Post on 27-Nov-2021
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-
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iew m
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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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