The Gerasimov-Drell-Hearn Sum Rule and the
Spin Struture of the Proton
G.Krein
Instituto deFsiaTeoria,UniversidadeEstadual Paulista,
RuaPamplona145, 01405-900,S~aoPaulo,SP
Reeivedon22February,2001
We present an overview of the present theoretial status of the Gerasimov-Drell-Hearn (GDH)
sumrule. Itisemphasizedtheimportaneofnearthresholds-wavephotoprodutionontributions
to the sumrule. We also disuss the experimental veriationof the sum rule, with partiular
emphasisonthereentexperimentattheMainzMirotron(MAMI).Thespinpolarizabilities and
thegeneralizedGDHsumrule,whihare ofinteresttoissuesrelated tothespinoftheprotonas
measuredathighenergies,aredisussed.
I Introdution
The responseof theinternal degreesof freedomof the
nuleontoanexternaleletromagnetieldanbe
de-sribed in terms of polarizabilities. Forreal photons,
polarizabilities and other ground-state properties an
berelated to integralsoverphotoabsorptionross
se-tionsby sumrules. Oneof themostprominent
exam-plesistheGerasimov-Drell-Hearn(GDH)sumrule[1℄,
whih provides a relationship between the anomalous
magneti moment
N
of the nuleon and the
pho-toabsorptionrosssetionsforparallelandantiparallel
alignments of the nuleon and photon heliities,
3=2
and
1=2
2
N
4 =
M 2
8 2
I
N
; (1)
where I
N
is theGDH integral,givenby
I
N =
Z
1
!
th
1=2 (!)
3=2 (!)
!
d!: (2)
The importane of the sum rule stems from the fat
that it is basedon generalpriniples of physis, suh
as Lorentz and gauge invariane, rossing symmetry,
ausalityandunitarity.
CloselyrelatedtotheGDHsumruleistheforward
spinpolarizability[2℄
= 1
4 2
Z
1
!th
1=2 (!)
3=2 (!)
! 3
d!: (3)
This quantity is related to the Ragusa
polarizabili-ties[3℄
i
; i=14,as
=
1
2 2
4
: (4)
The Ragusapolarizabilitiesappear aslow-energy
on-tantsofthetermsO(! 3
)intheexpansionofthe
Comp-Theuseofvirtualphotonsfromeletronsattering
proessesprovidesuswithevenmoredetailed
informa-tionon theinternaldegreesoffreedomofthenuleon.
Inpartiular,asweinreasethefour-momentumQ 2
of
thevirtualphotonfromthereal-photonpoint(Q 2
=0)
tolargevaluesofQ 2
,weaninvestigatethetransition
fromthenonperturbativetotheperturbativeregimeof
quantumhromodynamis(QCD).Thegeneralizations
ofreal-photonsumrulestovirtualphotonsprovidean
interestingpossibility to study this transition and the
varyingroleoftherelevantdegreesoffreedom.
In this ontribution we present an overview of the
presenttheoretialandexperimentalstatusoftheGDH
sumrule. Wealsodisussgeneralisationsofthespin
po-larizabilityand GDH sum rule. These generalizations
areofinteresttoissuesrelatedtothespinoftheproton
asmeasuredat high energies. Atthe end ofthis
on-tribution,wealsodisussageneralizationoftheBaldin
sumruleasafuntion of Q 2
and omparepreditions
basedonhiralperturbationtheoryanda
phenomeno-logialresonanemodel.
II The GDH sum rule
Untilreently,there wasnodiret measurementofthe
rosssetions
3=2 and
1=2
intheintegrandsofthe
in-tegralsabove. The situation hangedwith the reent
experiment at MAMI [4℄, whih measured the
ross-setion dierene = N
1=2
N
3=2
for photon lab.
energiesintherange200 800MeV.Beforethis
experi-ment,onehadtorelyonestimatesusingpion
The deomposition of in Eqs. (2) and (3) in
termsofphotoprodutionmultipolesisgivenas[6℄
= 8
q
k
jE
0+ j
2
+3jE
1+ j
2
+6E
1+ M
1+ jM
1+ j
2
+ jM
1 j
2
+
; (5)
whereqisthe.m. momentumofthepionandkinthe
.m. momentumofthephoton. Beauseofthe
weight-ingfator1=!inI
GDH
,itistobeexpetedthatalarge
frationofthethesumruleissaturatedbys-wave
near-threshold pion photoprodution and by (1232)
reso-nane prodution. This seemspartiularly true for ,
beauseofthe1=! 3
in theintegrandofEq.(3).
ThestatusoftheGDHsumruleuntilreentlywas,
asdisussed in Refs.[7℄ and [8℄, that there seemed to
bea severe disrepany betweenthe preditionof the
sumruleandthesaturationoftheintegralwith
photo-produtiondata. Fortheproton,forexample,thesum
rulepredits
I
p
= 204b; (6)
while the SAID multipoles, together with an old
esti-mate [9℄ of the NN ontribution, predited[7℄ I
p =
289b. AmorereentSAID analysis,SAID-SP97K,
gave[8℄
I SAID
p
= 281b: (7)
Thedisrepanybetweenthepreditionofthesumrule
andtheSAIDmultipoleswasontheorderof30%.
Veryreently, thepresentauthor together with D.
Drehsel[10℄showedthatthedisrepanyisdrastially
redued when using the HDT [11℄ multipoles, instead
oftheSAIDmultipoles. TheHDTmultipolesarebased
on xed-t dispersion relations and unitarity. In the
HDT dispersive approah the threshold region is not
inludedinthettingoffreeparameters;thethreshold
values for the s-wave amplitudes are genuine
predi-tions. The HDTanalysis islimitedto photonenergies
upto400MeV,s, p,andd wavesfor isospin1/2,and
sandpwavesforisospin3/2. Althoughthedierenes
betweentheSAID-SP97KandHDTmultipolesarevery
smallforenergiesabove400MeV,largedierenes
o-ur for the E
0+
multipole for +
prodution lose to
threshold. In Fig.1 we show the results for the
inte-grandofthesumruleusingtheHDT(solid)andSAID
(dashed)multipoles.
Figure 1. Multipole ontributions to the integrand of Ip:
(a)E
0
+,(b)M
1
+,and()all(uptoL=5)partialwaves.
Thesolidlinesorrespondto theHDTmultipoles andthe
dottedtotheSP97K-SAIDmultipoles.
The HDT multipoles lead to I
p
(1) = 196b.
Thisimpliesinahangeof20b,omparedtoSAID.If
weuse the(very)old estimateestimateofKarliner [9℄
for the two-pion ontribution, I
p
(2) = 65b, the
earlier disrepany [7℄ redues from 38 % to 28 %.
Moreover, if one uses the estimate of Ref. [12℄ of the
ontributionsbeyondone-pionprodution, 32b,
to-gether with the single-pion HDT predition, the
dis-repanyfallstoonly12%.
ThisvalueisnottoofartheresultofthereentGDH
experiment at Mainz [4℄, for energies upto 800 MeV.
Conerningtheremainingdisrepany,itwillbe
inter-esting to see the outome of the GDH experiment at
ELSAathigherenergies.
Regardingthe spinpolarizability, Eq. (3), beause
ofthe1=! 3
in thedenominator,oneexpetsthat
near-threshold s-wave pion photoprodution and (1232)
resonaneprodutionsaturate the aboveintegraleven
morerapidlythantheGDHintegral. Thiswasatually
veriedinRef.[13℄: theintegrandisalreadynegligible
at!' 500MeV.
TheHDT multipolesleadfortheproton
HDT
p
= 0:610 4
fm 4
; (8)
whiletheSP97K-SAIDgive
SAID
= 1:310 4
fm 4
The reentMainz experiment[4℄ of =
1=2
3=2
forenergieslargerthan200MeVgivesavalueof
p 1:710 4 fm 4
. TheHDTmultipolespreditthatthe
region very lose to threshold, !
th
< ! < 200 MeV,
ontributesto
p
with apositive value equalto +0:9.
Therefore,to theextentthattheHDT multipoles
pro-videanaurantedesriptionofthelow-energyregion,
one anpreditthat
p
= 0:810 4
fm 4
.
Itisinstrutivetoompare[13℄thesenumberswith
preditions of hiral perturbation theory. Two reent
alulations, Refs. [14℄and [15℄, in O(p 4
)of HBChPT
nd
p
= 3:910 4
fm 4
. Anotherreent O(p 4
)
al-ulation [16℄, whih diers from the previous two in
the waythe single-partilereduible ontributions are
treated, leads to
p
= 1:010 4
fm 4
. A
alula-tion [17℄using dispersionrelationstogetherwith large
N
argumentsleadsto
p
= 0:110 4
fm 4
.
III Generalized GDH integral
and spin polarizability
One of themany possiblegeneralizations of theGDH
sumrule[18℄tovirtualphotonsistheexpression
I 1 (Q 2 ) = M 2 8 2 Z 1 0 1 x 1+ 2 1=2 ( ;Q
2
)
3=2 ( ;Q
2
)
2 0
LT ( ;Q
2 ) d ; (10) where 0 =m +(m 2 +Q 2
)=2m is the thresholdlab
energyof one-pionprodution. Here,x
0
is the
thresh-oldvalueofx=Q 2
=2M,x
0 =Q 2 =(Mm +m 2 +Q 2 ),
and = Q=. Also,
3=2 and
1=2
are the ross
se-tionsforthesatteringof polarizedeletronson
polar-izednuleonswithparalleland antiparallelalignments
of the eletron and nuleon heliities, and 0
LT is the
longitudinal-transverserosssetion.
Contatwith deep inelastisattering experiments
an bemadebyexpressingI
1 (Q
2
)in termsofan
inte-graloverthespinstruturefuntiong
1 (x;Q 2 ) I 1 (Q 2 ) = 2M 2 Q 2 Z x 0 0 g 1 (x;Q 2 )dx = 2M 2 Q 2 1 (Q 2 ): (11)
A generalized polarizability an also be dened in
termsofeletroprodutionrosssetions
3=2 and 1=2 as[19℄ (Q 2 )= 1 4 2 Z 1 0 1=2 ( ;Q
2
)
3=2 ( ;Q
2
)
3
d : (12)
Thereisverylittleexperimentalinformationonthe
eletroprodution ross-setions in the resonane
re-gion. Thesituationwillhangein thenearfuture with
being, one has to rely on phenomenologial models.
Here, we review our reent work [20℄ [21℄ on
general-izedGDH sumrule and spinpolarizabilities. In these
worksweusedthereentlydevelopedisobarmodelfor
eletroprodution,knownasMAID. Themodeliswell
doumentedand aessible viathe Internet [22℄. It is
basedonaneetivephenomenologialLagrangianfor
Born terms (\bakground") and resonanes up to the
thirdresonaneregion. Foreah partialwavethe
mul-tipoles satisfy the onstraints of gaugeinvariane and
unitarity,andin thereal photonasetheresultsagree
wellwiththepreditionsofdispersiontheory[11℄. The
modelisabletodesribetheorretenergydependene
ofthemultipolesforphotonenergiesupto!'1GeV,
anditprovidesagooddesriptionofallexperimentally
measureddierentialrosssetionsandpolarization
ob-servables.
Figure2. Comparisonof resultsfor
1
from MAID(solid)
andHBChPT(dashed).
The preditionof MAID isshowm in Fig.??. We
ompareresultswithareentHBChPTalulation[23℄
at O(p 4
). For the proton, the result obtained in
Ref.[23℄for
1
isgivenby
1 (Q 2 )= p 4
+6:85Q 2
(GeV 2
): (13)
Beingthis aresultof aHBChPT alulation,it is
ex-peted to bevalid onlyfor small valuesof Q 2
. Inthe
FigureweomparethepreditionsofMAID(solidline)
andEq. (13)(dashed line). Is is woth mentioningthe
theMAIDresultisverylosetoreentSLACdata[20℄
atQ 2
=0:5GeV 2
. Fromthegure, itisseenthat for
Q 2
>0:05GeV 2
,theHBChPTresultbeomes
onsid-erablydierentfromtheMAIDpredition.
Forthegeneralizedspinpolizability,areent
alu-lation[19℄hasbeenmadeintheframeworkofthe
one-loopapproximationto(relativisti)hiralperturbation
theory (ChPT), supplemented by tree graphs for the
Figure3. Comparisonof resultsfor p fromMAID(solid)
andrelativistiChPT(dashed).
InFig.3weompareresultsagainwithMAID[21℄.
The most striking dierene refers to the slope of
p (Q
2
)losetotherealphotonpoint. Thepronouned
slope in
p (Q
2
) observed in the MAID result is due
to the interferene between bakground and (1232)
terms,whihhasitsthephysialorigininthe
dynami-aldressingoftheNvertex[24℄.
Insummary,oneofthestrikingfeaturesofthe
gen-eralizedGDH integralisits rapid utuationwith Q 2
andinpartiularahangeofsignat Q 2
'0:5(GeV) 2
,
whihimposessevereonstraintsonanymodelforthe
nuleonstruture. This zero-rossingseparates the
re-giondominatedbyresonane-drivenoherentproesses
fromaregionofessentiallyinoherentsatteringothe
nuleon's onstituents. A similar zero-rossingis
pre-ditedbyChPTforthegeneralizedspinpolarizability,
(Q 2
)[19℄,whilewefoundthatMAIDexludessuha
ross-overforQ 2
(1GeV) 2
.
Aknowledgments Work partially supported
FAPESPandCNPq.
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