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The pion form factor from first principles

The pion form factor from first principles
The pion form factor from first principles

a r X i v :h e p -l a t /0309183v 1 29 S e p 2003

The pion form factor from ?rst principles

J.van der Heide

National Institute for Nuclear Physics and High-Energy Physics (NIKHEF),1009DB Amsterdam,

The Netherlands

Abstract.We calculate the electromagnetic form factor of the pion in quenched lattice QCD.The non-perturbatively improved Sheikoleslami-Wohlert lattice action is used together with the O (a )improved current.We calculate form factor for pion masses down to m π=360MeV .We compare the mean square radius for the pion extracted from our form factors to the value obtained from the ’Bethe Salpeter amplitude’.Using (quenched)chiral perturbation theory,we extrapolate our results towards the physical pion mass.

INTRODUCTION

The pion,being the lightest and simplest particle in the hadronic spectrum has been stud-ied intensively in the https://www.sodocs.net/doc/0212184703.html,ing a variety of effective and phenemenological models,the properties of the pion have been desribed with varying succes.However,these mod-els make assumptions,for example,con?nement is put in by hand in contrast to being the result of the underlying https://www.sodocs.net/doc/0212184703.html,ttice QCD (LQCD)doesn’t have this drawback since it is solves QCD directly.

Using LQCD,global properties of the pion such as the mass and the decay width have been calculated to satisfying accuracy.The form factor,which directly re?ects the internal structure,is clearly an important challenge.The ?rst lattice results were obtained by Martinelli and Sachrajda [1].It was followed by a more detailed study by Draper et al.[2],who showed that the form factor could be described by a simple monopole form as suggested by vector meson dominance [3].We extend [4]these studies by adopting improved lattice techniques [10–14],which means that we include extra operators in order to systematically eliminate all the O (a )discretisation errors.

Some aspects of the pion structure have been obtained [5–9]using ’the Bethe-Salpeter method’.We also use this approach and compare its predictions to the results of our direct calculation of the pion form factor.

Finally we study a chiral extrapolation to reach the physical limit.

THE METHOD

To extract the form factor we calculate the two-and three point functions of the pion,analogously to [2].The two point function is given by

G 2(t ,p )=∑x

φ(t ,x )φ?

(0,0) e i p ·x ,

(1)

whereφ?is an operator creating a state with the quantum numbers of the pion.By varying the interquark distance at the sink,t f,we can improve the overlap with the physical pion and obtain information on the’Bethe-Salpeter amplitudes’.The three point

function is calculated as

G3(t f,t;p f,p i)=∑

x f,x φ(x f)j4(x)φ?(0) e?i p f·(x f?x)?i p i·x(2)

with j4the fourth component of the current,inserted at time t.Since the local current

j Lμ(x)=ˉψ(x)γμψ(x),(3) is not conserved on the lattice,one can construct the Noether current belonging to our action

j Cμ=κ ˉψ(x)(1?γμ)Uμ(x)ψ(x+?μ)?ˉψ(x+?μ)(1+γμ)U?μ(x)ψ(x) .(4)

This current however,still has O(a)corrections for Q2>0.A conserved and improved current can be constructed[12–14]

j Iμ=Z V{j Lμ(x)+a c V?νTμν},(5) with

Tμν=ˉψ(x)iσμνψ(x),(6)

Z V=Z0V(1+ab V m q).

The bare-quark mass is de?ned as m q=1

Z n R(p)Z n0(p) e?E n p t+e?E n p(Nτ?t) ,(7)

including the contribution of the ground state(n=0)and a?rst excited one(n=1).The Z n R denote the matrix elements,

Z n R(p)≡| ?|φR|n,p |2,(8) and are related to the’Bethe-Salpeter amplitude’,Φ(R)=

Z0R(p f)Z00(p i)e?E0p f(t f?t)?E0p i t

+

}?1,(10)

M2ρ

which is suggested by the vector meson dominance ansatz.Fitting our data to this model, we extract a vector meson mass which is within5%of the corresponding rho mass on the lattice[15].From the behaviour of the form factor at low Q2,we can extract the mean-square charge radius of the pion,

dF(Q2)

6 r2 FF=?1

0.20.40.60.8100.5

1 1.52

F (Q 2

)

Q 2

(GeV 2

)

FIGURE 1.Form factors as a function of Q 2for the ?ve pion masses.Curves:monopole ?ts to the data.

where in the last step we assume Eq.10and use the ?tted parameter M V .The results are shown in Fig.2as a function of the pion mass.Previously,the ’Bethe-Salpeter-amplitude’Φ(R )has been used to obtain estimates of the charge radius,

r 2 BS :=

1

d 3 r Φ2(| r |)

.

(12)

The results based on this procedure are also shown in Fig.2.As can be seen these val-ues are much lower than the actual values obtained from the form factor.Moreover,the

Bethe-Salpeter results are almost mass independent,in accordance with the observations of Refs.[5–7,9].This is a known[7]de?cit of the approach,which Fig.2makes quan-titative.We extrapolate our results obtained with Eq.11using three different parametri-sations.First,we try chiral perturbation theorie (χpt).At one-loop order the prediction for the radius [18]is

r 2 one ?loop

χPT =c 1+c 2log m 2

π(13)In quenched χpt the rms is constant to this order.There are however indications that a

mass dependence appears at higher order [19].For our masses we restrict ourselves to

a term linear in m 2π

[20].Lastly,we have also used the VMD prescription and assume a linear dependence of M V on m 2π

.These three expectations are also plotted in Fig.2.

00.050.10.150.20.250.30.350.40.450.50

0.2

0.4

0.60.81 1.2

R M S 2

(f m 2

)

m π2

(GeV 2

)

FIGURE 2.Radius of the pion as obtained from Bethe-Salpeter amplitudes,Eq.12,and from the form factor,Eq.11.Also shown are three different parametrizations of r 2 (see text).

ACKNOWLEDGEMENTS

This work has been done in collaboration with Justus Koch and Edwin Laermann.

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