Asympto(c safety v. strings: UV comple(on on the world line Steven - PowerPoint PPT Presentation

Asympto(c safety v. strings: UV comple(on on the world line Steven Abel (Durham) w/ Nicola Dondi (CP3), Daniel Lewis

Overview • Thoughts on asymptoDc safety in a messy UV • RG in a messy UV: the string case • UV compleDon on the world line

Asympto(c safety in a messy UV?

Gastmans et al ’78 Weinberg ’79 Peskin Reuter, WeXerich Gawedski, Kupiainen AS as a UV comple(on Kawai et al, de Calan et al ’, LiDm Morris Weinberg et al’s basis for a proposal of UV complete theories Gaussian IR fixed point => perturbaDve InteracDng UV fixed point => finite anomalous dimensions In a field theory replace 1/e with 1/c => divergences of marginal operators (which affect the fixed point), some cured

Categorise the possible content of a theory as follows: Irrelevant operators: would disrupt the fixed point - therefore asymptoDcally safe theories have to emanate precisely from UV fixed point where they are assumed zero (exactly renormalizable trajectory) Marginal operators: can be involved in determining the UV fixed point where they become exactly marginal. Or can be marginally relevant (asymptoDcally free) or irrelevant. Relevant operators: become “irrelevant” in the UV but may determine the IR fixed point. Dangerously irrelevant operators: grow in both the UV and IR (common in e.g. SUSY) Harmless relevant operators: shrink in both the UV and IR Note relevant or marginally relevant operators sDll have “infiniDes” at the FP - just as quark masses, they sDll run at the FP just like any other relevant operator: but being relevant they do not affect the FP. (By definiDon they become unimportant at in the UV.)

Normally try to think about such UV fixed point behaviour within field theory: but is string theory already asympto(cally free?

Normally try to think about such UV fixed point behaviour within field theory: but is string theory already asympto(cally free? A) No! (Distler) String theory doesn’t need such behaviour to make itself finite. The massless spectrum doesn’t control finiteness, and in any case it doesn’t resemble any known field theory with a UV fixed point.

Normally try to think about such UV fixed point behaviour within field theory: but is string theory already asympto(cally free? A) No! (Distler) String theory doesn’t need such behaviour to make itself finite. The massless spectrum doesn’t control finiteness, and in any case it doesn’t resemble any known field theory with a UV fixed point. B) Yes! (WeEerich) String theory has only one dimensionful parameter (which goes into defining the units by which we measure energy). A second energy scale is needed to observe scale violaDon. This could be the Planck scale, or the dynamical scale of some field theory. But well above the physics at which this second scale is generated, the theory should return to scale invariance(a.k.a. a UV fixed point for operators)

Normally try to think about such UV fixed point behaviour within field theory: but is string theory already asympto(cally free? A) No! (Distler) String theory doesn’t need such behaviour to make itself finite. The massless spectrum doesn’t control finiteness, and in any case it doesn’t resemble any known field theory with a UV fixed point. B) Yes! (WeEerich) String theory has only one dimensionful parameter (which goes into defining the units by which we measure energy). A second energy scale is needed to observe scale violaDon. This could be the Planck scale, or the dynamical scale of some field theory. But well above the physics at which this second scale is generated, the theory should return to scale invariance(a.k.a. a UV fixed point for operators) It would be interesHng to know if it is B) and if so how string theory does it.

• A meaningful RG procedure with a messy UV: aIempt 1) IR cut-off 16 ⇡ 2 � µ 2 gauge ( s ) = � 22 C A ✓ 1 ✓ ◆◆ g 2 A (2) ( p µ p ν � p 2 g µ ν ) ✏ � � E + log 4 ⇡ + log , 3 s ! 16 ⇡ 2 ✏ � � E + log 4 ⇡ + log µ 2 1 + 2 m 2 ferm ( s ) = 4 N f 1 ✓ ◆ g 2 A (2) 3 ( p µ p ν � p 2 g µ ν ) s + Λ ( s ; m f , m f ) , m 2 s f 16 ⇡ 2 ✏ � � E + log 4 ⇡ + log µ 2 1 � 4 m 2 ✓ 1 ✓ ◆ ◆ scalar ( s ) = 2 N s g 2 A (2) 3 ( p µ p ν � p 2 g µ ν ) s + Λ ( s ; m s , m s ) , m 2 s s Interested in s dependence at a parDcular mu. Normally count UV divergences

• A meaningful RG procedure with a messy UV: aIempt 1) Instead count branch cuts as a funcDon of s 16 ⇡ 2 � µ 2 gauge ( s ) = � 22 C A ✓ 1 ✓ ◆◆ g 2 A (2) ( p µ p ν � p 2 g µ ν ) ✏ � � E + log 4 ⇡ + log , 3 s ! 16 ⇡ 2 ✏ � � E + log 4 ⇡ + log µ 2 1 + 2 m 2 ferm ( s ) = 4 N f 1 ✓ ◆ g 2 A (2) 3 ( p µ p ν � p 2 g µ ν ) s + Λ ( s ; m f , m f ) , m 2 s f 16 ⇡ 2 ✏ � � E + log 4 ⇡ + log µ 2 1 � 4 m 2 ✓ 1 ✓ ◆ ◆ scalar ( s ) = 2 N s g 2 A (2) 3 ( p µ p ν � p 2 g µ ν ) s + Λ ( s ; m s , m s ) , m 2 s s The most physical picture: Total s branch cuts just tell us how many states above threshold (s > 4m^2) (but hard to get without doing the actual integral)  16 ⇡ 2 g 2 ( s ) = � 1 � g 2 Im ˜ A (2) ( s ) � 16 π 2 ⇡

• A meaningful RG procedure with a messy UV: aIempt 1) 16 ⇡ 2 � µ 2 gauge ( s ) = � 22 C A ✓ 1 ✓ ◆◆ g 2 A (2) ( p µ p ν � p 2 g µ ν ) ✏ � � E + log 4 ⇡ + log , 3 s ! 16 ⇡ 2 ✏ � � E + log 4 ⇡ + log µ 2 1 + 2 m 2 ferm ( s ) = 4 N f 1 ✓ ◆ g 2 A (2) 3 ( p µ p ν � p 2 g µ ν ) s + Λ ( s ; m f , m f ) , m 2 s f 16 ⇡ 2 ✏ � � E + log 4 ⇡ + log µ 2 1 � 4 m 2 ✓ 1 ✓ ◆ ◆ scalar ( s ) = 2 N s g 2 A (2) 3 ( p µ p ν � p 2 g µ ν ) s + Λ ( s ; m s , m s ) , m 2 s s Or impose IR cut-off on Schwinger integral: equivalent to deep Euclidean s, and then.. ⇣ A (2) ⌘ 16 π 2 ˜ @ g 2 g 2 ( s ) = Re � 16 π 2 @ log s reality on the beta function. These

• Toy example: KK theory m/R Z 1 Z ∞ � ✓ ◆ ⌧ ( s x (1 � x ) � ~ m · ~ m d ⌧ dx ⌧ − 1 ∆ b g 2 ( s ) = � ( non-KK ) X � 8 π 2 + Im exp 8 π 2 R 2 g 2 0 0 m ~ Z 1 Z ∞ ! ~ ` · ~ 1  � X ` R d ⇡ d/ 2 exp s ) = � ( non-KK ) ⌧ ⇡ 2 R 2 + Im d ⌧ dx 2 ∆ b ⌧ ( s x (1 � x ) � 8 π 2 ⌧ 1+ d 0 0 g 2 ~ ` Poisson resum then to get the branch cut expand the exponenDal unDl you get the pole —> log —> power law running beta funcDon: ⇡ ( d +3) / 2 ∆ b R p s � d + O R p s ⇣� � d − 1 ⌘ g 2 ( s ) = � ( non-KK ) � + � 8 π 2 8 π 2 2 d +1 Γ (3 + d/ 2) g 2 ✓ ◆ ⇣ ⌘

• Toy example: KK theory m/R Note that the answer averages over the UV states and is not the same as a naive rigid cut-off at the scale s. (e.g. can introduce Scherk-Schwarz splikng of N=4 theory — the KK modes sDll give zero, even though the naive beta funcDon R p s � d would oscillate as ~ +- ) � b

RG in a messy UV: the string case

• Can we do the same thing in a string theory? • Kaplunovsky + \inOy … calculate threshold correc(ons by doing the same diagram: Z 1 Constrains model r 2 F r 1 x x g 2 d 2 τ Z 1 16 π 2 ( k µ X B int Z α , β ,Z 2 Π µ ν 1 k ν 2 � k 1 .k 2 η µ ν ) Z Z 2 Y M ⇡ (1.7) F 4 π 2 | η ( τ ) | 4 τ 2 F α , β ,Z 2  k Z d 2 z ✓ 4 π i ∂ τ log( ϑ αβ (0 | τ ) ◆  2 π � � | ϑ 1 ( z ) | 2 k 1 .k 2 exp = ( z ) 2 δ ab Tr 4 π 2 ∂ 2 z ) + Q 2 � k 1 .k 2 z log ϑ 1 (¯ ⇥ ¯ τ 2 η ( τ ) τ 2 Z d τ 2 ν ⇡ g 2 e � π s τ 2 1 ✓ 4 π i ∂ τ log ϑ αβ (0 | τ )  1 �◆ 16 π 2 δ ab ( k µ 1 k ν 2 � k 1 .k 2 η µ ν ) Y M + Q 2 4 π 2 Tr � η ( τ ) 4 πτ 2 τ 2

This is the scale s — the answer will go like log(s) — so this gives the correct running in the field theory limit (s << 1) where the cut-off is at tau_2 >> 1. g 2 d 2 τ Z 1 16 π 2 ( k µ X B int Z α , β ,Z 2 Π µ ν 1 k ν 2 � k 1 .k 2 η µ ν ) Z Z 2 Y M ⇡ (1.7) F 4 π 2 | η ( τ ) | 4 τ 2 F α , β ,Z 2  k Z d 2 z ✓ 4 π i ∂ τ log( ϑ αβ (0 | τ ) ◆  2 π � � | ϑ 1 ( z ) | 2 k 1 .k 2 exp = ( z ) 2 δ ab Tr 4 π 2 ∂ 2 z ) + Q 2 � k 1 .k 2 z log ϑ 1 (¯ ⇥ ¯ τ 2 η ( τ ) τ 2 Z d τ 2 ν ⇡ g 2 e � π s τ 2 1 ✓ 4 π i ∂ τ log ϑ αβ (0 | τ )  1 �◆ 16 π 2 δ ab ( k µ 1 k ν 2 � k 1 .k 2 η µ ν ) Y M + Q 2 4 π 2 Tr � η ( τ ) 4 πτ 2 τ 2

Note the importance of e − k 1 · k 2 G 12 ≡ e − sG 12 / 2 − → e − πτ 2 s

Note the importance of e − k 1 · k 2 G 12 ≡ e − sG 12 / 2 − → e − πτ 2 s The parDcle limit of the world-sheet Green’s funcDon gives a natural cut-off in s: .