Vector Meson Dominance as a first step in a systematic approximation: the pion vector form factor
aa r X i v : . [ h e p - ph ] J u l Vector Meson Dominanceas a first step in a systematic approximation:the pion vector form factor
P. Masjuan, S. Peris and
J.J. Sanz-Cillero
Grup de F´ısica Te`orica and IFAEUniversitat Aut`onoma de Barcelona, 08193 Barcelona, Spain.
Abstract
Pad´e Approximants can be used to go beyond Vector Meson Dominance ina systematic approximation. We illustrate this fact with the case of the pionvector form factor and extract values for the first two coefficients of its Taylorexpansion. Pad´e Approximants are shown to be a useful and simple tool forincorporating high-energy information, allowing an improved determination ofthese Taylor coefficients.
Introduction
It has been known for a long time that the pion vector form factor (VFF) in the space-like region is very well described by a monopole ansatz of the type given by VectorMeson Dominance (VMD) in terms of the rho meson. However, it has remained unclearwhether there is a good reason for this from QCD or it is just a mere coincidence and,consequently, it is not known how to go about improving on this ansatz.To begin our discussion, let us define the form factor, F ( Q ), by the matrix element h π + ( p ′ ) | uγ µ u − dγ µ d − sγ µ s | π + ( p ) i = ( p + p ′ ) µ F ( Q ) , (1)where Q = − ( p ′ − p ) , such that Q > F ( Q ) starts at twice thepion mass, the form factor can be approximated by a Taylor expansion in powers ofthe momentum for | Q | < (2 m π ) . At low momentum, Chiral Perturbation Theory isthe best tool for organizing the pion interaction in a systematic expansion in powers ofmomenta and quark masses [1, 2, 3]. With every order in the expansion, there comesa new set of coupling constants, the so-called low-energy constants (LECs), whichencode all the QCD physics from higher energies. This means, in particular, that thecoefficients in the Taylor expansion can be expressed in terms of these LECs and powersof the quark masses. Consequently, by learning about the low-energy expansion, onemay indirectly extract important information about QCD.In principle, the coefficients in the Taylor expansion may be obtained by means ofa polynomial fit to the experimental data in the space-like region below Q = 4 m π .However, such a polynomial fit implies a tradeoff. Although, in order to decreasethe (systematic) error of the truncated Taylor expansion, it is clearly better to goto a low-momentum region, this also downsizes the set of data points included inthe fit which, in turn, increases the (statistical) error. In order to achieve a smallerstatistical error one would have to include experimental data from higher energies, i.e.from Q > m π . Since this is not possible in a polynomial fit, the use of alternativemathematical descriptions may be a better strategy.One such description, which includes time-like data as well, is based on the useof the Roy equations and Omn´es dispersion relations. This is the avenue followedby [4, 5], which has already produced interesting results on the scalar channel [6],and which can also be applied to the vector channel. Other procedures have reliedon conformal transformations for the joint analysis of both time-like and space-likedata [7], or subtracted Omn´es relations [8, 9]. Further analyses may be found in Ref.[10].On the other hand, as already mentioned above, one may also consider an ansatzof the type F ( Q ) VMD = (cid:18) Q M VMD (cid:19) − . (2) Time-like data is provided by ππ production experiments and, consequently, they necessarilycorrespond to values of the momentum above the ππ cut, i.e. | Q | > m π with Q < Q ≫ m π . If this fact is not merely a fluke, it could certainly be interestingto consider the form (2) as the first step in a systematic approximation, which wouldthen allow improvement on this VMD ansatz.In this article, we would like to point out that the previous VMD ansatz for theform factor (2) can be viewed as the first element in a sequence of Pad´e Approximants(PAs) which can be constructed in a systematic way. By considering higher-order termsin the sequence, one may be able to describe the space-like data with an increasinglevel of accuracy . Of course, whether this is actually the case and the sequence isa convergent one in the strict mathematical sense or, on the contrary, the sequenceeventually diverges, remains to be seen. But the important difference with respectto the traditional VMD approach is that, as a Pad´e sequence, the approximation iswell-defined and can be systematically improved upon.Although polynomial fitting is more common, in general, rational approximants(i.e. ratios of two polynomials) are able to approximate the original function in amuch broader range in momentum than a polynomial [11]. This will be the greatadvantage of the Pad´es compared to other methods: they allow the inclusion of lowand high energy information in a rather simple way which, furthermore, can in principlebe systematically improved upon. In certain cases, like when the form factor obeys adispersion relation given in terms of a positive definite spectral function (i.e. becomes aStieltjes function), it is known that the Pad´e sequence is convergent everywhere on thecomplex plane, except on the physical cut [12]. Another case of particular interest is inthe limit of an infinite number of colors in which the form factor becomes a meromorphicfunction. In this case there is also a theorem which guarantees convergence of the Pad´esequence everywhere in a compact region of the complex plane, except perhaps at afinite number of points (which include the poles in the spectrum contained in thatregion) [13]. In the real world, in which a general form factor has a complicatedanalytic structure with a cut, and whose spectral function is not positive definite, wedo not know of any mathematical result assuring the convergence of a Pad´e sequence[14]. One just has to try the approximation on the data to learn what happens.In this work we have found that, to the precision allowed by the experimentaldata, there are sequences of PAs which improve on the lowest order VMD result in arather systematic way. This has allowed us to extract the values of the lowest-ordercoefficients of the low-energy expansion.We would like to emphasize that, strictly speaking, the Pad´e Approximants to agiven function are ratios of two polynomials P N ( z ) and Q M ( z ) (with degree N and M ,respectively), constructed such that the Taylor expansion around the origin exactlycoincides with that of f ( z ) up to the highest possible order, i.e. f ( z ) − R NM ( z ) = O ( z M + N +1 ). However, in our case the Taylor coefficients are not known. They are, infact, the information we are seeking for. Our strategy will consist in determining thesecoefficients by a least-squares fit of a Pad´e Approximant to the vector form factor data Obviously, unlike the space-like data, one should not expect to reproduce the time-like data sincea Pad´e Approximant contains only isolated poles and cannot reproduce a time-like cut.
2n the space-like region.There are several types of PAs that may be considered. In order to achieve afast numerical convergence, the choice of which one to use is largely determined bythe analytic properties of the function to be approximated. In this regard, a glanceat the time-like data of the pion form factor makes it obvious that the form factoris clearly dominated by the rho meson contribution. The effect of higher resonancestates, although present, is much more suppressed. In these circumstances the naturalchoice is a P L Pad´e sequence [11], i.e. the ratio of a polynomial of degree L over apolynomial of degree one . Notice that, from this perspective, the VMD ansatz in (2)is nothing but the P Pad´e Approximant.However, to test the aforementioned single-pole dominance, one should check thedegree to which the contribution from resonances other than the rho may be neglected.Consequently, we have also considered the sequence P L , and the results confirm thosefound with the PAs P L . Furthermore, for completeness, we have also considered the so-called Pad´e-Type approximants (PTs) [15, 16]. These are rational approximants whosepoles are predetermined at some fixed values, which we take to be the physical massessince they are known. Notice that this is different from the case of the ordinary PAs,whose poles are left free and determined by the fit. Finally, we have also consideredan intermediate case, the so-called Partial-Pad´e approximants (PPs) [15], in whichsome of the poles are predetermined (again at the physical masses) and some are leftfree. We have fitted all these versions of rational approximants to all the availablepion VFF space-like data [17]-[22]. The result of the fit is rather independent of thekind of rational approximant sequence used and all the results show consistency amongthemselves.The structure of this letter is the following. In section 2 we begin by testing theefficiency of the P L Pad´es with the help of a model. In section 3 we apply this very samemethod to the experimental VFF. Firstly, in sec. 3.1, we use the Pad´e Approximants P L ; then, in Sec. 3.2, this result is cross-checked with a P L PA. Finally, in sec. 3.3, westudy the Pad´e-Type and Partial-Pad´e approximants. The outcome of these analysesis combined in section 4 and some conclusions are extracted.
In order to illustrate the usefulness of the PAs as fitting functions in the way we proposehere, we will first use a phenomenological model as a theoretical laboratory to checkour method. Furthermore, the model will also give us an idea about the size of possiblesystematic uncertainties.We will consider a VFF phase-shift with the right threshold behavior and withapproximately the physical values of the rho mass and width. The form-factor is Conventionally, without loss of generality, the polynomial in the denominator is normalized tounity at the origin. F ( Q ) = exp (cid:26) − Q π Z ∞ m π dt δ ( t ) t ( t + Q ) (cid:27) , (3)where δ ( t ), which plays the role of the vector form factor phase-shift [8, 9, 23], is givenby δ ( t ) = tan − " ˆ M ρ ˆΓ ρ ( t )ˆ M ρ − t , (4)with the t -dependent width given byˆΓ ρ ( t ) = Γ t ˆ M ρ ! σ ( t ) σ ( ˆ M ρ ) θ (cid:0) t − m π (cid:1) , (5)and σ ( t ) = p − m π /t . The input parameters are chosen to be close to their physicalvalues: Γ = 0 .
15 GeV , ˆ M ρ = 0 . , m π = 0 . . (6)We emphasize that the model defined by the expressions (3-5) should be consideredas quite realistic. In fact, it has been used in Ref. [8, 9, 23] for extracting the valuesfor the physical mass and width of the rho meson through a direct fit to the (timelike)experimental data.Expanding F ( Q ) in Eq. (3) in powers of Q we readily obtain F ( Q ) = 1 − a Q + a Q − a Q + ... , (7)with known values for the coefficients a i . In what follows, we will use Eq. (7) as thedefinition of the coefficients a i . To try to recreate the situation of the experimentaldata [17]-[22] with the model, we have generated fifty “data” points in the region0 . ≤ Q ≤ .
25, thirty data points in the interval 0 . ≤ Q ≤
3, and seven pointsfor 3 ≤ Q ≤
10 (all these momenta in units of GeV ). These points are taken withvanishing error bars since our purpose here is to estimate the systematic error derivedpurely from our approximate description of the form factor.We have fitted a sequence of Pad´e Approximants P L ( Q ) to these data points and,upon expansion of the Pad´es around Q = 0, we have used them to predict the valuesof the coefficients a i . The comparison may be found in Table 1. The last PA we havefitted to these data is P . Notice that the pole position of the Pad´es differs from thetrue mass of the model, given in Eq. (6).A quick look at Table 1 shows that the sequence seems to converge to the exactresult, although in a hierarchical way, i.e. much faster for a than for a , and this onemuch faster than a , etc... The relative error achieved in determining the coefficients a i by the last Pad´e, P , is respectively 0 . a , a and a . Naively, onewould expect these results to improve as the resonance width decreases since the P L contains only a simple pole, and this is indeed what happens. Repeating this exercisewith the model, but with a Γ = 0 .
015 GeV (10 times smaller than the previous one),4 P P P P P P F ( Q )(exact) a (GeV − ) 1.549 1.615 1.639 1.651 1.660 1.665 1.670 1.685 a (GeV − ) 2.399 2.679 2.809 2.892 2.967 3.020 3.074 3.331 a (GeV − ) 3.717 4.444 4.823 5.097 5.368 5.579 5.817 7.898 s p (GeV ) 0 .
646 0 .
603 0 .
582 0 .
567 0 .
552 0 .
540 0 .
526 0 . Results of the various fits to the form factor F ( Q ) in the model, Eq. (3). Theexact values for the coefficients a i in Eq. (7) are given on the last column. The last rowshows the predictions for the corresponding pole for each Pad´e ( s p ), to be compared to thetrue mass ˆ M ρ = 0 . in the model. the relative error achieved by P for the same coefficients as before is 0 . . . five times bigger than the first oneproduces, respectively, differences of 2 . .
4% and 37 . P L pole at s p = ˆ M ρ and found a similar pattern as in Table 1. For P , thePad´e-Type coefficient a differs a 2 .
5% from its exact value, a by 16% and a by 40%.Based on the previous results, we will take the values in Table 1 as a rough estimateof the systematic uncertainties when fitting to the experimental data in the followingsections. Since, as we will see, the best fit to the experimental data comes from thePad´e P , we will take the error in Table 1 from this Pad´e as a reasonable estimate andadd to the final error an extra systematic uncertainty of 1 .
5% and 10% for a and a (respectively). We will use all the available experimental data in the space-like region, which maybe found in Refs. [17]-[22]. These data range in momentum from Q = 0 .
015 up to10 GeV .As discussed in the introduction, the prominent role of the rho meson contributionmotivates that we start with the P L Pad´e sequence. P L Without any loss of generality, a P L Pad´e is given by P L ( Q ) = 1 + L − X k =0 a k ( − Q ) k + ( − Q ) L a L a L +1 a L Q , (8)where the vector current conservation condition P L (0) = 1 has been imposed and thecoefficients a k are the low-energy coefficients of the corresponding Taylor expansion of5 Q H GeV L Q F H Q L H
GeV L Figure 1:
The sequence of P L PAs is compared to the available space-like data [17]-[22]: P (brown dashed), P (green thick-dashed), P (orange dot-dashed), P (blue long-dashed), P (red solid). the VFF (compare with (7) for the case of the model in the previous section).The fit of P L to the space-like data points in Refs. [17]-[22] determines the coeffi-cients a k that best interpolate them. According to Ref. [24], the form factor is supposedto fall off as 1 /Q (up to logarithms) at large values of Q . This means that, for anyvalue of L , one may expect to obtain a good fit only up to a finite value of Q , but notfor asymptotically large momentum. This is clearly seen in Fig. 1, where the Pad´esequence P L is compared to the data up to L = 4.Fig. 2 shows the evolution of the fit results for the Taylor coefficients a and a forthe P L PA from L = 0 up to L = 4. As one can see, after a few Pad´es these coefficientsbecome stable. Since the experimental data have non zero error it is only possible tofit a P L PA up to a certain value for L . From this order on, the large error bars inthe highest coefficient in the numerator polynomial make it compatible with zero and,therefore, it no longer makes sense to talk about a new element in the sequence. Forthe data in Refs. [17]-[22], this happened at L = 4 and this is why our plots stopat this value. Therefore, from the PA P we obtain our best fit and, upon expansionaround Q = 0, this yields a = 1 . ± .
03 GeV − , a = 3 . ± .
26 GeV − ; (9)with a χ / dof = 117 / P L PA is determined by the ratio s p = a L /a L +1 .This ratio is shown in Fig. 3, together with a gray band whose width is given by ± M ρ Γ ρ for comparison. From this figure one can see that the position of the pole of the PA isclose to the physical value M ρ [25], although it does not necessarily agree with it, aswe already saw in the model of the previous section.6 L a H GeV - L L a H GeV - L Figure 2: a and a Taylor coefficients for the P L PA sequence. L s p H GeV L Figure 3:
Position s p of the pole for the different P L . The range with the physical values M ρ ± M ρ Γ ρ is shown (gray band) for comparison. P L Pad´es
Although the time-like data of the pion form factor is clearly dominated by the ρ (770)contribution, consideration of two-pole P L PAs will give us a way to assess any possiblesystematic bias in our previous analysis, which was limited to only single-pole PAs.We have found that the results of the fits of P L PAs to the data tend to reproducethe VMD pattern found for the P L PAs in the previous section. The P L PAs place thefirst of the two poles around the rho mass, while the second wanders around the complexmomentum plane together with a close-by zero in the numerator. This association of apole and a close-by zero is what is called a “defect” in the mathematical literature[26].A defect is only a local perturbation and, at any finite distance from it, its effect isessentially negligible. This has the net effect that the P L Pad´e in the euclidean regionlooks just like a P L approximant and, therefore, yields essentially the same results.For example, for the P , one gets a = 1 . ± .
029 GeV − , a = 3 . ± .
14 GeV − , (10)with a χ / dof = 120 / Besides the ordinary Pad´e Approximants one may consider other kinds of rationalapproximants. These are the Pad´e Type and Partial Pad´e Approximants [13, 15, 16].7 L a H GeV - L Figure 4: Low-energy coefficient a from the T L Pad´e-Type sequenceIn the Pad´e Type Approximants (PTAs) the poles of the Pad´e are fixed to certainparticular values, which in this context are naturally the physical masses. On theother hand, in the Partial Pad´e Approximants (PPAs) one has an intermediate situationbeytween the PAs and the PTAs in which some poles are fixed while others are left asfree parameters to fit.Since the value of the physical rho mass is known ( M ρ = 775 . T L . This has the obvious advantage that the number of param-eters in the fit decreases by one and allows one to go a little further in the sequence.Our best value is then given by the Pad´e Type Approximant T , whose expansionaround Q = 0 yields the following values for the Taylor coefficients: a = 1 . ± .
03 GeV − , a = 3 . ± .
09 GeV − , (11)with a χ / dof = 118 / M ρ = 775 . M ρ ′ = 1459 MeV and M ρ ′′ = 1720 MeV, we may construct further Pad´e-Type se-quences of the form T L and T L .In the PTA sequence T L one needs to provide the value of two poles. For the firstpole, the natural choice is M ρ . For the second pole, we found that choosing either M ρ ′ or M ρ ′′ (the second vector excitation) does not make any difference. Both outcomesare compared in Fig. (5). Using M ρ ′ , we found that the T PTA yields the best valuesas a = 1 . ± .
024 GeV − , a = 3 . ± .
07 GeV − , (12)with a χ / dof = 118 / M ρ ′′ as the second pole one also gets the best value from the T PTA, withthe following results: a = 1 . ± .
023 GeV − , a = 3 . ± .
06 GeV − , (13) As will be seen, results do not depend on the precise value chosen for these masses. L a H GeV - L L a H GeV - L Figure 5:
Low energy coefficient a for the T L Pad´e-Type sequence with M ρ and M ρ ′ (left),and with M ρ and M ρ ′′ (right). with a χ / dof = 119 /
92. We find the stability of the results for the coefficients a , quite reassuring.We have also performed an analysis of the PTA sequence T L , with similar conclu-sions. From the T we obtain the following values for the coefficients: a = 1 . ± .
023 GeV − , a = 3 . ± .
09 GeV − , (14)with a χ / dof = 119 / P L , in which the first pole is given by M ρ and the other is left free.The best determination of the Taylor coefficients is given by P , , and they yield a = 1 . ± .
029 GeV − , a = 3 . ± .
09 GeV − , (15)with the free pole of the PPA given by M free = (1 . ± . and a χ / dof = 119 / Combining all the previous rational approximants results in an average given by a = 1 . ± . stat ± . syst GeV − , a = 3 . ± . stat ± . syst GeV − . (16)The first error comes from combining the results of the different fits by means of aweighted average. On top of that, we have added what we believe to be a conservativeestimate of the theoretical (i.e. systematic) error based on the analysis of the VFFmodel in Sec. 2. We expect the latter to give an estimate for the systematic uncertaintydue to the approximation of the physical form factor with rational functions. Forcomparison with previous analyses, we also provide in Table 2 the value of the quadraticradius, which is given by h r i = 6 a . See Ref. [13] for notation. r i (fm ) a (GeV − )This work 0 . ± . stat ± . syst . ± . stat ± . syst CGL [4, 5] 0 . ± .
005 ...TY [7] 0 . ± .
001 3 . ± . . ± .
016 3 . ± . . ± .
012 3 . ± . . ± .
031 ...Table 2:
Our results for the quadratic radius h r i and second derivative a are comparedto other determinations [4, 5, 7, 27, 9, 28]. Our first error is statistical. The second one issystematic, based on the analysis of the VFF model in section 2. In summary, in this work we have used rational approximants as a tool for fittingthe pion vector form factor. Because these approximants are capable of describing theregion of large momentum, they may be better suited than polynomials for a descriptionof the space-like data. As our results in Table 2 show, the errors achieved with theseapproximants are competitive with previous analyses existing in the literature.
Acknowledgements
We would like to thank G. Huber and H. Blok for their help with the experimentaldata. This work has been supported by CICYT-FEDER-FPA2005-02211, SGR2005-00916, the Spanish Consolider-Ingenio 2010 Program CPAN (CSD2007-00042) and bythe EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”.
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