Bruno El-Bennich

LFTC, Universidade Cruzeiro do Sul, São Paulo

Institute for Theoretical Physics, State University of São Paulo

Charm Physics in a unified nonperturbative approach

Charming motivation ...

2

• Important hadronic effects stem from non-perturbative QCD contributions to strong and weak decay amplitudes, whether in beauty and charm decays.

• Charm decays are particularly afflicted by hadronic uncertainties since corrections to the decay

amplitudes of order ΛQCD ⁄mc are not under control.

• Charm physics actively studied by LHCb, BaBar, Belle, BES, CLEO, FOCUS, RHIC ...

• PANDA at GSI will contribute to further development of charm physics.

• Charmonium production rate at RHIC is sensitive to existence and properties of the intermediate “quark-gluon plasma” as well as to final-state interactions ( decays).J/

A few examples of hadronic charmed decays

k + Q!

Q! = PD! ! PD

k

PD

PD!

k + PD!

A(D� ⇥ D⇥) = ��D�µ (pD�)Mµ(p2

D, p2D�) := ��D�

µ (pD�)pµD gD�D⇤

Mµ(p2D, p2

D�) = Nc tr� � d4k

(2⇥)4�D(k;�PD)Sc(k + PD�)i�µ

D�(k;PD�)Su(k)�⇤(k;�Q⇤)Su(k + Q⇤)

Strong decay D* → Dπ

The coupling gH*H! can be calculated even if decay is kinematically forbidden: B*: mB* – mB ≈ 46 MeV

Coupling yields D* width

4

5

Unphysical decay B* → Bπ to extract effective heavy quark coupling

• Dynamics is constrained by heavy quark symmetry.

• Blind to the heavy quark flavor and spin.

• Heavy pseudoscalar and vector mesons are

mass degenerate.

• Can be improved upon — take into account light

degrees of freedom, chiral symmetry breaking

⇒ HMChPT.

Heavy Quark Effective Theory

R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Rept. 281, 145 (1997)

Lheavy = �traTr[Haiv ·DbaHb] + g traTr[HaHb�µAµba�5]

DµbaHb = ⇤µHa �Hb

12[⇥†⇤µ⇥ + ⇥⇤µ⇥†]ba ;

Aabµ =

i

2�⇥†⇤µ⇥ � ⇥⇤µ⇥†

�ab

;

Ha(v) =1 + /v

2�P �a

µ (v)�µ � P a(v)�5

�;

⇥ = exp(i�/f0�) ;

� is matrix of N2f � 1 pseudo-Goldstone boson.

• D* ➝ Dπ can be generalized to three-point functions for vector mesons with off-shell momenta.• Couplings D!D, D*!D, D*!D* needed in final-state interactions with intermediate D – D* states.

Generalization to higher spin structures

D*ρD coupling :

with M ⌘ ✏⌫⇢0(q)✏µD⇤(p1)✏

⇢D⇤(p2)M

µ⌫⇢(p21, p

22, q

2) =

= ✏⌫⇢0(q)✏µD⇤(p1)✏

⇢D⇤(p2)Nc tr

Z ⇤ d4k

(2⇡)4¯

�

⇢D⇤(k;�p2)Sc(k + p1)�

µD⇤(k; p1)Su(k)¯�

⌫⇢0(k;�q)Su(k + q)

�

µD⇤ : D⇤

-meson Bethe-Salpeter amplitude

�

⌫⇢0 : ⇢0-meson Bethe-Salpeter amplitude

✏µi : Polarization vector

D*ρD* coupling :

✽

✽

M ⌘ "⇢↵"D⇤

� M↵�(pD⇤, p⇢) = ✏↵�µ⌫"↵(p

D⇤)"�(p

⇢) pD⇤

µ q⇢⌫gD⇤⇢D

mD⇤

Mµ⌫⇢(p1, p2) = G1 gµ⌫p⇢2 +G2 g

µ⇢p⌫2 +G3 g⌫⇢pµ2 +G4 g

µ⌫p⇢1 +G5 gµ⇢p⌫1 +G6 g

⌫⇢pµ1 +G7 pµ2p

⌫2p

⇢2

+ G8 pµ1p

⌫2p

⇢2 +G9 p

µ2p

⌫1p

⇢2 +G10 p

µ2p

⌫2p

⇢1 +G11 p

µ1p

⌫1p

⇢2 +G12 p

µ1p

⌫2p

⇢1 +G13 p

µ2p

⌫1p

⇢1 +G14 p

µ1p

⌫1p

⇢1

7

X(3872)X(3872) X(3872)

X(3872)X(3872)X(3872)

Why higher spin structures?

Example: decay of recently the discovered resonance X(3872) ( )

Main decay modes are X(3872)! D0D0⇤(D0⇡0), J/ ⇡+⇡�, J/ �, J/ ⇡+⇡�⇡+...

➪ clear isospin violation

JPC = 1?+

Exotic nuclear bound states: mass shift in nuclear matter

G. Krein, A.W. Thomas & K. Tsushima (2010)

• J/Ψ mass decrease in nuclear matter estimated

to 4 – 7 MeV with QCD sum rules.

• An interesting challenge are properties of

D and D(*) mesons in medium:

Possible formation of meson-nuclear bound states.

Enhanced dissociation of J/Ψ in nuclear matter (heavy nuclei).

Enhancement of production in antiproton-nucleus collisions.

D(D)

D(D)

K. Saito, K. Tsushima and A. W. Thomas, Prog. Part. Nucl. Phys. 58, 1 (2007)

L⌅DD = ig⌅DD �µ[D(⇤µD)� (⇤µD)D],

L⌅DD� =g⌅DD�

m⌅⇥�⇥µ⇤(⇤��⇥)

�(⇤µD�⇤)D + D(⇤µD�⇤)

�,

L⌅D�D� = ig⌅D�D���

�µ[(⇤µD�⇤)D�⇤ � D�⇤(⇤µD�

⇤)�

+�(⇤µ�⇤)D�

⇤ � �⇤(⇤µD�⇤)

�D�µ + D�µ [�⇤(⇤µD�

⇤)� (⇤µ�⇤)D�⇤ ]

�

8

J/

D-meson interaction with nucleons

9

Antiproton annihilation on the deuteron (PANDA @ FAIR) Meson exchange — effective Lagrangians

10

SU(4) symmetry used ....

Jülich model:A. Müller-Groeling et al. NPA 513, 557 (1990) M. Hoffmann et al. NPA 593, 341 (1995)D. Hadjimichef, J. Haidenbauer and G. Krein, PRC 66, 055214 (2002)

SU(4) symmetry: gD⇥D = gD⇤D =gKK⇥ = 12g��⇥

?

PS → S decays and rescattering

• So far there are no BSAs for scalar mesons

• More modeling and assumptions involved

• Scalar mesons tend to be broad (background)

Important in Dalitz plot analyses of non-leptonic three-body D and B decays

D(s) � f0(980)D(s) � K�

0 (1430)

B(s) � f0(980)B(s) � K�

0 (1430)

D(s) � (��)S (KK)S

D(s) � (K�)S

B(s) � (��)S (KK)S

B(s) � (K�)S

11

B.E. , O. Leitner, J.-P. Dedonder and B. Loiseau (2009)

Dyson-Schwinger and Bethe-Salpeter framework

Dyson-Schwinger equations

• Well suited to Relativistic Quantum Field Theory

• Non-perturbative continuum approach to QCD

• Hadrons as composites of Quarks and Gluons

• Qualitative and quantitative importance for:

Dynamical Chiral Symmetry Breaking

Generation of fermion mass from nothing

Quark & Gluon Confinement

• Method yields Schwinger functions (Euclidean Green’s Functions) ≡ Propagators

• Confinement can be related to the analytic properties of QCD’s Schwinger functions

• No assumption of heavy-quark symmetry is made.

• In particular, pseudoscalar and vector meson masses are not degenerate.

• Weak decay constants are not degenerate.

• Although impulse approximation and truncations are employed,

contributions are systematically included.

Caveat

14

⇤QCD/mc

The large current-quark mass of the b quark almost entirely suppresses momentum-dependent dressing, so that Mb(p2) is nearly constant on a substantial domain. This is true to a lesser extent for the c quark.

(ME)2 = {s|s+M2(s) = 0}

Euclidean mass definition

DSE mass functions

Define a single quantitative measure, namely the renormalization-point invariant ratio

with the constituent-quark 𝞂  term:

⇣f :=�f

MEf

�f := mf (⇣)@ME

f

@mf (⇣)�f

m!0�! 0

Euclidean mass, sigma term

The ratio ζf thus quantifies the effect of explicit chiral symmetry breaking on the dressed-quark mass function compared with the sum of the effects of explicit and dynamical chiral symmetry breaking.

The solutions of in GeV: f chiral u, d s c bME

f 0.42 0.42 0.56 1.57 4.68

Reasonable approximation : SQ=c,b ⇡1

i� · p+ MQ

(ME)2 = {s|s+M2(s) = 0}

f u, d s c b⇣f 0.02 0.23 0.65 0.8

Strong decays: D* → Dπ

A(D� ⇥ D⇥) = ��D�µ (pD�)Mµ(p2

D, p2D�) := ��D�

µ (pD�)pµD gD�D⇤

Mµ(p2D, p2

D�) = Nc tr� � d4k

(2⇥)4�D(k;�PD)Sc(k + PD�)i�µ

D�(k;PD�)Su(k)�⇤(k;�Q⇤)Su(k + Q⇤)

Coupling yields D* width

17

B. E., M.A. Ivanov and C.D. Roberts (2012)

CLEO DSE QCDSR LatticegD⇤D⇡ 17.9± 0.3± 1.9 16.5± 2 14.0± 1.5 20± 2

Strong decays: D* → Dπ

A(D� ⇥ D⇥) = ��D�µ (pD�)Mµ(p2

D, p2D�) := ��D�

µ (pD�)pµD gD�D⇤

Mµ(p2D, p2

D�) = Nc tr� � d4k

(2⇥)4�D(k;�PD)Sc(k + PD�)i�µ

D�(k;PD�)Su(k)�⇤(k;�Q⇤)Su(k + Q⇤)

Coupling yields D* width

18

B. E., M.A. Ivanov and C.D. Roberts (2012)

CLEO DSE QCDSR LatticegD⇤D⇡ 17.9± 0.3± 1.9 16.5± 2 14.0± 1.5 20± 2

Similarly: D⇤s ! DK

gD⇤sDK = 20+2.5

�1.7

B. E., M.A. Ivanov and C.D. Roberts (2012)

DSE: B. E., M.A. Ivanov and C.D. Roberts (2011)LQCD: D. Bećirević, B. Blossier, E. Chang and B. Haas (2009)

Strong decays: B* → Bπ (analogy)

This amplitude can be used for to extract at leading order in HMChPT:

m2� � 0 g

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g =gB�B�

2�

mBmB�f�

DSE model Lattice in static limit (nf = 2)

g 0.37± 0.04 0.44± 0.03+0.07�0.0

Lheavy = �traTr[Haiv ·DbaHb] + g traTr[HaHb�µAµba�5]

DµbaHb = ⇤µHa �Hb

12[⇥†⇤µ⇥ + ⇥⇤µ⇥†]ba ;

Aabµ =

i

2�⇥†⇤µ⇥ � ⇥⇤µ⇥†

�ab

;

Ha(v) =1 + /v

2�P �a

µ (v)�µ � P a(v)�5

�;

⇥ = exp(i�/f0�) ;

� is matrix of N2f � 1 pseudo-Goldstone boson.

The value obtained from D* decay is: corrections important! gc = 0.56+0.07�0.03 ⇤QCD/mc

-0.5 0 0.5q2 [GeV2]

0

2

4

6

8

10

12

g PlP

gKlKg/l/gDlD

Flavour SU(3), SU(4), sensible symmetries?

Define ⇣⇢ :=gD⇢D(q2)

gK⇢K(q2)

Ratio measures the effect of SU(4) breaking ≈ 300%

gD⇢D 6= gK⇢K 6= 1

2g⇡⇢⇡

SU(3) breaking ≈ 20-30%

B. E., G. Krein, L. Chang, C.D. Roberts and D. Wilson (2012)

Consequences?

The integrated D!D interaction is enhanced by about 40% compared with an SU(4) prediction for the coupling/form factor.

Large value value for the interaction strength entails an enhancedcross section in DN scattering (I = 1 cross section inflated by a factor 4–5).

Possible novel charmed resonances or bound states in nuclei?

Higher-spin couplings

-0.5 0 0.5q2 [GeV2]

0

5

10

15

g lD

D* /

mD

*

D ! D*

Higher-spin couplings

-0.5 0 0.5q2 [GeV2]

5

10

15

20

25

g lD

*D*

[GeV

-2]

G3G6G5G1=G5

D* ! D*

Form factors are the single important source of uncertainties and much work is left to pin down theoretical uncertainties to a few percent in charm physics.

One aspect learned over and over again is: heavy-quark symmetry is not applicable in charmed mesons and corrections are not negligible.

Other (flavor) symmetries often invoked to simplify calculations are not justified.

DSE/BSA treatments of heavy mesons must go beyond the rainbow-ladder approximation.

Epilogue

24

⇤QCD/mc