arXiv:hep-lat/0211017 v2 13 Nov 2002ADP-02-97/T

535

Quark

Contrib

utio

nsto

BaryonMagnetic

Moments

inFull,

QuenchedandPartia

llyQuenchedQCD

Derek

B.Lein

weber �

Departm

entofPhysic

sandMathematica

lPhysic

sandSpecialResea

rchCentre

fortheSubatomic

Stru

cture

ofMatte

r,University

ofAdelaide5005,Austra

lia

Anew

intuitive

meth

odfor

therap

idcalcu

lationofthenonanaly

ticbehavior

ofhadron

icobserv

-ables

instan

dard

,quenched

andpartially

-quenched

chiral

pertu

rbation

theory

ispresen

ted.After

prov

ingthetech

niquein

acon

sidera

tionof

octet

bary

onmasses,

thequark

- avor

contrib

ution

sto

themagn

eticmom

ents

ofoctet

baryon

sare

calculated

infull,

quenched

andpartially

-quenched

QCD.Thetech

niqueprov

ides

asep

arationof

quark

-sectormagn

etic-mom

entcon

tribution

sinto

direct

sea-quark

loop,valen

ce-quark

,indirect

sea-quark

loop

andquenched

valence

contrib

ution

s,thelatter

bein

gthecon

vention

alview

ofthequenched

approx

imation

.Both

meson

andbaryon

mass

violation

sofSU(3)-

avorsymmetry

areaccou

nted

for.Acom

preh

ensiv

eexam

ination

ofthe

individual

quark

-sectorcon

tribution

sto

octet

bary

onmagn

eticmom

entsrev

ealsnumerou

sexcitin

gopportu

nities

toobserve

andunderstan

dtheunderly

ingstru

cture

ofbaryon

sandthenatu

reof

chiral

nonanaly

ticbehavior

inQCD

andits

quenched

variants.

Inparticu

lar,thevalen

ceu-quark

contrib

ution

totheproton

magn

eticmom

entprov

ides

theoptim

alopportu

nity

todirectly

view

nonanaly

ticbehavior

associated

with

themeson

cloudof

fullQCDandthequenched

meson

cloud

ofquenched

QCD.Theuquark

in�+prov

ides

thebest

opportu

nity

todisp

laytheartifacts

ofthe

quenched

approx

imation

.

PACSnumbers:

12.39.Fe,12.38.Gc,13.40.Em

I.IN

TRODUCTION

Separatio

nofthevalen

ceandsea-q

uark

-loop

contri-

bution

sto

themeso

nclo

udoffullQCD

hadron

sis

anon-triv

ialtask.Early

calculatio

nsaddressin

gtheme-

sonclo

udof

meso

nsem

ployed

adiagrammatic

meth

od

[1].Theform

altheory

ofquenched

chira

lpertu

rbation

theory

(Q�PT)was

subseq

uently

establish

edin

Ref.

[2].There,

meson

properties

were

examined

inaform

ula-

tionwhere

extra

commutin

gghost-q

uark

�eld

sare

intro-

duced

toelim

inate

thedependence

ofthepath

integral

ontheferm

ion-m

atrix

determ

inant.

Thisapproach

was

exten

ded

tothebaryon

sectorin

Ref.

[3].

While

theform

alismofQ�PTisessen

tialto

establish

-ingthe�eld

theoretic

properties,

itisdesira

ble

tofor-

mulate

aneÆ

cientandperh

apsmore

intuitiv

eapproach

tothecalcu

lation

ofquenched

chira

lcoeÆ

cients.

Rath

erthan

intro

ducin

gextra

degrees

offreed

omto

remove

the

e�ects

ofsea

-quark

loops,theapproa

chdescrib

edherein

intro

duces

aform

alism

fortheidenti�

cationandcalcu

-lation

ofsea

-quark

-loop

contrib

ution

sto

hadron

prop

er-ties,

allowingthesystem

aticsep

aration

ofvalen

ce-and

sea-quark

contrib

utio

nsto

baryon

form

factorsingen

eral.Uponrem

ovingthecontrib

utio

nsofsea

-quark-lo

ops,one

arrivesatthecon

vention

alview

ofquenched

chiral

per-

turbatio

ntheory.

Abrief

acco

untof

these

meth

odsis

publish

edin

Ref.

[4].Since

thispresen

tation,there

hasbeen

aresu

r-

�Electro

nic

address:

dleinweb@physic

s.adelaide.edu.au;

URL:

http://www.physics.adelaide.edu.au/theory/staff/

leinweber/

gence

inquenched

andpartially

-quenched

�PTcalcu

la-tion

sof

baryon

magn

eticmom

ents[5,

6].In

particu

lar,themagn

eticmom

entsofoctet

bary

onshave

been

exam

-ined

[5]usin

gtheform

alapproach

ofQ�PT

[3].There

thelead

ing-n

onanaly

tic(LNA)behavior

ofthemagn

eticmom

entfor

eachbaryon

oftheoctet

iscalcu

lated.The

formalap

proach

completely

eliminates

allsea-q

uark

-loop

contrib

ution

sto

quenched

baryon

mom

ents.

How

ever,sea-q

uark

loopsdomake

acon

tribution

tomatrix

elements

inthequenched

approx

imation

.Inser-

tionof

thecurren

tin

calculatin

gthethree-p

ointcorrela-

tionfunction

prov

ides

pair(s)

ofquark

-creationandan-

nihilation

operators.

These

canbecon

tractedwith

the

quark

�eld

operators

ofthehadron

interp

olating�eld

sprov

iding\con

nected

insertion

s"of

thecurren

t,or

self-con

tractedto

formadirect

sea-quark

-loop

contrib

ution

or\discon

nected

insertion

"of

thecurren

t.Thelat-

tercon

tribution

sto

baryon

electromagn

eticform

factorsare

under

inten

seinvestigation

inquenched

simulation

s[7,

8,9,10].

Hence

inform

ulatin

gquenched

chiralp

ertur-

bation

theory

itisim

portan

tto

prov

idean

opportu

nity

toinclu

dethese

particu

larsea-q

uark

-loop

contrib

ution

s.Amore

exibleapproach

tothecalcu

lationof

quenched

chiral

coeÆ

cients

isdesirab

le.

Moreover,

thepresen

tcalcu

lations[5,

6,11]

ofchi-

ralnonanaly

ticbehavior

inbary

onmagn

eticmom

ents

focuson

bulk

baryon

prop

erties.Theform

alismpre-

sented

here

prov

ides

ameth

odfor

theisolation

ofindi-

vidualquark

sectorcon

tribution

s[12,

13,14,

15,16,

17]to

formfactors

infull,

quenched

andpartially

-quenched

QCD.Individual

quark

sectorcon

tribution

sto

thenu-

cleonmagn

eticmom

entsare

under

experim

ental

investi-

gation[18,

19,20]

where

thestran

ge-quark

contrib

ution

tothenucleon

mom

entisof

param

ountinterest.

Under-

brought to you by C

OR

EV

iew m

etadata, citation and similar papers at core.ac.uk

provided by CE

RN

Docum

ent Server

2

standing the manner in which quarks compose baryonsis essential to a complete understanding of QCD [16].In contrast to conventional calculations of chiral non-

analytic structure, SU (3)- avor violations in both mesonand baryon masses are accounted for in the following.These are particularly important for K-meson dressingsof hyperons where the baryon mass splitting can be pos-itive or negative, suppressing or enhancing contributionsdepending on whether the intermediate baryon is heavieror lighter respectively.Hence, the purpose of this study is three-fold:

1. To provide the �rst calculation of the leading non-analytic behavior of quark-sector contributions tobaryon magnetic moments in full, quenched andpartially-quenched QCD.

2. To extend previous calculations to include both me-son and baryon mass splittings in SU (3)- avor vi-olations.

3. To introduce a new method for calculating the chi-ral coeÆcients of quenched and partially-quenchedQCD which is rapid and transparent, allowing com-plete exibility in the consideration of quark con-tributions to baryon form factors.

In the process, we will see that it is possible to separatevalence- and sea-quark contributions to baryon form fac-tors in full QCD and in quenched QCD. Moreover, it willbecome apparent that the technique and most of the re-sults may be applied to other baryon form factor studiesin general.Sec. II presents the essential concepts for isolating

and calculating sea-quark loop contributions to baryonproperties and proves the technique via a considera-tion of baryon masses. The derivation of the quenchedchiral coeÆcients for the quark-sector contributions tothe quenched magnetic moments of octet baryons is de-scribed in Sec. III. The technique provides a separationof magnetic moment contributions into \total" full-QCDcontributions, \direct sea-quark loop" and \valence" con-tributions of full-QCD. The latter are obtained by re-moving the direct-current coupling to sea-quark loopsfrom the total contributions. Upon further removing\indirect sea-quark loop" contributions, one obtains the\quenched valence" contributions, the conventional viewof the quenched approximation. We will use the quotedterms for reference to these contributions in the following.Sec. III also accounts for both baryon mass and mesonmass violations of SU (3)- avor symmetry. The quenched�0 gives rise to new nonanalytic behavior [5] and this isbrie y reviewed in Sec. IIID.A comprehensive examination of the individual quark-

sector contributions to octet baryon magnetic momentsis presented in Sec. IV. General expressions are accom-panied by numerical evaluations to identify channels ofparticular interest. Partially-quenched results are pre-sented in Sec. V. Sec. VI provides a summary of thehighlights of the �ndings.

II. QUENCHED BARYON MASSES

A. Formalism

The SU (3)- avor invariant couplings are described inthe standard notation by de�ning

B =

0BBBBBB@

�0

p2+

�p6

�+ p

�� ��0

p2+

�p6

n

�� �0 � 2�p6

1CCCCCCA

; (1)

Poct =

0BBBBB@

�0p2+

�p6

�+ K+

�� � �0p2+

�p6

K0

K� K0 � 2�p6

1CCCCCA ; (2)

and

Psin =1p3diag(�0; �0; �0) : (3)

The SU (3)-invariant combinations are�BBP

�F

= Tr(BPoctB)� Tr(BBPoct); (4)�BBP

�D

= Tr(BPoctB) + Tr(BBPoct); (5)�BBP

�S

= Tr(BB)Tr(Psin) : (6)

The following calculations are simpli�ed through the useof the corresponding interaction Lagrangians [21]. Theoctet interaction Lagrangian is

Loctint = �fNN� (N�TN )�� + if���(���)���f���(�� + ��)�� � f���(��

T�)���f�NK

�(NK)� + �(KN )

��f��K

�(�Kc)� + �(Kc�)

��f�NK

���(K�TN ) + (N�TK)���

�f��K���(Kc�

T�) + (��TKc)���

�fNN� (NN )� � f���(��)�

�f���(���)� � f���(��)�: (7)

and the singlet interaction Lagrangian is

Lsinint = �fNN�0 (NN )�0 � f���0(��)�0

�f���0(���)�0 � f���0 (��)�0; (8)

where

N =

�pn

�; � =

��0

��

�;

K =

�K+

K0

�; Kc =

�K0

�K�

�:

(9)

3

The octet meson-baryon couplings are expressed in termsof the F and D coupling coeÆcients as follows:

fNN� = F +D; f�NK = � 1p3(3F +D);

fNN� =1p3(3F �D); f��� = 2F;

f��K = 1p3(3F �D); f��� = � 2p

3D;

f��� =2p3D; f�NK = D � F;

f��� =2p3D; f��� = F �D;

f��K = �(F +D); f��� = � 1p3(3F +D);

(10)and the singlet couplings satisfy

fNN�0 = f���0 = f���0 = f���0 : (11)

The light quark content of

j�0 i = 1p3

�juu i+ jdd i+ jss i� ; (12)

and

j� i = 1p6

�juu i+ jdd i � 2 jss i� ; (13)

mesons suggests

fNN�0 =p2fNN� : (14)

This relation between nucleon octet and singlet couplingsis commonly used in Q�PT calculations to estimate �0

couplings to octet baryons.In the following, numerical estimates are based on the

tree-level axial couplings F = 0:50 and D = 0:76 withf� = 93 MeV. We focus on the leading-nonanalytic be-havior which provides the most rapid variation in baryonmagnetic moments.

B. Baryon Mass

To calculate the quenched chiral coeÆcients, we be-gin by calculating the total full-QCD contribution in thelimit where the � and �0 mesons are taken to be degener-ate with the pion. In the quenched approximation, quarkloops which otherwise break this degeneracy are absent.The quark ow diagrams of Fig. 1 illustrate the processeswhich give rise to the LNA behavior of proton observ-ables. Table I summarizes the contributions of the �-,�-, �0- and K-cloud diagrams of Fig. 1 labeled by thecorresponding quark- ow diagrams. Summation of thesecouplings and incorporation of the factors from the loopintegral provides the LNA term proportional to m3

� of

� (3F 2 +D2)m3�

8�f2�: (15)

The separation of the meson cloud into valence andsea-quark contributions is shown in the quark- ow di-agrams labeled by letters. To quench the theory, one

TABLE I: Meson-cloud contributions of Fig. 1 in full QCD. �and �0 masses are set degenerate with the pion in anticipationof quenching the theory.

Fig. Channel Mass Coupling Coupling

a,b,c p�0 N� f2NN� (F +D)2

a,b,c p � N� f2NN� (3F �D)2=3

a,b,c p �0 N� f2NN�0 2(3F �D)2=3

f,g n�+ N� 2f2NN� 2(F +D)2

h �K+ �K f2�NK (3F +D)2=3

h �0K+ �K f2�NK (D � F )2

i �+K0 �K 2f2�NK 2(D� F )2

TABLE II: Sea-quark-loop contributions of Fig. 1.

Fig. Channel Mass Coupling Coupling

b �K+ N� f2�NK (3F +D)2=3

b �0K+ N� f2�NK (D� F )2

c �+K0 N� 2f2�NK 2(D� F )2

e �+K0 N� 2f2�NK 2(D� F )2

g �K+ N� f2�NK (3F +D)2=3

g �0K+ N� f2�NK (D� F )2

h �K+ �K f2�NK (3F +D)2=3

h �0K+ �K f2�NK (D� F )2

i �+K0 �K 2f2�NK 2(D� F )2

must understand the chiral behavior of the valence-quarkloops of Figs. 1(a), (d) and (f) and the sea-quark loops ofFigs. 1(b), (c), (e), (g), (h) and (i) separately. If one canisolate the behavior of the diagrams involving a quarkloop, then one can use the known LNA behavior of thefull meson-based diagrams to extract the correspondingvalence-loop contributions.For example, Fig. 1(b) involves a u-quark loop where

no exchange term is possible. Thus the u-quark in theloop is distinguishable from all the other quarks in thediagram. The chiral structure of this diagram is thereforeidentical to that for a \strange" quark loop, as illustratedin Fig. 1(h), (p ! K+�0 or p ! K+�) provided the\strange" quark in this case is understood to have thesame mass as the u-quark.

Similarly, the corresponding hadron diagram whichgives rise to the LNA structure of Fig. 1(c) is thereforethe K0-loop diagram of Fig. 1(i), with the distinguish-able \strange" quark mass set equal to the mass of thed-quark. That is, the intermediate �-baryon mass ap-pearing in the K0-loop diagram is degenerate with thenucleon. Similarly, the \kaon" mass is degenerate withthe pion.The sum of the �rst two lines of Table II provides the

contribution of diagramFig. 1(b). The third line providesthe contribution of Fig. 1(c). Similar arguments allow

4

FIG. 1: The pseudo-Goldstone meson cloud of the proton and associated quark ow diagrams.

one to establish the remaining loop contributions to thelight-meson cloud. Summing the couplings of Table IIindicates sea-quark-loops contribute a term

� (9F 2 � 6FD + 5D2)m3�

24�f2�; (16)

to the LNA behavior of the nucleon such that the netquenched contribution proportional to m3

� is

� (3FD �D2)m3�

12�f2�; (17)

in agreement with the formal approach of Labrenz andSharpe [3]. We note that the kaon-cloud contributionsof Fig. 1(h) and (i) are pure sea contributions and triv-ially vanish in subtracting sea-contributions from totalcontributions as outlined above.

III. BARYON MAGNETIC MOMENTS

A. Quark-Sector Contributions to the Proton

The LNA contribution to baryon magnetic momentsproportional to m� or mK has its origin in couplings ofthe electromagnetic (EM) current to the meson propa-gating in the intermediate meson-baryon state. In order

to pick out a particular quark- avor contribution, onesets the electric charge for the quark of interest to oneand the charge of all other avors to zero.Tables III through V report results for the u-quark

in the proton. The total contributions are calculated inthe standard way, but with charge assignments for theintermediate mesons (indicated in the Charge column)re ecting in this case qu = 1 and qd = qs = 0. The extrabaryon subscripts on the meson masses are a reminderof the baryons participating in the diagram to facilitatemore accurate treatments of the loop integral in whichbaryon mass splittings are taken into account. The LNAcontribution is

�mN

8�f2�m� � �m� (18)

with � and � indicated in the last two columns. Through-out the following, the units of � are �N=GeV, such thatwhen multiplied by the pion mass in GeV, one obtainsmagnetic moment contributions in units of the nuclearmagneton, �N .The \direct sea-quark-loop contributions" indicated in

Table IV are contributions in which the EM current cou-ples to a sea-quark loop, in this case a u quark. Usingthe techniques described in Sec. II, one can calculate thecontributions of these loops alone to the baryon magneticmoment. The Mass column of Table IV is a reminderthat the mass of the \kaon" considered in determining

5

TABLE III: Determination of the total u-quark contribution to the proton magnetic moment as illustrated in Fig. 1.

Diagram Channel Mass Charge Term � �

f,g n �+ N� +1 +2f2NN�m� �(F +D)2 �6:87

h �0K+ �K +1 +f2�NKmN�K �(D � F )2=2 �0:15

h �K+ �K +1 +f2�NKmN�K �(3F +D)2=6 �3:68

TABLE IV: Determination of direct u-quark sea-quark loop contributions to the proton magnetic moment as illustrated inFig. 1.

Diagram Channel Mass Charge Term � �

b �K+ N� �1 �f2�NKm� (3F +D)2=6 +3:68

b �0K+ N� �1 �f2�NKm� (D� F )2=2 +0:15

e �+K0 N� �1 �2f2�NKm� (D� F )2 +0:29

Total +4:12

TABLE V: Indirect sea-quark loop contributions from u valence quarks to the proton magnetic moment. Here, the u-valencequark forms a meson composed with a sea-quark loop as illustrated in Fig. 1.

Diagram Channel Mass Charge Term � �

b �K+ N� +1 +f2�NKm� �(3F +D)2=6 �3:68

b �0K+ N� +1 +f2�NKm� �(D � F )2=2 �0:15

g �K+ N� +1 +f2�NKm� �(3F +D)2=6 �3:68

g �0K+ N� +1 +f2�NKm� �(D � F )2=2 �0:15

h �0K+ �K +1 +f2�NKmN�K �(D � F )2=2 �0:15

h �K+ �K +1 +f2�NKmN�K �(3F +D)2=6 �3:68

the coupling is actually the pion mass for Figs. 1(b) and(e). These diagrams will contribute, even in the quenchedapproximation, when disconnected insertions of the EMcurrent are included in simulations [7, 8, 9, 10].Subtraction of these sea-quark-loop contributions from

the total contributions of III leaves a net valence contri-bution of �11:0m� � 0:15mN�K � 3:68mN�K in full

QCD.Table V focuses on diagrams in which the EM current

couples to a valence quark in a meson composed with asea-quark loop. These are the \indirect sea-quark loop"contributions. Subtracting o� these couplings from thevalence contribution provides the net quenched valencecontribution of �3:33m�.Tables VI through VIII provide a similar analysis of

the d quark in the proton, where qd = 1 and qu = qd = 0.Subtraction of the direct sea-quark loop contributions ofTable VII from the total contributions of Table VI leavesa net valence contribution of 2:75m� � 0:29mN�K. Fur-ther removal of the indirect sea-quark loops of Table VIIIprovides the �nal net d-quark quenched valence contri-bution to the proton moment of +3:33m�.Table IX describes the s-quark contributions to the

proton magnetic moment, where qs = 1 and qu = qd = 0.As there are no s valence quarks in the proton, the con-

tributions are purely sea-quark-loop contributions. Thenet valence contribution is zero and there are no furtherquenching considerations.Charge symmetry provides the quark-sector contribu-

tions to the neutron magnetic moment. For unit chargequarks, dn = up, un = dp and sn = sp.The QCD Lagrangian is avor blind in the SU (3)-

avor symmetry limit. This independence from quark avor is manifest in Tables IV, VII and IX for the directsea-quark loop contributions to the proton form factor.In each case there are three channels for the coupling,�. Indeed the u and d direct sea-quark loop contribu-tions are exactly equal. However SU (3)- avor symmetrybreaking due to the massive s quark requires one to trackthe masses of intermediate mesons and baryons, and thisintroduces the K, � and � masses in Table IX.SU (3)- avor symmetry is also manifest in the indirect

sea-quark loop contributions to the proton magnetic mo-ment. For example, the u-quark indirect sea-quark loopresult receives contributions from each of u, d and s quarkloops in Figs. 1(b), 1(g) and 1(h). Each of these contri-butions appearing in Table V are equal up to symmetrybreaking in the meson and baryon masses. Similar re-sults hold for the d-valence quark of the proton in TableVIII.

6

TABLE VI: Determination of the total d-quark contribution to the proton magnetic moment as illustrated in Fig. 1.

Diagram Channel Mass Charge Term � �

f,g n�+ N� �1 �2f2NN�m� (F +D)2 +6:87

i �+K0 �K +1 +2f2�NKmN�K �(D� F )2 �0:29

TABLE VII: Determination of direct d-quark sea-quark loop contributions to the proton magnetic moment as illustrated inFig. 1.

Diagram Channel Mass Charge Term � �

c �+K0 N� �1 �2f2�NKm� (D� F )2 +0:29

g �K+ N� �1 �f2�NKm� (3F +D)2=6 +3:68

g �0K+ N� �1 �f2�NKm� (D� F )2=2 +0:15

Total +4:12

TABLE VIII: Indirect sea-quark loop contributions from d valence quarks to the proton magnetic moment. Here the d-valencequark forms a meson composed with a sea-quark loop as illustrated in Fig. 1.

Diagram Channel Mass Charge Term � �

c �+K0 N� +1 2f2�NKm� �(D� F )2 �0:29

e �+K0 N� +1 2f2�NKm� �(D� F )2 �0:29

i �+K0 �K +1 +2f2�NKmN�K �(D� F )2 �0:29

TABLE IX: Determination of the total s-quark contribution to the proton magnetic moment as illustrated in Fig. 1. As thereare no s valence quarks in the proton, the contributions are purely sea-quark-loop contributions.

Diagram Channel Mass Charge Term � �

h �0K+ �K �1 �f2�NKmN�K (D� F )2=2 0:15

h �K+ �K �1 �f2�NKmN�K (3F +D)2=6 3:68

i �+K0 �K �1 �2f2�NKmN�K (D� F )2 0:29

The avor-blind nature of QCD makes it trivial to ex-tend this calculation of quenched quark-sector magneticmoments to the partially-quenched theory. As new a-vors are introduced through the use of dynamically gen-erated gauge �elds, one simply adds the direct and indi-rect sea-quark loop contributions evaluated here to thequenched results, keeping track of the meson mass of thevalence-sea meson. The latter is simple to do as we havealready isolated each valence quark avor contribution tothe baryon moment. This is described in further detailin Sec. V.

B. Quark-Sector Contributions to �+

Tables X through XII describe the various LNA con-tributions of the u quark to the �+ magnetic momentderived from Fig. 2. The total contribution of the uquark alone is isolated by setting the charge of the sand d quarks to zero, and otherwise using standard tech-

niques. The LNA behavior of the u-quark contribution tothe �+ magnetic moment is �2:16m��� � 1:67m��� �6:87m��K. Sea-quark-loop contributions are isolated byusing meson-baryon couplings where quark loops of u ands quark avors (the valence avors of �+) are replaced bya d quark. Table XI summarizes the direct sea-quark loopcontributions. Subtracting these contributions leaves anet valence contribution of �0:29m�NK � 4:32m��� �3:33m����6:87m��K in full QCD. Table XII describesindirect sea-quark loop contributions from u valencequarks in mesons formed with a sea-quark loop. Re-moving these contributions provides the net quenchedu-valence contribution of �0:29m�NK � 3:04m��K.

The d-quark contributions to the LNA behavior of the�+ magnetic moment are pure sea in origin. Thereforethe total contributions are the sea contributions suchthat the valence d-quark contributions vanish. Table XIIIsummarizes the contributions.

s-quark contributions to the �+ magnetic momentare summarized in Tables XIV through XVI. Removal

7

FIG. 2: The pseudo-Goldstone meson cloud of �+ and associated quark ow diagrams.

of the direct sea-quark-loop contributions from the to-tal contributions provides an s-valence contribution of�0:29m�NK+3:04m��K�0:29m���s in full QCD. Fur-ther removal of the indirect sea-quark loop contributionsof Table XVI provides the net quenched s-valence contri-bution of +0:29m�NK + 3:04m��K.Charge symmetry provides the quark sector contribu-

tions to the �� baryon, while the �0-baryon results areobtained from the isospin average of �+ and ��.The SU (3)- avor symmetry of the direct sea-quark-

loop contributions to the �+ baryon magnetic momentis manifest throughout Tables XI, XIII and XV. How-ever, the implementation of SU (3)- avor breaking via thehadron masses hides the avor symmetry in the resultssummarized in Sec. IV, where both meson and baryonmass splittings are maintained. SU (3)- avor breakinggives rise to very di�erent behaviors for these contribu-tions. This is particularly true in the common applicationof holding the strange-quark mass �xed while varying thelight u and d masses. In this case the �s meson mass isconstant.

C. � and � Baryons

The derivation of the quark sector contributions to �baryons proceeds in precisely the same manner as that forthe � baryons. As there are no new concepts, derivationis left as an exercise for the interested reader. �-baryonresults are summarized in Sec. IV.However, the avor singlet structure of the � baryon

presents a problem to the approach described thus far.The necessary presence of u-, d- and s-quark avors si-multaneously, appears to require the introduction of a

fourth quark avor and its associated SU(4) couplings todescribe the disconnected sea-quark loop contributions.Fortunately one can exploit the SU (3)- avor symme-

try relation among octet baryons. Such a relation is man-ifest in two- and three-point correlation functions for the� [12]. Denoting the two-point correlation function for�0 as �0

s(x), one has

�(x) =1

3

�2�0

u(x) + 2�0d(x)� �0

s(x)�; (19)

where �0s(x) has symmetry between u and d quarks,

�0u(x) has symmetry between s and d quarks, and simi-

larly �0d(x) has symmetry between u and s quarks. Just

as

�0s(x) =

1

2

��+(x) + ��(x)

�; (20)

one also has

�0u(x) =

1

2

�n(x) + �0(x)

�; (21)

and

�0d(x) =

1

2

�p(x) + ��(x)

�; (22)

in the SU (3)- avor limit, such that

�(x) =1

3

�p(x) + n(x) + �0(x) + ��(x)

�1

2

��+(x) + ��(x)

��: (23)

Note that this relation holds for any electric-charge as-signments to the quark avors, such that individualquark- avor contributions can be resolved.

8

TABLE X: Determination of the total u-quark contribution to the �+ magnetic moment as illustrated in Fig. 2.

Diagram Channel Mass Charge Term � �

c �0 �+ �� +1 f2���m��� �2F 2 �2:16

c ��+ �� +1 f2���m��� �2D2=3 �1:67

g,h �0K+ �K +1 2f2��Km��K �(F +D)2 �6:87

TABLE XI: Determination of direct u-quark sea-quark loop contributions to the �+ magnetic moment as illustrated in Fig. 2.

Diagram Channel Mass Charge Term � �

b �0 �+ �� �1 �f2���m��� 2F 2 +2:16

b ��+ �� �1 �f2���m��� 2D2=3 +1:67

e pK0 NK �1 �2f2�NKm�NK (D� F )2 +0:29

TABLE XII: Indirect sea-quark loop contributions from u valence quarks to the �+ magnetic moment. Here the u-valencequark forms a meson composed with a sea-quark loop as illustrated in Fig. 2.

Diagram Channel Mass Charge Term � �

b �0 �+ �� +1 f2���m��� �2F 2 �2:16

b � �+ �� +1 f2���m��� �2D2=3 �1:67

c �0 �+ �� +1 f2���m��� �2F 2 �2:16

c � �+ �� +1 f2���m��� �2D2=3 �1:67

h �0 �+ �K +1 f2���m��K �2F 2 �2:16

h � �+ �K +1 f2���m��K �2D2=3 �1:67

TABLE XIII: Determination of the total d-quark contribution to the �+ magnetic moment as illustrated in Fig. 2. d-quarkcontributions are purely sea in origin such that the valence contribution vanishes.

Diagram Channel Mass Charge Term � �

c �0 �+ �� �1 �f2���m��� 2F 2 +2:16

c ��+ �� �1 �f2���m��� 2D2=3 +1:67

f pK0 NK �1 �2f2�NKm�NK (D� F )2 +0:29

Of course it is essential to recover SU (3)- avor viola-tions. To do this one begins exactly as for the protonor �+ described above by constructing the quark- owdiagrams describing the one-loop meson cloud of the �.The couplings of all sea-quark loop contributions can berelated to the three quark- ow diagrams of Fig. 3. Un-known couplings f2u, f

2d and f

2s are introduced to describe

the couplings of diagrams (a), (b) and (c) respectively.Our working approximation of exact SU (2)-isospin sym-metry at the current-quark level provides f2d = f2u in �,leaving two parameters, f2u and f2s , to be determined viathe SU (3) relation of Eq. (23). As both the light- andstrange-quark contributions to the � moment can be re-solved, there are two SU (3) relations to constrain the twoparameters f2u and f2s . This is particularly easy, whenone recalls that the indirect sea-quark loop contributionfrom a u or s valence quark participating in a meson con-

structed with a sea-quark loop are proportional to eitherf2u or f2s alone. Results are summarized in Sec. IV.

D. Quenched Exotics

The double hair-pin graph of Fig. 4 associated withthe quenched-�0 meson gives rise to new singular log(m�)behavior in the chiral limit [5]. This logarithmic termprovides a correction to the tree-level term. The con-tribution has its origin in the loop integral of Fig. 4(a)corresponding to

�i16�2

3

Zd4q

(2�)4q2 � (v � q)2

[v � q + i�]2[q2 �m21 + i�][q2 �m2

2 + i�]:

(24)For equal singlet-meson masses m1 = m2, as in the quark ow of Fig. 4(b), this integral provides the nonanalytic

9

TABLE XIV: Determination of the total s-quark contribution to the �+ magnetic moment as illustrated in Fig. 2.

Diagram Channel Mass Charge Term � �

f pK0 NK +1 2f2�NKm�NK �(D� F )2 �0:29

g,h �0K+ �K �1 �2f2��Km��K (F +D)2 +6:87

TABLE XV: Determination of direct s-quark sea-quark loop contributions to the �+ magnetic moment as illustrated in Fig. 2.�s denotes the ss � meson.

Diagram Channel Mass Charge Term � �

h �0 �+ �K �1 �f2���m��K 2F 2 +2:16

h ��+ �K �1 �f2���m��K 2D2=3 +1:67

i pK0 ��s �1 �2f2�NKm���s (D � F )2 +0:29

TABLE XVI: Indirect sea-quark loop contributions from s valence quarks to the �+ magnetic moment. Here the s-valencequark forms a meson composed with a sea-quark loop as illustrated in Fig. 2. �s denotes the ss � meson.

Diagram Channel Mass Charge Term � �

e pK0 NK +1 2f2�NKm�NK �(D� F )2 �0:29

f pK0 NK +1 2f2�NKm�NK �(D� F )2 �0:29

i pK0 ��s +1 2f2�NKm���s �(D� F )2 �0:29

behavior of

log

�m2

�2

�: (25)

However, Fig. 4(c) indicates that the meson masses in thedouble hair-pin graph need not be equal when consideringhyperon magnetic moments. Here the �0-meson massesm1 and m2 may correspond to di�erent quark-antiquarksources; e.g. �0(uu) versus �0(ss). When m1 6= m2, one

FIG. 3: Key quark- ow diagrams for the � baryon. Diagrams(a), (b) and (c) are proportional to the introduced couplingsf2u, f

2d and f2s respectively.

FIG. 4: Diagrams giving rise to the logarithmic divergence ofa baryon magnetic moment in the quenched approximation.The cross on the meson propagator in (a) denotes the doublehairpin graph of the quark- ow diagrams of (b) and (c).

�nds a nonanalytic contribution of

m21 log

�m21

�2

��m2

2 log�m22

�2

�m2

1 �m22

: (26)

Hence, for the hyperons, one must isolate the doubly-and singly-represented quark sector couplings to �0

mesons. Consider for example, �0 couplings for �+ in-volving u quarks. The transition �+ ! �+�0 involves u

10

quarks alone at one loop and can be used to determinethe �0u coupling. The �+-�0 coupling is f��� = 2F .Since

j�0 i =1p2

�juu i� jdd i� ;j� i =

1p6

�juu i+ jdd i � 2 jss i� ;j�0 i =

1p3

�juu i+ jdd i + 2 jss i� ;one has

juu i = 1p6

�p3 j�0 i+ j� i+

p2 j�0 i

�: (27)

The �0u coupling isp2=3 of the pion coupling to j uu i;

i.e. 2p2=3F .

For the singly represented quark sector, consider �0 !�0�0 involving u-quarks alone at one loop. Here the u-quark coupling is f��� = (F � D), such that the �0ucoupling to � baryons is

p2=3 (F �D).

To check this separation of quark sector contributionsto �0 contributions, consider the proton. Here the contri-bution to intermediate �0 states is

2

3

�4F 2 I(�u; �u) + 2 � 2F (F �D)I(�u; �d)

+(F �D)2I(�d; �d); (28)

where I(�u; �d) denotes the loop integral of Fig. 4(a) forquark ow diagram Fig. 4(c). For equal �u and �d massesone recovers

2

3(3F �D)2 log

�m2

�2

�; (29)

where the leading factor is the standard NN�0 coupling.Double-hairpin �0 contributions to � and � baryons maybe obtained from Eq. (28) with the appropriate quark- avor assignments. For example, the factor multiply-ing the tree level contribution to ��-baryon quark-sectormagnetic moments is

1 � �02

3

�4F 2 log

�m2ss

�2

�

+2 � 2F (F �D)m2ss log

�m2ss

�2

��m2

� log�m2�

�2

�m2ss �m2

�

+(F �D)2 log

�m2�

�2

��; (30)

where remaining loop-integral factors have been incorpo-rated in

�0 =M2

0

16�2 f2�; (31)

with the double hair-pin interaction strength M0 � 0:75GeV [22, 23]. While this logarithmic divergence domi-nates the chiral expansion near the chiral limit, applica-tion of these results to the extrapolation of the quenchedproton magnetic moment [24] reveals that the curvatureassociated with this term is small for m2

� & 0:1 GeV2.

IV. RESULTS

This approach allows one to separate an individualquark- avor contribution to a baryon form factor into�ve categories, namely: \total" full-QCD contributions,\direct sea-quark loop," and \valence" contributions offull-QCD, obtained by removing the direct current cou-pling to sea-quark loops from the total contributions.Upon further removing \indirect sea-quark loop" con-tributions, one obtains the \quenched valence" contribu-tions. Tables XVII and XVIII report the axial couplingsfor these quark-sector contributions to baryon magneticmoments.The LNA \direct sea-quark loop" contribution is rele-

vant to disconnected insertions of the EM current in ei-

ther full or quenched QCD, whereas the LNA \valence"contribution is relevant to connected insertions of the EMcurrent only in full QCD. The �nal category of \quenchedvalence" contributions is relevant to connected insertionsof the EM current in quenched QCD. The latter is com-monly referred to as the quenched QCD result.The channels denoted K in Tables XVII, XVIII and

in the following actually involve the propagation of anoctet sss baryon; i.e. the �� baryon with md = ms.In separating valence and sea-quark loop contributions,the cancellation of valence and sea-quark loop octet-sss-baryon contributions does not occur. Figure 5 providesquark ow diagrams for �0 ! �K+ which illustratethis phenomenon.In the SU (3)- avor symmetry limit, Figs. 5(a) and (b)

are equivalent due to the avor-blindness of QCD inter-

FIG. 5: Quark- ow diagrams illustrating the presence of ansss-octet baryon propagating in each of diagrams (a) and (c).Further discussion is provided in the text.

11

actions. Fig. 5(b) certainly has overlap with an octet-�� �+ intermediate state. Hence the diagramof Fig. 5(a)also has an octet baryon propagating in the interme-diate state. Of course, we know there is no sss octetbaryon and this problem is solved by the contributionof Fig. 5(c) which must be equal but opposite in signto Fig. 5(a) when an octet baryon propagates in the in-termediate state, thus eliminating the octet sss baryoncontribution in full QCD. In separating valence and seacontributions, each quark ow graph must be taken onits own such that octet baryons are not necessarily elim-inated. Indeed, in quenched QCD, only Fig. 5(c) sur-vives, and this quark ow graph has an sss-octet baryonpropagating in the intermediate state. To some extentthis physics has already been seen in Figs. 1(d) plus (e)where only a decuplet baryon can contribute in full QCD,but octet baryons provide contributions in the process ofseparating valence and sea sectors.

Baryon moments are constructed from the quark sectorcoeÆcients by multiplying the u, d and s results by theirappropriate charge factors and summing. For example,the proton moment is

�p =2

3up � 1

3dp � 1

3sp ; (32)

and the neutron moment is

�n = �1

3up +

2

3dp � 1

3sp : (33)

Similarly, the �+ moment is

��+ =2

3u�+ �

1

3d�+ �

1

3s�+ ; (34)

and the �� moment is

��� = �1

3u�+ +

2

3d�+ �

1

3s�+ : (35)

Tables XIX and XX report the axial couplings for theintermediate meson-baryon channels contributing to thenonanalytic behavior of baryon magnetic moments. Wenote that upon neglecting the baryon mass splittings, onerecovers the full QCD results of Ref. [11] summarized intheir Eqs. (A.2) and (A.4), and the quenched results ofRef. [5] summarized in their Table 2.

Table XXI reports values for the coeÆcient, �, provid-ing the LNA contribution to baryon magnetic moments(�m� or �mK or �m�s as appropriate) by quark sec-tors with each quark avor normalized to unit charge.Charge symmetry provides the contributions for otherbaryons. Values are based on the tree-level axial cou-plings F = 0:50 and D = 0:76 with f� = 93 MeV.Similar results for bulk baryon moments are providedin Table XXII. For convenience, values using the one-loop corrected values [11] of F = 0:40 and D = 0:61 areprovided in Tables XXIII and XXIV respectively.

V. PARTIAL QUENCHING

A. Hadron Masses

The avor-blind nature of QCD makes it trivial to ex-tend this calculation of quenched baryon magnetic mo-ments to the partially-quenched theory. As new avorsare introduced through the use of dynamically gener-ated gauge �elds, one simply adds the direct and indirectsea-quark loop contributions evaluated in Sec. III to thequenched results of Sec. IV. To incorporate hadron massviolations of SU (3)- avor symmetry, one must track themeson mass of the valence-sea meson. As we have al-ready isolated each valence quark avor contribution tothe baryon moment, the mass of the meson is identi�edby the valence- and sea-quark mass composing the me-son.It should be noted that the double hair-pin graph of the

�0 meson remains anomalous in the partially-quenchedtheory [27]. However, the contribution of the �0 prop-agator is suppressed by the di�erence in valence- andsea-quark masses.

B. Sea- and Ghost-Quark Electric Charge

Assignments

There has been some discussion on the electric chargeassignments that may be applied to the various quarksectors of partially-quenched e�ective �eld theory [6]. Inthe conventional view of quenched chiral perturbationtheory, the charges of the commuting ghost-quark �eldsare tied to the valence quark charges in order to eliminateboth the direct and indirect sea-quark loop contributionsof the valence sector. Similarly, for partially-quenchedchiral perturbation theory, it is usually argued that theghost quarks are identical to the valence quarks, exceptfor their statistics.However, it has been indicated that when the num-

ber of sea quarks matches the number of valence quarks,more general charge assignments are possible [6]. Theidea is that when the masses and charges of the sea- andghost-quarks match, these contributions cancel leavingthe theory of full QCD. In this case the charges of the seaand ghost quarks need not be related to the the valencequarks. However, it is essential that the quark masses ofthe valence and ghost sectors match, such that the in-direct sea-quark loop contributions of the valence sectorcontinue to be quenched.We have already argued in the Introduction that it is

important to provide an opportunity to include discon-nected insertions of the electromagnetic current in thequenched approximation. These insertions can be cal-culated in the quenched approximation and give rise todirect sea-quark loop contributions. It is now clear thatthis goal can be realized in the formal theory of quenchedchiral perturbation theory by assigning neutral electriccharges to the ghost-quark �elds. Indirect sea-quark loop

12

TABLEXVII:CoeÆcients,�,providingtheLNAcontributiontonucleon,�and�-baryonmagneticmomentsbyquarksectorswithquarkchargesnormalizedtounit

charge.Intermediate(Int.)meson-baryonchannelsareindicatedtoallowforSU(3)- avorbreakinginboththemesonandbaryonmasses.Total,directsea-quark

loop,valence,indirectsea-quarkloopandquenchedvalencecoeÆcientsareindicated.

q

Int.

TotalQuarkSector

DirectSea-QuarkLoop

ValenceSector

IndirectLoop

QuenchedValence

up

N�

�(D+F)2

(5D2�6DF+9F2)=3

�4(2D2+3F2)=3

�4(D2+3F2)=3

�4D2=3

�K

�(D+3F)2=6

0

�(D+3F)2=6

�(D+3F)2=6

0

�K

�(D�F)2=2

0

�(D�F)2=2

�(D�F)2=2

0

dp

N�

(D+F)2

(5D2�6DF+9F2)=3

�2(D2�6DF+3F2)=3

�2(D�F)2

4D2=3

�K

�(D�F)2

0

�(D�F)2

�(D�F)2

0

s p

�K

(D+3F)2=6

(D+3F)2=6

0

0

0

�K

3(D�F)2=2

3(D�F)2=2

0

0

0

u�+

��

�2F2

2F2

�4F2

�4F2

0

��

�2D2=3

2D2=3

�4D2=3

�4D2=3

0

NK

0

(D�F)2

�(D�F)2

0

�(D�F)2

�K

�(D+F)2

0

�(D+F)2

�2(D2+3F2)=3

(�D2�6DF+3F2)=3

d�+

��

2F2

2F2

0

0

0

��

2D2=3

2D2=3

0

0

0

NK

(D�F)2

(D�F)2

0

0

0

s �+

��s

0

(D�F)2

�(D�F)2

�(D�F)2

0

NK

�(D�F)2

0

�(D�F)2

�2(D�F)2

(D�F)2

�K

(D+F)2

2(D2+3F2)=3

(D2+6DF�3F2)=3

0

(D2+6DF�3F2)=3

u�0

jd�0

��

0

2F2

�2F2

�2F2

0

��

0

2D2=3

�2D2=3

�2D2=3

0

NK

(D�F)2=2

(D�F)2

�(D�F)2=2

0

�(D�F)2=2

�K

�(D+F)2=2

0

�(D+F)2=2

�(D2+3F2)=3

(�D2�6DF+3F2)=6

u�

jd�

��

0

2D2=3

�2D2=3

�2D2=3

0

��l

0

2(2D�3F)2=9

�2(2D�3F)2=9

�2(2D�3F)2=9

0

NK

(D+3F)2=6

(D+3F)2=9

(D+3F)2=18

0

(D+3F)2=18

�K

�(D�3F)2=6

0

�(D�3F)2=6

(�7D2+12DF�9F2)=9

(11D2�6DF�9F2)=18

s �

��s

0

(D+3F)2=9

�(D+3F)2=9

�(D+3F)2=9

0

NK

�(D+3F)2=3

0

�(D+3F)2=3

�2(D+3F)2=9

�(D+3F)2=9

�K

(D�3F)2=3

2(7D2�12DF+9F2)=9

(�11D2+6DF+9F2)=9

0

(�11D2+6DF+9F2)=9

13

TABLEXVIII:CoeÆcients,�,providingtheLNAcontributionto�-baryonmagneticmomentsbyquarksectorswithquarkchargesnormalizedtounitcharge.

Intermediate(Int.)meson-baryonchannelsareindicatedtoallowforSU(3)- avorbreakinginboththemesonandbaryonmasses.Total,directsea-quarkloop,valence,

indirectsea-quarkloopandquenchedvalencecoeÆcientsareindicated.

q

Int.

TotalQuarkSector

DirectSea-QuarkLoop

ValenceSector

IndirectLoop

QuenchedValence

u�0

��

�(D�F)2

(D�F)2

�2(D�F)2

�2(D�F)2

0

�K

0

(D�3F)2=6

�(D�3F)2=6

0

�(D�3F)2=6

�K

(D+F)2

(D+F)2=2

(D+F)2=2

0

(D+F)2=2

K

0

0

0

�(D�F)2

(D�F)2

d�0

��

(D�F)2

(D�F)2

0

0

0

�K

(D�3F)2=6

(D�3F)2=6

0

0

0

�K

(D+F)2=2

(D+F)2=2

0

0

0

s �0

�K

�(D�3F)2=6

0

�(D�3F)2=6

�(D�3F)2=3

(D�3F)2=6

�K

�3(D+F)2=2

0

�3(D+F)2=2

�(D+F)2

�(D+F)2=2

K

0

(D�F)2

�(D�F)2

0

�(D�F)2

��s

0

2(D2+3F2)=3

�2(D2+3F2)=3

�2(D2+3F2)=3

0

TABLEXIX:CoeÆcients,�,providingtheLNAcontributiontonucleonmagneticmoments.Intermediate(Int.)meson-baryonchannelsareindicatedtoallowfor

SU(3)- avorbreakinginboththemesonandbaryonmasses.

Baryon

Int.

TotalQuarkSector

DirectSea-QuarkLoop

ValenceSector

IndirectLoop

QuenchedValence

p

N�

�(D+F)2

(5D2�6DF+9F2)=9

2(�7D2�6DF�9F2)=9

�2(D+3F)2=9

�4D2=3

�K

�(D+3F)2=6

�(D+3F)2=18

�(D+3F)2=9

�(D+3F)2=9

0

�K

�(D�F)2=2

�(D�F)2=2

0

0

0

n

N�

(D+F)2

(5D2�6DF+9F2)=9

4D(D+6F)=9

8D(�D+3F)=9

4D2=3

�K

0

�(D+3F)2=18

(D+3F)2=18

(D+3F)2=18

0

�K

�(D�F)2

�(D�F)2=2

�(D�F)2=2

�(D�F)2=2

0

14

TABLEXX:CoeÆcients,�,providingtheLNAcontributionto�-,�-and�-baryonmagneticmoments.Intermediate(Int.)meson-baryonchannelsareindicatedto

allowforSU(3)- avorbreakinginboththemesonandbaryonmasses.

Baryon

Int.

TotalQuarkSector

DirectSea-QuarkLoop

ValenceSector

IndirectLoop

QuenchedValence

�+

��

�2F2

2F2=3

�8F2=3

�8F2=3

0

��

�2D2=3

2D2=9

�8D2=9

�8D2=9

0

NK

0

(D�F)2=3

�(D�F)2=3

2(D�F)2=3

�(D�F)2

�K

�(D+F)2

�2(D2+3F2)=9

(�7D2�18DF�3F2)=9

�4(D2+3F2)=9

(�D2�6DF+3F2)=3

��s

0

�(D�F)2=3

(D�F)2=3

(D�F)2=3

0

�0

��

0

2F2=3

�2F2=3

�2F2=3

0

��

0

2D2=9

�2D2=9

�2D2=9

0

NK

(D�F)2=2

(D�F)2=3

(D�F)2=6

2(D�F)2=3

�(D�F)2=2

�K

�(D+F)2=2

�2(D2+3F2)=9

(�5D2�18DF+3F2)=18

�(D2+3F2)=9

(�D2�6DF+3F2)=6

��s

0

�(D�F)2=3

(D�F)2=3

(D�F)2=3

0

��

��

2F2

2F2=3

4F2=3

4F2=3

0

��

2D2=3

2D2=9

4D2=9

4D2=9

0

NK

(D�F)2

(D�F)2=3

2(D�F)2=3

2(D�F)2=3

�(D�F)2

�K

0

�2(D2+3F2)=9

2(D2+3F2)=9

2(D2+3F2)=9

0

��s

0

�(D�F)2=3

(D�F)2=3

(D�F)2=3

0

�

��

0

2D2=9

�2D2=9

�2D2=9

0

��l

0

2(2D�3F)2=27

�2(2D�3F)2=27

�2(2D�3F)2=27

0

NK

(D+3F)2=6

(D+3F)2=27

7(D+3F)2=54

2(D+3F)2=27

(D+3F)2=18

�K

�(D�3F)2=6

2(�7D2+12DF�9F2)=27

(19D2+6DF�45F2)=54

(�7D2+12DF�9F2)=27

(11D2�6DF�9F2)=18

��s

0

�(D+3F)2=27

(D+3F)2=27

(D+3F)2=27

0

�0

��

�(D�F)2

(D�F)2=3

�4(D�F)2=3

�4(D�F)2=3

0

�K

0

(D�3F)2=18

�(D�3F)2=18

(D�3F)2=9

�(D�3F)2=6

�K

(D+F)2

(D+F)2=6

5(D+F)2=6

(D+F)2=3

(D+F)2=2

K

0

�(D�F)2=3

(D�F)2=3

�2(D�F)2=3

(D�F)2

��s

0

�2(D2+3F2)=9

2(D2+3F2)=9

2(D2+3F2)=9

0

��

��

(D�F)2

(D�F)2=3

2(D�F)2=3

2(D�F)2=3

0

�K

(D�3F)2=6

(D�3F)2=18

(D�3F)2=9

(D�3F)2=9

0

�K

(D+F)2=2

(D+F)2=6

(D+F)2=3

(D+F)2=3

0

K

0

�(D�F)2=3

(D�F)2=3

(D�F)2=3

0

��s

0

�2(D2+3F2)=9

2(D2+3F2)=9

2(D2+3F2)=9

0

15

TABLE XXI: CoeÆcients, �, providing the LNA contribution to baryon magnetic moments by quark sectors with quark chargesnormalized to unit charge. Intermediate (Int.) meson-baryon channels are indicated to allow for SU(3)- avor breaking in boththe meson and baryon masses. Total, direct sea-quark loop (Direct Loop), Valence, indirect sea-quark loop (Indirect Loop)and Quenched Valence coeÆcients are indicated. The axial couplings take the tree-level values F = 0:50 and D = 0:76 withf� = 93 MeV. Note � = 0:0004.

q Int. Total Direct Loop Valence Indirect Loop Quenched Valence

up N� �6:87 +4:12 �11:0 �7:65 �3:33

�K �3:68 0 �3:68 �3:68 0

�K �0:15 0 �0:15 �0:15 0

dp N� +6:87 +4:12 +2:75 �0:59 +3:33

�K �0:29 0 �0:29 �0:29 0

sp �K +3:68 +3:68 0 0 0

�K +0:44 +0:44 0 0 0

u�+ �� �2:16 +2:16 �4:32 �4:32 0

�� �1:67 +1:67 �3:33 �3:33 0

NK 0 +0:29 �0:29 0 �0:29

�K �6:87 0 �6:87 �3:83 �3:04

d�+ �� +2:16 +2:16 0 0 0

�� +1:67 +1:67 0 0 0

NK +0:29 +0:29 0 0 0

s�+ NK �0:29 0 �0:29 �0:59 +0:29

�K +6:87 +3:83 +3:04 0 +3:04

��s 0 +0:29 �0:29 �0:29 0

u�0 j d�0 �� 0 +2:16 �2:16 �2:16 0

�� 0 +1:67 �1:67 �1:67 0

NK +0:15 +0:29 �0:15 0 �0:15

�K �3:43 0 �3:43 �1:91 �1:52

u� j d� �� 0 +1:67 �1:67 �1:67 0

��l 0 � �� �� 0

NK +3:68 +2:45 +1:23 0 +1:23

�K �0:40 0 �0:40 �0:83 +0:44

s� ��s 0 +2:45 �2:45 �2:45 0

NK �7:36 0 �7:36 �4:91 �2:45

�K +0:79 +1:67 �0:88 0 �0:88

u�0 �� �0:29 +0:29 �0:59 �0:59 0

�K 0 +0:40 �0:40 0 �0:40

�K +6:87 +3:43 +3:43 0 +3:43

K 0 0 0 �0:29 +0:29

d�0 �� +0:29 +0:29 0 0 0

�K +0:40 +0:40 0 0 0

�K +3:43 +3:43 0 0 0

s�0 �K �0:40 0 �0:40 �0:79 +0:40

�K �10:3 0 �10:3 �6:87 �3:43

K 0 +0:29 �0:29 0 �0:29

��s 0 +3:83 �3:83 �3:83 0

16

TABLE XXII: CoeÆcients, �, providing the LNA contribution to baryon magnetic moments. Intermediate (Int.) meson-baryonchannels are indicated to allow for SU(3)- avor breaking in both the meson and baryon masses. Total, direct sea-quark loop(Direct Loop), Valence, indirect sea-quark loop (Indirect Loop) and Quenched Valence coeÆcients are indicated. The axialcouplings take the tree-level values F = 0:50 and D = 0:76 with f� = 93 MeV. Note � = 0:0001.

Baryon Channel Total Direct Loop Valence Indirect Loop Quenched Valence

p N� �6:87 +1:37 �8:24 �4:91 �3:33

�K �3:68 �1:23 �2:45 �2:45 0

�K �0:15 �0:15 0 0 0

n N� +6:87 +1:37 +5:49 +2:16 +3:33

�K 0 �1:23 +1:23 +1:23 0

�K �0:29 �0:15 �0:15 �0:15 0

�+ �� �2:16 +0:72 �2:88 �2:88 0

�� �1:67 +0:56 �2:22 �2:22 0

NK 0 +0:10 �0:10 +0:20 �0:29

�K �6:87 �1:28 �5:59 �2:55 �3:04

��s 0 �0:10 +0:10 +0:10 0

�0 �� 0 +0:72 �0:72 �0:72 0

�� 0 +0:56 �0:56 �0:56 0

NK +0:15 +0:10 +0:05 +0:20 �0:15

�K �3:43 �1:28 �2:16 �0:64 �1:52

��s 0 �0:10 +0:10 +0:10 0

�� �� +2:16 +0:72 +1:44 +1:44 0

�� +1:67 +0:56 +1:11 +1:11 0

NK +0:29 +0:10 +0:20 +0:20 0

�K 0 �1:28 +1:28 +1:28 0

��s 0 �0:10 +0:10 +0:10 0

� �� 0 +0:56 �0:56 �0:56 0

��l 0 � �� �� 0

NK +3:68 +0:82 +2:86 +1:64 +1:23

�K �0:40 �0:56 +0:16 �0:28 +0:44

��s 0 �0:82 +0:82 +0:82 0

�0 �� �0:29 +0:10 �0:39 �0:39 0

�K 0 +0:13 �0:13 +0:26 �0:40

�K +6:87 +1:14 +5:72 +2:29 +3:43

K 0 �0:10 +0:10 �0:20 +0:29

��s 0 �1:28 +1:28 +1:28 0

�� �� +0:29 +0:10 +0:20 +0:20 0

�K +0:40 +0:13 +0:26 +0:26 0

�K +3:43 +1:14 +2:29 +2:29 0

K 0 �0:10 +0:10 +0:10 0

��s 0 �1:28 +1:28 +1:28 0

17

contributions are removed while leaving direct sea-quarkloop contributions from the valence sector unaltered.

C. Examples

Consider for example the quark sector contributions toa baryon magnetic moment in a partially quenched the-ory with two degenerate light quarks and one heavy seaquark, labeled u0, d0 and s0. Electric charge assignmentsare qu, qd, qs for the valence sector of the theory and q0u,q0d, q

0s for the ghost- and sea-quark sectors.

1. Proton Magnetic Moment

The quenched quark-sector results for the proton arecomplemented by direct sea-quark loop contributionsfrom the valence- and ghost-quark sectors plus both di-rect and indirect contributions from the sea-quark sector.As discussed in Sec. III A such loop contributions are a-vor blind and the couplings are easily extracted from Ta-bles IV, VII or IX for the direct contributions and TableV for the indirect contribution. For simplicity, we willsuppress baryon mass splittings in the following. How-ever, they may be introduced in a transparent manner.For the u-quark sector in the proton, one has

up = �

��qu 4

3D2m� + qu

1

3(5D2 � 6DF + 9F 2)m�

+q0u1

3(5D2 � 6DF + 9F 2) ( em� �m�)

�qu 43(D2 + 3F 2) em�

�qu 23(D2 + 3F 2) emK

�; (36)

where � � mN=(8�f2�). em� denotes a �-meson composedof a light-valence and light-sea quark, and emK denotesa K-meson composed of a light-valence and heavy-seaquark. The second term is a direct u sea-quark loopcontribution associated with the valence sector, cancelledby the ghost-quark contribution in the third term whenq0u = qu. The third term also includes the direct u0 sea-quark loop contribution associated with the sea-quarksector and originates from Table IV for Figs. 1(b) and(e). The last two terms are indirect u sea-quark loopcontributions and originate from Table V for diagrams(b), (g) and (h) respectively.Similarly

dp = �

�+qd

4

3D2m� + qd

1

3(5D2 � 6DF + 9F 2)m�

+q0d1

3(5D2 � 6DF + 9F 2) ( em� �m�)

�qd 2 (D � F )2 em� � qd (D � F )2 emK

�;

(37)

and

sp = �

�qs

1

3(5D2 � 6DF + 9F 2)mK (38)

+q0s1

3(5D2 � 6DF + 9F 2) ( emK �mK )

�:

The LNA behavior of the proton magnetic moment inthe partially quenched theory is

�p = �

��4

3D2m�

+1

9(5D2 � 6DF + 9F 2) (m� �mK )

+ (q0u + q0d)1

3(5D2 � 6DF + 9F 2) ( em� �m�)

+q0s1

3(5D2 � 6DF + 9F 2) (emK �mK)

�2

9(D + 3F )2 em� � 1

9(D + 3F )2 emK

�: (39)

We note that this �nal expression agrees with that ofEq. (48) in Ref. [6].

2. �+ Magnetic Moment

To clearly establish the method for constructingpartially-quenched chiral coeÆcients, we consider the �+

hyperon. The direct and indirect sea-quark loop con-tributions are avor blind and the couplings are easilyextracted from Tables XI, XIII or XV for the direct con-tributions and Table XII for the indirect contribution.For the u-quark sector in �+, one has

u�+ = �

��qu 4

3D2mK

+qu2

3(D2 + 3F 2)m� + qu (D � F )2mK

+q0u2

3(D2 + 3F 2) ( em� �m�)

+q0u (D � F )2 ( bmK �mK )

�qu 43(D2 + 3F 2) em�

�qu 23(D2 + 3F 2) emK

�; (40)

where emK denotes a K-meson composed of a light-valence and heavy-sea quark and bmK denotes a K-mesoncomposed of a strange-valence and light-sea quark. Thesecond and third terms are direct u sea-quark loop con-tributions associated with the valence sector, cancelledby the ghost-quark contribution in the fourth and �fthterms when q0u = qu. The fourth and �fth terms also in-clude the direct u0 sea-quark loop contribution associatedwith the sea-quark sector and originate from Table XI for

18

Figs. 2(b) and (e). The last two terms are indirect u sea-quark loop contributions and originate from Table XIIfor Figs. 2(b) + (c) and (h) respectively. Similarly

s�+ = �

�+qs

4

3D2mK (41)

+qs2

3(D2 + 3F 2)mK + qs (D � F )2m�s

+q0s2

3(D2 + 3F 2) (emK �mK)

+q0s (D � F )2 ( em�s �m�s )

�qs 2 (D � F )2 bmK � qs (D � F )2 em�s

�;

where em�s denotes an s s 0 �-meson composed of astrange-valence and anti-heavy sea-quark, and

d�+ = �

�+qd

2

3(D2 + 3F 2)m� + qd (D � F )2mK

+q0d2

3(D2 + 3F 2) ( em� �m�)

+q0d (D � F )2 ( bmK �mK )

�: (42)

Thus, the LNA behavior of the �+ magnetic moment inthe partially quenched theory is

��+ = �

��4

3D2mK

+2

9(D2 + 3F 2)m� +

1

3(D � F )2mK

�2

9(D2 + 3F 2)mK � 1

3(D � F )2m�s

+(q0u + q0d)2

3(D2 + 3F 2) ( em� �m�)

+ (q0u + q0d) (D � F )2 ( bmK �mK )

+q0s2

3(D2 + 3F 2) ( emK �mK)

+q0s (D � F )2 ( em�s �m�s)

+1

3(D � F )2 (2 bmK + em�s)

�4

9(D2 + 3F 2) (2 em� + emK)

�; (43)

again in agreement with Eq. (51) of Ref. [6].Partially-quenched results may be similarly obtained

for the remainder of the quark-sector contributions tooctet baryon magnetic moments using the approach de-scribed here in detail. Since the results require speci�cknowledge of the number and nature of dynamical a-vors, we defer writing further speci�c results.

VI. SUMMARY

The diagrammatic method for separating valence andsea-quark-loop contributions to the meson cloud of

hadrons provides a transparent approach to the calcula-tion of quenched chiral coeÆcients. The origin of chiralnonanalytic structure is obvious, and facilitates the in-corporation of the correct nonanalytic structure match-ing today's numerical simulations. In the process, thecoeÆcients for partially-quenched QCD are derived; nonew calculations are required.

The valence sector of full QCD contains the largestcoeÆcients for the leading nonanalytic behavior of mag-netic moments. The u-quark contribution to the protonmagnetic has a coeÆcient of �11:0 for the rapidly varyingpion-cloud contribution, which is complemented furtherby the kaon cloud. These are connected insertions of theelectromagnetic current in full QCD and should revealsigni�cant curvature in the approach to the chiral limit.It is also encouraging to note that the u-quark sector isknown to have relatively small statistical uncertaintiesin the quenched approximation [12] compared to that forthe d quark.

The coeÆcients of the leading nonanalytic terms of fullQCD change signi�cantly upon quenching. Some chan-nels still hold excellent promise for revealing the non-analytic behavior of meson-cloud physics even in thequenched approximation. For example, the u or d-quark in the proton have large coeÆcients for the non-analytic term proportional to m� , with opposite signsrespectively. Similarly, both the proton and neutronmagnetic moments have large coeÆcients surviving inquenched QCD. Because the u-quark in the proton hassigni�cantly smaller statistical errors than that for thed quark in the proton [12], the u-quark contributionto the proton magnetic moment provides the optimalopportunity to directly view nonanalytic behavior as-sociated with the quenched meson cloud of baryonsin the quenched approximation. Figure 6 illustratesthe anticipated curvature [25] associated with the term�(4=3)D2mN m�=(8� f

2� ) surviving in the quenched ap-

proximation.

There are other interesting opportunities. Considerfor example the s-quark contribution to the quenched �magnetic moment. Table XXI indicates that the coeÆ-cient of the NK contribution to the � magnetic momentis large. Because the nucleon is signi�cantly lighter thanthe �, the NK loop can contribute enhanced nonlinearbehavior. Hence, there is a prediction of signi�cant cur-vature in the extrapolation of the s-quark contribution,even when the mass of the strange quark is held �xed asis commonly done in lattice QCD simulations. As such,the e�ect is purely environmental [12, 13, 16, 17]. The ef-fect arises from the extrapolation of light u and d quarksin the �.

The s-quark in � provides another opportunity to ob-serve a purely environmental e�ect in quenched QCD.Here the coeÆcient of the �K channel is very large, againpredicting curvature in the s-quark contribution to themagnetic moment, even when the s-quark mass is held�xed. The mass of � is less than the mass of � allowingthe kaon to provide enhanced nonlinear behavior.

19

TABLE XXIII: One-loop corrected coeÆcients, �, providing the LNA contribution to baryon magnetic moments by quarksectors with quark charges normalized to unit charge. Total, direct sea-quark loop (Direct Loop), Valence, indirect sea-quarkloop (Indirect Loop) and Quenched Valence coeÆcients are indicated. Here, the axial couplings take the one-loop correctedvalues F = 0:40 and D = 0:61 with f� = 93 MeV. Note � = 0:0004.

q Int. Total Direct Loop Valence Indirect Loop Quenched Valence

up N� �4:41 +2:65 �7:06 �4:91 �2:15

�K �2:36 0 �2:36 �2:36 0

�K �0:10 0 �0:10 �0:10 0

dp N� +4:41 +2:65 +1:76 �0:38 +2:15

�K �0:19 0 �0:19 �0:19 0

sp �K +2:36 +2:36 0 0 0

�K +0:29 +0:29 0 0 0

u�+ �� �1:38 +1:38 �2:77 �2:77 0

�� �1:07 +1:07 �2:15 �2:15 0

NK 0 +0:19 �0:19 0 �0:19

�K �4:41 0 �4:41 �2:46 �1:95

d�+ �� +1:38 +1:38 0 0 0

�� +1:07 +1:07 0 0 0

NK +0:19 +0:19 0 0 0

s�+ NK �0:19 0 �0:19 �0:38 +0:19

�K +4:41 +2:46 +1:95 0 +1:95

��s 0 +0:19 �0:19 �0:19 0

u�0 j d�0 �� 0 +1:38 �1:38 �1:38 0

�� 0 +1:07 �1:07 �1:07 0

NK +0:10 +0:19 �0:10 0 �0:10

�K �2:21 0 �2:21 �1:23 �0:98

u� j d� �� 0 +1:07 �1:07 �1:07 0

��l 0 � �� �� 0

NK +2:36 +1:57 +0:79 0 +0:79

�K �0:25 0 �0:25 �0:54 +0:29

s� ��s 0 +1:57 �1:57 �1:57 0

NK �4:72 0 �4:72 �3:15 �1:57

�K +0:50 +1:07 �0:57 0 �0:57

u�0 �� �0:19 +0:19 �0:38 �0:38 0

�K 0 +0:25 �0:25 0 �0:25

�K +4:41 +2:21 +2:21 0 +2:21

K 0 0 0 �0:19 +0:19

d�0 �� +0:19 +0:19 0 0 0

�K +0:25 +0:25 0 0 0

�K +2:21 +2:21 0 0 0

s�0 �K �0:25 0 �0:25 �0:50 +0:25

�K �6:62 0 �6:62 �4:41 �2:21

K 0 +0:19 �0:19 0 �0:19

��s 0 +2:46 �2:46 �2:46 0

20

TABLE XXIV: One-loop corrected coeÆcients, �, providing the LNA contribution to baryon magnetic moments. Total, directsea-quark loop (Direct Loop), Valence, indirect sea-quark loop (Indirect Loop) and Quenched Valence coeÆcients are indicated.Here, the axial couplings take the one-loop corrected values F = 0:40 and D = 0:61 with f� = 93 MeV. Note � = 0:000128.

Baryon Channel Total Direct Loop Valence Indirect Loop Quenched Valence

p N� �4:41 +0:88 �5:29 �3:15 �2:15

�K �2:36 �0:79 �1:57 �1:57 0

�K �0:10 �0:10 0 0 0

n N� +4:41 +0:88 +3:53 +1:38 +2:15

�K 0 �0:79 +0:79 +0:79 0

�K �0:19 �0:10 �0:10 �0:10 0

�+ �� �1:38 +0:46 �1:85 �1:85 0

�� �1:07 +0:36 �1:43 �1:43 0

NK 0 +0:06 �0:06 +0:13 �0:19

�K �4:41 �0:82 �3:59 �1:64 �1:95

��s 0 �0:06 +0:06 +0:06 0

�0 �� 0 +0:46 �0:46 �0:46 0

�� 0 +0:36 �0:36 �0:36 0

NK +0:10 +0:06 +0:03 +0:13 �0:10

�K �2:21 �0:82 �1:39 �0:41 �0:98

��s 0 �0:06 +0:06 +0:06 0

�� �� +1:38 +0:46 +0:92 +0:92 0

�� +1:07 +0:36 +0:72 +0:72 0

NK +0:19 +0:06 +0:13 +0:13 0

�K 0 �0:82 +0:82 +0:82 0

��s 0 �0:06 +0:06 +0:06 0

� �� 0 +0:36 �0:36 �0:36 0

��l 0 � �� �� 0

NK +2:36 +0:53 +1:84 +1:05 +0:79

�K �0:25 �0:36 +0:11 �0:18 +0:29

��s 0 �0:53 +0:53 +0:53 0

�0 �� �0:19 +0:06 �0:25 �0:25 0

�K 0 +0:08 �0:08 +0:17 �0:25

�K +4:41 +0:74 +3:68 +1:47 +2:21

K 0 �0:06 +0:06 �0:13 +0:19

��s 0 �0:82 +0:82 +0:82 0

�� �� +0:19 +0:06 +0:13 +0:13 0

�K +0:25 +0:08 +0:17 +0:17 0

�K +2:21 +0:74 +1:47 +1:47 0

K 0 �0:06 +0:06 +0:06 0

��s 0 �0:82 +0:82 +0:82 0

21

FIG. 6: Chiral extrapolation via the Pad�e of [25, 26]. Thesolid curve displays the self-consistent extrapolation of thequenched simulation results of [12], whereas the dashed curveshows the curvature of full QCD. One-loop corrected axialcouplings are used in the Pad�e.

A few channels hold tremendous potential for revealingquenched artifacts in quenched simulations. Despite theprediction of curvature for both the s and d-quark sec-tors of �� in both full and quenched QCD, the extrapo-lation of the total ��-baryon magnetic moment receivesno leading nonanalytic contribution from neither the �-nor the K-meson cloud in the quenched approximation.Similar results hold for the �� magnetic moment.It is particularly diÆcult to directly determine the loop

contribution to baryon magnetic moments in numericalsimulations [7, 8, 9, 10]. As such, it is of particular in-terest to compare the coeÆcients of the valence quarkcontributions in full QCD (column \Valence" of Tables

XXI and XXII) to that for the valence quark contribu-tions of quenched QCD (column \Quenched Valence" ofTables XXI and XXII). Here, the u-quark in �+ standsout with the signi�cant curvature of the �� and �� chan-nels completely suppressed from -4.32 to 0 and -3.33 to0 respectively. Only the �K channel has a signi�cantcoupling for the u quark in the quenched �+, but curva-ture in this channel is suppressed by the large excitationenergy required to form the intermediate state. The uquark in the proton is also worthy of note, with the co-eÆcient of the rapidly-varying �N channel dropping sig-ni�cantly from �11:0 in full QCD to �3:33 in quenchedQCD and the kaon contribution vanishing completely.

In summary, this study of quark-sector contributionsto baryon magnetic moments in quenched, partially-quenched and full QCD indicates there are numerous ex-citing opportunities to observe and understand the un-derlying structure of baryons and the nature of chiralnonanalytic behavior in QCD and its quenched variants.Numerical simulations of the observables discussed hereinare currently in production on the Australian Partner-ship for Advanced Computing (APAC) National Facilityusing FLIC fermions [28] which provide eÆcient accessto the light quark-mass regime. It will be fascinatingto confront these predictions with numerical simulationresults.

Acknowledgments

Thanks to Matthias Burkardt, Ian Cloet, Ben Crouch,Martin Savage, Tony Thomas, Tony Williams, StewartWright and Ross Young for bene�cial discussions. Thisresearch is supported by the Australian Research Coun-cil.

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