Table Of ContentIFUP-TH 62/97
The effective potential in three-dimensional O(N) models.
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1 Andrea Pelissetto and Ettore Vicari
n Dipartimento di Fisica dell’Universita` and I.N.F.N., I-56126 Pisa, Italy
a
(February 1, 2008)
J
2
1
Abstract
]
h
c
e
m We consider the effective potential in three-dimensional models with O(N)
- symmetry. For generic values of N, and in particular for the physically inter-
t
a esting cases N = 0,1,2,3, we determine the six-point and eight-point renor-
t
s malizedcouplingconstantswhichparametrizeitssmall-fieldexpansion. These
.
t
a estimates are obtained from the analysis of their ǫ-expansion, taking into ac-
m
count the exact results in one and zero dimensions, and, for the Ising model
-
(i.e. N = 1), the accurate high-temperature estimates in two dimensions.
d
n They are compared with the available results from other approaches. We also
o
obtain corresponding estimates for the two-dimensional O(N) models.
c
[
Keywords: Fieldtheory,Criticalphenomena,O(N)models,Isingmodel,
1
Effective potential, n-point renormalized coupling constants, ǫ-expansion,
v
8 1/N expansion.
9
0 PACS numbers: 05.70.Jk, 64.60.Fr, 05.50.+q, 75.10.Hk, 11.10.Kk,
1
11.15.Tk.
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t
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1
I. INTRODUCTION
The effective potential is widely used in the field-theoretic description of fundamental
interactions and phase transitions. In field theory the effective potential is the generating
functional of the one-particle irreducible correlation functions at zero external momenta. In
statisticalphysicsitrepresentsthefree-energydensity asafunctionoftheorderparameter,
F
which, for spin models, is the magnetization M. Its global minimum determines the value of
theorderparameter,whichcharacterizesthephaseofthesystem. Inthehigh-temperatureor
symmetric phase the minimum is unique with M = 0. As the temperature decreases below
the critical value, the effective potential takes a double-well shape: the order parameter
does not vanish anymore and the system is in the low-temperature or broken phase1. The
equation of state is closely related to the effective potential. It relates the magnetization M
(i.e. the order parameter), the magnetic field H and the reduced temperature t T T .
c
∝ −
It is simply given by
∂
H = F . (1)
∂M
In this paper we study the effective potential of O(N) models. We recall that O(N)
models describe many important critical phenomena in nature: liquid-vapour transitions
in classical fluids, the helium superfluid transition, the critical properties of isotropic ferro-
magnetic materials and long polymers. We will focus mainly on the small-renormalized-field
expansion of the effective potential in the symmetric phase. Its coefficients are directly re-
lated to the zero-momentum n-point renormalized coupling constants g . The four-point
n
coupling g g plays an important role in the field-theoretic perturbative expansion at
4
≡
fixed dimension [3], which provides accurate estimates of critical indices and universal ra-
tios in the symmetric phase. In this approach any universal quantity is obtained from a
series in powers of g (g-expansion) which is then resummed and evaluated at the fixed-point
value of g, g∗. Accurate estimates of g∗ have been obtained by calculating the zero of the
Callan-Symanzik β-function associated with g (see e.g. Refs. [4–9]). These results have been
substantiallyconfirmedbycomputationsusingdifferentapproaches, suchasǫ-expansion[10],
high-temperature expansion (see, e.g., Refs. [11–14,10]), Monte Carlo simulations (see, e.g.,
Refs. [15–17]), 1/N expansion [11,10].
Recentlytherehasbeenalotofinterestintheproblemofcomputingthehigher-ordercou-
pling constants g , g , ..., for the Ising model (see e.g. Refs. [18,15,19,13,20,9,14,21,17,22]).
6 8
Here we will study the issue for generic values of N. We will obtain satisfactory estimates of
g andg , or, equivalently, ofthe ratiosr g /gj−1, fromtheir ǫ-expansion [23](ǫ 4 d)
6 8 2j 2j
≡ ≡ −
within the φ4 theory defined by the action
1 1 1
S = ddx ∂ φ(x)∂ φ(x)+ m2φ2 + g (φ2)2 . (2)
2 µ µ 2 0 4! 0
Z (cid:20) (cid:21)
1Actually in the broken phase the double-well shape is not correct because the effective potential
must always be convex. In this phase it should present a flat region around the origin. For a
discussion see e.g. Refs. [1,2].
2
The ǫ-expansion of the fixed-point value of r can be derived from the ǫ-expansion of
2j
the equation of state, which is known to O(ǫ3) for the Ising model [24,25], i.e. N = 1,
and to O(ǫ2) for generic values of N [26], thus leading to series of the same length for
r . Since the available series are short and have large coefficients increasing with j, their
2j
straightforward analysis does not provide reliable estimates, but gives only an indication of
the order of magnitude. A considerable improvement can be achieved if one uses the results
for d = 0,1,2, whenever they are available. This idea was employed in Ref. [27] to improve
the estimates of the critical exponents of the Ising and self-avoiding walk models, and in
Ref. [28] it was used to the study of the two-point function. In Ref. [10] it was generalized
andsuccessfully appliedtothedeterminationofthezero-momentum four-pointrenormalized
coupling. In the present case the basic assumption is that the zero-momentum n-point
renormalized couplings g , and therefore the ratios r , are analytic and quite smooth in
2j 2j
the domain 4 > d > 0 (thus 0 < ǫ < 4). This can be verified in the large-N limit. One may
then perform a polynomial interpolation among the values of d where the constants r are
2j
known (d = 0,1) or for which good estimates are available (d = 2 for N = 1, obtained from
a high-temperature analysis), and then analyze the series of the difference. This procedure
leads to more accurate estimates, which are consistent with those obtained by the direct
analysis of the original ǫ-series, but have a much smaller uncertainty. As a by-product of
our analysis we also obtain relatively good estimates of r and r for two-dimensional O(N)
6 8
models.
For N = 1, most of the published results concern the renormalized four-point coupling
6
constant g. As far as we know, estimates of g and g have only been obtained from
6 8
approximate solutions of the renormalization group equation [18], and from the analysis
of high-temperature series [12]. The latter results present a large uncertainty. The former
are reported without errors — which, in any case, are very difficult to estimate — and their
reliability is unclear. For instance, the estimates of g that are obtained using this method
are in disagreement with the results of other computations. Therefore our new independent
estimates of r and r for N = 1 represent our main results, and provide an important check
6 8
6
of the above-mentioned calculations.
For the Ising model we can compare our results with the estimates obtained using dif-
ferent approaches: the g-expansion at fixed dimension d = 3 [20,9] that apparently provides
the most precise results; a different analysis of the ǫ-expansion based on the parametric
representation of the equation of state [9]; approximate solutions of the exact renormaliza-
tion group equation [18,21]; high-temperature expansions [14,12,13]; dimensional expansion
around d = 0 [29,19]; Monte Carlo simulations [15,17]. Our final results are in good agree-
ment with these estimates and their precision is comparable with that of the analysis of the
g-expansion.
The paper is organized as follows. In Sec. II we introduce our notation and give some
general formulae for the small-field expansion of the effective potential. In Sec. III we
compute the effective potential in the large-N limit, the ratios r to O(1/N), and give the
2j
exact values of r and r for d = 1,0. Furthermore we present a high-temperature analysis
6 8
of the two-dimensional Ising model, which provides accurate estimates of the first few r . In
2j
Sec. IV we present our analysis of the ǫ-expansion of r . In Sec. V we compare our results
2j
with other approaches. In App. A we give some useful formulae relating the ratios r to
2j
the connected Green’s functions. In App. B we report the ǫ-expansion of r derived from
2j
3
the ǫ-expansion of the equation of state. In App. C some details of the calculations in one
dimension are given.
II. SMALL-FIELD EXPANSION OF THE EFFECTIVE POTENTIAL
The free energy per unit volume can be expanded in powers of the renormalized magne-
tization ϕ (i.e. the expectation value of the renormalized field φ = φ/√Z):
r
Γ(ϕ) 1 1 1
(ϕ) = = (0)+ m2ϕ2 + m4−dgϕ4 + m2j+(1−j)d g ϕ2j, (3)
2j
F V F 2 4! (2j)!
j=3
X
where Γ(ϕ) is the generating functional of one-particle irreducible correlation functions at
zero external momenta, i.e. the effective potential of the renormalized theory. The mass
scale m is the inverse of the second-moment correlation length, i.e. m = ξ−1 and
1 dx x2G(x)
ξ2 = , (4)
2d dx G(x)
R
R
where the function G(x) is defined by
φ (0)φ (x) = δ G(x). (5)
α β αβ
h i
By rescaling ϕ as
m(d−2)/2
ϕ = z (6)
√g
in Eq. (3), the free energy can be written as
md
(ϕ) (0) = A(z), (7)
F −F g
where
1 1 1
A(z) = z2 + z4 + r z2j, (8)
2j
2 4! (2j)!
j=3
X
and
g
2j
r = . (9)
2j
gj−1
In App. A we give some useful formulae to derive the constants r from the connected
2j
Green’s functions.
One can show that z t−βM, and that the equation of state can be written in the form
∝
∂A(z)
H tβδ . (10)
∝ ∂z
4
This relation can be exploited in order to derive A(z) from the equation of state, which is
usually written in the form (see e.g. Ref. [4])
H = Mδf(x) (11)
where x = tM−1/β. The function A(z) is thus given by
∂A(z)
= h zδf x z−1/β , (12)
0 0
∂z
(cid:16) (cid:17)
where the normalization constants h and x are fixed by the requirement that
0 0
1 1
A(z) = z2 + z4 +O(z6). (13)
2 4!
The ratiosr are obtained by expanding A(z) in powers of z. Notice that, since the function
2j
f(x) in Eq. (11) is regular at x = 0 and nonzero, Eq. (12) implies A(z) zδ+1 for z .
∼ → ∞
In the following we will be interested in the rescaled effective potential A(z) and we will
calculate the fixed-point values of the first few coefficients r of its small-z expansion.
2j
III. EXACT AND HIGH-TEMPERATURE RESULTS
In order to get a qualitative idea of the properties of the effective potential, we consider
its large-N limit. It is easy to derive the large-N limit of the rescaled effective potential
A(z) from the corresponding equation of state [30]:
H = Mδ(1+x)2/(d−2) , (14)
where δ = (d+2)/(d 2), x = tM−1/β and β = 1/2. One finds
−
d/(d−2)
6 d 2
A(z) = 1+ − z2 1 , (15)
d 12 ! −
from which
(2j)! j−2 i 1
r = ǫ 2 − . (16)
2j 22j−23j−1j(j 1) − i
− iY=1(cid:18) (cid:19)
The constants r are (j 2)th-order polynomials in ǫ 4 d that have j 2 real zeros at
2j
− ≡ − −
ǫ = 2(i 1)/i with i = 1,...,j 2. Notice that in the limit j the zeros have ǫ = 2 (i.e.
− − → ∞
d = 2) as an accumulation point. It is interesting to note the form of the large-N limit of
A(z) for integer d:
A(z) = 1z2 + 1 z4 for d = 4, (17)
2 24
A(z) = 1z2 + 1 z4 + 1 z6 for d = 3, (18)
2 24 864
A(z) = 3 ez2/6 1 for d = 2, (19)
−
A(z) = 6z2(cid:16)(12 z2)(cid:17)−1 for d = 1. (20)
−
5
Notice that in the large-N limit, for d = 3,4 (and in general for d = 2n/(n 1) with
−
integer n 2) the effective potential is a polynomial in z, or equivalently in ϕ. In four
≥
dimensions only the first two terms are present so that the effective potential (in the variable
z) coincides with the phenomenological expression of Ginzburg and Landau. Notice however
that the rescaling (6) is not strictly defined in four dimensions since g 0 in the critical
→
limit. Therefore this simple expression is not valid in the original variable ϕ and indeed
logarithms of the magnetization appear in the four-dimensional effective potential [31]. In
three dimensions also the ϕ6 term is present so that δ = 5. As d 2 all terms become
→
relevant. Of course this simple behaviour is peculiar of the large-N limit. For finite values
of N all terms are present in the small-field expansion, and δ can only be determined after
resumming the series.
One can also derive the O(1/N) correction to Eq. (16) from the corresponding O(1/N)
correction to the equation of state calculated2 in Ref. [30]. In d = 3 one obtains
5 12.2556 1
r = 1+ +O , (21)
6 6 N N2
(cid:20) (cid:18) (cid:19)(cid:21)
67.3140 1
r = +O , (22)
8 − N N2
(cid:18) (cid:19)
1406.83 1
r = +O , (23)
10 N N2
(cid:18) (cid:19)
etc.... For large values of j the coefficients of the O(1/N) term in the 1/N expansion of r
2j
behave approximately as (2j)!( c)j, where c is a constant: c 0.40. The large coefficient
∼ − ≃
of the O(1/N) correction in r indicates that the region where Eq. (21) may be a good
6
>
approximation corresponds to very large values of N, say N 100. This is expected to be
∼
true also for r with larger values of j.
2j
The constants r can be computed exactly in one and zero dimensions. These results
2j
will be useful in our analysis of the ǫ-expansion of r as explained in the introduction.
2j
Some details of the calculations for d = 1 can be found in App. C. Here we only give
the results for r and r . For N 1 we have
6 8
≥
5N(N 1)2(8N +7)
r = 5 − , (24)
6 − (N +1)(N +4)(4N 1)2
−
175 35N(N 1)2(256N3 +3037N2 +1705N 588)
r = − − . (25)
8
3 − 3(N +1)(N +4)(N +6)(4N 1)3
−
For N 1 instead
≤
r = 5, (26)
6
175
r = . (27)
8
3
2 We mention the presence of a misprint in the final expression of the O(1/N) equation of state
given in Eq. (29) of Ref. [30]: in the third term of the first line (2π)2−ǫ/2 should be replaced by
2π2−ǫ/2.
6
For N = 1 these expressions agree with the results of Ref. [19]. For N = they reproduce
∞
Eq. (16). For the Ising model we will also need the value of r , which has been computed
10
in Ref. [19]: r = 1225.
10
For d = 0 and N 1, using the formulae reported in App. A, it is easy to obtain3
≥
10(N +8)
r = , (28)
6
3(N +4)
70(N2 +14N +120)
r = , (29)
8
3(N +4)(N +6)
280(10752+3136N +256N2 +30N3 +N4)
r = . (30)
10 (N +4)2(N +6)(N +8)
For N = 1 these results agree with those of Ref. [19]. It is not clear how to determine the
value of r for N = 0. Unlike the case d = 1, we cannot prove that setting N = 0 in the
2j
formulae obtained for N 1 provides the correct answer.
≥
Forthe two-dimensional Ising model reliableestimates ofthe first few r canbeobtained
2j
from the analysis of their high-temperature expansion on the lattice. The basic reason
is that the leading correction to scaling is analytic, since the subleading exponent ∆ is
expected to belarger thanone [32,33]. Therefore the traditionalmethods of analysis of high-
temperature series shouldwork well. The series published inRefs. [34,35]for thelattice Ising
model with nearest-neighbor interactions allow us to calculate r (more precisely, a high-
2j
temperature series whose value at the critical point is r ) to 17th-order on the square lattice
2j
(for which β = ArcTanh(√2 1)) and to 14th-order on the triangular lattice (for which
c
−
β = ArcTanh(2 √3)). In the analysis of these series we followed Ref. [10], using several
c
−
types of approximants, Pad`e, Dlog-Pad`e and first-order integral approximants. Table I
reports the results obtained on the square and on the triangular lattice. They are consistent
with each other. Assuming universality, as final estimates for the two-dimensional Ising
model we take
r = 3.678(2), (31)
6
r = 26.0(2), (32)
8
r = 275(15). (33)
10
The error on the estimate of r increases with j and thus the analysis of the higher-order
2j
coefficients does not lead to reliable estimates. We mention that the high-temperature
analysis of Ref. [13] led4 to r = 3.679(8), which is perfectly consistent with our estimate.
6
3The calculation is easily done for the N-vector model: in this case one has a single field ~s with
~s ~s = 1 and the Gibbs measure is simply d~s δ(~s ~s 1).
· · −
4 Actually Ref. [13] gives an estimate of R χ2/(χ χ ) = (10 r )−1.
0 ≡ 4 2 6 − 6
7
IV. ANALYSIS OF THE ǫ-EXPANSION
In this Section we will compute r for j = 3,4,5 using the ǫ-expansion. The series in ǫ
2j
of r , r and r are reported in App. B. They were obtained from the ǫ-expansion of the
6 8 10
equation of state [26,24,25]. Since the ǫ-expansion is asymptotic, it requires a resummation
to get estimates for d = 3, i.e. ǫ = 1, which is usually performed assuming its Borel
summability.
The main point of our analysis is the use of the exact values of r for d = 0,1 we have
2j
reported in the previous Section, and, for N = 1, of the precise two-dimensional estimates
which have been obtained from the analysis of high-temperature series, see Eqs. (31), (32),
(33). Indeed the constants r are expected to be analytic in the domain 4 > d > 0. This
2j
can be explicitly verified in the large-N limit where the constants r are polynomials in
2j
ǫ, cf. Eq. (16). Moreover it was implicitly assumed in the dimensional expansion around
d = 0 done in Refs. [29,19].
The idea of the method is the following: consider a generic observable and let R(ǫ) be
its expansion in ǫ. Moreover suppose that the values of R are known for a set of dimensions
ǫ ,...,ǫ . In this case one may use as zeroth order approximation the value for ǫ = 1 of the
1 k
polynomial interpolation through ǫ = 0, ǫ ,...,ǫ and then use the series in ǫ to compute the
1 k
deviations. More precisely, let us suppose that exact values R (ǫ ), ..., R (ǫ ) are known
ex 1 ex k
for the set of dimensions ǫ , ..., ǫ , k 2. Then define
1 k
≥
k R (ǫ ) k
Q(ǫ) = ex i (ǫ ǫ )−1 (34)
i j
(ǫ ǫ ) −
Xi=1 − i j=Y1,j6=i
and
R(ǫ)
S(ǫ) = Q(ǫ), (35)
k (ǫ ǫ ) −
i=1 − i
Q
and finally
k
R (ǫ) = [Q(ǫ)+S(ǫ)] (ǫ ǫ ). (36)
imp i
−
i=1
Y
One can easily verify that the expression
k
[Q(ǫ)+S(0)] (ǫ ǫ ) (37)
i
−
i=1
Y
is the k-order polynomial interpolation through the points ǫ = 0,ǫ ,...,ǫ . The resummation
1 k
procedure is applied to S(ǫ) and the final estimate is obtained by computing R (ǫ = 1).
imp
Since the series of r begins with a term of order ǫ, we analyze the quantity r /ǫ.
2j 2j
Notice that, as a consequence, the interpolation formula (37) actually uses the value of the
derivative of r in four dimensions. If the interpolation is a good approximation one should
2j
find that the series which gives the deviations has smaller coefficients than the original one.
Consequently one expects that also the errors in the resummation are reduced. We find
8
that, as expected, the coefficients of the corresponding series S(ǫ) decrease in size with k,
the number of exact values that are used to constrain the series. This fact was also shown
in Ref. [10] for the case of the four-point renormalized coupling.
The large-N results of Eqs. (21-23) provide further support to our constrained analysis.
Indeed one may consider the simple polynomial interpolation (which uses the values of r
2j
in d = 0,1 and the value of its derivative in d = 4) evaluated at d = 3, rint, and compare its
2j
large-N expansion with the exact one. One finds (for N 1)
≥
5(16N5 +402N4 +1734N3 +539N2 669N +84)
rint = −
6 6(N +1)(N +4)(N +8)(4N 1)2
−
5 101 87 1
= 1+ +O (38)
6 8N − N2 N3
(cid:20) (cid:18) (cid:19)(cid:21)
and
75.65 988.4 1
rint = + +O . (39)
8 − N N2 N3
(cid:18) (cid:19)
Comparing Eq. (38) with Eq. (21), one sees that rint gives the exact result for N = .
6 ∞
Moreover, also the O(1/N) correction is closely reproduced: indeed the coefficient of the
O(1/N)terminrint is10.52tobecomparedwiththeexactvalue10.21ofEq.(21). Therefore
6
in the large-N limit rint provides an estimate of r with a relative error which behaves as
6 6
0.37/N: rint becomes increasingly accurate as N . The same discussion applies to r :
6 → ∞ 8
the coefficient of the 1/N term in Eq. (39) is very close to the exact one 67.31, cf. Eq.
−
(22). Therefore also in this case the interpolation rint provides good estimates of r : for
8 8
N the relative error is 12%.
→ ∞
The analysis of the series S(ǫ), cf. Eq. (35), can be performed by using the method
proposed in Ref. [6], which is based on the knowledge of the large-order behaviour of the
series. Itisindeedknownthatthen-thcoefficient oftheseries behaveas ( a)nΓ(n+b +1)
0
∼ −
for large n. The constant a, which characterizes the singularity of the Borel transform does
not depend on the specific observable; it is given by [36,37] a = 3/(N +8). The coefficient
b depends instead on the series one considers. Given a quantity R with series
0
R(ǫ) = R ǫk, (40)
k
k=0
X
we have generated new series R (α,b;ǫ) according to
p
p ∞ u(ǫt)k
R (α,b;ǫ) = B (α,b) dt tb e−t (41)
p k [1 u(ǫt)]α
kX=0 Z0 −
where
√1+ax 1
u(x) = − . (42)
√1+ax+1
The coefficients B (α,b) are determined by the requirement that the expansion in ǫ of
k
R (α,b;ǫ) coincides with the original series. For each α, b and p an estimate of R is simply
p
given by R (α,b;ǫ = 1).
p
9
For the Ising model, where the available series are of order O(ǫ3), we followed Ref. [10]
in order to derive the estimates and their uncertainty. We determine an integer value of b,
b , such that
opt
R (α,b ;ǫ = 1) R (α,b ;ǫ = 1) (43)
3 opt 2 opt
≈
for α < 1. b is the value of b such that the estimate from the series to order O(ǫ3) is
opt
essentially identical to the estimate from the series to order O(ǫ2). In a somewhat arbitrary
way we have then considered as our final estimate the average of R (α,b;ǫ = 1) with
p
1 < α 1 and 2 + b b 2 + b . The error we report is the variance of the
opt opt
− ≤ − ≤ ≤
values of R (α,b;ǫ = 1) with 1 < α 1 and b /3 1 b 4b /3 + 1 . This
3 opt opt
− ≤ ⌊ − ⌋ ≤ ≤ ⌈ ⌉
procedure is ad hoc, but provides estimates that are all consistent among each other. In
order to test the method, in Ref. [10], the procedure was applied to the determination of the
critical indices and it provided estimates and error bars in substantial agreement with the
results of other authors. Therefore we believe that our error bars are reasonable, although
one should be cautious in giving them the standard statistical meaning.
The results of our analysis for the Ising model, corresponding to N = 1, are presented
in Table II. We report various estimates of r , r and r obtained from an unconstrained
6 8 10
analysis and constrained analyses in various dimensions. They are all consistent. As ex-
pected, the error decreases when additional lower dimensional values are used to constrain
the analysis: the error of the unconstrained analysis is approximately an order of magnitude
larger than the error of our best result that uses the known values at d = 0,1,2. In Figs. 1
and 2 we show respectively r and r as a function of d. There we plot the polynomial
6 8
interpolations through the known values of r /ǫ at d = 4,2,1,0, and the results of our con-
6
strained (d = 2,1,0) ǫ-expansion analysis. Their comparison shows how well the polynomial
interpolation works.
We have also repeated the analysis in two dimensions. In this case, of course, it is
more difficult to get precise estimates: the unconstrained expansion gives results with large
errors and it is therefore practically useless. Better estimates are obtained constraining
the expansion in one and zero dimensions. The results for these two cases are reported in
Table I. The final estimates for r and r are in very good agreement with the much more
6 8
precise results obtained from the high-temperature analysis. The result for r is instead
10
significantly lower than the high-temperature estimate. It should be noted however that the
series in ǫ for r has very large coefficients and the estimates show large fluctuations with
10
the parameters b and α. Therefore it is not clear if our algorithm to determine the error
bars is working properly here. For this reason, in this case we believe the high-temperature
estimate to be more reliable than the ǫ-expansion result.
For generic values of N, the series is one order shorter and we have only two non-trivial
terms. The procedure we presented above cannot beapplied andwe used a different method.
For each value of b, averaging over 1 α 1, we obtain three estimates: the first one
− ≤ ≤
is the result of the analysis of the unconstrained series, the second and the third one the
results from the series constrained respectively in d = 1 and in d = 1,0. We then compute
the weighted average and the corresponding χ2. The optimal value b is chosen as the
opt
value of b with the smallest χ2. Once b has been chosen, we calculated mean values and
opt
relative spreads varying b and α according to the algorithm we presented above. Again, we
10
