21 2 Domenico Orlando conformal invariance. fmuctuations of σ give the (massive) Goldstone for the broken f 2 2 An EFT for a CFT Introduction to the large charge expansion ( χ is a Goldstone so it is dimensionless.) Start with two derivatives: An action for χ 2 ∂ μ χ ∂ μ χ − C 3 L [ χ ] = f π We want to describe a CFT: we can dress with a dilaton L [ σ , χ ] = f π e − 2 f σ ∂ μ χ ∂ μ χ − e − 6 f σ C 3 + e − 2 f σ � � ∂ μ σ ∂ μ σ − ξ R The fmuctuations of χ give the Goldstone for the broken U ( 1 ) , the
22 An EFT for a CFT Domenico Orlando The fjeld ϕ is some complicated function of the original φ . Scale invariance is manifest. We get 2 f 1 and rewrite the action in terms of a complex scalar We can put together the two fjelds as Linear sigma model Introduction to the large charge expansion Σ = σ + if π χ e − f Σ √ ϕ = L [ ϕ ] = ∂ μ ϕ ∗ ∂ μ ϕ − ξ R ϕ ∗ ϕ − u ( ϕ ∗ ϕ ) 3 Only depends on dimensionless quantities b = f 2 f π and u = 3 ( Cf 2 ) 3 .
23 1 Domenico Orlando An EFT for a CFT Introduction to the large charge expansion Centrifugal barrier The O ( 2 ) symmetry acts as a shift on χ . Fixing the charge is the same as adding a centrifugal term ∝ | ϕ | 2 . V o r centrifugal i g i n a l | barrier | 6 ϕ n e w v a c u u m 2 | φ
24 where Domenico Orlando The classical energy is 1 An EFT for a CFT Introduction to the large charge expansion We can fjnd a fjxed-charge solution of the type Ground state χ ( t , x ) = μ t σ ( t , x ) = 1 f log ( v ) = const. , μ ∝ Q 1 / 2 + . . . v ∝ Q 1 / 2 E = c 3 / 2 VQ 3 / 2 + c 1 / 2 RVQ 1 / 2 + O � Q − 1 / 2 �
25 An EFT for a CFT Domenico Orlando nothing to do with a light dilaton in the full theory) (This is a heavy fmuctuation around the semiclassical state. It has 2 μ p 2 The fmuctuations over this ground state are described by two modes. 1 Fluctuations Introduction to the large charge expansion • A universal “ conformal Goldstone ”. It comes from the breaking of the U ( 1 ) . √ ω = • The massive dilaton . It controls the magnitude of the quantum fmuctuations. All quantum effects are controled by 1 / Q . ω = 2 μ + p 2
26 An EFT for a CFT Non-linear sigma model Since σ is heavy we can integrate it out and write a non-linear sigma model (NLSM) for χ alone. These are the leading terms in the expansion around the classical Domenico Orlando Introduction to the large charge expansion L [ χ ] = k 3 / 2 ( ∂ μ χ ∂ μ χ ) 3 / 2 + k 1 / 2 R ( ∂ μ χ ∂ μ χ ) 1 / 2 + . . . solution χ = μ t . All other terms are suppressed by powers of 1 / Q .
27 An EFT for a CFT State-operator correspondence Δ H Protected by conformal invariance: a well-defjned quantity. Domenico Orlando Introduction to the large charge expansion The anomalous dimension on R d is the energy in the cylinder frame. R d R × S d − 1 S d − 1 S d − 1
28 2 Domenico Orlando This is the unique contribution of order Q 0 . An EFT for a CFT 2 Introduction to the large charge expansion 1 Goldstone . We know the energy of the ground state. Conformal dimensions The leading quantum effect is the Casimir energy of the conformal E G = √ ζ ( − 1 2 | S 2 ) = − 0 . 0937 . . . Final result: the conformal dimension of the lowest operator of charge Q in the O ( 2 ) model has the form π Q 3 / 2 + 2 √ π c 1 / 2 Q 1 / 2 − 0 . 094 + O � Q − 1 / 2 � Δ Q = c 3 / 2 2 √
29 now matrix-valued and has the form Domenico Orlando mutually commuting Cartan generators H I . An EFT for a CFT Introduction to the large charge expansion under The O ( 2 N ) model Next step: O ( 2 N ) . We take 2 N fjelds and an action that is invariant φ a → M a M T M = 1 . b φ b , The conserved current associated to the global O ( 2 n ) symmetry is ( j μ ) ab = ( φ a ∂ μ φ b − φ b ∂ μ φ a ) . we can only fjx the rank ( O ( 2 n )) coeffjcients in the directions of the � H I , H J � 2 ⟨ QH I ⟩ , ⟨ H I H J ⟩ = 2 δ IJ . q I = 1 = 0 ,
30 An EFT for a CFT Domenico Orlando where ... 0 0 0 0 q 0 q 0 Introduction to the large charge expansion n system is invariant. transformation M such that conjugacy class of Q . The energy of a state of fjxed charge Q can only depend on the The O ( 2 N ) model The q I transform under the action of O ( 2 n ) , while the spectrum of the There exists a homogeneous ground state . There is always an O ( 2 n ) ˆ − ˆ MQM − 1 = q I H I = ∑ ˆ . I = 1 q = q 1 + . . . q N ˆ
31 π Domenico Orlando (but e.g. in large- N ). The coeffjcients depend on N and cannot be computed in the EFT An EFT for a CFT Introduction to the large charge expansion and takes the same form The ground-state energy only depends on the sum of the charges The O ( 2 N ) model q = q 1 + · · · + q N ˆ q 3 / 2 + 2 √ E = c 3 / 2 ( N ) q 1 / 2 + O � q − 1 / 2 � 2 √ ˆ π c 1 / 2 ( N ) ˆ ˆ
32 spont. Domenico Orlando We have singled out the time. The system is non-relativistic. 2 p 1 An EFT for a CFT freedom (DOF). exp. The symmetry breaking pattern is Fluctuations Introduction to the large charge expansion O ( 2 N ) − → U ( N ) − → U ( N − 1 ) and there are dim ( U ( N ) / U ( N − 1 )) = 2 N − 1 degrees of • One singlet, the universal conformal Goldstone ω = √ • One vector of U ( N − 1 ) , with quadratic dispersion ω = p 2 2 μ + . . . antiferromagnet ω ∝ p ferromagnet ω ∝ p 2 (count double)
33 1 Domenico Orlando and the dispersion relation An EFT for a CFT Introduction to the large charge expansion The inverse propagator for the type-II is Type II Goldstones 2 ( ∇ 2 − ∂ 2 � 1 0 ) � μ ∂ 0 D − 1 = 2 ( ∇ 2 − ∂ 2 − μ ∂ 0 0 ) � p 2 + μ 2 ± μ . ω = Each type-II Goldstone counts for two DOF: 1 + 2 × ( N − 1 ) = 2 N − 1 . Only the type-I has a Q 0 contribution: it is universal .
34 An EFT for a CFT Domenico Orlando Introduction to the large charge expansion O ( 4 ) on the lattice 12 10 8 D(j, j) 6 4 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 j π ( 2 j ) 3 / 2 + 2 √ π c 1 / 2 ( 2 j ) 1 / 2 − 0 . 094 + O � j − 1 / 2 � Δ j = c 3 / 2 2 √
35 An EFT for a CFT What happened? We started from a CFT. We picked a sector. plus massless fmuctuations. The full theory has no small parameters but we can study this sector with a simple EFT . observables using perturbation theory . Domenico Orlando Introduction to the large charge expansion There is no mass gap, there are no particles , there is no Lagrangian . In this sector the physics is described by a semiclassical confjguration We are in a strongly coupled regime but we can compute physical
36 Large N vs. Large Charge Large N vs. Large Charge Domenico Orlando Introduction to the large charge expansion
37 d t d Σ Domenico Orlando d Σ j 0 where Large N vs. Large Charge N We compute the partition function at fjxed charge Introduction to the large charge expansion N The model φ 4 model on R × Σ for N complex fjelds � � i ϕ i ) 2 � g μν ( ∂ μ ϕ i ) ∗ ( ∂ ν ϕ i ) + r ϕ ∗ 2 ( ϕ ∗ ∑ S θ [ ϕ i ] = i ϕ i + u i = 1 It fmows to the WF in the IR limit u → ∞ when r is fjne-tuned. � � e − β H δ ( ˆ ∏ Z ( Q 1 , . . . , Q N ) = Tr Q i − Q i ) i = 1 � � ˆ ϕ ∗ i ϕ i − ϕ ∗ Q i = i = i d Σ [ ˙ ϕ i ] . i ˙ Dimensions of operators of fjxed charge Q on R 3 (state/operator): Δ ( Q ) = − 1 β log Z S 2 ( Q ) .
38 N Domenico Orlando d θ d θ well understood. Q depends on the momenta, the integration is not trivial but Large N vs. Large Charge Q i N Introduction to the large charge expansion 2 π N Fix the charge Explicitly d θ i � � � π e − i θ i ˆ e i θ i Q i Tr e − β H ∏ ∏ ∏ Z = . − π i = 1 i = 1 i = 1 Since ˆ � π � 2 π e − i θ Q D ϕ i e − S [ ϕ ] Z Σ ( Q ) = − π ϕ ( 2 πβ )= e i θ ϕ ( 0 ) � π � 2 π e − i θ Q D ϕ i e − S θ [ ϕ ] = − π ϕ ( 2 πβ )= ϕ ( 0 )
38 N Domenico Orlando d θ d θ well understood. Q depends on the momenta, the integration is not trivial but Large N vs. Large Charge Q i N Introduction to the large charge expansion 2 π N Fix the charge Explicitly d θ i � � � π e − i θ i ˆ e i θ i Q i Tr e − β H ∏ ∏ ∏ Z = . − π i = 1 i = 1 i = 1 Since ˆ boundary condition � π � 2 π e − i θ Q D ϕ i e − S [ ϕ ] Z Σ ( Q ) = − π ϕ ( 2 πβ )= e i θ ϕ ( 0 ) � π � 2 π e − i θ Q D ϕ i e − S θ [ ϕ ] = − π ϕ ( 2 πβ )= ϕ ( 0 )
38 N Domenico Orlando d θ d θ well understood. Q depends on the momenta, the integration is not trivial but Large N vs. Large Charge Q i N Introduction to the large charge expansion 2 π N Fix the charge Explicitly d θ i � � � π e − i θ i ˆ e i θ i Q i Tr e − β H ∏ ∏ ∏ Z = . − π i = 1 i = 1 i = 1 covariant derivative Since ˆ � π � 2 π e − i θ Q D ϕ i e − S [ ϕ ] Z Σ ( Q ) = − π ϕ ( 2 πβ )= e i θ ϕ ( 0 ) � π � 2 π e − i θ Q D ϕ i e − S θ [ ϕ ] = − π ϕ ( 2 πβ )= ϕ ( 0 )
39 β ϕ Domenico Orlando λ 2 1 Expand around the VEV μ ϕ i D i μ ϕ i D i d t d Σ Large N vs. Large Charge N rewrite the action as Stratonovich transformation: introduce Lagrange multiplier λ and Introduction to the large charge expansion where Effective actions The covariant derivative approach: N d t d Σ � � � ( D μ ϕ i ) ∗ ( D μ ϕ i ) + R 8 ϕ ∗ i ϕ i + 2 u ( ϕ ∗ ∑ S θ [ ϕ ] = i ϕ i ) 2 i = 1 � D 0 ϕ = ∂ 0 ϕ + i θ D i ϕ = ∂ i ϕ � � ∗ � �� � �� � + ( r + λ ) ϕ ∗ ∑ − i θ i Q i + S Q = i ϕ i i = 1 m 2 − r � � + ˆ ϕ i = √ A i + u i , λ =
40 Large N vs. Large Charge Saddle point equations With some massaging, we fjnd the fjnal equations Domenico Orlando Introduction to the large charge expansion � F Σ ( Q ) = mQ + N ζ ( − 1 2 | Σ , m ) , 2 | Σ , m ) = − Q m ζ ( 1 N . The control parameter is actually Q / N .
41 2 N Domenico Orlando 2 N 4 π V 12 2 N V 4 π 3 F Σ 2 N Large N vs. Large Charge V 24 4 π Introduction to the large charge expansion 4 π kernel coeffjcients) V The zeta function is written in terms of the geometry of Σ (heat Large Q / N If Q / N ≫ 1 we can use Weyl’s asymptotic expansion. e Δ Σ t � = ∞ K n t n / 2 − 1 . � ∑ Tr n = 0 � − 1 / 2 � � 1 / 2 � � Q � Q m Σ = + R + . . . � � 3 / 2 � � 1 / 2 � Q � Q 2 N = 2 + R + . . .
42 3 Domenico Orlando 2 N 90720 2 N Large N vs. Large Charge 2 N 360 Introduction to the large charge expansion 2 N Order N 3 � 3 / 2 � 1 / 2 � Q � Q F S 2 ( Q ) = 4 N + N e − √ � − 1 / 2 � − 3 / 2 � Q � Q � Q / ( 2 N ) � − 7 N − 71 N + O
42 2 N Domenico Orlando 2 N 90720 2 N Large N vs. Large Charge 360 Introduction to the large charge expansion 3 2 N Order N 3 leading Q 3 / 2 � 3 / 2 � 1 / 2 � Q � Q F S 2 ( Q ) = 4 N + N e − √ � − 1 / 2 � − 3 / 2 � Q � Q � Q / ( 2 N ) � − 7 N − 71 N + O
42 2 N Domenico Orlando 2 N 90720 2 N Large N vs. Large Charge 360 Introduction to the large charge expansion 3 2 N Order N 3 1 / Q expansion � 3 / 2 � 1 / 2 � Q � Q F S 2 ( Q ) = 4 N + N e − √ � − 1 / 2 � − 3 / 2 � Q � Q � Q / ( 2 N ) � − 7 N − 71 N + O
42 2 N Domenico Orlando 2 N 90720 2 N Large N vs. Large Charge 360 Introduction to the large charge expansion 3 Order N 2 N 3 EFT coeffjcients EFT coeffjcients � 3 / 2 � 1 / 2 � Q � Q F S 2 ( Q ) = 4 N + N e − √ � − 1 / 2 � − 3 / 2 � Q � Q � Q / ( 2 N ) � − 7 N − 71 N + O
42 2 N Domenico Orlando Large N vs. Large Charge 2 N 90720 2 N 360 Introduction to the large charge expansion 3 3 Order N 2 N Q 3 / 2 0.500 1 Q 9 / 2 Q asymptotic expansion 0.100 1 0.050 Q 1 0.010 Q 7 / 2 1 0.005 1 Q 5 / 2 Q 3 / 2 1 2 3 4 5 6 7 � 3 / 2 � 1 / 2 � Q � Q F S 2 ( Q ) = 4 N + N e − √ � − 1 / 2 � − 3 / 2 � Q � Q � Q / ( 2 N ) � − 7 N − 71 N + O
43 Large N vs. Large Charge Where is the universal Goldstone? Domenico Orlando Introduction to the large charge expansion
44 Large N vs. Large Charge Was it worth it? Domenico Orlando Introduction to the large charge expansion
45 Large N vs. Large Charge Domenico Orlando would you like to know more? 2 N Introduction to the large charge expansion Final result 2 N � 3 / 2 � 1 / 2 � 4 N �� Q � N �� Q Δ ( Q ) = 3 + O ( 1 ) + 3 + O ( 1 ) + . . . − 0 . 0937 . . .
45 2 N Domenico Orlando would you like to know more? Large N vs. Large Charge 2 N Introduction to the large charge expansion Final result � 3 / 2 � 1 / 2 � 4 N �� Q � N �� Q Δ ( Q ) = 3 + O ( 1 ) + 3 + O ( 1 ) + . . . − 0 . 0937 . . . 0.7 0.6 0.4 0.5 0.3 0.4 c 3 2 c 1 2 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 N N
46 Large charge and supersymmetry Large charge and supersymmetry Domenico Orlando Introduction to the large charge expansion
47 Large charge and supersymmetry And Now for Something Completely Different All the models that you have seen have something in common: isolated vacuum. No moduli space. What happens when there is a fmat direction? dimensions. Coulomb branch with a dimension-one moduli space: all the physics We will write an effective action for a canonically-normalized dimension-one vector multiplet Φ . Domenico Orlando Introduction to the large charge expansion Many known examples of (non-Lagrangian) N ≥ 2 SCFT in four is encoded in a single operator O and every chiral operator is just O n .
from anomaly 2, everything else is a D-term and does not 48 L K Domenico Orlando theory (with one-dimensional moduli space). 2 Claim : at large R-charge this action is all you need for any anomaly in the UV. The coeffjcient α fjxes the a -anomaly of the EFT. It has to match the α L WZ L EFT Large charge and supersymmetry contribute to protected quantities. Because of There will also be a WZ term for the Weyl symmetry and U 1 charge. We have a single vector multiplet. The kinetic term is just Effective action Introduction to the large charge expansion � d 4 θ Φ 2 + c.c. = | ∂ φ | 2 + fermions + gauge fjelds L k =
from anomaly 2, everything else is a D-term and does not 48 L K Domenico Orlando theory (with one-dimensional moduli space). 2 Claim : at large R-charge this action is all you need for any anomaly in the UV. The coeffjcient α fjxes the a -anomaly of the EFT. It has to match the α L WZ L EFT Large charge and supersymmetry contribute to protected quantities. Because of We have a single vector multiplet. The kinetic term is just Effective action Introduction to the large charge expansion � d 4 θ Φ 2 + c.c. = | ∂ φ | 2 + fermions + gauge fjelds L k = There will also be a WZ term for the Weyl symmetry and U ( 1 ) charge.
from anomaly 48 The coeffjcient α fjxes the a -anomaly of the EFT. It has to match the Domenico Orlando theory (with one-dimensional moduli space). 2 Claim : at large R-charge this action is all you need for any anomaly in the UV. contribute to protected quantities. Large charge and supersymmetry We have a single vector multiplet. The kinetic term is just Effective action Introduction to the large charge expansion � d 4 θ Φ 2 + c.c. = | ∂ φ | 2 + fermions + gauge fjelds L k = There will also be a WZ term for the Weyl symmetry and U ( 1 ) charge. Because of N = 2, everything else is a D-term and does not L EFT = L K + α L WZ
48 Large charge and supersymmetry Domenico Orlando theory (with one-dimensional moduli space). anomaly in the UV. The coeffjcient α fjxes the a -anomaly of the EFT. It has to match the contribute to protected quantities. We have a single vector multiplet. The kinetic term is just Effective action Introduction to the large charge expansion � d 4 θ Φ 2 + c.c. = | ∂ φ | 2 + fermions + gauge fjelds L k = from anomaly There will also be a WZ term for the Weyl symmetry and U ( 1 ) charge. Because of N = 2, everything else is a D-term and does not L EFT = L K + α L WZ Claim : at large R-charge this action is all you need for any N = 2
The OPE of Φ with itself is regular , so we can set x 2 x 1 and the three-point function is actually a two-point function . Φ n x 1 Φ n x 2 n Δ is the controlling parameter (it’s the R-charge ) The coeffjcients satisfy a Toda lattice equation that can be solved n n Domenico Orlando using as boundary condition the one loop EFT computation. Q 2 n Δ x 2 x 1 49 C n n Large charge and supersymmetry Three-point function of the Coulomb branch operators Observables Introduction to the large charge expansion C n 1 , n 2 , n 1 + n 2 � � Φ n 1 + n 2 ( x 3 ) Φ n 1 ( x 1 ) Φ n 2 ( x 2 ) ¯ = | x 1 − x 3 | 2 n 1 Δ | x 2 − x 3 | 2 n 2 Δ
The coeffjcients satisfy a Toda lattice equation that can be solved 49 Large charge and supersymmetry Domenico Orlando using as boundary condition the one loop EFT computation. Introduction to the large charge expansion Observables Three-point function of the Coulomb branch operators C n 1 , n 2 , n 1 + n 2 � � Φ n 1 + n 2 ( x 3 ) Φ n 1 ( x 1 ) Φ n 2 ( x 2 ) ¯ = | x 1 − x 3 | 2 n 1 Δ | x 2 − x 3 | 2 n 2 Δ The OPE of Φ with itself is regular , so we can set x 2 = x 1 and the three-point function is actually a two-point function . C n ′ , n − n ′ , n = | x 1 − x 2 | 2 n Δ � Φ n ( x 1 ) ¯ Φ n ( x 2 ) � Q = n Δ is the controlling parameter (it’s the R-charge )
49 Large charge and supersymmetry Domenico Orlando using as boundary condition the one loop EFT computation. Introduction to the large charge expansion Three-point function of the Coulomb branch operators Observables C n 1 , n 2 , n 1 + n 2 � � Φ n 1 + n 2 ( x 3 ) Φ n 1 ( x 1 ) Φ n 2 ( x 2 ) ¯ = | x 1 − x 3 | 2 n 1 Δ | x 2 − x 3 | 2 n 2 Δ The OPE of Φ with itself is regular , so we can set x 2 = x 1 and the three-point function is actually a two-point function . C n ′ , n − n ′ , n = | x 1 − x 2 | 2 n Δ � Φ n ( x 1 ) ¯ Φ n ( x 2 ) � Q = n Δ is the controlling parameter (it’s the R-charge ) The coeffjcients satisfy a Toda lattice equation that can be solved
50 Large charge and supersymmetry Final result This result is valid for any rank-one theory, Lagrangian or not. Domenico Orlando Introduction to the large charge expansion The fjnal result for the generator O of the Coulomb branch is: � = C n ( τ , ¯ τ ) Γ ( 2 n Δ + α + 1 ) � O n ( x 1 ) ¯ O n ( x 2 ) | x 1 − x 2 | 2 n Δ The coeffjcient C n is scheme-dependent. The gamma term is universal , only depends on α . We have completely resummed the 1 / Q expansion.
51 localization. arXiv:1602.05971 Domenico Orlando Large charge and supersymmetry Introduction to the large charge expansion How well does this work? Comparison with localization For the special case of SU ( 2 ) SQCD with N f = 4 we can compare with 1 4 5 10 2 15 0 20 25 - 2 30 - 4 40 50 - 6 n 5 10 15 20 25 30 35
52 Large charge and supersymmetry Domenico Orlando would you like to know more? Taken from arXiv:2006.01847 Introduction to the large charge expansion Comparison with boostrap For strongly coupled theories one can use bootstrap to place bounds This is the worst possible situation for us. And still… on the three-point coeffjcients with n = 1. λ 2 ϕ 2 4 H 0 H 1 H 2 SU ( N ) 3 MFT 2 1 0 ∆ ϕ 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0
53 Asymptotically safe QFT An asymptotically safe QFT Domenico Orlando Introduction to the large charge expansion
54 Asymptotically safe QFT IR vs. UV We have discussed an IR fjxed point. We need a hierarchy for the scale Λ of the EFT 1 The situation improves if we consider a ultraviolet (UV) fjxed point. 1 and we can take the charge as large as we like. Domenico Orlando Introduction to the large charge expansion The fjxed charge induces a scale Λ Q = Q 1 / d r . r ≪ Λ ≪ Λ Q ≪ Λ UV r ≪ Λ UV ≪ Λ ≪ Λ Q
55 Asymptotically safe QFT Domenico Orlando 19 Perturbatively-controlled UV fjxed point fjxed, this theory is asymptotically safe. Introduction to the large charge expansion An asymptotically safe theory DQ � + y Tr � ¯ � � ¯ Q L HQ R + ¯ Q R H † Q L L = − 1 2 Tr ( F μν F μν ) + Tr Qi / − v ( Tr H † H ) 2 − R � ∂ μ H † ∂ μ H � � � 2 � � H † H H † H − u Tr + Tr 6 Tr . In the Veneziano limit of N F → ∞ , N C → ∞ with the ratio N F / N C √ 23 − 1 α ∗ α ∗ α ∗ α ∗ v = − 0 . 13 ε . g = 26 57 ε , y = 4 19 ε , h = ε ,
56 Asymptotically safe QFT An asymptotically safe theory New features from our point of view Domenico Orlando Introduction to the large charge expansion • H is a matrix. There is a large non-Abelian global symmetry • there are fermions • there are gluons • it’s a four-dimensional system • we have a trustable effective action
57 0 Domenico Orlando 0 0 0 0 The ground state is Asymptotically safe QFT 0 Introduction to the large charge expansion For simplicity The scalar sector and the equations of motion (EOM) reduce to Inspired by the O ( 2 ) model we use a homogeneous ansatz H 0 = e 2 iMt B , 2 M 2 = uB 2 + v Tr � B 2 � − R 12 . � 1 � Q L = −Q R = J , − 1 where 1 is the N F / 2 × N F / 2 identity matrix. � 1 � � 1 � M = μ B = b , . − 1 1
58 36 Domenico Orlando We have again an expansion in powers of the charge. N F which is a natural expansion in 2 π 2 6 144 1 Asymptotically safe QFT 2 π 2 Introduction to the large charge expansion Ground state energy and fmuctuations V F The ground state has energy N 2 2 � 1 / 3 � � 2 / 3 � 2 π � V J 4 / 3 + R J 2 / 3 E = 3 α h + α v J − 2 / 3 �� � 4 / 3 � R � 2 � V J 0 + O � − J = 2 J α h + α v ≫ 1 The leading exponent is 4 / 3 because we are in four dimensions.
59 Type-I and type-II Goldstones . Domenico Orlando Total count: α h 2 μ Asymptotically safe QFT p Introduction to the large charge expansion spont. Goldstones The symmetry-breaking pattern is quite involved exp. SU ( N F ) × SU ( N F ) × U ( 1 ) − → C ( M ) × SU ( N F ) − → C ( M ) . where C ( M ) = SU ( N F / 2 ) × SU ( N F / 2 ) × U ( 1 ) 2 . • One conformal Goldstone ω = √ 3 , which is a singlet of C ( M ) • One bifundamental with ω = p 2 • One fjeld in the ( Adj , 1 ) and one in the ( 1 , Adj ) with � ω = 3 α h + 2 α v p 1 + 2 × ( N F / 2 ) 2 + 2 × ( N 2 F / 4 − 1 ) = N 2 F − 1 = dim ( SU ( N F ))
60 Asymptotically safe QFT Summing it up theories dispersions. would you like to know more? Domenico Orlando Introduction to the large charge expansion • We can use the large-charge expansion for asymptotically safe • Being in the UV, the large-charge condition is more natural • For the QCD-inspired model that we have considered: • Fermions and gluons decouple. • 1 / J expansion of the anomalous dimensions, starting at J 4 / 3 • Rich spectrum of Goldstone modes , with linear and quadratic
61 Conclusions In conclusion systems perturbatively . semiclassical state . Domenico Orlando Introduction to the large charge expansion • With the large-charge approach we can study strongly-coupled • Select a sector and we write a controllable effective theory . • The strongly-coupled physics is (for the most part) subsumed in a • Compute the CFT data. • Very good agreement with lattice (supersymmetry, large N ). • Precise and testable predictions .
62 Large N vs. Large Charge Large N vs. Large Charge Domenico Orlando Introduction to the large charge expansion
63 d t d Σ Domenico Orlando d Σ j 0 where Large N vs. Large Charge N We compute the partition function at fjxed charge Introduction to the large charge expansion N The model φ 4 model on R × Σ for N complex fjelds � � i ϕ i ) 2 � g μν ( ∂ μ ϕ i ) ∗ ( ∂ ν ϕ i ) + r ϕ ∗ 2 ( ϕ ∗ ∑ S θ [ ϕ i ] = i ϕ i + u i = 1 It fmows to the WF in the IR limit u → ∞ when r is fjne-tuned. � � e − β H δ ( ˆ ∏ Z ( Q 1 , . . . , Q N ) = Tr Q i − Q i ) i = 1 � � ˆ ϕ ∗ i ϕ i − ϕ ∗ Q i = i = i d Σ [ ˙ ϕ i ] . i ˙ Dimensions of operators of fjxed charge Q on R 3 (state/operator): Δ ( Q ) = − 1 β log Z S 2 ( Q ) .
64 N Domenico Orlando d θ d θ well understood. Q depends on the momenta, the integration is not trivial but Large N vs. Large Charge Q i N Introduction to the large charge expansion 2 π N Fix the charge Explicitly d θ i � � � π e − i θ i ˆ e i θ i Q i Tr e − β H ∏ ∏ ∏ Z = . − π i = 1 i = 1 i = 1 Since ˆ � π � 2 π e − i θ Q D ϕ i e − S [ ϕ ] Z Σ ( Q ) = − π ϕ ( 2 πβ )= e i θ ϕ ( 0 ) � π � 2 π e − i θ Q D ϕ i e − S θ [ ϕ ] = − π ϕ ( 2 πβ )= ϕ ( 0 )
64 N Domenico Orlando d θ d θ well understood. Q depends on the momenta, the integration is not trivial but Large N vs. Large Charge Q i N Introduction to the large charge expansion 2 π N Fix the charge Explicitly d θ i � � � π e − i θ i ˆ e i θ i Q i Tr e − β H ∏ ∏ ∏ Z = . − π i = 1 i = 1 i = 1 Since ˆ boundary condition � π � 2 π e − i θ Q D ϕ i e − S [ ϕ ] Z Σ ( Q ) = − π ϕ ( 2 πβ )= e i θ ϕ ( 0 ) � π � 2 π e − i θ Q D ϕ i e − S θ [ ϕ ] = − π ϕ ( 2 πβ )= ϕ ( 0 )
64 N Domenico Orlando d θ d θ well understood. Q depends on the momenta, the integration is not trivial but Large N vs. Large Charge Q i N Introduction to the large charge expansion 2 π N Fix the charge Explicitly d θ i � � � π e − i θ i ˆ e i θ i Q i Tr e − β H ∏ ∏ ∏ Z = . − π i = 1 i = 1 i = 1 covariant derivative Since ˆ � π � 2 π e − i θ Q D ϕ i e − S [ ϕ ] Z Σ ( Q ) = − π ϕ ( 2 πβ )= e i θ ϕ ( 0 ) � π � 2 π e − i θ Q D ϕ i e − S θ [ ϕ ] = − π ϕ ( 2 πβ )= ϕ ( 0 )
65 β ϕ Domenico Orlando λ 2 1 Expand around the VEV μ ϕ i D i μ ϕ i D i d t d Σ Large N vs. Large Charge N rewrite the action as Stratonovich transformation: introduce Lagrange multiplier λ and Introduction to the large charge expansion where Effective actions The covariant derivative approach: N d t d Σ � � � ( D μ ϕ i ) ∗ ( D μ ϕ i ) + R 8 ϕ ∗ i ϕ i + 2 u ( ϕ ∗ ∑ S θ [ ϕ ] = i ϕ i ) 2 i = 1 � D 0 ϕ = ∂ 0 ϕ + i θ D i ϕ = ∂ i ϕ � � ∗ � �� � �� � + ( r + λ ) ϕ ∗ ∑ − i θ i Q i + S Q = i ϕ i i = 1 m 2 − r � � + ˆ ϕ i = √ A i + u i , λ =
66 θ 2 Domenico Orlando λ , that can be expanded order-by-order λ i λ μ D i Large N vs. Large Charge i A 2 i Introduction to the large charge expansion V β N λ λ alone Effective action for ˆ We can now integrate out the u i and get an effective action for ˆ � � � � � μ + m 2 + ˆ �� S θ [ ˆ ∑ λ ] = β 2 + m 2 2 + Tr − D i log i = 1 � � ˆ λΔ ˆ − A 2 � 2 Tr . This is a non-local action for ˆ in 1 / N . Today we will only look at the leading order (saddle point).
67 N Domenico Orlando β β p where i θ 2 s Large N vs. Large Charge Introduction to the large charge expansion Saddle point equations � V β � ∂ S Q ∑ ∂ m 2 = i + ζ ( 1 | θ i , Σ , m ) = 0 , 2 A 2 i = 1 � ∂ S Q ∂ � = − iQ + θ i i + 1 ζ ( s | θ i , Σ , m ) = 0 � ∂ θ i ∂ θ i β VA 2 � s = 0 � � ∂ S Q = V β β 2 + m 2 A i = 0 . ∂ A i � − s �� 2 π n � 2 + E ( p ) 2 + m 2 ζ ( s | θ , Σ , m ) = ∑ n ∈ Z ∑ + θ .
68 Large N vs. Large Charge Saddle point equations With some massaging, we fjnd the fjnal equations Domenico Orlando Introduction to the large charge expansion � F Σ ( Q ) = mQ + N ζ ( − 1 2 | Σ , m ) , 2 | Σ , m ) = − Q m ζ ( 1 N . The control parameter is actually Q / N .
69 π 2 Domenico Orlando one-loop calculation 3 π 4 N 2 Q 2 Large N vs. Large Charge Q Introduction to the large charge expansion Q Result: Small Q / N The zeta function can be expanded in perturbatively in small Q / N . π 2 − 12 � � Δ ( Q ) = 1 2 + 4 N + 16 + . . . • Expansion of a closed expression • Start with the engineering dimension 1 / 2 • Reproduce an infjnite number of diagrams from a fjxed-charge
70 2 N Domenico Orlando 2 N 4 π V 12 2 N V 4 π 3 F Σ 2 N Large N vs. Large Charge V 24 4 π Introduction to the large charge expansion 4 π kernel coeffjcients) V The zeta function is written in terms of the geometry of Σ (heat Large Q / N If Q / N ≫ 1 we can use Weyl’s asymptotic expansion. e Δ Σ t � = ∞ K n t n / 2 − 1 . � ∑ Tr n = 0 � − 1 / 2 � � 1 / 2 � � Q � Q m Σ = + R + . . . � � 3 / 2 � � 1 / 2 � Q � Q 2 N = 2 + R + . . .
71 3 Domenico Orlando 2 N 90720 2 N Large N vs. Large Charge 2 N 360 Introduction to the large charge expansion 2 N Order N 3 � 3 / 2 � 1 / 2 � Q � Q F S 2 ( Q ) = 4 N + N e − √ � − 1 / 2 � − 3 / 2 � Q � Q � Q / ( 2 N ) � − 7 N − 71 N + O
71 2 N Domenico Orlando 2 N 90720 2 N Large N vs. Large Charge 360 Introduction to the large charge expansion 3 2 N Order N 3 leading Q 3 / 2 � 3 / 2 � 1 / 2 � Q � Q F S 2 ( Q ) = 4 N + N e − √ � − 1 / 2 � − 3 / 2 � Q � Q � Q / ( 2 N ) � − 7 N − 71 N + O
71 2 N Domenico Orlando 2 N 90720 2 N Large N vs. Large Charge 360 Introduction to the large charge expansion 3 2 N Order N 3 1 / Q expansion � 3 / 2 � 1 / 2 � Q � Q F S 2 ( Q ) = 4 N + N e − √ � − 1 / 2 � − 3 / 2 � Q � Q � Q / ( 2 N ) � − 7 N − 71 N + O
71 2 N Domenico Orlando 2 N 90720 2 N Large N vs. Large Charge 360 Introduction to the large charge expansion 3 Order N 2 N 3 EFT coeffjcients EFT coeffjcients � 3 / 2 � 1 / 2 � Q � Q F S 2 ( Q ) = 4 N + N e − √ � − 1 / 2 � − 3 / 2 � Q � Q � Q / ( 2 N ) � − 7 N − 71 N + O
71 2 N Domenico Orlando Large N vs. Large Charge 2 N 90720 2 N 360 Introduction to the large charge expansion 3 3 Order N 2 N Q 3 / 2 0.500 1 Q 9 / 2 Q asymptotic expansion 0.100 1 0.050 Q 1 0.010 Q 7 / 2 1 0.005 1 Q 5 / 2 Q 3 / 2 1 2 3 4 5 6 7 � 3 / 2 � 1 / 2 � Q � Q F S 2 ( Q ) = 4 N + N e − √ � − 1 / 2 � − 3 / 2 � Q � Q � Q / ( 2 N ) � − 7 N − 71 N + O
72 Large N vs. Large Charge Universal term: integrate all but one Domenico Orlando Introduction to the large charge expansion
73 σ Domenico Orlando λ out. 2 m At low energies we can approximate the non-local term as 2 Large N vs. Large Charge Introduction to the large charge expansion d t d Σ Order N 0 The order N 0 terms are � � σ ) ∗ ( D μ ˆ σ ) + ( m 2 + ˆ σ ∗ ˆ σ , ˆ S θ [ ˆ λ ] = ( D μ ˆ λ ) ˆ ˆ σ ∗ ) λ v ( ˆ σ + ˆ � + ( N − 1 ) 1 / 2 � d x 1 d x 2 ˆ λ ( x 1 ) ˆ + 1 λ ( x 2 ) D ( x 1 − x 2 ) 2 where D ( x − y ) is the propagator ( D μ D μ + m 2 ) − 1 . � � d t d Σ ˆ d t d Σ ˆ λ ( x ) 2 ζ ( 2 | θ , Σ , m ) ≈ V λ ( x ) 2 and we can integrate ˆ
74 Large N vs. Large Charge Domenico Orlando 2 1 2 Its contribution to the partition function is This is the conformal Goldstone that we have seen in the EFT. Introduction to the large charge expansion m ω It describes a massive mode and a massless mode with dispersion The inverse propagator for σ is Order N 0 � 1 / 2 ( ω 2 + p 2 + 4 m 2 ) � 1 / 2 ( ω 2 + p 2 ) − m ω ω 2 + 1 2 p 2 + . . . = 0 ω 2 + 8 m 2 + 3 2 p 2 + . . . = 0 √ E G = 1 ζ ( 1 / 2 | S 2 ) = − 0 . 0937 . . . This is universal . Does not depend on N or Q .
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