symmetry, and medium modifications

Thomas Hilger

Karl-Franzens University Graz

EriceSeptember 22, 2015

together with:

Maria Gomez-Rocha (ECT*, Trento), Carina Popovici,Andreas Krassnigg (Univ. Graz), Wolfgang Lucha (HEPHY,Vienna)

Sergey Dorkin (JINR, Dubna & Univ. Dubna), Leonid Kaptari(JINR, Dubna & HZDR, Dresden), Burkhard Kampfer(HZDR, Dresden & TU Dresden)

Stefan Leupold (Univ. Uppsala)

supported by:Austrian Science Fund (FWF) project no. P25121-N27,Heisenberg-Landau program of the JINR-FRG collaboration,GSI-FE, BMBF

Covariant.ModelsOfHadrons.com

T. Hilger

Covariant.ModelsOfHadrons.com

The Problem

study hadrons as compositesof quarks and gluons

issues

chiral symmetry andDχSBPoincare covarianceconfinementperturbative limit

→ calculate observables

→ comprehensivephenomenology

u±±±du±±±u±±±

u±±±ddd

u

du

T. Hilger

Incitement

[Bottomonium by Godfrey, Isgur, 1985]

T. Hilger

The Tool

Dyson-Schwinger equations can be used to solve QCD

Bethe-Salpeter and Faddeev-type equations allow covariantand symmetry-preserving study of bound-state problems

Infinite set of coupled (and nonlinear) integral equations

Numerical studies: Truncation ↔ numerical effort

Make the truncation respect symmetries

Construct sophisticated models

Perform reliable calculations of hadron properties

T. Hilger

Dyson-Schwinger and Bethe-Salpeter EquationsDyson-Schwinger equation for quark propagator

= + ΓBνigtAγµ

rainbow approximation

= +

S−1(p) = S (0)−1+

∫dq Gµν(p − q)γµS(q)γν

Bethe-Salpeter equation

K = + G

K = + GK

homogeneous, rainbow-ladder aproximation

K = K

Γ(p;P) = Γ(0)(p)−∫

dq Gµν(p − q)γµS (q+) Γ(q;P)S (q−) γν

T. Hilger

Model and Strategy

. . . so far no comprehensive attempt at RL meson phenomenology

s [GeV2]

G(s)s

Maris-Tandy interaction

= 4π2Dω6 se−s/ω2

+4π2γmF(s)

1/2 ln[τ+(1+s/Λ2

QCD)2]

application to systems wherecorrections to RL are expectedto be least important →bottomonium

leave functional and UV formunchanged

allow for more freedom in theeffective interaction → quarkmass dependence, vary ω and Dindependently

include lowest radial excitations

J = 0, 1, 2, . . .

. . . is that good enough?

T. Hilger

Bottomonium

evaluate splittings at (ω − D)-grid

find minimal χ2(ω,D) =∑

splittings(∆Mexp −∆Mth)2

find minimal χ2(mq) =∑

groundstates(Mexp −Mth)2 foroptimal (ω,D)

ω [GeV]

0.4

0.5

0.6

0.7

0.8

D[GeV 2]

0.00.10.3

0.50.60.70.80.91.1

1.31.5

1.7

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

χ2

0.4 0.5 0.6 0.7 0.8

ω [GeV]

0.0

0.1

0.3

0.5

0.6

0.7

0.8

0.9

1.1

1.3

1.5

1.7

D[GeV

]

[C. Popovici, T. Hilger, M. Gomez-Rocha, A. Krassnigg, Few Body Syst. 56: 481, 2015.]

T. Hilger

Bottomonium

0−+ 1−− 0+ + 1+ + 1+− 2+ + 2−+ 2−−

JPC

9.4

9.6

9.8

10.0

10.2

10.4

10.6

M [

GeV

]

ηb (1S)

Υ(1S)

χb0(1P)χb1(1P) hb (1P) χb2(1P)

ηb (2S)Υ(2S)

Υ(1D)

χb0(2P)χb1(2P) hb (2P)χb1(2P) hb (2P) χb2(2P)

Υ(3S)

Υ(4S)

× experiment

goodidentificationof states

wellreproducedsplittings(excitations,levelorderings)

[T. Hilger, C. Popovici, M. Gomez-Rocha, A. Krassnigg, Phys. Rev. D 91: 034013, 2015.]

mb = 3.635 GeV at µ = 19 GeV, ω = 0.7 GeV, D = 1.3 GeV2

T. Hilger

Charmonium

0−+ 1−− 0+ + 1+ + 1+− 2+ + 2−+ 2−−

JPC

3.0

3.5

4.0

4.5

M [

GeV

]

ηc (1S)

J/Ψ(1S)

χc0(1P)

χc1(1P) hc (1P)χc2(1P)

ηc (2S)Ψ(2S)

Ψ(3770)

χc1(2P)χc0(2P) χc2(2P)

Ψ(4040)

Ψ(4160)

Ψ(4415)

× experiment

no extrastates

excellentlyreproducedsplittings, inparticular1−−

[T. Hilger, C. Popovici, M. Gomez-Rocha, A. Krassnigg, Phys. Rev. D 91: 034013, 2015.]

mc = 0.855 GeV at µ = 19 GeV, ω = 0.7 GeV, D = 0.5 GeV2

T. Hilger

Light Isovector Quarkonium

0�+ 1�� 0++ 1++ 1+� 2++ 2�+ 2�� 0�� 0+� 1�+ 2+�

JPC

0.0

0.5

1.0

1.5

M[G

eV]

⇡

⇢

a0(980)

a1(1260)b1(1235)

a2(1320)

⇡(1300)

⇡1(1400)

a0(1450)⇢(1450)

⇡1(1600)⇡2(1670)⇢(1700)

a2(1700)

⇡(1800)

[T. Hilger, M. Gomez-Rocha, A. Krassnigg, arXiv:1508.07183]

mq = 0.003 GeV at µ = 19 GeV, ω = 0.4 GeV, D = 1.7 GeV2

T. Hilger

Exotics: Light Isovector Quarkonium

0�+ 1�� 0++ 1++ 1+� 2++ 2�+ 2�� 0�� 0+� 1�+ 2+�

JPC

0.0

0.5

1.0

1.5

M[G

eV]

⇡

⇢

a0(980)

a1(1260)b1(1235)

a2(1320)

⇡(1300)

⇡1(1400)

a0(1450)⇢(1450)

⇡1(1600)⇡2(1670)⇢(1700)

a2(1700)

⇡(1800)

[T. Hilger, M. Gomez-Rocha, A. Krassnigg, Phys. Rev. D 91: 114004, 2015.]


T. Hilger

Exotics: Bottomonium

0�+ 1�� 0++ 1++ 1+� 2++ 2�+ 2�� 0�� 0+� 1�+ 2+�

JPC

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10.0

10.1

10.2M

[GeV

]

⌘b(1S)

⌥(1S)

�b0(1P )

�b1(1P ) hb(1P )�b2(1P )

⌘b(2S)⌥(2S)

⌥(1D)


mb = 3.635 GeV at µ = 19 GeV, ω = 0.7 GeV, D = 0.8 GeV2

T. Hilger

Exotics: Charmonium

0�+ 1�� 0++ 1++ 1+� 2++ 2�+ 2�� 0�� 0+� 1�+ 2+�

JPC

3.0

3.2

3.4

3.6

3.8

4.0

M[G

eV]

⌘c(1S)

J/ (1S)

�c0(1P )

�c1(1P )hc(1P )

�c2(1P )

⌘c(2S)

(2S)

(3770)

X(3872)

�c0(2P ) �c2(2P )

(4040)


mc = 0.855 GeV at µ = 19 GeV, ω = 0.6 GeV, D = 0.9 GeV2

T. Hilger

Exotics: Strangeonium

0�+ 1�� 0++ 1++ 1+� 2++ 2�+ 2�� 0�� 0+� 1�+ 2+�

JPC

0.0

0.5

1.0

1.5

2.0

M[G

eV]

�(1020)

h1(1380)f1(1420)

f2(1430)

⌘(1475)

�(1680)

[T. Hilger, M. Gomez-Rocha, A. Krassnigg, arXiv:1508.07183]


T. Hilger

Outlook: Decay Constants

radial excitations

ground states 1st 2nd

state JPC f calc. f exp. f calc. f exp. f calc. f exp.

ηc 0−+ 401 338 244(12) 243() 145(145) −J/Ψ 1−− 450 416 30(3) 296() 118(91) 100()ηb 0−+ 773 − 419(8) − 534(57) −Υ 1−− 768 701 467(17) 497() 41(7) 430()

T. Hilger

A Path to Open Flavor Meson Spectroscopy

-1 1 2 3 4 5 6 7 8 9

-0.3

0.3

= 0.14 GeVM qq

Im p2

-2 2 4 6 8

-6

-4

-2

2

4

6 = 2 GeVM qq

Re p2

Im p2

[S. Dorkin, L. Kaptari, T. Hilger, B. Kampfer, Phys. Rev. C 89: 034005, 2014.]

numerics technically involved

Outlook: AVWTI construction

estimate BRL corrections

Munczek-Nemirovsky model: G(s) ∝ sδ(4)(s)

integral equations reduce to algebraic equations

infinite dressing of quark-gluon vertex with gluon loopspossible

T. Hilger

Quark-Gluon Vertex and Quark Propagator

-4 -2 0 2 4

1

2

3

4

5

A(s)

n=∞

n=4

n=3

n=2

n=1

n=0

-4 -2 0 2 4

0.0

0.5

1.0

1.5

2.0

2.5M(s)

n=∞

n=4

n=3

n=2

n=1

n=0

[M. Gomez-Rocha, T. Hilger, A. Krassnigg, accepted by PRD,

arXiv:1506.03686] [M. Gomez-Rocha, T. Hilger, A. Krassnigg, Few

Body Syst. 56: 475, 2015.]

Rn :=Mn(s = 0)− M∞(s = 0)

M∞(s = 0)

qualitative differences on timelikedomain

weaker effect for heavier quarks

T. Hilger

Quark-Gluon Vertex Dressing and Meson Masses

[M. Gomez-Rocha, T. Hilger, A. Krassnigg, accepted by PRD, arXiv:1506.03686]

momentum partitioningdependence

minimize dressing correction

error estimate

sizeable but notoverwhelming dressingeffects

careful, comprehensiveRL phenomenologyworthwhile

0.125

0.13

0.135

0.14

0.145

0.15

0 1 2 3 4∞π 0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0 1 2 3 4∞K1.5

1.6

1.7

1.8

1.9

2.0

0 1 2 3 4∞D 4.6

4.8

5.0

5.2

5.4

5.6

0 1 2 3 4∞B

1.7

1.8

1.9

2.0

2.1

0 1 2 3 4∞Ds4.8

5.0

5.2

5.4

5.6

0 1 2 3 4∞Bs 6.0

6.1

6.2

6.3

6.4

6.5

6.6

0 1 2 3 4∞Bc 2.9

2.95

3.0

3.05

3.1

3.15

3.2

0 1 2 3 4∞ηc

T. Hilger

Reminder: Chiral Symmetry

two flavor Lagrangian:

L =(ud

)T (iγµ∂

µ −[mu 00 md

])(ud

)

invariant for mu,d = 0

chiral transformations:

T : ψR,L ≡1±γ5

2

(ud

)→ e−i~λ2~ΘR,LψR,L

P = −1 meson current:jV,τµ (x) = ψγµτψ

P = +1 meson current:jA,τµ (x) = ψγ5γµτψ

current-current correlator:

ΠXµν (q) = i

∫d4xe−iqx 〈0|T

[jX,τµ (x)

(jX,τν (0)

)†]|0〉

conversion by set of finite {ΘR,L}

chirally symmetric |0〉

parity blind correlatorsdegenerate spectra

T. Hilger



L =(ud

)T (iγµ∂

µ −[mu 00 md

])(ud

)



T : ψR,L ≡1±γ5

2

(ud





ΠXµν (q) = i


[jX,τµ (x)

(jX,τν (0)

)†]|0〉




T. Hilger



L =(ud

)T (iγµ∂

µ −[mu 00 md

])(ud

)



T : ψR,L ≡1±γ5

2

(ud





ΠXµν (q) = i


[jX,τµ (x)

(jX,τν (0)

)†]|0〉




T. Hilger



L =(ud

)T (iγµ∂

µ −[mu 00 md

])(ud

)



T : ψR,L ≡1±γ5

2

(ud





ΠXµν (q) = i


[jX,τµ (x)

(jX,τν (0)

)†]|0〉




T. Hilger



L =(ud

)T (iγµ∂

µ −[mu 00 md

])(ud

)



T : ψR,L ≡1±γ5

2

(ud





ΠXµν (q) = i


[jX,τµ (x)

(jX,τν (0)

)†]|0〉




T. Hilger

Reminder: Chiral Symmetry and Open FlavorMesonsthree flavor Lagrangian:

L =

udh

T iγµ∂µ −

mu 0 00 md 00 0 mh

udh



T : ψR,L ≡1±γ5

2

udh

→ exp

− i2

Θ3 Θ1 − iΘ2 0Θ1 + iΘ2 −Θ3 0

0 0 0

R,L

ψR,L




ΠXµν (q) = i


[jX,τµ (x)

(jX,τν (0)

)†]|0〉




T. Hilger


L =

udh

T iγµ∂µ −

mu 0 00 md 00 0 mh

udh



T : ψR,L ≡1±γ5

2

udh

→ exp

− i2

Θ3 Θ1 − iΘ2 0Θ1 + iΘ2 −Θ3 0

0 0 0

R,L

ψR,L




ΠXµν (q) = i


[jX,τµ (x)

(jX,τν (0)

)†]|0〉




T. Hilger


L =

udh

T iγµ∂µ −

mu 0 00 md 00 0 mh

udh



T : ψR,L ≡1±γ5

2

udh

→ exp

− i2

Θ3 Θ1 − iΘ2 0Θ1 + iΘ2 −Θ3 0

0 0 0

R,L

ψR,L




ΠXµν (q) = i


[jX,τµ (x)

(jX,τν (0)

)†]|0〉




T. Hilger


L =

udh

T iγµ∂µ −

mu 0 00 md 00 0 mh

udh



T : ψR,L ≡1±γ5

2

udh

→ exp

− i2

Θ3 Θ1 − iΘ2 0Θ1 + iΘ2 −Θ3 0

0 0 0

R,L

ψR,L




ΠXµν (q) = i


[jX,τµ (x)

(jX,τν (0)

)†]|0〉




T. Hilger


L =

udh

T iγµ∂µ −

mu 0 00 md 00 0 mh

udh



T : ψR,L ≡1±γ5

2

udh

→ exp

− i2

Θ3 Θ1 − iΘ2 0Θ1 + iΘ2 −Θ3 0

0 0 0

R,L

ψR,L




ΠXµν (q) = i


[jX,τµ (x)

(jX,τν (0)

)†]|0〉




T. Hilger

Open Flavor Mesons in the Medium- A Window to DCSB

mediummodifications

s−0 s+0

−m− m+0

∆Π(s)

s [GeV]

DχSB

properties ofmatter under

restoration

orderparameters〈qq〉

ccc---

ddd

vacuum

u

d

d

u

d

u

u

duu

du

ccc----

ddd

d

du

u

d

u

d

d

u

u

d

d

u d

u

u

d

d

medium

T. Hilger


mediummodifications

s−0 s+0

−m− m+0

∆Π(s)

s [GeV]

DχSB


restoration


ccc---

ddd

vacuum

u

d

d

u

d

u

u

duu

du

ccc----

ddd

d

du

u

d

u

d

d

u

u

d

d

u d

u

u

d

d

medium

T. Hilger


mediummodifications

s−0 s+0

−m− m+0

∆Π(s)

s [GeV]

DχSB


restoration


ccc---

ddd

vacuum

u

d

d

u

d

u

u

duu

du

ccc----

ddd

d

du

u

d

u

d

d

u

u

d

d

u d

u

u

d

d

medium

T. Hilger


mediummodifications

s−0 s+0

−m− m+0

∆Π(s)

s [GeV]

DχSB


restoration


ccc---

ddd

vacuum

u

d

d

u

d

u

u

duu

du

ccc----

ddd

d

du

u

d

u

d

d

u

u

d

d

u d

u

u

d

d

medium

T. Hilger


mediummodifications

s−0 s+0

−m− m+0

∆Π(s)

s [GeV]

DχSB


restoration


ccc---

ddd

vacuum

u

d

d

u

d

u

u

duu

du

ccc----

ddd

d

du

u

d

u

d

d

u

u

d

d

u d

u

u

d

d

medium

T. Hilger


mediummodifications

s−0 s+0

−m− m+0

∆Π(s)

s [GeV]

DχSB


restoration


ccc---

ddd

vacuum

u

d

d

u

d

u

u

duu

du

ccc----

ddd

d

du

u

d

u

d

d

u

u

d

d

u d

u

u

d

d

medium

T. Hilger


mediummodifications

s−0 s+0

−m− m+0

∆Π(s)

s [GeV]

DχSB


restoration


ccc---

ddd

vacuum

u

d

d

u

d

u

u

duu

du

ccc----

ddd

d

du

u

d

u

d

d

u

u

d

d

u d

u

u

d

d

medium

T. Hilger

Hadron physics and QCD sum rulescurrent-current correlator

Πµν(q) = i∫d4xe iqx 〈T

[jµ(x) (jν(0))†

]〉

spectraldensity

↔ hadronicproperties

ρH(s)

s[GeV2]

operator product expansion= C1(q) + C2(q)〈qq〉 + C3(q)〈qgσG q〉 + . . .

= + + + . . .

QCD condensates:

encode medium dependence

order parameters of chiralsymmetry phase transition

separationof scalesdispersion relation

Π(q2) = 1π

∫∞0 ds ∆Π(s)

s−q2

Req2

Imq2

OPE

×q2

Γ

∞∫

0

= + + + . . .

T. Hilger

Order Parameters and Light Quark Currents

〈qq〉 suppression in light-quark meson operator productexpansion (e.g. ρ meson sum rules): mq〈qq〉〈qq〉 influence via assumptions/models: e.g.

〈qΓqqΓq〉 ∝ 〈qq〉2→ fragile transition to mediumcontinuum threshold s0 ↔ fπ ↔ 〈qq〉

determination of other order parameters (e.g. four-quarkcondensates 〈qΓqqΓq〉) is model dependent

T. Hilger

Properties of the ρ Meson under Chiral SymmetryRestoration

impact of DχSB order parameters onthe ρ meson and implications of chirallysymmetric sum rules[TH, R. Thomas, B. Kampfer, S. Leupold, Phys. Lett. B 709 (2012)

200]

0.0 0.1 0.2 0.3

0.65

0.70

0.75

0.80

mpe

ak [G

eV]

FWHM [GeV]

T. Hilger

Spin-0 Open Flavor Mesons at Finite Density

particle-antiparticle splitting + shift

0.00 0.05 0.10 0.15-0.04

-0.03

-0.02

-0.01

0.00

m [G

eV]

n [fm-3]0.00 0.05 0.10 0.15

1.84

1.86

1.88

1.90

1.92

1.94

m [G

eV]

n [fm-3]

〈qq〉 amplification due to heavy charm quark mass: mc〈qq〉[TH, R. Thomas, B. Kampfer, Phys. Rev. C 79 (2009) 025202][S. Zschocke, TH, B. Kampfer, Eur. Phys. J. A 47 (2011) 151][TH, B. Kampfer, Nucl. Phys. B Proc. Suppl. 207-208 (2010) 025202][TH, B. Kampfer, Conf. Proc. Italian Phys. Soc. 99 (2010)][B. Kampfer, TH, H. Schade, R. Schulze, G. Wolf, PoSBormio 2010][R. Rapp et al., In-medium excitations, Lect. Notes Phys. 814 335 (2011)]

T. Hilger

Chiral QCD Sum Rules for Open FlavorSpin-0 and -1 Mesons at Finite Density

Weinberg-Kapusta-Shuryak sum rules: 〈qq〉-suppression bylight quark mass

open flavor chiral partner sum rules: spectral differencesdriven by order parameters only

amplification of order parameters due to heavy charm quarkmass

hierarchy of order parameters

[TH, B. Kampfer, S. Leupold, Phys. Rev. C 84 (2011) 045202][TH, T. Buchheim, B. Kampfer, S. Leupold, Prog. Part. Nucl. Phys.67 (2012) 188][TH, R. Schulze, B. Kampfer, J. Phys. G: Nucl. Part. Phys. 37(2010) 094054][TH, B. Kampfer, Nucl. Phys. Proc. Suppl. 207-208 (2010) 277] 0 2 4 6 8 10

-0.020

-0.015

-0.010

-0.005

0.000

V-A

OP

E [

GeV

2 ]

M [ GeV ]

T. Hilger

Four-Quark Condensates and Open Flavor Mesons[T. Buchheim, TH, B. Kampfer,

Phys. Rev. C 91 (2015)] [T. Buchheim,

TH, B. Kampfer, arXiv:1509.06144]

→ order parameters

→ spin-0 and -1

→ in-medium

→ chiral sum rules

1

factorization

heavy-quark massexpansion

vacuumlimits

〈ψστΓψψσ′τ ′Γ′ψ〉

[T. Buchheim, TH, B. Kampfer, J. Phys. Conf. Ser. 503 (2014)][T. Buchheim, TH, B. Kampfer, E.P.J. WoC 81 (2014)]

[T. Buchheim, TH, B. Kampfer, Nucl. Phys. Proc. Suppl.258-259 (2015)]

T. Hilger





→ spin-0 and -1

→ in-medium


1

factorization


vacuumlimits




T. Hilger





→ spin-0 and -1

→ in-medium


1

factorization


vacuumlimits




T. Hilger





→ spin-0 and -1

→ in-medium


1

factorization


vacuumlimits




T. Hilger





→ spin-0 and -1

→ in-medium


1

factorization


vacuumlimits




T. Hilger





→ spin-0 and -1

→ in-medium


1

factorization


vacuumlimits




T. Hilger

Summary I

quark mass dependence of effective interaction

optimized rainbow-ladder DS-BS study describes groundstates and lowest radial excitations

extra states in vector- and axial-vector channel forbottomonium

exotic charmonium, bottomonium (and light isovector)spectrum

1st BRL study of open flavor mesons in Munczek-Nemirovskymodel

stay tuned: . . . decay constants . . . open flavor mesons . . .

[M. Gomez-Rocha, T. Hilger, A. Krassnigg, accepted by PRD, arXiv:1506.03686][T. Hilger, M. Gomez-Rocha, A. Krassnigg, arXiv:1508.07183]

[T. Hilger, M. Gomez-Rocha, A. Krassnigg, Phys. Rev. D 91: 114004, 2015.][T. Hilger, C. Popovici, M. Gomez-Rocha, A. Krassnigg, Phys. Rev. D 91: 034013, 2015.]

[M. Gomez-Rocha, T. Hilger, A. Krassnigg, Few Body Syst. 56: 475, 2015.][C. Popovici, T. Hilger, M. Gomez-Rocha, A. Krassnigg, Few Body Syst. 56: 481, 2015.]

T. Hilger

Summary II

light four-quark condensates for D mesons in medium[T. Buchheim, TH, B. Kampfer, Phys. Rev. C 91 (2015)]

heavy-quark mass expansion and factorization of four-quarkcondensates in the medium

continuous transition from medium to vacuum → algebraicvacuum limits

[T. Buchheim, TH, B. Kampfer, J. Phys. Conf. Ser. 503 (2014)][T. Buchheim, TH, B. Kampfer, Nucl. Phys. Proc. Suppl. 258-259 (2015)][T. Buchheim, TH, B. Kampfer, E. P. J. WoC 81 (2014)]

T. Hilger

Thank You!

Your fundingagencycould be here!

T. Hilger

Documents

Exotic and non-exotic meson spectroscopy, chiral symmetry ...crunch.ikp.physik.tu-darmstadt.de/erice/2015/sec/talks/tuesday/hilge… · Model and Strategy:::so far no comprehensive