[PPT] - Quark-gluon hybrids (meson- and baryon-like) within a constituent PowerPoint Presentation

SLIDE 1

Quark-gluon hybrids (meson- and baryon-like) within a constituent quark-gluon model (CQGM)

Jorge Segovia

Physik-Department T30f Technische Universit¨ at M¨ unchen

T30f Theoretische Teilchen- und Kernphysik

Nanjing University and Nanjing Normal University

May 2017

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 1/38

SLIDE 2

The subatomic zoo

With the advent of the first particle accelerators, a large number of (lighter) hadrons were discovered in the 1950s and 1960s ☞ Wolfgang Pauli: “Had I foreseen that, I would have gone into botany”. ☞ Enrico Fermi: “If I’d remember the names of these particles, I’d have been a botanist”.

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 2/38

SLIDE 3

The quark model (I)

☞ A classification scheme for hadrons in terms of their valence quarks and antiquarks: ☞ The quarks and antiquarks give rise the quantum numbers of the hadrons:

d u s c b t Q - Electric charge

1/3

+2/3

1/3

+2/3

1/3

+2/3 I - Isospin +1/2 +1/2 Iz - Isospin z-component

1/2

+1/2 S – strangeness

1

C – charm +1 B – Bottomness

1

T – Topness +1

☞ Underlies “flavor SU(3)” symmetry

Murray Gell-Mann George Zweig 3 and ¯ 3 representations

− − − − − − − − − − − − − − − − − →

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SLIDE 4

The quark model (II)

Successful classification scheme organizing the large number of conventional hadrons Baryons Mesons

3 ⊗ 3 ⊗ 3 = 10S ⊕ 8M ⊕ 8M ⊕ 1A 3 ⊗ ¯ 3 = 8 ⊕ 1

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SLIDE 5

The heavy quarkonia before 2003

Charmonium and bottomonium states were discovered in the 1970s. Experimentally clear spectrum of narrow states below the open-flavor threshold

Eichten et al., Rev. Mod. Phys. 80 (2008) 1161

Heavy quarkonia are bound states made of a heavy quark and its antiquark (c¯ c charmonium and b¯ b bottomonium). They can be classified in terms of the quantum numbers of a nonrelativistic bound state → Reminds positronium [(e+e−)-bound state] in QED. Heavy quarkonium is a very well established multiscale system which can serve as an ideal laboratory for testing all regimes of QCD.

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 5/38

SLIDE 6

The discovery of the X(3872)

In 2003, Belle observed an unexpected enhancement in the π+π−J/ψ invariant mass spectrum while studying B+ → K +π+π−J/ψ. It was later confirmed by BaBar in B-decays and by both CDF and D0 at Tevatron in prompt production from p¯ p collisions. Its quantum numbers, mass, and decay patterns make it an unlikely conventional charmonium candidate.

)

2

(GeV/c

X

m 3.8 3.82 3.84 3.86 3.88 3.9 3.92 3.94 3.96 3.98 4 )

2

Events / ( 0.005 GeV/c 10 20 30 40 50 60 70 80 )

2

(GeV/c

X

m 3.8 3.82 3.84 3.86 3.88 3.9 3.92 3.94 3.96 3.98 4 )

2

Events / ( 0.005 GeV/c 10 20 30 40 50 60 70 80

BaBar; PRD 77, 111101 (2008)

)

2

Mass (GeV/c

π

+

π ψ J/ 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

2

Candidates/ 5 MeV/c 500 1000 1500 2000 2500 3000

3.80 3.85 3.90 3.95 900 1000 1100 1200 1300 1400

CDF II

CDF; PRL 93, 072001 (2004) )

2

(GeV/c

µ

+

µ

M
π

+

π

µ

+

µ

M 0.6 0.7 0.8 0.9 1

2

Candidates / 10 MeV/c 200 400 600 800 DØ

(2S) ψ X(3872) )

2

(GeV/c

µ

+

µ

M

2.9 3 3.1 3.2 3.3 2

Candidates / 10 MeV/c

10000 20000

ψ J/

D0; PRL 93, 162002 (2004)

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SLIDE 7

Many experiments around the world

The scientific community has witnessed an explosion of related experimental activity

BELLE@KEK (Japan) BABAR@SLAC (USA) CLEO@CORNELL (USA) PANDA@GSI (Germany) BES@IHEP (China) LHCb@CERN (Switzerland) GLUEX@JLAB (USA)

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SLIDE 8

The XYZ particles – A new particle zoo?

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SLIDE 9

Summary of the XYZ particles

S.L. Olsen, Front. Phys. 10 (2015) 101401

Similar structures have been discovered in other quark sectors such as the JPC = 1−+ candidates π1(1400) [PLB 657 (2007) 27-31] and π1(1600) [PRL 104 (2010) 241803]

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 9/38

SLIDE 10

Exotic matter

The states that do not fit into the quark model are called Exotics (Keep in mind that QCD’s spectrum will inevitably be richer than baryons and mesons) ☞ Glueball (only gluons) An hypothetical composite particle which consists solely of gluon particles, without valence quarks. ☞ Hybrid (Q ¯ Qg) Exotic properties are due to gluonic excitations. ☞ Molecule (Q ¯ q − ¯ Qq) Shallow bound states of heavy mesons analogous to the deuteron. ☞ Diquarkonium (Qq − ¯ Q ¯ q) The constituent quarks are assumed to be clustered into color triplet diquarks. ☞ Hadroquarkonium (Q ¯ Q − q¯ q) A compact core that is a color-singlet Q ¯ Q surrounded by light mesons. And so on...

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SLIDE 11

QCD’s Key feature

Quantum Electrodynamics (QED) Theory of the electroweak interaction. d.o.f: electrons and photons. No Photon self-interactions. Quantum Chromodynamics (QCD) Theory of the strong interaction. d.o.f.: quarks and gluons. GLUON SELF-INTERACTIONS. Origin of confinement, DCSB, ... → How does glue manifest itself in low energy regime? ☞ Possible clues looking at hadrons with explicit gluonic d.o.f. Same role played by gluons and quarks in making matter!! ☞ Hybrid mesons with a heavy-quark pair are the most amenable to theoretical treatment. ☞ LHCb@CERN, GlueX@JLab12 and PANDA@FAIR are producing a rich environment of gluons in order to promote the formation

f glueballs and quark-gluon hybrids.

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SLIDE 12

Theoretical study of glueballs and hybrids

☞ Glueballs. It is difficult to single out which states of the hadronic spectrum are glueballs because we lack the necessary knowledge to determine their decay properties. The strong expected mixing between glueballs and conventional quarkonia leads to a broadening of the possible glueball states, not simplifying their isolation. ☞ Hybrids. Valence gluonic degrees of freedom increase the quantum numbers that are available to fermion-antifermion systems. Some of them cannot be confused with ordinary quark-antiquark mesons and, moreover, they do not mix with conventional quark model states. At lowest order, hybrids decay into a pair of mesons with a valence gluon decaying into a quark-antiquark pair followed by a color re-arrangement process. ☞ Two broad ideas concerning soft glue: A local quasi-particle degree of freedom.

(MIT) bag model. Quasi-gluon model. Constituent gluon model.

Collective non-local degrees of freedom.

Flux-tube model. EFT description and Born-Oppenheimer approximation. Hybrid static energies in Lattice QCD.

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SLIDE 13

(MIT) bag model for hybrid mesons

☞ The quarks in the interior of a bag have small (current) masses and feel only weak forces whereas in the exterior the quarks are not allowed to propagate. ☞ A gluonic field is also placed in the interior of a bag with appropriate boundary

conditions. This gives (1−−,TM) or (1+−,TE) solutions, with TE modes the lightest.

M.S. Chanowitz and S.R. Sharpe, Nucl. Phys. B222 (1983) 211-244

☞ General relevant features: The bag energy density B is taken to be universal because it is a property of the complex structure of the QCD vacuum exterior to the hadronic bag. The energy is given by E = 4 3πR3B +

constituents

Emode R + Z0 R + ∆E(αn

s ) ,

Some difficulties are: (i) computation of gluon self-energies, (ii) existence of spurious degrees of freedom associated with the center of mass, and (iii) determination of the bag’s response when quarks and gluons are present. ☞ Consequences in the meson sector: The lowest-lying hybrid multiplet JPC = 1−−, (0, 1, 2)−+ is constructed from a q¯ q color octet with JPC = 0−+ or 1−− and a TE gluon with JPC = 1+−. Spin splittings follow the pattern observed by lattice QCD which is due to the interaction of the valence gluon with the valence (anti-)quark through a TE field.

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SLIDE 14

Constituent gluon model for hybrid mesons

The massless gluon is one more constituent besides quarks and antiquarks meaning that hybrid mesons and baryons are, respectively, 3- and 4-body bound state systems

D. Horn and J. Mandula, Phys. Rev. D17 (1978) 898

☞ Consequences in the meson sector: The effective Hamiltonian:

Heff. = mq + m¯

q +

p 2

q

2mq +

p 2

¯ q

2m¯

q

+ | pg| + G (| rq − rg| + | r¯

q −

rg|) + V0 The heavy quarks are treated non-relativistically whereas the gluon is massless and thus its kinetic energy is the absolute value of its momentum. The interaction between the quark (antiquark) and the gluon is represented by an attractive linear potential. A weak repulsive quark-antiquark force is lumped with other short-range effects into the undetermined constant. |K| η ξ l lq¯

q

lg sq¯

q = 0;

JPC sq¯

q = 1;

JPC + + 1+− (0, 1, 2)++ + + 1 1 (0, 1, 2)−+ (0, 1, 1, 1, 2, 2, 3)−− 1 − + + − 1 1

1

(0, 1, 2)−− (0, 1, 1, 1, 2, 2, 3)−+ The constituent gluon has JPC |g = 1−− quantum numbers instead of the JPC |g = 1+− for the gluon field: JPC |g = 1−− : {1+−; (0, 1, 2)++} − → JPC |g = 1+− : {1−−; (0, 1, 2)−+}

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SLIDE 15

Quasi-gluon model for hybrid mesons

A non-relativistic reduction of Coulomb gauge QCD in which the gluon d.o.f. are treated in the mean field approximation calibrated to the gluelump spectrum

P. Guo et al., Phys. Rev. D78 (2008) 056003

☞ Consequences in the meson sector: The Schr¨

dinger equation for the radial wave function (in momentum space):
2m + q2

m + k2 4m + Σg (k) + Σq

Ψα(k, q) −

1 2NC

α′
q′2dq′

(2π)3 VQ ¯

Q(q, α; q′, α′) Ψα(k, q′)

+

α′
k′2dk′

(2π)3 q′2dq′ (2π)3 VQ ¯

Qg(k, q, α; k′, q′, α′)Ψα(k′, q′) = MΨα(k, q)

☞ Kinetic and self-energy terms for the quarks and gluons. ☞ The quark-antiquark potential in color octet. ☞ The attractive (anti-)quark-gluon interactions. ☞ Irreducible three-body interaction. In the quasi-particle approximation, the gluelumps are the constituent gluons and they should be coupled to the Q ¯ Q-pair. The irreducible 3-body interaction is responsible for producing the inverted parity

rdering of the gluelump spectra and thus of the Q ¯

Qg quarkonium spectra. The two lowest gluelumps have quantum numbers 1+− and 1−− that coupled with Q ¯ Q in an S-wave: {1−−, {0−+, 1−+, 2−+}} and {1+−, {0++, 1++, 2++}}.

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SLIDE 16

Flux-tube model for hybrid mesons

Hamiltonian approach in which the degrees of freedom are quarks (antiquarks) connected by gluons that are condensed into collective string-like flux-tubes

N. Isgur and J. Paton, Phys. Rev. D31 (1985) 2910

☞ Consequences in the meson sector: The lowest excitations of the string will correspond to nonrelativistic, small, transverse displacements oscillations: HS = b0r + 1 b0r

m

2

i=1
πi

mπi m + 1

4ω2

mb2 0r2qi mqi m

Quantization of the equation above:

E0 −b0r =

N

m=1

2

i=1

1 2ωm = √ 2 a sin(πN/4(N + 1)) sin(π/4(N + 1))

≃

4 πa2

r − 1

a − π 12r +. . . The eigenenergy of the string eigenstate will trace out an adiabatic potential, E S(r), that describes the physics of a hybrid meson: H(1) = − 1 2µ ∂2 ∂r2 + l(l + 1) − Λ2 + L2

S⊥

2µr2 + E (1)(r) E (1)(r) = − 4αs 3r + c + br + π r (1 − e−fb1/2r ) Hybrid mesons are constructed by specifying the gluonic states via phonon

perators and combining these with quark operators. The lowest multiplet

predicted is 1±±, (0, 1, 2)±∓, in contradiction with lattice QCD computations.

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SLIDE 17

EFT description for hybrid mesons

Heavy quarkonium hybrids are characterized by the vast dynamical difference between the slow and massive quarks and the fast and massless gluons

E. Braaten, C. Langmack and D.H. Smith, Phys. Rev. D90 (2014) 014044
M. Berwein, N. Brambilla, J. Tarrus Castella and A. Vairo, Phys. Rev. D92 (2015) 114019

☞ Consequences in the meson sector: The Born-Oppenheimer approximation can be employed to replace the fast gluon field by an effective potential between the nonrelativistic quarks. The effective potentials originate from excited gluon configurations and can be reliably computed using lattice gauge simulations (next slide). The motion of the Q and ¯ Q can be described by the Schr¨

dinger equation with

potential VΓ(r).  − 1 mr2 ∂r r2 ∂r + 1 mr2   l(l + 1) + 2 2

l(l + 1)

2

l(l + 1)

l(l + 1) l(l + 1)   +   VΣ(r) V−Π V+Π     Ψ = EΨ The eigenstates of the gluonic system must be

rganized in representations of the cylindrical

symmetry group D∞h.

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SLIDE 18

Hybrid static energies in Lattice QCD

They are Nonperturbative quantities! E (0)

Σ+

g (r) is the ground state potential that

generates the standard Quarkonium states. The rest of the static energies correspond to gluonic excitations that generate hybrids. The two lowest hybrid static energies are E (0)

Πu (r) and E (0) Σ−

u (r).

→ Nearly degenerate at short distances. Good agreement was found between quenched and unquenched computations.

K.J. Juge et al. PRL 90 (2003) 161601

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SLIDE 19

Why do we want to construct a phenomenological model?

The construction of a reliable model of hybrid meson structure and dynamics is important for the interpretation of experimental results ☞ Among its advantages: Solving the 3- and 4-body bound state problem is quite standard nowadays. The quark-quark and quark-gluon interactions are known. The three-body interaction can be discussed. Particular decay models for hybrids have been proposed and their computational method is quite similar to the one presented for the 3P0 decay model. Other hadron properties like electromagnetic properties are easy to characterized as soon as hybrid wave functions are computed. ☞ Against Lattice computations It is expensive to compute large numbers of experimentally relevant quantities on the lattice. It is also likely that the computation of complicated amplitudes involving hybrids will remain out of reach of lattice methods for a long time. ☞ Against EFT computations It is mostly based on the fact that valence quarks (antiquarks) are heavy and thus

ne can separate gluonic degrees of freedom from quark dynamics.

They rely on the computation of hybrid static energies in Lattice QCD making this approach dependent on the one above. It is not yet clear how to deal with spin-dependent corrections, mixture of different adiabatic surfaces, and how hybrids should decay.

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 19/38

SLIDE 20

Short-range potentials of QCD (I)

Scattering diagram Annihilation diagram

☞ Scattering case: V (r) = C1αs

1

r + π

6 − 4S2

+

3m1m2 − 1 2m2

1

− 1 2m2

2

δ3(r) −

1 2m1m2 p1 · p2 r + (p1 · r)(p2 · r) r3

−

m2

2 + m2 1 + 4m1m2

4m2

1m2 2

L · S+ r3 + 1 4m1m2 S− · Ω r3 + m2

1 − m2 2

4m2

2m2 1

L · S− r3 + (m2 − m1) 4(m1 + m2)m1m2 S+ · Ω r3 + 1 2m1m2

S2

+

r3 − 3(S+ · r)2 r5

☞ Annihilation case:

V (r) = C4 2παs m2 3 4 + S1 · S2

δ3(r) = C4

παs m2 S2

+δ3(r)

V. Mathieu and F. Buisseret, J. Phys. G: Nucl. Part. Phys. 35 (2008) 025006

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 20/38

SLIDE 21

Short-range potentials of QCD (II)

Scattering diagram Compton diagrams

☞ Scattering case: V sca(r) = C7αs 1 r − 1 2mgmq pg · pq r + (pg · r)(pq · r) r3

+ π
11

3mg mq − 2 3m2

g

− 1 2m2

q

− 4 3mgmq S2

+

δ3(r) −
m2

q + m2 g + 4mgmq

4m2

gm2 q

L · S+

r3 + 1 4mgmq S− · Ω r3 + (mq − mg) 4mqmg(mq + mg) S+ · Ω r3 +

m2

g − m2 q

4m2

qm2 g

L · S−

r3 + 1 2mg mq

S2

+

r3 − 3(S+ · r)2 r5

+ mq − mg

m2

gmq

Q(r)

☞ Compton case:

V com(r) = 2πC8αs mg(mg + 2mq) 15 4 − S2

+

δ3(r) + C10αs

mq 2mg

S2

+ − 7

4 e−mqr r

V. Mathieu and F. Buisseret, J. Phys. G: Nucl. Part. Phys. 35 (2008) 025006

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SLIDE 22

Large-range potentials of QCD (I)

G.S. Bali et al. Phys. Rep. 343 (2001) 1.

3
2
1

1 2 3 4 0.5 1 1.5 2 2.5 3 [V(r)-V(r0)]r0 r/r0 Σg

+

Πu 2 mps mps + ms quenched κ = 0.1575

G.S. Bali et al. Phys. Rev. D71 (2005) 114513.

0.8
0.6
0.4
0.2

0.2 2 4 6 8 10 12 14 16 18 [E(r) - 2 mB]a r/a

state |1>

state |2>

☞ Linear screened potential: VCON(r ij) = −ac(1 − e−µcrij ) + ∆ (λi · λj) rij → 0 ⇒ VCON(r ij) → −acµcrij + ∆ (λi · λj) ⇒ Linear. rij → ∞ ⇒ VCON(r ij) → (−ac + ∆)(λi · λj) ⇒ Threshold.

J. Segovia, PhD thesis

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 22/38

SLIDE 23

Large-range potentials of QCD (II)

☞ Same central parts for the scalar and vector Lorentz structures of the confinement:

V C,scalar

CON

(r ij) = V C,vector

CON

(r ij) =

−ac(1 − e−µc rij ) + ∆
(λc

i · λc j )

There are different spin-dependent corrections related with the scalar or vector Lorentz character of the confinement

V SS

CON(r ij) =

1 6mimj (σi · σj)∇2V C

CON(r ij)

V T

CON(r ij) =

1 12mimj

1

r dV C

CON(r ij)

drij − d2V C

CON(r ij)

dr2

ij

Sij

V SO

CON(r ij) =

1 4m2

i m2 j

1 r dV C(r ij) drij

((mi + mj)2 + 2mimj)(S+ · L) + (m2

j − m2 i )(S− · L)

☞ The final expressions are:

V SO

CON(r ij) = −(λc i · λc j ) acµce−µc rij

4m2

i m2 j rij

((m2

i + m2 j )(1 − 2as)

+4mimj(1 − as))(S+ · L) + (m2

j − m2 i )(1 − 2as)(S− · L)

V T

CON(r ij) = −(λc i · λc j ) acµce−µc rij

12mimjrij (1 − as)(1 + µcrij)Sij V SS

CON(r ij) = −(λc i · λc j ) acµce−µc rij

6mimjrij (1 − as)(2 − µcrij)(σi · σj)

J. Segovia, PhD thesis

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 23/38

SLIDE 24

Light quark sector: Goldstone-boson exchange potentials

☞ QCD Lagrangian invariant under the chiral transformation Chiral symmetry is spontanously broken L = ¯ ψ iγµ∂µ − M(q2)Uγ5 ψ ☞ Pseudo-Goldstone Bosons ( π, Ki and η8) Uγ5 = exp (iπaλaγ5/fπ) ∼ 1 + i fπ γ5λaπa − 1 2f 2

π

πaπa + . . . ☞ Constituent quark mass M(q2) = mqF q2 = mq

Λ2

Λ2 + q2 1/2

J. Segovia, PhD thesis

1 2 3 p [GeV] 0.1 0.2 0.3 0.4 M(p) [GeV]

m = 0 (Chiral limit) m = 30 MeV m = 70 MeV

effect of gluon cloud Rapid acquisition of mass is

C.D. Roberts, arXiv:1109.6325v1 [nucl-th]

+ + π σ . . .

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 24/38

SLIDE 25

Gaussian Expansion Method (GEM). Two-body bound state problem (I)

[E. Hiyama, Y. Kino and M. Kamimura, Prog. Part. Nucl. Phys. 51 (2003) 223-307]

☞ Two-body Schr¨

dinger equation:
− 1

2µ∇2 + V (r) − E

ψlm (r) = 0

☞ Expand ψlm(r) in terms of a set of Gaussian basis functions: ψlm(r) =

nmax

n=1

cnl φG

nlm(r) ,

φG

nlm(r) = φG nl(r)Ylm(ˆ

r) where φG

nl(r) = Nnlrle−νnr2 ,

Nnl =

2l+2 (2νn)l+ 3

2

√π (2l + 1)!! 1

2

☞ The best set of Gaussian size parameters are those in geometric progression: νn = 1 r2

n

, rn = r1an−1 (n = 1, . . . , nmax) Dense distribution in the short-range region ⇒ Short-range correlations Coherent superposition in the asymptotic region ⇒ Exponentially-damped tails ☞ Rayleigh-Ritz variational principle and a generalize matrix eigenvalue problem:

nmax

n′=1

[(Tnn′ + Vnn′) − ENnn′] cn′l = 0 (n = 1, . . . , nmax)

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 25/38

SLIDE 26

Gaussian Expansion Method (GEM). Two-body bound state problem (II)

☞ Some examples of matrix elements are: Nnn′ = 2√νnνn′ νn + νn′ l+ 3

2

Tnn′ = φG

nlm| − 1

2µ ∇2|φG

n′lm = 1

µ (2l + 3)νnνn′ νn + νn′ 2√νnνn′ νn + νn′ l+ 3

2

Vnn′ = φG

nlm|V (r)|φG n′lm = NnlNn′l

∞ r2l e−(νn+νn′ )r2V (r)r2dr ☞ For three explicit forms of V (r): φG

nlm|r2|φG n′lm =

l + 3

2

νn + νn′ 2√νnνn′ νn + νn′ l+ 3

2

φG

nlm| 1

r |φG

n′lm =

2 √π 2ll! (2l + 1)!!

νn + νn′

2√νnνn′ νn + νn′ l+ 3

2

φG

nlm|e−µr2|φG n′lm =

2√νnνn′

νn + νn′ + µ l+ 3

2

☞ The generalization for coupled-channels can be read as

nmax

n′=1
T α

nn′ − ENα nn′

cα

n′l +

no. channels
α′

V αα′

nn′ cα′ n′l = 0

n = 1, . . . , nmax

α = 1, . . . , no. channels where T α

n′n and Nα n′n are diagonal and the mixing is given by V αα′ n′n .

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SLIDE 27

Complex-range Gaussian expansion method

The complex-range Gaussian expansion method is introduced in order to reproduce highly oscillatory functions. Such oscillating functions can appear in highly excited vibrational states of few-body systems. Eigenfunctions of a harmonic oscillator Hamiltonian potential? → They are not useful because the calculation of the matrix elements with them is very hard. ☞ The complex-range Gaussian basis functions are given by: φGC

nl

= NGC

nl

rl e−νnr2 cos(ανnr) , φGS

nl

= NGS

nl

rl e−νnr2 sin(ανnr) , The parameter α is a free parameter in principle, but numerical tests suggest α ∼ π/2. ☞ The reason why the functions φGC

nl

and φGS

nl

are easy to be used in numerical calculations is as follows: φGC

nl

= NGC

nl

rl e−ηnr2 + e−η∗

n r2

2 , φGS

nl

= NGS

nl

rl e−ηnr2 + e−η∗

n r2

2i with complex size parameters: ηn = (1 + iα)νn , η∗

n = (1 − iα)νn

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 27/38

SLIDE 28

Three-body bound state problem

☞ The 3-body Schr¨

dinger equation is given by (for the case of central forces alone):
T + V (1)(r1) + V (2)(r2) + V (3)(r3) − E
ΨJM = 0

☞ The wave function is described as a sum of amplitudes of 3 rearrangement channels ΨJM = Φ(i=1)

JM

(r 1, R1) + Φ(i=2)

JM

(r 2, R2) + Φ(i=3)

JM

(r 3, R3) ☞ Each amplitude is expanded in terms of the Gaussian basis functions written in Jacobian coordinates r i and Ri: Φ(i)

JM(r i, Ri) =

ni li mi ,Ni Li Mi

A(i)

ni li mi ,Ni Li Mi

φG

ni li mi (r i)ψG Ni Li Mi (Ri)

JM

(i = 1 − 3) The li and Li are restricted to 0 ≤ li ≤ lmax and |J − li| ≤ Li ≤ J + li. Eigenenergy and coefficients → Rayleigh-Ritz variational principle.

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 28/38

SLIDE 29

Rearrangement channels and Jacobi coordinates (I)

☞ Given a particular reference system with the coordinates of the three particles as x1, x2 and x3, the corresponding Jacobi coordinates are: r 1 = x2 − x3, R1 = x1 − m2x2 + m3x3 m2 + m3 , r 2 = x3 − x1, R2 = x2 − m1x1 + m3x3 m1 + m3 , r 3 = x1 − x2, R3 = x3 − m1x1 + m2x2 m1 + m2 , RCM = m1x1+m2x2+m3x3

m1+m2+m3

. ☞ In the computation of matrix elements we need to express the Jacobi coordinates of

ne channel in function of the others:

r i = αijr j + βijRj Ri = γijr j + δijRj

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 29/38

SLIDE 30

Rearrangement channels and Jacobi coordinates (II)

α12 = − m1 m1 + m3 , β12 = +1, γ12 = − m3(m1 + m2 + m3) (m1 + m3)(m2 + m3), δ12 = − m2 m2 + m3 ; α13 = − m1 m1 + m2 , β13 = −1, γ13 = + m2(m1 + m2 + m3) (m1 + m2)(m2 + m3), δ13 = − m3 m2 + m3 ; α21 = − m2 m2 + m3 , β21 = −1, γ21 = + m3(m1 + m2 + m3) (m1 + m3)(m2 + m3), δ21 = − m1 m1 + m3 ; α23 = − m2 m1 + m2 , β23 = +1, γ23 = − m1(m1 + m2 + m3) (m1 + m2)(m1 + m3), δ23 = − m3 m1 + m3 ; α31 = − m3 m2 + m3 , β31 = +1, γ31 = + m2(m1 + m2 + m3) (m1 + m2)(m2 + m3), δ31 = + m1 m1 + m2 ; α32 = − m3 m1 + m3 , β32 = −1, γ32 = + m1(m1 + m2 + m3) (m1 + m2)(m1 + m3), δ32 = − m2 m1 + m2 .

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 30/38

SLIDE 31

Computing three-body matrix elements (I)

☞ We consider how to calculate a central-potential matrix element of the following type (Spin and isospin parts are omitted for simplicity of expressions): Ψ(k)

JM,αknk Nk (r k, Rk)|V (rj)|Ψ(i) JM,αi ni Ni (r i, Ri)

in which the ket- and bra-vectors are from different channels i and k and the potential is a function of rj. ☞ We transform both the i-channel and k-channel functions into j-channel functions and perform the integration over r j and Rj. r i = αijr j + βijRj , Ri = γijr j + δijRj r k = αkjr j + βkjRj , Rk = γkjr j + δkjRj ☞ Using the formula rli

i Yli mi (ˆ

ri) =

li

λ=0

2li 2λ

4π(2li + 1)

(2λ + 1)(2(li − λ) + 1) 1

2

(αijrj)li −λ(βijRj)λ× ×

Yli −λ(ˆ

rj) ⊗ Yλ( ˆ Rj)

li mi

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 31/38

SLIDE 32

Computing three-body matrix elements (II)

☞ We can rewrite the i-channel three body basis function as a function of r j and Rj

φG

ni li (ri )Yli (ˆ

ri) ⊗ φG

Ni Li (Ri)YLi ( ˆ

Ri)

LML

= Nni li NNi Li e−νni r2

i −λNi R2 i ×

×

lj,Lj

li +Li

K=0

liLiL|ljLjL; K

i→j rli +Li −K j

RK

j

Ylj (ˆ

rj)YLj ( ˆ Rj)

LML

where liLiL|ljLjL; K

i→j = (2li + 1)(2Li + 1) li

λ=0

Li

Λ=0

2li 2λ 2Li 2Λ 1

2

αli −λ

ij

βλ

ij γLi −Λ ij

δΛ

ij ×

×    li − λ Li − Λ lj λ Λ Lj li Li L    (li − λ0Li − Λ0|lj0)(λ0Λ0|Lj 0) and can be calculated and stored prior to the computation. ☞ We still have to deal with the Gaussian part of the expression above: e−νni r2

i −λNi R2 i = e−ηijr2 j −ζij R2 j

∞

l=0

(4π) √ 2l + 1 Il(2ξijrjRj)

Yl(ˆ

rj)Yl( ˆ Rj)

where Il(z) = (−i)ljl(iz) is the modified spherical Bessel function of the first kind and

ηij = νni α2

ij + λNi γ2 ij ,

ζij = νni β2

ij + λNi δ2 ij ,

ξij = νni αijβij + λNi γijδij

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 32/38

SLIDE 33

Four-body bound state problem

The Four-body total wave function ΨJM is described as a sum

f the components of 18 rearrangement channels

Calculation of the Hamiltonian matrix elements becomes much laborious!

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 33/38

SLIDE 34

Infinitesimally-shifted Gaussian basis functions

☞ In order to make the calculation tractable, replace the Gaussian basis function by a superposition of infinitesimally-shifted Gaussians: φG

nlm(r) = Nnlrle−νnr2Ylm(ˆ

r) = Nnl lim

ε→0

1 (νnε)l

kmax

k=1

Clm,k e−νn(r−εDlm,k)2 where the limit ε → 0 must be carried out after the matrix elements have been calculated analytically. ☞ This new set of basis functions make the calculation of three- and four-body matrix elements very easy: Advantages of using the usual Gaussians remain with the new basis functions. No complicated angular-momentum algebra is needed. ☞ More explicitly: Nnlrle−νnr2Ylm(ˆ r) = Nνl lim

ε→0

l

4νnε l

l−m 2

j=0

Alm,j

p

s=0

q

t=0

j

u=0

p s q t j u

e−ν(r−εD)2

where Alm,j = (2l + 1)(l − m)! 4π(l + m)! 1

2 (l + m)!

2m (−1)! 4jj!(m + j)!(l − m − 2j)! and with p = l − m − 2j , q = l + m , D = 2 l (2s − p)az + (2t − q)axy + (2u − j)a∗

xy

E. Hiyama, Y. Kino and M. Kamimura, Prog. Part. Nucl. Phys. 51 (2003) 223-307

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 34/38

SLIDE 35

Hybrid decay (I)

To lowest order the decay is described by the matrix element of the QCD interaction Hamiltonian between a hybrid wave function and a two-mesons wave function: Hybrid → meson + meson ☞ The Hamiltonian annihilating a gluon and creating a quark pair is: Hint = g

dx ¯

ψ(x)γµ λa 2 ψ(x)Aµ

a (x) .

☞ We expand at t = 0: ψ(x) =

2

s=1
dp

(2π)3 eipx upsbps + v−psd†

−ps

,

Aµ

a (x) = 2

λ=1
dk

√ 2ω(2π)3 ϕaεµ

kλ

akλeikx + a†

kλe−ikx

, ☞ Then, the transition operator is given by H = g

ss′λ0
dpdkdp′

√ 2ω(2π)3 δ(3)(p − k − p′)¯ upsγµ λa 2 u−p′s′b†

psd† −p′s′akλϕaεµ kλ

☞ Once the meson and hybrid wave functions are expanded in function of quark, antiquark and gluon creator operators, the matrix element is: BC|H|A = g f (A, B, C) (2π)3 δ(3)(pA − pB − pC )

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 35/38

SLIDE 36

Hybrid decay (II)

[A. Le Yaouanc el al. Z. Phys. C28, 309 (1985)], similar formulae appear in J. Segovia PhD thesis

with f (A, B, C) =

mq¯

q,mg ,mB ,mC ,µq¯ q,mug ,µB ,µC

Ω X(µq¯

qµg; µB, µC ) I(mq¯ q, mg ; mB, mC , m) ×

× C

jg Mg lg mg 1µg C Lm lq¯

qmq¯ qJg Mg C JM

LmSq¯

qµq¯ q C JB MB

lBmB SB µB C JC MC lC mC SC µC Φ

Color factor: Ω = 1 24

a

Tr(λa)2 = 2 3 Spin factor: X(µq¯

qµg; µB, µC ) =

s

√ 2

(2SB + 1)(2SC + 1)3(2Sq¯

q + 1)

   1/2 1/2 SB 1/2 1/2 SC Sq¯

q

1 S    × C

Sµq¯

q+µg

Sq¯

qµq¯ q1µg C SµB +µC

SB µB SC µC

Isospin factor: Φ =

(2IA + 1)(2IB + 1)(2IC + 1)

   i1 i3 IB i2 i4 IC IA IA    η ε

The I’s (i’s) labels the hadron (quark) isospins. η = 1 if the gluon goes into strange quarks and η = √ 2 if it goes into non-strange ones. ε is the number of diagrams contributing to the decay.

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 36/38

SLIDE 37

Hybrid decay (III)

Spatial overlap factor: I(mq¯

q, mg , mB, mC , m) =

dpdk √ 2ω(2π)6 Ψmq¯

q

q¯ q (pB − p)Ψmg g (k)

× ΨmB ∗

B

pB 2 − p − k 2

Ψmc ∗

C

− pB

2 + p − k 2

dΩBY m∗

l

(ΩB) ☞ Selection rules: Factor δlg 0. TE hybrid cannot decay into two mesons which have both zero

rbital angular momentum between their quarks.

Factor δlg 0. TM hybrids with jg ≥ 2 cannot decay into ground mesons pseudoscalar-pseudoscalar, pseudoscalar-vector and vector-vector. Factor δlq¯

ql. The inter-meson orbital momentum is a direct measure of the

interquark orbital momentum in the hybrid. Keep in mind that the selection rules are model dependent and this applies to other theoretical statements that appear in the literature and are cosidered dogma ☞ Other models like the flux-tube decay model (3P0 model + Gaussian vertex) or the PSS hybrid decay model (3S1 model) can be explored. Common predictions: Low-lying hybrids do not decay to two identical mesons. The pair-creation is in a spin-triplet. It appears to be a universal feature in all nonrelativistic decay models.

Jorge Segovia (jorge.segovia@tum.de) Quark-gluon hybrids within a constituent quark-gluon model 37/38

SLIDE 38

Epilogue

☞ A new generation of facilities are scheduled for the next two decades: run II LHCb@CERN, GlueX@JLab12 and PANDA@FAIR. One of their main goals is to produce in the primary collision point a rich environment of gluons in order to promote the formation of glueballs and quark-gluon hybrids. ☞ If a low energy gluonic spectroscopy can be discovered and decoded, this will change the way we think that matter is constructed. For the first time, it will be shown that the QCD’s gauge boson (gluon) participates at the same level than the basic fermions

f the theory (quarks) in building matter.

A new breed of models that is capable of reproducing central lattice results is required These models will reproduce the gluonic adiabatic potentials and the spectrum of heavy and light hybrids reasonably well. This will require a formalism that captures short range and long range dynamics in an approximate fashion without double counting or other conceptual issues. Such a model should also be able to describe strong and electromagnetic decays reasonably accurately. Experimental outcomes and theoretical insights should hopefully lead to a quantitative and qualitative understanding of the soft gluonic sector of the Standard Model

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