Inclusive decays at order v7

International Workshop on Heavy Quarkonium

Brookhaven National Laboratory

Phenomenological MotivationP-wave charmonium inclusive decays

• Electromagnetic width: error on c0 reduced to 20%

error on c2 reduced to 10%

• Hadronic width: error on c0,2 l.h. reduced to 10%

• The ratios of electromagnetic and hadronic widths are clearly sensitive to the NLO corrections in s(M); but, being for charmonium systems s(M) ~ v2 ~ 0.3, the relativistic corrections are of the same order of the pertubative ones and, potentially, of 30%. Vector S-wave charmonium and bottomonium decays into leptons and light hadrons

• The J/and electromagnetic decay in two leptons and the hadronic decay in light hadrons are both known with an error less than 5%. The (2S) and (2S) decays are known with an accuracy of ~ 5%

• Corrections of order v4 , s(M) v2 and s(M) are needed to reach the same accuracy.

Pseudoscalar S-wave charmonium into two photons or light hadron

• c electromagnetic decay in two photons is known with an accuracy 35%

• c hadronic decay in light hadrons ~ 15%.

For reference:

G. T Bodwin and A. Petrelli, Phys. Rev. D66, 094011 (2002)

J.P. Ma and Q. Wang, Phys Lett. B537, 233 (2002)

NRQCD Decay widths – Power Counting

• NRQCD provides the theoretical framework for the study of electromagnetic and hadronic inclusive

G. T. Bodwin, E. Braaten and G.P. Lepage, Phys. Rev. D51, 1125 (1995)

The matching coefficients

• are evaluated comparing scattering amplitudes in QCD e NRQCD

• have an expansion in s(M)

The matrix elements of four fermion operators

• have an expansion in the velocity of the heavy quark.

Power Counting

• In NRQCD several different scales are still dynamical: Mv (momentum of the heavy quark), E ~ Mv2 , QCD .

• The matrix elements of four fermion operators will receive contributions from these different scales, so it is impossible to give them a definite power counting.

• We choose a “conservative” power counting: each operator of mass dimension d scales as (Mv)d.

• Further suppression come if we consider operators acting on subleading component of the quarkonium Fock state

• The set of operators defined in this way is not minimal, and through a field redefinition it is possible to

eliminate from the NRQCD Lagrangian the operators T8(0,2)(3S1), writing them as linear combinations of the

operators of mass dimension 10.

S-wave decay into

The contributions up to order v7 to the 3S1 decay into e+e- are:

• relativistic corrections to the dimension 6 and 8 operators

• spin flip operators of mass dimension 8 containing a chromomagnetic field

• operators of mass dimension 9 containing a chromoelectric field

~ v ~ v2

~ v~ v3

The decay width reads:

S-wave and P-wave decays in With the same analysis done for the S-wave decays into leptons, we find that the decays width of a pseudoscalar

quarkonium into two photons is given by:

The contributions up to order v7 to the P-wave decays in two photons are

• relativistic corrections to the dimension 8 operators

• operators of mass dimension 9 containing a chromoelectric field

~ v7

• In this case, the spin flip operators scale as v8

• It is not possible to find a field redefinition that eliminates the operator containing the chromoelectric field

The decay widths read:

• gauge invariance of color octet operators

Hadronic decay widths

If we consider the component of the Fock in the ingoing quarkonium, in the center of mass rest frame the quark-antiquark pair has total momentum different from 0 and proportional to the momentum of the gluon: the matrix element is different from 0.

The order of the color matrices and the covariant derivatives becomes relevant. To guarantee gauge invariance, we have to prescribe the ordering:

• operators proportional to the total derivative of the quark-antiquark pair

• color octet operators

Electromagnetic situation always correspond to the lhs: the total derivative acts on the quark, antiquark and the gluon.

In the hadronic case, we have to first situation, but the gluon can also travel until the outgoing pair, in this case the total derivative acts only on the quark-antiquark pair, and the operator is different from 0

S-wave decays into light hadrons. Decay widths

• c (1S0) decay into light hadrons

• J3S1) decay into light hadrons

P-wave decays into light hadrons. Decay widths

• c0 decay into light hadrons

• c2 decay into light hadrons

S-wave decay into leptons: matching coefficients of singlet operators

G. T Bodwin and A. Petrelli, Phys. Rev. D66, 094011 (2002)

• The matching can only determine the sum, not the individual matching coefficients V.A. Novikov, L.B. Okun, M.A. Shifman, A.I. Vainshtein, M.B. Voloshin and V.I. Zakharov, Phys. Rept.41, 1-133, (1978)

Matching

• order by order in inverse power of the heavy quark mass M.

• color singlet operators:

• singlet-octet transition operator:

• since the QCD scattering amplitude doesn’t have a definite angular momentum, the matching determines more coefficients than needed in the decay widths.

p2 p’2

p2 p’2

p4 + p’4

p4

S-wave decay into leptons: matching coefficients of octet operators

In QCD we evaluate the imaginary part of the diagrams

In NRQCD, we must consider different contributions

• four fermions operators with chromoelectric or chromomagnetic field

• four fermion operators in which we consider the gluon field in the covariant derivative

• four fermion operators with insertions of quark-gluon vertices of the bilinear NRQCD Lagrangian, up to order 1/M5

Decay into photons: matching coefficients of singlet operators

• the coefficients of dimension 10 operators agree with the literature

• it is not possible to find the individual coefficients of the

1S0 operators

J.P. Ma and Q. Wang, Phys Lett. B537, 233 (2002)

Agrees with the one in Disagree with the ones

We cannot find the origin of such a discrepancy

Decay into photons: matching coefficients of octet operators

As in the S-wave decay into leptons we must consider

• four fermions operators with chromoelectric field

• four fermion operators in which we consider the gluon field in the covariant derivative

• four fermion operators with insertions of quark-gluon vertices of the bilinear NRQCD Lagrangian, up to order 1/M5

Disagree with J.P. Ma and Q. Wang. In particular, being the coefficient of the operator

T8(3P2) equal to zero, we find we need another matrix element less than in that paper.

Matching coefficients of singlet operators

As far as the singlet operators are concerned, we find the coefficients

Agree with the literature Original results of this work

Matching coefficients of octet operators

The same QCD Feynman diagrams showed in the previous slide give the matching coefficients of the octet operators that don’t contain a chromoelectric o chromomagnetic field

Matching of operators containing chromoelectric field

Work in progress.

ConclusionsElectromagnetic decays

• complete calculation of the matching coefficients of dimension 10 operators involved in heavy quarkonium decay in two photon or leptons to order v7 in the velocity expansion

• S-wave decays: we confirm the result of Bodwin and Petrelli for the sum of the coefficients of the operators

Q’( 3S1) , Q’’(3S1) , Q’( 1S0) and Q’’( 1S0), we can fix the individual coefficients of the same operators, we find the coefficients of dimension 8 operators contaning a chromomagnetic field ( S8(1S0) and S8(3S1) ) and of one operator containing a chromoelectric field, T(1)

8(3S1).

• P-wave decays: we confirm the result of Ma and Wang for the coefficient P(3P0). We calculated the coefficients of the operators P(3P2), T8(3P0) and T8(3P2). Discrepancy with the results of Ma and Wang. In particular: the vanishing of the coefficient of T8(3P2) and a different decomposition on spherical tensor allow us to write the decay width of the 3P2 decay with two matrix element less.

• D-wave decays: we confirm the results of Novikov et. al.Hadronic decays

• we calculated the matching for the singlet and octet operators, confirming the result of Bodwin and Petrelli for S-wave, of Novikov et. al. for D-wave and finding original results for the P-wave.

Perspectives

Before the phenomenological applications is necessary to reduce the number of non perturbative matrix elements.

• lattice evaluation, fitting from data.

• integration of the mv scale through a pNRQCD analysis.

Cose che potrebbero servire: le trasformazioni di gauge

Our expression doesn’t contain the operator P(3P2,3F2)

The electromagnetic situation always correspond to the lhs, the total derivative acts on the quark, antiquark and the gluon.

In the hadronic case, we have to first situation, but the gluon can also travel until the outgoing pair, in this case the total derivative acts only on the quark-antiquark pair, and the operator is different from 0

Conclusion:

One possibility is to repeat the conclusion of the paper, saying that for the perturbative calculation in the 3S1 decay the bigger uncertainty comes from missing corrections in alphas v^2, in the 1S_0 case from alpha_s^2, alpha_s v^2 besides the uncertainties coming from the the coefficients we have not yet evaluated.

Then we have to note that the number of matrix elements involved in the decay widths is pretty big and these matrix elements are not well known. (see, for example, QWG, the matrix element known by fitting or by direct lattice evaluation are O_1 (1P_1), O_8 (^1S_0) and I think also, from Bodwin, Lee and Sinclair, 2005, O_1(3S_1), O_1(1S0) )

So a possibility to reduce the non perturbative is to switch to pNRQCD, where only the mv^2 scale is still dynamical. This has also the advantage that a definite power counting can be assigned to the different matrix elements.