BB and B ¯B static potentials and heavy tetraquarks from lattice

BB and B B¯ static potentials and heavy
tetraquarks from lattice QCD
Bethe Forum on methods in lattice field theory – Rheinische
Friedrich-Wilhelms-Universit¨at Bonn
Marc Wagner
Goethe-Universit¨at Frankfurt am Main, Institut fu¨r Theoretische Physik
[email protected]
http://th.physik.uni-frankfurt.de/∼mwagner/
in collaboration with Pedro Bicudo, Krzystof Cichy, Antje Peters,
Jonas Scheunert, Annabelle Uenver, Bj¨orn Wagenbach
March 25, 2015
Goals, motivation (1)
• Study tetraquarks/mesonic molecules by combining lattice QCD and
phenomenology/model calculations.
• Basic idea:
(1) Compute the potential of two heavy quarks in the presence of two light
quarks using lattice QCD.
(2) Explore, whether the potentials are sufficiently attractive to generate a
bound state (a rather stable tetraquark/mesonic molecule) using
phenomenology/model calculations.
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
Goals, motivation (2)
• Why are such investigations important?
Quite a number of mesons are only poorly understood.
– Charged bottomonium states, e.g. Zb (10610)+ and Zb(10650)+ ... must
be four quark states.
– Charged charmonium states, e.g. Zc (3940)± and Zc (4430)± ... must be
four quark states.
– X(3872) (¯
cc state): mass not as expected from quark models; could be
a D-D∗ molecule, a bound diquark-antidiquark, ...
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
Outline
• A brief introduction to lattice QCD hadron spectroscopy.
– QCD (quantum chromodynamics).
– Hadron spectroscopy.
– Lattice QCD.
• Heavy-heavy-light-light tetraquarks.
• BB static potentials.
• BB tetraquarks.
¯ static potentials.
• BB
• Inclusion of B/B ∗ mass splitting.
• Outlook.
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
QCD (quantum chromodynamics)
• Quantum field theory of quarks (six flavors u, d, s, c, t, b, which differ in
mass) and gluons.
• Part of the standard model explaining the formation of hadrons
(usually mesons = q q¯ and baryons = qqq/¯
q q¯q¯) and their masses; essential for
decays involving hadrons.
• Definition of QCD simple:
Z
X
1 (f )
(f )
(f )
4
¯
γµ ∂µ − iAµ + m
ψ
ψ + 2 Tr Fµν Fµν
dx
S =
2g
Fµν
f ∈{u,d,s,c,t,b}
= ∂µ Aν − ∂ν Aµ − i[Aµ, Aν ].
• However, no analytical solutions for low energy QCD
observables, e.g. hadron masses, known, because of
the absence of any small parameter (i.e. perturbation
theory not applicable).
→ Solve QCD numerically by means of lattice QCD.
D meson
proton
d
c¯
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
u
u
u
example: D meson
Hadron spectroscopy
c¯
u
• Proceed as follows:
(1) Compute the temporal correlation function C(t) of a suitable hadron
creation operator O (an operator O, which generates the quantum
numbers of the hadron of interest, when applied to the vacuum |Ωi).
(2) Determine the corresponding hadron mass from the asymptotic
exponential decay in time.
• Example: D meson mass mD (valence quarks c¯ and u, J P = 0− ),
Z
O ≡
d3 r c¯(r)γ5u(r)
0.09
t→∞
∝
0.07
〈 O+(t) O(0) 〉
C(t) ≡ hΩ|O† (t)O(0)|Ωi
t→∞
∝ exp − mD t .
〈 O+(t) O(0) 〉
A exp(-mmeson t)
points included in the fit
0.08
0.06
0.05
0.04
0.03
0.02
0.01
0
0
2
4
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
6
t
8
10
12
Lattice QCD (1)
• To compute a temporal correlation function C(t), use the path integral
formulation of QCD,
C(t) = hΩ|O† (t)O(0)|Ωi =
Z
1 Y
(f ) ¯(f )
(f )
(f )
¯
Dψ Dψ
DAµ O† (t)O(0)e−S[ψ ,ψ ,Aµ ] .
=
Z
f
– |Ωi: ground state/vacuum.
– O† (t), O(0): functions of the quark and gluon fields (cf. previous slides).
R Q
– ( f Dψ (f ) Dψ¯(f ) )DAµ : integral over all possible quark and gluon field
configurations ψ (f ) (x, t) and Aµ (x, t).
– e−S[ψ
(f )
,ψ¯(f ) ,Aµ ]
: weight factor containing the QCD action.
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
Lattice QCD (2)
• Numerical implementation of the path integral formalism in QCD:
– Discretize spacetime with sufficiently small lattice spacing
a ≈ 0.05 fm . . . 0.10 fm
→ “continuum physics”.
– “Make spacetime periodic” with sufficiently large extension
L ≈ 2.0 fm . . . 4.0 fm (4-dimensional torus)
→ “no finite volume effects”.
xµ = (n0 , n1 , n2 , n3 ) ∈ Z4
x0
a
x1
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
Lattice QCD (3)
• Numerical implementation of the path integral formalism in QCD:
– After discretization the path integral becomes an ordinary
multidimensional integral:
Z
YZ
¯
¯
dψ(xµ ) dψ(xµ ) dU(xµ) . . .
Dψ Dψ DA . . . →
xµ
– Typical present-day dimensionality of a discretized QCD path integral:
∗ xµ: 324 ≈ 106 lattice sites.
a,(f )
∗ ψ = ψA : 24 quark degrees of freedom for every flavor
(×2 particle/antiparticle, ×3 color, ×4 spin), 2 flavors.
∗ U = Uµab : 32 gluon degrees of freedom (×8 color, ×4 spin).
∗ In total: 324 × (2 × 24 + 32) ≈ 83 × 106 dimensional integral.
→ standard approaches for numerical integration not applicable
→ sophisticated algorithms mandatory (stochastic integration
techniques, so-called Monte-Carlo algorithms).
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
Heavy-heavy-light-light tetraquarks (1)
¯ Qqq
¯ and QQ¯
¯ q q tetraquark states (q ∈ {u, d, s, c}):
• Study possibly existing Q
¯Q
¯ and QQ
¯ (reduces
– Use the static approximation for the heavy quarks Q
the necessary computation time significantly).
¯Q
¯ ≡ ¯b¯b and QQ
¯ ≡ ¯bb, e.g. Zb (10610)+ and
– Most appropriate for Q
Zb (10650)+.
¯Q
¯ ≡ c¯c¯ and QQ
¯ ≡ c¯c, e.g.
– Could also provide information about Q
Zc (3940)± and Zc (4430)±.
• Proceed in two steps:
¯Q
¯ and QQ
¯ in the presence
(1) Compute the potential of two heavy quarks Q
of two “light quarks” qq and q¯q (q ∈ {u, d, s, c}) using lattice QCD.
(2) Solve the non-relativistic Schr¨odinger equation for the relative coordinate
¯Q
¯ and QQ;
¯ the existence of a bound state would
of the heavy quarks Q
indicate a rather stable tetraquark state.
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
Heavy-heavy-light-light tetraquarks (2)
• Since heavy b quarks are treated in the static approximation, their spins are
irrelevant (mesons are labeled by the spin of the light degrees of freedom j).
• Consider only pseudoscalar/vector mesons (j P = (1/2)−, PDG: B, B ∗) and
scalar/pseudovector mesons (j P = (1/2)+, PDG: B0∗, B1∗), which are among
the lightest static-light mesons.
• Study the dependence of the mesonic potential V (R) on
– the “light” quark flavors u, d, s and/or c (isospin),
– the “light” quark spin (the static quark spin is irrelevant),
– the type of the meson B, B ∗ and/or B0∗ , B1∗ .
→ Many different channels/quantum numbers ... attractive, repulsive ...
V (R) =?
¯
Q
¯
Q
u
P=−
d
R
P=+
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
BB static potentials (1)
¯ Qqq,
¯
¯ q q, i.e. “B B”).
¯
• In the following Q
i.e. “BB” (not QQ¯
• To extract the potential(s) of a given sector (I, Iz , |jz |, P, Px ), compute the
temporal correlation function of the trial state
(1)
(2)
˜
¯
¯
(CΓ)AB (C Γ)CD QC (−R/2)qA (−R/2) QD (+R/2)qB (+R/2) |Ωi.
– C = γ0γ2 (charge conjugation matrix).
– q (1) q (2) ∈ {ud − du , uu , dd , ud + du , ss , cc} (isospin I, Iz ).
– Γ is an arbitrary combination of γ matrices (spin |jz |, parity P, Px ).
static quark
light quark
+
color trace
spin trace
R
T
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
BB static potentials (2)
• I = 0 (left) and I = 1 (right); |jz | = 0 (top) and |jz | = 1 (bottom).
singlet B (|jz| = 0, I = 0, P = +, Px = −)
1000
1000
1000
800
800
800
800
600
600
600
600
400
200
400
200
0
attractive P−P−
repulsive SP−
attractive SS
-400
0
0.1
0.2
0.3
0.4
0.5
r in fm
0.6
0.7
400
200
0
repulsive P−P−
attractive SP−
repulsive SS
-200
-400
0.8
V in MeV
1000
V in MeV
1200
-200
0
0.1
singlet C (|jz| = 0, I = 0, P = +, Px = +)
0.2
0.3
0.4
0.5
r in fm
0.6
0.7
200
0
attractive P−P−
repulsive SP−
attractive SS
-200
-400
0.8
400
0
0.1
singlet D (|jz| = 0, I = 0, P = −, Px = −)
0.2
0.3
0.4
0.5
r in fm
0.6
0.7
-400
0.8
0
1000
1000
1000
800
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800
800
600
600
600
600
200
V in MeV
1000
V in MeV
1200
400
400
200
0
0
-200
-200
-200
-200
-400
attractive SP−
0.1
0.2
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0.8
r in fm
-400
repulsive SP−
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
r in fm
doublet I (|jz| = 1, I = 0, P = +, Px = +/−)
-400
attractive SP−
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1000
1000
800
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600
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repulsive P−P−
attractive SP−
repulsive SS
-200
-400
0
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0.4
0.5
r in fm
0.6
0.7
400
200
0
0
-200
-200
-400
0.8Marc
repulsive SP−
V in MeV
1000
800
V in MeV
1000
V in MeV
1200
200
0.2
0.3
0.4
0.5
0.6
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0.8
400
200
0
attractive P−P−
repulsive SP−
attractive SS
-400
¯0.4static
0
0.1
0.5 potentials
0.6
0.7 and
0.8 heavy tetraquarks
0
0.1
0.2
0.3
0.4 QCD”,
0.5
0.6
0.7 25,
0.82015
Wagner,
“BB0.2and0.3B B
from
lattice
March
r in fm
0.8
sextet L (|jz| = 1, I = 1, P = +, Px = −/+)
1200
200
0.1
sextet K (|jz| = 1, I = 1, P = −, Px = +/−)
1200
400
0.7
r in fm
1200
400
0.6
repulsive SP−
0
r in fm
doublet J (|jz| = 1, I = 0, P = −, Px = −/+)
0.4
0.5
r in fm
200
0
0
0.3
400
0
-400
0.2
triplet H (|jz| = 0, I = 1, P = −, Px = +)
1200
200
0.1
triplet G (|jz| = 0, I = 1, P = +, Px = −)
1200
400
repulsive P−P−
attractive SP−
repulsive SS
-200
1200
V in MeV
V in MeV
triplet F (|jz| = 0, I = 1, P = +, Px = +)
1200
0
V in MeV
triplet E (|jz| = 0, I = 1, P = −, Px = −)
1200
V in MeV
V in MeV
singlet A (|jz| = 0, I = 0, P = −, Px = +)
1200
r in fm
-200
-400
attractive SP−
0
0.1
0.2
0.3
0.4
0.5
r in fm
0.6
0.7
0.8
BB static potentials (3)
Why are certain channels attractive and others repulsive? (1)
• Wave function of two identical fermions (light quarks q (1) q (2) ) must be
antisymmetric (Pauli principle); in qft/QCD automatically realized on the
level of states.
• q (1) q (2) isospin: I = 0 antisymmetric, I = 1 symmetric.
• q (1) q (2) spin: j = 0 antisymmetric, j = 1 symmetric.
• q (1) q (2) color:
– (I = 0, j = 0) and (I = 1, j = 1): must be antisymmetric, i.e. a triplet ¯3.
– (I = 0, j = 1) and (I = 1, j = 0): must be symmetric, i.e. a sextet 6.
¯ Qq
¯ (1) q (2) must form a color singlet:
• The four quarks Q
– q (1) q (2) in a color triplet ¯3
– q (1) q (2) in a color sextet 6
→
→
¯Q
¯ also in a triplet 3.
static quarks Q
¯Q
¯ also in a sextet ¯6.
static quarks Q
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
BB static potentials (4)
Why are certain channels attractive and others repulsive? (2)
¯Q
¯ at small separations r is mainly due to
• Attractive/repulsive behavior of Q
¯Q
¯ are rather close, inside a large light
1-gluon exchange (the static quarks Q
quark cloud formed by q (1) q (2) , i.e. no color screening of the color charges
¯Q
¯ due to q (1) q (2) ):
Q
– color triplet 3 is attractive, V (r) = −2αs /3r,
– color sextet ¯6 is repulsive, V (r) = +αs /3r
(easy to calculate in LO perturbation theory).
• Summary:
– (I = 0, j = 0) and (I = 1, j = 1)
– (I = 0, j = 1) and (I = 1, j = 0)
→
→
¯Q
¯ potential V (r).
attractive Q
¯Q
¯ potential V (r).
repulsive Q
This expectation is consistent with the obtained lattice results.
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
BB static potentials (5)
• Focus on the two attractive channels between ground state static-light
mesons “B and B ∗ ” (probably the best candidates to find a tetraquark):
– Scalar isosinglet (more
attractive):
√
qq = (ud − du)/ 2, Γ = γ5 + γ0γ5,
quantum numbers (I, |jz |, P, Px ) = (0, 0, −, +).
– Vector isotriplet (less √
attractive):
qq ∈ {uu, (ud + du)/ 2, dd}, Γ = γj + γ0γj ,
quantum numbers (I, |jz |, P, Px ) = (1, {0, 1}, −, ±).
• Computations for qq = ll, ss, cc (l ∈ {u, d}) to study the mass dependence.
(b) vector isotriplet
2m(S)
2m(S)
2m(S) - 0.5
2m(S) - 0.5
V [GeV]
V [GeV]
(a) scalar isosinglet
2m(S) - 1.0
2m(S) - 1.5
2m(S) - 1.0
2m(S) - 1.5
2m(S) - 2.0
0
charm quarks
charm quarks
strange quarks
strange quarks
2m(S) - 2.0
light quarks
light quarks
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
0.25
0.3
R [fm]
R [fm]
BB static potentials (6)
• Two competing effects:
– The potential for light quarks is wider/deeper, i.e. favors the existence of
a bound state (a tetraquark).
– Heavier quarks correspond to heavier mesons
(m(B) < m(Bs ) < m(Bc)), which form more readily a bound state (a
tetraquark), i.e. require a less wide/deep potential for a bound state.
[M.W., PoS LATTICE 2010, 162 (2010) [arXiv:1008.1538]]
[M.W., Acta Phys. Polon. Supp. 4, 747 (2011) [arXiv:1103.5147]]
[B. Wagenbach, P. Bicudo, M.W., arXiv:1411.2453]
(b) vector isotriplet
2m(S)
2m(S)
2m(S) - 0.5
2m(S) - 0.5
V [GeV]
V [GeV]
(a) scalar isosinglet
2m(S) - 1.0
2m(S) - 1.5
2m(S) - 1.0
2m(S) - 1.5
2m(S) - 2.0
0
charm quarks
charm quarks
strange quarks
strange quarks
2m(S) - 2.0
light quarks
light quarks
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
0.25
0.3
R [fm]
R [fm]
BB tetraquarks (1)
• Solve the non-relativistic Schr¨odinger equation for the relative coordinate of
¯ Q,
¯
the heavy quarks Q
1
− △ + V (r) ψ(r) = Eψ(r) , µ = m(B(s,c) )/2;
|{z}
2µ
=R(r)/r
a bound state, i.e. E0 < 0, would be an indication for a tetraquark state.
(a) scalar isosinglet: α = 0.29 ± 0.03, p = 2.7 ± 1.2, d/a = 4.5 ± 0.5
0
-0.1
Va
• There is a bound state for the scalar isosinglet and
qq = ll (i.e. BB), binding energy E ≈ −50 MeV,
confidence level ≈ 2 σ.
-0.2
-0.3
• No binding for the vector isotriplet or for qq = ss, cc
(i.e. Bs Bs , Bc Bc ).
-0.4
-0.5
0
3
4
5
6
7
8
(b) vector isotriplet: α = 0.20 ± 0.08, d/a = 2.5 ± 0.7 (p = 2.0 fixed)
0
µ = mb/2, a = 0.079 fm
µ = mb/2, a = 0.096 fm
-0.1
µ = mB/2, a = 0.079 fm
µ = mB/2, a = 0.096 fm
Va
probability density in 1/fm
2
2
r/a
probability to find the b antiquark pair at separation r
2.5
1
-0.2
-0.3
1.5
-0.4
1
-0.5
0
1
2
3
4
r/a
0.5
0
¯ static
0
0.2
0.6
0.8 lattice QCD”,
1
1.2 25, 2015
Marc Wagner, “BB and B B
potentials
and 0.4
heavy tetraquarks
from
March
r in fm
5
6
7
8
BB tetraquarks (2)
• To quantify “no binding”, we list for each channel the factor, by which the
effective mass µ in Schr¨odinger’s equation has to be multiplied, to obtain
binding with confidence level 1 σ and 2 σ (the potential is not changed).
flavor
confidence level for binding
scalar isosinglet
vector isotriplet
light
1σ 2σ
0.8 1.0
1.9 2.1
strange
1σ 2σ
1.9 2.2
2.5 2.7
charm
1σ 2σ
3.1 3.2
3.4 3.5
– Factors ≤ 1.0 indicate binding.
– Light quarks (u/d) are unphysically heavy (correspond to
mπ ≈ 340 MeV); physically light u/d quarks are expected to yield
stronger binding for the scalar isosinglet, might lead to binding also for
the vector isotriplet (computations in progress).
– Mass splitting m(B ∗ ) − m(B) ≈ 50 MeV, neglected at the moment, is
expected to weaken binding (coupled channel analysis; see later slides).
[P. Bicudo, M.W., Phys. Rev. D 87, 114511 (2013) [arXiv:1209.6274]]
[B. Wagenbach, P. Bicudo, M.W., arXiv:1411.2453]
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
BB tetraquarks (3)
What are the quantum numbers of the ¯b¯bll tetraquark (light scalar
isosinglet)?
• Light scalar isosinglet: I = 0, j = 0, ll in a color ¯3, ¯b¯b in a color 3
(antisymmetric) ... as discussed above.
• Wave function of ¯b¯b must also be antisymmetric (Pauli principle); in the
lattice QCD computation not automatically realized (static quarks are
spinless color charges, which can be distinguished by their positions).
– ¯b¯b is flavor symmetric.
– ¯b¯b spin must also be symmetric, i.e. jb = 1.
• The ¯b¯bll tetraquark has quantum numbers isospin I = 0, spin J = 1,
parity P = + (parity not obvious).
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
B B¯ static potentials
¯ q q, i.e. “B B”,
¯ trial states
• Experimentally more interesting case: QQ¯
(1)
(2)
˜
¯
ΓAB ΓCD QC (−R/2)qB (−R/2) q¯A (+R/2)QD (+R/2) |Ωi.
• At the moment only preliminary results for q¯q = c¯c, “I = 1”.
¯ Qqq:
¯
¯ Qqq
¯ half of
all channels are attractive (for Q
• Qualitative difference to Q
them are attractive, half of them are repulsive).
¯
• Can again be understood by the 1-gluon exchange potential of QQ:
– No Pauli principle for q¯(1) q (2) (particle and antiparticle are not identical).
– q¯(1) q (2) can be in a symmetric color singlet 1 for any isospin/spin
orientation.
¯ also in a singlet 1.
– q¯(1) q (2) in a color singlet 1 → static quarks QQ
– Color singlet is attractive, V (r) = −4αs /3r (LO perturbation theory).
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
Inclusion of B/B ∗ mass splitting (1)
• Mass splitting mB ∗ − mB ≈ 50 MeV has been neglected so far.
• Mass splitting mB ∗ − mB is, however, of the same order of magnitude as the
previously obtained binding energy E ≈ −50 MeV.
• Moreover, two competing effects:
¯ Qqq
¯ channel correspond to a linear combination of BB,
– An attractive Q
BB ∗ and/or B ∗B ∗ , e.g.
scalar isosinglet ≡ BB + Bx∗Bx∗ + By∗ By∗ + Bz∗Bz∗ .
¯ Qqq
¯
– The BB interaction is a superposition of attractive and repulsive Q
potentials.
• Goal: take mass splitting mB ∗ − mB ≈ 50 MeV into account
¯ Qqq
¯ potentials.
→ refined model calculation with the computed Q
• Will there still be a bound state?
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
Inclusion of B/B ∗ mass splitting (2)
Solve a coupled channel Schr¨
odinger equation (1)
• Previously:
– A wave function ψ with 1 component corresponding to BB (B ≡ B ∗ ).
• Now:
– A static light meson can correspond to B or B ∗ = (Bx∗, By∗, Bz∗).
~ with 16 components corresponding to
– Therefore, a wave function ψ
(BB, BBx∗, BBy∗, BBz∗, Bx∗B, Bx∗Bx∗ , Bx∗By∗, Bx∗Bz∗ , . . . , Bz∗Bz∗).
~ 1, r2) = E ψ(r
~ 1 , r2),
• Coupled channel Schr¨odinger equation H ψ(r
H
p21
p22
−1
−1
= M ⊗ 1 + 1 ⊗ M + (M ⊗ 1) + (1 ⊗ M) + V (|r1 − r2|),
2
2
where M = diag(mB , mB ∗ , mB ∗ , mB ∗ ) and V is a 16 × 16 non-diagonal
¯ Qqq
¯ potentials (both attractive and repulsive).
matrix containing the Q
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
Inclusion of B/B ∗ mass splitting (3)
Solve a coupled channel Schr¨
odinger equation (2)
~ 1, r2) = E ψ(r
~ 1 , r2),
• Coupled channel Schr¨odinger equation H ψ(r
H
p21
p22
−1
−1
= M ⊗ 1 + 1 ⊗ M + (M ⊗ 1) + (1 ⊗ M) + V (|r1 − r2|),
2
2
where M = diag(mB , mB ∗ , mB ∗ , mB ∗ ) and V is a 16 × 16 non-diagonal
¯ Qqq
¯ potentials (both attractive and repulsive).
matrix containing the Q
• Specific limits:
– V = 0, i.e. no interactions:
q
q
q
q
2
2
2
2
2
2
E =
mB + p1 + mB + p2 , mB + p1 + m2B ∗ + p22 , . . .
– mB ∗ = mB , i.e. “old 1-component SE calculation”:
E ≈ 2mB − 50 MeV.
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
Inclusion of B/B ∗ mass splitting (4)
Solve a coupled channel Schr¨
odinger equation (3)
• Transform the 16 × 16 Schr¨odinger equation to block diagonal structure:
– Total spin J = 0: 2 × 2 structure.
– Total spin J = 1: 3 × 3 structure (3× due to Jz degeneracy).
– Total spin J = 2: 1 × 1 structure (5× due to Jz degeneracy).
• Work in progress ...
– First very preliminary results indicate that for J = 0 the bound state
does not exist anymore (however, still very close to a bound state).
– However:
∗ More realistic/relevant J = 1 equation not yet investigated.
∗ Unphysically heavy u/d quarks (mπ ≈ 340 MeV) ... physically light
¯ Qqq
¯ potentials.
quarks will lead to more attractive Q
¯Q
¯ separation M = mb
∗ M = diag(mB , mB ∗ , mB ∗ , mB ∗ ); for small Q
would be more appropriate ... should enhance binding.
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
Outlook (1)
¯
• Future plans for BB and B B:
– Computations with light u/d quarks of physical mass (mπ ≈ 140 MeV
instead of mπ ≈ 340 MeV).
– Light quarks of different mass: BBs, BBc and Bs Bc potentials.
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B
Outlook (2)
¯
• Future plans for BB and B B:
– Study the structure of the states corresponding to the computed
potentials:
∗ In a lattice computation two different creation operators generating
the same quantum numbers yield the same potential.
∗ At the moment exclusively creation operators of mesonic molecule
type.
∗ For BB use also
· creation operators of diquark-antidiquark type.
¯ use also
∗ For B B
· creation operators of diquark-antidiquark type,
¯ string + π),
· creation operators of bottomonium + pion type (QQ
¯ string).
· for I = 0 creation operators of bottomonium type (QQ
∗ Resulting correlation matrices provide information about the
structure.
¯ static potentials and heavy tetraquarks from lattice QCD”, March 25, 2015
Marc Wagner, “BB and B B