The pion mass dependence of the nucleon formfactors
of the energy momentum tensor in the chiral quarksoliton model
Abstract
The nucleon form factors of the energymomentum tensor are studied in the large limit in the framework of the chiral quarksoliton model for model parameters that simulate physical situations in which pions are heavy. This allows for a direct comparison to lattice QCD results.
pacs:
11.15.Pg, 12.39.Fe, 12.38.Lg, 12.38.Gcpreprint RUBTP2082006
I Introduction
The nucleon energy momentum tensor (EMT) form factors Pagels contain valuable information on the nucleon structure. They carry information on, e.g., how the quark and gluon degrees of freedom share the total momentum, and angular momentum of the nucleon Ji:1996ek , or on the distribution of strong forces inside the nucleon Polyakov:2002yz . The first is known from deeply inelastic lepton nucleon scattering experiments. The latter two can be deduced from generalized parton distribution functions (GPDs) Muller:1998fv ; Ji:1996nm ; Collins:1996fb ; Radyushkin:1997ki accessible in hard exclusive reactions Saull:1999kt ; Adloff:2001cn ; Airapetian:2001yk ; Stepanyan:2001sm ; Ellinghaus:2002bq ; Chekanov:2003ya , see Ji:1998pc ; Radyushkin:2000uy ; Goeke:2001tz ; Diehl:2003ny ; Belitsky:2005qn for reviews.
Nucleon EMT form factors were studied in lattice QCD calculations Mathur:1999uf ; Gadiyak:2001fe ; Hagler:2003jd ; Gockeler:2003jf ; Negele:2004iu . In principle, lattice QCD provides a rigorous and modelindependent approach to compute the nucleon EMT form factors. In practice, however, present day technics and computing power allow to simulate on lattices “worlds” with pions of typically . The situation is extected to improve in the future, see Schroers:2007qf for status reports on selected topics.
For the time being, however, it is necessary to use chiral extrapolation in order to relate lattice results to the real world situation. Chiral perturbation theory (PT) provides a modelindependent tool for that, and the chiral behaviour of the nucleon EMT form factors was studied in Chen:2001pv ; Belitsky:2002jp ; Diehl:2006ya . Experience with chiral extrapolations of other nucleon properties indicates that PT is applicable up to the lowest presently available lattice values of Bernard:2003rp ; Procura:2003ig ; AliKhan:2003cu although the issue is not yet settled Beane:2004ks .
In this situation it is worth looking on what one can learn about the chiral behaviour of nucleon properties from other effective approaches, e.g. the “finite range regulator” (FRR) approach. There chiral loops are regulated by suitably chosen vertex form factors in order to phenomenologically simulate the effects of the pion cloud which has a finite range due to Leinweber:1998ej ; Thomas:2005rz . However, model calculations Matevosyan:2005fz ; Schweitzer:2003sb ; Goeke:2005fs are equally of interest in this context.
A phenomenologically successful and theoretically consistent model for the description of nucleon properties at the physical point and in the chiral limit, is the chiral quark soliton model (CQSM) Diakonov:yh ; Diakonov:1987ty . This model describes the nucleon as a soliton of a static background pion field in the limit of a large number of colours , and hence provides a particular realization of the general large picture of the nucleon Witten:1979kh . The CQSM describes numerous nucleonic properties without adjustable parameters – including among others form factors Christov:1995hr ; Christov:1995vm ; Silva:2005qm , usual parton distribution functions Diakonov:1996sr ; Diakonov:1997vc ; Diakonov:1998ze ; Wakamatsu:1998rx ; Goeke:2000wv and GPDs Petrov:1998kf ; Penttinen:1999th ; Schweitzer:2002nm ; Schweitzer:2003ms ; Ossmann:2004bp ; Wakamatsu:2005vk — within an accuracy of as far as those quantities are known.
That it is possible to extend the CQSM to the description of the nucleon at large pion masses was shown in Goeke:2005fs , where the model was demonstrated to provide a good description of lattice data on the dependence of the nucleon mass up to . An important prerequisite for that is that the CQSM formally contains the correct heavy quark limit result for the nucleon mass Goeke:2005fs .
In this work we present a study of the dependence of the nucleon EMT form factors in the CQSM up to pion masses as large as . The present study extends the study of Ref. accompanyingpaperI where the nucleon EMT form factors were studied at the physical point and in the chiral limit, and its purpose is threefold.
First, we provide an important supplement for the study in Ref. Goeke:2005fs . There soliton solutions were obtained numerically for model parameters corresponding to pion masses in the range . Here we provide a cross check demonstrating that the numerical solutions found in Goeke:2005fs correspond, in fact, to stable solitons.
Second, with the results obtained for large we are in the position to confront the model predictions for the nucleon EMT form factors directly to lattice QCD results Mathur:1999uf ; Gadiyak:2001fe ; Hagler:2003jd ; Gockeler:2003jf ; Negele:2004iu . In view of the early stage of art of the experimental situation of hard exclusive reactions, such a comparison provides the only presently available test for our results.
Third, though the model — as discussed in detail in Goeke:2005fs — cannot be used as a quantitative guideline for the chiral extrapolation, our study still allows to gain several interesting qualitative insights with this respect.
The note is organized as follows. In Sec. II we introduce the nucleon EMT form factors and discuss their properties. In Sec. III we briefly review how the nucleon EMT form factors are described in the CQSM. In Secs. IV and V we describe respectively the model results for the densities associated with the form factors and the form factors themselves. In Sec. VI we compare the model results with lattice QCD data, and in Sec. VII we discuss which qualitative observations from our study could be of interest in the context of the chiral extrapolation of lattice data. Sec. VIII contains the conclusions. A remark on different notations for the EMT form factors is posed in App. A.
Ii Form factors of the energymomentum tensor
The nucleon matrix element of the symmetric EMT of QCD is characterized by three scalar form factors Pagels . The quark and gluon parts, and , of the EMT are separately gaugeinvariant and can be parameterized as Ji:1996ek ; Polyakov:2002yz , see App. A for an alternative notation,
(1)  
Here the nucleon states and spinors are normalized by and , and spin indices are suppressed for brevity. The kinematical variables are defined as , , . The form factor accounts for nonconservation of the separate quark and gluon parts of the EMT, and enters the quark and gluon parts with opposite signs such that the total (quark+gluon) EMT is conserved.
The nucleon form factors of the EMT are related to the second Mellin moments of the unpolarized GPDs and as (we use the notation of Ref. Goeke:2001tz )
(2) 
where denotes the socalled skewedness parameter Ji:1996ek . The sum rules in Eqs. (2) are special cases of the socalled polynomiality property of GPDs Ji:1998pc . The second sum rule in (2) provides the possibility to access , i.e. the total (spin+orbital angular momentum) contribution of quarks to the nucleon spin, through the extraction of GPDs from hard exclusive processes and extrapolation to the unphysical point . The sensitivity of different observables to the total quark angular momenta was investigated in model studies Goeke:2001tz ; Ellinghaus:2005uc . For gluons there are analog definitions and expressions. Suffice to remark that the full GPDs contain far more information ReffurtherinformationinGPDs .
The form factors of the EMT in Eq. (1) can be interpreted Polyakov:2002yz in analogy to the electromagnetic form factors Sachs in the Breit frame characterized by . In this frame one can define the static energymomentum tensor for quarks
(3) 
and analogously for gluons. The initial and final polarization vectors of the nucleon and are defined such that in the respective restframe they are equal to with the unit vector denoting the quantization axis for the spin.
The components of and correspond respectively to the distribution of quark momentum and quark angular momentum inside the nucleon. The components of characterize the spatial distribution of “shear forces” experienced by quarks inside the nucleon. The respective form factors are related to by
(4)  
(5)  
(6) 
where the prime denotes derivative with respect to the Mandelstam variable . Note that for a spin1/2 particle only the components are sensitive to the polarization vector. Note also that Eq. (6) holds for the sum with and and defined analogously, but not for the separate quark and gluon contributions – since otherwise the form factor would not cancel out.
The form factor at is connected to the fractions of the nucleon momentum carried respectively by quarks and gluons. More precisely
(7) 
where are the unpolarized parton distributions accessible in inclusive deeply inelastic scattering.
The form factors , and are renormalization scale dependent (the indication of the renormalization scale is suppressed for brevity). Their quark+gluon sums, however, are scale independent form factors, which at satisfy the constraints,
(8) 
which mean that in the rest frame the total energy of the nucleon is equal to its mass, and that its spin is 1/2. The value of is not known a priori and must be determined experimentally. However, being a conserved quantity it is to be considered on the same footing as other basic nucleon properties like mass, anomalous magnetic moment, etc. Remarkably, determines the behaviour of the term Polyakov:1999gs (and thus the unpolarized GPDs) in the asymptotic limit of renormalization scale Goeke:2001tz .
The form factor is connected to the distribution of pressure and shear forces experienced by the partons in the nucleon Polyakov:2002yz which becomes apparent by recalling that is the static stress tensor which (for spin 0 and 1/2 particles) can be decomposed as
(9) 
Hereby describes the radial distribution of the “pressure” inside the hadron, while is related to the distribution of the “shear forces” Polyakov:2002yz . Both are related due to the conservation of the EMT by the differential equation
(10) 
Another important consequence of the conservation of the EMT is the socalled stability condition
(11) 
Let us review briefly what is known about — which in terms of the pressure or shear forces is given by
(12) 
For the pion it can be calculated exactly using soft pion theorems with the result Polyakov:1999gs , see also Teryaev:2001qm . Also for the nucleon was found in calculations in CQSM accompanyingpaperI ; Petrov:1998kf ; Kivel:2000fg . For the nucleon the large limit predicts the flavourdependence Goeke:2001tz . Lattice calculations Mathur:1999uf ; Gadiyak:2001fe ; Hagler:2003jd ; Gockeler:2003jf ; Negele:2004iu confirm this flavour dependence and yield a negative , see Sec. VI. In a simple “liquid drop” model can be related to the surface tension of the “liquid” and comes out negative Polyakov:2002yz . Such a model is in particular applicable to nuclei and predictions from this picture Polyakov:2002yz were confirmed in calculations assuming realistic nuclear models Guzey:2005ba . In Ref. accompanyingpaperI it was conjectured on the basis of plausible physical arguments that the negative sign of is dictated by stability criteria. This conclusion, however, remains to be proven for the general case.
Iii Nucleon EMT form factors in the CQSM
The effective theory underlying the CQSM was derived from the instanton model of the QCD vacuum Diakonov:1983hh ; Diakonov:1985eg ; Diakonov:1995qy which assumes that the basic properties of the QCD vacuum are dominated by a strongly interacting but dilute instanton medium, see the reviews Diakonov:2000pa . In this medium light quarks acquire a dynamical (“constituent”) quark mass due to interactions with instantons. At low momenta below a scale set by , where denotes the average instanton size, the dynamics of the effective quark degrees of freedom is given by the partition function Diakonov:1984tw ; Dhar:1985gh
(13) 
Here we restrict ourselves to two flavours and neglect isospin breaking effects with denoting the current quark mass, while denotes the chiral pion field with . The dynamical mass is strictly speaking momentum dependent, i.e. . However, in practical calculations it is convenient to work with a constant following from the instanton vacuum Diakonov:2000pa , and to regularize the effective theory by means of an explicit regularization with a cutoff of whose precise value is fixed to reproduce the physical value of the pion decay constant . In this work we will use the propertime regularization.
The quark degrees of freedom of the effective theory (13) correspond to QCD quark degrees of freedom up to corrections which are small in the instanton packing fraction , where denotes the average separation of instantons. The same parameter suppresses the contribution of gluon degrees of freedom Diakonov:1995qy .
The CQSM is an application of the effective theory (13) to the description of baryons Diakonov:yh ; Diakonov:1987ty . While the Gaussian path integral over fermion fields in (13) can be solved exactly, the path integral over pion field configurations can be solved only in the large limit by means of the saddlepoint approximation (in the Euclidean formulation of the theory). In the leading order of the large limit the pion field is static, and one can determine the spectrum of the oneparticle Hamiltonian of the effective theory (13)
(14) 
The spectrum of (14) consists of an upper and a lower Dirac continuum, distorted by the pion field as compared to continua of the free DiracHamiltonian (given by with replaced by ), and of a discrete bound state level of energy , if the pion field is strong enough. By occupying the discrete level and the lower continuum states each by quarks in an antisymmetric colour state, one obtains a state with unity baryon number. The soliton energy
(15) 
is a functional of the pion field. It is logarithmically divergent, see e.g. Christov:1995vm for explicit expressions in the propertime regularization. By minimizing one obtains the selfconsistent solitonic pion field . This procedure is performed for symmetry reasons in the socalled hedgehog ansatz with the radial (soliton profile) function and , . The nucleon mass is given by .
In the large limit the path integral over in Eq. (13) is solved by evaluating the expression at and integrating over translational and rotational zero modes of the soliton solution. In order to include corrections in the expansion one considers time dependent pion field fluctuations around the solitonic solution. In practice hereby one restricts oneself to time dependent rotations of the soliton field in spin and flavourspace which are slow because the corresponding soliton moment of inertia
(16) 
is large, . It is logarithmically divergent and has to be regularized. In (16) one has to sum over occupied (“occ”) states which satisfy , and over nonoccupied (“non”) states which satisfy .
The model expressions for the nucleon EMT form factors in the effective theory (13) were derived explicitly in accompanyingpaperI . The gluon part of the EMT is zero in the effective theory (13), because there are no explicit gluon degrees of freedom. (So we omit the index when discussing the model results in this and in Secs. IV, V but restore it later.)
Consequently, in the model the quark part of the EMT is conserved by itself, and the formfactor in Eq. (1) vanishes accompanyingpaperI . The model expressions for the other form factors read
(17)  
(18)  
(19) 
with the Bessel functions and . The Fourier transforms of the form factors, which are radial functions and to which we refer as “densities” in the following, are defined as
(20)  
(21)  
(22) 
The expressions in (20, 21, 22) are logarithmically UVdivergent. Here we use the propertime method to regularize them, see accompanyingpaperI for explicit expressions in this regularization. In Ref. accompanyingpaperI analytical proofs were given that

the stability condition (11) is satisfied in the model,

the form factors satisfy the constraints at in Eq. (8), and

the same expressions for EMT form factors follow in the model from unpolarized GPDs via the sum rules (2).
Notice that the latter is a special case of the “polynomiality property” of GPDs Ji:1998pc satisfied in the CQSM Schweitzer:2002nm ; Schweitzer:2003ms . The fieldtheoretic character of the model is a crucial prerequisite which allows to formulate and analytically prove such and other Diakonov:1996sr general QCD requirements. This in turn provides important cross checks for the theoretical consistency of the approach.
For the numerical calculation we employ the socalled KahanaRipka method Kahana:1984be , whose application to calculations of the nucleon EMT form factors in the CQSM is briefly described in Ref. accompanyingpaperI . The use of the propertime regularization has the advantage (over, e.g., the PauliVillars method Kubota:1999hx ) that it is possible to include explicitly chiral symmetry breaking effects due to a finite current quark mass in the effective action (13).
In Ref. Goeke:2005fs it was shown that it is possible to obtain soliton solutions and compute nucleon masses for current quark masses up to which corresponds to pion masses up to . What provides a certain justification for the application of the model up such large is the fact that the model formally contains the correct heavy quark limit result for the nucleon mass Goeke:2005fs . The proof that in the limit , where is the heavy quark mass, the nucleon mass tends to given in Goeke:2005fs is formal because in this proof it was taken for granted that stable soliton solutions do exist up to such large pion mass values.
Therefore, it was of importance in Ref. Goeke:2005fs to demonstrate numerically the existence of soliton solutions for large up to, at least, . These soliton solutions were found by using a standard iteration procedure for the calculation of the selfconsistent profile function described in detail, e.g. in Christov:1995vm . Here, as a byproduct of our study of EMT form factors, we will be in the position to provide an important and valuable cross check. Namely, do the pressures computed with the respective large soliton profiles really satisfy the stability condition (11)? The answer is yes, see below Sec. IV.4.
Before discussing the dependence of the densities (20, 21, 22) and the form factors (17, 18, 19) we have to establish which model parameters are allowed to vary and which are kept fixed while is varied. Here we shall use the choice of Ref. Goeke:2005fs to keep and fixed. Then the propertime cutoff is adjusted such that for a given and one reproduces the physical value of ( and are related to each other in the effective theory (13) by a relation which for small corresponds to the GellMann—Oakes—Renner relation, see Goeke:2005fs ).
This way of parameter handling in the model was found to provide a good description of lattice data on the variation of with , once one takes into account the generic overestimate of the nucleon mass in the soliton approach Pobylitsa:1992bk . However, the above way of parameter handling is just one possible choice and other choices are possible as well. An investigation whether other choices of parameter handling yield equally satisfactory results will be presented elsewhere.
Iv The densities of the EMT
In this Section we shall focus our attention on the densities (20, 21, 22) which are interesting objects by themselves, before we discuss the form factors (17, 18, 19) in the next Section. The study of the densities will enable us to address the question whether the model provides a satisfactory description of the nucleon in (fictious) worlds with pion masses up to . As we shall see, the answer is yes.
iv.1 Energy density
The energy density is just in the static EMT (3), and is normalized as for any where we explicitly indicate the pion mass dependence. as function of was studied in Goeke:2005fs .
Fig. 2a shows as function of for pion masses in the range . In the following we focus on the region , and include only for completeness the results for discussed in detail in accompanyingpaperI .
In the physical situation with the energy density in the center of the nucleon is or . This corresponds roughly to 13 times the equilibrium density of nuclear matter. As increases becomes larger and reaches
These observations mean that with increasing the nucleon becomes “smaller”. To quantify this statement we consider the mean square radius of the energy density defined as
(23) 
which decreases with increasing . The pion mass dependence of is shown in Fig. 2a, see also Table 1 where many results are summarized. We observe an approximately linear growth of with . The pion mass dependence of is shown in Fig. 2b, see also Table 1. Up to we observe a roughly linear decrease of with which proceeds at a slower rate for . (Throughout we choose a linear or quadratic in presentation of the dependence of the quantities — depending on which one is more convenient.)
The above observations can be intuitively understood. With increasing the range of the “pion cloud” decreases. This results in a less wide spread nucleon. The above observations are also consistent with what one expects from the heavy quark limit point of view. The heavier the constituents building up a hadron, the smaller is the size of that hadron. Thus, the model results for are in agreement with what one expects for increasing .
We remark that, being a chiral model, the CQSM correctly describes the behaviour of in the chiral limit accompanyingpaperI .
iv.2 Angular momentum density
The angular momentum density is related to the components of the static EMT as . Fig. 3a shows as function of for different pion masses. For any we find that at small , it reaches then a maximum around and goes slowly to zero at large .
As increases we observe that the density becomes larger in the small region at the price of decreasing in the region of larger . The increase in one and decrease in another region of (as is varied) occurs in a precisely balanced way, because — in contrast to the energy density — is always normalized as . These observations can be understood within the picture of a rigidly rotating soliton as follows. For large pion masses the “matter” inside the soliton is localized more towards its center, as we have observed above, such that the inner region of the soliton plays a more important role for its rotation. As decreases, and hence the range of the pion cloud increases, the energy density in the soliton becomes more strongly delocalized, and then the “outer regions” play a more and more important role for the rotation of the soliton. , independently of
These findings can be quantified by considering the mean square radius of the angular momentum density defined analogously to (23). Fig. 3b shows as function of , and we see that decreases with increasing . Notice that in the chiral limit at large such that diverges accompanyingpaperI . Our numerical results for in Fig. 3b indicate this effect.
iv.3 Pressure and shear forces
Next we turn to the discussion of the distributions of pressure and shear forces, and , which are related to the components of the static EMT.
Figs. 5a and b show the distributions of pressure and shear forces as functions of for different . For all the distributions of pressure and shear forces exhibit the same qualitative behaviour. The pressure takes at its global maximum, decreases monotonically becoming zero at some point till reaching its global minimum at some point , and decreases then monotonically tending to zero but remaining always negative. The distribution of shear forces is never negative. It starts at a zero value at , increases monotonically till reaching a global maximum at some point , and decreases then monotonically tending to zero.
The positive sign of the pressure for corresponds to repulsion, while the negative sign in the region means attraction. This is intuitive because in the inner region we expect repulsion among quarks due to the Pauli principle, while the attraction in the outer region is an effect of the pion cloud which is responsible for binding the quarks to form a nucleon accompanyingpaperI .
With increasing the pressure in the center of the nucleon increases. At the same time also the absolute value of its (negative) minimum increases. Also the maximum of becomes larger with increasing , while the characteristic positions , and move towards smaller , see Fig. 5 for dependence of and .
These observations can be understood as follows. The repulsive forces in the center of the nucleon increase as a response to the higher density at larger . At the same time the size of the nucleon decreases requiring stronger binding forces — and a “movement” of characteristic length scales of and towards the center. At any repulsive and attractive forces are precisely balanced due to (11), see next Sec. IV.4.
iv.4 Stability
While the densities, and , are normalized with respect to and , for the pressure the corresponding analogon is the stability criterion (11). In Ref. accompanyingpaperI it was proven analytically that (11) is satisfied in the model — provided one evaluates the pressure with the selfconsistent profile, i.e. with that profile which for a given provides the true minimum of the soliton energy (15).
For given model parameters the soliton profiles are obtained by means of an iteration procedure which is described in detail for example in Christov:1995vm . The profiles used here were computed in Ref. Goeke:2005fs where a good convergence of the iteration procedure was observed. However, what precisely means that the convergence of the iteration was good? In other words, how to test the quality of the numerical results? One could, for example, slightly modify the obtained profiles and check that they yield larger soliton masses than the respective true selfconsistent profile. But the probably most elegant method is provided by the stability criterion (11). If, and only if, we found the soliton profile which truely minimizes the soliton energy (11), the pressure computed with that profile will satisfy (11).
One way to check to which numerical accuracy our results satisfy (11) is as follows. Let us consider as function of , see Fig. 6a, and compute the integrals from to and from to .^{1}^{1}1 The numerical calculations are carried carried out in a finite spherical volume — here of the size . For most quantities the densities decay fast enough at large such that it is sufficient to integrate up to . This is what we did in Eq. (24). However, for certain quantities given by integrals over the densities weighted by a higher power of the integrands may happen not to be negligibly small at large in particular for small . Then it is necessary to explore the analytically known large asymptotics of the densities, see accompanyingpaperI , and to include the contribution of the regions not covered in the numerical calculation. Examples of such quantities are or . The latter is divergent in the chiral limit, see Sec. IV.2. We obtain
(24) 
and see that the stability criterion is satisfied to within a satisfactory numerical accuracy.
Finally, we may test another sort of stability. We may ask the question how do pressure and energy density in the center of nucleon depend on each other for varying . In fact, with and at hand, we may eliminate at and express as function of . This is shown in Fig. 6b which demonstrates how the center of the nucleon responds to changes of . Understanding the center of the nucleon for a moment as a “medium which is subject to variations of the external parameter ” we observe that for any we have . This is a criterion for stability of a system which must respond with an increase of pressure if the density is increased.
V Results for the form factors
Fig. 7 shows the form factors of the EMT as functions of for for different . All EMT form factors (with the exception of and in the chiral limit, see below) can be well approximated by dipole fits of the kind
(25) 
It is instructive to compare within the model the EMT form factors to the electromagnetic form factors — for example to the electric form factor of the proton Christov:1995hr . Interestingly, and show a similar dependence. But falls off with increasing slower than , while exhibits a faster fall off, see accompanyingpaperI for more details.
The dipole masses of the different form factors exhibit different dependences, see Fig. 8a and Table 1. For all form factors the dipole masses increase with increasing . It is an interesting observation that the dipole masses of and exhibit for to a good approximation a linear dependence on . But the dependence of the dipole mass of follows a different pattern. We shall comment more on that in Sec. VII
That the dipole approximation for and fails in the chiral limit, is due to fact that the slopes of and at diverge in this limit accompanyingpaperI . For this is clear because its derivative at is related to the mean square radius of the angular momentum density as , and diverges for , see Sect. IV.2.
The slope of at becomes infinitely steep in the chiral limit because it is related as
and are normalized at for any as accompanyingpaperI . In the CQSM these constraints are consistent for they mean that entire momentum and spin of the nucleon are carried by quark degrees of freedom. The numerical results satisfy these constraints within a numerical accuracy of better than , see Figs. 7a and 7b.
In contrast, no principle fixes the normalization of the form factor at neither in the model nor in QCD. For all we find . This condition has been conjectured to be dictated by stability requirements accompanyingpaperI . Fig. 8b shows the dependence of which is rather strong. This is due to the fact that receives a large leading nonanalytic contribution proportional to . (The “nonanalyticity” refers to the current quark mass .) The chiral expansion of reads
(26) 
where denotes the chiral limit value of , and the dots indicate subleading terms in the chiral limit. Since the limits and do not commute Dashen:1993jt ; Cohen:1992uy one has in Eq. (26) for finite Belitsky:2002jp and in the large limit accompanyingpaperI . The latter corresponds to the situation in the CQSM.
It is interesting to observe that the leading nonanalytic term in the chiral expansion of in Eq. (26) dominates the chiral behaviour of up to the physical point, see Fig. 8b. But for larger higher orders in the chiral expansion become important, and change the qualitative behaviour of . We shall come back to this point in Sec. VII.
Finally, we discuss dependence of the mean square radius of the trace of the total EMT operator given due to the trace anomaly Adler:1976zt by
(27) 
Let denote the form factor of the operator (27) which is normalized as . Its slope at defines which can be related as
(28) 
to and , see accompanyingpaperI . Fig. 9 shows how depends on the pion mass. In the chiral limit is the mean square radius of the operator and its large value there is in contrast to what is known about the mean square radii of other gluonic operators Braun:1992jp .
dipole masses in GeV for  

0  1.54  0.82  0.195  0.59  3.46  0.867  undef.  undef.  1.04  
50  1.57  0.76  1.88  0.202  0.59  3.01  0.873  0.692  0.519  0.95  
140  1.70  0.67  1.55  0.232  0.57  2.35  0.906  0.745  0.646  0.81  
300  2.14  0.53  1.11  0.298  0.54  1.81  0.990  0.844  0.872  0.62  
500  3.10  0.40  0.77  0.377  0.51  1.66  1.111  0.986  1.069  0.46  
700  4.50  0.32  0.59  0.450  0.49  1.60  1.228  1.120  1.214  0.37  
900  6.86  0.26  0.48  0.553  0.46  1.55  1.334  1.237  1.337  0.29  
1200  9.53  0.22  0.38  0.597  0.42  1.47  1.473  1.390  1.492  0.24 
Vi Comparison to lattice results
It is instructive to compare the results for the form factors of the EMT to lattice QCD data Mathur:1999uf ; Gadiyak:2001fe ; Hagler:2003jd ; Gockeler:2003jf ; Negele:2004iu . Presently, this offers actually the only available test for the model results. For the comparison it is necessary to evolve the model results from a low initial scale to typically in the lattice calculations which we shall do to leading logarithmic accuracy. Under evolution the quark (flavoursinglet) form factors mix with the corresponding gluon form factors. We set the latter to zero at the initial scale.
In this context it is worthwhile recalling that in early parameterizations of unpolarized parton distributions , the gluon (and sea quark) distribution(s) and thus were assumed to be zero at a low initial scale Gluck:1977ah . One success of these approaches was that they were able to explain the observation at . I.e. starting with and at a low scale, which is the situation in the CQSM, it is possible to reproduce the observation at several . However, with the advent of more data (especially at low ) it became clear Gluck:1994uf that nonzero gluon (and see quark) distributions are required already at low initial scales. Modern parameterizations performed at low scales require a sizeable gluon distribution and Gluck:1998xa . This is not in disagreement with the instanton picture where twist2 gluon operators are suppressed with respect to quark operators by the instanton packing fraction which is numerically of order Diakonov:1995qy . Thus in some sense the phenomenologically required “portion of gluons” is within the accuracy of the model Diakonov:1998ze . With these remarks in mind we conclude that the CQSM result at the low scale of the model is in agreement with phenomenology — within the accuracy of this model.
Next, before we compare the results from the model to those from lattice QCD, let us confront the lattice results Mathur:1999uf ; Gadiyak:2001fe ; Hagler:2003jd ; Gockeler:2003jf ; Negele:2004iu with predictions from the large limit. In this limit one has for independently of the scale Goeke:2001tz
(29) 
cf. App. A for the explanation of the notation. For completeness let us quote that the gluon form factors satisfy
(30) 
which is the same large behaviour as the corresponding quarkflavoursinglet form factors.
Remarkably, although in the real world does not seem to be large, nevertheless lattice data Mathur:1999uf ; Gadiyak:2001fe ; Hagler:2003jd ; Gockeler:2003jf ; Negele:2004iu reflect the large flavour dependence of the quark form factors (29). In fact, large relations of the type (29, 30) are observed to be satisfied in phenomenology Efremov:2000ar and serve within their range of applicability as useful guidelines Pobylitsa:2003ty . The soliton approach is justified in the large limit Witten:1979kh and satisfies general large relations of the type (29). The observation that the lattice results Mathur:1999uf ; Gadiyak:2001fe ; Hagler:2003jd ; Gockeler:2003jf ; Negele:2004iu are compatible with (29) is therefore an encouraging prerequisite for our study.
Let us first compare the model results to the lattice data computed by the LHPC and SESAM Collaborations Hagler:2003jd . There unquenched SESAM Wilson configurations on a lattice at with were used. This corresponds to and a lattice spacing of with physical units fixed by extrapolating the nucleon mass. The form factor was computed omitting disconnected diagrams at a scale for . (The different notations are discussed in App. A.) The lattice data for , which can be fit to the dipole form, are shown in Fig. 11a. In Fig. 11a we also show the CQSM results evolved to the same scale for . We observe that the model results agree with the lattice data Hagler:2003jd to within which is within the accuracy to which the CQSM results typically agree with phenomenology.
Fig. 11b shows the form factor from Ref. Hagler:2003jd which was computed in the range and was found consistent with zero within the statistical accuracy of the simulation. Also in the CQSM we find close to zero — in reasonable agreement with the lattice data, see Fig. 11b. Notice that in the model, at the low scale, we have since (and equal unity), see Secs IV.1 and IV.2. This implies a vanishing quark contribution to the ”gravitomagnetic moment” of the nucleon Ossmann:2004bp conjectured in Teryaev:1998iw . The smallness of implies that to a good approximation . We shall come back to this point below.
Next, let us compare to the quenched lattice results by the QCDSF Collaboration Gockeler:2003jf which were obtained using nonperturbatively