text: 2 MATTHEW A WILLIAMS the p-adic valuation and parity of the numbers S(m, n), and to formulate and prove a generalization of Wilson’s Theorem (Proposition 14). 1. THE REAL SOLUTIONS OF S(m, n, z) = S(m, n). We ﬁrst observe a classical formula from combinatorics [1]: Theorem 1. The number of ways of partitioning a set of m elements into n nonempty sub- sets is given by (cid:88) (cid:18) (cid:19) 1 n n − − (1) S(m, n) = ( 1)k(n k)m. n! k k=0 It was discovered independently(cid:88)by(cid:18)Ru(cid:19)iz [1,2] that 1 n n − − ∈ R (2) S(n, n) = ( 1)k(z k)n (z ). n! k k=0 Indeed, (2) is an evident consequence of the Mean Value Theorem. Katsuura [1] no- ticed that (2) holds even if z is an arbitrary complex value, as did Vladimir Dragovic (in- dependently). The following proposition extends (2) to the case d > 0. ∈ C ≤ Proposition 1. The equation S(m, n, z) =(cid:18)S(m(cid:19), n) holds for all z if d 0, and for only (cid:88) the roots of the polynomial d m − − P (z) = S(m j, n)( z)j (m,n) j j=1 in the case d > 0. Proof. L(cid:88)et z(cid:18)∈(cid:19)C. One easily veriﬁes tha(cid:88)t (cid:18) (cid:19) (cid:88) (cid:18) (cid:19) 1 n n (−1)k(z − k)m = 1 n (cid:88)n(cid:18)(−(cid:19)1)(cid:34)k m(cid:88)m(cid:18) (cid:19)zj(−k)m−j (cid:35) n! k n! k j k=0 k=0 j=0 − m m 1 n n − − − − = ( 1)d ( 1)n kkm j ( z)j. j n! k j=0 k=0 In view o(cid:18)f Th(cid:19)e(cid:34)orem 1,(cid:18)we(cid:19)have by symm(cid:35)etry (cid:18) (cid:19) (cid:88) (cid:88) (cid:88) − m m 1 n n − − − − − d m − − ( 1)d ( 1)n kkm j ( z)j = ( 1)d S(m j, n)( z)j j n! k j j=0 k=0 j=0 − (3) = ( 1)d(S(m, n) + P (z)). (m,n) Hence by (3) − (4) S(m, n) = S(m, n, z) P (z). (m,n) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: UCXYVC+Utopia-Regular

text: MATTHEW

text: A

text: WILLIAMS

text: the

text: p-adic

text: valuation

text: and

text: parity

text: of

text: the

text: numbers

text: S(m,

text: n),

text: and

text: to

text: formulate

text: and

text: prove

text: a

text: generalization

text: of

text: Wilson’s

text: Theorem

text: (Proposition

label: section

text: THE

text: REAL

text: SOLUTIONS

text: OF

text: S(m,

text: n,

text: z)

text: S(m,

text: n).

text: We

text: ﬁrst

text: observe

text: a

text: classical

text: formula

text: from

text: combinatorics

text: Theorem

font: MSOBBC+Utopia-Bold

text: The

font: ZUETKN+Utopia-Italic

text: number

text: of

text: ways

text: of

text: partitioning

text: a

text: set

text: of

text: m

text: elements

text: into

text: n

text: nonempty

text: sub-

text: sets

text: is

text: given

text: by

text: n

text: n

text: S(m,

text: n)

text: k)m.

text: n!

text: k

text: It

text: was

text: discovered

text: that

text: n

text: n

text: R

text: S(n,

text: n)

text: k)n

text: (z

text: n!

text: k

text: Indeed,

text: is

text: an

text: evident

text: consequence

text: of

text: the

text: Mean

text: Value

text: Theorem.

text: Katsuura

text: no-

text: ticed

text: that

text: holds

text: even

text: if

text: z

text: is

text: an

text: arbitrary

text: complex

text: value,

text: as

text: did

text: Vladimir

text: Dragovic

text: (in-

text: dependently).

text: The

text: following

text: proposition

text: extends

text: to

text: the

text: case

text: d

text: C

text: Proposition

text: The

text: equation

text: S(m,

text: n,

text: z)

text: n)

text: holds

text: for

text: all

text: z

text: if

text: d

text: and

text: for

text: only

text: the

text: roots

text: of

text: the

text: polynomial

text: d

text: m

text: P

text: (z)

text: S(m

text: j,

text: n)(

text: z)j

text: (m,n)

text: j

text: in

text: the

text: case

text: d

text: Proof.

text: One

text: easily

text: veriﬁes

text: n

text: n

text: k)m

text: n

text: n!

text: k

text: n!

text: k

text: j

text: m

text: m

text: n

text: n

text: kkm

text: j

text: z)j.

text: j

text: n!

text: k

text: In

text: view

text: by

text: m

text: m

text: n

text: n

text: d

text: m

text: kkm

text: j

text: z)j

text: S(m

text: j,

text: n)(

text: z)j

text: j

text: n!

text: k

text: j

text: n)

text: P

text: (z)).

text: (m,n)

text: Hence

text: by

text: S(m,

text: n)

text: S(m,

text: n,

text: z)

text: P

text: (z).

text: (m,n)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:18) (cid:19) STIRLING FUNCTIONS AND A GENERALIZATION OF WILSON’S THEOREM 7 (cid:48) (cid:48) N ∈ O − (cid:48) S(j) m, N , = 0 (j , j < m N ). (cid:18) (cid:19) 2 (cid:18) (cid:19) and by Proposition 1 (cid:48) (cid:48) − (cid:48) (cid:48) N − − (cid:48) (cid:48) (cid:48) N S(m N ) m, N , = ( 1)m N (m) − (cid:48)S N , N , = (m) − (cid:48) > 0. 2 m N 2 m N (cid:48) (cid:18) (cid:19) Thus S(m, N , z) may be written as(cid:88) − (cid:48) m N (cid:48) (cid:48) (cid:48) (cid:48) 2 S(2j)(m, N , N /2) − N 2j S(m, N , z) = z (2j)! 2 j=0 (cid:48) where each coefﬁcient of the above expansion at z = N /2 is positive. Since m is arbitrary, (cid:3) the Proposition follows by induction. FIGURE 3. Plots of S(6, 4, z), S(8, 4, z) and S(10, 4, z). Note that each function achieves its global minimum (a positive value) at z = 2. Corollary 2. Let k ∈ Z . Then, |P(k) (z)| > 0 holds for every z ∈ R if (d − k) ∈ E. + (m,n) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: STIRLING

text: FUNCTIONS

text: AND

text: A

text: GENERALIZATION

text: OF

text: WILSON’S

text: THEOREM

text: N

text: O

text: S(j)

text: m,

text: N

text: (j

text: j

text: m

text: N

text: and

text: by

text: Proposition

text: N

text: N

text: S(m

text: N

text: m,

text: N

text: N

text: (m)

text: N

text: N

text: (m)

text: m

text: N

text: m

text: N

text: Thus

text: S(m,

text: N

text: z)

text: may

text: be

text: written

text: m

text: N

text: N

text: N

text: N

text: S(m,

text: N

text: z)

text: z

text: where

text: each

text: coefﬁcient

text: of

text: the

text: above

text: expansion

text: at

text: z

text: N

text: is

text: positive.

text: Since

text: m

text: is

text: arbitrary,

text: the

text: Proposition

text: follows

text: by

text: induction.

text: FIGURE

text: Plots

text: of

text: z),

text: z)

text: and

text: z).

text: Note

text: that

text: each

text: function

text: achieves

text: its

text: global

text: minimum

text: (a

text: positive

text: value)

text: at

text: z

text: Corollary

text: Let

text: k

text: Z

text: Then,

text: |P(k)

text: (z)|

text: holds

text: for

text: every

text: z

text: R

text: if

text: (d

text: k)

text: E.

text: (m,n)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 1 Introduction In the previous paper[1], the present authors studied the proposal[2] that a d + 1 dimen- sional induced metric can be constructed from a d dimensional ﬁeld theory using gradient ﬂow[3–6], applying the method to the O(N) ϕ4 model. We have shown that in the large N limit the induced metric becomes classical and describes Euclidean Anti-de-Sitter (AdS) space in both ultra-violet (UV) and infra-red (IR) limits of the ﬂow direction. The method proposed in Ref. [2] may provide an alternative way to understand the AdS/CFT (or more generally Gravity/Gauge theory) correspondence[7], and the result in Ref. [1] might be related to the correspondence between O(N) vector models in d-dimensions and (generalized) gravity theories in d + 1 dimensions[8]. To further investigate a possible connection between Ref. [1] and Ref. [8] at the quantum level, one must calculate, for example, the anomalous dimension of the O(N) invariant operator φ2(x), which requires the next-to-leading order (NLO) of the 1/N expansion for the ﬂow equation to evaluate necessary quantum corrections. Since the method employed in Refs. [1, 2] is a speciﬁc one adopted for the large N limit, some systematic way to solve the ﬂow equation in the 1/N expansion is needed. In this paper, we employ the Schwinger-Dyson equation (SDE) to solve the ﬂow equation in the 1/N expansion for the O(N) invariant ϕ4 model in d dimensions. Using this method we explicitly calculate the 2-pt and 4-pt functions at the NLO. As the ﬁrst application of the NLO calculations, we deﬁne a running coupling from the connected 4-pt function of ﬂowed ﬁelds, which runs with the ﬂow time t such that t = 0 ∞ corresponds to the UV limit while t = is the IR limit. This property establishes that the ﬂow equation can be interpreted as a renormalization group transformation. In particular at d = 3, we show that the running coupling so deﬁned has not only the asymptotic free UV ﬁxed point but also a Wilson-Fisher IR ﬁxed point for the massless case. As the second application, we investigate the NLO correction to the induced metric in 3 + 1 dimensions from the massless scalar model in 3 dimensions. In the massless limit, the whole 4-dimensional space becomes AdS at the leading order, as shown in Ref. [1]. The NLO corrections give a small perturbation to the metric, which makes the space asymptotically ∞ AdS in UV (t = 0) and IR (t = ) limits only. A remarkable thing is that, while the NLO corrections do not change the AdS radius in the UV limit, the AdS radius is reduced by the NLO correction in the IR limit, which corresponds to the Wilson-Fisher IR ﬁxed point of the original theory. In other words, a nontrivial ﬁxed point in the ﬁeld theory leads to a change of the AdS radius in the geometry at the NLO. The induced metric at NLO 2

text: Introduction

text: In

text: the

text: previous

text: the

text: present

text: authors

text: studied

text: the

text: that

text: a

text: d

text: dimen-

text: sional

text: induced

text: metric

text: can

text: be

text: constructed

text: from

text: a

text: d

text: dimensional

text: ﬁeld

text: theory

text: using

text: gradient

text: applying

text: the

text: method

text: to

text: the

text: O(N)

text: ϕ4

text: model.

text: We

text: have

text: shown

text: that

text: in

text: the

text: large

text: N

text: limit

text: the

text: induced

text: metric

text: becomes

text: classical

text: and

text: describes

text: Euclidean

text: Anti-de-Sitter

text: (AdS)

text: space

text: in

text: both

text: ultra-violet

text: (UV)

text: and

text: infra-red

text: (IR)

text: limits

text: of

text: the

text: ﬂow

text: direction.

text: The

text: method

text: proposed

text: in

text: Ref.

text: may

text: provide

text: an

text: alternative

text: way

text: to

text: understand

text: the

text: AdS/CFT

text: (or

text: more

text: generally

text: Gravity/Gauge

text: theory)

text: and

text: the

text: result

text: in

text: Ref.

text: might

text: be

text: related

text: to

text: the

text: correspondence

text: between

text: O(N)

text: vector

text: models

text: in

text: d-dimensions

text: and

text: (generalized)

text: gravity

text: theories

text: in

text: d

text: To

text: further

text: investigate

text: a

text: possible

text: connection

text: between

text: Ref.

text: and

text: Ref.

text: at

text: the

text: quantum

text: level,

text: one

text: must

text: calculate,

text: for

text: example,

text: the

text: anomalous

text: dimension

text: of

text: the

text: O(N)

text: invariant

text: operator

text: which

text: requires

text: the

text: next-to-leading

text: order

text: (NLO)

text: of

text: the

text: expansion

text: for

text: the

text: ﬂow

text: equation

text: to

text: evaluate

text: necessary

text: quantum

text: corrections.

text: Since

text: the

text: method

text: employed

text: in

text: Refs.

text: is

text: a

text: speciﬁc

text: one

text: adopted

text: for

text: the

text: large

text: N

text: limit,

text: some

text: systematic

text: way

text: to

text: solve

text: the

text: ﬂow

text: equation

text: in

text: the

text: expansion

text: is

text: needed.

text: In

text: this

text: paper,

text: we

text: employ

text: the

text: Schwinger-Dyson

text: equation

text: (SDE)

text: to

text: solve

text: the

text: ﬂow

text: equation

text: in

text: the

text: expansion

text: for

text: the

text: O(N)

text: invariant

text: ϕ4

text: model

text: in

text: d

text: dimensions.

text: Using

text: this

text: method

text: we

text: explicitly

text: calculate

text: the

text: and

text: functions

text: at

text: the

text: NLO.

text: As

text: the

text: ﬁrst

text: application

text: of

text: the

text: NLO

text: calculations,

text: we

text: deﬁne

text: a

text: running

text: coupling

text: from

text: the

text: connected

text: function

text: of

text: ﬂowed

text: ﬁelds,

text: which

text: runs

text: with

text: the

text: ﬂow

text: time

text: t

text: such

text: that

text: t

text: corresponds

text: to

text: the

text: UV

text: limit

text: while

text: t

text: is

text: the

text: IR

text: limit.

text: This

text: property

text: establishes

text: that

text: the

text: ﬂow

text: equation

text: can

text: be

text: interpreted

text: as

text: a

text: renormalization

text: group

text: transformation.

text: In

text: particular

text: at

text: d

text: we

text: show

text: that

text: the

text: running

text: coupling

text: so

text: deﬁned

text: has

text: not

text: only

text: the

text: asymptotic

text: free

text: UV

text: ﬁxed

text: point

text: but

text: also

text: a

text: Wilson-Fisher

text: IR

text: ﬁxed

text: point

text: for

text: the

text: massless

text: case.

text: As

text: the

text: second

text: application,

text: we

text: investigate

text: the

text: NLO

text: correction

text: to

text: the

text: induced

text: metric

text: in

text: dimensions

text: from

text: the

text: massless

text: scalar

text: model

text: in

text: dimensions.

text: In

text: the

text: massless

text: limit,

text: the

text: whole

text: space

text: becomes

text: AdS

text: at

text: the

text: leading

text: order,

text: as

text: shown

text: in

text: Ref.

text: The

text: NLO

text: corrections

text: give

text: a

text: small

text: perturbation

text: to

text: the

text: metric,

text: which

text: makes

text: the

text: space

text: asymptotically

text: AdS

text: in

text: UV

text: (t

text: and

text: IR

text: (t

text: limits

text: only.

text: A

text: remarkable

text: thing

text: is

text: that,

text: while

text: the

text: NLO

text: corrections

text: do

text: not

text: change

text: the

text: AdS

text: radius

text: in

text: the

text: UV

text: limit,

text: the

text: AdS

text: radius

text: is

text: reduced

text: by

text: the

text: NLO

text: correction

text: in

text: the

text: IR

text: limit,

text: which

text: corresponds

text: to

text: the

text: Wilson-Fisher

text: IR

text: ﬁxed

text: point

text: of

text: the

text: original

text: theory.

text: In

text: other

text: words,

text: a

text: nontrivial

text: ﬁxed

text: point

text: in

text: the

text: ﬁeld

text: theory

text: leads

text: to

text: a

text: change

text: of

text: the

text: AdS

text: radius

text: in

text: the

text: geometry

text: at

text: the

text: NLO.

text: The

text: induced

text: metric

text: at

text: NLO


text: (cid:20) (cid:18) (cid:19) (cid:21) −(cid:3) (cid:20) u 2(cid:18) (cid:19) (cid:21) 0 = ( + µ2) + 1 + Γ(0) K(x , x ; x , x ) x 1 2 3 4 1 3! N u (cid:20)(cid:18) (cid:19) 2 2 (cid:21) + Γ(x ) 2Γ(x )Γ(x ) + 1 + K(x , x ; x , x ) + K(x , x ; x , x ) 12 13 14 1 1 3 4 1 3 1 4 3! N N u 2 2 + 1 + H(x , x ; x , x ; x , x ) + H(x , x ; x , x ; x , x ) 1 1 1 2 3 4 1 2 1 3 1 4 3!N N N 2u + [Γ(x )K(x , x ; x , x ) + Γ(x )K(x , x ; x , x )] . (A13) 13 1 2 1 4 14 1 2 3 1 3!N A.2 The leading order in the 1/N expansion We introduce the 1/N expansion as (cid:88) (cid:88) ∞ ∞ − − Γ(x ) = N iΓ (x ), K(x , x ; x , x ) = N iK (x , x ; x , x ), (A14) 12 i 12 1 2 3 4 i 1 2 3 4 (cid:88) i=0 i=0 ∞ − and so on, together with µ2 = N iµ2. i i=0 At the leading order (LO) of the 1/N expansion, the eq. (A12) in momentum space becomes (cid:18) (cid:90) (cid:19) (cid:90) (cid:101) (cid:101) (cid:101) u 1 = p2 + µ2 + Dq Γ (q) Γ (p), Γ (x) = Dp Γ (p) eipx, (A15) 0 6 0 0 0 0 which can easily be solved as (cid:101) 1 u Γ (p) = , m2 = µ2 + Z(m), (A16) 0 p2 + m2 0 6 ≥ where m 0 is the renormalized mass and Z(m) is given in eq. (6). Thus the 2-pt function at the LO becomes (cid:90) − (cid:104) (cid:105) δab eip(x y) ϕa(x)ϕb(y) = Dp . (A17) N p2 + m2 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: u

text: K(x

text: x

text: x

text: x

text: x

text: N

text: u

text: Γ(x

text: )Γ(x

text: K(x

text: x

text: x

text: x

text: K(x

text: x

text: x

text: x

text: N

text: N

text: u

text: H(x

text: x

text: x

text: x

text: x

text: x

text: H(x

text: x

text: x

text: x

text: x

text: x

text: N

text: N

text: [Γ(x

text: )K(x

text: x

text: x

text: x

text: Γ(x

text: )K(x

text: x

text: x

text: x

text: The

text: leading

text: order

text: in

text: the

text: expansion

text: We

text: introduce

text: the

text: expansion

text: as

text: Γ(x

text: N

text: iΓ

text: (x

text: K(x

text: x

text: x

text: x

text: N

text: iK

text: (x

text: x

text: x

text: x

text: i

text: i

text: and

text: so

text: on,

text: together

text: with

text: µ2

text: N

text: i

text: At

text: the

text: leading

text: order

text: (LO)

text: of

text: the

text: expansion,

text: the

text: eq.

text: in

text: momentum

text: space

text: becomes

text: u

text: µ2

text: Dq

text: Γ

text: (q)

text: Γ

text: (p),

text: Γ

text: (x)

text: Dp

text: Γ

text: (p)

text: eipx,

text: which

text: can

text: easily

text: be

text: solved

text: as

text: u

text: Γ

text: (p)

text: µ2

text: Z(m),

text: where

text: m

text: is

text: the

text: renormalized

text: mass

text: and

text: Z(m)

text: is

text: given

text: in

text: eq.

text: Thus

text: the

text: function

text: at

text: the

text: LO

text: becomes

text: δab

text: eip(x

text: y)

text: ϕa(x)ϕb(y)

text: Dp

text: N

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:16) (cid:17) − Z u u Z = 1 1 , µ2 = 1 + B(0) Z m2 + C, (A36) R N 1 6 1 6 where (cid:34) (cid:35) (cid:90) − DQ dB(Q2) 6 + B(0) C = + 2 u . (A37) 6 + B(Q2) dm2 Q2 + m2 u The renormalization condition for the coupling, which ﬁrst appears at the NLO of the 1/N − expansion, is given by G (Q2 = s) = u (s)/3, so that u (s) is regarded as the renormalized 0 r r coupling at the scale s. Eq. (A20) thus leads to u u (s) = , (A38) r 1 + uB(s) 6 ≥ where B(Q2) is divergent at d 4. Therefore the renormalized coupling goes to zero as (cid:39) 6 → → ∞ u (s) 0, Λ (A39) r B(s) ≥ ≥ at d 4. This indicates the triviality of the ϕ4 theory at d 4. A.5 Renormalization constants We here explicitly evaluate the renormalization constants. A.5.1 d = 1 At d = 1, µ2 is ﬁnite as (cid:18) (cid:19) 0 1 Λ Z(m) = arctan (A40) πm m is ﬁnite, and the coupling is also ﬁnite and nonzero since 1 (cid:39) 1 · · · → ∞ B(Q2) = + , Q2 , (A41) m(Q2 + 4m2) mQ2 25 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Z

text: u

text: u

text: Z

text: µ2

text: Z

text: C,

text: R

text: N

text: where

text: DQ

text: C

text: u

text: u

text: The

text: renormalization

text: condition

text: for

text: the

text: coupling,

text: which

text: ﬁrst

text: appears

text: at

text: the

text: NLO

text: of

text: the

text: expansion,

text: is

text: given

text: by

text: G

text: s)

text: u

text: so

text: that

text: u

text: (s)

text: is

text: regarded

text: as

text: the

text: renormalized

text: r

text: r

text: coupling

text: at

text: the

text: scale

text: s.

text: Eq.

text: thus

text: leads

text: to

text: u

text: u

text: (s)

text: r

text: uB(s)

text: where

text: is

text: divergent

text: at

text: d

text: Therefore

text: the

text: renormalized

text: coupling

text: goes

text: to

text: zero

text: as

text: u

text: (s)

text: Λ

text: r

text: B(s)

text: at

text: d

text: This

text: indicates

text: the

text: triviality

text: of

text: the

text: ϕ4

text: theory

text: at

text: d

text: Renormalization

text: constants

text: We

text: here

text: explicitly

text: evaluate

text: the

text: renormalization

text: constants.

text: d

text: At

text: d

text: µ2

text: is

text: ﬁnite

text: as

text: Λ

text: Z(m)

text: arctan

text: πm

text: m

text: is

text: ﬁnite,

text: and

text: the

text: coupling

text: is

text: also

text: ﬁnite

text: and

text: nonzero

text: since

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ˙ where F means a t-derivative of F . Then, the SDE (20) becomes F˙ (t, p) −u f = Γ (t), (B5) 0 F (t, p) 6 which tells us that F (t, p) is independent of p, so we put F (t, p) = F (t). The above equation is thus reduced to −u − F˙ (t) = f F 3(t)e 2µ2ftζ0(t), (B6) 6 where ζ (t) is deﬁne(cid:90)d in eq. (24), whose solution is given by 0 − u t − ≡ − ζ(t) F 2(t) = 1 + f dsζ0(s)e 2µ2fs e 2µ2ft , ζ(t) = ζ0(t) + ∆(t) (B7) 3 Z(m ) 0 f where m is deﬁned in eq. (6) and (cid:32) (cid:33) f (cid:90) (cid:0) (cid:1) − − ∆(t) = e2tµ2f Z(m ) − Z(m) + Dp p2 + m2f e2tµ2f e 2tp2 . (B8) f p2 + m2 p2 + µ2 f → ∞ In the case of the interacting ﬂow with u > 0, µ2 negatively diverges as Z(m ) + →f ∞ f f in the continuum limit at d > 1 or as u + in the NLSM limit. In these limits, ∆(t) f vanishes as (cid:0) (cid:1) − lim ∆(t) (cid:39) −m2fζ0(t) ζ˙0(t)/2 + O 1/µ4 (B9) µ2→−∞ µ2 f f f for t > 0. In the case of free ﬂow (u = 0), we simply have F (t) = 1. f We then obtain  (cid:90)     (cid:112)    − − Z(m ) e p2(t1+t2)eip(x1 x2) (cid:54) f Dp , u = 0  (cid:90) ζ(t1)ζ(t2) p2 + m2 f  Γ0(12) =  . (B10) − − e (p2+µ2f)(t1+t2)eip(x1 x2) Dp , u = 0 p2 + m2 f 32 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: F

text: means

text: a

text: t-derivative

text: of

text: F

text: Then,

text: the

text: SDE

text: becomes

text: F˙

text: (t,

text: p)

text: −u

text: f

text: Γ

text: (t),

text: F

text: (t,

text: p)

text: which

text: tells

text: us

text: that

text: F

text: (t,

text: p)

text: is

text: independent

text: of

text: p,

text: so

text: we

text: put

text: F

text: (t,

text: p)

text: F

text: (t).

text: The

text: above

text: equation

text: is

text: thus

text: reduced

text: to

text: −u

text: F˙

text: (t)

text: f

text: F

text: where

text: ζ

text: (t)

text: is

text: in

text: eq.

text: whose

text: solution

text: is

text: given

text: by

text: u

text: t

text: ζ(t)

text: F

text: f

text: e

text: ζ(t)

text: ∆(t)

text: Z(m

text: f

text: where

text: m

text: is

text: deﬁned

text: in

text: eq.

text: and

text: f

text: ∆(t)

text: Z(m

text: Z(m)

text: Dp

text: e

text: f

text: µ2

text: f

text: In

text: the

text: case

text: of

text: the

text: interacting

text: ﬂow

text: with

text: u

text: µ2

text: negatively

text: diverges

text: as

text: Z(m

text: →f

text: f

text: f

text: in

text: the

text: continuum

text: limit

text: at

text: d

text: or

text: as

text: u

text: in

text: the

text: NLSM

text: limit.

text: In

text: these

text: limits,

text: ∆(t)

text: f

text: vanishes

text: as

text: lim

text: ∆(t)

text: O

text: µ2→−∞

text: µ2

text: f

text: f

text: f

text: for

text: t

text: In

text: the

text: case

text: of

text: free

text: ﬂow

text: (u

text: we

text: simply

text: have

text: F

text: (t)

text: f

text: We

text: then

text: obtain

text: Z(m

text: e

text: f

text: Dp

text: u

text: f

text: e

text: Dp

text: u

text: f

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:48)| (cid:48) | | X(τ, τ 12; 34) = X(τ , τ 43; 21), X(τ, 0 12; 34) = 0, (B17) | | | Y (τ 12; 34) = Y (τ 12; 43), Y (0 12; 34) = 0. (B18) | Since g(12; 34 12; 34) agrees with the amputated connected 4-pt function in the d dimensional theory at τ = 0 (i = 1, 2, 3, 4), we obtain i | − 2 Z( p , p , p , p ) = δˆ | , (B19) 1 2 3 4 6/u + B(0 p ) 34 | where B(t Q) is deﬁned in eq. (31). Then one can easily check that g satisﬁes the required symmetries | | | | g(12; 34 12; 34) = g(21; 34 21; 34) = g(12; 43 12; 43) = g(34; 12 34; 12). (B20) B.2.1 Solution for Y Terms which depend only on t in eq. (B15) can be written as 1 (cid:34) (cid:35) | −u (cid:90)− − ∂tY (t 21; 34) = 3f F 2(t)e 2tµ2f (p21 + m2)et(p21 p22)δˆ − | × | e t(q12+q22)Y (t q1, q2; 34) ρ(t 34) + Dq Dq , (B21) 1 2 (q2 + m2)(q2 + m2) 1 2 | where ρ(t 34) is deﬁned in eq. (35). To solve (cid:90)this equation, we set | t − | Y (t 21; 34) = δˆ(p2 + m2) ds es(p21 p22)ψ(s 34), (B22) 1 0 satisfying eq. (B18). Eq. (B21) is reduc(cid:20)ed to (cid:90) (cid:21) | −u − | t | | ψ(t 34) = f F 2(t)e 2tµ2f ρ(t 34) + ds K(t, s p34)ψ(s 34) , (B23) 3 0 34 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: X(τ,

text: τ

text: X(τ

text: τ

text: X(τ,

text: Y

text: (τ

text: Y

text: (τ

text: Y

text: Since

text: agrees

text: with

text: the

text: amputated

text: connected

text: function

text: in

text: the

text: d

text: dimensional

text: theory

text: at

text: τ

text: (i

text: we

text: obtain

text: i

text: Z(

text: p

text: p

text: p

text: p

text: δˆ

text: p

text: where

text: B(t

text: Q)

text: is

text: deﬁned

text: in

text: eq.

text: Then

text: one

text: can

text: easily

text: check

text: that

text: g

text: satisﬁes

text: the

text: required

text: symmetries

text: Solution

text: for

text: Y

text: Terms

text: which

text: depend

text: only

text: on

text: t

text: in

text: eq.

text: can

text: be

text: written

text: as

text: −u

text: ∂tY

text: (t

text: F

text: e

text: (t

text: ρ(t

text: Dq

text: Dq

text: where

text: ρ(t

text: is

text: deﬁned

text: in

text: eq.

text: To

text: solve

text: equation,

text: we

text: set

text: t

text: Y

text: (t

text: ds

text: satisfying

text: eq.

text: Eq.

text: is

text: to

text: −u

text: t

text: ψ(t

text: f

text: F

text: ρ(t

text: ds

text: K(t,

text: s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: which shows ψ does not depend on p , p , where K is deﬁned in eq. (34). Since 1 2 − u F 2(t)e 2tµ2f = u Z(m )/ζ(t) goes to inﬁnity in the continuum limit at t > 0 and d > 1 or f f f → ∞ in the NLSM limit u , eq. (32) must hold in either of the two limits. f B.2.2 Solution for X We next consider the solution for X. Terms depending on both t and t in eq. (B15), 1 3 ↔ and thereafter replacing t by t and interchanging p p , give(cid:90)s 3 2 1 2 | −u − − ∂t1X(t1, t2 12; 34) = 6f F 2(t1)e 2t1µ2f (p22 + m2)et1(p22 p21)δˆ Dq1Dq2 − × e t1(q12+q22) { | | } 2X(t , t q , q ; 34) + Y (t 43; q , q ) ,(B24) (q2 + m2)(q2 + m2) 1 2 1 2 2 1 2 1 2 where (cid:90) | t − | Y (t 43; q1, q2) = (2π)dδ(p34 + q12)(p23 + m2) ds es(p23 p34)ψ(s q1, q2). (B25) 0 We deﬁne | − − | ∂t2∂t1X(t1, t2 12; 34) = δˆ(p22 + m2)(p23 + m2)et1(p22 p21)et2(p23 p24)β(t1, t2 12; 34),(B26) | | | where properties of X imply β(t , t 12; 34) = β(t , t 43; 21) and β(t, 0 12; 34) = 1 2 2 1 | β(0, t 12; 34) = 0. Then the above equation becomes (cid:90) (cid:90) (cid:104) β(t1, t2|12; 34) = −uf F 2(t1)e−2t1µ2f g(t1, t2|p34) + 2 (cid:105)t1 ds1 Dq1Dq2(2π)dδ(q12 + p34) 6 0 − − − × e (t1+s1)q12 (t1 s1)q22 | β(s , t q , q ; 34) , (B27) q2 + m2 1 2 1 2 1 where (cid:90) − | e t1(q12+q22) | g(t , t Q) = Dq Dq (2π)dδ(q + Q) ψ(t q , q ). (B28) 1 2 1 2 12 (q2 + m2)(q2 + m2) 2 1 2 1 2 35 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: which

text: shows

text: ψ

text: does

text: not

text: depend

text: on

text: p

text: p

text: where

text: K

text: is

text: deﬁned

text: in

text: eq.

text: Since

text: u

text: F

text: u

text: Z(m

text: )/ζ(t)

text: goes

text: to

text: inﬁnity

text: in

text: the

text: continuum

text: limit

text: at

text: t

text: and

text: d

text: or

text: f

text: f

text: f

text: in

text: the

text: NLSM

text: limit

text: u

text: eq.

text: must

text: hold

text: in

text: either

text: of

text: the

text: two

text: limits.

text: f

text: Solution

text: for

text: X

text: We

text: next

text: consider

text: the

text: solution

text: for

text: X.

text: Terms

text: depending

text: on

text: both

text: t

text: and

text: t

text: in

text: eq.

text: and

text: thereafter

text: replacing

text: t

text: by

text: t

text: and

text: interchanging

text: p

text: p

text: −u

text: F

text: e

text: t

text: q

text: q

text: Y

text: (t

text: q

text: q

text: where

text: t

text: Y

text: (t

text: ds

text: We

text: deﬁne

text: where

text: properties

text: of

text: X

text: imply

text: β(t

text: t

text: β(t

text: t

text: and

text: β(t,

text: t

text: Then

text: the

text: above

text: equation

text: becomes

text: −uf

text: F

text: e

text: β(s

text: t

text: q

text: q

text: where

text: e

text: g(t

text: t

text: Q)

text: Dq

text: Dq

text: Q)

text: ψ(t

text: q

text: q

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:3)− φa(t, x) = et( µ2f)ϕa(x). (7) (cid:54) We therefore consider the interacting ﬂow (u = 0) hereafter unless otherwise stated. f The above ﬂow equation leads to the Schwinger-Dyson equation (SDE)[9] as (cid:104) O(cid:105) −u (cid:104) O(cid:105) ≡ ∂ − (cid:3) − Dfφa(z) = f φa(z)φ2(z) , Df ( µ2), (8) z 6 z ∂t f O (cid:104)O(cid:105) where z = (t, x), is an arbitrary operator and the expectation value should be calculated in d dimensions as (cid:90) (cid:90) (cid:104)O (cid:105) ≡ 1 D O − D − (ϕ) [ ϕ] (ϕ)e S(µ2,u), Z = [ ϕ] e S(µ2,u). (9) (cid:89) Z − 2n 1 O If we take = φai(z ) the SDE becomes i (cid:88) i=1 Df Γaa1···a2n−1(z, z , · · · , z − ) = − uf Γabba1···a2n−1(z, z, z, z , · · · , z − ), (10) z 2n 1 2n 1 6N2 2n+2 1 2n 1 b where Γ is the n-point function, deﬁned by n (cid:89) n ··· · · · − (cid:104) (cid:105) ≡ · · · Γa1 an(z , , z ) = Nn 1 φai(z ) Γ [12 n], (11) n 1 n i n i=1 which is analogous to the d dimensional counterpart in eq. (A3). We consider only the symmetric phase in this paper, where Γ − = 0 for all positive integers n. 2n 1 We consider the next-to-leading order of the 1/N expansion, so that the following two SDE’s need to be considered. (cid:88) − u f f (cid:88) D Γ [12] = Γ [1bb2], (12) 1 2 6N2 4 b − u f f D Γ [1234] = Γ [1bb234], (13) 1 4 6N2 6 b 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: φa(t,

text: x)

text: et(

font: default

text: We

text: therefore

text: consider

text: the

text: interacting

text: ﬂow

text: (u

text: hereafter

text: unless

text: otherwise

text: stated.

text: f

text: The

text: above

text: ﬂow

text: equation

text: leads

text: to

text: the

text: Schwinger-Dyson

text: equation

text: as

text: −u

text: Dfφa(z)

text: f

text: Df

text: z

text: z

text: ∂t

text: f

text: O

text: where

text: z

text: (t,

text: x),

text: is

text: an

text: arbitrary

text: operator

text: and

text: the

text: expectation

text: value

text: should

text: be

text: calculated

text: in

text: d

text: dimensions

text: as

text: D

text: O

text: D

text: (ϕ)

text: ϕ]

text: (ϕ)e

text: Z

text: ϕ]

text: e

text: Z

text: O

text: If

text: we

text: take

text: φai(z

text: the

text: SDE

text: becomes

text: i

text: Df

text: z

text: z

text: uf

text: z,

text: z,

text: z

text: z

text: z

text: b

text: where

text: Γ

text: is

text: the

text: n-point

text: function,

text: deﬁned

text: by

text: n

text: n

text: an(z

text: z

text: Nn

text: φai(z

text: Γ

text: n],

text: n

text: n

text: i

text: n

text: which

text: is

text: analogous

text: to

text: the

text: d

text: dimensional

text: counterpart

text: in

text: eq.

text: We

text: consider

text: only

text: the

text: symmetric

text: phase

text: in

text: this

text: paper,

text: where

text: Γ

text: for

text: all

text: positive

text: integers

text: n.

text: We

text: consider

text: the

text: next-to-leading

text: order

text: of

text: the

text: expansion,

text: so

text: that

text: the

text: following

text: two

text: SDE’s

text: need

text: to

text: be

text: considered.

text: u

text: f

text: f

text: D

text: Γ

text: Γ

text: b

text: u

text: f

text: f

text: D

text: Γ

text: Γ

text: b

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Similarly (cid:90) (cid:90) (cid:90) (cid:90) √ − R(2)(t) = 16 2π9 Dp Dq 1 √dx x √dy T| | G(Q2x)G(Q2y)e 2p2 e(p2−q2)(x−y). κ x y (T + Q )2 p2 0 0 (D50) Doing the q2 integ(cid:90)rations ﬁrst, we h(cid:90)ave √ ∞ ∞ R(2)(t) = 64 2π (cid:90) dQ (cid:90)QT dpe−2p2 (cid:90) κ (T + Q)2 p 0 0 (D51) × 1 x (Q+p)2 − − dx dy G(Q2x2) G(Q2y2) e(p2 q2)(x2 y2)dq2. − 0 0 (Q p)2 (cid:90) The q2 integral can be done analytically and we ﬁnd √ ∞ QT (2) (2) R (t) = 128 2π dQ ρ (Q), (D52) κ κ (T + Q)2 0 where (cid:90) (cid:90) (cid:90) ∞ − dp − Q z ew2 z2 2p − ρ(2)(Q) = e 2p2 dz dw G(z2) G(w2) − sinh (z2 w2). (D53) κ p z2 w2 Q 0 0 0 (cid:90) (cid:90) (cid:90) (2) Thus ρ = O(Q) for small Q, while κ ∞ ∞ ∼ 1 − z − ρ(2)(Q) dp e 2p2 dz dw G(z2) G(w2)ew2 z2 (D54) κ Q −∞ 0 0 (cid:90) (cid:90) for large Q, and ∞ ≡ ∞ z − r(2) R(2)( ) = 128π dz dw G(z2) G(w2)ew2 z2 = 0.45846. (D55) κ κ 0 0 (3) D.4.5 The R contribution κ (cid:90) (3) For R we ﬁnd κ √ ∞ QT (3) (3) R (t) = 32 2π dQ ρ (Q) (D56) κ κ (T + Q)2 0 48 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Similarly

text: Dp

text: Dq

text: √dx

text: x

text: √dy

text: T|

text: κ

text: x

text: y

text: (T

text: Q

text: Doing

text: the

text: ﬁrst,

text: we

text: 2π

text: dQ

text: κ

text: (T

text: p

text: x

text: dx

text: dy

text: (Q

text: The

text: integral

text: can

text: be

text: done

text: analytically

text: and

text: we

text: ﬁnd

text: QT

text: R

text: (t)

text: 2π

text: dQ

text: ρ

text: (Q),

text: κ

text: κ

text: (T

text: where

text: dp

text: Q

text: z

text: e

text: dz

text: dw

text: sinh

text: κ

text: p

text: Q

text: Thus

text: ρ

text: O(Q)

text: for

text: small

text: Q,

text: while

text: κ

text: z

text: dp

text: e

text: dz

text: dw

text: κ

text: Q

text: for

text: large

text: Q,

text: and

text: z

text: dz

text: dw

text: κ

text: κ

text: The

text: R

text: contribution

text: κ

text: For

text: R

text: we

text: ﬁnd

text: κ

text: QT

text: R

text: (t)

text: 2π

text: dQ

text: ρ

text: (Q)

text: κ

text: κ

text: (T

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: with (cid:90) (cid:90) (cid:90) (cid:90) ∞ − 1 1 − (Q+p)2 e q2(x2+y2) ρ(3)(Q) = dx dy pdp e 2p2+p2(x2+y2)G(Q2x2)G(Q2y2) dq2. κ − q2 0 0 0 (Q p)2 (D57) After rescaling (cid:90) (cid:90) (cid:90) (cid:18) (cid:19) ∞ 1 Q Q − p ρ(3)(Q) = dz dw G(z2) G(w2) pdp e 2p2Z , z2 + w2 , (D58) κ Q2 Q 0 0 0 (cid:90) where − 1+ε e Aξ2 ≈ − → Z(ε, A) = 2eAε2 dξ 4εe A, ε 0. (D59) | − | ξ 1 ε (3) Thus ρ (Q) = O(Q) for small Q, while κ (cid:90) (cid:90) (cid:90) ∞ ∞ ∞ ρ(3)(Q) ∼ 4 dz (cid:114)dw(cid:18)G(cid:90)(z2) G(w2) p(cid:19)2dp e−2p2−z2−w2 Q3 0 0 0 ∞ (D60) π − 2 1 = dz G(z2)e z2 , 8 Q3 0 for large Q, which leads to ≡ ∞ (3) (3) r R ( ) = 0. (D61) κ κ Thus the total relative correction is negative: − (1) (2) (3) r = r + r + r + r = 0.41869. (D62) b κ κ κ 49 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: with

text: e

text: dx

text: dy

text: pdp

text: e

text: κ

text: (Q

text: After

text: rescaling

text: Q

text: Q

text: p

text: dz

text: dw

text: pdp

text: e

text: κ

text: Q

text: where

text: e

text: Z(ε,

text: A)

text: dξ

text: A,

text: ε

text: ξ

text: ε

text: Thus

text: ρ

text: (Q)

text: O(Q)

text: for

text: small

text: Q,

text: while

text: κ

text: dz

text: π

text: dz

text: for

text: large

text: Q,

text: which

text: leads

text: to

text: r

text: R

text: κ

text: κ

text: Thus

text: the

text: total

text: relative

text: correction

text: is

text: negative:

text: r

text: r

text: r

text: r

text: r

text: b

text: κ

text: κ

text: κ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − |{ } − g(t) = 3gˆ(t, t; t, t p )t2 d/2, (37) sym { } ∼ where p is given by p2t = 3∆/4 (i = 1 4) and p2 t = p2 t = ∆ (p = p + p ), sym i 12 34 ij i j − which is the symmetric point for d > 2, and t2 d/2 is introduced to make the coupling dimensionless. Here ∆ is an arbitrary dimensionless constant but we can take ∆ = 1 without → loss of generality by the rescaling t ∆t. Explicitly we have |{ } |{ } |{ } |{ } ˆ ˆ ˆ gˆ(t, t; t, t p ) = 4X(t, t p ) + 4Y (t p ) + Z( p ), (38) sym sym sym sym O O O ˆ ˆ ˆ where we remove δ by deﬁning = δ(cid:90) for (cid:90)= g, X, Y, Z, and Xˆ(t1, t2|12; 34) = (p22 + m2)((cid:90)p23 + m2) t1 ds1 t2 ds2 es1(p22−p21)es2(p23−p24)ω(s1, s2|p34), (39) 0 0 | t − | Yˆ (t 12; 34) = (p2 + m2) ds es(p22 p21)ψ(s 34), (40) 2 0 | −1 u Zˆ( 12; 34) = | . (41) 3 1 + uB(0 p ) 6 34 3.2 Free ﬂow |{ } |{ } ˆ For simplicity, we ﬁrst consider the free ﬂow, where gˆ(t, t; t, t p ) = Z( p ). sym sym Taking ∆ = 1, the running coupling is given by − ut2 d/2 g(t) = , (42) u 1 + B (1/t) 6 | where B(p2) = B(0 p). 3.2.1 d = 2 In 2-dimensions, we obtain (cid:18) (cid:19) ut g(t) = , (43) √ ut − √ 1 1 + tanh 1 6π 1 + 4m2t 1 + 4m2t 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: g(t)

text: t;

text: t,

text: t

text: p

text: sym

text: where

text: p

text: is

text: given

text: by

text: (i

text: and

text: t

text: t

text: (p

text: p

text: p

text: sym

text: i

text: ij

text: i

text: j

text: which

text: is

text: the

text: symmetric

text: point

text: for

text: d

text: and

text: is

text: introduced

text: to

text: make

text: the

text: coupling

text: dimensionless.

text: Here

text: is

text: an

text: arbitrary

text: dimensionless

text: constant

text: but

text: we

text: can

text: take

text: without

text: loss

text: of

text: generality

text: by

text: the

text: rescaling

text: t

text: ∆t.

text: Explicitly

text: we

text: have

text: ˆ

text: ˆ

text: ˆ

text: gˆ(t,

text: t;

text: t,

text: t

text: p

text: t

text: p

text: (t

text: p

text: Z(

text: p

text: sym

text: sym

text: sym

text: sym

text: O

text: O

text: O

text: ˆ

text: ˆ

text: ˆ

text: where

text: we

text: remove

text: δ

text: by

text: deﬁning

text: for

text: g,

text: X,

text: Y,

text: Z,

text: and

text: t

text: Yˆ

text: (t

text: ds

text: −1

text: u

text: Zˆ(

text: p

text: Free

text: ﬂow

text: ˆ

text: For

text: simplicity,

text: we

text: ﬁrst

text: consider

text: the

text: free

text: ﬂow,

text: where

text: gˆ(t,

text: t;

text: t,

text: t

text: p

text: Z(

text: p

text: sym

text: sym

text: Taking

text: the

text: running

text: coupling

text: is

text: given

text: by

text: g(t)

text: u

text: B

text: where

text: p).

text: d

text: In

text: we

text: obtain

text: ut

text: g(t)

text: ut

text: tanh

text: 6π

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: with y , z , z¯ and y , z , z¯ replaced by (3.8) and (3.7). The delta function has support 2 2 2 3 3 3 on √ √ R = ε 2 + ε cos2 θ y + εy cos θ . (3.11) 1 1 So far we have not taken any limit on the masses 2(1 + ε)m, m, m of the three particles. → Now let us consider the near extremal limit ε 0. In this limit the three momenta m pˆ i i become c√ollinear and the corresponding points (y , z , z¯ ) in H become coincident. To leading i i i 3 order in ε, the solution of R can be approximated by √ O R = 2εy + (ε) . (3.12) 1 ∼ In the near extremal limit we have R 0, hence the three bulk points y , z , z¯ in H i i i 3 become coincident as commented above. We can then bring the three bulk-to-boundary propagators outside the R, θ, φ-integral. Next, by a simple power counting of R, we ﬁnd that the three-point amplitude of the conformal primary wavefunction is zero when ε = 0, which is related to the fact that the the phase space vanishes for marginal decay process. We should proceed to the subleading order in the near extremal expansion to obtain a nonzero (cid:90) (cid:90) (cid:89) answer, which is9 ∞ A i(2(cid:90)π)5λ d(cid:90)y 3 ˜(w , w¯ ) = 1 dz dz¯ (cid:0) G (y ,(cid:1)z , z¯ ; w , w¯ ) i i m4 y3 1 1 ∆i 1 1 1 i i ∞ 0 1 i=1 × 1 (cid:32)(cid:90) (cid:90) π − O (cid:33) dRR2 dθ sin(cid:89)θ δ R2 2εy2 + (ε) (3.13) y 1 1 0 0 √ ∞ i211 π5λ dy 3 O 2 1 = ε dz dz¯ G (y , z , z¯ ; w , w¯ ) + (ε) m4 y3 1 1 ∆i 1 1 1 i i 0 1 i=1 √ − − − − i292 π6λΓ(∆1+∆2+∆3 2)Γ(∆1+∆2 ∆3 )Γ(∆1 ∆2+∆3 )Γ( ∆1+∆2+∆3 ) ε O = 2| − | 2 − | − 2| − | 2− | − + (ε) , m4Γ(∆1)Γ(∆2)Γ(∆3) w1 w2 ∆1+∆2 ∆3 w2 w3 ∆2+∆3 ∆1 w3 w1 ∆3+∆1 ∆2 where the term in the parentheses in the second to last line is precisely the tree-level three- point Witten diagram in H , which was evaluated in [28]. Hence the near extremal massive 3 three-point amplitude takes the form of the three-point function of scalar primaries with conformal dimensions ∆ in a two-dimensional CFT. i → 9Note that in the near extremal limit ε 0, the upper bound θ∗(y , R) of the angular coordinate becomes 1 π on the support of the delta function. 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: with

text: y

text: z

text: z¯

text: and

text: y

text: z

text: z¯

text: replaced

text: by

text: and

text: The

text: delta

text: function

text: has

text: support

text: on

text: R

text: ε

text: ε

text: θ

text: y

text: εy

text: cos

text: θ

text: So

text: far

text: we

text: have

text: not

text: taken

text: any

text: limit

text: on

text: the

text: masses

text: ε)m,

text: m,

text: m

text: of

text: the

text: three

text: particles.

text: Now

text: let

text: us

text: consider

text: the

text: near

text: extremal

text: limit

text: ε

text: In

text: this

text: limit

text: the

text: three

text: momenta

text: m

text: pˆ

text: i

text: i

text: become

text: c√ollinear

text: and

text: the

text: corresponding

text: points

text: (y

text: z

text: z¯

text: in

text: H

text: become

text: coincident.

text: To

text: leading

text: i

text: i

text: i

text: order

text: in

text: ε,

text: the

text: solution

text: of

text: R

text: can

text: be

text: approximated

text: by

text: O

text: R

text: (ε)

text: In

text: the

text: near

text: extremal

text: limit

text: we

text: have

text: R

text: hence

text: the

text: three

text: bulk

text: points

text: y

text: z

text: z¯

text: in

text: H

text: i

text: i

text: i

text: become

text: coincident

text: as

text: commented

text: above.

text: We

text: can

text: then

text: bring

text: the

text: three

text: bulk-to-boundary

text: propagators

text: outside

text: the

text: R,

text: θ,

text: φ-integral.

text: Next,

text: by

text: a

text: simple

text: power

text: counting

text: of

text: R,

text: we

text: ﬁnd

text: that

text: the

text: three-point

text: amplitude

text: of

text: the

text: conformal

text: primary

text: wavefunction

text: is

text: zero

text: when

text: ε

text: which

text: is

text: related

text: to

text: the

text: fact

text: that

text: the

text: the

text: phase

text: space

text: vanishes

text: for

text: marginal

text: decay

text: process.

text: We

text: should

text: proceed

text: to

text: the

text: subleading

text: order

text: in

text: the

text: near

text: extremal

text: expansion

text: to

text: obtain

text: a

text: nonzero

text: answer,

text: which

text: A

text: ˜(w

text: w¯

text: dz

text: dz¯

text: G

text: (y

text: z¯

text: w

text: w¯

text: i

text: i

text: ∆i

text: i

text: i

text: π

text: O

text: dθ

text: δ

text: (ε)

text: y

text: π5λ

text: dy

text: O

text: ε

text: dz

text: dz¯

text: G

text: (y

text: z

text: z¯

text: w

text: w¯

text: (ε)

text: ∆i

text: i

text: i

text: ∆3

text: )Γ(

text: ε

text: O

text: 2−

text: (ε)

text: ∆3

text: ∆1

text: ∆2

text: where

text: the

text: term

text: in

text: the

text: parentheses

text: in

text: the

text: second

text: to

text: last

text: line

text: is

text: precisely

text: the

text: tree-level

text: three-

text: point

text: Witten

text: diagram

text: in

text: H

text: which

text: was

text: evaluated

text: in

text: Hence

text: the

text: near

text: extremal

text: massive

text: three-point

text: amplitude

text: takes

text: the

text: form

text: of

text: the

text: three-point

text: function

text: of

text: scalar

text: primaries

text: with

text: conformal

text: dimensions

text: in

text: a

text: two-dimensional

text: CFT.

text: i

text: that

text: in

text: the

text: near

text: extremal

text: limit

text: ε

text: the

text: upper

text: bound

text: θ∗(y

text: R)

text: of

text: the

text: angular

text: coordinate

text: becomes

text: π

text: on

text: the

text: support

text: of

text: the

text: delta

text: function.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Acknowledgements We are grateful to T. Dumitrescu, P. Mitra, M. Pate, B. Schwab, D. Simmons-Duﬃn, and A. Zhiboedov for useful conversations. This work was supported in part by NSF grant 1205550. S.P. is supported by the NSF and by the Hertz Foundation through a Harold and Ruth Newman Fellowship. S.H.S. is supported by the National Science Foundation grant PHY-1606531. A Klein-Gordon Inner Product In this section we evaluate the Klein-Gordon inner product between two conformal pri- C mary solutions with the same mass m and generic complex weights ∆ . SL(2, ) implies (cid:54) ∗ 1,2 ∗ this must vanish for ∆ = ∆ , while some kind of delta function is expected at ∆ = ∆ . 1 2 1 2 The Klein-Gordon inner product between two outgoing wavefunctions φ+ (Xµ; w , w¯ ) (cid:90) ∆1,m 1 1 and φ+ (Xµ; w , w¯ )(cid:2)evaluated on the slice X0 = 0 is ∆2,m 2 2 (cid:3) − ∗ (φ+, φ+) = i d3X(cid:126) φ+ (Xµ; w , w¯ )∂ φ+ (Xµ; w , w¯ ) 1 2 (cid:32) (cid:90) ∆1,m (cid:90) 1 1(cid:33) X0 ∆2,m 2 2 (cid:89)− ∗ ∂ φ+ (Xµ; w , w¯ )φ+ (Xµ; w , w¯ ) X0 ∆1,m 1 1 ∆2,m 2 2 ∞ 2 (cid:18) − dy (cid:19) (cid:18) (cid:19) (cid:18) ∗ (cid:19) =(2π)3m 2 i dz dz¯ G (y , z , z¯ ; w , w¯ )G (y , z , z¯ ; w , w¯ ) y3 i i ∆1 1 1 1 1 1 ∆2 2 2 2 2 2 |i=1| 0 i | | − − | | − − | | (cid:90) (cid:90) × 1 + y2 + z 2 1 + y2 + z 2 z − z 1 y2 z 2 − 1 y2 z 2 1 1 + 2 2 δ(2) 1 2 δ 1 1 2 2 2y 2y y y 2y 2y 1 ∞ 2 1 2 1 2 − dy ∗ =2(2π)3m 2 dzdz¯G (y, z, z¯; w , w¯ )G (y, z, z¯; w , w¯ ) . (A.1) y3 ∆1 1 1 ∆2 2 2 0 Using the Feynman trick, (cid:90) − − − 1 Γ(a + b) 1 αa 1(1 α)b 1 = dα − , (A.2) AaBb Γ(a)Γ(b) (αA + (1 α)B)a+b 0 we can perform the integrals in y, z, z¯ to obtain (cid:90) ∗− ∗ (φ+, φ+) = 2(2π)3m−2 πΓ(∆1+∆2∗2 |2)Γ−(∆1+2|∆2 ) ∗ 1 dαα∆1−2∆∗2 −1(1 − α)−∆12+∆∗2 −1 . (A.3) 1 2 2Γ(∆1)Γ(∆2) w1 w2 ∆1+∆2 0 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Acknowledgements

text: We

text: are

text: grateful

text: to

text: T.

text: Dumitrescu,

text: P.

text: Mitra,

text: M.

text: Pate,

text: B.

text: Schwab,

text: D.

text: Simmons-Duﬃn,

text: and

text: A.

text: Zhiboedov

text: for

text: useful

text: conversations.

text: This

text: work

text: was

text: supported

text: in

text: part

text: by

text: NSF

text: grant

text: S.P.

text: is

text: supported

text: by

text: the

text: NSF

text: and

text: by

text: the

text: Hertz

text: Foundation

text: through

text: a

text: Harold

text: and

text: Ruth

text: Newman

text: Fellowship.

text: S.H.S.

text: is

text: supported

text: by

text: the

text: National

text: Science

text: Foundation

text: grant

text: A

text: Klein-Gordon

text: Inner

text: Product

text: In

text: this

text: section

text: we

text: evaluate

text: the

text: Klein-Gordon

text: inner

text: product

text: between

text: two

text: conformal

text: pri-

text: C

text: mary

text: solutions

text: with

text: the

text: same

text: mass

text: m

text: and

text: generic

text: complex

text: weights

text: implies

text: this

text: must

text: vanish

text: for

text: while

text: some

text: kind

text: of

text: delta

text: function

text: is

text: expected

text: at

text: The

text: Klein-Gordon

text: inner

text: product

text: between

text: two

text: outgoing

text: wavefunctions

text: φ+

text: (Xµ;

text: w

text: w¯

text: and

text: φ+

text: (Xµ;

text: w

text: w¯

text: on

text: the

text: slice

text: is

text: (φ+,

text: φ+)

text: i

text: φ+

text: (Xµ;

text: w

text: w¯

text: φ+

text: (Xµ;

text: w

text: w¯

text: φ+

text: (Xµ;

text: w

text: w¯

text: )φ+

text: (Xµ;

text: w

text: w¯

text: dy

text: i

text: dz

text: dz¯

text: G

text: (y

text: z

text: z¯

text: w

text: w¯

text: )G

text: (y

text: z

text: z¯

text: w

text: w¯

text: i

text: i

text: ∆1

text: ∆2

text: i

text: z

text: z

text: z

text: z

text: z

text: z

text: δ

text: y

text: y

text: dy

text: dzdz¯G

text: (y,

text: z,

text: z¯;

text: w

text: w¯

text: )G

text: (y,

text: z,

text: z¯;

text: w

text: w¯

text: ∆1

text: ∆2

text: Using

text: the

text: Feynman

text: trick,

text: Γ(a

text: b)

text: αa

text: α)b

text: dα

text: AaBb

text: Γ(a)Γ(b)

text: (αA

text: α)B)a+b

text: we

text: can

text: perform

text: the

text: integrals

text: in

text: y,

text: z,

text: z¯

text: to

text: obtain

text: (φ+,

text: φ+)

text: πΓ(∆1+∆2∗2

text: dαα∆1−2∆∗2

text: α)−∆12+∆∗2

text: −1

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [8] C. Cheung, A. de la Fuente, and R. Sundrum, “4D Scattering Amplitudes and Asymptotic Symmetries from 2D CFT,” 1609.00732. [9] S. He, Y.-t. Huang, and C. Wen, “Loop Corrections to Soft Theorems in Gauge Theories and Gravity,” JHEP 12 (2014) 115, 1405.1410. [10] M. Bianchi, S. He, Y.-t. Huang, and C. Wen, “More on Soft Theorems: Trees, Loops and Strings,” Phys. Rev. D92 (2015), no. 6, 065022, 1406.5155. [11] Z. Bern, S. Davies, and J. Nohle, “On Loop Corrections to Subleading Soft Behavior of Gluons and Gravitons,” Phys. Rev. D90 (2014), no. 8, 085015, 1405.1015. [12] Z. Bern, S. Davies, P. Di Vecchia, and J. Nohle, “Low-Energy Behavior of Gluons and Gravitons from Gauge Invariance,” Phys. Rev. D90 (2014), no. 8, 084035, 1406.6987. [13] S. W. Hawking, M. J. Perry, and A. Strominger, “Superrotation Charge and Supertranslation Hair on Black Holes,” 1611.09175. [14] T. He, D. Kapec, A.-M. Raclariu, and A. Strominger, “Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity,” to appear. [15] F. Cachazo and A. Strominger, “Evidence for a New Soft Graviton Theorem,” 1404.4091. [16] A. Strominger, “Asymptotic Symmetries of Yang-Mills Theory,” JHEP 07 (2014) 151, 1308.0589. [17] T. He, P. Mitra, and A. Strominger, “2D Kac-Moody Symmetry of 4D Yang-Mills Theory,” JHEP 10 (2016) 137, 1503.02663. [18] A. Ashtekar and J. D. Romano, “Spatial inﬁnity as a boundary of space-time,” Class. Quant. Grav. 9 (1992) 1069–1100. [19] M. Campiglia and A. Laddha, “Asymptotic symmetries of QED and Weinberg’s soft photon theorem,” JHEP 07 (2015) 115, 1505.05346. [20] M. Campiglia and A. Laddha, “Asymptotic symmetries of gravity and soft theorems for massive particles,” JHEP 12 (2015) 094, 1509.01406. [21] M. Campiglia, “Null to time-like inﬁnity Green’s functions for asymptotic symmetries in Minkowski spacetime,” JHEP 11 (2015) 160, 1509.01408. [22] L. Cornalba, M. S. Costa, and J. Penedones, “Deep Inelastic Scattering in Conformal QCD,” JHEP 03 (2010) 133, 0911.0043. 14

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text: Sundrum,

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text: and

text: Asymptotic

text: Symmetries

text: from

text: CFT,”

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text: He,

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text: Huang,

text: and

text: C.

text: Wen,

text: “Loop

text: Corrections

text: to

text: Soft

text: Theorems

text: in

text: Gauge

text: Theories

text: and

text: Gravity,”

text: JHEP

text: M.

text: Bianchi,

text: S.

text: He,

text: Y.-t.

text: Huang,

text: and

text: C.

text: Wen,

text: “More

text: on

text: Soft

text: Theorems:

text: Trees,

text: Loops

text: and

text: Strings,”

text: Phys.

text: Rev.

text: no.

text: Z.

text: Bern,

text: S.

text: Davies,

text: and

text: J.

text: Nohle,

text: “On

text: Loop

text: Corrections

text: to

text: Subleading

text: Soft

text: Behavior

text: of

text: Gluons

text: and

text: Gravitons,”

text: Phys.

text: Rev.

text: no.

text: Z.

text: Bern,

text: S.

text: Davies,

text: P.

text: Di

text: Vecchia,

text: and

text: J.

text: Nohle,

text: “Low-Energy

text: Behavior

text: of

text: Gluons

text: and

text: Gravitons

text: from

text: Gauge

text: Invariance,”

text: Phys.

text: Rev.

text: no.

text: S.

text: W.

text: Hawking,

text: M.

text: J.

text: Perry,

text: and

text: A.

text: Strominger,

text: “Superrotation

text: Charge

text: and

text: Supertranslation

text: Hair

text: on

text: Black

text: Holes,”

text: T.

text: He,

text: D.

text: Kapec,

text: A.-M.

text: Raclariu,

text: and

text: A.

text: Strominger,

text: “Loop-Corrected

text: Virasoro

text: Symmetry

text: of

text: Quantum

text: Gravity,”

text: to

text: appear.

text: F.

text: Cachazo

text: and

text: A.

text: Strominger,

text: “Evidence

text: for

text: a

text: New

text: Soft

text: Graviton

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text: A.

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text: “Asymptotic

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text: Theory,”

text: JHEP

text: T.

text: He,

text: P.

text: Mitra,

text: and

text: A.

text: Strominger,

text: Kac-Moody

text: Symmetry

text: of

text: Yang-Mills

text: Theory,”

text: JHEP

text: A.

text: Ashtekar

text: and

text: J.

text: D.

text: Romano,

text: “Spatial

text: inﬁnity

text: as

text: a

text: boundary

text: of

text: space-time,”

text: Class.

text: Quant.

text: Grav.

text: M.

text: Campiglia

text: and

text: A.

text: Laddha,

text: “Asymptotic

text: symmetries

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text: and

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text: soft

text: photon

text: theorem,”

text: JHEP

text: M.

text: Campiglia

text: and

text: A.

text: Laddha,

text: “Asymptotic

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text: JHEP

text: M.

text: Campiglia,

text: “Null

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text: asymptotic

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text: Minkowski

text: spacetime,”

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text: Cornalba,

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text: S.

text: Costa,

text: and

text: J.

text: Penedones,

text: “Deep

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text: Scattering

text: in

text: Conformal

text: QCD,”

text: JHEP


text: [23] S. Weinberg, “Six-dimensional Methods for Four-dimensional Conformal Field Theories,” Phys. Rev. D82 (2010) 045031, 1006.3480. [24] M. S. Costa, J. Penedones, D. Poland, and S. Rychkov, “Spinning Conformal Correlators,” JHEP 11 (2011) 071, 1107.3554. [25] M. S. Costa, J. Penedones, D. Poland, and S. Rychkov, “Spinning Conformal Blocks,” JHEP 11 (2011) 154, 1109.6321. [26] D. Simmons-Duﬃn, “Projectors, Shadows, and Conformal Blocks,” JHEP 04 (2014) 146, 1204.3894. [27] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998) 253–291, hep-th/9802150. [28] D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, “Correlation functions in the CFT(d) / AdS(d+1) correspondence,” Nucl. Phys. B546 (1999) 96–118, hep-th/9804058. 15

text: S.

text: Weinberg,

text: “Six-dimensional

text: Methods

text: for

text: Four-dimensional

text: Conformal

text: Field

text: Theories,”

text: Phys.

text: Rev.

text: M.

text: S.

text: Costa,

text: J.

text: Penedones,

text: D.

text: Poland,

text: and

text: S.

text: Rychkov,

text: “Spinning

text: Conformal

text: Correlators,”

text: JHEP

text: M.

text: S.

text: Costa,

text: J.

text: Penedones,

text: D.

text: Poland,

text: and

text: S.

text: Rychkov,

text: “Spinning

text: Conformal

text: Blocks,”

text: JHEP

text: D.

text: Simmons-Duﬃn,

text: “Projectors,

text: Shadows,

text: and

text: Conformal

text: Blocks,”

text: JHEP

text: E.

text: Witten,

text: “Anti-de

text: Sitter

text: space

text: and

text: holography,”

text: Adv.

text: Theor.

text: Math.

text: Phys.

text: D.

text: Z.

text: Freedman,

text: S.

text: D.

text: Mathur,

text: A.

text: Matusis,

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text: Rastelli,

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text: Phys.


text: constrained to be ‘Virasoro-ready’.3 This suggests that they should resemble a subset of 2D CFT correlators, likely involving complex and continuous conformal dimensions. Indeed it has already been observed [16,17] that soft photon amplitudes take the form of a 2D current algebra. Here we seek to understand the 2D description of 4D scattering amplitudes away from the soft limit. This paper considers massive scalar three-point functions, and recasts them in the stan- CS dard form of 2D CFT correlators on the celestial sphere 2. To summarize the result, let CS Xµ (µ = 0, 1, 2, 3) be the ﬂat coordinates on Minkowski space. A natural coordinate on 2 where XµX = 0 is µ X1 + iX2 w = . (1.1) X0 + X3 CS Lorentz transformations act as the global conformal group on 2 → aw + b w . (1.2) cw + d − Here the complex parameters a, b, c, d obey ad bc = 1. We will construct a three-parameter CS C family of solutions labelled by a point w on 2 and an SL(2, ) weight ∆ (rather than the three components of spatial momenta p(cid:126) which label plane waves) which transform as conformal primaries. We will ﬁnd below they are naturally displayed in a hyperbolic slicing of Minkowski space, in harmony with the form of ﬂat space holography advocated in [2]. C SL(2, ) then implies that the 4D tree amplitude takes the form A C ˜(wi, w¯i) = |w1 − w2|∆1+∆2−∆3|w2 − w3|∆2+∆3−∆1|w3 − w1|∆3+∆1−∆2 , (1.3) where the ‘OPE coeﬃcients’ C depends on the masses, conformal weights, and cubic coupling of the three asymptotic scalars. An integral formula for C involving Witten-like diagrams on the hyperbolic slices is derived. In general this integral may not be computable in closed form, but it simpliﬁes in the near-extremal case when the incoming particle is only slightly heavier than the sum of the outgoing particles and is explicitly given below. Other tractable examples would be of great interest. In particular, the beautiful struc- N ture of = 4 amplitudes suggest they may take a particularly nice 2D form rewritten as CS correlators on 2. In [8] one contribution to such amplitudes (from the interior of the forward lightcone) was expressed as a Witten diagram, but to obtain the full amplitude 3This generalizes the well known constraint that any QFT that can be coupled to gravity must have a local conserved stress tensor. 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: constrained

text: to

text: be

text: This

text: suggests

text: that

text: they

text: should

text: resemble

text: a

text: subset

text: of

text: CFT

text: correlators,

text: likely

text: involving

text: complex

text: and

text: continuous

text: conformal

text: dimensions.

text: Indeed

text: it

text: has

text: already

text: been

text: observed

text: that

text: soft

text: photon

text: amplitudes

text: take

text: the

text: form

text: of

text: a

text: current

text: algebra.

text: Here

text: we

text: seek

text: to

text: understand

text: the

text: description

text: of

text: scattering

text: amplitudes

text: away

text: from

text: the

text: soft

text: limit.

text: This

text: paper

text: considers

text: massive

text: scalar

text: three-point

text: functions,

text: and

text: recasts

text: them

text: in

text: the

text: stan-

text: CS

text: dard

text: form

text: of

text: CFT

text: correlators

text: on

text: the

text: celestial

text: sphere

text: To

text: summarize

text: the

text: result,

text: let

text: CS

text: Xµ

text: (µ

text: be

text: the

text: ﬂat

text: coordinates

text: on

text: Minkowski

text: space.

text: A

text: natural

text: coordinate

text: on

text: where

text: XµX

text: is

text: µ

text: w

text: CS

text: Lorentz

text: transformations

text: act

text: as

text: the

text: global

text: conformal

text: group

text: on

text: aw

text: b

text: w

text: cw

text: d

text: Here

text: the

text: complex

text: parameters

text: a,

text: b,

text: c,

text: d

text: obey

text: ad

text: bc

text: We

text: will

text: construct

text: a

text: three-parameter

text: CS

text: C

text: family

text: of

text: solutions

text: labelled

text: by

text: a

text: point

text: w

text: on

text: and

text: an

text: weight

text: (rather

text: than

text: the

text: three

text: components

text: of

text: spatial

text: momenta

text: which

text: label

text: plane

text: waves)

text: which

text: transform

text: as

text: conformal

text: primaries.

text: We

text: will

text: ﬁnd

text: below

text: they

text: are

text: naturally

text: displayed

text: in

text: a

text: hyperbolic

text: slicing

text: of

text: Minkowski

text: space,

text: in

text: harmony

text: with

text: the

text: form

text: of

text: ﬂat

text: space

text: holography

text: advocated

text: in

text: C

text: then

text: implies

text: that

text: the

text: tree

text: amplitude

text: takes

text: the

text: form

text: A

text: C

text: ˜(wi,

text: w¯i)

text: where

text: the

text: ‘OPE

text: coeﬃcients’

text: C

text: depends

text: on

text: the

text: masses,

text: conformal

text: weights,

text: and

text: cubic

text: coupling

text: of

text: the

text: three

text: asymptotic

text: scalars.

text: An

text: integral

text: formula

text: for

text: C

text: involving

text: Witten-like

text: diagrams

text: on

text: the

text: hyperbolic

text: slices

text: is

text: derived.

text: In

text: general

text: this

text: integral

text: may

text: not

text: be

text: computable

text: in

text: closed

text: form,

text: but

text: it

text: simpliﬁes

text: in

text: the

text: near-extremal

text: case

text: when

text: the

text: incoming

text: particle

text: is

text: only

text: slightly

text: heavier

text: than

text: the

text: sum

text: of

text: the

text: outgoing

text: particles

text: and

text: is

text: explicitly

text: given

text: below.

text: Other

text: tractable

text: examples

text: would

text: be

text: of

text: great

text: interest.

text: In

text: particular,

text: the

text: beautiful

text: struc-

text: N

text: ture

text: of

text: amplitudes

text: suggest

text: they

text: may

text: take

text: a

text: particularly

text: nice

text: form

text: rewritten

text: as

text: CS

text: correlators

text: on

text: In

text: one

text: contribution

text: to

text: such

text: amplitudes

text: (from

text: the

text: interior

text: of

text: the

text: forward

text: lightcone)

text: was

text: expressed

text: as

text: a

text: Witten

text: diagram,

text: but

text: to

text: obtain

text: the

text: full

text: amplitude

text: generalizes

text: the

text: well

text: known

text: constraint

text: that

text: any

text: QFT

text: that

text: can

text: be

text: coupled

text: to

text: gravity

text: must

text: have

text: a

text: local

text: conserved

text: stress

text: tensor.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∈ C − C where a, b, c, d with ad bc = 1 and Λµ is its associated SL(2, ) group element ν in the four-dimensional representation.5 Note that, in contrast to the situation in AdS/CFT, the mass m and the conformal dimension ∆ are not related. 2.1 Integral Representation Conformal primary wavefunctions for a massive scalar ﬁeld can be constructed from the Fourier transform of the bulk-to-boundary propagator in three-dimensional hyperbolic space H . Let (y, z, z¯) be the Poincar´e coordinates of the H with metric, 3 3 dy2 + dzdz¯ ds2 = . (2.3) H3 y2 ∞ C Here 0 < y < and y = 0 is the boundary of the H . This geometry has an SL(2, ) 3 isometry that acts as → (cid:48) (az + b)(c¯z¯ + d¯) + ac¯y2 z z = | | | | , cz + d 2 + c 2y2 → (cid:48) (a¯z¯ + ¯b)(cz + d) + a¯cy2 z¯ z¯ = | | | | , cz + d 2 + c 2y2 → (cid:48) y y y = | | | | , (2.4) cz + d 2 + c 2y2 ∈ C − R with a, b, c, d and ad bc = 1. The H can be embedded into 1,3 as either one of 3 the two branches (pˆ0 > 0 or pˆ0 < 0) of the unit hyperboloid − − (pˆ0)2 + (pˆ1)2 + (pˆ2)2 + (pˆ3)2 = 1 . (2.5) 5There is no canonical way to embed the celestial sphere into the lightcone in Minkowski space. It follows C that there is also no canonical way to associate a M¨obius action on w with an SL(2, ) element Λµ in the C ν four-dimensional representation. In fact, any two Λµ ’s that diﬀer by an SL(2, ) conjugation are equally ν good for our deﬁnition. Below we will make a choice of the map Λµ (a, b, c, d) by ﬁxing a reference frame for C ν pˆµ in (2.6). More explicitly, Λµ is the SL(2, ) transformation matrix acting on pˆµ given by plugging (2.4) ν into (2.6). 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: C

text: C

text: where

text: a,

text: b,

text: c,

text: d

text: with

text: ad

text: bc

text: and

text: Λµ

text: is

text: its

text: associated

text: group

text: element

text: ν

text: in

text: the

text: four-dimensional

text: Note

text: that,

text: in

text: contrast

text: to

text: the

text: situation

text: in

text: AdS/CFT,

text: the

text: mass

text: m

text: and

text: the

text: conformal

text: dimension

text: are

text: not

text: related.

text: Integral

text: Representation

text: Conformal

text: primary

text: wavefunctions

text: for

text: a

text: massive

text: scalar

text: ﬁeld

text: can

text: be

text: constructed

text: from

text: the

text: Fourier

text: transform

text: of

text: the

text: bulk-to-boundary

text: propagator

text: in

text: three-dimensional

text: hyperbolic

text: space

text: H

text: Let

text: (y,

text: z,

text: z¯)

text: be

text: the

text: Poincar´e

text: coordinates

text: of

text: the

text: H

text: with

text: metric,

text: dzdz¯

text: C

text: Here

text: y

text: and

text: y

text: is

text: the

text: boundary

text: of

text: the

text: H

text: This

text: geometry

text: has

text: an

text: isometry

text: that

text: acts

text: as

text: (az

text: b)(c¯z¯

text: d¯)

text: z

text: z

text: cz

text: d

text: c

text: (a¯z¯

text: ¯b)(cz

text: d)

text: z¯

text: z¯

text: cz

text: d

text: c

text: y

text: y

text: y

text: cz

text: d

text: c

text: C

text: R

text: with

text: a,

text: b,

text: c,

text: d

text: and

text: ad

text: bc

text: The

text: H

text: can

text: be

text: embedded

text: into

text: as

text: either

text: one

text: of

text: the

text: two

text: branches

text: or

text: of

text: the

text: unit

text: hyperboloid

text: is

text: no

text: canonical

text: way

text: to

text: embed

text: the

text: celestial

text: sphere

text: into

text: the

text: lightcone

text: in

text: Minkowski

text: space.

text: It

text: follows

text: C

text: that

text: there

text: is

text: also

text: no

text: canonical

text: way

text: to

text: associate

text: a

text: M¨obius

text: action

text: on

text: w

text: with

text: an

text: element

text: Λµ

text: in

text: the

text: C

text: ν

text: four-dimensional

text: representation.

text: In

text: fact,

text: any

text: two

text: Λµ

text: ’s

text: that

text: diﬀer

text: by

text: an

text: conjugation

text: are

text: equally

text: ν

text: good

text: for

text: our

text: deﬁnition.

text: Below

text: we

text: will

text: make

text: a

text: choice

text: of

text: the

text: map

text: Λµ

text: (a,

text: b,

text: c,

text: d)

text: by

text: ﬁxing

text: a

text: reference

text: frame

text: for

text: C

text: ν

text: pˆµ

text: in

text: More

text: explicitly,

text: Λµ

text: is

text: the

text: transformation

text: matrix

text: acting

text: on

text: pˆµ

text: given

text: by

text: plugging

text: ν

text: into

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ≡ − A where pˆµ pˆµ(y , z , z¯ ) is given by (2.6) satisfying pˆ2 = 1. By construction ˜ ··· (w , w¯ ) j j j j C i ∆1, ,∆n i i transforms covariantly under SL(2, ), (cid:32) (cid:33) (cid:18) (cid:19) (cid:89) A aw + b a¯w¯ + ¯b n | | A ˜ ··· i , i = cw + d 2∆i ˜ ··· (w , w¯ ) . (2.14) ∆1, ,∆n cw + d c¯w¯ + d¯ i ∆1, ,∆n i i i i i=1 2.2 Analytic Continuation and the Massless Wavefunction The integral expression (2.10) is only a formal deﬁnition for the conformal primary wave- function since the integral is divergent for real mass m. More rigorously, we should deﬁne our conformal primary wavefunction by analytic continuation of the integral expression (2.10) ∈ − R from an unphysical region. When the mass is purely imaginary m i and Xµ lies inside + the future lightcone, the outgoing wavefunction (2.10) is convergent and can be evaluated as (cid:16) (cid:17) √ − − √ 4π ( X2)∆ 1 | | − φ+ (Xµ; w, w¯) = | | − K − m X2 , ∆,m m ( Xµq )∆ ∆ 1 µ ∈ − R if X0 > 0 , XµX < 0 , m i , (2.15) µ + R where qµ is a null vector in 1,3 deﬁned as (cid:0) (cid:1) | | − − − | | qµ = 1 + w 2, w + w¯, i(w w¯), 1 w 2 . (2.16) After landing on the Bessel function expression (2.15), we can then analytically continue R (cid:16) (cid:17) it to real mass m and other regions in 1,3, √ − − √ ± 4π ( X2)∆ 1 − φ (Xµ; w, w¯) = − ∓ K − im X2 . (2.17) ∆,m im ( Xµq i(cid:15))∆ ∆ 1 µ We have introduced an i(cid:15) prescription to regularize the integral (2.10) in the case of real mass m. In practice, the integral representation (2.10) will however prove to be more convenient to compute the scattering amplitudes of these conformal primary wavefunctions. We note that at late times inside the future lightcone, the wave equation takes the asymptotic form − − − 3 − · · · − → ∞ (∂µ∂ m2)φ = ( ∂2 ∂ m2 + )φ , τ2 = XµX . (2.18) µ τ τ τ µ 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: where

text: pˆµ

text: pˆµ(y

text: z

text: z¯

text: is

text: given

text: by

text: satisfying

text: By

text: construction

text: (w

text: w¯

text: j

text: j

text: j

text: j

text: C

text: i

text: ,∆n

text: i

text: i

text: transforms

text: covariantly

text: under

text: A

text: aw

text: b

text: a¯w¯

text: ¯b

text: n

text: A

text: i

text: i

text: cw

text: d

text: (w

text: w¯

text: ,∆n

text: cw

text: d

text: c¯w¯

text: d¯

text: i

text: ,∆n

text: i

text: i

text: i

text: i

text: Analytic

text: Continuation

text: and

text: the

text: Massless

text: Wavefunction

text: The

text: integral

text: expression

text: is

text: only

text: a

text: formal

text: deﬁnition

text: for

text: the

text: conformal

text: primary

text: wave-

text: function

text: since

text: the

text: integral

text: is

text: divergent

text: for

text: real

text: mass

text: m.

text: More

text: rigorously,

text: we

text: should

text: deﬁne

text: our

text: conformal

text: primary

text: wavefunction

text: by

text: analytic

text: continuation

text: of

text: the

text: integral

text: expression

text: R

text: from

text: an

text: unphysical

text: region.

text: When

text: the

text: mass

text: is

text: purely

text: imaginary

text: m

text: i

text: and

text: Xµ

text: lies

text: inside

text: the

text: future

text: lightcone,

text: the

text: outgoing

text: wavefunction

text: is

text: convergent

text: and

text: can

text: be

text: evaluated

text: as

text: 4π

text: φ+

text: (Xµ;

text: w,

text: w¯)

text: K

text: m

text: ∆,m

text: m

text: Xµq

text: µ

text: R

text: if

text: XµX

text: m

text: i

text: µ

text: R

text: where

text: qµ

text: is

text: a

text: vector

text: in

text: deﬁned

text: as

text: qµ

text: w

text: w

text: w¯,

text: i(w

text: w¯),

text: w

text: After

text: landing

text: on

text: the

text: Bessel

text: function

text: expression

text: we

text: can

text: then

text: analytically

text: continue

text: R

text: it

text: to

text: real

text: mass

text: m

text: and

text: other

text: regions

text: in

text: 4π

text: φ

text: (Xµ;

text: w,

text: w¯)

text: K

text: im

text: ∆,m

text: im

text: Xµq

text: µ

text: We

text: have

text: introduced

text: an

text: prescription

text: to

text: regularize

text: the

text: integral

text: in

text: the

text: case

text: of

text: real

text: mass

text: m.

text: In

text: practice,

text: the

text: integral

text: representation

text: will

text: however

text: prove

text: to

text: be

text: more

text: convenient

text: to

text: compute

text: the

text: scattering

text: amplitudes

text: of

text: these

text: conformal

text: primary

text: wavefunctions.

text: We

text: note

text: that

text: at

text: late

text: times

text: inside

text: the

text: future

text: lightcone,

text: the

text: wave

text: equation

text: takes

text: the

text: asymptotic

text: form

text: (∂µ∂

text: ∂2

text: )φ

text: τ2

text: XµX

text: µ

text: τ

text: τ

text: τ

text: µ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − ⊗ δ = tr[(ρAsR ρAs ρR)Φs(O )] s s s 0 AR − ⊗ = tr[(ρAsR ρAs ρR)Φs ΦrA(O )] s − s ⊗ s (cid:88)rA 0 AR = tr[(ρAsR ρAs ρR)Φs (O )] s s s rA ArAR d2 − ⊗ ⊗ = tr[(ρAsR ρAs ρR)( Φs (O ) O )], s s s rA ArA,j R,j j=1 where O = ΦrA(O ) and r is chosen such that O(cid:80) is supported on one of the ArAR 0 AR A ArAR elementary blocks and the purifying space. This operator can, without loss of generality, ⊗ be decomposed into the operator Schmidt decomposition d2 O O , where the (cid:107) (cid:107) j=1 ArA,j R,j norm of each of the terms is bounded by O . Here d is the dimension of the elementary AR block, which is bounded by some constant. − ⊗ ⊗ The unique ﬁxed point of Φs is the identity operator, and tr[(ρAsR ρAs ρR)1 O] = 0 rA s s s for any operator O. Therefore, (cid:88) d2 − ⊗ δ (cid:88)= tr[ρAsR ρAs ρR] s s s j=1 − ⊗ λs rAtr[L O ]R O (cid:54) k k ArA,j k R,j k=0 ≤ (cid:107) (cid:107) − − 2d2 O 2 ν(s rA). (4.4) AR One can see that Eq.4.1 holds with a choice of constant C = 2d2. By invoking Theorem 1, we can easily show that the region A is correctable. C Corollary 1. For an RG-regular MERA code , and for any simply connected region A, R H s there exists a CPTP acting on such that 0 (cid:107)R − (cid:107) ≤ − − | | (ρAcR) ρAAcR 1 C2 ν(s log2 A )/2, (4.5) where C is a numerical constant, and Ac denotes the complement of A. The proof simply follows by applying Theorem 1 and then using the relation between the Bures distance and the trace distance (Eq.3.3). Our ﬁndings support the conclusion of Ref. [20], in which it was suggested that low energies of the critical systems should have a certain error correction property. In partic- ular, our work provides a satisfying answer to the question: how does an error correcting code emerge in these systems? It arises from the fact that the ground state can be well- approximated by a MERA state. – 11 – ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: δ

text: tr[(ρAsR

text: ρAs

text: ρR)Φs(O

text: s

text: s

text: s

text: AR

text: tr[(ρAsR

text: ρAs

text: ρR)Φs

text: ΦrA(O

text: s

text: s

text: s

text: AR

text: tr[(ρAsR

text: ρAs

text: ρR)Φs

text: (O

text: s

text: s

text: s

text: rA

text: ArAR

text: tr[(ρAsR

text: ρAs

text: ρR)(

text: Φs

text: (O

text: O

text: s

text: s

text: s

text: rA

text: ArA,j

text: R,j

text: where

text: O

text: ΦrA(O

text: and

text: r

text: is

text: chosen

text: such

text: that

text: is

text: supported

text: on

text: one

text: of

text: the

text: ArAR

text: AR

text: A

text: ArAR

text: elementary

text: blocks

text: and

text: the

text: purifying

text: space.

text: This

text: operator

text: can,

text: without

text: loss

text: of

text: generality,

text: be

text: decomposed

text: into

text: the

text: operator

text: Schmidt

text: decomposition

text: O

text: O

text: where

text: the

text: ArA,j

text: R,j

text: norm

text: of

text: each

text: of

text: the

text: terms

text: is

text: bounded

text: by

text: O

text: Here

text: d

text: is

text: the

text: dimension

text: of

text: the

text: elementary

text: AR

text: block,

text: which

text: is

text: bounded

text: by

text: some

text: constant.

text: The

text: unique

text: ﬁxed

text: point

text: of

text: Φs

text: is

text: the

text: identity

text: operator,

text: and

text: tr[(ρAsR

text: ρAs

text: O]

text: rA

text: s

text: s

text: s

text: for

text: any

text: operator

text: O.

text: Therefore,

text: δ

text: tr[ρAsR

text: ρAs

text: ρR]

text: s

text: s

text: s

text: λs

text: rAtr[L

text: O

text: ]R

text: O

text: k

text: k

text: ArA,j

text: k

text: R,j

text: O

text: ν(s

text: rA).

text: AR

text: One

text: can

text: see

text: that

text: holds

text: with

text: a

text: choice

text: of

text: constant

text: C

text: By

text: invoking

text: Theorem

text: we

text: can

text: easily

text: show

text: that

text: the

text: region

text: A

text: is

text: correctable.

text: C

text: Corollary

text: For

text: an

text: RG-regular

text: MERA

text: code

text: and

text: for

text: any

text: simply

text: connected

text: region

text: A,

text: R

text: H

text: s

text: there

text: exists

text: a

text: CPTP

text: acting

text: on

text: such

text: that

text: (ρAcR)

text: ρAAcR

text: ν(s

text: A

text: where

text: C

text: is

text: a

text: numerical

text: constant,

text: and

text: Ac

text: denotes

text: the

text: complement

text: of

text: A.

text: The

text: proof

text: simply

text: follows

text: by

text: applying

text: Theorem

text: and

text: then

text: using

text: the

text: relation

text: between

text: the

text: Bures

text: distance

text: and

text: the

text: trace

text: distance

text: Our

text: ﬁndings

text: support

text: the

text: conclusion

text: of

text: Ref.

text: in

text: which

text: it

text: was

text: suggested

text: that

text: low

text: energies

text: of

text: the

text: critical

text: systems

text: should

text: have

text: a

text: certain

text: error

text: correction

text: property.

text: In

text: partic-

text: ular,

text: our

text: work

text: provides

text: a

text: satisfying

text: answer

text: to

text: the

text: question:

text: how

text: does

text: an

text: error

text: correcting

text: code

text: emerge

text: in

text: these

text: systems?

text: It

text: arises

text: from

text: the

text: fact

text: that

text: the

text: ground

text: state

text: can

text: be

text: well-

text: approximated

text: by

text: a

text: MERA

text: state.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 2 1 0 -1 -100 -80 -60 -40 -20 0 20 40 60 80 100 x 3 2 1 0 -1 -100 -80 -60 -40 -20 0 20 40 60 80 100 x Figure 2: On top the step-like initial data and on bottom the solution at time t = 12. One can clearly see the soliton region containing two solitons and the collision-less shock region where modulated oscillations are formed. 3.2 Small ε asymptotic The idea of the formation of an oscillatory structure in the limit of small dis- persion of a dispersive equation belongs to Sagdeev [48]. Gurevish and Pitaevskii in 1973 called the oscillations, arising in the small dispersion limit of KdV, disper- sive shock waves in analogy with the shock waves appearing in the zero dissipation limit of the Burgers equation. A very recent experiment in a water tank has been set up where the dispersive shock waves have been reproduced [55]. The main steps for the description of the dispersive shock waves are the fol- lowing: • as long as the solution of the Cauchy problem for Hopf equation vt +6vvx = 0 with the initial data v(x,0) = f(x) exists, then the solution of the KdV equation u(x,t;ε) = v(x,t) + O(ε2). Generically the solution of the Hopf equation obtained by the method of characteristics v(x,t) = f(ζ), x = f(ζ)t +ζ, (45) develops a singularity when the function ζ = ζ(x,t) given implicitly by the map x = f(ζ)t + ζ is not uniquely deﬁned. This happens at the ﬁrst time (cid:48) (cid:48)(cid:48) when f (ζ)t + 1 = 0 and f (ζ) = 0 (see Figure 3). These two equations and (45) ﬁx uniquely the|poin→t (xc,tc) and uc = v(xc,tc). At this point, the gradient blow up: vx(x,t) xc,tc ∞. • The solution of the KdV equations remains smooth for all positive times. Around the time when the solution of the Hopf equation develops its ﬁrst 14 u u ##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##

text: x

text: x

text: Figure

text: On

label: caption

text: top

text: the

text: step-like

text: initial

text: data

text: and

text: on

text: bottom

text: the

text: solution

text: at

text: time

text: t

text: One

text: can

text: clearly

text: see

text: the

text: soliton

text: region

text: containing

text: two

text: solitons

text: and

text: the

text: collision-less

text: shock

text: region

text: where

text: modulated

text: oscillations

text: are

text: formed.

text: Small

text: ε

text: asymptotic

text: The

text: idea

text: of

text: the

text: formation

text: of

text: an

text: oscillatory

text: structure

text: in

text: the

text: limit

text: of

text: small

text: dis-

text: persion

text: of

text: a

text: dispersive

text: equation

text: belongs

text: to

text: Sagdeev

text: Gurevish

text: and

text: Pitaevskii

text: in

text: called

text: the

text: oscillations,

text: arising

text: in

text: the

text: small

text: dispersion

text: limit

text: of

text: KdV,

text: disper-

text: sive

text: shock

text: waves

text: in

text: analogy

text: with

text: the

text: shock

text: waves

text: appearing

text: in

text: the

text: zero

text: dissipation

text: limit

text: of

text: the

text: Burgers

text: equation.

text: A

text: very

text: recent

text: experiment

text: in

text: a

text: water

text: tank

text: has

text: been

text: set

text: up

text: where

text: the

text: dispersive

text: shock

text: waves

text: have

text: been

text: reproduced

text: The

text: main

text: steps

text: for

text: the

text: description

text: of

text: the

text: dispersive

text: shock

text: waves

text: are

text: the

text: fol-

text: lowing:

text: as

text: long

text: as

text: the

text: solution

text: of

text: the

text: Cauchy

text: problem

text: for

text: Hopf

text: equation

text: vt

text: with

text: the

text: initial

text: data

text: f(x)

text: exists,

text: then

text: the

text: solution

text: of

text: the

text: KdV

text: equation

text: u(x,t;ε)

text: v(x,t)

text: Generically

text: the

text: solution

text: of

text: the

text: Hopf

text: equation

text: obtained

text: by

text: the

text: method

text: of

text: characteristics

text: v(x,t)

text: f(ζ),

text: x

text: f(ζ)t

text: +ζ,

text: develops

text: a

text: singularity

text: when

text: the

text: function

text: ζ

text: ζ(x,t)

text: given

text: implicitly

text: by

text: the

text: map

text: x

text: f(ζ)t

text: ζ

text: is

text: not

text: uniquely

text: deﬁned.

text: This

text: happens

text: at

text: the

text: ﬁrst

text: time

text: when

text: f

text: (ζ)t

text: and

text: f

text: (ζ)

text: (see

text: Figure

text: These

text: two

text: equations

text: and

text: ﬁx

text: uniquely

text: the|poin→t

text: (xc,tc)

text: and

text: uc

text: v(xc,tc).

text: At

text: this

text: point,

text: the

text: gradient

text: blow

text: up:

text: vx(x,t)

text: xc,tc

text: The

text: solution

text: of

text: the

text: KdV

text: equations

text: remains

text: smooth

text: for

text: all

text: positive

text: times.

text: Around

text: the

text: time

text: when

text: the

text: solution

text: of

text: the

text: Hopf

text: equation

text: develops

text: its

text: ﬁrst

text: u

text: u

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text: the oscillations are linear, while inside the Whitham zone, it gives a qualitative description of the oscillations [29]. Another interesting asymptotic regime is obtained when one wants to describe the ﬁrst few oscillations of the KdV solution in the small dispersion limit. In this case the so called Painleve´ I2 asymptotics should be used. Furthermore we point out that it is simpler to solve one ODE, rather then the Whitham equations. For example, near the critical point x and near the critical time the following c asymptotic behavi(cid:18)our ha(cid:19)s been c(cid:32)onjectured in [20] and proved in(cid:33)[9] (cid:16) (cid:17) u(x,t,ε) (cid:39) uc + 2σε22 1/7U x−xc(8−σ6εu6c)(17t −tc); (64(σt3−ε4tc))17 +O ε4/7 , (61) − (cid:48)(cid:48)(cid:48) where σ = hL (uc),(cid:20)and U = U(X,T) is the unique real sm(cid:21)ooth solution to the fourth order ODE [13] − U3 1 1 X = T U 6 + 24(UX2 +2UUXX)+ 240UXXXX , (62) which is the second member of the Painleve´ I hierarchy (PI2 ). The relevant solution is uniquely [?] characterized by the asymptotic behavior ∓ | | ∓ 1 | |− | |− → ± U(X,T) = (6 X )1/3 62/3T X 1/3 +O( X 1), as X ∞, (63) 3 ∈ R for each ﬁxed T . Such Painleve´ solution matches, the elliptic solution (54) for the cubic inital data f(x) = −x13 for large times [8]. Such solution of the PI2 has been conjectured to describe the initial time of the formation of dispersive shock waves for general Hamiltonian perturbation of hyperbolic equations [21]. We conclude by stressing that the asymptotic descriptions reviewed in this chapter for the KdV equation can be developed for other integrable equations like the nonlinear Schro¨dinger equation, [39] the Camass-Holm equation [6] or the modiﬁed KdV equation [41]. Acknowledgements T.G. acknowledges the support by the Leverhulme Trust Research Fellowship RF-2015-442 from UK and PRIN Grant eometric and analytic theory of Hamilto- nian systems in ﬁnite and inﬁnite dimensionsf Italian Ministry of Universities and Researches. References [1] Ablowitz, M. J. and Newell, A. C. The decay of the continuous spectrum for solutions of the Korteweg-de Vries equation. J. Mathematical Phys. 14 (1973), 1277 - 1284. [2] Ablowitz, M. J. and Segur, H. Asymptotic solutions of the Korteweg-de Vries equation. Stud. Appl. Math. 57 (1977), 13 - 44. 22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: oscillations

text: are

text: linear,

text: while

text: inside

text: the

text: Whitham

text: zone,

text: it

text: gives

text: a

text: qualitative

text: description

text: of

text: the

text: oscillations

text: Another

text: interesting

text: asymptotic

text: regime

text: is

text: obtained

text: when

text: one

text: wants

text: to

text: describe

text: the

text: ﬁrst

text: few

text: oscillations

text: of

text: the

text: KdV

text: solution

text: in

text: the

text: small

text: dispersion

text: limit.

text: In

text: this

text: case

text: the

text: so

text: called

text: Painleve´

text: asymptotics

text: should

text: be

text: used.

text: Furthermore

text: we

text: point

text: out

text: that

text: it

text: is

text: simpler

text: to

text: solve

text: one

text: ODE,

text: rather

text: then

text: the

text: Whitham

text: equations.

text: For

text: example,

text: near

text: the

text: critical

text: point

text: x

text: and

text: near

text: the

text: critical

text: time

text: the

text: following

text: c

text: asymptotic

text: been

text: in

text: and

text: proved

text: u(x,t,ε)

text: uc

text: −tc);

text: +O

text: where

text: σ

text: hL

text: U

text: U(X,T)

text: is

text: the

text: unique

text: real

text: solution

text: to

text: the

text: fourth

text: order

text: ODE

text: X

text: T

text: U

text: which

text: is

text: the

text: second

text: member

text: of

text: the

text: Painleve´

text: I

text: hierarchy

text: The

text: relevant

text: solution

text: is

text: uniquely

text: characterized

text: by

text: the

text: asymptotic

text: behavior

text: U(X,T)

text: X

text: X

text: +O(

text: X

text: as

text: X

text: R

text: for

text: each

text: ﬁxed

text: T

text: Such

text: Painleve´

text: solution

text: matches,

text: the

text: elliptic

text: solution

text: for

text: the

text: cubic

text: inital

text: data

text: f(x)

text: for

text: large

text: times

text: Such

text: solution

text: of

text: the

text: has

text: been

text: conjectured

text: to

text: describe

text: the

text: initial

text: time

text: of

text: the

text: formation

text: of

text: dispersive

text: shock

text: waves

text: for

text: general

text: Hamiltonian

text: perturbation

text: of

text: hyperbolic

text: equations

text: We

text: conclude

text: by

text: stressing

text: that

text: the

text: asymptotic

text: descriptions

text: reviewed

text: in

text: this

text: chapter

text: for

text: the

text: KdV

text: equation

text: can

text: be

text: developed

text: for

text: other

text: integrable

text: equations

text: like

text: the

text: nonlinear

text: Schro¨dinger

text: equation,

text: the

text: Camass-Holm

text: equation

text: or

text: the

text: modiﬁed

text: KdV

text: equation

text: Acknowledgements

text: T.G.

text: acknowledges

text: the

text: support

text: by

text: the

text: Leverhulme

text: Trust

text: Research

text: Fellowship

text: from

text: UK

text: and

text: PRIN

text: Grant

text: eometric

text: and

text: analytic

text: theory

text: of

text: Hamilto-

text: nian

text: systems

text: in

text: ﬁnite

text: and

text: inﬁnite

text: dimensionsf

text: Italian

text: Ministry

text: of

text: Universities

text: and

text: Researches.

text: References

text: Ablowitz,

text: M.

text: J.

text: and

text: Newell,

text: A.

text: C.

text: The

text: decay

text: of

text: the

text: continuous

text: spectrum

text: for

text: solutions

text: of

text: the

text: Korteweg-de

text: Vries

text: equation.

text: J.

text: Mathematical

text: Phys.

text: Ablowitz,

text: M.

text: J.

text: and

text: Segur,

text: H.

text: Asymptotic

text: solutions

text: of

text: the

text: Korteweg-de

text: Vries

text: equation.

text: Stud.

text: Appl.

text: Math.

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