The coupling constant gNDΛc from QCD sum rules

Nuclear Physics B (Proc. Suppl.) 74 (1999) 214-217

ELSEVIER

The Coupling F.S. Navarra

PROCEEDINGS SUPPLEMENTS

Constant

and M. Nielsen

gNDA,

from QCD Sum Rules

a

aInstituto de Fisica, Universidade C.P. 66318, 05315-970 S&o Paulo,

de %io Paulo, SP, Brazil

The NDA, coupling constant is evaluated in a full QCD sum rule calculation. We study the Bore1 sum rule for the three point function of one pseudoscalar one nucleon and one AC current up to order seven in the operator product expansion. The Bore1 transform is performed with respect to the nucleon and A, momenta, which are taken to be equal, whereas the momentum q2 of the pseudoscalar vertex is taken to be zero. This coupling constant is relevant in the meson cloud description of the nucleon which has been recently used to explain exotic events observed by the Hl and ZEUS Collaborations at HERA.

1. Introduction

The Hl and ZEUS Collaborations at HERA measured recently an excess of events at large z and Q2 in deep inelastic scattering (DIS). One of the explanations for these events was suggested by Melnitchouk and Thomas [l]. They pointed out that an enhanced charm quark distribution at large z could account for the observed effect. This hard component of the c distribution has to be generated non-perturbatively, since the usual charm quark distribution arising from gluon bremsstrahlung alone looks like a typical soft sea distribution. The charmed sea is then taken to arise partially from the quantum fluctuation of the nucleon to a virtual DA, configuration, as first proposed in the works [2,3]. In the framework of this meson cloud model the anti-charm quark distribution in the nucleon (which is the main responsible for the observed effect) is given by the convolution of the valence e momentum distribution in the D meson with the momentum distribution of this meson in the nucleon. The function EN(Z) turns out to be directly proportional to gaDA, [l]. The value used in ref. [1] was gN* = $+$ = -3.795. Plugging the calculated i?N (z), toghether with parton distributions of all flavors, in the standard DIS cross sections expressions, d CY/ d z d Q2, Melnitchouk and Thomas were able to reproduce the observed rise of the cross sections at large values of M (M N 200 GeV) and Q2. 0920-5632/99/S - see front matter 0 1999 Elsevier Science B.V. PI1 SO920-5632(99)00165-6

In this work we would like to concentrate on the coupling constant QDNA, since it enters directly in the calculation of the DIS cross section. The value used to reproduce the HERA events in [l] , while a reasonable guess, has neither a solid theoretical justification nor a phenomenological basis. It may, however, be calculated in the framework of QCD Sum Rules (QCDSR). The coupling gDNh may also be useful in the context of relativistic heavy ion physics, specially in the study of quark-gluon plasma signatures. As it is well known, the suppression of J/Q is one of the proposed signatures. There is however an important hadronic background of nuclear absorption. One of the reactions that contribute to J/$J absorption in a nuclear environment is J/q + p + D + A,. In this process the J/G exchanges a D meson with the proton and becomes a D. The proton couples to the virtual D and becomes a AC and therefore it is necessary to know A'DNA, . 2. The phenomenological

side

To calculate the nucleon-meson D-A, coupling constant using the QCD sum rule approach [4] we consider the three-point function

A(p,p’,d

= x

s

&.-&

eiP'xe-iPU

(01T{111\,(Z)j5(Y)~N(0)}10)

constructed with two baryon currents, nN, for A, and the nucleon respectively, All rights reserved.

(l)

VA, and and the

215

IX Nauarm, M. Nielsen/Nuclear Physics B (Proc. Suppl.) 74 (1999) 214-217

pseudoscalar

meson

D current,

js, given by [5,6]

VA, =

~&@~ddQc

(2)

7lN =

u&~w~b)~5y~~c

(3)

to p2 = -p2 = -p/2, and subtracting uum contribution,

the contin-

we get

= -_i--~ [pz(M” q2)lpeTt ;

,~--I,,

c

js = Qiysu

(4)

where Q, u and d are the charm, up and down quark fields respectively, C is the charge conjugation matrix. In the phenomenological side the above amplitude is given by

,A,(e-

CJ Acphen)(p, p’,4) = AA,AN(z+;c) /osdm2

p.p’)iys

+

2X(%u, q2)

MAc l MN$i75 +(MAP - MN)$%ys

--

q2 - mf, ~'2 -

x + -

3 me = -2 (2.rr)2 dm

x

[(MA,MN

otivy5p,pL]

1

1

QNDA,

-

1

with P(%s,q2)

X

u/M2 _ (p/M=

x p(u,s,q2)-

{m2

1

(a

M& p2 - M;G

+ higher

resonances

(5)

We will follow refs.[7,8] and write a sum rule for the structure $iys. The coefficient functions of this structure will be called Fs. As we are interested in the value of the coupling constant to at 4 2 = 0, we will make a Bore1 transform both p2 = p/2 -+ M2. The contribution from excited baryons will be taken into account as usual through the standard form of ref.[9] As in any QCD sum rule calculation, our goal is to make a match between the two representations of the correlation function (1) at a certain region of M2: the OPE side and the phenomenological side. 3. The OPE side In the OPE side only odd dimension operators contribute to the &ys structure, since the dimension of Eq.( 1) is four and $ takes away one dimension. Therefore, the perturbative diagram contributes through the m, operator. To evaluate the perturbative contribution we write a double dispersion relation to the amplitude Fs and use the Cutkosky’s rules to evaluate the double discontinuity. After Bore1 transforming with respect

fip’ (u-pb&-m:)>

m4 2

L 2s + X(s,lq2) 1

(I-

$I’]}

x o((coseK)2- 1) where pb = (s + u - q”)/(2&

(7) and

~=2sU+m2-mZ-~~(s+m2)l~

(8)

(s - m2>JGZ7 In Eq.(6) F2 stands for the Bore1 transformation of the amplitude Fz and ~0 gives the continuum threshold for A,. The next lowest dimension operator is the quark condensate with dimension three. In ref.[7] the pion mass was neglected and, therefore, only terms proportional to l/q2 were considered. Since in our case mD will not be neglected we have also to consider the diagrams in which the light quark lines condensate. Their contributions are given by

[i%(M2,q2)](Tqj = -$&M4EI a

with El = 1 - eMUolM2 (1 + uo/M2)

c

and

(9)

216

ES. Naoarm, M. Nielsen/Nuclear Physics B (Pmt. Suppl.) 74 (1999) 214-217

(2- 6+ ab + (1+ ab

(p/M2

_ e-s/M2)

(l(j)

(u - s)2 where 6 = l/2 - o(u + s)/(2s)

and

+-@(m:+y)

(11)

The last class of diagrams which we will consider is the dimension 7 operators of the type m, (ifqijq) (q being a u or d quark). Other dimension 7 operators come from graphs which contain at least one loop and are suppressed by factors l/4& The expressions for these contributions are [F,(M2,

q2)] (94Tig)+b =

@p

(12)

*

a

c

(qqqq) [~2(M2J12&__q),

=

1 -

-y---

(13)

m,

The Bore1 transformation logical side gives

Mb + MN(e -M;/M=

e-d/M2

of the phenomeno-

_ ,M;,/M~

“M;/M;

>

(14)

For XA, and AN we use the values obtained from the respective mass sum rules for the nucleon [5,7] and for A, [6]:

I/j~,12e-Mic/M2 =

m4

uo -u/M’

C

du

512~~ s mZ e [(l-s)

(I--$+$)

-121,(z)]

(15) and (16)

where Es = 1 - e -so/M’ (1 + so/M2 + s;/(2M4)) accounts for the continuum contribution with se being the continuum threshold for the nucleon.

We have neglected the contribution of the gluon condensate in the mass sum rules and in the three point function since it is of little influence. In order to obtain gND& we identify eq. (14) with the sum of eqs. (6, 9, 10, 12, 13) and using eqs. (15, 16) we solve the resulting equation for QND& as a function of the Bore1 mass squared. Since eqs. (15, 16) determine only the absolute value of AN and XA~ we can not determine the sign of QNDA,. In this calculation there is no free parameter. There are, instead, some numbers which are heavily constrained by other theoretical or phenomenological analyses. They are the sources of uncertainties in our result. The quark condensate was taken to be (ijq) = -(O.23)3 GeV3. The continuum thresholds were chosen to be = (MA + 0.75)2 GeV2 and SO = (MN + ;fp75)2 GeVi. The hadron masses are MN = 0.938GeV, n/i,= = 2.285 GeV and mD = 1.8 GeV. The charm quark mass was taken to be m, = 1.4 GeV. In order to take into account possible deviations from the factorization hypothesis [lo] we introduce a correction factor K in the formula (i&N) = K(@!)2

(17)

and consider the cases K = 1 and K = 2. Finally we need the D meson decay constant fD. This constant will be probably evaluated in the next generation of experiments on fully leptonic D decays. So far it has been usually estimated [ll] to be fD = 200f30 MeV. On the other hand, new estimates [12] point to ( fD = 1.35f0.07) fir. In view of this we take fD in the interval 140- 180 MeV. The relevant Bore1 mass here is M 2~MN:MAc and therefore we analyse the sum rule in the interval 1.5 5 M2 5 3.5 GeV2. A plot gNDI\, as a function of M2 for different values of fD and K shows a rather flat behaviour. For QCD sum rules standards, this sum rule is stable. The uncertainties both in fD and K lead to the main spread in the value of QNDA, . Changing the charm quark mass from 1.3 GeV to 1.5 GeV leads to a corresponding VariatiOn in gNDA, of the order of only 15%. We have also varied the continuum thresholds in the interval MN + 0.5 I

FS. Naoarra. M. Nielsen/Nuclear

Physics B (Pnx.

SO5 MN + 1.0 GeV and MA, + 0.5 5 ug 5 MA, + 1.0 GeV. Under these variations the final result does not change appreciably. This indicates that the continuum contribution is under control.

-]gND& t = 3.0 -f 0.5

W. Melnitchoukand

4.

&+

This value is smaller than the one used in the calculations of ref. [I]. At first sight, since the normalization of the c distribution in the nucleon is proportional to the square of gND&, using our value would reduce the contribution of the non-perturbative nucleon charm sea (employed to study the HERA events) by a factor two. However there are many other possible intermediate states apart from D -A, which contribute to generate this charm sea like, for example, D’ - A,, D - C and D’ - C. These higher mass states are neglected in some calculations but, as shown in ref. [3], already in the strange sector they may give important contributions for the evaluation of strange form factors and strange quark momentum distributions. In the charm sector we expect the contribution of these states to be even more important. Our conclusion is therefore that the non-perturbatively generated charm sea remains qualitatively a good candidate to explain the observed exotic events. Quantitative calculations should, however, include the contribution of other charm mesonic clouds, specially in view of the value found here for QNDA,.

and A.W. Thomas, Phys.

Let%. B414, 134 (1997).

discussed

W

217

REFERENCES

4. Result and Discussion Considering all the uncertainties above our final result is

Suppl.) 74 (1999) 214-217

5. 6.

10. 11.

12.

F.S. Navarra, M. Nielsen, C.A.A. Nunes and M. Teixeira, Phys.Rev. D54, 842 (1996). Virtual Meson Cloud of the Nucleon and Intrinsic Strangeness and Charm, S. Paiva, M. Nielsen, F.S. Navarra, F.O. Duraes, L.L. Barz, hep-ph/9610310 , to appear in Mod. Phys. Lett. A. M.A. Shifman, AI. Vainshtein and V.I. Zakharov, Nucl. Phys. B120, 316 (1977). B.L. Ioffe, Nucl. Phys. B188, 317 (1981). E. Bagan, M. Chabab, H.G. Dosch and S. Narison, Phys. Lett. B278, 367 (1992); B287, 176 (1992); B301, 243 (1993). L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rep. 127, 1 (1985). S. Choe, M.K. Cheoun and S.H. Lee, Phys. Rec. C53, 1363 (1996). B.L. Ioffe and A.V. Smilga, Nucl. Phys. B216, 373 (1983); Phys. Lett. B147, 353 (1982). S. Narison, &CD Spectral Sum Rules , World Scientific, Singapore 1989. S.P. Ratti, Proceedings of the LISHEP 95, ed. F. Caruso et al., Editions Frontieres. (1995) p. 203. S. Narison, in these proceedings.