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 [A.Strominger ’13,…] Memory Effect Fourier Conservation transform Law Asymptotic Soft Symmetry theorem Ward-Takahashi identity

Triangle of the Infrared Structures [A.Strominger ’13,…] Memory Effect Fourier Conservation transform Law Main topic of this talk Asymptotic Soft Symmetry theorem Ward-Takahashi identity in QED

Asymptotic Soft Symmetry theorem Ward-Takahashi identity Q 1. Why the AS transformation is physical symmetry rather than gauge redundancies…? Q 2. Are there any symmetry associated with subleading soft theorem in QED…?

Asymptotic Soft Symmetry theorem Ward-Takahashi identity Results A 1. Asymptotic symmetry in QED is physical symmetry rather than gauge redundancy because it transforms physical (BRST inv.) state in BRST formalism. A 2. We obtained the expression of conserved charge associated with subleading soft theorem in massive scalar QED.

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

Asymptotic symmetry in QED i + Lorenz gauge : I + ∂ µ A µ ( x ) = 0 u Residual local U(1) trnsf.: i 0 � A µ ( x ) = @ µ ✏ ( x ) I − with . ⇤ ✏ ( x ) = 0 AS transformation in QED = “Large” residual local U(1) transformation r → ∞ ✏ ( u, r, Ω ) − → O (1) with .

QED in BRST formalism L QED = L EM + L matter + L GF + L F P L EM = − 1 L matter = − 1 2 D µ φ D µ φ − 1 4 F µ ν F µ ν 2 m 2 φφ , L GF = B ∂ µ A µ + 1 2 B 2 L F P = i ∂ µ c ∂ µ c , No first class constraint i.e all gauge d.o.f are fixed. Therefore, we can successfully quantize the theory.

Residual symmetry BRST symmetry = Gauge redundancy in QM δ B φ ( x ) = ie λ c ( x ) φ ( x ) δ B A µ ( x ) = λ∂ µ c ( x ) , δ B c ( x ) = 0 δ B B ( x ) = 0 δ B c ( x ) = i λ B ( x ) , , BRST cond. [ Q BRST , O phys ] = 0 Q BRST | Ψ i phys = 0 , Residual local U(1) symmetry � g � ( x ) = ie ✏ ( x ) � ( x ) , � g � ( x ) = − ie ✏ ( x ) � ( x ) , � g A µ ( x ) = @ µ ✏ ( x ) with ⇤ ✏ ( x ) = 0

Asymptotic behavior of the fields d 3 k Z ⇣ k ) e ikx + a † k ) e − ikx ⌘ a µ ( ~ µ ( ~ A µ ( u, r, Ω B ) = ( µ = t, x, y, z ) (2 ⇡ ) 3 2 ! k , infrared cutoff: u = fixed , r → ∞ λ , Z ∞ i r − 2 log r x ) e − i ω u − ( h.c. ) ⇥ ⇤ � � = − a µ ( ω ˆ + O d ω 8 π 2 r λ [He, Mitra, Porfyriadis, Strominger ’14] A u ( x ) = A (1) u ( u, Ω ) coord. r − 2 log r � � + O transf. r ( µ, r, Ω B ) A r ( x ) = A (1) r ( u, Ω ) I + r − 2 log r � � + O r r − 1 log r A B ( x ) = A (0) � � B ( u, Ω ) + O

Physical d.o.f. on null infinity I ± [H.H&Sugishita ’18] If we choose Fourier expanded solution of e.o.m ⇤ c ( x ) = 0 (comments will r − 1 � � r →∞ ,u : fixed c ( x ) = O lim be given later.) In this case, BRST transformation on null infinity is δ BRST A B ( u, r, Ω ) = λ∂ B c ( u, r, Ω ) = O ( r − 1 ) δ BRST A (0) B ( u, Ω ) = 0 n o A (0) is physical d.o.f. , i.e. Cauchy data, on . I + B ( u, Ω )

Physical d.o.f. on null infinity I ± [H.H&Sugishita ’18] Asymptotic � g A (0) B ( u, Ω ) = @ B ✏ (0) ( Ω ) symmetry AS transformation changes the physical operators on . I + 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. AS is physical symmetry . We actually rederived the equivalence between the W-T id of AS and leading soft theorem in QED in BRST formalism.

Physical fall-off condition of ghost fields [H.H&Sugishita ’18] Whether the AS in QED is physical symmetry or not does depends on the fall-off condition of the ghost field in c ( x ) our logic. We chose by hand. r − 1 � � r →∞ ,u : fixed c ( x ) = O lim Q. What happens if we choose ? r →∞ ,u : fixed c ( x ) = O (1) lim δ BRST A B ( u, r, Ω ) = λ∂ B c ( u, r, Ω ) = O (1) n o In this case, is not physical d.o.f. but gauge redundancy. A (0) B ( u, Ω ) Then, AS is not physical symmetry…? This fall-off condition is formally allowed, but that is not consistent with real physics in following sense.

Physical fall-off condition of ghost fields [H.H&Sugishita ’18] BRST transf local U(1) transf λ c ( x ) ✏ ( x ) BRST condition: Q BRST | Ψ i phys = 0 r →∞ ,u : fixed c ( x ) = O (1) lim Physical states : LG | Ψ i e, � = 0 Q ✏ 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. r − 1 � is justified. � r →∞ ,u : fixed c ( x ) = O lim

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

Soft theorem in QED out q out p k j p k j · · · · · · Soft factor S S + O ( ω ) × = in · · · · · · p k i p k i h ω 0 �i + S (sub) S (lead) � ω − 1 � � h out | a B ( ω ˆ x ) S| in i = h out |S| in i + O ( ω ) B e k ~ p k · ✏ B e k ~ p k · ✏ B X X S (lead) � ! − 1 � = − p k · q p k · q k ∈ out k ∈ in e k q µ J k ✓ ◆ ∂ ∂ µB X S (sub) � ω 0 � +(loop corr.) J k = − i µ ν = − i p kµ − p k ν ∂ p µ p k · q ∂ p ν , k k k [Low 1954] Subleading soft factor gets non-universal loop corrections. [Bern, Davies, Nohle ’14]

Subleading soft theorem in QED out q out p k j p k j · · · · · · Soft factor S S + O ( ω ) × = in · · · · · · p k i p k i h ω 0 �i + S (sub) S (lead) � ω − 1 � � h out | a B ( ω ˆ x ) S| in i = h out |S| in i + O ( ω ) B (1 + ω∂ ω ) O ( ω − 1 ) = 0 (1 + ω∂ ω ) 1 = 1 , Subleading soft theorem e k q µ J k µB X ω → 0 h out | (1 + ω∂ ω ) a B ( ω ˆ lim x ) S| in i = � i h out |S| in i +( corr. ) p k · q k

Leading x ) S| in i = S (lead) � ω − 1 � ω → 0 h out | a B ( ω ˆ lim h out |S| in i soft thrm: � in � � Q + S − S Q − � W-T id of AS : ⌦ ↵ out = 0 in massless and massive scalar QED. [He, Mitra, Porfyriadis, Strominger ’14] [Campiglia, Laddha ’15]. Subleading e k q µ J k µB X ω → 0 h out | (1 + ω∂ ω ) a B ( ω ˆ lim x ) S| in i = � i h out |S| in i soft thrm: p k · q k Symmetry � Q (sub)+ S − S Q (sub) − � � D E out � in = 0 � � of S -matrix i + in massless scalar QED. [Lysov, Pasterski, Strominger ’14] I + I + [Campiglia, Laddha ’16] [Conde, Mao ’17] Generalization to massive scalar QED. [ H.H&Sugishita ’18] I − I − i −

Sub leading soft theorem as conservation law Around timelike infinity, [Campiglia, Laddha ’15] i + ( τ , ρ , Ω B ) ds 2 = − d τ 2 + τ 2 dl 2 H 3 I + where, dl 2 H 3 = h αβ d σ α d σ β d ρ 2 1 + ρ 2 + ρ 2 γ AB d Ω A d Ω B = I − ρ = const : trajectory of a free massive particle i − In this coordinate, we can study the asymptotic behavior of massive fields around timelike infinity . τ → ∞

Sub leading soft theorem as conservation law [ H.H&Sugishita ’18] e k q µ J k µB X ω → 0 h out | (1 + ω∂ ω ) a B ( ω ˆ lim x ) S| in i = � i h out |S| in i p k · q k New symmetry of S -matrix � ⌘� D ⇣ ⌘ ⇣ E Q (sub)+ + Q (sub)+ Q (sub) − + Q (sub) − S − S out � in = 0 � � H S H S � = � 1 Z I ± dud 2 Ω p �✏ (0) u @ u ∆ S 2 r B A (0) Q (sub) S ± B 2 p p 1 + ⇢ 2 = 1 Z h i Q (sub) H 3 d 3 � ⇢ 2 h αβ ( r ( h ) α r ( h ) + 2 ⇢ h αβ ( r ( h ) ρ ✏ H 3 ) I mat α ✏ H 3 ) I mat h β β f,i H 2 ⇢ in scalar QED.

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

Discussions Whether the subleading soft theorem including loop correction can be associated with symmetry of S-matrix? 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, “Infrared finite S-matrix” Faddeev, Kulish 1970] Asymptotic symmetry in “infrared finite S-matrix formalism” ?

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Asymptotic symmetry as physical symmetry [H.H&Sugishita ’18] δ BRST A B ( u, r, Ω ) = λ∂ B c ( u, r, Ω ) = O ( r − 1 ) r − 1 � � r →∞ ,u : fixed c ( x ) = O lim δ BRST A (0) ・・ Physical d.o.f B ( u, Ω ) = 0 Asymptotic symmetry changes the physical d.o.f on null infinity � g A (0) B ( u, Ω ) = @ B ✏ (0) ( Ω )