Strange magnetism in the nucleon

1 October 1998

Ž .Physics Letters B 437 1998 184–190

Strange magnetism in the nucleon

Thomas R. Hemmert a,1, Ulf-G. Meißner a,2, Sven Steininger a,b,3,4

a ( )Forschungszentrum Julich, IKP Theorie , D-52425 Julich, Germany¨ ¨b Department of Physics, HarÕard UniÕersity, Cambridge, MA 02138, USA

Received 15 June 1998Editor: H. Georgi

Abstract

Using heavy baryon chiral perturbation theory to one loop, we derive an analytic and parameter-free expression for theŽ s.Ž 2.momentum dependence of the strange magnetic form factor of the nucleon G Q and its corresponding radius. ThisM

should be considered as a lower bound. We also derive a model-independent relation between the isoscalar magnetic and theŽ .strange magnetic form factors of the nucleon based on chiral symmetry and SU 3 only. This gives an upper bound on the

strange magnetic form factor. We use these limites to derive bounds on the strange magnetic moment of the proton from theŽ s.Ž 2 2.recent measurement of G Q s0.1 GeV by the SAMPLE collaboration. We further stress the relevance of this result forM

the on-going and future experimental programs at various electron machines. q 1998 Published by Elsevier Science B.V. Allrights reserved.

PACS: 13.40.Cs; 12.39.Fe; 14.20.Dh

1. There has been considerable experimental andtheoretical interest concerning the question: Howstrange is the nucleon? Despite tremendous efforts,we have not yet achieved a detailled understandingabout the strength of the various strange operators inthe proton. These are ss, as extracted from theanalysis of the pion-nucleon S-term, sg g s as mea-m 5

sured e.g. in deep-inelastic lepton scattering off pro-

1 Email: [email protected] Email: [email protected] Email: [email protected] Work supported in part by the Graduiertenkolleg ‘‘Die Er-

forschung subnuklearer Strukturen der Materie’’ at Bonn Univer-sity and by the DAAD.

tons and the vector current sg s, which is accesiblem

e.g. in parity-violating electron-nucleon scattering. Adedicated program at Jefferson Laboratory preceded

Ž . Ž .by experiments at BATES MIT and MAMI Mainzis aimed at measuring the form factors related to thestrange vector current. In fact, the SAMPLE collabo-ration has recently reported the first measurement of

w xthe strange magnetic moment of the proton 1 . To beprecise, they give the strange magnetic form factor at

Ž s.Ž 2 2 .a small momentum transfer, G q sy0.1 GeVM

s q0.23 " 0.37 " 0.15 " 0.19 nuclear magnetonsŽ .n.m. . The rather sizeable error bars document thedifficulty of such type of experiment. On the theoret-ical side, there is as much or even more uncertainty.To document this, let us pick one particular ap-

w xproach. Jaffe 2 deduced rather sizeable strange

0370-2693r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 00889-2

( )T.R. Hemmert et al.rPhysics Letters B 437 1998 184–190 185

vector current matrix elements from the dispersion-theoretical analysis of the nucleon electromagneticform factors, assuming that the isoscalar spectralfunctions are dominated at low momentum transferby the v and f mesons. This analysis was updated

w xin 3 with similar results. However, if one improvesthe isoscalar spectral function by considering also

w x w xthe correlated pr-exchange 4 or kaon loops 5 , thecorresponding strange matrix elements can changedramatically. Also, the spread of the theoretical pre-dictions for the strange magnetic moment, y0.8FmŽ s.F0.5 n.m. underlines clearly the abovemadep

Ž w x.statement for a review, see Ref. 6 . As we willdemonstrate in the following, there is, however, onequantity of reference, namely we can make a param-eter-free prediction for the momentum dependence

Ž .of the nucleons’ strange magnetic Sachs form fac-tor based on the chiral symmetry of QCD solely. Inaddition, we derive a leading order model-indepen-dent relation between the strange and the isoscalarmagnetic form factors, which allows to give an upper

Ž s.Ž 2 .bound on the momentum dependence of G Q .M

These two different results can then be combined toextract a range for the strange magnetic moment ofthe proton from the SAMPLE measurement of theform factor at low momentum transfer.

2. The strangeness vector current of the nucleon isdefined as

0 8 '² < < : ² < < :N sg s N s N qg l r3yl r 3 q NŽ .m m

0 8's 1r3 J y 1r 3 J , 1Ž . Ž .Ž .m m

Ž .with qs u,d,s denoting the triplet of the light0 Ž a. Žquark fields and l s I l the unit the as8

. Ž .Gell-Mann SU 3 matrix. Assuming conservation ofall vector currents, the corresponding singlet andoctet vector current for a spin-1r2 baryon can then

Žbe written as from here on, we mostly consider the.nucleon

X Ž .0,8 0,8 2J su p F q gŽ . Ž .m N 1 m

nis qmnŽ .0,8 2qF q u p . 2Ž . Ž .Ž .2 N2mN

Here, q spX yp corresponds to the four-momen-m m m

tum transfer to the nucleon by the external singletŽ Ž0. 0. Ž Ž8. 8.Õ sÕ l and the octet Õ sÕ l vectorm m m m

source Õ , respectively. The strangeness Dirac andm

Pauli form factors are defined via

11Ž s. 2 Ž0. 2 Ž8. 2F q s F q y F q , 3Ž .Ž . Ž . Ž .1,2 1,2 1,23 '3

Ž s.Ž .subject to the normalization F 0 sS , with S1 B BŽthe strangeness quantum number of the baryon SN

. Ž s.Ž . Ž s. Ž s. Ž .s0 and F 0 sk , with k the anomalous2 B B

strangeness moment. In the following we concentrateour analysis on the ‘‘magnetic’’ strangeness form

Ž s .Ž 2 .factor G q , which in analogy to theMŽ .electro magnetic Sachs form factor is defined as

GŽ s. q2 sF Ž s. q2 qF Ž s. q2 . 4Ž .Ž . Ž . Ž .M 1 2

It is this ‘‘strangeness’’ form factor for which chiralŽ .perturbation theory CHPT gives the most interest-

Ž s.Ž 2 .ing predictions. Furthermore, G q is also theM

strangeness form factor that figures prominently inw xthe recent Bates measurement 1 .

3. Heavy baryon chiral perturbation theoryŽ .HBCHPT is a precise tool to investigate the low-energy properties of the nucleon. It has, however,been argued that due to the appearance of higherorder local contact terms with undetermined coeffi-cents, CHPT can not be used to make any predictionfor the strange magnetic moment or the strangeelectric radius without additional, model-dependent

w xassumptions 7 . However, the analysis of the nucle-Ž .ons electromagnetic form factors in SU 2 CHPT

shows that to one loop the slope of the isovectorPauli form factor can be predicted in a parameter-freemanner. Furthermore, one obtains a constant piece

Ž .for the isoscalar form factor in SU 2 CHPT and aparameter-free prediction for the magnetic Sachsform factor of the proton and the neutron, see Refs.w x8–10,12 . It is thus natural to extend this calculationto the three flavor case with the appropriate singlet

Ž .and octet currents as defined in Eq. 2 .We give here the relevant HBCHPT Lagrangians

needed for the calculation. The baryon octet isparametrized in the matrix B, which has the usualtransformation properties of any matter field under

( )T.R. Hemmert et al.rPhysics Letters B 437 1998 184–190186

chiral transformation. We utilize the chiral covariantderivative D ,m

Ž .0D Bs E qG y iÕ B 5Ž .Ž .m m m m

i1 Ž . Ž .† † i iG s u ,E u y u Õ qa uŽ .m m m m2 2

iŽ . Ž .i i †y u Õ ya u , is3,8 6Ž . Ž .Ž .m m2

the chiral vielbein u ,m

u s i u†= U u† 7Ž .m m

Ž . Ž .i i= UsE Uy i Õ qa UŽ .m m m m

Ž . Ž .i iq iU Õ ya , is3,8 8Ž . Ž .Ž .m m

Ž x .Ž Ž x ..where the quantities Õ a , xs0,3,8 correspondm m

Ž . Ž .to external vector axial-vector sources and U x s2Ž .u x parametrizes the octet of Goldstone bosons.

The three flavor HBCHPT Lagrangian then readsŽwe only give the terms relevant to the calculations

.presented here

Ž1. m m² : ² :LL s B iÕ D B qD B S u , B� 4MB m m q

m² :qF B S u , B 9Ž .m y

Fib6 aŽ2. m n Ž3.² :w xLL sy B S ,S f , BMB qmn4mN

ib D6 a m n Ž3.² :w xy B S ,S f , B� 4qm n4mN

Fib6 b m n Ž8.² :w xy B S ,S f , Bqm n4mN

ib F6 b m n Ž8.² :w xy B S ,S f , B� 4qm n4mN

2 ib6 c m n Ž0.² :w xy B S ,S Õ B q . . . 10Ž .mn4mN

with

f Ž i. su†F R Ž i.uquF L Ž i.u† 11Ž .qm n mn mn

Ž . Ž .L , R Ž i. L , R Ž i. L , R Ž i. L , R i L , R iF sE F yE F y i F ,Fmn m n n m m n

12Ž .

F R Ž i.sÕŽ i.qaŽ i. , F L Ž i.sÕŽ i.yaŽ i. 13Ž .m m m m m m

ÕŽ0.sE ÕŽ0.yE ÕŽ0. , 14Ž .mn m n n m

² :where . . . denotes the trace in flavor space andm the nucleon mass. Furthermore, F,1r2 andN

Ž .D,3r4 are the conventional SU 3 axial couplingŽ 5.constants in the chiral limit, to be precise . The

dimension two terms are accompanied by finiteŽ . D , F D , Flow-energy constants LECs , called b ,b ,b .6 a 6 b 6 c

Their precise meaning will be discussed later.

4. To be specifc, we now consider the strangemagnetic form factor to one loop order in CHPT.The strange magnetic moment of the nucleon getsrenormalized by the kaon cloud, completely analo-gous to the renormalization of the nucleon isovectormagnetic moment m by the pion cloudNw x13,10,12,14,15 . It can be written as

mŽ s.smŽ s.smŽ s.N p n

1 1Ž0. D Fs G 0 q b ybŽ .M 6 b 6 b3 3

m MN K 2 2q 5D y6DFq9F , 15Ž . Ž .224p Fp

Ž .with M the kaon mass and F ' F qF r2,K p p K

102 MeV the average pseudoscalar decay constant.We use this value because the difference between thepion and the kaon decay constants only shows up at

Ž 3.higher order. One finds that to OO p in the chiralcalculation the strange magnetic moments of theproton and the neutron are predicted to be equalŽbecause symmetry breaking only starts by insertions

.from the dimension two meson-baryon Lagrangianand consist of three distinct contributions. First, the

Ž0.Ž .singlet magnetic moment G 0 is parametrized inM

terms of the unknown singlet coupling b . It cannot6 c

5 To the order we are working, it is sufficient to identify thephysical with the chiral limit values.


Ž 0 . Ž 8.Fig. 1. Coupling of the singlet ; l and octet ; l vectorŽ . Ž . .currents wiggly line to the nucleon solid line . a is a one kaon

Ž . .dashed line loop graph and b a dimension two contact term. Thelatter only contributes to the strange magnetic moment.

be predicted without additional experimental input asw xhas already been noted in 7 . The counterterms

b D , F, however, can be extracted from the anomalous6 b

magnetic moments k ,k of the proton and thep n

neutron. Third, there is a strong renormalization ofŽ s. Ž 3.m due to the kaon cloud. To OO p we findN

mŽ s. K - loops s2.0, which is large and positiÕe. ThisNw xresult is in agreement with the calculation of Ref. 7 .

It is well-known that such large leading order kaonloop effects generally are diminished by higher order

Ž . w xcorrections unitarization , see e.g. 4 . We also notethat in some models the strange magnetic moment isassumed to be generated exclusively by the kaon

w xcontributions 16 , which is already ruled out toŽ .leading order chiral analysis of Eq. 15 .

To obtain the complete strange magnetic formfactors one only has to consider the diagrams shownin Fig. 1. The important observation to make is thatto leading order in the chiral analysis, the singletcurrent only couples to the strange magnetic mo-ment, whereas the Q2-dependence is entirely gov-erned by the octet current mediated by the mesoncloud. For the loop graph 1a, in case of an incomingnucleon, the only allowed intermediate states areK L and K S, i.e. the pion and the h cloud do not

Ž .contribute to this order. For the proton p and theŽ .neutron n one finds

GŽ s. Q2 sGŽ s. p Q2 sGŽ s. n Q2Ž . Ž . Ž .M M M

p m MN KŽ s.sm qN 24p FŽ .p

= 2 2 2 25D y6DFq9F f Q , 16Ž . Ž .Ž .3

with Q2 syq2. The momentum dependence is givenentirely in terms of

2 2 2(4qQ rM QK12f Q sy q arctan . 17Ž .Ž . 2 2 2 ž /2 M(4 Q rM KK

Ž 2 .The function f Q is shown in Fig. 2. For smalland moderate Q2, it rises almost linearly with in-

2 Ž .creasing Q . We note from Eq. 16 that the slope ofŽ s.Ž 2 .G Q is uniquley fixed in terms of well-knownM

low energy parameters,

d GŽ s. q2Ž .M22<r sŽ . q s0M 2dq

p mN 1 2 2sy 5D y6DFq9FŽ .1824p F MŽ .p K

sy0.027 fm2 . 18Ž .The slope is identical for a proton or a neutron targetŽ .for the reason given above , it is negatiÕe and tothis order independent of the the strange magneticmoment mŽ s.. The radius has the very reasonableN

behavior that in the limit of very heavy kaons,M ™`, it goes to zero, whereas it explodes in theK

chiral limit M ™0. This quantity allows one toK

obtain the strange magnetic moment measured at asmall value of Q2 by linear extrapolation to Q2 s0.Note, however, that the value given should be con-sidered as a lower limit. From an analysis of theelectromagnetic form factors of the nucleon we knowthat at low momentum transfer the leading CHPT

Ž 2 .Fig. 2. The function f Q for small and moderate momentumtransfer squared.


predictions are already quite satisfactory. However,Ž .in the SU 2 ‘‘small scale expansion framework’’

w x w x11 it was found 12 that the radius of the isovectorIs1Ž 2 .magnetic Sachs form factor G q is increasedM

by 15-20% due to intermediate Dp cloud effects. AsŽ .similar analysis in the SU 3 ‘‘small scale expan-

sion’’ framework is in preparation to see whetherthere are similarly sizeable corrections for the mag-netic strangeness form factor due to intermediatedecuplet-octet states. In addition, there are other

Žmechanisms like e.g. contributions from vector.mesons not covered at this order.

5. We can also give an upper limit for the strangemagnetic form factor as the following argumentsshows. For that, we consider the electromagneticcurrent

2 1 1EM ² < < :J s N ug uy dg dy sg s Nm m m m3 3 3

11² < < : ² < < :s N qg l q N q N qg l q Nm 8 m 32'2 3

118 3s J q J , 19Ž .m m2'2 3

8 Ž .where J corresponds to the octet current of Eq. 1 .m

Ž . 3The conserved triplet current J parameterizes them

response of a nucleon coupled to an external tripletvector source ÕŽ3.sÕ l3. The calculation proceedsm m

Ž .as before. We find that while in an SU 3 calculationthe magnetic form factors of the proton and theneutron both have a pion and a kaon cloud contribu-tion, the pion cloud terms drop out to leading orderfor the isoscalar magnetic form factor of the nucleon.

Ž . w xAlso, in contrast to the SU 2 calculations 10,12 ,Ž . Is0Ž 2 .the leading one loop SU 3 contribution to G QM

picks up a momentum dependence given again en-Ž 2 . Ž .tirely in terms of the function f Q , see Eq. 17 ,

Ž .SU 3Is0 2 p 2 n 2G Q sG Q qG QŽ . Ž . Ž .M M M

M m pK Nsm ys 24p FŽ .p

= 2 2 2 25 D y6 D Fq9 F f Q ,Ž . Ž .3

20Ž .

with m s0.88 n.m. the isoscalar nucleon magnetics

moment. We note that to this order in the chiral

expansion the prediction is again free of countert-erms for the momentum dependence. Interestingly,

Ž 3.this means that to OO p the isoscalar magneticform factor of the nucleon is completely dominatedby the kaon cloud, as all virtual pion contributionscancel exactly to this order. The result is of course

Ž . w xconsistent with the SU 2 analyses of 10,12 as oneIs0Ž 2 .can check that G Q ™m in the limit M ™`,M s K

i.e. the kaon cloud contribution shows up via higherŽ .order counterterms in the SU 2 calculation. For the

leading chiral contribution to the isoscalar magneticradius one finds

6 d G Is0 q2Ž .2 MIs02<r sŽ . q s0M 2m dqs

5D2 y6DFq9F 2 mŽ . N 2s s0.18 fm ,248F m p Mp s K

21Ž .

which is about 27% of the radius derived from theŽempirical dipole parametrization notice that for the

accuracy discussed here, we do not need to employmore sophisticated parametrizations as e.g. given in

w x.Ref. 17 . It should now be clear that the isoscalarmagnetic form factor and the strangeness magneticform factor of the nucleon are closely related. InCHPT one can establish this connection on a firmground. Based on the results obtained so far, we canderive in addition to the counterterm-free prediction

2 Ž s. Ž .of the low Q -dependence of G in Eq. 16 an-M

other model-independent relation between theIs0Ž 2 .isoscalar magnetic form factor G q of the nu-M

cleon and the strange magnetic form factor:

GŽ s. Q2 smŽ s.qm yG Is0 Q2 qOO p4 22Ž .Ž . Ž . Ž .M N s M

Ž 3. Ž .This relation is exact to OO p in SU 3 heavybaryon CHPT. Possible corrections in higher orderscan be calculated systematically. This relation does

Ž s.Ž . Ž s.not constrain G 0 sm , but makes new predic-M N

tions on its Q2-dependence. Utilizing the dipoleIs0Ž 2 .parameterization for G Q one findsM

d G Is0, dip q2Ž .2 MŽ .s , dip 22<r sy sy0.10 fm .Ž . q s0M 2dq

23Ž .


Fig. 3. The strange magnetic form factor derived from the isoscalarmagnetic one with mŽ s.s0.N

This number is roughly three times larger than theŽ .leading chiral estimate of Eq. 18 . Given that there

are also non-strange contributions in the isoscalarmagnetic form factor, which will start to manifest at

4 Ž .order q , we consider Eq. 23 as an upper boundon the strange magnetic radius. The correspondingstrange magnetic form factor is shown in Fig. 3 for avanishing strange magnetic moment and using thedipole parametrization for the isoscalar magneticform factor. Any finite value for mŽ s. can be accom-N

modated by simply shifting the curve up or down theabzissa. Note that a similar dipole-like behaviour

Žwith a much smaller slope corresponding to the.lower bound discussed before was found in the

vector meson dominance type analysis supplementedw xby regulated kaon loops presented in Ref. 18 . It is

also important to note that chiral physics dominatesthe strange magnetic form factor at low momentum

Ž s.Ž 2 .transfer. However, the steady increase in G QM

will eventually be taken over by pole contributionsw xas e.g. exploited in Refs. 2,3 leading to a fall-off at

large Q2. At which momentum transfer that willhappen depends on the detailed dynamics and can

w xonly be worked out in specific models, see e.g. 18 .Note that the G0 collaboration will probe this partic-

w xular range of momentum transfer 19 .One can now utilize the Q2-dependence from the

Ž . Ž .two bounds, Eqs. 16 , 23 , to extract the strangemagnetic moment from the SAMPLE result for thestrange magnetic form factor. For Q2 s0.1 GeV 2,

Ž .the correction is y0.06 using F s102 MeV andp

y0.20, respectively, i.e. for the mean value of Ref.w x1 we get

mŽ s.s0.03 . . . 0.18 n.m. , 24Ž .p

which even for the upper value is a sizeable correc-tion. It is amusing to note that the small value formŽ s. is in agreement with the analysis presented inp

w xRef. 4 . Clearly, these numbers should only beŽ .considered indicative since a the experimental er-

Ž .rors are bigger than the correction and b higherorder corrections to the relations derived here shouldbe worked out.

6. In summary, we have derived two novel rela-tions which constrain the momentum dependence ofthe strange magnetic form factor in the low energyregion. The first one is based on the observation thatto one loop oder in three flavor chiral perturbationtheory, the strange form factor picks up a momentumdependence which is free of unknown coupling con-

Ž .stants, see Eq. 16 . The second one rests upon theobservation that the isoscalar magnetic form factor

Ž .calculated in SU 3 also acquires a momentum de-pendence which can be related to the one of thestrange magnetic form factor. This gives the model-

Ž .independent relation shown in Eq. 23 . These re-sults, which should be considered as a lower and anupper bound, respectively, should help to sharpen theextraction of the strange magnetic moment from themeasurement of the form factor at small and moder-ate momentum transfer. Clearly, the leading orderresults discussed here also need to be improved by asystematic calculation of the corresponding correc-tions. Such efforts are under way.

Acknowledgements

We would like to thank the participants of the N )

Ž ) .workshop at Trento ECT for helpful comments,especially Steve Pollock.

References

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Strange magnetism in the nucleon