Soliton Model analysis

for Baryons with a Heavy Quark

of SU(3) symmetry breaking

H. Weigel(Stellenbosch University, Institute for Theoretical Physics)

Bled Mini–Workshop : July 3rd − 10th, 2016

Presentation mainly based on:J.P. Blanckenberg & HW, Phys. Lett. B750 (2015) 230

see also: J. Schechter, A. Subbaraman, S. Vaidya & HW, Nucl. Phys. A590 (1995) 655

M. Harda, A. Qamar, F. Sannino, J. Schechter & HW, Nucl. Phys. A625 (1997) 789

Framework

⋆ substantial empirical information on baryon spectrum

⋆ baryon spectrum dictated by QCD

⋆ probe our understanding of QCD

⋆ approximation schemes to QCD:

• chiral symmetry: up, down (and strange) quarks

• heavy spin/flavor symmetry: charm and bottom quarks

• large NC: baryons as solitons (Skyrmions) [merely motivation, take NC = 3]

⋆ strange quark is neither light nor heavy

mπ ≪ mK ≪ mD

requires particular treatment

Strategy

⋆ point of departure: chiral soliton model in up–down sector

⋆ incorporate heavy mesons according to heavy spin–flavor symmetry

⋆ soliton induces attractive potential for heavy mesons

⋆ occupied bound state represents heavy baryon

⋆ strangeness as collective excitation of soliton in flavor SU (3)

⋆ include flavor symmetry breaking:mπ 6= mK , fπ 6= fK , MD 6= MD∗ ....

⋆ treat collective excitations beyond perturbation expansion (Yabu–Ando)

⋆ coupling between collective excitations and bound heavy meson−→ hyperfine splitting

Emerging picture

Heavy baryon: bound system of heavy quark and light diquark

• light diquark represented by soliton

• soliton spin ∼ WZ and heavy meson contributons

• heavy quark represented by heavy meson

• heavy meson bound to soliton

Model

⋆ chiral Lagrangian for pseudoscalar and vector meson octet

⋆ hedgehog profile functions (in SU(2) isospin subspace)

U0(r) = exp [τ · rF (r)] , ω(0)µ (r) = ω(r) gµ0 , ρ(0)(r) = τ × r

G(r)

r

⋆ boundary conditions with unit baryon number, e.g. F (0) = π

(w/o heavy fields)

⋆ soliton from minimizing classical energy functional, Ecl[F, ω,G]

⋆ generate good baryon quantum numbers byquantizing time dependent configuarations

⋆ collective coordinates to approximate time dependent solution

U(r, t) = A(t)U0(r)A†(t) A(t) ∈ SU(3)

8∑

a=1

λaV(a)µ (r, t) = A(t)

[ω(0)µ (r)

(2 +

√3λ8

)/3 + τ · ρ(0)

µ (r) + induced components]A†(t)

[Meißner et al., Park & HW]

⋆ time dependence measured by angular velocities

A†(t)dA(t)

dt=

i

2

8∑

a=1

λaΩa

⋆ collective coordinate Lagrange functionfrom WZ–term with NC = 3

L = −Ecl +1

2α2

3∑

i=1

Ω2i +

1

2β2

7∑

a=4

Ω2a −

√3

2Ω8

⋆ induced components from extremizing moments of inertia α2 and β2

SU(3) quantization

⋆ right generators

Ra = − ∂L

∂Ωa=

−α2Ωa = −Ja, a = 1, 2, 3

−β2Ωa, a = 4, .., 7√32, a = 8

⋆ commutators: [Ra, Rb] = −ifabcRc

⋆ HamiltonianH = Ecl +

1

2α2~J

2+

1

2β2

7∑

α=4

R2α

= Ecl +1

2

(1

α2− 1

β2

)~J

2+

1

2β2C2[SU (3)]− 3

8β2

quad. Casimir op.

⋆ constraint: YR = 2√3R8 = 1 octet, anti-decuplet,... J = 1/2, decuplet,... J = 3/2

SU(3) quantization

⋆ right generators

Ra = − ∂L

∂Ωa=

−α2Ωa = −Ja, a = 1, 2, 3

−β2Ωa, a = 4, .., 7√32 , a = 8

⋆ commutators: [Ra, Rb] = −ifabcRc

⋆ HamiltonianH = Ecl +

1

2α2~J

2+

1

2β2

7∑

α=4

R2α

= Ecl +1

2

(1

α2− 1

β2

)~J

2+

1

2β2C2[SU(3)]− 3

8β2

⋆ constraint: YR = 2√3R8 = 1 octet, anti-decuplet,... J = 1/2, decuplet,... J = 3/2

⋆ flavor (left generators): La =∑8

b=1Dab(A)Rb (Dab(A) = (1/2)tr(λaAλbA†))

L1,2,3: isospin L4,5,6,7: U and V spin L8: hypercharge

Heavy mesons

⋆ single heavy (anti)quark inside heavy meson: D ∼ uc or B ∼ ub

⋆ pseudoscalar (P ) and vector mesons (Qµ) must be considered(P , Qµ flavor spinors)

π–ρ K–K∗ D–D∗ B–B∗M∗−MM∗+M 0.70 0.07 0.03 0.004

⋆ relativistic Lagrangian (Qµν = DµQν −DνQµ)

Lheavy = (DµP )†DµP − 1

2Q†

µν(Qµν)−M2P †P +M∗2Q†

µQµ

+ 2iMd1(P †pµQ

µ −Q†µp

µP)− d2

2ǫαβµν

[(Qνα)

†pµQβ +Q†βpµQνα

]+ . . .

⋆ coupling of light components via chiral symmetry

Dµ = ∂µ − iαgρµ − i (1− α) vµ vµ, pµ =i

2

(√U∂µ

√U † ±

√U †∂µ

√U)

⋆ large mass expansion: single multiplet for P and Qµ

P = eiMV ·xP Qν = eiM∗V ·xQν

(V µ reference velocity of heavy quark)

H =1

2(1 + γµV

µ)(iγ5P + γνQν

)

⋆ heavy components follow heavy flavor/spin symmetry (M = M ∗)

1

MLheavy −→ iV µTr

(H†DµH

)− dTr

(H†γµγ5p

µH)+ . . .

requires d = d1 ≡ d2 ≈ 0.53 (from D∗s pole in semi–leptonic decay D → K)

⋆ P–wave coupling to soliton background (V µ = (1, 0))

Ha =

(0 Ha

0 0

)Ha

lh = eiωtu(r)√2M

(r · τ )ad ǫdlχh .

u2(r) ∼ δ(r) =⇒ ω =3

2dF ′(0) + . . .

⋆ P–wave heavy meson bound state

P =eiωt√4π

Φ(r)r · τχ , Q0 =eiωt√4π

Ψ0(r)χ , Qi =eiωt√4π

[iΨ1(r)ri +

12Ψ2(r)ǫijkrjτk

]χ

⋆ Fourier amplitude χ = χ(ω) (to be quantized as harmonic oscillator)

⋆ via Lheavy, chiral soliton produces attractive potential

• DEQ for Φ(r) and Ψi(r); (similar for S–wave)

• eigenvalue problem for bound state energy ω (MeV)

heavy limit 1016

M(GeV) M∗(GeV) P S

50.0 50.0 869 769

40.0 40.0 853 74330.0 30.0 831 706

20.0 20.0 796 64610.0 10.0 721 519

5.279 5.325 595 3381.865 2.007 314 29

⋆ significant deviation from heavy limit

Normalization

⋆ unit heavy charge

⋆ Noether charge for phase transformation of heavy mesons

⋆ equivalent to infinitesmal shift of bound state energy

∣∣∣∣∂

∂ω

∫d3rLH

∣∣∣∣ = 1

(solution to bound state equation substituted in LH)

⋆ negative integral: heavy pentaquarks (buudd)(negative energy eigenvalue)

Pentaquarks

⋆ heavy limit wave–function

Hald =

u(r)√2M

(r · τ )adΨdl (g, g3; r)χh g = I +L + S′

⋆ light grand spin

Ψdl (g, g3; r) =∑

r3k3

Cgg3rr3,kk3

Yrr3 (r) ξdl(k, k3) k = I + S′

ξdl(1, 0) =1√2[sl(↑)id(↓) + sl(↓)id(↑)] etc.

⋆ binding energy: ǫ(p)B = 1

2dF′(0) +

√2c

gmVG′′(0) + α

2ω(0) (k = 1)

⋆ total grand spin: G = g + S = I +L + S′ + S G = 12 (Schechter, Subbaraman)

=⇒ must be S and/or P–wave finite mass wave–functions

⋆ relation for finite mass wave–functions

(g, r) = (0, 1) and (1, 0) −→ S–wave

(g, r) = (1, 0): Φ = −3Ψ1 = −3Ψ2/2 (g, r) = (1, 2): Φ = 0, Ψ1 = −Ψ2

⋆ boundary conditions at small distances: 2Ψ1 − Ψ2 − rΨ′2 ∼ 0

=⇒ case (g, r) = (1, 2) requires Ψ1 ∼ r−3 not normalizable

⋆ numerical resultsheavy limit 339

M(GeV) M∗(GeV) (g, r) = (0, 1) (g, r) = (1, 1) (g, r) = (1, 0)

50.0 50.0 169 231 260

40.0 40.0 153 220 25230.0 30.0 130 206 241

20.0 20.0 96 183 22210.0 10.0 35 136 182

5.279 5.325 – 71 1181.865 2.007 – – –

⋆ Pentaquarks disappear in physical limit

Bound state in SU(3)

⋆ SU (3) embedding P,Qµ =

(2–dim from SU (2)

0

)

⋆ collective rotations: P → A(t)P and Qµ → A(t)Qµ with A(t) ∈ SU (3)

⋆ time derivative

P = A(t)[iω + A†(t)A(t)

] eiωt√4π

Φ(r)

(r · τχ

0

)

= iA(t)

[ω +

1

2

8∑

a=1

Ωaλa

]eiωt√4π

Φ(r)

(r · τχ

0

)

= iA(t)

[ω +

1

2√3Ω8 +

1

2

7∑

a=1

Ωaλa

]eiωt√4π

Φ(r)

(r · τχ

0

)

∂

∂Ω8=

1

2√3

∂

∂ω

⋆ SU (3) constraint: R8 = − ∂L∂Ω8

=√32

(1− 1

3χχ†)

SU(3) Symmetry breaking and collective coordinates

⋆ flavor symmetry breaking Lagrangian

Lsb ∝ Tr

1 0 0

0 1 0

0 0 x

(U + U † − 2

) + . . . x =

2ms

mu +md≫ 1

⋆ addition to collective coordinate Lagrange function

Lsb = −1

2γ [1−D88(A)] = −x

2γ [1−D88(A)] Dab =

12Tr

[λaAλbA

†]

⋆ γ: functional of (all) profile functions.

⋆ Hamilton operator

H = Ecl +1

2

(1

α2− 1

β2

) 3∑

i=1

R2i +

1

2β2

8∑

a=1

R2a +

1

2γ [1−D88(A)]−

3

8β2

(1− 1

3N

)2

+ |ω|N +Hhf

• Ra =∂L∂Ωa

right SU(3) generator: [Ra, Rb] = −i∑8

c=1 fabcRc

• N = χχ† number operator for heavy quarks: N = 0, 1

• Hhf hyperfine splitting, later

SU(3) Diagonalization

⋆ non–perturbative treatment of x 6= 1 in H

⋆ eigenvalue equation for collective coordinates

8∑

a=1

R2a + (xγβ2) [1−D88(A)]

Ψ(A) = ǫΨ(A)

subject to constraintYR = 2√

3R8 = 1− N

3

• light baryons, N = 0:

Ψ(A) describes octet, decuplet,..., with half–integer spin

• heavy baryons, N = 1:

Ψ(A) describes anti–triplet, sextet,..., diquarks with integer spin

⋆ Euler angle decomposition of Aeigenvalue eq. −→ coupled ODE in strangeness changing angle ν

⋆ symmetry breaking reduces excitation of strange degrees of freedom

⋆ admixture of higher dim. SU (3) reps.: 3 → 3⊕ 15⊕ . . .

6 → 6⊕ 15⊕ 24 . . .

Hyperfine splitting

⋆ Lhf ∼ ρχ(Ω · τ2

)χ† −→ Hhf ∼ ρ R

α2· χτ

2χ†

⋆ coeff. ρ determined from soliton and heavy meson profiles

⋆ ρ → 0 as M,M ∗ → ∞⋆ total spin from spatial rotations

• soliton: spatial rotation ⇔ isospace rotation (hedgehog)

• heavy meson: spatial rotation ⇔ isospace rotation & unitary transf. of χ

J = −R− χτ

2χ†

quartic in heavy meson operator

⋆ R · χτχ† = j(j + 1)− r(r + 1)− 34∼ j(j + 1)− r(r + 1)

Mass formula

E =

(1

α2− 1

β2

)r(r + 1)

2+ǫ(x)

2β2− 3

8β2

(1− N

3

)2

+ |ω|N +ρ

2α2[j(j + 1)− r(r + 1)]N

⋆ spin of the baryon: j

⋆ spin of the (light) diquark: r

• symmetric: r = 1

• anti–symmetric: r = 0

⋆ isospin of the baryon: I enters eigenvalue equation for ǫ(I = r for zero strangeness)

⋆ S–wave heavy meson −→ negative parity heavy baryons

Results

⋆ only consider mass differences (quantum corrections to Ecl)

⋆ symmetry breaking x ∼ 25 only adjustable parameter

⋆ light baryons and hyperons (up, down, strange)

M −MN

Λ Σ Ξ ∆ Σ∗ Ξ∗ Ω

x = 25 134 218 320 324 438 551 661

x = 30 162 253 404 323 461 601 740

Data (PDG) 177 254 379 293 446 591 733

⋆ charm baryons, positive parity

x = 25 x = 30 Data (PDG)

Bary. (I, j, r) M −MN M −MΛcM −MN M −MΛc

M −MN M −MΛc

Λc (0, 1/2, 0) 1230 0 1233 0 1347 0

Σc (1, 1/2, 1) 1423 193 1425 192 1515 168

Ξc (1/2, 1/2, 0) 1446 216 1486 253 1529 182

Ωc (0, 1/2, 1) 1693 463 1756 523 1756 409

Ξc (1/2, 1/2, 1) 1557 328 1588 355 1637 290

Σc (1, 3/2, 1) 1464 234 1466 233 1579 232

Ξc (1/2, 3/2, 1) 1598 369 1629 396 1706 359

Ωc (0, 3/2, 1) 1734 504 1797 564 1831 484

⋆ charm baryons, negative parity

x = 25 x = 30 Data (PDG)

Bary. (I, j, r) M −MN M −MΛcM −MN M −MΛc

M −MN M −MΛc

Λc (0, 1/2, 0) 1479 249 1482 249 1653 306

Σc (1, 1/2, 1) 1664 434 1666 433 - -

Ξc (1/2, 1/2, 0) 1695 465 1735 502 1851 504

Ωc (0, 1/2, 1) 1934 704 1997 764 - -

Ξc (1/2, 1/2, 1) 1798 569 1829 596 - -

Σc (1, 3/2, 1) 1717 487 1719 486 - -

Ξc (1/2, 3/2, 1) 1851 622 1882 649 1876 529(?)

Ωc (0, 3/2, 1) 1987 757 2050 817 - -

⋆ observed Λc(0, 3/2,−) must be D–wave resonancePDG: Ξc(1/2, 3/2,−) in same SU (4) multiplet?

⋆ bottom baryons, positive parity

x = 25 x = 30 Data (PDG)

Bary. (I, j, r) M −MN M −MΛbM −MN M −MΛb

M −MN M −MΛb

Λb (0, 1/2, 0) 4391 0 4394 0 4681 0

Σb (1, 1/2, 1) 4601 210 4603 209 4872 191

Ξb (1/2, 1/2, 0) 4608 216 4647 253 4855 174

Ωb (0, 1/2, 1) 4871 480 4935 540 5110 429

Ξb (1/2, 1/2, 1) 4736 345 4766 372 - -

Σb (1, 3/2, 1) 4617 226 4619 225 4893 212

Ξb (1/2, 3/2, 1) 4751 360 4782 387 5006 325

Ωb (0, 3/2, 1) 4887 496 4950 556 - -

⋆ bottom baryons, negative parity

• binding overestimated: MΛ′c−MΛc = 168MeV vs. 292MeV (expt.)

• insufficient deviation from heavy flavor/spin symmetry?

⋆ Ξ – Σ anomaly

MΞ −MΣ

x = 25 x = 30 Data (PDG)

s 101 151 125

c 23 62 14

b 6 44 -17

⋆ moderate role of SU (3) symmetry breaking

Conclusions

⋆ chiral soliton model for baryons

⋆ heavy meson coupling motivated by heavy flavor/spin symmetry

⋆ heavy baryons as bound states of heavy meson and soliton

⋆ soliton to be quantized as diquark in flavor SU (3)

⋆ flavor symmetry breaking beyond perturbation expansion

⋆ comprehensive picture of baryon spectrum

⋆ agreement with data as expected within chiral soliton models

⋆ future work

• kinematical corrections

• further deviations from heavy limit

• additional SU(3) symmetry breaking operators