PION COUPLING TO CONSTITUENT QUARKSVERSUS COUPLING TO NUCLEON

Talk presented at the International WorkshopFLAVOR STRUCTURE OF THE NUCLEON SEA

Trento, July 1-5, 2013

MITJA ROSINA

Faculty of Mathematics and Physics, University of Ljubljana,and Jozef Stefan Institute, Ljubljana

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ABSTRACT

The importance of the pion cloud in the nucleon has been

demonstrated for several processes and nucleon observables.

The most direct observations come from deep inelastic scat-

tering (for example from the forward neutron production in

the ep collisions at 300 GeV measured by the H1 and ZEUS

Collaborations at DESY). They should be supplemented by

forward proton production and we call for more substantial

experiments and analyses.

References

[1] S. Baumgartner, H.J. Pirner, K. Konigsmann and B. Povh, Zeitschrift

fur Physik A 353 (1996) 397.

[2] H.J. Pirner, Prog. Part. Nucl. Phys. 36 (1996) 19-28.

[3] A. Bunyatyan and B. Povh, Eur. Phys. J. A27 (2006) 359-364.

[4] B. Povh and M. Rosina, Bled Workshops in Physics 12/1 (2011) 82.

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OUTLINE

1. INTRODUCTION

2. THE MODEL – PION IN CONSTITUENT QUARKS

3. DEFINITIONS

4. NUCLEON OBSERVABLES

5. ALTERNATIVE PARAMETRIZATION

6. PROTON HAS A NEUTRON+PION COMPONENT

7. EXPERIMENTAL TEST OF PION FLUCTUATION

8. LEADING PROTON PRODUCTION: p→ p π0

9. LEADING PROTON PRODUCTION: p→ p π+π−

10. CONCLUSION

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1. INTRODUCTION

Is it a better picture in which each constituent quark con-

tains its own pion cloud, or that with pions coupled directly

to a bare nucleon?

Since the amplitudes of pions from different quarks add up

(interfere) both pictures seem equivalent. If one takes, how-

ever, only the leading multi-pion configuration the descrip-

tions differ, like in comparing jj and LS coupling in nuclear

physics.

We show in which cases the ”private pion” description gives

a better agreement of different observables with experiment

and in which cases both descriptions yield the same values.

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Figure 1: Two different coupling schemes

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2. THE MODEL – PION IN CONSTITUENT QUARKS

|u〉 =

√√√√√(1− 3

2a) |u〉 −

√a|dπ+〉 +

√√√√a2|uπ0〉,

|d〉 =

√√√√√(1− 3

2a) |d〉 +

√a |uπ−〉 −

√√√√a2|dπ0〉.

versus – PION CLOUD AROUND BARE NUCLEON

|p〉 =

√√√√√(1− 3

2α) |p〉 −

√α|nπ+〉 +

√√√√α2|pπ0〉,

The basis of pure flavour quarks (and bare nucleons) is

denoted by red u, d, p and n.

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3. DEFINITIONS

The quantities without superscript p or n refer to proton.

Isospin symmetry is assumed: un = d, dn = u and sn = s.

u(x) = u(x) + u(x).

u = u↑ + u↓, d = d↑ + d↓, s = s↑ + s↓;

∆u = u↑ − u↓, ∆d = d↑ − d↓, ∆s = s↑ − s↓.F p

1 = 12(4

9u + 19d + 1

9 s), F n1 = 1

2(49d + 1

9u + 19 s).

Gottfried = IG = 2(F p1 − F n

1 ) = 13(u− d) = 1

3 + 23(u− d).

Ip = gp1 = 12(4

9∆u + 19∆d + 1

9∆s), In = gn1 = 12(4

9∆d + 19∆u + 1

9∆s),

Id = 12(Ip + In) = 5

18(∆u + 19∆d) + 1

9∆s.

∆Σ = ∆u + ∆d + ∆s.

gnpA = ∆u−∆d, gΣnA = ∆d−∆s.

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4. NUCLEON OBSERVABLES

observable definition model value

gA = 1.269± 0.003 ∆u−∆d 53(1− 4

3a) = input

IG = 0.216± 0.033 13 + 2

3(u− d) 13(1− 2a) = 0.214

Ip = 0.120± 0.017 12(4

9∆u + 19∆d) 5

18(1− 53a) = 0.194

Id = 0.043± 0.006 536(∆u + ∆d) 5

36(1− 2a) = 0.089

∆Σ = 0.330± 0.064 ∆u + ∆d (1− 2a) = 0.642

Table 1: The π+ probability a = 0.179 is used.

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Notes

• Both models, (q-π)3 and (N-π), give equal results for the

listed observables.

• The strange sea is neglected in this model

• gA and the Gottfried sum rule IG do not feel the strange sea and are

in excellent agreement with each other. Other observables obviously

lack the strange sea contribution and something else.

• In polarized observables antiquarks do not appear since they appear in

pions with equal probability with spin up and spin down.

• We use IG with screening corrections [?]. Uncorected values are some-

what higher( about 0.227-0.240 from deep inelastic scattering and 0.255

from Drell-Yan).

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5. ALTERNATIVE PARAMETRIZATION

Some authors ( Eichten+Hinchliffe+Quigg; Garvey+Peng; Pirner+Povh

presented somewhat different relations for the polarized observables . Prob-

ably they tacitly included some contribution of the strange sea. The π+

probability has to be fitted again. Some observables like IG come worse,

while the observables sensitive to the strange sea come better.

observable definition model value

gA = 1.269± 0.003 ∆u−∆d 53(1− a) = input

IG = 0.216± 0.033 13 + 2

3(u− d) 13(1− 2a) = 0.174

Ip = 0.120± 0.017 12(4

9∆u + 19∆d) 5

18(1− 2a) = 0.145

Id = 0.043± 0.006 536(∆u + ∆d) 5

36(1− 3a) = 0.039

∆Σ = 0.330± 0.064 ∆u + ∆d (1− 3a) = 0.283

Table 2: The π+ probability a = 0.239 is used.

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6. PROTON CONTAINS A NEUTRON PLUS PION COM-

PONENT

The final state 〈nπ+| has an overlapp with a Fock component

of the proton. The result of the explicit calculation is

|〈nπ+|p〉|2 = |〈dπ+|u〉|2 = (1− 3

2a) a = 0.13.

Although the proton has two u quarks there is no factor 2

in the amplitude, due to the flavor-spin-color structure of

the nucleon. The flavor-spin wavefunction of the proton has

a mixed symmetry combined into a symmetric flavor-spin

function:

|p〉 =

√√√√√1

213

2f× 1

32s

+

√√√√√1

212

3f× 1

23s.

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ISOSPIN FORMALISM

In the act of producing a positive pion, the corresponding u

quark loses one unit of charge, it becomes a d quark. This can

be described with the operator∑i t−(i) = T− where T− = Tx−iTy.

We conveniently took the sum over all three quarks since the

third quark, d, contributes zero anyway.

< TM − 1|T−|TM >=√T (T + 1)−M(M − 1)

which for proton (T = 1/2,M = 1/2) gives in fact the factor 1.

It is instructive to compare with ∆+ (T = 3/2,M = 1/2) in the

process ep→ e∆X where one gets the factor 2, pointing out

that the two u quarks are always symmetric and interfere

constructively.

For the squared amplitude, we get the additional factors:

a ... from the π+-dressed component of the u-quark ,

(1− 32a) ... for the naked component of the final d-quark.

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7. EXPERIMENTAL TEST OF PION FLUCTUATION

The analysis of the forward neutron production in the e +

p → e + n + X collisions at 300 GeV measured by the H1 and

ZEUS Collaborations at DESY suggests

〈nπ+|p〉2 = 0.17± 0.01

in reasonable agreement with the above theoretical value

|〈nπ+|p〉|2 = |〈dπ+|u〉|2 = (1− 3

2a) a = 0.13.

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8. LEADING PROTON PRODUCTION: p→ p π0

Due to the isospin symmetry,

|〈pπ0|p〉|2 = 12|〈nπ

+|p〉|2 = 12(1− 3

2a) a = 0.065.

In the analysis of leading proton spectrum from DIS at

HERA, A. Szczurek, N.N. Nikolaeva and J. Speth ( 1997) de-

duced the contributions of pion exchange, reggeon exchange

and pomeron exchange.

Our π0 contribution to the leading proton production is,

however, too small. This suggests the importance of two-

pion contributions.

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Figure 2: Analysis of Szczurek, Nikolaeva and Speth ( 1997)

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9. LEADING PROTON PRODUCTION: p→ p π+π−

We again use the isospin formalism. In the act of producing

a positive pion, the corresponding u quark loses one unit of

charge, it becomes a d quark, and viceversa.

∑i 6=jt+(i)t−(j) = T+T− −

∑it+(i)t−(i) = ~T 2 − T 2

z −∑

(~t 2 − t2z) ,

For proton, the expectation value is 34 −

14 − 3× (3

4 −14) = −1.

|〈pπ+π−|p〉|2 = (1− 3

2a)2a2 = 0.017.

This result is even smaller than the probability for the

p→ p π0 production. One possibility is to extend the model.

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THE EXTENDED MODEL

We add two-pion configurations to constituent quarks

|u〉 =

√√√√√(1− 3

2a−−3

2a2) |u〉−

√a|dπ+〉+

√√√√a2|uπ0〉+a |u(π+π−−π0π0/

√2)〉,

|d〉 =

√√√√√(1− 3

2a−−3

2a2) |d〉+

√a |uπ−〉−

√√√√a2|dπ0〉+a |d(π+π−−π0π0/

√2)〉.

In principle, the amplitude of the two-pion configuration

should be a model parameter to be fitted. However, since

we are interested only in the qualitative effect, we assumed

that each addition of a charged pion brings tha same factor√a in the amplitude.

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Since the added sea is flavor symmetric (isoscalar) it does

not change the Gottfried sum rule. Since it is also isotropic

(scalar) it does not contribute to polarization observables.

Therefore, no refitting of a is needed.

The main point is, that the amplitudes for kicking two

pions from different quarks and from the same quark add

coherently.

Ai6=j = (1− 3

2a−−3

2a2) a = 0.122 ,

Aii = 3× (1− 3

2a−−3

2a2) a = 0.366 ,

A = Ai6=j + Aii = 0.488, A2 = 0.24 .

Such calculation of the two-pion contribution is consitent

with their result in the reggeon region. This agreement sug-

gests that their reggeon in fact consists of a scalar two-pion

state present in our model. The pomeron exchange is not rel-

evant for our study since it is related to gluon interaction.

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10. CONCLUSION

The pion fluctuation p→n+π+ and p→p+π0 is an effect of

the quark-antiquark pairs of the constituent quarks. The

agreement between the measured and the calculated values

is a strong support of the constituent quark model.

For the value a = 〈dπ+|u〉2 = 0.18 each quark contains 0.27

qq →∼ 0.8 qq per nucleon. Using this value of a gives 〈nπ+|p〉2 =

0.13, to be compared with the experimental value 0.17 ± 0.01.

It follows that in ≈ 0.26 cases the proton is a neutron+ π+ or

a proton+π0. This means that about one quater of the nu-

cleon’s quark-antiquark pairs show up as pion fluctuations.

All listed observables (asymmetry IG, polarization observ-

ables gA, Ip, In, ∆Σ) as well as the one-pion content of proton

are the same in both models, the private pion cloud, and the

communal pion cloud.

However, the two-pion content as seen in the forward pro-

ton process will distinguish the extended model with two-

pion admixtures in the constituent quark.

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MINI-WORKSHOP BLED 2013:

LOOKING INTO HADRONS

Bled (Slovenia), July 7-14, 2013

LINK: http://www-f1.ijs.si/Bled2013

CONTACT: Mitja.Rosina@ijs.si

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ACKNOWLEDGEMENT

I am pleased to acknowledge the inspiration for this study

and the fruitful collaboration with BOGDAN POVH (Max

Planck Institute for Nuclear Physics at Heidelberg)

THANKS FOR YOUR ATTENTION!

ISHALL APPRECIATEYOUR CRITICISM AND SUGGESTIONS

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