Non-perturbative pion dynamics for the X (3872) Vadim Baru Institut - - PowerPoint PPT Presentation

Non-perturbative pion dynamics for the X(3872)

Vadim Baru

Institut für Theoretische Physik II, Ruhr-Universität Bochum Germany Institute for Theoretical and Experimental Physics, Moscow, Russia

in collaboration with

• E. Epelbaum, A. Filin, C. Hanhart,
• Yu. Kalashnikova, A. Kudryavtsev, U.-G. Meißner

and A. Nefediev

related articles: PLB 726, 537 (2013), PRD 84, 074029 (2011)

Meson 2014, Krakow

X(3872). Known facts

⟹ isospin violation

• Quantum numbers: 1++ and 2-+ from BABAR/Belle angular distributions
• LHCb (2013): 2-+ is rejected with CL = 8 σ

Discovered by the Belle: B± → K±J/Ψ π+π− Observed: CDF, D0, BABAR, LHCb, BESIII MX= 3871.68 ± 0.17 MeV PDG (2013)

• MeV Belle

ΓX < 2.3

• X → J/Ψγ

= ⇒ C = +

• Br(X → J/Ψω)/Br(X → J/Ψρ) ≈ 0.8 ± 0.3 Belle (2003), BABAR(2010)

Belle

) (GeV)

• M(J/

3.82 3.84 3.86 3.88 3.9 3.92 Events / ( 0.005 GeV ) 5 10 15 20 25 30 35

b)

• ⟹ X is 1++

X(3872). Mechanisms

⟹ X is the S-wave bound state of with EB= 0.12 ± 0.26 MeV MX= 3871.68 ± 0.17 MeV; MD0D0*= 3871.80 ± 0.35 MeV

• Conventional charmonium:
• Exotics: tetraquark, molecular, mixture....

tetraquarks Maiani,... molecular

D

q̅ q

D*

q̅ q

q̅ q q̅ q

ηc2(11D2) ruled out by LHCb

compact state

☛ Strong hadronic interactions with

large s-wave scatt. length

☛ natural effective range ☛ specific analytic properties:

unitarity cut

D* ̅ D

̅

DD*

Okun, Voloshin (1976), many works Weinberg (1963-65)

χ0

c1(23P1)

From NN system to charm sector

π + heavy mesons NN force ̅ Charmonium force π + heavy mesons D* D D*

̅

D

Prediction of the molecular state in analogy to the deuteron D* ̅ D

Okun, Voloshin (1976), Törnqvist (1991)

From NN system to charm sector

π + heavy mesons NN force ̅ Charmonium force π + heavy mesons D* D D*

̅

D

Prediction of the molecular state in analogy to the deuteron D* ̅ D

Okun, Voloshin (1976), Törnqvist (1991)

binding momentum range of interaction static OPE 45 MeV mπ good approximation 20-30 MeV can go on shell ⟹ Im part questionable ⟹ 3-body unitarity is spoiled

Small scales << mπ ⟹ NR kinematics

µ = p 2mπ(mD∗ mD mπ) ' 45

X 𝜌0D0D0 ̅ ̅ D0D0* 𝜌∓D0D± 𝜌0D-D+

Thresholds

• 0.5
• 7

D∓D*± 2.5 3 8 NNthr d NNπ

• 2

140 MeV MeV

𝜌0D0D0

̅

NN X(3872) MeV

X(3872) as a bound state. Status.

Pionless:

• Asymptotic behavior of the X w.f. Voloshin (2004)
• Contact theory AlFiky et al. (2006), Nieves, Valderrama (2012)

D*

̅

D

X(3872) as a bound state. Status.

Pionless:

• Asymptotic behavior of the X w.f. Voloshin (2004)
• Contact theory AlFiky et al. (2006), Nieves, Valderrama (2012)

Pionful:

• X-EFT: resummation of LO contact operators + perturbative pions

—similar to KSW in NN sector

Flemming et al. (2007), Hammer et al. (2014), Mehen, Braaten et al. (2010-2011)

• Phenomenological deuteron-like studies with nonperturb. static OPE and formfactors

For example: Liu et al. (2008), Thomas and Close (2008), Törnqvist (2004)

D*

̅

D

X(3872) as a bound state. Status.

Pionless:

• Asymptotic behavior of the X w.f. Voloshin (2004)
• Contact theory AlFiky et al. (2006), Nieves, Valderrama (2012)

Pionful:

• X-EFT: resummation of LO contact operators + perturbative pions

—similar to KSW in NN sector

Flemming et al. (2007), Hammer et al. (2014), Mehen, Braaten et al. (2010-2011)

• Phenomenological deuteron-like studies with nonperturb. static OPE and formfactors

For example: Liu et al. (2008), Thomas and Close (2008), Törnqvist (2004)

D*

̅

D

̅ 𝜌DD Goals of our study

• Investigate the role of 3-body dynamics on a near threshold resonance

D0 ¯ D∗0, D+ ¯ D∗−

• Investigate the role of coupled channel effects:
• Check the validity of the static pion approximation
• Study the dependence of the X binding energy on the light quark masses

(chiral extrapolations)

Needed to explain isospin violation: Br(J/ψρ) ≃ Br(J/ψω) Gamermann, Oset (2009)

• Formalism. Faddeev-type integral Eqs. for 1++

Channels: D* ̅ D D* ̅ D D* ̅ D = + V V a00 a00

|0i = D0 ¯ D∗0, |¯ 0i = ¯ D0D∗0, |ci = D+D∗−, |¯ ci = D−D∗+

ann0

00 (p, p0, E) = λ0V nn0 00 (p, p0) −

X

i=0,c

λi Z d3s ∆i(s)V nm

0i (p, s)amn0 i0

(s, p0, E) ann0

c0 (p, p0, E) = λcV nn0 c0 (p, p0) −

X

i=0,c

λi Z d3s ∆i(s)V nm

ci

(p, s)amn0

i0

(s, p0, E),

• Partial waves of DD*: ⟹ projection operators

3S1,3D1

• Δ0 and Δc — propagators of the states ∣0> and ∣c>

a0 = (a00 − ac0)/2

• ̅

DD* — the X-amplitude

isospin coefficients

potential within chiral EFT

• Same 3-body cut appears due to dressing the D* propagator by πD loops

VEF T = V (0) + χV (1) + χ2V (2) + · · ·

χ = q ΛχP T

small scale: bind. momentum, range large scale: mρ

C0

3S1 3S1

• p

p p’

• p’
• p-p’

+

VOP E(~ p, ~ p 0) = g2 (~ p · ~ "d)(~ p 0 · ~ "d) M − (2mD + m⇡) − (~

p+~ p0)2 2mπ

−

p2 2mD − p02 2mD + i0

• Coupling constant g from Γ(D∗0 → D0π0) = 42 KeV

OPE LO:

3-body DD̅π prop.

3S1 3D1 3S1 3D1

V(0) =

̅

DD*

Renormalization of OPE

2 4 6

[GeV]

• 2
• 1

1 2

C0 [ GeV

• 2]
• Similar to NN

Lepage (1997), Nogga et al. (2005), Epelbaum et al. (2006-2009), ....

• VOPE const ⟹ divergent integrals ⟹ regularize, e.g., with cutoff Λ ⟹

renormalize tuning C0(Λ) to reproduce the binding energy

p→∞

− − − − →

• Renormalization group limit cycle Nogga et al. (2005), Bedaque et al.(1999), Braaten and Phillips (2004)

Once renormalized, observables should not depend on Λ within the range of applicability!

0.2 0.4 0.6 0.8 1 1.2

EB [MeV]

0.02 0.04 0.06 0.08 0.1 0.12

[MeV]

Partial width of the X(3872) due to the decay to

𝜌0D0D0

̅

∆−1

0 (s)

aSS

00 (s, p)

D∗0( ¯ D∗0) D0( ¯ D0) π0 ¯ D0(D0) D0( ¯ D0) π0 ¯ D0(D0) D∗0( ¯ D∗0) D∗0( ¯ D∗0) ¯ D0(D0)

• 0.8
• 0.6
• 0.4
• 0.2

E [MeV] 1 2 3 4 d Br/d E x 10

• 4

static OPE

• ur full with nonpert. pions

X-EFT band at NLO, Flemming et al.(2007)

LO, Voloshin (2004)

Conclusions:

• Perturbative inclusion of pions is justified
• Static approx. with nonpert. pions fails!
• Keeping 3-body dynamics in pionfull

schemes is mandatory production rate near the X(3872) pole 𝜌0D0D0

̅

EB=0.5 MeV

VB, Filin, Hanhart, Kalashnikova, Kudryavtsev, Nefediev (2011)

for a study of the FSI see Guo et al (2014)

D ̅ D

Chiral Extrapolations of the X(3872) with mπ

Strategy: vary all quantities with mπ, expand around physical pion mass

mph

π

C0(Λ, mπ) = C0(Λ) 1 + f(Λ)m2

π − mph π 2

M2 ! + O ✓δm4

π

M4 ◆

• f (Λ) is needed to absorb extra Λ-dependence when mπ 6= mph

π

• Estimate f (Λ): from two-pion exchange ⟹ ∣f (Λ)∣∼1, more conservatively: f (Λ) ∈ [-5, 5]
• could be fixed if we knew the slope

(∂EB/∂mπ)

• mπ=mph

π

from lattice

VB, Epelbaum Filin, Hanhart, Meißner, Nefediev (2013)

Chiral Extrapolations of the X(3872) with mπ

Strategy: vary all quantities with mπ, expand around physical pion mass

mph

π

C0(Λ, mπ) = C0(Λ) 1 + f(Λ)m2

π − mph π 2

M2 ! + O ✓δm4

π

M4 ◆

• f (Λ) is needed to absorb extra Λ-dependence when mπ 6= mph

π

• Estimate f (Λ): from two-pion exchange ⟹ ∣f (Λ)∣∼1, more conservatively: f (Λ) ∈ [-5, 5]
• could be fixed if we knew the slope

(∂EB/∂mπ)

• mπ=mph

π

from lattice

VB, Epelbaum Filin, Hanhart, Meißner, Nefediev (2013) 0.5 1 1.5 2 ξ = mπ / mπ

ph

• 8
• 6
• 4
• 2
• EB [MeV]

negative slope, case 1), full positive slope, case 2), full

1) If ⇒ f (Λ)<0 ⟹ bound state disappears quickly with mπ

• growth. Dominated by short-range physics

2) If ⇒ f (Λ)>0 ⟹ The X is bound deeper with mπ growth. Strong influence of pion dynamics

(∂EB/∂mπ)

• mπ=mph

π < 0

(∂EB/∂mπ)

• mπ=mph

π > 0

positive slope, case 2), contact negative slope, case 1), contact

Conclusions:

physical pion mass

Chiral Extrapolations of the X(3872) with mπ

Strategy: vary all quantities with mπ, expand around physical pion mass

mph

π

C0(Λ, mπ) = C0(Λ) 1 + f(Λ)m2

π − mph π 2

M2 ! + O ✓δm4

π

M4 ◆

• f (Λ) is needed to absorb extra Λ-dependence when mπ 6= mph

π

• Estimate f (Λ): from two-pion exchange ⟹ ∣f (Λ)∣∼1, more conservatively: f (Λ) ∈ [-5, 5]
• could be fixed if we knew the slope

(∂EB/∂mπ)

• mπ=mph

π

from lattice

0.5 1 1.5 2

ξ = mπ / mπ

ph

• 8
• 6
• 4
• 2
• EB [MeV]

negative slope, case 1), full positive slope, case 2), full

Conclusions ctd :

▶ First lattice measurement of the X(3872) Prelovsek and Leskovec PRL(2013), talk on Saturday EB = -11 ± 7 MeV mπ =266(4) MeV ⟹ scenario 2) seems to be preferred

positive slope, case 2), contact negative slope, case 1), contact

VB, Epelbaum, Filin, Hanhart, Meißner, Nefediev (2013)

Comparison to other EFT studies

• chiral extrapol. of the binding energy with mπ to NLO in X-EFT

Conclusion : qualitative agreement with our results

M.Jansen et al. (2013)

Comparison to other EFT studies

• chiral extrapol. of the binding energy with mπ to NLO in X-EFT

Conclusion : qualitative agreement with our results

M.Jansen et al. (2013)

• chiral extrapol. of the binding energy with mπ with non perturbative pions
• P. Wang and X.G. Wang, PRL (2013)

Claims:

1) OPE alone is sufficient to bind D0D̅*0 2) Short range contr’n (CT) — a small correction 3) X pole disappears when mπ grows

Comparison to other EFT studies

• chiral extrapol. of the binding energy with mπ to NLO in X-EFT

Conclusion : qualitative agreement with our results

M.Jansen et al. (2013)

• chiral extrapol. of the binding energy with mπ with non perturbative pions
• P. Wang and X.G. Wang, PRL (2013)

Claims:

1) OPE alone is sufficient to bind D0D̅*0 2) Short range contr’n (CT) — a small correction 3) X pole disappears when mπ grows Example: in cutoff regularization X has the same EB if C0(Λ)=0 (Λ≃1 GeV) or C0(Λ)=∞ (Λ≃3 GeV)

• 3) is in conflict with recent lattice results Prelovsek and Leskovec PRL(2013)
• All claims are scheme dependent. OPE due to its short ranged part is always tuned with CT

Our opinion

Conclusions

• We studied the role of nonperturbative long-range dynamics on

the X(3872) within a molecular picture

▶ 3-body effects are important in pionful approaches but pion is

essentially perturbative

̅ 𝜌DD

▶ static pion approximation is not appropriate

• The dependence of the X-binding energy on mπ is highly nontrivial —

may provide insights on the binding mechanism

▶ the X bound state may either disappear or get more bound depending

• n the interplay of long- and short-range forces

▶ First lattice results: more bound ▶ Γ(X → D0 ¯

D0π0) = 44 KeV for EB=0.5 MeV

▶ More accurate lattice data with better controlled finite volume

corrections would be important