Constraining the low energy pion electromagnetic form factor with space-like data

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by  B. Ananthanarayan

EPJ manuscript No.
(will be inserted by the editor)
malization F (0) = 1, immediately gives a lower bound
π
of ≈ 1.6×10−9. Imposing the experimental value of r
π
further improves the bound to 2.9×10−9 [4]. Caprini [5]
has more recently shown that if one were to assume that
the pionic contribution is less than I = 75×10−9, then
these principles yield constraints on the higher expansion
Constraining the low energy coefficients, c, d which translate into an allowedellipse in
the c−d plane. These results, as well as the results we
pion electromagnetic form
present here, hold as long as the true pionic contribution
is less than the value of I we have assumed. If the pionic
8 factor with space-like data
0 contribution is indeed significantly smaller, then the true
0 allowed region would be a proper subset of the region we
2 B. Ananthanarayanand S. Ramanan isolate.
The inclusion of experimental data from the time-like
n Centre for High Energy Physics, Indian Institute of Science,
a region, the phase, the modulus, and the phase as well
Bangalore 560 012, India.
J e-mail: [email protected] as the modulus each improves the bound on the muon
4 e-mail: [email protected] anomaly [1]. Using I as an input, Caprini has used dis-
1 persive techniques and time-like data,and has considered
other physical observables such as certain QCD polariza-
February 2, 2008
] tion functions, in addition to the muon anomaly, to con-
h
strainfurther the allowedregionsin the c−dplane 1.We
p Abstract. Thepioniccontributiontothe
- note that the parameter c itself has been determined to
p g−2 of the muon involves a certain in- 2-loop accuracy in chiral perturbation theory [7] and has
e tegral over the the modulus squared of a value of 4.49 GeV−4, which is safely accommodated in
h Fπ(t),thepionelectromagneticformfac-
the allowed ellipse of Caprini [5].
[ tor. We extend techniques that use cut-
plane analyticity properties of Fπ(t) in An important source of experimental information are
1 ordertoaccountforpresentdayestimates thevaluesoftheelectromagneticformfactorinthespace-
v
ofthepioniccontributionandexperimen- likeregion.Thesehavebeenreportedinelectroproduction
3
talinformationatafinitenumberofpoints experimentsconductedinthe1970’s(seerefs.[8],[9],[10]).
2
in the space-like region. Using data from RainaandSingh[11]usedthisinformationtoproduceim-
0
2 severalexperimentsoveralargekinematic provedbounds on the muon anomaly (see also [12]). New
. range for |t|, we find bounds on the ex- experimentaldataisnowavailable(seerefs.[13],[14],[15]
1
pansion coefficients of Fπ(t), sub-leading and [16]). In this work, we will make use of some of the
0
to the charge radius. The value of one data sets above.
8
0 of these coefficients in chiral perturba- Our main purpose in this work is to demonstrate that
: tion theory respects these bounds. Fur- the frameworkof Raina andSingh can be effectively used
v thermore, we present a sensitivity analy-
to obtain constraints in the c−d plane, using the pionic
i sistotheinputs.Abriefcomparisonwith
X contributionto the muonanomalyas aninput. This is an
results in the literature that use observ-
algebraic framework, which is clear cut and transparent.
r ables other than the g−2 and timelike
a Itprovidesanimportantconsistencycheckontheallowed
data is presented.
regions isolated by Caprini.
In sec. 2 we briefly review the dispersive formalism
1 Introduction and describe the implementation of space-like constraints
andthemethodbywhichweisolatetheregioninthec−d
plane.Here,weprovidethetheoreticalframeworkandalso
General principles such as analyticity and unitarity have
a discussion on the present experimental status, as they
beenusedtoderiveusefulconstraintsonformfactors(for
are both required for this purpose. In sec. 3 we present
a review see ref. [1]). In particular, useful lower bounds
our results using data from [8], [9], [10] and more recent
were obtained on the pionic contribution to the (g−2) of
data from [15] and [16]. We present a detailed discussion
themuon(muonanomaly).Forrecentreviewsonthecur-
of our results and our conclusions in sec. 4.
rent status of this quantity, see refs. [2,3]. Consider the
following low-energy expansion for the pion electromag-
netic form factor F (t),
π
r2
F (t)=F (0)+ π t+ct2+dt3+··· , (1)
π π
6
where r is the pion chargeradius,c and d are Taylorco-
π
efficients, with units of GeV−4 and GeV−6 respectively. 1Applications to form factors in semi-leptonic decays of
For a recent discussion, refer [7]. From general princi- such general methods in a modern context have also been re-
ples, discussed in the next section, and using the nor- cently considered [6].
2 B. Ananthanarayan,S.Ramanan: Constraining the low energy pion electromagnetic form factor with space-like data
2 Formalism Theexpansioncoefficientsa canbeobtainedfromaTay-
n
lor expansion of the function h(z) in terms of f(z) and
We recall the formalism presented in ref. [11]. The pion w (z). The coefficients are given by
π
contribution to the muon anomaly is given by:
a =h(0)=w (0), (15)
0 π
∞
1
aµ(π+π−)= π Ztπ dtρ(t)|Fπ(t)|2 (2) a1 =h′(0)=wπ′(0)+ 23rπ2tπwπ(0), (16)
wanhderetπ =4m2π isthebranchpointofthepionformfactor a2 = h′2′(!0) = 12(cid:20)wπ(0)(cid:18)−38rπ2tπ+32ct2π(cid:19)(cid:21)
α2m2 (t−t )3/2
µ π
ρ(t)= K(t)≥0 (3) 1 2
12π t7/2 + 2w′(0) r2t +w′′(0) , (17)
2(cid:20) π (cid:18)3 π π(cid:19) π (cid:21)
where,
and
1 m2u2
K(t)= du(1−u)u2(1−u+ µ )−1. (4) h′′′(0) 1
Z0 t a3 = 3! = 6 wπ(0) 12rπ2tπ −384ct2π+384dt3π
(cid:2) (cid:0) (cid:1)(cid:3)
Using the following map from the t-plane, which is cut + 1 3w′(0) −8r2t +32ct2
from t along the real t axis, to the complex z-plane (re- 6(cid:20) π (cid:18) 3 π π π(cid:19)(cid:21)
π
gion |z|<1), 1
+ 2w′′(0)r2t +w′′′(0) . (18)
z−1 t−tπ 6 π π π π
=i , (5) (cid:2) (cid:3)
z+1 r t
π In our treatment, the expansion coefficients satisfy
and the definitions:
∞
a2 =I (19)
f(z)=F (t) (6) i
π Xi=0
p(z)=ρ(t) (7) GivenI, eqn. (19) yields constraints onthe expansionco-
efficients of the form factor. Including up to the second
the pioniccontributiontothe muonanomalycanbe writ-
(third) derivative for F (t) results in constraints for c (c
ten as π
and d). It may be pointed out that Caprini’s result on c
1 2π andd shownin Fig.1 (dashedlines) ofref.[5] is obtained
a (π+π−)= dθw(θ)|f(eiθ)|2 (8)
µ precisely in this manner.
2π Z
0
The constraints on the expansion coefficients of inter-
where, estmaybesignificantlyimprovedthroughtheinclusionof
experimental information on the form factor. Our objec-
w(θ)=4m2 sec2(θ/2)tan(θ/2) p(eiθ)≥0. (9) tive inthis workis to study the effect ofincluding experi-
π
mental information coming from the space-like region. In
We now consider a function h(z) defined as: ordertomeetthisobjective,wefirstextendtheformalism
that has been presented in ref. [11]. In that work, space-
h(z)=f(z)w (z) (10)
π like constraints were used to obtain lower bounds on the
muon anomaly, denoted as I .
where, min
We observe that N space-like constraints are linear
constraints that may be expressed as:
1 2π eiθ+z
wπ(z)=exp(cid:20)4π Z dθ eiθ−z lnw(θ)(cid:21). (11) ∞
0 A = a c(n) (20)
n i i
Then eqn. (8) can be written as: Xi=0
wheren=1,2,···N.Suchconstraintscanbeimplemented
1 2π
a (π+π−)= dθ|h(eiθ)|2 (12) through the technique of Lagrangemultipliers, by setting
µ
2π Z up the Lagrangian
0
Now h(z) is analytic within the unit circle |z| < 1 and 1 ∞ N ∞
for realz,h(z)is real.Therefore h(z)canbe expandedas L= 2 a2i + αn(An− aici(n)) (21)
follows: Xi=0 nX=1 Xi=0
h(z)=a +a z+a z2+··· (13)
0 1 2 where in eqn. (21), we consider only finite number of ex-
where a ,a ··· are realcoefficients. Therefore,in the an- pansion coefficients. The Lagrange equations yield:
0 1
alytic region, |z|≤1, a (π+π−) can be written as:
µ N
a = α c(n) (22)
a (π+π−)=a2+a2+··· (14) i n i
µ 0 1 nX=1
B. Ananthanarayan,S.Ramanan: Constraining the low energy pion electromagnetic form factor with space-like data 3
and to these experimental uncertainties as their dimensions
N ∞ increase. This sensitivity is particularly severe for small
I = α a c(n). (23)
n i i values of |t|. As a result, at such values we are able to
nX=1 Xi=0 implementatmosttwoconstraints,whiledatafromhigher
It then follows that energies allows us to implement up to three constraints.
However,the data from smaller values of |t| provide more
N stringentboundsforfixednumberofconstraints.Thuswe
Imin = αnAn, (24) see a fairly complex interplay between the energy regime
nX=1 thatwecanuseandthenumberofconstraintsweareable
to implement.
where
N ∞
A = α c(m)c(n). (25)
n m i i
mX=1 Xi=0
2.1 Space-like data
All the α s may be eliminated to yield a determinantal
n
equation for I
I A A ...
1 2 We will begin with the data in the space-like region that
(cid:12)(cid:12)A1 c(i1)c(i1) c(i1)c(i2) ...(cid:12)(cid:12) was used in the work of Raina and Singh [11], that came
(cid:12)A Pc(2)c(1) Pc(2)c(2) ...(cid:12) frommeasurementsintheseventies.Weshallrefertothese
(cid:12) 2 i i i i (cid:12)=0. (26)
(cid:12)(cid:12) . P ... P ... ...(cid:12)(cid:12) asthe BrowndataandBebek datarespectively.The data
(cid:12) . ... ... ...(cid:12) we use from these sets is given in Tables 1, 2. The ta-
(cid:12) (cid:12)
(cid:12) . ... ... ...(cid:12) bles also list the values of z(t) = x(t), which is the map
(cid:12) (cid:12)
(cid:12) (cid:12) from the t plane to the disc |z|≤1 (x(t) lies in the range
Forthecasea(cid:12)thand,wherewewishtosp(cid:12)ecifya , a , a ,a [−1≤x(t)≤0]),forthechosendatapointsandthecorre-
0 1 2 3
andthevalueatspace-likepointsh(x ), i=1,2,3...,where sponding value of h(z) = h(x) (refer eqn. (10)). To make
i
x is real, the determinantal equation reads, ournotationsclear,werefertothedatapointscorrespond-
i
ingtoaparticular|t|asx ,x andsoon,intheascending
1 2
I a a a a h(x ) h(x ) ... orderof magnitude of |t|. The tables alsoshow the exper-
0 1 2 3 1 2
(cid:12) a0 1 0 0 0 1 1 ...(cid:12) imental errors in the data.
(cid:12) (cid:12)
(cid:12) a1 0 1 0 0 x1 x2 ...(cid:12) More recent data come from four sources. The first
(cid:12)(cid:12) a2 0 0 1 0 x21 x22 ...(cid:12)(cid:12) of these is ref. [13] which is from a Fermilab experiment
(cid:12)(cid:12)(cid:12)h(ax3 ) 01 x0 x02 x13 (1−xx312)−1 (1−xx32x )−1 ......(cid:12)(cid:12)(cid:12)=0. (F1). Here the range is 0.03GeV2 ≤ −t ≤ 0.07GeV2.
(cid:12) 1 1 1 1 1 1 2 (cid:12) Even at the high end, we are way below the lowest en-
(cid:12)h(x ) 1 x x2 x3 (1−x x )−1 (1−x2)−1 ...(cid:12)
(cid:12) 2 2 2 2 2 1 2 (cid:12) ergy of Brown. The next set of data come from another
(cid:12) . ... ... ... ... ... ... (cid:12)
(cid:12) (cid:12) Fermi lab experiment (F2) (ref. [14]), covering the range
(cid:12)(cid:12) . ... ... ... ... ... ... (cid:12)(cid:12) of 0.037GeV2 ≤ −t ≤ 0.094GeV2. The CERN NA7 ex-
(cid:12) . ... ... ... ... ... ... (cid:12)
(cid:12) (cid:12) periment provides very accurate data [15] in the range
(cid:12)(cid:12) (cid:12)(cid:12) (27) 0.014GeV2 ≤−t≤0.26GeV2(byAmendoliaet.al).There
Intheabovemerelyretainingthefirsttworowsandcolumns
is an overlap region between this last set and the Brown
givesthefirstboundofPalmer,givenintheIntroduction,
data and the two are consistent. The final data we use
while retaining the first three rows and columns yields
comefromtheJLabexperiments(Tadevosyanetal.,ref.[16]).
the second of Palmer’s bounds. Raina and Singh [11] use
the value of F (0) = 1 and r to obtain the lower bound In our study we choose to work with recent data from
π π
toa (π+π−).Thisamountstodroppingoutthe rowsand Amendoliaet.al[15]thatcoverboththe lowandinterme-
µ
columnscorrespondingtoa anda intheeqn.(27)which diate energy range (note that the data from [13,14] cover
2 3
are related to the expansion coefficients c and d. Instead, a smaller energy range; hence our choice) and the data
from the JLab experiments (Tadevosyan et al., ref. [16])
providing an input to I in eqn. (27), gives us an allowed
that cover a higher range of |t|. Using really small |t| val-
regioninthec−dplane.Droppingtherowandcolumncor-
responding to a would result in determining an allowed ues lead to numerical instabilities as the entries in the
3
regionforc alone,whichwe pursueinthe nextsectionfor determinant (eqn. (27)) become small.
purposes of illustration. Tables 1, 2, 3, 4 show the data that is used in our
In the next subsection, we provide a discussion on the analysis to constrain the expansion coefficients c and d
present day experimental information that is utilized in of the form factor F (t). The data from various experi-
π
our study. This information spans an impressive range of ments are chosen so as to give reliable numerical results.
energies, viz. |t|. While in principle there is no limit to Wefinddatainlower|t|toconstraintheexpansioncoeffi-
the number of constraints, in practice the uncertainties cients better but we cannotinclude more than two space-
in the experimental determination and sensitivity of the like constraints; while the higher |t| region gives a weaker
determinantalequationlimitsthisnumber.Thereasonfor bound, which can be improved by increasing the number
suchrestrictionsisthe extreme sensitivityofthe matrices of space-like constraints.
4 B. Ananthanarayan,S.Ramanan: Constraining the low energy pion electromagnetic form factor with space-like data
Table 1. Space-likedata from Bebek et.al [10]
t(−Q2) [GeV2] Fπ(t) x(t) h(x)×10−5
1 -0.620 0.453 ± 0.014 -0.499 3.057
2 -1.216 0.292 ± 0.026 -0.606 2.035
3 -1.712 0.246 ± 0.017 -0.655 1.716
Table 2. Space-like data from Brown et.al [9]
t(−Q2) [GeV2] Fπ(t) x(t) h(x)×10−5
1 -0.294 0.606 ± 0.028 -0.372 3.775
2 -0.795 0.380 ± 0.013 -0.540 2.608
Table 3. Space-like data from Tadevosyan et al. [16]
t(−Q2) [GeV2] Fπ(t) x(t) h(x)×10−5
1 -0.600 0.433 ± 0.017 -0.494 2.915
2 -1.000 0.312 ± 0.016 -0.576 2.163
3 -1.600 0.233 ± 0.014 -0.645 1.626
Table 4. Space-like data from Amendolia et.al [15]
t(−Q2) [GeV2] Fπ(t) x(t) h(x)×10−5
1 -0.131 0.807 ± 0.015 -0.242 4.454
2 -0.163 0.750 ± 0.016 -0.275 4.286
3 Results paredtodatafromsmallervaluesof|t|forafixednumber
of space-like constraints.
Itisworthinvestigatingthesensitivityoftheboundsto
Inourresults,weuseI =75×10−9asaninput,asquoted the errors in the data. The Brown data shifts the bounds
in [5]. This value is certainly greater than a recent esti- onctonegativevalueswhentwospace-likeconstraintsare
mate for the hadronic contribution to ahad =69.2×10−9 used. This shift is sensitive to the error bounds on F (t).
µ π
given in [3]. Pionic contribution is expected to be around If we use as input F (t )+0.028 (table 2) for the first
π 1
70% of the total hadronic value. Our value for I gives a constraint and F (t )−0.013 for the second constraint,
π 2
conservativeupperboundtothepioniccontributiontoa . then the upper bound on c is positive. Here, t and t
µ 1 2
represent the first and second data point respectively in
We begin by constraining only the expansion coeffi-
table2.Wepresentfurthersensitivityanalysislaterinthis
cient c which amounts to dropping out the row and col-
section. Similarly for the data from Amendolia et.al [15],
umn corresponding to a in eqn. (27). We determine the
3
we see that one space-like constraint can be incorporated
bounds on c in stages, starting with only the pion charge
using central values for F (t); but varying this input over
radius r and the normalization of F (t) at t = 0 (i.e., π
π π
the error bounds given in table 4, we can use up to two
no space-like constraints)and then incorporate space-like
constraints (not shown here). Unless othewise specified,
constraints successively. The results are shown in fig. (1),
all results presented in this section use the central values
where we have used data from tables 1, 2, 3, and 4. For
of the form factor data given in tables 1, 2, 3, and 4..
any data set, we start with the smallest value of |t| for
one space-like constraint and include more constraints in Including constraints from a results in a relationship
3
increasing magnitude of |t|. All the data sets show the between the expansioncoefficients c and d which we shall
trend that inclusion of more space-like constraints in this nowexplore.Fig.(2),showstheallowedregioninthec−d
manner, improve the bounds on c. The largest range for planeusingonlythenormalizationconditiononF (t)and
π
c in units of GeV−4 is obtained when no space-like con- thevalueofr .Analogoustothestudyoftheboundsonc,
π
straints are used. This agrees with the result of Caprini we can include space-like constraints to impose stringent
(fig.(1) in [5]). Note however, that for data from larger bounds on the expansion coefficients c and d, that are
values of |t|, the bound on c is significantly weaker, com- reflected by smaller allowed regions in the c − d plane.
B. Ananthanarayan,S.Ramanan: Constraining the low energy pion electromagnetic form factor with space-like data 5
100
50
]
4
-
V
e 0
G
[

c -50
Brown Tadevosyan
-100
100
50
]
4
-
V
e 0
G
[

c -50
Amendiola
Bebek
-100
0 1 2 3 0 1 2 3
number of spacelike constraints number of spacelike constraints
Fig. 1. ConstraintsoncstartingwithjustinformationonFπ(0)=1andrπ2 andincorporatingspace-likeconstraints.Notethat
theallowed values of c decrease as we include more space-like constraints.
n = 0
space
1000
]
6
-
V
e
G
0
[

d
-1000
-40 -20 0 20 40 60 80
-4
c [GeV ]
Fig. 2. Constraints on c and d using thenormalization Fπ(0)=1 and thepion charge radius rπ.
6 B. Ananthanarayan,S.Ramanan: Constraining the low energy pion electromagnetic form factor with space-like data
500 h(x ) h(x )
1 500 1
h(x ) h(x )
] 2 2
-6V 0 0 h(x3)
e
G
d [ -500 -500
Tadevosyan
Brown
-1000 -1000
-30 -20 -10 0 10 20 30 -40 -20 0 20 40
400
h(x ) h(x )
200 1 1
h(x ) 500 h(x )
2 2
6] 0 h(x )
-V 0 3
e-200
G
d [-400 Amendolia -500
Bebek
-600
-1000
-800
-20 -10 0 10 -40 -20 0 20 40
-4 -4
c [GeV ] c [GeV ]
Fig. 3. Constraints on cand d usingonespace-likeconstraint. Herewestart with thesmallest |t| valueand thecorresponding
valueof h(z)=h(x1) and study thevariation of theboundson c and d as we usedata from higher the|t| region.
500 200
nspace = 1 n = 2 150 n = 3
0 space space
6] 0 0
-V -200
e -150
G -400
-500
[
d Bebek -600 Bebek -300 Bebek
-1000
-800 -450
-40 -20 0 20 -30-20-10 0 10 -10 0 10
500 200
n = 1 n = 2 150 n = 3
space space space
0
6] 0 0
-V -200
e -150
G -400
-500
[

d -600 -300
Tadevosyan Tadevosyan Tadevosyan
-1000
-800 -450
-40 -20 0 20 -30-20-10 0 10 -10 0 10
-4 -4 -4
c [GeV ] c [GeV ] c [GeV ]
Fig. 4. Constraints on c and d as the number of space-like constraints are increased. Here the space-like data is taken from
refs. [10], [16].
B. Ananthanarayan,S.Ramanan: Constraining the low energy pion electromagnetic form factor with space-like data 7
500 0
250 n = 1 n = 2
space -200 space
]
6 0
-
V
-400
e-250
G
[ Brown
d -500 -600
Brown
-750
-800
-30 -20 -10 0 10 -20 -10 0
-4
c [GeV ]
500
250 n = 1
space
]
6 0
-
V
e-250
G
[
d -500
Amendolia
-750
-30 -20 -10 0 10
-4
c [GeV ]
Fig. 5. Constraints on c and d as the number of space-like constraints are increased. Here the space-like data is taken from
refs. [9], [15].
Fig. (3) shows the variation in the bounds on c and d i.e.atthe upper limitofthe errorbound,centralrefersto
for one space-likeconstraint,where datafromdifferent |t| thecentralvaluesand“min”refersto centralvalue minus
region is used. In the figure, h(x ) is the value of h(x) as the error bound, i.e., the lower limit of the error bound.
1
definedineqn.(10)correspondingtothesmallestvalueof Weseethatanoverlapbetweentheboundsobtainedfrom
tineachdataset(refertables1-4).Similarlyh(x )isthe thedifferentdatasetoccurswhenthedataareclosetothe
2
constraintatahigher|t|valuetakenintheascendingorder upper error bound. In fact at the upper error bound, the
of magnitude. It may be readily observed that the most data from Tadevosyan et al. gives better bounds on the
stringent bounds are obtained when data from smallest expansion coefficients compared to the data from Bebek
values of |t| are used, and they improve as the number of et al. At the central value of F (t), the data from Bebek
π
constraints are increased, as noted earlier. et al. does better while at the lower end, the data from
Bebek etal.,no longeryields stable result,while the data
Fig. (4) (as well as fig. (5)) shows that increasing the from Tadevosyan et al. still gives reasonable bounds. We
number of space-like constraints improves the bounds on find that despite small differences in the allowed regions,
the expansion coefficients c and d. We observe that the the fact that the allowed regions are essentially the same
allowed range for c corresponds to that in fig. 1. Using offers an important consistency check on the form factor
the central values tabulated in tables 1- 4, we see that determinations by each of the experiments.
we are able to use up to three space-like constraint with
As seen from the our results so far, the bounds on
the data from Bebek et al. [10] and the recent data from
the Taylor coefficients vary due to the errors in the data
Tadevosyan et al. [16], while we are able to incorporate
presented in the tables 1 - 4; hence it is important to
only two space-likeconstraints with the data fromBrown
examine the sensitivity of our results to the inputs. We
etal.[9]andonewiththedatafromAmendoliaetal.[15].
havemadeasystematicstudyofthesensitivitybyvarying,
ThebestestimatefortheboundsontheTaylorcoefficient
(a)the input value ofr , (b) the input value ofI,and(c)
c and d is obtained for the data from Bebek et al. [10]. π
varying the experimental determinations of F (t) within
But varying the Bebek data within the error bounds, we π
theirquotederrors,andineachinstancekeepingallother
see that there is overlap with the bounds obtained with
inputs fixed.
thedatafromTadevosyanetal.[16],asseeninfig.(6).In
fig.(6), the label “max”refers to the data takenfrom the Variations in the bounds on c and d for one space-like
tables1and3withthecorrespondingerrorboundsadded, constraint from [10], [15], when the value of r is var-
π
8 B. Ananthanarayan,S.Ramanan: Constraining the low energy pion electromagnetic form factor with space-like data
Tadevosyan
200 200 200
Bebek
0 0 0
]
6
-
V
e
G
[ -200 -200 -200
d
-400 -400 -400
max central min
-10 0 10 -10 0 10 -10 0 10
-4 -4 -4
c [GeV ] c [GeV ] c [GeV ]
Fig. 6. Variations in theboundson cand dasdatafrom [10]and [16]arevaried within theirerror boundsfor threespace-like
constraints. The shaded region representsthe boundsfrom [16] and thesymbol representsthose from [10].
ied within the error bounds quoted therein, is shown in We explore this sensitivity of the bounds to the data
fig. (7). We see that the bounds do not vary much as taken from the low |t| region using one space-like con-
r is varied within the allowed errors of a few percent. straintfrom[15]andvaryingthisconstraintoverthegiven
π
Fig. (8) shows the variations in the bounds of c and d as errorbounds (Table 4) and study the corresponding vari-
the value of I is changed.LowerI values results in a sub- ations in the bounds on c alone. This is seen in fig. (10)
space of the region allowed for the coefficients at higher where the circles representthe maximumand the squares
I values. We expect this behavior as our method gives representthe minimum of the bounds. The filled symbols
only an allowed region for the coefficients. This explicitly standfortheboundsobtainedfromthedataatlowervalue
demonstrates that as long as the true pionic contribution of|t| and the open symbols representthose fromthe data
is lower than the values we have used, the allowed region at higher value of |t|. Fig. (11) shows similar variations
lies within the isolated ellipse. in the bounds on c when one as well as two space-like
Fig. (9) shows the variations bounds on c and d for constraints (ref. table 3) for the data taken from [16] are
the data from Amendolia et.al [15] and Bebek et.al [10] variedovertheerrorbounds.Notethatthedatafrom[16]
for one space-like constraint (smallest magnitude of |t| in lie at higher |t| value compared to the data from [15].
the tables 1 and 4). The data are varied within their ex- When two space-like constraints are used, we fix one con-
perimentalboundsandtheresultsaredepictedinfig.(9). straint at the central value and vary the other over the
We see that the bounds on the expansion coefficients are error bounds quoted in [16] and plot the maximum and
most sensitive to errors in the data from lower |t|. Hav- minimum of the bounds as a function of the constraint
ing saidthat, it is our view that one may read-offreliable that is varied. From figs. (10) and (11), we see that data
rangesforbothcanddinlightofthissensitivityanalysis. fromlower|t|value ineachsetconstraintsthe coefficients
Howeverananalogoussensitivitytestthatwehavecarried betterforonespace-likeconstraint.Alsothesensitivityto
outfortwospace-likeconstraintsforthedatafromAmen- the errorbounds is greaterfor a setat lower|t| compared
doliaetal.[15]leadstoacompletelossofcoherence.This to the set at higher value of |t|. We also note that two
is an unavoidable numerical difficulty in the determinant space-like constraints are more sensitive to the variations
that arises due to the fact that the entries are small and in Fπ(t) compared to one space-like constraint, as can be
closely spaced. Therefore we can conclude that for low |t| seen in fig. 11.
information we are unable to obtain reliable results for Relative errors in the bounds on c as one space-like
more than one constraint. constraint is varied over the quoted error bounds com-
B. Ananthanarayan,S.Ramanan: Constraining the low energy pion electromagnetic form factor with space-like data 9
400 400
Amendolia central
200 200 max
min
] 0 ] 0
6 6
-V -V Amendolia
e e
d [G--420000 <<rr22>>ππ == 00..442311 ffmm22 d [G--420000 n = 1
<r2> = 0.441 fm2 space
π
-600 -600
-20 -10 0 10 20 -30 -20 -10 0 10 20
500
500
Bebek Bebek
] ]
6 6
-V -V 0
e 0 e
d [G <<rr22>>π == 00..55 3f mfm22 d [G
π -500
-500 <r2>π = 0.57 fm2 nspace = 1
-20 0 20 40 -40 -20 0 20 40
-4 -4
c [GeV ] c [GeV ]
Fig. 7. Variations in the bounds on c and d when the value Fig. 9. Variationsintheboundsoncanddwhenthedatafor
of rπ is changed. This has been carried out for one space-like Fπ(t)fromtables1and4arevariedwithintheirerrorbounds.
constraint for thedata set byAmandolia et.al [15]. Here“central”referstothevaluewithouttheerrorsand“min”
tolowestvalueand“max”tothehighestvaluewithintheerror
bounds.Notethattheboundsaremoresensitivetotheerrors
in thedata from thesmall |t| region.
400
I = 60 x 10-09
200 I = 75 x 10-09
4 Discussion and Conclusions
I = 90 x 10-09
0
-6V] nspace = 1 Inthisworkwehaveconsideredtheconstraintsonthepion
d [Ge-200 Amendolia etilmecatrteosmoafgtnheeticpifoonrimc cfoancttroirbucotimoninsgtofrtohme mprueosennatndoamyaelsy-,
andfromspace-likedatawhichisavailableoverafairlyex-
-400
tendedkinematicregime.Wehaveadoptedtheframework
of Raina and Singh and have extended it to obtain con-
-600
straintson c andd, the expansioncoefficients sub-leading
tothe chargeradiusofthe pion.We haveuseddata avail-
-20 -10 0 10
c [GeV-4] able in the 1970’s (also used in ref. [11]) and more recent
data which spans significantly lower values of |t|, as well
Fig. 8. Variations in the bounds on c and d when the value
as very recent data coming from Tadevosyan et al. which
of I is changed. This has been carried out for one space-like
areat higher |t| values, which lie in the range of Bebek et
constraint for thedata set byAmandolia et.al [15].
al.
AsmentionedintheIntroduction,Caprini[5]hasused
timelike data to obtain constraints on the Taylor coeffi-
cients.Bysystematicallyconsideringtheinclusionofonly
pared to the bounds obtained from the central value is the phase, the ellipse with no constraints was found to
shown in fig. (12) for the data sets from Amendolia et shrink considerably using as an observable the QCD po-
al. and Tadevosyan et al.. We see that relative errors are larization function (see figs.(1) and (2) in [5]). This anal-
smaller for the data from Tadevosyan et al., once again ysis employs an optimal technique, resulting in an inte-
emphasizingourearlierobservationthatthe sensitivityof gral equation of the Fredholm type that was solved nu-
the Taylor coefficients to the errors in the determination merically. The ellipse was found to shrink even further
of F (t) is greater for the data from the low |t| region when both the phase as well as the modulus information
π
compared to those from the high |t| region. wereusedeventhoughthetechniquewasnon-optimal(see
10 B. Ananthanarayan,S.Ramanan: Constraining the low energy pion electromagnetic form factor with space-like data
20 1
ccmmianx Amendolia ccmin Tadevosyan
10 max
-4c, c [GeV]maxmin-100 Amandolia - one spacelike constraint rel error (c, c)maxmin 0.00.11
-20
one spacelike constraint
-30 0.001
0.74 0.76 0.78 0.8 0.82 0.74 0.76 0.78 0.8 0.82 0.3 0.35 0.4 0.45
f(t) f(t) f(t)
Fig. 10. Variations in the bounds on c when the data for Fig.12.Relativeerrorsintheboundsoncwithrespecttothe
Fπ(t) from table 4 are varied within their error bounds. The central value,for onespace-like constraint as theconstraint is
filled circles and squares refer to the maximum and minimum varied overthe error boundsfor thedata sets [15] and [16].
oftheboundsoncobtainedfromthefirstentryintable4and
theopen ones correspond to thesecond entry in the table.
600
40
c Constraining Taylor coeffn c and d
cmin 400
max
20
-4eV] one spacelike constraint two spacelike constraint -6V] 200
G e
[min 0 Tadevosyan Tadevosyan d [G 0
c
c, max -200
-20
Bebek
Caprini with phase
-400
-40 -20 -10 0 10 20 30 40 50
0.3 0.35 0.4 0.45 0.3 0.35 0.4 0.45 c [GeV]-4
f(t) f(t)
Fig. 13. Comparing ourbest ellipse obtained for threespace-
Fig.11.VariationsintheboundsoncwhenthedataforFπ(t) likeconstraintsusingdatafromBebeketal.[10](symbol)with
from table 3 are varied within their error bounds. The filled the result obtained using the phase of the form factor in the
circlesandsquaresrefertothemaximumandminimumofthe timelike region (see fig.(2) in [5]).
bounds on c obtained from the first entry in table 3 and the
openonescorrespondtothesecondentryinthetable.Theleft
panelshowsthevariationononespace-likeconstraintwhilethe
right panel shows variations on two space-like constraint. For interesting to note that the overlap region between the
two space-likeconstraints, we fixone constraint at thecentral two determinations, ours using space-like data and the
value and vary the other over the quoted error bounds [10] muon anomaly and Caprini’s using the phase of timelike
andplotthevariationsintheboundsoncasafunction ofthe dataandtheQCDpolarizationobservable,accommodates
constraint that is varied. comfortably the value of c from chiral perturbation the-
ory.We note here that similar bounds obtained using the
central values of the Tadevosyan data (ref. table 3) us-
ingthreespace-likeconstraintsdoesnotaccommodatethe
fig.(3) in [5]). Questions remain about its validity due to
value of c obtained from chiral perturbation theory, but
certainmathematicaldifficulties, as discussedby Caprini.
varying the data at the upper end of the error bound, as
The best estimates for the bounds on c and d is ob- in fig. 6, the allowed region does include this value.
tained using the data from Bebek et al. [10] for three Our conclusions are that the constraints are weaker
space-like constraints. In fig. (13), we compare our best thanthoseobtainedbyCapriniusingbothphaseandmod-
resultwiththoseobtainedbyCapriniusingonlythephase ulus of data from the timelike region (see fig.(3) in [5]),
fromthe timelike region.Using space-likedatawe getthe while they are more stringent than those obtained with
following range for the expansion coefficients: −1 GeV−4 only the phase of the timelike data. Here we note again,
. c . 11GeV−4 and −132 GeV−6 . d . 220 GeV−6. that the latter of these treatments is rigorous and the re-
On the other hand Caprini has a range of [−14 GeV−4, sults may be taken as reliable. On the other hand, the
44 GeV−4] and [−236 GeV−6, 594 GeV−6] for c and d results obtained from using both the modulus and phase
respectively using only the phase of timelike data. It is whileappearingmorestringentareonalessrigorousfoot-