b

nd chiraletizationthis pionisovector

Physics Letters B 586 (2004) 291–296

www.elsevier.com/locate/physlet

On the pion cloud of the nucleon

H.-W. Hammera,1, D. Drechselb, Ulf-G. Meißnera,c

a Universität Bonn, Helmholtz-Institut für Strahlen und Kernphysik (Theorie), Nußallee 14-16, D-53115 Bonn, Germanyb Universität Mainz, Institut für Kernphysik, J.-J. Becher Weg 45, D-55099 Mainz, Germany

c Forschungszentrum Jülich, Institut für Kernphysik (Theorie), D-52425 Jülich, Germany

Received 22 October 2003; accepted 18 December 2003

Editor: G.F. Giudice

Abstract

We evaluate the two-pion contribution to the nucleon electromagnetic form factors by use of dispersion analysis aperturbation theory. After subtraction of the rho-meson component, we calculate the distributions of charge and magnin coordinate space, which can be interpreted as the effects of the pion cloud. We find that the charge distribution ofcloud effect peaks at distances of about 0.3 fm. Furthermore, we calculate the contribution of the pion cloud to thecharges and radii of the nucleon. 2004 Elsevier B.V. All rights reserved.

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1. That the pion plays an important role in thlong-range structure of the nucleon is known sinlong. This can, for example, be deduced fromphenomenological analysis of the nucleon electromnetic form factors (for an early attempt within mestheory see, e.g., [1]). However, only with the adveof QCD and its spontaneously broken chiral symmtry, in which the pions emerge as pseudo-Goldstbosons, this concept could be put on a firmer basis.ploiting the chiral symmetry of QCD, the long-ranlow-momentum structure of the nucleon can be calated within chiral perturbation theory (CHPT), whicis the low-energy effective field theory of the Standa

E-mail addresses: hammer@itkp.uni-bonn.de(H.-W. Hammer), drechsel@kph.uni-mainz.de (D. Drechsel),meissner@itkp.uni-bonn.de (U.-G. Meißner).

1 Present address: Institute for Nuclear Theory, UniversityWashington, Seattle, WA 98195-1550, USA.

0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2003.12.073

Model. For calculations of these form factors withCHPT, see [2–10]. Furthermore, since vector mesplay an important role in the electromagnetic structof the nucleon, see, e.g., [11–20], care must be takeattributing a certain size or length scale to the varicontributions (this is discussed in some detail in Stion 2 of [21]). A new twist to this picture was givein the recent paper of Ref. [22], where an interpretion of the form factor data was given in terms ofphenomenological fit with an ansatz for the pion clobased on the old idea that the proton can be thougas virtual neutron-positively charged pion pair. A velong-range contribution to the charge distributionthe Breit-frame extending out to about 2 fm was fouand attributed to the pion cloud. While naively the piCompton wave length is of this size, these findingsindeed surprising if compared with the “pion cloucontribution due to the two-pion contribution for thisovector spectral functions of the nucleon form fa

.

292 H.-W. Hammer et al. / Physics Letters B 586 (2004) 291–296

aln-uc-wellon

re-ofc-iral

s.nedro-

heors

-se

andthef a

aly-ro-tion

ven-m

alarn-s.

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ar,the

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tors, which can be obtained from unitarity or chirperturbation theory. As it will turn out these latter cotributions to the long-range part of the nucleon strture are much more confined in space and agreewith earlier (but less systematic) calculations basedchiral soliton models, see, e.g., [23]. Therefore itmains to be shown how to reconcile the findingsRef. [22], based on a global fit to all nucleon form fators, with the results of dispersion analysis and chperturbation theory.

2. First, we must collect some basic definitionThe nucleon electromagnetic form factors are defiby the nucleon matrix element of the quark electmagnetic current,

⟨N(p′)

∣∣qγ µQq∣∣N(p)

⟩

= u(p′)[γ µF1

(q2)

(1)+ i

2mσµν(p′ − p)νF2

(q2)]u(p),

with q2 = (p′ − p)2 = t the invariant momentumtransfer squared,Q the quark charge matrix, andmthe nucleon mass.F1(q

2) andF2(q2) are the Dirac

and the Pauli form factors, respectively. Following tconventions of [17], we decompose the form factinto isoscalar (S) and isovector (V ) components,

(2)Fi

(q2) = FS

i

(q2) + τ3F

Vi

(q2), i = 1,2,

subject to the normalization

(3)FS1 (0) = FV

1 (0) = 1

2, F

S,V2 (0) = κp ± κn

2,

with κp (κn) = 1.793 (−1.913) the anomalous magnetic moment of the proton (neutron). We will also uthe Sachs form factors,

GIE

(q2) = FI

1

(q2) + q2

4m2FI2

(q2),

(4)GIM

(q2) = FI

1

(q2) + FI

2

(q2), I = S,V .

These are commonly referred to as the electricthe magnetic nucleon form factors. The slope ofform factors atq2 = 0 can be expressed in terms o

nucleon radius

(5)

⟨r2⟩I

i= 6

FIi (0)

dF Ii (q

2)

dq2

∣∣∣∣q2=0

, i = 1,2, I = S,V,

and analogously for the Sachs form factors. The ansis of the nucleon electromagnetic form factors pceeds most directly through the spectral representagiven by2

(6)

FIi

(q2) = 1

π

∞∫

(µI0)

2

σ Ii (µ

2) dµ2

µ2 − q2, i = 1,2, I = S,V,

in terms of the real spectral functionsσ Ii (µ

2) =ImFI

i (µ2), and the corresponding thresholds are gi

by µS0 = 3Mπ , µV

0 = 2Mπ . Since the isovector spectral function is non-vanishing for smaller momentutransfer (starting at the two-pion cut) than the isoscone (starting at the three-pion cut), we will mostly cosider the isovector spectral functions in what followWe consider the nucleon form factors in the spalike region. In the Breit-frame (where no energytransferred), any form factorF can be written as thFourier-transform of a coordinate space density,

(7)F(q2) =

∫d3r eiq·rρ(r),

with q the three-momentum transfer. In particulcomparison with Eq. (6) allows us to expressdensityρ(r) in terms of the spectral function

(8)ρ(r) = 1

4π2

∞∫

µ20

dµ2σ(µ2)e−µr

r.

Note that for the electric and the magnetic Saform factor,ρ(r) is nothing but the charge and thmagnetization density, respectively. For the Dirac aPauli form factors, Eq. (8) should be considered aformal definition. This equation expresses the denas a linear combination of Yukawa distributions, ea

2 We work here with unsubtracted dispersion relations. Sincenormalizations of all the form factors are known, one could awork with once-subtracted dispersion relations. For the topic stuhere, this is of no relevance.

H.-W. Hammer et al. / Physics Letters B 586 (2004) 291–296 293

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of massµ. The lightest mass hadron is the pion, afrom Eq. (8) it is evident that pions are responsiblethe long-range part of the electromagnetic structurthe nucleon. This contribution is commonly called t“pion cloud” of the nucleon and in fact this long-ranlow-q2 contribution to the nucleon form factors candirectly calculated on the basis of chiral perturbattheory, as will be discussed later.

3. Next, we wish to evaluate the two-pion contbution in a model-independent way and draw soconclusions on the spatial extent of the pion clofrom that. As pointed out long ago [11] and furthelaborated on [14], unitarity allows us to determthe isovector spectral functions from threshold upmasses of about 1 GeV in terms of the pion chaform factor Fπ(t) and theP -wave ππNN partialwaves, see Fig. 1. We use here the form

ImGVE(t) = q3

t

m√t

∣∣Fπ(t)∣∣2J+(t),

(9)ImGVM(t) = q3

t√2t

∣∣Fπ(t)∣∣2J−(t),

where qt = √t/4−M2

π . The functionsJ±(t) arerelated to thet-channelP -wave πN partial wavesf 1±(t) via f 1±(t) = Fπ(t)J±(t) in the conventionaisospin decomposition, with the tabulated values ofJi(t) from [24]. For the pion charge form factorFπ

we use the standard Gounaris–Sakurai form [13].stress that the representation of Eq. (9) gives the eisovector spectral functions for 4M2

π � t � 16M2π

but in practice holds up tot 50M2π . It has two

distinct features. First, as already pointed out in [1it contains the important contribution of theρ meson(see Fig. 1) with its peak att 30M2

π . Second, on theleft shoulder of theρ, the isovector spectral functiondisplay a very pronounced enhancement close totwo-pion threshold. This is due to the logarithmsingularity on the second Riemann sheet locatetc = 4M2

π − M4π/m

2 = 3.98M2π , very close to the

threshold. This pole comes from the projection ofnucleon Born graphs, or in modern language, frthe triangle diagram also depicted in Fig. 1. If owere to neglect this important unitarity correction, owould severely underestimate the nucleon isoveradii [15]. In fact, precisely the same effect is obtain

Fig. 1. Two-pion contribution to the isovector nucleon form factoOn the left side, the exact representation based on unitarity is shwhereas the triangle diagram on the right side leads to the stenhancement of the isovector spectral functions close to thresAlso shown is the dominantρ-meson contribution.

at leading one-loop accuracy in chiral perturbattheory, as discussed first in [2,25]. This topic wfurther elaborated on in the framework of heabaryon CHPT [4,9] and in a covariant calculatibased on infrared regularization [7]. Thus, the mimportant two-pion contribution to the nucleon forfactors can be determined by using either unitarityCHPT (in the latter case, of course, theρ-contributionis not included).

We now want to separate the (uncorrelated) pcontribution from theρ-contribution in the isovectospectral functions, that is we decompose the isovespectral functions as

(10)

ImGVI (t) = ImG

V,2πI (t) + ImG

V,ρI (t), I = E,M,

and analogously for ImFV1,2(t). Using Eq. (8), we

can then calculate the pion cloud contributionthe charge and magnetization density in the Brframe. Theρ-contribution in Eq. (10) can be werepresented by a Breit–Wigner form with a runniwidth [9],

(11)

ImGV,ρI (t) = bIM

2ρ

√t Γρ(t)

(M2ρ − t)2 + tΓ 2

ρ (t), I = E,M,

with the massMρ = 769.3 MeV and the widthΓρ(t) = g2(t − 4M2

π)3/2/(48πt), where the coupling

g = 6.03 is determined from the empirical valuΓρ(M

2ρ) = 150.2 MeV, and the parametersbI can be

adjusted to the height of the resonance peak. Theresponding expressions for the imaginary parts ofDirac and Pauli form factors can be obtained froEq. (4). It is clear that the separation into the (ucorrelated) pion contribution and theρ-contributionintroduces some model-dependence. To get an

294 H.-W. Hammer et al. / Physics Letters B 586 (2004) 291–296

ac

Fig. 2. The densities of charge and magnetization due to the pion cloud. Left panel: 4πr2ρ(r) for the isovector Pauli (upper band) and Dir(lower band) form factors. Right panel: 4πr2ρ(r) for the isovector magnetic (upper band) and electric (lower band) Sachs form factors.

ce-ent

b-ry

ion

gyof

n

ns

nion

d

cu-s

ntain

lar

odnant

utn

oreAs

inedp

ralm

rm

t is

ion.are(c)cepterracofw atndhtted

rm

about the theoretical error induced by this produre, we perform the separation in three differways:

(a) The two-pion contribution can be directly otained from the two-loop chiral perturbation theocalculation of [9]. Together with theρ-contribution ofEq. (11), this calculation gives a very good descriptof the empirical spectral functions.3 We will use theanalytical formulae given in [9] where the low-enerconstantc4 was readjusted to avoid double countingtheρ-contribution (see [27]).

(b) A lower bound on the two-pion contributiocan be obtained by settingFπ(t) = 1 in Eq. (9).This prescription does not only remove theρ-pole butalso some small uncorrelated two-pion contributiocontained in the pion form factor.

(c) To obtain the two-pion contribution, we caalso subtract Eq. (11) from the spectral functEq. (9) including the full pion form factor.4 The pa-rametersbE = 1.512 andbM = 5.114 are determinesuch that the two-pion contribution at theρ-resonancematches the two-loop chiral perturbation theory callation of [9]. Variation of thebI around these valuegives an additional error estimate.

3 Note that on the right side of theρ, the two-loop representatiois slightly larger than the empirical one, so that we expect to oban upper bound by employing this procedure.

4 A similar procedure was performed in [26] to extract scameson properties from the scalar pion form factor.

Using these three methods, we obtain a fairly gohandle on the theoretical accuracy of the non-resotwo-pion contribution.

4. We have now collected all pieces to work othe density distribution of the two-pion contributioto the nucleon electromagnetic form factors. Befshowing the results, some remarks are in order.stated above, the spectral functions are determby unitarity (or chiral perturbation theory) only uto some maximum value oft , denotedtmax in thefollowing. Thus, we have simply set the spectfunctions in the integral Eq. (8) to zero for momentutransfers beyond the valuetmax = 40M2

π . In Fig. 2,we show the resulting densities for the isovector fofactors weighted with 4πr2. The contribution of the“pion cloud” to the total charge or magnetic momenthen simply obtained by integration overr. The bandsreflect the theoretical uncertainty in the separatFor all form factors, the lower and upper boundsgiven by methods (b) and (a), respectively. Methodgenerally yields a result between these bounds, exfor the Dirac form factor where it gives the uppbound. The weighted densities for the isovector Diand Pauli form factors are shown in the left panelFig. 2. We see that these charge distributions shopronounced peak aroundr 0.3 fm, quite consistenwith earlier determinations (see, e.g., [23,28]), afall off smoothly with increasing distance. In the rigpanel of Fig. 2, we show the densities (again weighwith 4πr2) for the electric and magnetic Sachs fo

H.-W. Hammer et al. / Physics Letters B 586 (2004) 291–296 295

s and

Table 1Two-pion contribution to charges and radii (in fm2) for the various nucleon form factors. The radii are normalized to the physical chargemagnetic moments

FV1 (0) FV

2 (0) GVE(0) GV

M(0) 〈r2〉V1 〈r2〉V2 〈r2〉V

E〈r2〉V

M

0.07. . . 0.08 0.4. . .1.0 0.1. . .0.2 0.4. . .1.0 0.1. . .0.2 0.2. . .0.3 0.2. . .0.3 0.2. . . 0.3

ofithony a

ltsei-w-ndenaofo-ri-

ionic)is

tricndote

mthediisem.re-ins

onheromldt

centormtum

.

ntatedef-ti-

ure,entd bydis-the

asp-

ein-

6

88)

ys.

1

56

er,

er,

p-

ep-

2.

.

factors which come out very similar to the casethe Dirac and Pauli form factors. In comparison wRef. [22], we generally obtain much smaller picloud effects at distances beyond 1 fm, e.g., bfactor 3 forρV

E (r) at r = 1.5 fm.5

We have also studied the sensitivity of our resuto the cut-offtmax. While this may increase the valuof the “pion cloud” contribution, it leaves the postion of the maximum essentially unchanged. Hoever, it is obvious from Eq. (8) that masses beyo1 GeV and corresponding small-distance phenom(r � 0.2 fm) should not be related to the pion cloudthe nucleon. Finally, we show the corresponding twpion contribution to the charges and radii for the vaous nucleon form factors in Table 1. The contributof the pion cloud to the isovector electric (magnetcharge is 20% (10%) in the model of Ref. [22]. This consistent with our range of values for the eleccharge but a factor of 1.5 smaller than our lower boufor the magnetic one, see Table 1. Furthermore, nthat the pion cloud gives only a fraction of all forfactors at zero momentum transfer. Normalized tocontribution of the pion cloud, the corresponding raare of the order of 1 fm. In the model of [22], theradii are considerably larger, of the order of 1.5 fNote that if one shifts all the strength of the corsponding spectral functions to threshold, one obtaan upper limitrmax = √

3/2M−1π 1.7 fm, assuming

that the spectral functions are positive definite.

5. In this Letter, we have considered the two-picontribution to the nucleon isovector form factors. Tcorresponding spectral functions can be obtained funitarity or chiral perturbation theory from threshot0 = 4M2

π up to tmax 40M2π in a model-independen

5 Note that our results are not in disagreement with the reJefferson Lab data on the ratio of the proton electromagnetic ffactors [29,30]. The effect observed there occurs at momentransfers beyond 1 GeV2 and is thus not related to the pion cloud

way. Subtracting the contribution from the dominaρ-meson pole, we have constructed the uncorreltwo-pion component (loosely spoken the dominantfects of the nucleons’ pion cloud). To obtain an esmate about the theoretical error of such a procedwe have performed this subtraction in three differways. The corresponding charge densities obtaineFourier transforming the spectral functions peak attances of about 0.3 fm and show no structure atlarger distances.

Acknowledgements

We thank J. Friedrich and Th. Walcher as wellN. Kaiser for useful discussions. This work was suported in part by the Deutsche Forschungsgemschaft (SFB 443).

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