text: ℓ+ ν ℓ p2 p′2 m2 m′2 B , P X, P c f p1 p′1 m1 m′1 → FIG. 1: Feynman diagram of the semi-leptonic decay B X(cid:96)+ν , X denote X(3940) or X(4160). c (cid:96) II. THE FORMULATIONS OF SEMILEPTONIC DECAYS AND NONLEPTONIC DECAY OF B c In this section we present the formulations of semi-leptonic decay and nonleptonic decay of B mesons to X(3940) and X(4160) which are considered as η (3S) and η (4S) states. c c c A. Semileptonic decay of B c The feynman diagram of B semileptonic decay to X = X(3940) or X = X(4160) is c shown in Fig. 1. The corresponding amplitude for the decay can be written as G√ − (cid:104) | | (cid:105) T = F V u¯ γµ(1 γ )v X(P ) J B (P ) , (1) 2 bc ν(cid:96) 5 (cid:96) f µ c − where V is the CKM matrix element, G is the the Fermi constant, J = V A is the bc F µ µ µ charged weak current, P and P are the momentum of the initial meson B and the ﬁnal f c state, respectively. The hadronic part can be written as, (cid:104) | | (cid:105) − X(P ) V B (P ) = f (P + P ) + f (P P ) , f µ c + f µ − f µ (cid:104) | | (cid:105) X(P ) A B (P ) = 0, (2) f µ c where f , f are the Lorentz invariant form factors. + − ≡ ≡ − We deﬁne x E /M, y (P P )2/M2, where E is the energy of the ﬁnal charge (cid:96) f (cid:96) lepton, M is the mass of initial meson. The diﬀerential width of the decay can be reduced to: 4 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ℓ+

font: default

text: ν

text: ℓ

text: B

text: P

text: X,

text: P

text: c

text: f

text: FIG.

text: Feynman

text: diagram

text: of

text: the

text: semi-leptonic

text: decay

text: B

text: X

text: denote

text: or

text: c

text: II.

text: THE

text: FORMULATIONS

text: OF

text: SEMILEPTONIC

text: DECAYS

text: AND

text: NONLEPTONIC

text: DECAY

text: OF

text: B

text: c

text: In

text: this

text: section

text: we

text: present

text: the

text: formulations

text: of

text: semi-leptonic

text: decay

text: and

text: nonleptonic

text: decay

text: of

text: B

text: mesons

text: to

text: and

text: which

text: are

text: considered

text: as

text: η

text: and

text: η

text: states.

text: c

text: c

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text: A.

text: Semileptonic

text: decay

text: of

text: B

text: c

text: The

text: feynman

text: diagram

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text: B

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text: decay

text: to

text: X

text: or

text: X

text: is

text: c

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text: The

text: corresponding

text: amplitude

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text: the

text: decay

text: can

text: be

text: written

text: as

text: G√

text: T

text: F

text: V

text: u¯

text: γµ(1

text: γ

text: )v

text: X(P

text: J

text: B

text: (P

text: bc

text: f

text: µ

text: c

text: where

text: V

text: is

text: the

text: CKM

text: matrix

text: element,

text: G

text: is

text: the

text: the

text: Fermi

text: constant,

text: J

text: V

text: A

text: is

text: the

text: bc

text: F

text: µ

text: µ

text: µ

text: charged

text: weak

text: current,

text: P

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text: P

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text: the

text: momentum

text: of

text: the

text: initial

text: meson

text: B

text: and

text: the

text: ﬁnal

text: f

text: c

text: state,

text: respectively.

text: The

text: hadronic

text: part

text: can

text: be

text: written

text: as,

text: X(P

text: V

text: B

text: (P

text: f

text: (P

text: P

text: f

text: (P

text: P

text: f

text: µ

text: c

text: f

text: µ

text: f

text: µ

text: X(P

text: A

text: B

text: (P

text: f

text: µ

text: c

text: where

text: f

text: f

text: are

text: the

text: Lorentz

text: invariant

text: form

text: factors.

text: We

text: deﬁne

text: x

text: E

text: /M,

text: y

text: (P

text: P

text: where

text: E

text: is

text: the

text: energy

text: of

text: the

text: ﬁnal

text: charge

text: f

text: lepton,

text: M

text: is

text: the

text: mass

text: of

text: initial

text: meson.

text: The

text: diﬀerential

text: width

text: of

text: the

text: decay

text: can

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text: reduced

text: to:

text: ##LTFigure##

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text: (cid:34) (cid:35) (cid:90) (cid:54) (cid:104) | | (cid:105) d(cid:126)q P − X(P ) J B (P ) = Tr ϕ¯++((cid:126)q ) ϕ++((cid:126)q)γ (1 γ ) , (7) f µ c (2π)3 Pf f M P µ 5 where (cid:126)q ((cid:126)q ) is the relative three-momentum between the quark and anti-quark in the f − (cid:48) initial (ﬁnal) meson and (cid:126)q = (cid:126)q (cid:48)m1 (cid:48) P(cid:126) . M is the mass of B , P(cid:126) is the three dimensional f m +m f c f 1 2 momentum of X, ϕ++((cid:126)q) is the positive Salpeter wavefunction of B meson and ϕ++((cid:126)q ) P c Pf f is the positive Salpeter wavefunction of X meson, ϕ¯++ = γ (ϕ++)†γ . We will show the 0 0 Pf Pf Salpeter wavefunctions for the diﬀerent mesons in next section. IV. THE RELATIVISTIC WAVEFUNCTIONS OF PSEUDOSCALAR MESON A. For B meson with quantum numbers JP = 0− c The general form for the relativistic wavefunction of pseudoscalar meson B can be written c as [55]: (cid:104) (cid:105) (cid:54) (cid:54) (cid:54) (cid:54) P q ϕ −((cid:126)q) = f ((cid:126)q)P + f ((cid:126)q)M + f ((cid:126)q) q + f ((cid:126)q) ⊥ γ , (8) 0 1 2 3 ⊥ 4 M 5 | | where M is the mass of the pseudoscalar meson, and f ((cid:126)q) are functions of (cid:126)q 2. Due to i the last two equations of Eq. (A7): ϕ+−− = ϕ−−+ = 0, we have: 0 0 − f ((cid:126)q)M( ω + ω ) −f ((cid:126)q)M(ω + ω ) f ((cid:126)q) = 2 1 2 , f ((cid:126)q) = 1 1 2 . (9) 3 m ω + m ω 4 m ω + m ω (cid:113)2 1 1 2 (cid:113) 2 1 1 2 where m , m and ω = m2 + (cid:126)q2, ω = m2 + (cid:126)q2 are the masses and the energies of 1 2 1 1 2 2 − · −| | quark and anti-quark in B mesons, q = q (q P/M2)P , and q2 = (cid:126)q 2. c ⊥ ⊥ The numerical values of radial wavefunctions f , f and eigenvalue M can be obtained 1 2 by solving the ﬁrst two Salpeter equations in Eq. (A7). To show the numerical results of wavefunctions explicitly, we plot the wavefunctions of B meson in Fig. 3. According to the c Eq. (A6) the relativistic positive wavefunction of pseudoscalar meson B in C.M.S can be c written as [55]: (cid:34) (cid:35) (cid:54) (cid:54) (cid:54) P (cid:54) q P ϕ+−+((cid:126)q) = b b + + b q + b ⊥ γ , (10) 0 1 2 M 3 ⊥ 4 M 5 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

label: paragraph

text: P

text: X(P

text: J

text: B

text: (P

text: Tr

text: γ

text: f

text: µ

text: c

text: Pf

text: f

text: M

text: P

text: µ

text: where

text: is

text: the

text: relative

text: three-momentum

text: between

text: the

text: quark

text: and

text: anti-quark

text: in

text: the

text: f

text: initial

text: (ﬁnal)

text: meson

text: and

text: M

text: is

text: the

text: mass

text: of

text: B

text: is

text: the

text: three

text: dimensional

text: f

text: m

text: +m

text: f

text: c

text: f

text: momentum

text: of

text: X,

text: is

text: the

text: positive

text: Salpeter

text: wavefunction

text: of

text: B

text: meson

text: and

text: P

text: c

text: Pf

text: f

text: is

text: the

text: positive

text: Salpeter

text: wavefunction

text: of

text: X

text: meson,

text: ϕ¯++

text: γ

text: (ϕ++)†γ

text: We

text: will

text: show

text: the

text: Pf

text: Pf

text: Salpeter

text: wavefunctions

text: for

text: the

text: diﬀerent

text: mesons

text: in

text: next

text: section.

text: IV.

text: THE

text: RELATIVISTIC

text: WAVEFUNCTIONS

text: OF

text: PSEUDOSCALAR

text: MESON

text: A.

text: For

text: B

text: meson

text: with

text: quantum

text: numbers

text: JP

text: 0−

text: c

text: The

text: general

text: form

text: for

text: the

text: relativistic

text: wavefunction

text: of

text: pseudoscalar

text: meson

text: B

text: can

text: be

text: written

text: c

text: as

text: P

text: q

text: ϕ

text: f

text: f

text: f

text: q

text: f

text: γ

text: M

text: where

text: M

text: is

text: the

text: mass

text: of

text: the

text: pseudoscalar

text: meson,

text: and

text: f

text: are

text: functions

text: of

text: Due

text: to

text: i

text: the

text: last

text: two

text: equations

text: of

text: Eq.

text: ϕ+−−

text: ϕ−−+

text: we

text: have:

text: f

text: ω

text: ω

text: −f

text: ω

text: f

text: f

text: m

text: ω

text: m

text: ω

text: m

text: ω

text: m

text: ω

text: where

text: m

text: m

text: and

text: ω

text: ω

text: are

text: the

text: masses

text: and

text: the

text: energies

text: of

text: quark

text: and

text: anti-quark

text: in

text: B

text: mesons,

text: q

text: q

text: (q

text: and

text: c

text: The

text: numerical

text: values

text: of

text: radial

text: wavefunctions

text: f

text: f

text: and

text: eigenvalue

text: M

text: can

text: be

text: obtained

text: by

text: solving

text: the

text: ﬁrst

text: two

text: Salpeter

text: equations

text: in

text: Eq.

text: To

text: show

text: the

text: numerical

text: results

text: of

text: wavefunctions

text: explicitly,

text: we

text: plot

text: the

text: wavefunctions

text: of

text: B

text: meson

text: in

text: Fig.

text: According

text: to

text: the

text: c

text: Eq.

text: the

text: relativistic

text: positive

text: wavefunction

text: of

text: pseudoscalar

text: meson

text: B

text: in

text: C.M.S

text: can

text: be

text: c

text: written

text: as

text: P

text: q

text: P

text: b

text: b

text: b

text: q

text: b

text: γ

text: M

text: M

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: 6 the variation of the boundary with various molecular pa- 1.0 rameters (such as (cid:15) , E , M, and (cid:15)), indicating that the s b boundary is insensitive to (cid:15) but depends on other param- s eters. For instance, the area of region II shrinks slightly 0.8 M = 5 with increasing Eb or M and saturates for suﬃciently ǫ = 200 K large Eb or M, while elevating (cid:15) leads to a dramatic in- 0.6 Eb = 1000 K crease in the area of region II in the φ-T plane. The Es = 1000 K above trends are found to apply for the present model ǫs = −1500 K 0.4 φ = 0.001 with Es > 0 K (data not shown). The inﬂuence of the φ = 0.01 sticky bending rigidity parameter E on the boundary φ = 0.05 s in the φ-T plane is presented in Fig. 5, which indicates 0.2 φ = 0.1 φ = 0.3 that the boundary strongly depends on E . In particular, s φ = 0.9 the temperatures marking the boundary signiﬁcantly de- 0.0 100 200 300 400 500 scend with increasing E in the low φ regime but ascend s T in the high φ regime. Interestingly, these boundary lines intersect at a common point with φ = 0.35 and T = 90.7 K for various E . Unfortunately, the physical signiﬁcance s FIG. 6. Temperature variation of the order parameter Φ = ∗ ∗ for the presence of such a point is unclear at present. y /y of self-assembly for various polymer ﬁlling fractions max B. Inﬂuence of stiﬀness of the sticky bonds on the φ. Crosses indicate the positions of the inﬂection points of self-assembly transition the curves. The computations are performed for a melt of linear telechelic chains, where the molecular weight is M = 5, The transition temperature T of self-assembly is an the cohesive interaction energy parameter is (cid:15)−= 200 K, the p important quantity in the thermodynamic description of sticky interaction energy parameter is (cid:15)s = 1500 K, and self-assembly. As implied by the FH type theories of the ordinary and sticky bending rigidity parameters are Eb = self-assembly,46 one common deﬁnition for T employs 1000 K and Es = 1000 K. p the temperature variation of the order parameter Φ of self-assembly. Speciﬁcally, Tp is identiﬁed with the tem- 1.0 perature at which Φ(T, φ = const) exhibits an inﬂection point as a function of T, i.e., the temperature at which 0.8 the second derivative o(cid:12)f Φ with r(cid:12)espect to T vanishes, φ = 0.1 (cid:12) (cid:12) (cid:12) (cid:12) ǫ = 200 K (cid:12) (cid:12) 0.6 Eb = 1000 K ∗ ∂2Φ ∂2y Es = 1000 K = = 0. (10) ǫs = −1500 K ∂T2 φ ∂T2 φ 0.4 M = 5 M = 15 M = 50 Figures 6 and 7 present, respectively, the temperature 0.2 M = 100 variation of Φ for various φ and M, when the other pa- M = 300 rameters are ﬁxed. Since the present model considers M = 1000 0.0 examples where self-assembly of the telechelic chains is 100 200 300 400 500 promoted upon cooling, self-assembly ﬂourishes at low T T where Φ may approach unity. Analysis of Figs. 6 and 7 reveals the presence of an inﬂection point in each curve. Hence, Tp may likewise be identiﬁed in the LCT for self- F∗IG.∗7. Temperature variation of the order parameter Φ = y /y of self-assembly for various molecular weights M. assembling telechelic polymers from the inﬂection points max Crosses indicate the positions of the inﬂection points of the in Φ(T, φ = const), in agreement with previous calcu- curves. The computations are performed for a melt of linear lations for fully ﬂexible telechelic polymers.45 Moreover, telechelic chains, where the polymer ﬁlling fraction is φ = 0.1, the self-assembly transition in models of telechelic poly- the cohesive interaction energy parameter is (cid:15) = 200 K, the − mers is found to be very broad, and the broadness of the sticky interaction energy parameter is (cid:15)s = 1500 K, and transition grows with increasing polymer ﬁlling fraction the ordinary and sticky bending rigidity parameters are Eb = φ or decreasing molecular weight M. These trends also 1000 K and Es = 1000 K. accord with those for fully ﬂexible telechelic polymers.45 Previous work45 extensively examines the dependence on thermodynamic and molecular parameters of the tran- not shown). Figure 8 further reveals the strong inﬂuence sition temperature T for self-assembling telechelic poly- of the sticky bending rigidity parameter E on T . When p s p mers composed of fully ﬂexible chains. For instance, T φ is held constant, T is shown to ﬁrst grow with E , dis- p p s is found to increase with elevating φ or |(cid:15) | but decrease play a maximum, then to reduce with E , and eventually s s with growing M. These general trends also remain in the reach a constant for suﬃciently large E . Our calcula- s present model of semiﬂexible telechelic polymers (data tions also indicate a non-monotonic change of T with E p b Φ Φ ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: variation

text: of

text: the

text: boundary

text: with

text: various

text: molecular

text: pa-

text: rameters

text: (such

text: as

text: E

text: M,

text: and

text: indicating

text: that

text: the

text: s

text: b

text: boundary

text: is

text: insensitive

text: to

text: but

text: depends

text: on

text: other

text: param-

text: s

text: eters.

text: For

text: instance,

text: the

text: area

text: of

text: region

text: II

text: shrinks

text: slightly

text: M

text: with

text: increasing

text: Eb

text: or

text: M

text: and

text: saturates

text: for

text: suﬃciently

text: ǫ

text: K

text: large

text: Eb

text: or

text: M,

text: while

text: elevating

text: leads

text: to

text: a

text: dramatic

text: in-

text: Eb

text: K

text: crease

text: in

text: the

text: area

text: of

text: region

text: II

text: in

text: the

text: φ-T

text: plane.

text: The

text: Es

text: K

text: above

text: trends

text: are

text: found

text: to

text: apply

text: for

text: the

text: present

text: model

text: ǫs

text: K

text: φ

text: with

text: Es

text: K

text: (data

text: not

text: shown).

text: The

text: inﬂuence

text: of

text: the

text: φ

text: sticky

text: bending

text: rigidity

text: parameter

text: E

text: on

text: the

text: boundary

text: φ

text: s

text: in

text: the

text: φ-T

text: plane

text: is

text: presented

text: in

text: Fig.

text: which

text: indicates

text: φ

text: φ

text: that

text: the

text: boundary

text: strongly

text: depends

text: on

text: E

text: In

text: particular,

text: s

text: φ

text: the

text: temperatures

text: marking

text: the

text: boundary

text: signiﬁcantly

text: de-

text: scend

text: with

text: increasing

text: E

text: in

text: the

text: low

text: φ

text: regime

text: but

text: ascend

text: s

text: T

text: in

text: the

text: high

text: φ

text: regime.

text: Interestingly,

text: these

text: boundary

text: lines

text: intersect

text: at

text: a

text: common

text: point

text: with

text: φ

text: and

text: T

text: K

text: for

text: various

text: E

text: Unfortunately,

text: the

text: physical

text: signiﬁcance

text: s

text: FIG.

text: Temperature

text: variation

text: of

text: the

text: order

text: parameter

text: Φ

text: for

text: the

text: presence

text: of

text: such

text: a

text: point

text: is

text: unclear

text: at

text: present.

text: y

text: /y

text: of

text: self-assembly

text: for

text: various

text: polymer

text: ﬁlling

text: fractions

text: max

text: B.

text: Inﬂuence

text: of

text: stiﬀness

text: of

text: the

text: sticky

text: bonds

text: on

text: the

text: φ.

text: Crosses

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text: 9 280 280 (a) (a) 260 260 240 240 220 220 200 180 200 120 (b) 120 (b) Eb = 400 K E = 500 K b E = 600 K 110 110 b 100 100 90 90 ǫ = 150 K 80 M = 8 80 M = 8 E = 500 K ǫ = 175 K ǫ = 200 K b E = 500 K ǫ = 200 K ǫ = −1000 K s s 70 70 0 500 1000 1500 2000 2500 3000 101 102 103 104 |ǫs| Es FIG. 12. (a) Glass transition temperature Tg and (b) isobaric FIG. 13. (a) Glass transition temperature Tg and (b) isobaric fragility parameter m as a function of the absolute sticky fragility parameter m as a function of the sticky bending P | | P interaction energy parameter (cid:15)s for various cohesive interac- rigidity parameter Es for various ordinary bending rigidity tion energy parameters (cid:15). The computations are performed parameters Eb. The computations are performed for a melt of for a melt of linear telechelic chains at a constant pressure of linear telechelic chains at a constant pressure of P = 0.101 325 P = 0.101 325 MPa (i.e., 1 atm), where the molecular weight MPa (i.e., 1 atm), where the molecular weight is M = 8, the is M = 8, and the ordinary and sticky bending rigidity pa- cohesive interaction energy parameter is (cid:15) = 200 K, and the − rameters are Eb = 500 K and Es = 500 K. sticky interaction energy parameter is (cid:15)s = 1000 K. the high temperature activation energy from the empir- Figure 12 displays the dependence of T and m on ical relation ∆µ = 6kBTc.38 The GET then identiﬁes Tg the absolute sticky interaction energy paragmeter |(cid:15)P| for as the temperature at which τ = 100 s. Likewise, the s α various microscopic cohesive interaction energy param- isobaric fragility parameter m is determined from the standard deﬁnition,53 P (cid:12)(cid:12) eters (cid:15). Both Tg and mP increase with the sticky in- (cid:12) teraction strength and tend to saturate for suﬃciently (cid:12) strong sticky interactions. These trends are quite under- m = ∂ log(τα) . (15) standable since an increase in |(cid:15)s| elevates the “eﬀective” P ∂(T /T) molecular weight due to increases in the average degree g P,T=Tg of self-assembly. Growing molecular weight usually leads Illustrative computations of characteristic tempera- to increases in both Tg and mP in polymer melts (see tures and fragility parameters and more details concern- Fig. 14), and hence, both Tg and mP are expected to ing the GET can be found in previous works (e.g., see become larger for longer averaged chain lengths induced Refs. 38 and 54). by stronger sticky interactions. In addition, Tg appears The calculations are performed at a constant pressure to saturate at higher |(cid:15)s| than mP . Apparently, when of P = 0.101 325 MPa (i.e., 1 atm) and use the common |(cid:15)s| becomes large for a telechelic melt, Tg or mP can be parameters z = 6 and Vcell = (2.7)3˚A3. Here, Vcell is identical to that for a polymer melt with (cid:15)s = 0 K but introduced to describe the volume of a single lattice site, with a higher M, deﬁning an eﬀective molecular weight a parameter that is required in order to express the pres- for telechelics with a given sticky energy. This compli- sure in real units. A low molecular weight of M = 8 is cated point, however, is not relevant to the present paper. ﬁrst chosen for our illustrative calculations because the Figure 12 also indicates that a larger (cid:15) results in a higher quantitative inﬂuence of the sticky interaction energy on Tg but a lower mP for a ﬁxed |(cid:15)s|, trends that are the glass-formation becomes more signiﬁcant for smaller M. same as those in polymer melts lacking sticky interac- The inﬂuence of M on glass-formation in the model of tions.50,54–56 telechelic melts is then brieﬂy discussed. The inﬂuence of bending rigidity parameters on glass- T m g P T m g P ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: 2 TYLER HELMUTH 5 3 2 3 7 2 4 4 1 8 5 1 6 Figure 1. Left: a conﬁguration of hard spheres of equal radius in R 2. The adjective hard reﬂects that the interiors are disjoint. Right: R a branched polymer of disks of equal radius in 2. The conﬁguration has an underlying tree structure: neighbouring vertices in the tree correspond to tangent disks. All disks have disjoint interiors. The dotted gray lines indicate the coordinate axes. ≥ article is a constructive proof of dimensional reduction formulas for all d 2; invariance lemmas play a central role in the proof. More precisely, this article establishes that (i) given a central essential complex hyperplane arrangement, there are dimensional reduction formulas from 2d+2 dimensions to 2d dimensions, and (ii) given a central essential real hyperplane arrangement, there are dimensional reduction formulas from d+2 dimensions to d dimensions. These results involve two main objects. The H ﬁrst is a generalization of branched polymers called -polymers, which are an extension of the planar polymers in [3]. The second is a generalization of the pressure of the hard sphere gas: for any central hyperplane arrangement we deﬁne an analogue of the Mayer expansion, which is a power series representation of the pressure of the hard sphere gas. ?? contain the precise H deﬁnitions of -polymers and the generalized pressure, and ?? contains a more precise statement of the theorem. The remainder of this introductory section brieﬂy describes new results and perspectives that follow from these generalized dimensional reduction formulas and their proof. The proof presented in this article yields more information than the non-constructive proof in [1], and the following results are new even in the case of branched polymers. First, we obtain a precise description of the law of d-dimensional projections of (d + 2)-dimensional polymers, see Corollaries 13 and 14. This last corollary can be viewed as a generalization of [2, Theorem 7], which gave a description of 1-dimensional projections of 3-dimensional branched polymers. Secondly, our proof of dimensional reduction extends to some non-spherical bodies, see Theorem 16. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TYLER

text: HELMUTH

text: Figure

text: Left:

label: caption

text: a

text: conﬁguration

text: of

text: hard

text: spheres

text: of

text: equal

text: radius

text: in

text: R

text: The

text: adjective

text: hard

text: reﬂects

text: that

text: the

text: interiors

text: are

text: disjoint.

text: Right:

text: R

text: a

text: branched

text: polymer

text: of

text: disks

text: of

text: equal

text: radius

text: in

text: The

text: conﬁguration

text: has

text: an

text: underlying

text: tree

text: structure:

text: neighbouring

text: vertices

text: in

text: the

text: tree

text: correspond

text: to

text: tangent

text: disks.

text: All

text: disks

text: have

text: disjoint

text: interiors.

text: The

text: dotted

text: gray

text: lines

text: indicate

text: the

text: coordinate

text: axes.

text: article

text: is

text: a

text: constructive

text: proof

text: of

text: dimensional

text: reduction

text: formulas

text: for

text: all

text: d

text: invariance

text: lemmas

text: play

text: a

text: central

text: role

text: in

text: the

text: proof.

text: More

text: precisely,

text: this

text: article

text: establishes

text: that

text: (i)

text: given

text: a

text: central

text: essential

text: complex

text: hyperplane

text: arrangement,

text: there

text: are

text: dimensional

text: reduction

text: formulas

text: from

text: dimensions

text: to

text: dimensions,

text: and

text: (ii)

text: given

text: a

text: central

text: essential

text: real

text: hyperplane

text: arrangement,

text: there

text: are

text: dimensional

text: reduction

text: formulas

text: from

text: dimensions

text: to

text: d

text: dimensions.

text: These

text: results

text: involve

text: two

text: main

text: objects.

text: The

text: H

text: ﬁrst

text: is

text: a

text: generalization

text: of

text: branched

text: polymers

text: called

text: -polymers,

text: which

text: are

text: an

text: extension

text: of

text: the

text: planar

text: polymers

text: in

text: The

text: second

text: is

text: a

text: generalization

text: of

text: the

text: pressure

text: of

text: the

text: hard

text: sphere

text: gas:

text: for

text: any

text: central

text: hyperplane

text: arrangement

text: we

text: deﬁne

text: an

text: analogue

text: of

text: the

text: Mayer

text: expansion,

text: which

text: is

text: a

text: power

text: series

text: representation

text: of

text: the

text: pressure

text: of

text: the

text: hard

text: sphere

text: gas.

text: contain

text: the

text: precise

text: H

text: deﬁnitions

text: of

text: -polymers

text: and

text: the

text: generalized

text: pressure,

text: and

text: contains

text: a

text: more

text: precise

text: statement

text: of

text: the

text: theorem.

text: The

text: remainder

text: of

text: this

text: introductory

text: section

text: brieﬂy

text: describes

text: new

text: results

text: and

text: perspectives

text: that

text: follow

text: from

text: these

text: generalized

text: dimensional

text: reduction

text: formulas

text: and

text: their

text: proof.

text: The

text: proof

text: presented

text: in

text: this

text: article

text: yields

text: more

text: information

text: than

text: the

text: non-constructive

text: proof

text: in

text: and

text: the

text: following

text: results

text: are

text: new

text: even

text: in

text: the

text: case

text: of

text: branched

text: polymers.

text: First,

text: we

text: obtain

text: a

text: precise

text: description

text: of

text: the

text: law

text: of

text: d-dimensional

text: projections

text: of

text: (d

text: polymers,

text: see

text: Corollaries

text: and

text: This

text: last

text: corollary

text: can

text: be

text: viewed

text: as

text: a

text: generalization

text: of

text: Theorem

text: which

text: gave

text: a

text: description

text: of

text: projections

text: of

text: branched

text: polymers.

text: Secondly,

text: our

text: proof

text: of

text: dimensional

text: reduction

text: extends

text: to

text: some

text: non-spherical

text: bodies,

text: see

text: Theorem

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 TYLER HELMUTH (cid:97) R (3.3) dn = Γ , G ⊂ G E(MH) { ∈ R | (cid:107) (cid:107) ≤ ∈ } (3.4) Γ = x dn h (x) 1 if and only if e G , G e 2 with the disjoint union in (3.3) being over all subsets of the ground set of MH. Each summand H in the left-hand side of Equation (3.2) is a spanning set. The integral over any region Γ is zero when G is not a spanning set: G ∈ ∈ in this case there is a normal to a hyperplane e H with e / G, and the { | (cid:107) (cid:107) } integral of 1{(cid:107)he(x)(cid:107)2≤1} over (cid:90)x he(x) 2 > 1 vanishes. The left-hand side of Equation (3(cid:88).2) can th(cid:88)erefore be(cid:89)rewritten as: − (3.5) 1{(cid:107) (cid:107) ≤ } dx G⊂E(MH) H⊂E(M(cid:88)H) ΓG e(cid:88)∈H (cid:90) he(x) 2 1 r(G)=n r(H)=n − | | (3.6) = ( 1) H dx G⊂(cid:88)E(MH) H⊂G ΓG r(G)=n r(H)=n (3.7) = vol(Γ )χ (0). G G ⊂ G E(MH) r(G)=n The ﬁrst equality follows as if H is not a subset of G the integral vanishes. The second equality follows from (2.3). This concludes the manipulations of the left-hand side of Equation (3.2). The second step is to perform the integrals in the right-hand side of Equation (3.2). The idea of how to do this is simple. The volume is computed H by ﬁrst ﬁxing the last d coordinates of the points x in a -polymer, and then i integrating over the ﬁrst two coordinates of each point. The integral over the ﬁrst two coordinates can be expressed as the volume of a generalized planar polymer, and hence can be computed explicitly by an invariance lemma. Lastly we integrate over the last d coordinates. This idea is similar to what was done for branched polymers in three dimensions in [2]. ∈ As in Lemma 11, it will be convenient to write x PH(d+2) as x = (w, y), ∈ R ∈ R where w = (w , w , . . . , w ) 2n, y = (y , y , . . . , y ) dn, and x = ∈ R 1 2 n ∈ 1 2 n i (w , y ) d+2. To each point x PH(d + 2) associate a vector (R(cid:63)(y)) ∈H i i e e of non-negative radii: − (cid:107) (cid:107) ∧ (3.8) R(cid:63)(y)2 = (1 h (y) 2) 0, e e 2

text: TYLER

text: HELMUTH

text: R

text: dn

text: Γ

text: G

text: G

text: E(MH)

text: R

text: Γ

text: x

text: dn

text: h

text: (x)

text: if

text: and

text: only

text: if

text: e

text: G

text: G

text: e

text: with

text: the

text: disjoint

text: union

text: in

text: being

text: over

text: all

text: subsets

text: of

text: the

text: ground

text: set

text: of

text: MH.

text: Each

text: summand

text: H

text: in

text: the

text: left-hand

text: side

text: of

text: Equation

text: is

text: a

text: spanning

text: set.

text: The

text: integral

text: over

text: any

text: region

text: Γ

text: is

text: zero

text: when

text: G

text: is

text: not

text: a

text: spanning

text: set:

text: G

text: in

text: this

text: case

text: there

text: is

text: a

text: normal

text: to

text: a

text: hyperplane

text: e

text: H

text: with

text: e

text: G,

text: and

text: the

text: integral

text: of

text: over

text: he(x)

text: vanishes.

text: The

text: left-hand

text: side

text: of

text: Equation

text: can

text: as:

text: dx

text: G⊂E(MH)

text: ΓG

text: he(x)

text: r(G)=n

text: r(H)=n

text: H

text: dx

text: H⊂G

text: ΓG

text: r(G)=n

text: r(H)=n

text: vol(Γ

text: )χ

text: G

text: G

text: G

text: E(MH)

text: r(G)=n

text: The

text: ﬁrst

text: equality

text: follows

text: as

text: if

text: H

text: is

text: not

text: a

text: subset

text: of

text: G

text: the

text: integral

text: vanishes.

text: The

text: second

text: equality

text: follows

text: from

text: This

text: concludes

text: the

text: manipulations

text: of

text: the

text: left-hand

text: side

text: of

text: Equation

text: The

text: second

text: step

text: is

text: to

text: perform

text: the

text: integrals

text: in

text: the

text: right-hand

text: side

text: of

text: Equation

text: The

text: idea

text: of

text: how

text: to

text: do

text: this

text: is

text: simple.

text: The

text: volume

text: is

text: computed

text: H

text: by

text: ﬁrst

text: ﬁxing

text: the

text: last

text: d

text: coordinates

text: of

text: the

text: points

text: x

text: in

text: a

text: -polymer,

text: and

text: then

text: i

text: integrating

text: over

text: the

text: ﬁrst

text: two

text: coordinates

text: of

text: each

text: point.

text: The

text: integral

text: over

text: the

text: ﬁrst

text: two

text: coordinates

text: can

text: be

text: expressed

text: as

text: the

text: volume

text: of

text: a

text: generalized

text: planar

text: polymer,

text: and

text: hence

text: can

text: be

text: computed

text: explicitly

text: by

text: an

text: invariance

text: lemma.

text: Lastly

text: we

text: integrate

text: over

text: the

text: last

text: d

text: coordinates.

text: This

text: idea

text: is

text: similar

text: to

text: what

text: was

text: done

text: for

text: branched

text: polymers

text: in

text: three

text: dimensions

text: in

text: As

text: in

text: Lemma

text: it

text: will

text: be

text: convenient

text: to

text: write

text: x

text: as

text: x

text: (w,

text: y),

text: R

text: R

text: where

text: w

text: (w

text: w

text: w

text: y

text: (y

text: y

text: y

text: dn,

text: and

text: x

text: R

text: n

text: n

text: i

text: (w

text: y

text: To

text: each

text: point

text: x

text: PH(d

text: associate

text: a

text: vector

text: ∈H

text: i

text: i

text: e

text: e

text: of

text: non-negative

text: radii:

text: h

text: (y)

text: e

text: e


text: DIMENSIONAL REDUCTION FOR GENERALIZED CONTINUUM POLYMERS 21 Alternately, the formula for the partition function in Equation (4.3) can R − × R be rewritten as a hard sphere gas in the upper half space d 1 , and the (cid:107) (cid:107) ≥ + condition x + x 1 can be interpreted as a boundary condition. Since i j 2 the pressure of a hard sphere gas is independent of the boundary conditions, the pressure of this model can be represented as the partition function of branched polymers in d + 2 dimensions by ?? . This is veriﬁed explicitly in ?? . More interestingly, there is a D -polymer representation for the n ↑ R lowest-order ﬁnite volume corrections to ZD as Λ d; this is explained in Λ ?? . Remark. The generating function analysis in these sections follows from general results on exponential Dowling structures [18]; similarly the analysis of signed graphs is a special case of results on gain graphs [19]. We include the analyses for the beneﬁt of readers unfamiliar with these topics. − Writing 1A = 1 1(cid:88)Ac in ((cid:90)4.3) y(cid:88)ields (cid:89) zn − (4.4) ZD(z) = 1{(cid:107) ∓ (cid:107) ≤ } dx, Λ ≥ n! Λn ∈G± ± ∈ xi xj 2 1 n 0 G [n] (ij, ) E(G) ± where G [n] denotes the set of signed graphs on [n]. These are graphs → {±} G together with a signing σ : E(G) . Note that an edge (ij, +) (cid:107) − (cid:107) ≤ corresponds to the constraint x x 1; this convention makes signed i j 2 graphs with all signs + correspond to the ordinary graphs that arise in the non-symmetrized hard-sphere gas. Note that signed graphs may have multiple edges: it is possible for i and j − to be connected by an edge labelled + and an edge labelled . Loops are not permitted. A cycle of a signed graph will refer to a cycle of the underlying − (multi)-graph. This means that if (ij, +) and (ij, ) are edges in a signed graph, the underlying graph contains two copies of the edge ij, and there is a two-cycle that consists of these two edges. C Deﬁnition 17. A cycle in a signed graph (G, σ) is called balanced if the C product of the signs σ(e) of the edges in a cycle is +1. A signed graph (G, σ) is balanced if every cycle in the graph is balanced. A signed graph that is not balanced is unbalanced. A signed graph (G, σ) can be partitioned into two vertex disjoint signed subgraphs (G , σ ) and (G , σ ), the former balanced and the latter unbal- G±b b G±u u anced. Let [n] and [n] denote the sets of balanced and unbalanced b u signed graphs on [n], respectively. Then ##LTLine##

text: DIMENSIONAL

text: REDUCTION

text: FOR

text: GENERALIZED

text: CONTINUUM

text: POLYMERS

text: Alternately,

text: the

text: formula

text: for

text: the

text: partition

text: function

text: in

text: Equation

text: can

text: R

text: R

text: be

text: rewritten

text: as

text: a

text: hard

text: sphere

text: gas

text: in

text: the

text: upper

text: half

text: space

text: d

text: and

text: the

text: condition

text: x

text: x

text: can

text: be

text: interpreted

text: as

text: a

text: boundary

text: condition.

text: Since

text: i

text: j

text: the

text: pressure

text: of

text: a

text: hard

text: sphere

text: gas

text: is

text: independent

text: of

text: the

text: boundary

text: conditions,

text: the

text: pressure

text: of

text: this

text: model

text: can

text: be

text: represented

text: as

text: the

text: partition

text: function

text: of

text: branched

text: polymers

text: in

text: d

text: dimensions

text: by

text: This

text: is

text: veriﬁed

text: explicitly

text: in

text: More

text: interestingly,

text: there

text: is

text: a

text: D

text: -polymer

text: representation

text: for

text: the

text: n

text: R

text: lowest-order

text: ﬁnite

text: volume

text: corrections

text: to

text: ZD

text: as

text: Λ

text: d;

text: this

text: is

text: explained

text: in

text: Λ

text: Remark.

text: The

text: generating

text: function

text: analysis

text: in

text: these

text: sections

text: follows

text: from

text: general

text: results

text: on

text: exponential

text: Dowling

text: structures

text: similarly

text: the

text: analysis

text: of

text: signed

text: graphs

text: is

text: a

text: special

text: case

text: of

text: results

text: on

text: gain

text: graphs

text: We

text: include

text: the

text: analyses

text: for

text: the

text: beneﬁt

text: of

text: readers

text: unfamiliar

text: with

text: these

text: topics.

text: Writing

text: in

text: zn

text: ZD(z)

text: dx,

text: Λ

text: n!

text: Λn

text: ∈G±

text: xi

text: xj

text: n

text: G

text: [n]

text: (ij,

text: E(G)

text: where

text: G

text: [n]

text: denotes

text: the

text: set

text: of

text: signed

text: graphs

text: on

text: [n].

text: These

text: are

text: graphs

text: G

text: together

text: with

text: a

text: signing

text: σ

text: E(G)

text: Note

text: that

text: an

text: edge

text: (ij,

text: corresponds

text: to

text: the

text: constraint

text: x

text: x

text: this

text: convention

text: makes

text: signed

text: i

text: j

text: graphs

text: with

text: all

text: signs

text: correspond

text: to

text: the

text: ordinary

text: graphs

text: that

text: arise

text: in

text: the

text: non-symmetrized

text: hard-sphere

text: gas.

text: Note

text: that

text: signed

text: graphs

text: may

text: have

text: multiple

text: edges:

text: it

text: is

text: possible

text: for

text: i

text: and

text: j

text: to

text: be

text: connected

text: by

text: an

text: edge

text: labelled

text: and

text: an

text: edge

text: labelled

text: Loops

text: are

text: not

text: permitted.

text: A

text: cycle

text: of

text: a

text: signed

text: graph

text: will

text: refer

text: to

text: a

text: cycle

text: of

text: the

text: underlying

text: (multi)-graph.

text: This

text: means

text: that

text: if

text: (ij,

text: and

text: (ij,

text: are

text: edges

text: in

text: a

text: signed

text: graph,

text: the

text: underlying

text: graph

text: contains

text: two

text: copies

text: of

text: the

text: edge

text: ij,

text: and

text: there

text: is

text: a

text: two-cycle

text: that

text: consists

text: of

text: these

text: two

text: edges.

text: C

text: Deﬁnition

text: A

text: cycle

text: in

text: a

text: signed

text: graph

text: (G,

text: σ)

text: is

text: called

text: balanced

text: if

text: the

text: C

text: product

text: of

text: the

text: signs

text: σ(e)

text: of

text: the

text: edges

text: in

text: a

text: cycle

text: is

text: A

text: signed

text: graph

text: (G,

text: σ)

text: is

text: balanced

text: if

text: every

text: cycle

text: in

text: the

text: graph

text: is

text: balanced.

text: A

text: signed

text: graph

text: that

text: is

text: not

text: balanced

text: is

text: unbalanced.

text: A

text: signed

text: graph

text: (G,

text: σ)

text: can

text: be

text: partitioned

text: into

text: two

text: vertex

text: disjoint

text: signed

text: subgraphs

text: (G

text: σ

text: and

text: (G

text: σ

text: the

text: former

text: balanced

text: and

text: the

text: latter

text: unbal-

text: G±b

text: b

text: G±u

text: u

text: anced.

text: Let

text: [n]

text: and

text: [n]

text: denote

text: the

text: sets

text: of

text: balanced

text: and

text: unbalanced

text: b

text: u

text: signed

text: graphs

text: on

text: [n],

text: respectively.

text: Then

text: ##LTLine##


text: 22 TYLER HELMUTH (cid:90) (cid:88) (cid:88) (cid:88) (cid:88) zn (4.5) ZD(z) = w(G)w(H) dx, Λ (cid:81) n≥0 m,(cid:96) m!(cid:96)! Λn G∈Gb±[m] H∈Gu±[(cid:96)] m+(cid:96)=n − where w(G) = ± ∈ 1{(cid:107) ± (cid:107) ≤ }. This is the convolution of two (ij, ) E(G) xi xj 2 1 exponential generating functions, and hence (4.6) ZD(z) = Zu(z)Zb (z), Λ Λ Λ where the subscripts u and b indicated unbalanced and balanced, respec- (cid:90) tively, e.g., (cid:88) (cid:88) zn (4.7) Zu(z) = w(G) dx. Λ ≥ n! ∈G± Λn n 0 G u [n] G± G Lemma 18. The map from [n] to [n] given by forgetting the signing of − b a balanced graph is a 2n 1-to-1 map when restricted to connected graphs. Proof. First we note that this map is well-deﬁned: a balanced graph cannot − contain the edges (ij, +), and (ij, ), as this would be an unbalanced cycle. Forgetting the signing of a balanced graph therefore does yield an ordinary graph. Let G be a connected unsigned graph on [n]. There is a one-to-2n map ∈ {±} given by arbitrarily labelling each vertex i of G with σ , and then i assigning the edge ij the sign σ σ . Each such assignment results in a i j balanced graph: supposing that i is +, trace any cycle containing i. By − construction seeing a edge means the sign of the next vertex is diﬀerent; since the cycle ends at i there must be at least one change of vertex sign − − after observing the ﬁrst . This implies there are an even number of signs, so the cycle is balanced. The lemma follows by counting how many ways distinct signings of vertices can give rise to the same signing of edges. This is 2: for any connected balanced signed graph once the sign of a single vertex is ﬁxed, the sign of every other vertex is determined by σ = σ σ . That this rule signs each ij i j (cid:3) vertex consistently follows from the graph being balanced. It follows that from Lemma 18 that ##LTLine## ##LTLine##

text: TYLER

text: HELMUTH

text: zn

text: ZD(z)

text: w(G)w(H)

text: dx,

text: Λ

text: Λn

text: G∈Gb±[m]

text: where

text: w(G)

text: This

text: is

text: the

text: convolution

text: of

text: two

text: (ij,

text: E(G)

text: xi

text: xj

text: exponential

text: generating

text: functions,

text: and

text: hence

text: ZD(z)

text: Zu(z)Zb

text: (z),

text: Λ

text: Λ

text: Λ

text: where

text: the

text: subscripts

text: u

text: and

text: b

text: indicated

text: unbalanced

text: and

text: balanced,

text: respec-

text: tively,

text: e.g.,

text: zn

text: Zu(z)

text: w(G)

text: dx.

text: Λ

text: n!

text: ∈G±

text: Λn

text: n

text: G

text: u

text: [n]

text: G±

text: G

text: Lemma

text: The

text: map

text: from

text: [n]

text: to

text: [n]

text: given

text: by

text: forgetting

text: the

text: signing

text: of

text: b

text: a

text: balanced

text: graph

text: is

text: a

text: map

text: when

text: restricted

text: to

text: connected

text: graphs.

text: Proof.

text: First

text: we

text: note

text: that

text: this

text: map

text: is

text: well-deﬁned:

text: a

text: balanced

text: graph

text: cannot

text: contain

text: the

text: edges

text: (ij,

text: and

text: (ij,

text: as

text: this

text: would

text: be

text: an

text: unbalanced

text: cycle.

text: Forgetting

text: the

text: signing

text: of

text: a

text: balanced

text: graph

text: therefore

text: does

text: yield

text: an

text: ordinary

text: graph.

text: Let

text: G

text: be

text: a

text: connected

text: unsigned

text: graph

text: on

text: [n].

text: There

text: is

text: a

text: map

text: given

text: by

text: arbitrarily

text: labelling

text: each

text: vertex

text: i

text: of

text: G

text: with

text: σ

text: and

text: then

text: i

text: assigning

text: the

text: edge

text: ij

text: the

text: sign

text: σ

text: σ

text: Each

text: such

text: assignment

text: results

text: in

text: a

text: i

text: j

text: balanced

text: graph:

text: supposing

text: that

text: i

text: is

text: trace

text: any

text: cycle

text: containing

text: i.

text: By

text: construction

text: seeing

text: a

text: edge

text: means

text: the

text: sign

text: of

text: the

text: next

text: vertex

text: is

text: diﬀerent;

text: since

text: the

text: cycle

text: ends

text: at

text: i

text: there

text: must

text: be

text: at

text: least

text: one

text: change

text: of

text: vertex

text: sign

text: after

text: observing

text: the

text: ﬁrst

text: This

text: implies

text: there

text: are

text: an

text: even

text: number

text: of

text: signs,

text: so

text: the

text: cycle

text: is

text: balanced.

text: The

text: lemma

text: follows

text: by

text: counting

text: how

text: many

text: ways

text: distinct

text: signings

text: of

text: vertices

text: can

text: give

text: rise

text: to

text: the

text: same

text: signing

text: of

text: edges.

text: This

text: is

text: for

text: any

text: connected

text: balanced

text: signed

text: graph

text: once

text: the

text: sign

text: of

text: a

text: single

text: vertex

text: is

text: ﬁxed,

text: the

text: sign

text: of

text: every

text: other

text: vertex

text: is

text: determined

text: by

text: σ

text: σ

text: σ

text: That

text: this

text: rule

text: signs

text: each

text: ij

text: i

text: j

text: vertex

text: consistently

text: follows

text: from

text: the

text: graph

text: being

text: balanced.

text: It

text: follows

text: that

text: from

text: Lemma

text: that

text: ##LTLine##

text: ##LTLine##


text: 10 TYLER HELMUTH ∼ H R C ∈ H ∈ C We ﬁrst deﬁne -polymers in 2d (cid:88)= d. Given e , x dn, deﬁne ∈ C h (x) d as in (2.1): e n h (x) = a x , e i i i=1 where a x is the usual pointwise complex-scalar multiplication of a vector ∈ C i i x d. In the next deﬁnition notice that polymers are deﬁned without i modding out by translations, in contrast to what was done for branched polymers in ?? . Deﬁnition 6. Deﬁne { ∈ C | (cid:107) (cid:107) } Tan(e) = x nd h (x) = R , e 2 e { ∈ C | (cid:107) (cid:107) } Dis(e) = x nd h (x) > R . e 2 e H { } The space of 2d-dimensional -polymers with radii R ∈H is the set ⊂ C (cid:97) e e PH(2d) dn deﬁned by (2.5) PH(2d) = (cid:92) PHS(2(cid:92)d), ∈B S (MH) ∩ (2.6) PHS(2d) = Tan(e) Dis(e). ∈ ∈ e S e/S There is a natural probability measure on the space PH(2d), deﬁned as S − (cid:55)→ follows. Injectively map PHS(2d) into ( (2d 1))n by x φ(x) = (φ (x)) ∈ , e e S where φ (x) is the unit vector in the direction h (x). The fact that this is e e an injection on PHS follows from the hypothesis that S is a base. With this map in hand deﬁne { | ∈ } (2.7) vol(PHS(2d)) = vol( φ(x) x PHS(2d) ), where the volume measure on the right-hand side of Equation (2.7) is the S − n-fold product of surface measure on (2d 1). As each of the sets PHS(2d) are disjoint, the volume of PHS(2d) is the sum of the volumes of each PHS(2d). After normalization, this gives a probability measure on PH(2d). Note that this measure agrees with the measure on branched polymers given in ?? . In general an explicit integral formula for this measure is given by replacing trees in Equation (1.1) with bases S of MH, and the function IT with its H BP analogue IHS . The law of -polymers will be thought of as the law of the ∈ C locations of the points x d. H i To construct -polymers in d dimensions for d odd a further condition ∈ H is needed. An arrangement is called complexiﬁed if each hyperplane e

text: TYLER

text: HELMUTH

text: H

text: R

text: C

text: H

text: C

text: We

text: ﬁrst

text: deﬁne

text: -polymers

text: in

text: d.

text: Given

text: e

text: x

text: dn,

text: deﬁne

text: C

text: h

text: (x)

text: d

text: as

text: in

text: e

text: n

text: h

text: (x)

text: a

text: x

text: e

text: i

text: i

text: where

text: a

text: x

text: is

text: the

text: usual

text: pointwise

text: complex-scalar

text: multiplication

text: of

text: a

text: vector

text: C

text: i

text: i

text: x

text: d.

text: In

text: the

text: next

text: deﬁnition

text: notice

text: that

text: polymers

text: are

text: deﬁned

text: without

text: i

text: modding

text: out

text: by

text: translations,

text: in

text: contrast

text: to

text: what

text: was

text: done

text: for

text: branched

text: polymers

text: in

text: Deﬁnition

text: Deﬁne

text: C

text: Tan(e)

text: x

text: nd

text: h

text: (x)

text: R

text: e

text: e

text: C

text: Dis(e)

text: x

text: nd

text: h

text: (x)

text: R

text: e

text: e

text: H

text: The

text: space

text: of

text: -polymers

text: with

text: radii

text: R

text: ∈H

text: is

text: the

text: set

text: C

text: e

text: e

text: dn

text: deﬁned

text: by

text: ∈B

text: S

text: (MH)

text: Tan(e)

text: Dis(e).

text: e

text: S

text: e/S

text: There

text: is

text: a

text: natural

text: probability

text: measure

text: on

text: the

text: space

text: deﬁned

text: as

text: S

text: follows.

text: Injectively

text: map

text: into

text: by

text: x

text: φ(x)

text: (φ

text: (x))

text: e

text: e

text: S

text: where

text: φ

text: (x)

text: is

text: the

text: unit

text: vector

text: in

text: the

text: direction

text: h

text: (x).

text: The

text: fact

text: that

text: this

text: is

text: e

text: e

text: an

text: injection

text: on

text: PHS

text: follows

text: from

text: the

text: hypothesis

text: that

text: S

text: is

text: a

text: base.

text: With

text: this

text: map

text: in

text: hand

text: deﬁne

text: vol(

text: φ(x)

text: x

text: where

text: the

text: volume

text: measure

text: on

text: the

text: right-hand

text: side

text: of

text: Equation

text: is

text: the

text: S

text: n-fold

text: product

text: of

text: surface

text: measure

text: on

text: As

text: each

text: of

text: the

text: sets

text: are

text: disjoint,

text: the

text: volume

text: of

text: is

text: the

text: sum

text: of

text: the

text: volumes

text: of

text: each

text: After

text: normalization,

text: this

text: gives

text: a

text: probability

text: measure

text: on

text: Note

text: that

text: this

text: measure

text: agrees

text: with

text: the

text: measure

text: on

text: branched

text: polymers

text: given

text: in

text: In

text: general

text: an

text: explicit

text: integral

text: formula

text: for

text: this

text: measure

text: is

text: given

text: by

text: replacing

text: trees

text: in

text: Equation

text: with

text: bases

text: S

text: of

text: MH,

text: and

text: the

text: function

text: IT

text: with

text: its

text: H

text: BP

text: analogue

text: IHS

text: The

text: law

text: of

text: -polymers

text: will

text: be

text: thought

text: of

text: as

text: the

text: law

text: of

text: the

text: C

text: locations

text: of

text: the

text: points

text: x

text: d.

text: H

text: i

text: To

text: construct

text: -polymers

text: in

text: d

text: dimensions

text: for

text: d

text: odd

text: a

text: further

text: condition

text: H

text: is

text: needed.

text: An

text: arrangement

text: is

text: called

text: complexiﬁed

text: if

text: each

text: hyperplane

text: e