Quark Hadron Duality and Rein-Sehgal Model†

Krzysztof M. Graczyk, Cezary Juszczak, Jan T. Sobczyk

Institute of Theoretical PhysicsWrocław University, Poland

ABSTRACTQuark-Hadron duality in the neutrino-nucleon interactions is investigated in the context of predictions of the Rein-Sehgal model for single pion production. A detail analysis of structure functions is done. We consider structure functions for Charged Current (CC) and Neutral Current (NC) interactions of neutrino with nucleon. GRV94 parton distribution functions are used. 1-function is used to extract single pion contribution from the DIS structure functions. In the case of resonance local duality hypothesis is studied.

kgraczyk@ift.uni.wroc.pl http://wng.ift.uni.wroc.pl

† The article which contains the results of this presentation will appear soon.

2

Rein-Sehgal (RS) ModelThe Rein-Sehgal model describes Single Pion Production (SPP) in charged and neutral current neutrino-nucleon interactions. The pions are produced by excitations of 18 resonances with masses smaller than 2 GeV.

The RS model is described in the resonance rest frame. It is obtained by boosting from the nucleon Breit frame. In the nucleon Breit frame four-momentum transfer is the following:

),0,0,( 0 Qqq

The leptonic current matrix element are expressed by linear combination of three four-vectors:

, , SRL eee

*NlN 'N

It allows to decompose scattering amplitude into three independent contributions (assuming the lepton mass to be equal to zero).

3

RS - model

The differential cross section can be written in the form:

SRLC uuQ

qG

dqdq

d

v2v4

cos 222

2

2

22

02

z

z

jRESzzS

jRESzzRL

MWjNFjNq

Q

W

M

MWjNFjNM

W

)(,||,2

)(,||1,2

2*

02

2

2*

,

Fe

Q

qFFeFFeF LRL 2

2

0 , ,

M

qq

2

2

0

Structure Functions

TLS

TLS

TL

Q

MqF

Q

qqF

MF

2

22

2

03

2

20

2

1

One can obtain expressions for the structure functions

matrix elements ofhadronic current

E

QqEE

QqEu

2

2v

2

2

0

0

4

)cos(2)(

)cos(2 )(

)sincos(2 )(

)sincos(2 )(

23

22

223

222

sduxpxF

sduxpF

usdxpxF

usdxpF

CCC

CCC

CCCC

CCCC

Charged Current

)cos(2)(

)cos(2 )(

)sincos(2 )(

)sincos(2 )(

23

22

223

222

sudxnxF

sudxnF

dsuxnxF

dsuxnF

CCC

CCC

CCCC

CCCC

uuggddggxnnxF

suuggddggxnnF

ddgguuggxppxF

sddgguuggxppF

RLRLNC

RLRLNC

RLRLNC

RLRLNC

22223

22222

22223

22222

''2),(

2''2 ),(

''2),(

2''2 ),(

Neutral Current

Callan-Grossrelation is adopted

212 FxF

DIS Structure Functions

WR

WL

g

g

2

2

sin3

1'

sin3

1

2

1'

WR

WL

g

g

2

2

sin3

2

sin3

2

2

1

5

Quark-Hadron Duality

If FESR holds for some particular resonance (e.g. ) we say that there is a Local Duality.

Definition:The duality between quarks and hadrons (in the sense of Bloom and Gilman) appears when the following relation between resonance and scaling structure functions holds:

),(),(2max

min2

2max

min2 DIS

DISRES

RES QFdQFd

FESR equation

22DISRES QQ enough large to apply DIS formalismsmall

We define the region of integration by choosing of maximal an minimal hadronic mass then appropriate Nachtaman variable is obtained.

),(

),(2

2

RESMAXMIN

RESMINMAX

QW

QW

The same Q2RES in the DIS and RES integral!!!

mMWMIN

6

dependence on W at given Q2 [GeV] Fig.1

222

2

MWq

qx

In our analysis the scaling functions depend on Nachtman variable:

222

2

/411

2),(

qMx

xQx

7

Comparison of the RS structure functions with the scaling functions for charged current neutrino-nucleon interaction. Resonance structure function (at Q2=1 GeV2) is denoted by red line. Blue and

green lines correspond to scaling structure functions at Q2=1 GeV2 and Q2=10 GeV2.

Fig.2

8

Comparison of the RS structure functions with the scaling functions for neutral current neutrino-nucleon interaction. The resonance structure function (Q2=1 GeV2) is denoted by red line.

Blue and green lines correspond to scaling structure functions at Q2=1 GeV2 and Q2=10 GeV2.

Fig.3

9

Comparison of the RS structure functions with the scaling functions for charged current neutrino-nucleon interaction. Red line denotes resonance structure function calculated for Q2=1 GeV2. Blue and green lines correspond to sea and valence contribution to scaling functions at Q2=10 GeV2.

Fig.4

10

Fig.5

Comparison of the RS structure functions with the scaling functions for neutral current neutrino-nucleon interaction. The resonance structure function (at Q2=1 GeV2) is denoted by red line. Blue and green lines correspond to sea and valence contribution to scaling functions (at Q2=10 GeV2).

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1-function

DISdWd )(f11

We assume that DIS structure functions which describe the single pion production can be expressed by using 1-function:

ii FWF )(f11

1-functions for CC neutrino-nucleon interaction.

Fig.6

In order to evaluate SPP in the framework of the DIS formalism we introduce 1-function. They measure the probability that after fragmentation and hadronisation the final hadronic state is that of SPP. The functions are obtained from the Monte Carlo simulation which are based on the LUND algorithm. We show three 1-functions in Fig. 6.

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Fig.7

Comparison of the RS structure functions with the scaling functions for charged current neutrino-nucleon interaction. The resonance structure function (at Q2=0.4 GeV2) is denoted by red line. Blue line correspond to the scaling function (at Q2=0.4 GeV2). Structure functions multiplied by f1(W) are denoted by green line.

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Comparison of the RS structure functions with the scaling functions for charged current neutrino-nucleon interaction. The resonance structure function (at Q2=1 GeV2) is denoted by red line. Blue line correspond to the scaling function (at Q2=1 GeV2). Structure functions multiplied by f1(W) are denoted by green line.

Fig.8

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Comparison of resonance structure functions with scaling functions for charge current neutrino-nucleon interaction. Red, green and blue lines denote resonance structure functionscalculated at Q2=0.4, 1 and 2 GeV2. The scaling function is calculated at 10 GeV2 (pink line).

Fig.9

15

Fig.10

Comparison of resonance structure functions and scaling functions for neutral current neutrino-nucleon interaction. Red, green and blue lines denote resonance structure functionscalculated for Q2=0.4, 1 and 2 GeV2. The scaling function is calculated for 10 GeV2 (pink line).

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Fig.11

In above figures the solution of equation is presented.Namely for given value Q2

RES the limit of integrationWMAX is calculated. Green line denotes solution of equation (on left side). Red line denotes W=1.450 GeV.

red line correspondsto P11 mass

1)GeV 10,(

),(2)(max

)(min 2

2)(max)(min 2

mM

MAXWDIS

RESmM

MAXWRES

dF

QdF

17

Description

We start our analysis by presentation of F2 and xF3 structure functions for proton and neutron for charged and neutral current interactions (Fig.2 and Fig. 3). Resonance structure functions are calculated by taking into account all 18 resonances included in the RS model. We compare resonance structure functions calculated for small Q2 (1 GeV2) with scaling functions at Q2=10 GeV2.

In Figs. 4 and 5 we naively test possibility of occurrence of two-component duality in the neutrino-nucleon interaction. The resonance structure functions (Q2=1 GeV2) are compared with valence and sea scaling functions (at Q2=10 GeV2).

The RS model describes only single pion production, so to make our analysis more realistic we suplement the DIS formalism by 1function. It is done by using LUND algorithm. In Fig. 6 the 1function are presented for three channels for charged current neutrino-nucleus interaction. We multiply scaling function by F1(W) and compare with RS structure function (Fig. 7: Q2=0.4 GeV2 and Fig. 8: Q2=1 GeV2).

To investigate local duality in the case of (1232) we compare RS structure functions calculated for three values of Q2 (0.4, 1 and 2 GeV2) with scaling functions at Q2=10 GeV2. The analysis is performed for charged (Fig. 9) and neutral (Fig. 10) current neutrino processes.

In Fig. 11 the solution of FESR equation (F2 structure function) is presented. The minimal value of hadronic mass was kept and for each Q2

RES value the WMAX was calculated. We present solution for the case of four F2 scaling functions (proton, neutron, CC and NC type of processes).

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Some Comments• In general the RS and the DIS (GRV94) structure functions are comparable. In the

case of F2(CC) for proton the scaling function cuts across the peak in contrast to neutron where the scaling function is larger than all resonances. For NC processes the scaling functions intersects peak in the case of proton and neutron.

• The using of 1-function improves duality especially for larger values. However for neutrino-neutron processes the scaling functions are still bigger than the RS functions (about two times).

• As it was mentioned we made a naive test of two-component duality. Sea contribution to the scaling function becomes important in the area of large . It seems that taking into account only valence contribution does not improve the duality. So simple understanding of two-component duality hypothesis that resonances are described by valence contribution of scaling functions and nonresonant background corresponds to the sea contribution does not work here.

• The local duality in the case of resonance was investigated and results are presented in Figs. 9, 10, 11.The results presented in Fig. 11 are especially interested. We found the area of integration which satisfies the FESR equation. In the case of proton (CC) WMAX are much bigger than 1.45 GeV (mass of P11), which is a maximal possible limit of integration in the case of local duality. In the case of neutron (CC) WMAX is below 1.45, however as Q2 increase WMAX goes to 1.45 GeV. I means that the local duality does not appear for CC neutrino-proton interaction. One can observe that in the case of NC structure functions obtained values of WMAX are the closest to the P11 mass. Hence there is possibility occurrence of the local duality.

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• The duality occurs more in the NC than the CC neutrino interactions. • So, Is the duality characteristic feature of the CC neutrino-nucleon interactions? • May be answer lays in treatment of

nonresonant background. How to describe the nonresonant background.

Conclusions and Open Questions

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References1. D. Rein, L. M. Sehgal, Ann. Phys. 133, (1981) 79.2. E. D. Bloom, F. J. Gilman, Phys. Rev. Lett. 25 (1970) 1140; Phys.

Rev. D4 (1971) 2901.3. W. Milnitchouk, R. Ent, and C. E. Keppel, Phys. Rep. 406 (2005)

127.4. I. Niculescu et. al, Phys. Rev. Lett.B364 (1995) 107.5. H. Harari, Phys. Rev. Lett. 20 (1969) 1395; ibid. 22 (1969) 562; ibid

24 (1970) 286; Annals Phys. 63 (1971) 432.6. P. G. O. Freund, Phys. Rev. Lett. 20 (1968) 235; P. G. O. Freund

and R. J. Rivers, Phys. Lett. B29 (1969) 510.7. C. E. Carlson, hep-ph/0005169.8. E. Leader, E. Predazzi, An introduction to gauge theories and

modern particle physics, vol. I, Cambridge University Press 1996.9. S. Mohanty, Proc. of NuInt02, see http://nuint.ps.uci.edu/.

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