Shallow bound states and resonances from la3ce QCD Sasa Prelovsek

(1)

Shallow bound states and resonances from la3ce QCD

University of Ljubljana & Jozef Stefan InsAtute, Slovenia Theory & Jeﬀerson Lab

(All August 2015)

1

in collaboraAon with:

ChrisAan B. Lang, Luka Leskovec, Daniel Mohler, Richard Woloshyn

Sasa Prelovsek

Graz Ljubljana FERMILAB TRIUMF

S. Prelovsek, Bound states and resonances

College of William & Mary, Williamsburg 29^st April, 2015

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Outline

•  Experimental appeAzer:

most interesAng experimental states are located near threshold !

•  I will discuss hadronic states that have to be address by simulaAng the sca]ering of two mesons on the la3ce

•  States slightly below threshold :

D

_s0

, D

_s1

, B

_s0

, B

_{s1 ,}

X(3872)

Flavor according to quark model: s c s b c c “deuterium-‐like” bound states: ﬁrst la3ce simulaAons of such states in mesonic system

Only few such states in heavy-‐meson spectrum; none (to my knowledge) in light-‐meson spectrum

•  Resonances above threshold:

**K*, a**

₁

, b

₁

, D

₀^*

, D

₁

, Ψ(3770)

•  Conclusions

(3)

S. Prelovsek, Bound states and resonances 3

[BESIII, 2013, 1303.5949, PRL]

Z_c⁺(3900) g J/Ψ π⁺ cc du

state

conﬁrmed by BeSII, Belle, Cleo-‐c

MoAvaAon from experiment:

need to understand near-‐threshold states from QCD ! A charged charmonium-‐like state: Z

_c⁺

(3900)

DD* thr.

(4)

[review: Brambilla et al., 1404.3723]

Challenges for the theory community:

quarkonium-‐like states at threshold

(5)

Evaluation of Feynman path integrals in discretized space-time

Non-‐perturbaAve method: QCD on la3ce

L_QCD = −¹₄G_µν^a G_a^µν + qiγ_µ(

q=u,d,s,c,b,t

∑

^∂^µ ⁺îg^s^Gâ^µ^Tâ^)q⁻^m^q^qq

input: g_s , m_q

output: hadron properties

hadron interactions (if we are lucky)

V = N_L³ ×N_T a≈0.1 fm

a

€

Dx e

^{i S}^/^!

∫ ∫ DG Dq Dq e

^{i SQCD} ^/^!

€

S= ∫ dt L[x(t)]

€

S_QCD =

∫

d⁴x L_QCD[G(x), q(x)^,^q⁽^x^)]

quantum m. quantum ﬁeld theory

(6)

La3ce setup

•  Wilson-‐clover quarks

•  Fermilab method for c and b :

[El Khadra, Kronfeld et al, 1997]

Rest hadron energies have sizable discreAzaAon errors but these largely cancel in spli3ngs.

Only spli3ngs with respect to a chosen reference mass are compared to experiment.

•  evaluaAng Wick contr.: disAllaAon (Ensemble 1) [Peardon et. al., HSC, 2009]

stochasAc disAllaAon (Ensemble 2) [Morningstar et al., 2011]

PACS-‐CS

(7)

7

O = q Γq , (q Γ

₁

q )

_p^!₁

(q Γ

₂

q )

^!_p₂

, [qq ][qq]

C

_ij

(t ) = 0 O

_i

(t) O

_j⁺

(0) 0 =

n

∑ ^Z

ⁱⁿ

^Z

^j^n*

^e

^−Eⁿ ^t

^Z

ⁱⁿ

⁼ ⁰ ^O

ⁱ

ⁿ

Example: meson channel with given J^PC

Discrete energy spectrum from correlators

All physical states with given J^PC appear as energy levels E_n in principle : single parAcle, two-‐parAcle,...

channel : "eigenstates"

J^PC = 0⁺, sc: D_s₀ ,DK

J^PC =1⁺⁺, cc: χ_c1,X(3872), DD*, J^PC =1⁻⁻, su: K*, Kπ

J^PC =1⁺⁺, ud : a₁, ρπ

Two-‐meson states:

•  In experiment: two-‐meson decay products with conAnuous E.

•  On la3ce: discrete E due to ﬁnite L and periodic BC: p=n 2π/L Bound state and narrow resonance:

typically lead to extra energy level (in addiAon to two-‐meson levels)

Energies and overlaps extracted using GEVP

(8)

ExtracAng sca]ering matrix from discrete spectrum Luscher’s method

E (L ) = m

_M² ₁

+ !

p

₁ⁿⁱ²

+ m

_M² ₂

+ !

p

₂ⁿⁱ²

+ Δ E, !

p

_1,2ⁿⁱ

= 2 π L

n, ! n ∈ N

³

or

E (L ) = m

_M² ₁

+ !

p

₁²

+ m

_M² ₂

+ !

p

₂²

, !

p

_1,2

≠ 2 π L

n !

due to strong int.

(9)

9

x x x

x x

E(L)

ExtracAng sca]ering matrix from discrete spectrum:

Luscher’s method

δ(E) E(L)

energies from la3ce with spaAal extent L sca]ering phase shiws

at inﬁnite volume

(10)

ElasAc sca]ering of two mesons

E_n=E_M1(p)+E_M2(-‐p)

at total momentum P=0

E

_n

(L)

p δ(p)

Luscher’s eq.

cot (p) = 2 Z

⁰⁰

(1; (

^pL_2⇡

)

²

) Lp p

⇡

M₁(p)

M₂(-‐p)

or 1/K(p) inverse

K-‐matrix

Luscher’s formalism renders sca]ering informaAon on the real axis for p²>0: above threshold

p²<0: some region below threshold

(11)

ElasAc sca]ering of two mesons

E_n=E_M1(p)+E_M2(-‐p)

at total momentum P=0

E

_n

(L)

p δ(p)

Luscher’s eq.

Sca]ering matrix for parAal wave l:

Bound state:

cot[δ(p_B)] = i , p_B²<0

m_B=E_M1(p_B)+E_M2(-‐p_B)

Resonance (of Breit-‐Wigner type):

M₁(p)

M₂(-‐p)

inverse K-‐matrix

(12)

Bound states

slightly below threshold

(13)

Scalar meson D _s0 & historic single hadron approx.

O : s c

Almost all simulaAons before 2013 used

“single-‐hadron” approximaAon for D_s0:

One of the problems:

Is the level D_s0 or perhaps D(0)K(0) ?

Energies extracted from a la3ce simulaAon

(14)

Aims to extract also two-‐meson states E_n

Scalar meson D

_s0

and DK sca]ering J

^P

=0

⁺

€

^p^!⁼ ⁿ^!²_L^π

K ( − !

p )

€

C

_ij

(t ) = 0 O

_i

^(t ⁾ O

_j⁺

^{(0) 0}

€

C

_ij

(t) = A

_n^ij

e

^−Eⁿ^t

n

∑

€

D( ! p )

O=s c

O=D(0)K(0) ≈[d

γ

₅^c]₀ ^[s

γ

₅^d^]₀ O=D(−1)K(1)≈[d

γ

₅^c]₁ ^[s

γ

₅^d^]₋₁

(15)

•  δ for DK sca]ering in s-‐wave extracted using Luscher's relaAon

•  InterpolaAon: eﬀecAve range

a₀<0 indicates a state below th.

•  pole posiAon of D_s0^*(2317)

^D

_s0^*

(2317) and DK sca]ering

15

D. Mohler, C. Lang, L. Leskovec, S.P. , R. Woloshyn:

1308.3175, PRL : m_π≈156 MeV, L≈2.9 fm, Nf=2+1 pcotδ(p)= 1

a₀ +1 2r₀p² a₀ =−1.33±0.20 fm r₀ =0.27±0.17 fm

T ∝[cot

δ

−i]⁻¹ = ∞, cot

δ

⁽^p_B⁾ =i p_B cot

δ

= (i | p_B |)i = −| p_B |= 1

a₀ − 1

2 r₀ | p_B |² m_D^lat,_s₀^L→∞ = E_D(p_B)+ E_K(p_B)

•  E=E_K(p)+E_D(-‐p) renders p Energies from a la3ce

(16)

^D

_s0

and D

_s1

below DK and D

^*

K thresholds

[D. Mohler, C. Lang, L. Leskovec, S.P. , R. Woloshyn:

1308.3175, Phys. Rev. Le] 2013 1403.8103, PRD 2014]

•  D_s0 and D_s1 have been observed experimentally

•  Quark models expected them above thresholds but they were found below them

•  Our post-‐dicAons agree with measured masses

•  Threshold eﬀect lowers their masses

•  ComposiAon of resulAng D_s0 and D_s1 analyzed via Weinberg-‐type compositness condiAons:

[MarAnez Torres, E. Oset, S.P., A. Ramos:

1412.1706]

(17)

InteracAons of charmed mesons with light pseudoscalars

(1) Five channels that do not include Wick contracAons are simulated (2) Sca]ering lengths for four m_π extracted

(3) simultaneous ﬁt using SU(3) unitarized ChPT is performed and LEC's are determined (4) using these LEC's indirect predicAons for:

•  sca]ering length of two resonant-‐channels with contracAons

•  DK (S=1,I=0): pole in the ﬁrst Riemann sheet found

L. Liu, Orginos, Guo, Hanhart, Meissner, 1208.4535, PRD, m_π≈300-‐620 MeV, Nf=2+1

€

a=lim_p_→₀tanδ(p) p

€

DK (S=−1,I=1)

€

DK (−1, 0)

€

DK (S=2,I= ¹₂)

€

Dπ (0,³₂)

€

DK (1,1)

€

Dπ (S=0,I=¹₂)

D_s0*(2317) m Γ [D_s0*gD_sπ]

indirect lat 2315 ⁺¹⁸-‐28 MeV 133±22 keV

exp 2317.8 ±0.6 MeV < 3.8 MeV

points: Mohler et al, PRL 2013 line: Liu et al PRD 2012

(18)

Mass predicAon for missing B _s0 and B _s1

QuanAAes shown:

•  B_s1’ and B_s2 agree well with exp

•  B_s0and B_s1 are predicAons for yet unobserved states (errors contain staAsAcal and several sources of systemaAcal uncertainAes)

Ensemble (2), m_π=156 MeV

[C. Lang, D. Mohler, S.P. ,

(19)

X(3872) , J ^PC =1 ⁺⁺ , charmonium-‐like

•  First charmonium-‐like state discovered

[Belle, PRL, 2003]

•  sits within 1 MeV of D

⁰

D

^0*

threshold 8 MeV below D

⁺

D

^*-‐

threshold

•  believed to have a large molecular D

⁰

D

^0*

Fock component

•  Γ < 1.2 MeV

•  decays to I=0, 1 equally important X(3872) g J/Ψ ω ( I=0 )

X(3872) g J/Ψ ρ ( I=1 )

[LHCb, PRL 2013]

isospin breaking eﬀects my be important

(20)

X(3872), 1

⁺⁺

, I=0

O: c c, DD*=(cu)(uc)+(cd)(dc), J /

ψ ω

= (cc)(uu+dd)

[S.P. and L. Leskovec, Phys.Rev.Le]. 2013 ]

•  all Wick contracAons calculated using

disAllaAon method [Peardon et al. 2009]

•  charm annihilaAon

contracAons not used in analysis

charm annihilaAon

(21)

X(3872) below DD* threshold, I=0

O: c c, DD^*, J /

ψ ω

[S.P. and L. Leskovec : 1307.5172, Phys. Rev. Le]. 2013]

Ensemble (1), m_π≈266 MeV, Nf=2

X(3872) m -‐ (m_D0+m_D0*) lat -‐ 11 ± 7 MeV exp -‐ 0.14 ± 0.22 MeV

X(3872) appears only if both cc and DD* interp. used.

AssumpAons/approximaAons:

•  charm Wick annihilaAon omi]ed

•  DD* sca]ering analyzed assuming J/ψω is decoupled (good evidence for that from the la3ce data)

•  m_u=m_d

•  δ for DD* sca]ering in s-‐wave

extracted using Luscher's relaAon

•  δ interpolated near threshold

•  pole found in the sca]ering matrix

T ∝[cotδ−i]⁻¹ =∞, cotδ(p_BS)=i m_D

s0

lat,L→∞

= E_D(p_BS)+E_K(p_BS)

(22)

New evidence for X(3872) &

inﬂuence of diquark anA-‐diquark interpolators

O: c c, DD^*, J /

ψ ω

^,

χ

_c1

η

^,

η

_c

σ

^{, [}^cu^]_3c^[cu]_3c^{, [cu}^]₆_c^[cu]₆_c

no cc

X(3872) found

only if cc in the basis

X(3872) not found if cc not in the basis,

although [cu][cu] in the basis

[M. Padmanath, C.B. Lang, S.P.,

(23)

Mesons above threshold – resonances

O = q Γq,

(q Γ

₁

q)

_p^!₁

(q Γ

₂

q)

_p^!₂

=

_M1(^p^!_{1 ) M2 (}^p^!_{2 )}

(24)

Almost all hadrons are resonances above threshold

well below strong decay th.

€

u u

€

s u

€

c u

€

uud

€

c c

well below open

charm decay th.

rigorous treatment a]empted

(25)

P≠0: s=E

²

-‐P

², Luscher-‐type relaAon:

s g δ(s)

ρ resonance

[Lang, Mohler, S.P. , Vidmar, PRD 2011]

m_π≈266 MeV

[HSC, PRD 2013]

m_π≈400 MeV

SimulaAon also by CP-‐PACS, PACS-‐CS, QCDSF, ETMC

[Pelissier, Alexandru, PRD 2013],

m_π≈300 MeV

[Pelissier, Alexandru, 1111.2314]

(26)

^D

₀^*

(2400) resonance in Dπ sca]ering: J

^P

=0

⁺

, I =1/2

All states with J^P=0⁺ appear in lat. spectrum:

• 

D

_o^*

(2400)

•  D(p) π(-‐p)

with p=n 2π/L : "two-‐parAcle" states horizontal lines indicate their energies in absence of interacAon

€

O : u c

D(p )! π(-! p ) ≈[d γ₅c] [u γ₅d]

D(p) π(-‐p)

Rigorous relaAon [M. Luscher , 1991]:

E g δ(E) phase shiw for Dπ sca]ering in s-‐wave

m_R and g or Γ for D₀^*(2400)

"rigorous" treatment illustrated on this example

€

p = n2π L

pcot

ps = 1

g²(m²_R s)

Γ(s)= p s g²

(27)

D-‐meson resonances in Dπ and D*π

g is compared to exp instead of Γ (Γ depends on phase sp. and m_π)

D₀^*(2400) m -‐ 1/4(mD+3 mD*) g

lat 351 ± 21 MeV 2.55 ± 0.21 GeV exp 347 ± 29 MeV 1.92 ± 0.14 GeV

D₁(2430) m -‐ 1/4(mD+3 mD*) g

lat 381 ± 20 MeV 2.01 ± 0.15 GeV exp 456 ± 40 MeV 2.50 ± 0.40 GeV

€

Γ(E) ≡ g² p E²

ﬁrst la3ce result for strong decay width of a hadron containing charm quark

[D. Mohler, S.P. , R. Woloshyn: 1208.4059, PRD]

•  m_π≈266 MeV, L≈2 fm, Nf=2

J

^P

=0

+ :

D π J

^P

=1

+ :

D* π

(analysis of spectrum in this case is based on an assumpAon given in paper below)

(28)

Kπ , I=1/2: p-wave phase shift

[S.P. ,Lang, Leskovec, Mohler,

K*(892) resonance Kπ

irreps where p-‐wave does not mix with s-‐wave, Lusher-‐type rel.:

[Lekovec, S.P. PRD 2012]

p³cot δ / s^1/2

(29)

•  qq, Kπ, Kη interpolators

•  a number of diﬀerent 0<P≤2

•  for each E_n: one determinant equaAon for many unknowns

•  T-‐matrix parametrized to get around this problem

•  the locaAon of poles of T-‐matrix in complex plain is given below

•  K*(892) and κ are below threshold for this m_π

•  K₀^* , K₂^*are resonances

•  m_π=391 MeV, N_L=16, 20, 24

[Dudek, Edwards, Thomas, Wilson, HSC, 1406.4158, PRL; 1411.2004]

Resonances in Kπ, Kη coupled channels

locaAon of poles in T matrix in complex plane

(30)

•  SimulaAng sca]ering:

ρ π in 1

⁺⁺

channel to extract a

₁

(1260)

ω π in 1

^+-‐

channel to extract b

₁

(1235)

•  One moAvaAon: COMPASS claim for a

₁

’(1420) from f

₀

(980) π

[1312.3678]

Our spectrum supports only one a₁ below 1.8 GeV (not two)

•  m

_π

≈266 MeV, L≈2 fm, Nf=2 , P=0

[Lang, Leskovec, Mohler, S.P. , 1401.2088, JHEP]

PDG

exp 1.40(2) [Basdevant, Berger, 1501.04643]

•  ρ and ω assumed to be stable, good approx. for given simulaAon parameters : [Roca, Oset, 1201.0438]

•  going beyond that approximaAon will be very challenging

•  3-‐parAcles: [Hansen, Sharpe 1311.4848; Polejaeva, Rusetsky, 1203.1241; Briceno, Davoudi, 1212.3398,.... ]

Lightest axial resonances a ₁ and b ₁

€

Γ(E) ≡ g² p E²

(31)

Resonance ψ(3770) in p-‐wave DD sca]ering

31

¯

cc, J^{P C} = 1 : J/ , (2S) below DD¯ threshold

(3770) lowest state above threshold

“ψ(2S)” “Ψ(3770)” “D(1)D(-‐1)”

Γ^exp ~ 27 MeV

(s) = g² 6⇡

p³ s

“ψ(2S)” “Ψ(3770)” “D(1)D(-‐1)”

ﬁt (ii):

cubic ﬁt in p² taking ψ(2S) as pole in DD sca]ering

cubic ﬁt (ii) Mass [MeV] g (no unit)

Ensemble (1) 3774 ±6±10 19.7 ±1.4

Ensemble (2) 3789 ±68±10 28 ± 21

Experiment 3773.15± 0.33 18.7 ± 1.4

(32)

Conclusions

Meson spectrum from la3ce:

Evidence found for states with non-‐exoAc ﬂavor:

•  states well below th. : charmonium , D, π, K ... and all the others

•  shallow bound states : D

_s0

, D

_s1

, B

_s0

, B

_{s1 ,}

X(3872) with I=0

•  resonances via BW : ρ, K*, D

₀^*

, D

₁

, a

₁

, b

₁

, Ψ(3770)

require simulaAon of strong sca]ering

(33)

Backup slides

(34)

X(3872) channel: I=1, J

^PC

=1

⁺⁺

S. P. and L. Leskovec : 1307.5172 PRL 2013, m_π≈266 MeV, Nf=2

Only expected two-‐parAcle states observed.

No candidate for X(3872) with I=1 found.

In agreement with experiment that does not ﬁnd charged X either.

The simulaAon is done in the isospin limit m_u=m_d. The absence of I=1 state for m_u=m_dis in agreement with two interpretaAons:

(1)

(2) X(3872) pure I=0 state

isospin breaking decay X(3872) -‐> J/ψ ρ (I=1) is due to isospin spli3ng D⁰D^0* , D⁺D^-‐*

€

a_I₌₁(m_u = m_d) = 0 a_I₌₁(m_u ≠ m_d) << a_I₌₀

(35)

^D

_s0

from indirect simulaAon

(1) Five channels that do not include Wick contracAons are simulated (2) Sca]ering lengths for four m_π extracted

(3) simultaneous ﬁt using SU(3) unitarized ChPT is performed and LEC's are determined (4) using these LEC's indirect predicAons for

D_s0 channel with DK (S=1,I=0):

pole in the ﬁrst Riemann sheet found

L. Liu, Orginos, Guo, Hanhart, Meissner, 1208.4535, PRD, m_π≈300-‐620 MeV, Nf=2+1

€

a=lim_p_→₀tanδ(p) p

€

DK (S=−1,I=1)

€

DK (−1, 0)

€

DK (S=2,I= ¹₂)

€

Dπ (0,³₂)

€

DK (1,1)

D_s0*(2317) m Γ [D_s0*gD_sπ]

indirect lat 2315 ⁺¹⁸-‐28 MeV 133±22 keV

exp 2317.8 ±0.6 MeV < 3.8 MeV

€

DK (S=1,I=0)

points:

D. Mohler, S.P. et al 1308.3175, PRL

Shallow bound states and resonances from la3ce QCD Sasa Prelovsek