F. S. N. , M. Nielsen

IFUSP (São Paulo) BRAZIL

Charm hadronic form factors with QCD sum rules

Motivation

Conclusion

Results on form factors

Application: charmonium production

QCDSR

interactions at RHIC and LHC/J

Charm form factors Lin,Ko nucl-th/0210014

Charmonium decays in B factories

Liu, Zhang, Zhu, hep-ph/0610278

X (3872)

X (3872)

interactions at FAIRD

Haidenbauer, Krein, Meissner, Sibirtsev arXiv:0704.3668 [nucl-th]

Form factors in QCD sum rules

DD* /* JDD(data)

/JDD DD

/** JDD **DD

**DD DD*

2) Choose the currents:

1) Write the three-point correlation function:

The QCD side (Operator Product Expansion side):

3) Insert the currents in and make the contractions:

][ duQOPE SSSTr

DD*

4) Perform the OPE:

The hadronic side (phenomenological side):

+ higher resonances

5) Insert hadronic states in

7) Use an effective Lagrangian to compute the amplitude:

6) Use the matrix elements:

8) Write)( phen

qqgpHqpH HH )(),(*)(|´)( 2

*

9) Decompose in tensor structures and choose one of them:

10) Write a double dispersion relation:

On both sides:

11) IdentifyOPEphen

12) Apply a double Borel transform:

222 MPp 222 ´)(´)(´)( MPp

double discontinuity

13) Write the sum rule:

14) Numerical analysis:

Numbers :

Borel masses:

Continuum thresholds :GeVs 5.0 GeVu 5.0

or

15) Check Borel stability and OPE convergence:

total

perturbative

gluoncondensate

),( 22 MQg

off-shell

D

off-shell

off-shell

Exp.

16) Fix M , plot fit and extrapolate to the meson pole:)( 2Qg

)()( 2222* D

DDD mQgmQgg

Correlated extrapolation of the three form factors

D*

D

J/Psi

/* JDD

Varying M and M´ independently:

Good stability !

*D off-shell

/* JDD

Dependence on the continuum threshold

0.6

0.5

0.4

0.6

0.5

0.4

D off-shell

Couplings

3.03.4 DDg

Parametrizations:

Comments

Compatibility with HQET relations :

/**/ JDDJDD gg =

Coupling D*-D-pion compatible with data and with lattice QCD:

data

lattice

**DDDD gg 3.03.4 DDg

Oh, Lee, Song, PRC (2000)

Becirevic, Charles, Le Yaouanc, L.Olivier, Pene, Raynal, (2003)

Oh, Lee, Song, PRC (2000)

Application

Charmonium production and absorption in nuclear matter

Application

Charmonium production and absorption in nuclear matter

“Charmonium regeneration”

withwithout

without with

Conclusion

Charm form factors are still very usefull for phenomenology

We can calculate them with QCDSR (finished the first round)

The obtained coupling constants are of the same order of magnitude

The numbers roughly agree with previous phenomenological estimates

The form factors were used in one phenomenological application

Charm form factors change calculations by one order of magnitude

References

F. Carvalho et al., Phys. Rev. C72, 024902 (2005)

M.E. Bracco et al., Phys. Lett. B521, 1 (2001)

F.S. Navarra et al., Phys. Rev. D65, 037502 (2002)

M.E. Bracco et al., Phys. Lett. B605, 326 (2005)

M.E. Bracco et al., Phys. Lett. B659, 559 (2008)

B. Osorio Rodrigues et al., arXiv:1003.2604 [hep-ph]

R.D. Matheus et al., Int. J. Mod. Phys. E14, 555 (2005)

R.D. Matheus et al., Phys. Lett. B541, 265 (2002)

D*

D

J/Psi

Three different particles off-shell in the vertex

Pole versus continuum

off-shell*D off-shell

** DD

Errors

Truncation of the OPE

Pole + continuum Ansatz

Continuum threshold parameters s u

s corrections

Values of masses and condensates

Choice of Borel mass

Choice of tensor structure

Extrapolation to the mass shell

~ 20 %

?

Medium effects

Borel stability in different structures:

D

D

´p

´p

p

p

DD

/** JDD

14 tensor structures

Choose

Form factors in different structures :

*D

/J

*D

/J*D

/J

*D

versus

*D *D *D

** DD /** JDD

Couplings with vector mesons not very compatible with VDM estimates :

3.03.4 DDg

Matinyann, Muller, PRC (1998)

Oh, Lee, Song, PRC (2000)

P1

P3

P3

P4

P4

P4

P5

P5

Sergei G. Matinyann, Berndt Muller Phys.Rev.C58:2994-2997,1998. nucl-th/9806027

P5

P6

P6

Kodjamirian

P7

P8

P10

Frequent questions

Higher dimension condensates ?

Infinities ? Killed by Imaginary Corr + Cutkosky + Borel + s0 / QHD

Differences between on and off-shell ? Only Borel ?

Compatible with SU(4) ? HQET ? VMD?

Pole versus continuum (well defined ?) changes with Q2 ?

Extension to quartic couplings ?Why the restrictions in the Q2 region ?

Observable applications ?

Why not smaller Q2 ?Final errors ?

Back ups

** DD

Borel stability:*D off-shell

total

perturbative

quarkcondensate

bare couplings

B) Meson loops with

*D D Meson loops

Introduce the :

)( 2pCalculate the loops and compute the vertex function

Calculate the form factor of an off-shell D: )()( 22 pGqp

CCK fitted to adjust QCDSR points

16) Fix M , plot fit and extrapolate to the pion pole:)( 2Qg

How to reduce the uncertainties ?

Exp.

CLEO, PRL 87 (2001)

Coupling constant: )( 22* mQgg DD

/* JDD

sum rules

*D off-shell

/J off-shell

Three different particles off-shell in the vertex !

OPE convergence :

*D off-shell

total

perturbative

gluoncondensate

/* JDD

J/Psi

D

/JDD

/J

Parametrizations:

/JDD

DD

D

/J

/J

D

D D D

versus

/J

D

D D D

versus

off-shell

/J

D

DD

Parametrizations:

DD

Borel stability:

off-shell*D off-shell/J

/** JDD

dependence on the continuum threshold parameters

/** JDD

Parametrizations:

( other structure )

/** JDD

** DD

structure

Borel stability:

off-shell*D off-shell

total

perturbative

quark condensate

perturbative

Dependence on the continuum threshold parameters

** DD

Form factors

*D

Parametrizations:

** DD

*D

** DD

Parametrizations:

** DD

DD*

Borel stability:

off-shellD off-shell

perturbative

total

perturbative

quark condensate

Dependence on the continuum threshold parameters:

DD*

D

DD*

Parametrizations:

DD*

/** JDD ** DD

** DD

*D

/J*D

*D

*

List of form factors

The gluon condensate

The quark condensate

Oh, Song, Lee, Wong, nucl-th/0205065

Charm form factors

Oh, Song, Lee, Wong, nucl-th/0205065

Oh, Song, Lee, Wong, nucl-th/0010064

Liu,Ko nucl-th/070277Phys.Rev.C75,064903

Conclusion

Motivação

D* D pi : dados, rede, mais calculos feitos, mais leve!

D* D Psi: mais pesado!

D D Psi e D D rho: comparação de sondas diferente !

D* D* Psi e D* D* rho: comparação de sondas diferente !

D* D rho

D* D* pi

Table...

Motivation

Understand final state interactions in B decays and X(3872) decays

Understand J/psi and D interactions in hadronic and nuclear matter

In the strange sector: KN *KN ...Form factors: simple parametrizations fitted to data

Monopole:22

222 )(

Q

MgQF

22 qQ

gMF )( 2Coupling constant

Hyperon - nucleon interactions

Haidenbauer, Meissner, Nogga, Polinder, nucl-th/0702015

12) Fix M , plot fit and extrapolate to the pion pole:)( 2Qg

PLB (2000) PRD (2002)

Exp.

/JDDold

8) Check the pole dominance:

2

0

2

4

4

)(

)(

c

c

m

s

m

ssd

ssd

Pole

DDDDDDDDccX **00*00* ||||

É difícil acreditar...

BV

V

m

gV 2

202

Compute the Lagrangian, energy-momentum tensor and obtain the EOS :

But we can estimate the Laplacian :