Asymptotic Symmetry and Subleading Charges in QED Hayato Hirai - - PowerPoint PPT Presentation

Asymptotic Symmetry and Subleading Charges in QED

Hayato Hirai (Osaka univ.)

Based on 1805.05651 (JHEP07(2018)122) joint work with Sotaro Sugishita (previous speaker)

Strings and Fields 2018 July, 30, 2018 @YITP, Kyoto

Triangle of the Infrared Structures

Memory Effect Soft theorem Asymptotic Symmetry

Conservation Law Ward-Takahashi identity Fourier transform

[A.Strominger ’13,…]

Triangle of the Infrared Structures

Memory Effect Soft theorem Asymptotic Symmetry

Conservation Law Ward-Takahashi identity Fourier transform

[A.Strominger ’13,…]

in QED

Main topic of this talk

Soft theorem Asymptotic Symmetry

Ward-Takahashi identity Why the AS transformation is physical symmetry rather than gauge redundancies…?

Q1.

Are there any symmetry associated with subleading soft theorem in QED…?

Q2.

Soft theorem Asymptotic Symmetry

Ward-Takahashi identity Asymptotic symmetry in QED is physical symmetry rather than gauge redundancy because it transforms physical (BRST inv.) state in BRST formalism. We obtained the expression of conserved charge associated with subleading soft theorem in massive scalar QED.

A1. A2.

Results

Asymptotic symmetry as physical symmetry Charge associated with subleading soft theorem Introduction Discussion

Asymptotic symmetry in QED

∂µAµ(x) = 0

Aµ(x) = @µ✏(x)

⇤✏(x) = 0

Lorenz gauge:

Residual local U(1) trnsf.: with .

= “Large” residual local U(1) transformation

r → ∞

✏(u, r, Ω) − → O(1)

with .

AS transformation in QED

u i+ I− I+ i0

QED in BRST formalism

No first class constraint i.e all gauge d.o.f are fixed. Therefore, we can successfully quantize the theory.

LQED = LEM + Lmatter + LGF + LF P LEM = −1 4FµνF µν L matter = −1 2DµφDµφ − 1 2m2φφ LF P = i∂µc∂µc

,

, LGF = B∂µAµ + 1 2B2

Residual symmetry

g(x) = ie✏(x)(x) , g(x) = −ie✏(x)(x), gAµ(x) = @µ✏(x) ⇤✏(x) = 0

with

δBφ(x) = ieλc(x)φ(x) δBc(x) = 0 δBB(x) = 0

BRST symmetry = Gauge redundancy in QM

, , ,

Residual local U(1) symmetry

QBRST |Ψiphys = 0 [QBRST, Ophys] = 0

,

BRST cond.

δBAµ(x) = λ∂µc(x)

δBc(x) = iλB(x)

Asymptotic behavior of the fields

u = fixed , r → ∞

Au(x) = A(1)

u (u, Ω)

r + O

• r−2 log r
• Ar(x) = A(1)

r (u, Ω)

r + O

• r−2 log r
• AB(x) = A(0)

B (u, Ω) + O

• r−1 log r
• Aµ(u, r, ΩB) =

Z d3k (2⇡)32!k ⇣ aµ(~ k)eikx + a†

µ(~

k)e−ikx⌘

I+

coord. transf.

[He, Mitra, Porfyriadis, Strominger ’14]

= − i 8π2r Z ∞

λ

dω ⇥ aµ(ωˆ x)e−iωu − (h.c.) ⇤ + O

• r−2 log r
• infrared cutoff:

, λ

(µ = t, x, y, z)

,

(µ, r, ΩB)

Physical d.o.f. on null infinity I±

If we choose Fourier expanded solution of e.o.m

⇤c(x) = 0

lim

r→∞,u: fixed c(x) = O

• r−1

δBRST A(0)

B (u, Ω) = 0

δBRST AB(u, r, Ω) = λ∂Bc(u, r, Ω) = O(r−1) is physical d.o.f. , i.e. Cauchy data, on . n A(0)

B (u, Ω)

• I+

In this case, BRST transformation on null infinity is

(comments will be given later.)

[H.H&Sugishita ’18]

Physical d.o.f. on null infinity I±

gA(0)

B (u, Ω) = @B✏(0)(Ω)

AS transformation changes the physical operators on . Therefore, its W-T id is not trivialized by BRST condition, and gives non-trivial constraint on the theory, which is soft theorem in QED.

Asymptotic symmetry

I+

[H.H&Sugishita ’18]

We actually rederived the equivalence between the W-T id of AS and leading soft theorem in QED in BRST formalism. AS is physical symmetry.

Whether the AS in QED is physical symmetry or not does depends on the fall-off condition of the ghost field in

• ur logic. We chose by hand.

c(x)

lim

r→∞,u: fixed c(x) = O

• r−1

What happens if we choose ? lim

r→∞,u: fixed c(x) = O (1)

Q. δBRST AB(u, r, Ω) = λ∂Bc(u, r, Ω) = O (1) In this case, is not physical d.o.f. but gauge redundancy. Then, AS is not physical symmetry…?

n A(0)

B (u, Ω)

• This fall-off condition is formally allowed, but that is not consistent

with real physics in following sense.

[H.H&Sugishita ’18]

Physical fall-off condition of ghost fields

BRST transf local U(1) transf ✏(x) λc(x) BRST condition:

QBRST |Ψiphys = 0

Physical states : Physical states are restricted to the invariant state under the large gauge trnsfs. But the large gauge charges includes the global U(1) charge, i.e physical states are restricted to the states with zero global U(1) charge. This is too restricted condition. Q✏

LG|Ψie, = 0

lim

r→∞,u: fixed c(x) = O (1)

[H.H&Sugishita ’18]

lim

r→∞,u: fixed c(x) = O

• r−1 is justified.

Physical fall-off condition of ghost fields

Asymptotic symmetry as physical symmetry Charge associated with subleading soft theorem Introduction Discussion

Soft theorem in QED

hout |aB(ωˆ x)S| ini = h S(lead) ω−1 + S(sub)

B

• ω0i

hout|S|ini + O(ω)

S(lead) !−1 = X

k∈ out

ek~ pk · ✏B pk · q − X

k∈ in

ek~ pk · ✏B pk · q S(sub) ω0 = −i X

k

ekqµJk

µB

pk · q

Jk

µν = −i

✓ pkµ ∂ ∂pν

k

− pkν ∂ ∂pµ

k

◆

· · · · · ·

S

pki

pkj

· · · · · ·

S

pki

pkj

in

• ut

q

• ut

= Soft factor × +O(ω) , +(loop corr.) Subleading soft factor gets non-universal loop corrections.

[Low 1954] [Bern, Davies, Nohle ’14]

Subleading soft theorem in QED

hout |aB(ωˆ x)S| ini = h S(lead) ω−1 + S(sub)

B

• ω0i

hout|S|ini + O(ω)

· · · · · ·

S

pki

pkj

· · · · · ·

S

pki

pkj

in

• ut

q

• ut

= Soft factor × +O(ω)

lim

ω→0 hout |(1 + ω∂ω) aB(ωˆ

x)S| ini = i X

k

ekqµJk

µB

pk · q hout|S|ini

(1 + ω∂ω) O(ω−1) = 0 (1 + ω∂ω) 1 = 1 , +(corr.)

Subleading soft theorem

lim

ω→0 hout |aB(ωˆ

x)S| ini = S(lead) ω−1 hout |S|ini

⌦

• ut
• Q+S − SQ−

in ↵ = 0 W-T id of AS : Leading soft thrm:

in massless and massive scalar QED.

Subleading soft thrm:

lim

ω→0 hout |(1 + ω∂ω) aB(ωˆ

x)S| ini = i X

k

ekqµJk

µB

pk · q hout|S|ini

D

• ut
• Q(sub)+S − SQ(sub)−
• in

E = 0

Symmetry

• f S-matrix

in massless scalar QED.

i+ I+ i− I− I+ I−

[Lysov, Pasterski, Strominger ’14] [Campiglia, Laddha ’16] [Conde, Mao ’17] [He, Mitra, Porfyriadis, Strominger ’14] [Campiglia, Laddha ’15].

Generalization to massive scalar QED.

[H.H&Sugishita ’18]

i−

ds2 = −dτ 2 + τ 2dl2

H3

dl2

H3 = hαβdσαdσβ

= dρ2 1 + ρ2 + ρ2γABdΩAdΩB ρ = const: trajectory of a free massive particle where, Around timelike infinity, In this coordinate, we can study the asymptotic behavior of massive fields around timelike infinity .

i+ I+ I−

τ → ∞

(τ, ρ, ΩB)

[Campiglia, Laddha ’15]

Subleading soft theorem as conservation law

Subleading soft theorem as conservation law

lim

ω→0 hout |(1 + ω∂ω) aB(ωˆ

x)S| ini = i X

k

ekqµJk

µB

pk · q hout|S|ini in scalar QED.

D

• ut
• ⇣

Q(sub)+

H

+ Q(sub)+

S

⌘ S − S ⇣ Q(sub)−

H

+ Q(sub)−

S

⌘

• in

E = 0

Q(sub)

±

= 1 2 Z

I± dud2Ωp✏(0)u@u∆S2rBA(0) B

S

Q(sub)

f,i

= 1 2 Z

H3d3

p h p 1 + ⇢2 ⇢ h ⇢2hαβ(r(h)

α r(h) ρ ✏H3)Imat β

+ 2⇢hαβ(r(h)

α ✏H3)Imat β

i

H

[H.H&Sugishita ’18]

New symmetry of S-matrix

Summary Soft theorem Asymptotic Symmetry

Leading soft theorem Universal part of subleading soft theorem W-T id of AS Symmetry of S-matrix (in classical limit) since it transforms BRST invariant state under appropriate falls-off condition of ghost fields.

AS = physical symmetry

Discussions

Whether the subleading soft theorem including loop correction can be associated with symmetry of S-matrix? Asymptotic symmetry in “infrared finite S-matrix formalism” ? We treated charged matter fields as “free” fields around timelike

• infinity. (This is traditional treatment in scattering formalism.)

But, asymptotic charged state should be surrounded by the Coulomb filed and radiation created by themselves. Dressed state [Chung 1965. Kibble 1968,

Faddeev, Kulish 1970]

“Infrared finite S-matrix”

Thank you for your attention!

Back up

Asymptotic symmetry as physical symmetry

[H.H&Sugishita ’18]

δBRST A(0)

B (u, Ω) = 0

δBRST AB(u, r, Ω) = λ∂Bc(u, r, Ω) = O(r−1) gA(0)

B (u, Ω) = @B✏(0)(Ω)

lim

r→∞,u: fixed c(x) = O

• r−1

・・Physical d.o.f Asymptotic symmetry changes the physical d.o.f on null infinity

lim

ω→0 hout |aB(ωˆ

x)S| ini = S(lead) ω−1 hout |S|ini

⌦

• ut
• Q+S − SQ−

in ↵ = 0 W-T id of AS : Leading soft thrm:

in massless and massive scalar QED.

Subleading soft thrm:

lim

ω→0 hout |(1 + ω∂ω) aB(ωˆ

x)S| ini = i X

k

ekqµJk

µB

pk · q hout|S|ini

D

• ut
• Q(sub)+S − SQ(sub)−
• in

E = 0

Symmetry

• f S-matrix

Q(sub)+

H

= Z

I+ d2zdu

• uDzYzJM

u + YzJM z

• in massless scalar QED.

i+ I+ i− I− I+ I−

Q(sub)+ = 1 e2 Z

I+ dud2z∂u

• uDzY z

r2Furγzz + Fzz

• + 2r2YzFzr
• S

[Lysov, Pasterski, Strominger ’14] [Campiglia, Laddha ’16] [Conde, Mao ’17] [He, Mitra, Porfyriadis, Strominger ’14] [Campiglia, Laddha ’15].

i− i+ I+

I−

Hard charges in terms of asymptotic fields

jτ

(mat)(τ, ρ, Ω) =

jτ(3)

(mat)(ρ, Ω)

τ 3 + O(τ −4) ✏H3(⇢, Ω) ≡ lim

τ→∞ ✏(⌧, ⇢, Ω)

GH3 (σ; Ω0) = 1 4π h − p 1 + ρ2 + ρˆ q (Ω0) · ˆ x(Ω) i2

= Z d2Ω0p (Ω0)GH3 (⇢, Ω; Ω0) ✏(0) (Ω0) Green function of from to

I+ i+

⇤✏ = 0

(⌧, ⇢, Ω) = √m 2(2⇡⌧)3/2 ⇣ b(~ p)e−im⌧−3⇡i/4 + d†(~ p)eim⌧+3⇡i/4⌘

• ~

p=m⇢ˆ x(Ω)

+O ⇣ τ − 3

2 −ε⌘

jµ

mat(x) = ie

• Dµφ(x)φ(x) − φ(x)Dµφ(x)
• ( )

Imat

α

(σ) ≡ lim

τ→∞

 1 4m2 ∂2

τ + 1

• τ 3jmat

α

(τ, σ) = iem 4(2π)3 ⇥ ∂αb†b − b†∂αb − ∂αd†d + d†∂αd ⇤ Key equations rB " ∂Bˆ x(Ω) · ˆ y(˜ Ω) q(Ω) · Y (ρ, ˜ Ω) # = 4π ρ GH3(ρ, ˜ Ω; Ω) 1 ρ τ 2 = t2 − r2 ρ = r √ t2 − r2 ,