Untangling scaling dimensions of fixed charge operators in Higgs Theories
Oleg Antipin, Jahmall Bersini, Francesco Sannino, Zhi-Wei Wang, Chen Zhang
UUntangling scaling dimensions of fixed charge operators in Higgs Theories
Oleg A ntipin ♣ , ∗ Jahmall B ersini ♣ , † Francesco S annino ♦ , ♥ , ‡ Zhi-Wei Wang ♦ , § and Chen Zhang ♠¶ ♣ Rudjer Boskovic Institute, Division of Theoretical Physics, Bijeniˇcka 54, 10000 Zagreb, Croatia ♦ CP -Origins & the Danish Institute for Advanced Study Danish IAS ,University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark. ♥ Dipartimento di Fisica “E. Pancini”, Universit`a di Napoli Federico II — INFN sezione di NapoliComplesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy. ♠ Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300
We go beyond a systematic review of the semiclassical approaches for determining the scalingdimensions of fixed-charge operators in U (1) and O ( N ) models by introducing a general strategy aptat determining the relation between a given charge configuration and the associated operators for moreinvolved symmetry groups such as the U ( N ) × U ( M ). We show how, varying the charge configuration,it is possible to access anomalous dimensions of di ff erent operators transforming according to a varietyof irreducible representations of the non-abelian symmetry group without the aid of diagrammaticalcomputations. We illustrate our computational strategy by determining the anomalous dimensionsof several composite operators to the next-to-leading order in the semiclassical expansion for the U ( N ) × U ( M ) conformal field theory (CFT) in 4 − (cid:15) dimensions. Thanks to the powerful interplaybetween semiclassical methods and group theory we can, for the first time, extract scaling dimensionsfor a wide range of operators. Preprint: RBI-ThPhys-2021-8, CP -Origins-2021-01 DNRF90 ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] a r X i v : . [ h e p - t h ] F e b CONTENTS
I. Introduction 2II. Review of semiclassical methods at fixed charge in weakly coupled theories 3A. The U (1) model in 4 − (cid:15) and 3 − (cid:15) dimensions 3B. The O ( N ) model in 4 − (cid:15) and 3 − (cid:15) dimensions 7III. From group theory to operators: The map 11A. The power of symmetries 11B. The nature of charge fixing 13C. Group-theoretic analysis: U (1) and O ( N ) vector models 14D. Group-theoretic analysis: the U ( N ) × U ( M ) linear sigma model 151. Introduction 152. The correspondence between weight and charge configuration 173. Scaling dimension and operator construction 20IV. Semiclassics and anomalous dimensions in the U ( N ) × U ( M ) model 22A. Charging the system 23B. On how to identify the fixed-charge operators 28V. Conclusions 33Acknowledgements 34A. The functions ρ ( x ∗ , M , N , s , A h ∗ , A ∗ v ) and σ ( (cid:96), x ∗ , M , N , s , A h ∗ , A ∗ v ) 34References 35 I. INTRODUCTION
Recently there has been a flurry of interest in studying conformal field theories with continuous global symmetriesin the limit of a large conserved charge ¯ Q in order to access non-perturbative corners of Quantum Field Theories(QFT)s. By identifying the emergence of e ff ective field theories (EFT)s stemming from the large charge dynamics[1–4] one can use them to extract relevant data in inverse powers of the charge as reviewed in [5]. Typically oneis led to determine the scaling dimension of certain fixed-charge operators for a subgroup of QFTs that displayconformal invariance and denoted as CFTs. One can, however, go beyond the CFT limit [6, 7] which is relevantfor establishing the spectrum and dynamics of near conformal dynamics emerging from quantum phase transitions[8, 9]. The approach is often referred to as semiclassical in the sense that the path-integral is typically dominated bytrajectories near the solution of the classical equation of motion.The hurdle in the strongly coupled regime is that the identification and construction of the specific operatorsassociated to a given charge configuration are impossible before solving the theory. This is the reason why such anidentification, except perhaps for some symmetry-protected operators, is left unspecified in the literature.There is, however, another relevant limit in which the semiclassical approach is useful. This is the one in whichthe CFT is perturbative and controlled by a small parameter (cid:15) emerging because there is a non-trivial interactingfixed point near the loss of asymptotic freedom of either perturbatively safe [10] or infrared nature of the Banks-Zakstype [11]. For the safe case this was investigated first in [12]. Another way to introduce a small parameter is toslightly modify the number of space-time dimensions typically injecting, for UV free theories, perturbative infraredfixed points in lower than four dimensions. Here, the charge expansion captures higher orders in the ordinaryperturbative coupling corrections [13–17]. The reason being that the presence of a small parameter allows studyingthe fixed-charge sectors of a CFT by defining a ’t Hooft-like coupling A = (cid:15) ¯ Q in which one can take the limit (cid:15) → A fixed. In fact, one can now resum ordinary perturbation series by providing all-order results inthe A coupling. In particular, much attention has been paid to the time-honoured O ( N ) model, first investigated forany N in 4 − (cid:15) dimensions in [15]. The results were recently successfully tested against ordinary perturbation theoryin [18] to four loops. Later the O(N) model was investigated via the semiclassical approach at large N in variousdimensions in [19, 20].Another, yet unexplored possibility, is to introduce two independent small parameters, one that takes into accountthe deformation of the number of space-time dimensions and the other that controls the original fixed point [21].This last case has not yet been investigated in the literature and will be considered elsewhere.Compared to conventional perturbation theory according to which one chooses a specific composite operatorand then diagrammatically determines its scaling dimension, in the semiclassical fixed-charged framework oneneeds to reverse engineer the given charge configuration to determine the irreducible representation of the relatedcomposite operator. This has, so far, restricted the semiclassical method to the highest weight representation operatorswhere there is no ambiguity with respect to the chosen charge configuration. In this work we show how to accessdi ff erent operators belonging to distinct irreducible representations. This is achieved by means of group-theoreticalconsiderations applied to the semiclassical approach. The resulting e ffi cient procedure will entail:1. Establishing the mathematical connection between classical operator dimensions and group-theoretical weights;2. Varying the charge configuration and using the first point to arrive at di ff erent operators transforming in avariety of irreducible representations and determine their scaling dimensions;3. Developing the strategy to deal with charge configurations that give rise to non-trivial chemical potentials.To test the power of our strategy we investigate several non-abelian global symmetries in various space-timedimensions culminating in the general U ( N ) × U ( M ) global symmetry case.The work is organized as follows. In section II, we introduce the semiclassical methods at fixed charge focusingon their applications and limitations. In doing this, we review a series of results obtained in the literature for U (1)and O ( N ) invariant theories. We then move to section III in which we provide the map between operators and theirgroup structure. we show how to identify the fixed charge operators in the U (1), O ( N ) and we then generalise it to thecase of the U ( N ) × U ( M ) symmetry group. In section IV we study various charge configurations in the U ( N ) × U ( M )model, compute the associated scaling dimensions, and identify the corresponding fixed charge operators. Theresults are used to establish the connection between charge configuration and fixed charge operators. We o ff er ourconclusions in section V. The appendix contains details related to the scaling dimensions of the U ( N ) × U ( M ) model.Readers already familiar with the basics of fixed-charge semiclassical methods may start reading from section III,while readers who wish to quickly get to the main results of this work may directly start from section IV. II. REVIEW OF SEMICLASSICAL METHODS AT FIXED CHARGE IN WEAKLY COUPLED THEORIESA. The U (1) model in − (cid:15) and − (cid:15) dimensions We start this section with a brief introduction to semiclassical methods at fixed charge in QFT by consideringthe Abelian U (1) theory in both 4 − (cid:15) and 3 − (cid:15) spacetime dimensions. These two cases have been investigated in[13, 14, 22, 23] and [17, 24], respectively. The Lagrangian reads L = ∂ ¯ φ∂φ + V (cid:16) ¯ φφ (cid:17) , (1)where V d = − (cid:15) = N λ (cid:16) ¯ φφ (cid:17) and V d = − (cid:15) = N λ (cid:16) ¯ φφ (cid:17) , with λ the bare coupling and N , the normalizations.By virtue of the Noether theorem, the U (1) symmetry implies the existence of a conserved charge ¯ Q given by Q = (cid:90) d d − x j , with j µ = ¯ φ∂ µ φ − φ∂ µ ¯ φ . (2)We adopt conventions such that φ and ¯ φ have charge ¯ Q = + Q = −
1, respectively. This model exhibits aninfrared Wilson-Fisher (WF) fixed point (FP) λ ∗ = λ ∗ ( (cid:15) ) in both d = − (cid:15) and d = − (cid:15) dimensions [25]. Thecorresponding fixed point theory is scale-invariant, and we assume it to be invariant under the full set of conformaltransformations. Furthermore, for (cid:15) (cid:28) ∆ φ ¯ Q of the φ ¯ Q operators, which we define to be the charge ¯ Q operators with the smallest scaling dimension . The CFT associated with the WF fixed point defined in a flat spacetime can be mapped to a QFT defined on acylinder geometry in a Weyl-invariant manner. Weyl invariance then dictates a correspondence between correlationfunctions of the two theories . Considering polar coordinates ( r , Ω d − ) for R d , the map reads R d → R × S d − , ( r , Ω d − ) → ( τ, Ω d − ) , r = Re τ/ R , ds = d τ + R d Ω d − = R r ds (3)with R the radius of the sphere and τ the time coordinate on the cylinder. According to the state-operator correspon-dence [29, 30], the action of an operator O at τ = −∞ creates a state with the same quantum numbers and with energygiven by E O ≡ ∆ O R . Since we are looking for the smallest scaling dimension, our goal turned to the computation of theground state energy (at fixed charge ¯ Q ) on the cylinder, E φ ¯ Q . This can be calculated by considering the expectationvalue of the evolution operator e − HT (with H the Hamiltonian and T = τ f − τ i ) in an arbitrary state | ¯ Q (cid:105) with fixedcharge ¯ Q and then taking the limit T → ∞ to project out the ground state from it. That is (cid:104) ¯ Q | e − HT | ¯ Q (cid:105) = T →∞ ˜ N e − E φ ¯ Q T . (4)Notice that we only require | ¯ Q (cid:105) have a non-zero overlap with the lowest-lying state in the fixed charge sector. Thenwhen we insert a complete set of energy eigenstates in the left-hand side (LHS) of the above equation, only thecontribution of the lowest energy state survives, with a prefactor ˜ N that is independent of T but depends on theoverlap between the states. Therefore, we may always extract the ground state energy from the T -dependent part ofthe expectation value. We consider polar coordinates for the field: φ = ρ √ e i χ , ¯ φ = ρ √ e − i χ . (5)Then a convenient choice for | ¯ Q (cid:105) yields [13] (cid:104) ¯ Q | e − HT | ¯ Q (cid:105) = Z − (cid:90) D ρ D χ e − S ef f , (6)where S e f f = (cid:90) T / − T / d τ (cid:90) d Ω d − (cid:34)
12 ( ∂ρ ) + ρ ( ∂χ ) + ˜ V ( ρ ) + i ¯ QR d − Ω d − ˙ χ (cid:35) , ˜ V ( ρ ) = V ( ρ ) + m ρ . (7)The prefactor Z is a T -independent constant that does not a ff ect the determination of the scaling dimension. The massterm m = d − d − R in (7) arises from the conformal coupling to the Ricci scalar R of S d − [31]. On a d − R , we have R = ( d − d − R and thus m = (cid:16) d − R (cid:17) . Rescaling the field as ρ → ρ/λ / and collectingan overall λ − as loop counting parameter, we see that, at small coupling, this path integral can be computed via asaddle point expansion, resulting in RE φ ¯ Q = (cid:88) k = − λ k e k ( A , d ) = (cid:88) k = − λ k ¯ e k ( A , RM , d ) . (8)where M is the renormalization group (RG) scale, ¯ e k ( A , RM , d ) the renormalized coe ffi cients of the semiclassicalexpansion and we introduced the ’t Hooft coupling A ≡ λ ¯ Q ( A ≡ λ ¯ Q as the renormalized one). In the last equality,we have renormalized the result by separating the divergent part in every term of the expansion and absorbing it inthe ˜ N coe ffi cient in (4). At the fixed point, the dependence on RM drops and we obtain ∆ φ ¯ Q = λ ∗ ∆ − ( A ∗ ) + ∆ ( A ∗ ) + λ ∗ ∆ ( A ∗ ) + · · · , (9) Since in the free theory limit derivatives increase the scaling dimension, in the perturbative regime φ ¯ Q is the lowest-lying operator with charge¯ Q . On the other hand, at large coupling, level crossing with other operators can, in principle, occur. We refer the reader to Refs. [26, 27] for introductory accounts of the Weyl map, and especially Ref. [28] for conceptual clarification betweenconformal invariance and Weyl invariance including the implication for correlation functions. Since | ¯ Q (cid:105) can be chosen arbitrarily as long as it has a nonzero overlap with the lowest lying fixed charge state, we do not impose any boundarycondition on ρ . This allows us to compute the saddle point expansion for the path integral. where the star notation “ ∗ ” denotes the value of the coupling at the FP, and the ∆ k are the ( k + as alarge charge expansion in 1 / Q in the limit Q → ∞ and A fixed, i.e. ∆ φ ¯ Q = (cid:88) k = − Q k ˜ ∆ k ( A ∗ ) , ˜ ∆ k ≡ ∆ k A k , (10)which is akin to the large number of flavor expansions in gauge theories [32–45].To compute the leading order (LO) contributions ∆ − ( A ∗ ), we need to solve the classical system and evaluate S e f f on the solution. The solution of the EOM with the lowest energy at fixed ¯ Q is spatially homogeneous and reads ρ = f , χ = − i µτ + const. , (11)where µ = f ∂ ˜ V ( ρ ) ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = f ¯ QR d − Ω d − = µ f . (12)Fixing the charge produces spontaneous symmetry breaking and the fields take a non-zero vev with µ playing therole of a fixed chemical potential. ∆ − ( A ∗ ) is then given by Eq.(7) evaluated on the solution (11) at the fixed point1 λ ∗ ∆ − ( A ∗ ) R = S e f f T = µ Q + ˜ V ( f ) . (13)Then, by plugging the second equation in (12) into the first one, we have( R µ ) d − d − − (cid:32) d − (cid:33) ( R µ ) d − = CA ∗ d − , C = N , M d − d − d d π − dd − Γ (cid:16) d (cid:17) − d − ( d − , (14)where N , and M are the normalizations of the potential and the kinetic term, respectively. Inserting the solution ofEq.(14) into Eq.(13) one obtains ∆ − as a function of the ’t Hooft coupling A ∗ .The case d = − (cid:15) (with N = /
4) has been considered in [13]. The results read ∆ − A ∗ = F d ( x ) , F d ( x ) ≡ x + x + (cid:16) + x (cid:17) x , x = A ∗ (4 π ) + (cid:115) − + A ∗ (4 π ) . (15)Notice that this classical result resums an infinite series of Feynman diagrams in the conventional perturbativeexpansion. In particular, it resums the leading power of ¯ Q at every perturbative order.The next leading order (NLO) contributions ∆ is given by the functional determinant of the fluctuation aroundthe classical solution. Its bare form can be written in terms of the dispersion relations of the fluctuations, w + and w − ,as e ( A , d ) = R ∞ (cid:88) (cid:96) = n (cid:96) [ ω + ( (cid:96) ) + ω − ( (cid:96) )] . (16)where (cid:96) labels the eigenvalues of the Laplacian on the sphere J (cid:96) = (cid:96) ( (cid:96) + d − / R and n (cid:96) = (2 (cid:96) + d − Γ ( (cid:96) + d − Γ ( (cid:96) + Γ ( d − is theLaplacian multiplicity on S d − . By expanding around the classical solution as ρ ( x ) = f + r ( x ) , χ ( x ) = − i µτ + f √ π ( x ) , (17) These two dual forms are equivalent to each other. It can be easily shown that the ’t Hooft coupling A plays the role as a standard “ruler”. Forany chosen ’t Hooft coupling (no matter in the perturbative regime A (cid:28)
A (cid:29) λ or the equivalent lower bound of ¯ Q simply through ¯ Q = A /λ . A key di ff erence of 1 / N f expansion in gauge theories is that it has a finite radius convergence in the ’t Hooft coupling which is determined bythe pole structure. For example in QED, the radius convergence is A = λ N f = /
2, and if we fix
A ≤ / N f to make the 1 / N f expansion under control [34]. However, less is known about the pole structure of the charge expansion. we obtain the action at the quadratic order in the fluctuations S (2) = (cid:90) T / − T / d τ (cid:90) d Ω d − (cid:20)
12 ( ∂π ) +
12 ( ∂ r ) + d − µ − m ) r − i µ r ˙ π (cid:21) . (18)where ˙ π = ∂π∂τ . From the quadratic action we can then easily obtain the dispersion relations of the spectrum, whichread ω ± ( l ) = (cid:118)(cid:116) ( d − J (cid:96) + d − µ − m ± (cid:113) m + µ (cid:16) ( d − J (cid:96) − d − m (cid:17) + ( d − µ d − . (19)The spectrum contains one relativistic Goldstone boson (the conformal mode) and one massive state with mass2 (cid:113) ( d − µ − m ( d − .We proceed by fixing d = − (cid:15) and N = /
4. Eq.(16) needs to be renormalized. This is achieved by expanding λ = M (cid:15) λ Z λ in Eq.(8) and keeping the terms of order λ . Then, to obtain ∆ we set d = e ( A , RM , d ) and addthe expansion of the LO term ¯ e − /λ to first order in (cid:15) . Notice that the procedure mixes di ff erent orders of the bareexpansion. In particular, ¯ e ( A , RM , d ) contains two (cid:15) terms that have to cancel each other in order to be able to take d = ∆ . The first comes from renormalizing e − while the second can be isolated by regularizing the sumover (cid:96) in e which formally diverges. Thus, since these two terms come from di ff erent orders of the bare expansion,their cancellation can be used as a non-trivial self-consistency check of the correctness of the calculations which canbe particularly useful when dealing with more complicated models. We have: ∆ ( A ∗ ) = − µ R + µ R − + ∞ (cid:88) (cid:96) = σ ( (cid:96) ) + (cid:112) µ R − √ , (20)where σ ( (cid:96) ) = R (1 + (cid:96) ) [ ω + ( (cid:96) ) + ω − ( (cid:96) )] − (cid:96) − (cid:96) − (cid:16) µ R + (cid:17) (cid:96) − µ R + (cid:96) (cid:16) µ R − (cid:17) , (21)with R µ given by Eq.(14) in d =
4. Summing ∆ − and ∆ , expanding the result for small A ∗ , and using the FP value λ ∗ = (cid:15) + (cid:15) + O (cid:16) (cid:15) (cid:17) , we have ∆ φ ¯ Q = ¯ Q (cid:32) d − (cid:33) + (cid:15)
10 ¯ Q ( ¯ Q − − (cid:15)
50 ¯ Q ( ¯ Q − Q ) + O (cid:16) (cid:15) ¯ Q , (cid:15) ¯ Q (cid:17) . (22)This result has been obtained in [13] and checked via a Feynman diagram calculation. Similarly, the expansion forlarge A is ∆ φ ¯ Q = (cid:15) (cid:18) (cid:15) ¯ Q (cid:19) − (cid:15) − (cid:15) (cid:20) + (cid:15) (cid:18) − . + (cid:19) + O (cid:16) (cid:15) (cid:17)(cid:21) + (cid:15) (cid:18) (cid:15) ¯ Q (cid:19) − (cid:15) − (cid:15) (cid:20) + (cid:15) (cid:18) − . − (cid:19) + O (cid:16) (cid:15) (cid:17)(cid:21) + O (cid:16) ( (cid:15) ¯ Q ) (cid:17) , (23)The above expression is of the form predicted by the large charge EFT approach, which in arbitrary dimensions reads[5] ∆ O ¯ Q = ¯ Q dd − (cid:104) α + α ¯ Q − d − + α ¯ Q − d − + . . . (cid:105) + ¯ Q (cid:104) β + β ¯ Q − d − + . . . (cid:105) + . . . . (24)In [13], the authors have compared Eq.(23) with the results of lattice studies of the 3-dimensional U (1) model in thelarge charge limit [46], with mild results compatible with the limitations related to taking (cid:15) =
1. A similar comparisonhas been performed in [22] via a slightly di ff erent approach and reaching a similar conclusion.We now move to consider the d = − (cid:15) case with N = /
36. Unlike the previous case, the beta function of themodel starts at two loops, and thus the 1-loop theory is conformal invariant in exactly three dimensions. This allowsa more direct comparison to the predictions of the large charge EFT of three-dimensional CFT. In particular, we cancompare the coe ffi cient of the ¯ Q term in (24), which in three dimensions is calculable and a theory-independentnumber related to the sound speed and the 1-loop Casimir energy on the sphere [1]. The value of this coe ffi cient ascomputed in the EFT approach reads: β d = = − . ∆ − ( A ∗ ) A ∗ = F d (cid:32) A ∗ π (cid:33) , F d ( x ) ≡ + √ + x + x / √ + √ + x ) / , (25)while ∆ is obtained by regularizing Eq.(16). Since the 1-loop beta function vanishes, the result is finite in d = ∆ ( A ∗ ) = − R µ ) + (cid:113) R µ − + ∞ (cid:88) (cid:96) = σ ( (cid:96) ) , (26)where σ ( (cid:96) ) = (1 + (cid:96) ) R [ ω + ( (cid:96) ) + ω − ( (cid:96) )] − (cid:96) ( (cid:96) + − R µ ) + . (27)Combining the previous results and expanding in the perturbative regime, we obtain ∆ φ ¯ Q = ¯ Q + κ (cid:34) ¯ Q − Q + O ( ¯ Q ) (cid:35) − κ (cid:34) ¯ Q − ¯ Q (64 − π )72 + O ( ¯ Q ) (cid:35) + κ (cid:34) Q + (cid:26) − + π + π (cid:27) ¯ Q + O ( ¯ Q ) (cid:35) + O (cid:16) κ (cid:17) , (28)where κ = (cid:16) λ π (cid:17) . This 6-loops result has been verified via diagrammatic computations in [17].We now proceed by analyzing the large A ∗ limit which is captured by the large charge EFT. ∆ − can be expandedanalytically while ∆ can be computed numerically and then fitted to the expected functional form (24). The value ofthe coe ffi cients can be found in [17, 24]. Here, we just report the result for β which reads β d = = − . , (29)with an error of 3 on the last digit. The universal coe ffi cient β agrees to high accuracy with the value obtained in theEFT approach and Montecarlo simulations.It has been recently pointed out that in 4 dimensions the large charge EFT predicts the existence of a universal log ¯ Q term with calculable coe ffi cient δ d = = − √ [48] . It would be interesting to test this prediction in the semiclassicalframework as done for the three-dimensional case. B. The O ( N ) model in − (cid:15) and − (cid:15) dimensions In this section, we analyse the large charge expansion in the non-abelian O ( N ) vector model, which constitutes thenatural generalization of the U (1) model investigated in the previous section. For N = , ,
3, it defines respectivelythe
Ising , XY , and Heisenberg universality classes while for N =
4, it describes the standard model Higgs. In euclideanspacetime, the O ( N ) theory is defined by the action S = (cid:90) d d x ∂ µ φ a ∂ µ φ a + V (cid:16) φ a φ a (cid:17) a = , . . . , N . (30)As before, we consider the massless theory in d = − (cid:15) and d = − (cid:15) with potentials V d = − (cid:15) = N g (cid:16) ¯ φ a φ a (cid:17) and V d = − (cid:15) = N g (cid:16) ¯ φ a φ a (cid:17) with g the bare coupling and N , the normalization. The conserved Noether currentassociated with the global O ( N ) symmetry transforms in the adjoint representation of O ( N ) and it is given by( j µ ) ab = (cid:16) φ a ∂ µ φ b − φ b ∂ µ φ a (cid:17) . (31) This term arises in the renormalization of β , which features a pole for d = The corresponding conserved charge is matrix-valued and can be decomposed in terms of the generators of thealgebra T A Q ab = (cid:90) d d − x (cid:16) j (cid:17) ab = (cid:88) A Q A (cid:16) T A (cid:17) ab . (32)The O ( N ) group with even or odd N has rank N or N − respectively, which corresponds to the number of “charges” Q A we can fix. Without loss of generality, we focus on the even- N case and we fix k < N / k constraints Q i = ¯ Q i , where { ¯ Q i } is a set of fixed constants and i = , . . . , k . Using the fact that the O ( N ) model has a SU ( N / × U (1)subalgebra, it is useful to introduce N / ϕ = √ (cid:16) φ + i φ (cid:17) = √ σ e i χ , ϕ = √ (cid:16) φ + i φ (cid:17) = √ σ e i χ , ϕ = . . . . (33)such that ϕ i or ¯ ϕ i has charge ¯ Q i = + − O ( N ) theory on a cylinder S d − × R where the action reads S cyl = (cid:90) d d x √ g (cid:20) g µν ∂ µ σ i ∂ ν σ i + σ i σ i g µν ( ∂ µ χ i ∂ ν χ i ) + ˜ V ( σ i σ i ) (cid:21) , ˜ V ( σ i σ i ) = V ( σ i σ i ) + m σ i σ i , i = , . . . , N / . (34)In analogy with the abelian case, we look for a spatially homogeneous solution of the EOM. This has the lowestenergy at fixed charge and reads σ i = A i , χ i = − i µ t i = , . . . , k ,ϕ k + j = , j = , . . . , N / − k . (35)The chemical potential µ is the same for all the χ i even if the charges ¯ Q i are all di ff erent. For i = , . . . , k this groundstate describes circular motion in the plane spanned by the real and imaginary parts of ϕ i . The motions in di ff erentplanes are synchronous with the same angular velocity µ and di ff erent radii of the circles A i . These parameters arefixed by the EOM and Eq.(31) as µ = F ( v ) ¯ QR d − Ω d − = µ v , (36)where we have defined v ≡ k (cid:88) i = A i , ¯ Q ≡ k (cid:88) i = ¯ Q i , F ( v ) = A j ∂ ˜ V ( σ i σ i ) ∂σ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ j = A j , (37)with ¯ Q the sum of the charges.It can be shown that the symmetry breaking pattern induced by fixing the charges can be seen as an explicitsymmetry breaking O ( N ) −→ O ( N − k ) × U ( k ) followed by a spontaneous symmetry breaking (SSB) U ( k ) −→ U ( k − O ( N ) rotation to rotate the ground state to1 √ A , ..., A k , , ..., −→ (cid:16) , ..., (cid:124)(cid:123)(cid:122)(cid:125) k − , v √ , , ..., (cid:124)(cid:123)(cid:122)(cid:125) N / − k (cid:17) . (38)This analysis shows that we can organize the saddle point computation as a single coupling ’t Hooft expansionin A = g ¯ Q in full analogy with the U (1) case. The sum of the charges acts as a single U (1) charge while the chargeconfiguration plays no role at all. In order to access more general charge configurations is necessary to considernon-homogeneous ground states as done for the O (4) critical model in [49–51]. The above considerations lead directlyto the path integral expression for the ground state energy, which reads (cid:104) ¯ Q | e − HT | ¯ Q (cid:105) = Z (cid:90) D k σ D k χ e −S ef f , (39)where S e f f = (cid:90) T / − T / dt (cid:90) d Ω d − (cid:32) ∂σ i ∂σ i + σ i ( ∂χ i ∂χ i ) + ˜ V ( σ i σ i ) + iR d − Ω d − ¯ Q ˙ χ N / (cid:33) . (40)The sums over i run from 1 to N /
2, i.e. we fixed the all the N / A ∗ ≡ g ∗ ¯ Q of the lowest-lying operator carrying a totalcharge ¯ Q takes the form ∆ T ¯ Q = E T ¯ Q R = ∞ (cid:88) j = − g ∗ j ∆ j ( A ∗ ) = ∞ (cid:88) j = − Q j ˜ ∆ j ( A ∗ ) . (41)As in the U (1) case, the leading term ∆ − ( A ∗ ) is given by Eq.(13) with R µ and A related by Eq.(14).To compute the leading quantum corrections ∆ we consider the ground state in (38) and parametrize the fluctua-tions around it as χ i = − i µ t + v p i ( x ) , i = , . . . , N / − ,χ N / = − i µ t + v π ( x ) ,σ i = s i ( x ) , i = , . . . , N / − ,σ N / = v + r ( x ) , (42)The Lagrangian at the quadratic order in the fluctuations reads L =
12 ( ∂π ) +
12 ( ∂ r ) + d − µ − m ) r − i µ r ˙ π + ∂ s i ∂ s i + ∂ p i ∂ p i − i µ s i ˙ p i . (43)The spectrum contains states that are already present in the U (1) case, i.e. the conformal mode χ N / and onemassive state σ N / with dispersion relations given by Eq.(19).Additionally, we now have also N − χ i and as many massive states σ i with mass 2 µ and dispersion relations ω ±± ( (cid:96) ) = (cid:113) J (cid:96) + µ ± µ . (44)According to the Nielsen-Chada theorem [52], Type II Goldstone bosons count double with respect to the numberof broken generators. Thus we have1 + × (cid:18) N − (cid:19) = N − = dim (cid:18) U (cid:18) N (cid:19) / U (cid:18) N − (cid:19)(cid:19) . (45) ∆ is again given by the fluctuation functional determinant. It can be easily shown that the generalization of Eq.(16)to general non-Abelian scalar theories is ∆ = R ∞ (cid:88) (cid:96) = n (cid:96) (cid:88) i g i ω i ( (cid:96) ) , (46)where the sum over i runs over all the fluctuations’ dispersion relations ω i , each counted with its multiplicity g i . Inthe O ( N ) case, we have ∆ = R ∞ (cid:88) (cid:96) = n (cid:96) (cid:104) ω + ( (cid:96) ) + ω − ( (cid:96) ) + ( N − ω ++ ( (cid:96) ) + ω −− ( (cid:96) )) (cid:105) . (47)It is instructive to analyze what happens to our computation if we don’t fix all the N / Q i but only k < N / k −
1, whereas the spectrum is completed by 2 × [( N / − − ( k − = N − k new massive states with mass µ and dispersion relation ω ∗ ( (cid:96) ) = (cid:113) J (cid:96) + µ . (48)0Accordingly, the expression for ∆ becomes ∆ = R ∞ (cid:88) (cid:96) = n (cid:96) [ ω + ( (cid:96) ) + ω − ( (cid:96) ) + ( k − ω ++ ( (cid:96) ) + ω −− ( (cid:96) )) + ( N − k ) ω ∗ ] . (49)Since ω ++ ( (cid:96) ) + ω −− ( (cid:96) ) = ω ∗ ( (cid:96) ), ∆ does not depend on the number of charges that are fixed. This result is consistentwith the scaling dimension not being sensitive to the charge configuration but only to the sum of the charges.In parallel with the previous section, we now proceed by providing explicit results starting from the case d = − (cid:15) and N = (4 π ) which has been considered by us in [15]. As his Abelian relative, this theory features an infrared WFFP, which for small (cid:15) can be expressed as a power series in (cid:15) .The computation of the leading order ∆ − is analogous to the U (1) case and leads to the same result ∆ − A ∗ = F d ( x ) , x ≡ A ∗ + √− + A ∗ , (50)where F d has been defined in Eq. (15). To compute the leading quantum correction, we start from Eq.(47) and wefollow the procedure of Sec.II A in order to regularize and renormalize the fluctuation determinant. As a result, weobtain ∆ ( A ∗ ) = − µ R + µ R − + ∞ (cid:88) (cid:96) = σ ( (cid:96) ) + (cid:112) µ R − √ − (cid:18) N − (cid:19) (cid:104) + R µ (cid:16) − + R µ + R µ (cid:17)(cid:105) . (51)where σ ( (cid:96) ) = R (1 + (cid:96) ) (cid:104) ω + ( (cid:96) ) + ω − ( (cid:96) ) + ( N − ω ++ ( (cid:96) ) + ω −− ( (cid:96) )) (cid:105) +
18 ( N + (cid:16) R µ − (cid:17) (cid:96) + (cid:16) − N − ( N + R µ (cid:17) + (cid:16) − N − (2 + N ) R µ (cid:17) (cid:96) − N (cid:96) − N (cid:96) . (52)Again all the quantities are evaluated in d = A ∗ = g ∗ Q , obtaining ∆ T ¯ Q = ¯ Q + (cid:32) − ¯ Q + ¯ Q ( ¯ Q − + N (cid:33) (cid:15) − (cid:34) + N (14 − N )4(8 + N ) ¯ Q + ( N − N + + N ) ¯ Q + + N ) ¯ Q (cid:35) (cid:15) + (cid:34) + N ) ¯ Q + − − N + N + + N )(14 + N ) ζ (3)(8 + N ) ¯ Q + − N − N + N − N + N + N + ζ (3) + N + N + ¯ Q + − + N [ − + N (184 + N (86 + N ))] + + N )(178 + N (37 + N )) ζ (3)16( N + ¯ Q (cid:35) (cid:15) + (cid:34) − + N ) ¯ Q + − N − N + N + ζ (5) − N + N + ζ (3) + N + + N ) ¯ Q + + N ) (cid:16) π N + N + π N + N + π N − N − N − N + ζ (5) − N + N ( N (3 N − − − ζ (3) + π N − N + π − (cid:17) ¯ Q − N + (cid:16) N + π N + N + π N + N + π N + N + π N − N − N + ( N (65 N + + ζ (5) − N + N ( N ( N ( N ( N + + − − − ζ (3) + π N − N + π − (cid:17) ¯ Q + + N ) (cid:16) N + π N + N + π N + N + π N + N + π N − N − N + ( N (25 N + + ζ (5) − N + N ( N ( N ( N ( N + + − − − ζ (3) + π N − N + π − (cid:17) ¯ Q (cid:35) (cid:15) + O (cid:16) (cid:15) (cid:17) , (53)where the terms highlighted in red stem from the semiclassical computation. To two loops, they agree with theknown 2-loop anomalous dimension of the ¯ Q -index traceless symmetric O ( N ) tensor with classical dimension ¯ Q [53],1which can be depicted as a ¯ Q -boxes Young tableau with one row. In [15], we also obtained all the black terms at threeand four loops by combining the knowledge of the red ones with the known perturbative results for the ¯ Q = Q = Q = Q = φ field, while for ¯ Q =
2, it is the bilineartraceless symmetric O ( N ) tensor φ a φ b − N φ c φ c , which is of interest to many critical phenomena being responsible forcrossover behaviour in the O ( N ) theory. Its anomalous dimension defines a so-called crossover exponent describingthe instability of the theory against anisotropy [59]. As we will see in the next section, in the perturbative regimethe operator identification can be proven via group-theoretical arguments. For sake of completeness, we report herealso the large A expansion of ∆ T ¯ Q [18] ∆ T ¯ Q = (cid:15) (cid:32) (cid:15) ¯ QN + (cid:33) dd − (cid:34) N + + (cid:15) (cid:32) − . − . N + N + N + (cid:33) + O ( (cid:15) ) (cid:35) + (cid:15) (cid:32) (cid:15) ¯ QN + (cid:33) d − d − (cid:34) N + + (cid:15) (cid:32) − . + . N − N + N + (cid:33) + O ( (cid:15) ) (cid:35) + O [( (cid:15) ¯ Q ) ] . (54)We end this review section with the O ( N ) sextic theory in d = − (cid:15) dimension with N = which has been studiedin [17]. The leading order energy is once again given by Eqs.(13) and (14) and reads ∆ − ( A ∗ ) A ∗ = F d (cid:32) A ∗ π (cid:33) , (55)where F d has been defined in Eq. (25).The regularized version of Eq.(47) provides the 1-loop correction in the semiclassical expansion as ∆ ( A ∗ ) = − R µ ) + (cid:113) R µ − − (cid:18) N − (cid:19) (cid:18) + ( R µ ) − R µ (cid:19) + ∞ (cid:88) (cid:96) = σ ( (cid:96) ) , (56)where σ ( (cid:96) ) = (1 + (cid:96) ) R (cid:20) ω + ( (cid:96) ) + ω − ( (cid:96) ) + (cid:18) N − (cid:19) ω ++ ( (cid:96) ) + ω −− ( (cid:96) ) (cid:21) − (cid:96) ( (cid:96) + − (cid:18) R µ ) − (cid:19) − (cid:18) N − (cid:19) (cid:18) (cid:96) ( (cid:96) + + R µ ) + (cid:19) , (57)which is again constructed so that the sum is convergent in d =
3. In [17], this result has been verified via conventionaldiagrammatic techniques at the 6-loop level. The large A ∗ expansion can be studied numerically, as in the U (1) case.The outcome is corrections (proportional to N −
2) to the U (1) values of the α ’s coe ffi cients in (24), while the β ’sreceives no new contributions. III. FROM GROUP THEORY TO OPERATORS: THE MAPA. The power of symmetries
Our knowledge of a quantum field theory (QFT) is generally encoded in the correlation functions of local operators.If the QFT under consideration has some internal global compact symmetry group G which is neither explicitly brokennor spontaneously broken, then without loss of generality we may restrict ourselves to local operators that transformunder definite unitary irreducible representations of G , since any other local operator should be able to be expressedas a linear combination of those local operators with definite transformation properties.Therefore let us consider a set of local operators O p , O p , ..., O pd p that transform under a d p -dimensional unitaryirreducible representation Γ p of G . This implies they have implicitly the same spacetime (Lorentz) transformation Much of the basic group theory introduced in this section is based on the textbooks by J. F. Cornwell [60, 61] and by B. C. Hall [62], which leadsus to the proofs of several important results needed for application in the fixed-charge semiclassical approach to CFT. V p for Γ p , that is, for i = , , ..., d p and all T ∈ G Φ ( T ) O pi = j = d p (cid:88) j = Γ p ( T ) ji O pj (58)where Φ ( T ) denotes the linear transformation operator corresponding to T ∈ G that acts on V p , and Γ p ( T ) denotesthe representation matrix corresponding to T ∈ G . It is important to note that the complete symmetry property ofan operator is encoded in two indices. For the set of operators O pi , one index is p , which refers to the irreduciblerepresentation the operator belongs to, up to equivalence. The other index is i , referring to which row of Γ p theoperator O pi transforms according to. There is a counterpart of Eq. (58) in Lie algebra representation theory. Suppose (cid:101) L is the complexification of the real Lie algebra of G , then for i = , , ..., d p and all a ∈ (cid:101) L Φ ( a ) O pi = j = d p (cid:88) j = Γ p ( a ) ji O pj (59)where now Φ ( a ) denotes the linear transformation operator corresponding to a ∈ (cid:101) L that acts on V p and Γ p ( a ) denotesthe representation matrix corresponding to a ∈ (cid:101) L . Let H be a Cartan subalgebra of (cid:101) L . Without loss of generalitywe may assume that, the set of operators O pi are chosen such that Γ p ( h ) is diagonal for all h ∈ H . This fact can berepresented by the following equation: Φ ( h ) O pi = λ i ( h ) O pi (60)for i = , , ..., d p . This defines d p linear functionals λ i ( h ) that act on H , which are the weights of the irreduciblerepresentation Γ p in mathematical terms. Therefore, the index i plays the role of labeling the weights of Γ p . From aphysical point of view, a weight when acting on a set of elements in H , gives the Cartan charges associated with theelements, and thus specifies a charge configuration.We state two important consequences of the intact (i.e. neither explicitly nor spontaneously broken) symmetry G . Consequence 1:
Operators of di ff erent symmetry properties (i.e. belonging to inequivalent irreducible representa-tions of G , or belonging to equivalent irreducible representations of G but correspond to di ff erent weights) do notmix under the renormalization group. Consequence 2:
In a conformal field theory (CFT), operators that transform in the same irreducible representationsof G but correspond to di ff erent weights have identical scaling dimensions, if they do not mix with operators that donot belong to their carrier space under renormalization.Here and hereafter, we always assume that for equivalent irreducible representations an appropriate similaritytransformation has been applied to make them identical. The first consequence above is simply the requirement thatrenormalization of the theory preserves its global symmetry. The second consequence can be verified by examiningthe two-point correlator of the operators in question and making use of the Wigner-Eckart theorem, which statesthat matrix elements of irreducible tensor operators can be factorized into two parts, with the first part solelydetermined by the corresponding Clebsch-Gordan coe ffi cients, and the remaining part called reduced matrix elementswhich are independent of the magnetic quantum numbers (weights). When we consider two-point correlators like (cid:104) Ω |O pi ( x ) O qj ( y ) | Ω (cid:105) ( | Ω (cid:105) being the vacuum), we may view O qj ( y ) | Ω (cid:105) , O pi ( x ) | Ω (cid:105) as a whole, and the identity operator as theirreducible tensor operator, in order to apply the Wigner-Eckart theorem. The Clebsch-Gordan coe ffi cients are trivialfor p = q and the Wigner-Eckart theorem tells us for i = j the two-point correlator is independent of the weight label i . With the further assumption that this set of operators do not mix with other operators, we deduce that their scalingdimensions must be the same (since in a CFT scaling dimension ∆ of an operator O can be completely determinedfrom its two-point correlator as (cid:104) Ω |O ( x ) O ( y ) | Ω (cid:105) = | x − y | − ∆ ). Note that it is important to require the two operatorsto transform in the same irreducible representation, i.e. they live in the same irreducible carrier space. If they bothmerely transform according to some irreducible representation Γ p , but are not in the same irreducible carrier space,then we cannot claim anything about their scaling dimensions.3 B. The nature of charge fixing
The fixed-charge approach has proven to be very powerful in probing the dynamics of a QFT with global symmetriesin regimes that are di ffi cult to access by conventional methods. In most applications so far a CFT is considered sinceone can employ the Weyl invariance of the theory to map the CFT to a cylinder, with the computation of scalingdimensions of fixed-charge operators turned into the computation of the ground state energies in the correspondingfixed-charge sectors of the cylinder theory. Obtaining results for non-CFTs may also be possible in various cases [16],however for the moment we will restrict our presentation to the case of CFTs for simplicity. Conventional perturbationtheory can probe the small-charge regime, up to a certain power in the coupling expansion, limited by computationalcapabilities, while the large-charge regime is beyond its validity range. In the fixed-charge approach, however,both the small-charge and large-charge regimes are dealt with by a semiclassical expansion around a nontrivialfixed-charge trajectory in the path integral.An important feature of the fixed-charge approach to scaling dimension computation is that a priori it does not fix the full symmetry properties of the fixed-charge operator under consideration. This can be inferred from thegeneral discussion of the fixed-charge path integral (see Sec. II), in which only the eigenvalues corresponding to aset of Cartan charges are required. Put it simpler, only weights are known and fixed, and we do not know whichirreducible representation the operator belongs to. Multiple irreducible representations can share the same weight,while a given irreducible representation can be realized by di ff erent sets of local operators. This is where the Liealgebraic theory cannot tell us more and we need some dynamical information.The dynamical information is hidden in the starting point of the derivation of a fixed-charge path integral. Ina Euclidean field theory, one considers the expectation of the evolution operator e − HT in an arbitrary state | ψ (cid:105) witha given fixed charge (i.e. weight). In the limit T → ∞ the expectation gets saturated by the lowest energy statecontained in | ψ (cid:105) which typically has a nonzero overlap with the lowest-lying energy state corresponding to the givencharge. Therefore by using state-operator correspondence, the scaling dimension one obtains for the fixed-chargeoperator should correspond to the scaling dimension of the lowest-lying operators with the given fixed charge.Unfortunately, for a given fixed charge, we still do not know a priori which operator in which irreducible represen-tation is lowest-lying. Nevertheless, progress can be made by1. Assuming that the lowest-lying operator has the minimal classical scaling dimension (MCSD) with which anoperator can be constructed corresponding to a given charge configuration. This will be called the MCSDassumption. If the MCSD assumption is valid with a unique operator O MCSD saturating the MCSD for agiven charge configuration, then O MCSD must have a definite scaling dimension (i.e. it does not mix withother operators). However, in more general cases multiple operators may saturate the MCSD for a given chargeconfiguration, and some appropriate linear combination of them will become the genuine lowest-lying operatorand have a definite scaling dimension. For spin-0 fixed-charge operators (corresponding to homogeneousground states in the cylinder theory) the MCSD assumption obviously requires we consider non-derivativeoperators only, as extra spacetime derivatives necessarily increase the classical scaling dimension.2. Carrying out semiclassical computations for various weights of a given irreducible representation. Then, foreach weight, list all the irreducible representations that contain it. If semiclassical computation gives di ff erentresults of scaling dimensions for di ff erent weights, and the MCSD assumption and Consequence 2 are used, itmight be possible to pin down the correspondence between weights and representations in the semiclassicalcomputation.To summarize, an important feature of the fixed-charge semiclassical computation is that a priori it only fixesthe weight, while the correspondence between the weight and the irreducible representation is hidden in the factthat only the lowest-lying state is projected out. Further progress in disentangling weights and representationscan be made by making the MCSD assumption and carrying out semiclassical computations for multiple weightsin question. To illustrate the main idea, in the following we will first review the simpler case of U (1) and O ( N )vector models, and then turn to the more complicated U ( N ) × U ( M ) linear sigma model which entails a sophisticatedgroup-theoretic analysis. In this work we are only concerned with spin-0 fixed-charge operators. Operators with nonzero spin would correspond to inhomogeneousground states on the cylinder [49]. existence of the fixed-chargeoperators. Simply obtaining the result from a fixed-charge semiclassical computation does not guarantee that theresults obtained are correct. Second, when comparing the results of the fixed-charge semiclassical computation withresults obtained by other methods (e.g. conventional perturbation theory), it is relevant to know the correspondencebetween the weight and the irreducible representation, or the explicit form of the fixed-charge operator. Third,for theory and application purposes we might just wish to know the scaling dimension of certain operators thattransform according to given irreducible representations.
C. Group-theoretic analysis: U (1) and O ( N ) vector models The simplest QFT with an internal continuous global symmetry is the theory of a complex scalar field φ that hasthe U (1) symmetry transformation φ → e − i α φ with α being an arbitrary constant real phase. To allow for a nontrivialfixed point we consider the theory in d = − (cid:15) Euclidean spacetime dimensions. The Lagrangian of the theory andthe Noether charge Q associated with the global symmetry can be read o ff from in Eq. (1) and Eq. (2). One can derivethe commutation relation [ Q , φ ] = φ (61)from canonical commutation relations for the field operator φ . The integrated form of Eq. (61) reads e − i α Q φ e i α Q = e − i α φ (62)for an arbitrary real phase constant α . It is from Eq. (61) and Eq. (62) that we deduce the U (1) charge of φ to be + φ n ( n is a positive integer) has U (1) charge n . The normalization of the U (1) charge changes bymultiplying via a nonzero real number. Nevertheless, one can always compute the U (1) charge of a local operator O by computing the parameter q in the commutator equation[ Q , O ] = q O (63)If O carries a definite U (1) charge, then there should exist a real number q such that Eq. (63) holds. If the chargenormalization is such that φ carries the charge +
1, then q must be an integer, as long as O can be written as a linearcombination of products of φ and its derivatives (non-integer powers operators are ill-defined). This simple U (1)example illustrates the well known fact that the charge is discretized for well-defined local operators of the theory,regardless of the normalization convention while, after a Weyl map to the cylinder, the charge density can be adjustedcontinuously by changing the compactification volume.For a given U (1) charge n >
0, the operator with MCSD is obviously φ n . Any additional ( ¯ φφ ) factor or derivativewould necessarily increase the classical scaling dimension. Therefore, with the MCSD assumption we expect asemiclassical computation in the charge- n sector with a homogeneous ground state to deliver the scaling dimensionof the operator φ n .From a group-theoretic point of view, the next-to-simplest case turns out to be the critical O ( N ) vector model in d = − (cid:15) dimensions (c.f. Section II B). This model features a N -component real scalar field φ = ( φ , φ , ..., φ N ). ItsLagrangian density in Euclidean spacetime is given in Eq. (30). For definiteness, let us consider the case where N is even. Extension to the case of odd N is straightforward. The maximal commuting set of charges we canfix corresponds to the maximal set of Cartan generators, which can be made explicit by defining the complex fields ϕ = √ (cid:16) φ + i φ (cid:17) , ϕ = √ (cid:16) φ + i φ (cid:17) , ...ϕ N / = √ (cid:16) φ N − + i φ N (cid:17) . For each j = , , ..., N /
2, there exists an independentphase rotation ϕ j → ϕ j e − i α j as a symmetry transformation of the theory corresponding to a Cartan generator,with α j being an arbitrary real phase. A generic charge configuration (i.e. weight) can thus be characterized by[ m ] ≡ ( m , m , ..., m N / ), with m i representing the charge associated with the i th Cartan generator. The normalizationof the Cartan charges can be chosen such that ϕ i corresponds to (0 , , ..., m i = + , , ...,
0) for i = , , ..., N /
2, which weadopt. This implies for a generic charge configuration [ m ] = ( m , m , ..., m N / ), m i ’s are all integers. Without loss ofgenerality we may consider only the case in which all m i ’s are nonnegative since the sign of the Cartan charge is a5matter of convention. The operator with MCSD that corresponds to [ m ] = ( m , m , ..., m N / ) is then easily constructed: O [ m ] ≡ i = N / (cid:89) i = ( ϕ i ) m i (64)If some m i is negative, we may simply use ϕ ∗ i instead of ϕ i for the corresponding factor. As in the U (1) case, anyadditional factor of ( ¯ ϕ i ϕ i ) or derivative would necessarily increase the classical scaling dimension.Let us note O [ m ] thus constructed live in the traceless fully symmetric subspace of O ( N ) transformations. It isfully symmetric because it is a product of commuting scalar fields. It is traceless because otherwise it would containsome factor like φ which would violate the MCSD assumption. Therefore, O [ m ] corresponds to an irreduciblerepresentation of O ( N ) and has a definite scaling dimension. The argument also shows that operators that have thesame value of (cid:80) i = N / i = | m i | and MCSD all belong to the same irreducible O ( N ) representation and thus have the samescaling dimension, in agreement with the expectation that by an O ( N ) rotation we can associate all charges to a singleCartan generator. D. Group-theoretic analysis: the U ( N ) × U ( M ) linear sigma model
1. Introduction
The O ( N ) vector model is simple in the fixed-charge semiclassical approach as charge fixing can always be associatedwith a single Cartan charge by virtue of a symmetry rotation. Thus all charge configurations are similar and solelycharacterized by (cid:80) i = N / i = | m i | .To allow for more variations in the charge configuration, here we consider the U ( N ) × U ( M ) linear sigma model in d = − (cid:15) dimensions, with N > M > L = Tr( ∂ µ H † ∂ µ H ) + u Tr( H † H ) + v (Tr H † H ) (65)Here H denotes an N × M complex matrix scalar field. Without loss of generality we may assume N ≤ M . The modelhas the global symmetry G ≡ SU ( N ) L × SU ( M ) R × U (1) A (66)in which U (1) A is the universal phase rotation of the H field . Under SU ( N ) L × SU ( M ) R , the H and H † fields transformas H → LHR † , H † → RH † L † (67)with L being an arbitrary N × N constant special unitary matrix, and R being an arbitrary M × M constant specialunitary matrix.Depending on the value of N and M , the model may feature fully-interacting real or complex fixed points. At sucha fixed point, we perform a Weyl map to a cylinder of radius R (i.e. R d → R × S d − ), with the cylinder action given by S cyl = (cid:90) d d x √ g (cid:104) Tr( ∂ µ H † ∂ µ H ) + u Tr( H † H ) + v (Tr H † H ) + m Tr( H † H ) (cid:105) . (68)Here g denotes the metric determinant and m = (cid:16) d − R (cid:17) is the coe ffi cient of the conformal coupling required by Weylinvariance.We consider a homogeneous ground state with the ansatz H ( τ ) = e iM E τ B , (69) It does not matter whether this universal U (1) rotation acts from the left or from the right. Therefore precisely speaking the global symmetryshould be written as Eq. (66). Writing it as U ( N ) × U ( M ) is less rigorous but more convenient, see [63] for example. τ denotes the cylinder time and M E is an N × N diagonal matrix. B is an N × M matrix in the form B N × M = (cid:16) B N × N N × ( M − N ) (cid:17) (70)in which B is an N × N diagonal matrix. The Noether charges associated with Cartan generators are encoded in thefollowing charge configuration matrices: Q L = − V ˙ H H † , Q R = VH † ˙ H (71)with V = R d − Ω d − being the volume of S d − . Plugging in the ansatz Eq. (69), it is straightforward to show Q L = − iVM E B † B , Q R = iVM E (cid:32) B † B N × ( M − N ) ( M − N ) × N ( M − N ) × ( M − N ) (cid:33) (72)If we parametrize Q R as Q R = (cid:32) Q R N × ( M − N ) ( M − N ) × N ( M − N ) × ( M − N ) (cid:33) (73)Then from Eq. (72) we find the constraint Q L + Q R = B is diagonal, Q L , Q R are also diagonal. We will restrict our attention to the sector neutral under U (1) A , whichimplies Tr Q L = Tr Q R = Q to denote Q L , that is Q ≡ Q L = −Q R (76)In the following, we will first determine what are the admissible charge configuration matrices Q and then we willdisentangle which irreducible representations they correspond to. Although we work in d = − (cid:15) dimensions weindicate the classical scaling dimensions (CSD) with the corresponding one in 4 dimensions. For example, the CSDof the field H is 1 and the one of the operator Tr( H † H ) is 2. The transition to d = − (cid:15) dimensions is straightforward.All the discussion will be restricted to the homogeneous ground state ansatz in Eq. (69) and the associated tracelesscharge configuration of Q .We first introduce and prove several propositions that underlay the determination of the irreducible representationassociated with a given charge configuration from Lie algebraic considerations. Proposition 1:
Suppose O is a fixed-charge operator that corresponds to a traceless charge configuration withMCSD. Let us denote the CSD of O by D . Let us also suppose O belongs to some irreducible representation ( Γ L , Γ R )of SU ( N ) L × SU ( M ) R in the U (1) A -neutral sector. Then Γ L must appear in ( Adj L ) D / , with Adj L being the adjointrepresentation of SU ( N ) L ; Γ R must appear in ( Adj R ) D / , with Adj R being the adjoint representation of SU ( M ) R . Proof of proposition 1:
Since we are considering a homogeneous ground state, the corresponding fixed-chargeoperator O must be a Lorentz scalar. Within the MCSD assumption, this implies no derivative can appear in theconstruction of O , and thus the operator O with CSD D must be built out of the product of D / H fields and D / H † fields (so that O is also neutral under U (1) A ). Now, under SU ( N ) L × SU ( M ) R H ∼ ( F L , ¯F R ) , H † ∼ ( ¯F L , F R ) (77)Here F L denotes the fundamental representation of SU ( N ) L , and ¯F L denotes the anti-fundamental representation of SU ( N ) L . The notation for representations of SU ( M ) R is self-explanatory. Therefore O must transform as an irreduciblecomponent inside the reducible representation( Γ L , Γ R ) , with Γ L ≡ ( F L ⊗ ¯F L ) D / , Γ R ≡ ( F R ⊗ ¯F R ) D / (78)7Now for representations of special unitary groups we know that F L ⊗ ¯F L = L ⊕ Adj L , F R ⊗ ¯F R = R ⊕ Adj R (79)and thus Γ L = ( L ⊕ Adj L ) D / , Γ R = ( R ⊕ Adj R ) D / (80)All singlet components in L ⊕ Adj L and R ⊕ Adj R can actually be dropped because O corresponds to an MCSDoperator. If a singlet component contributes then one would be able to construct another operator that correspondsto the same charge configuration with less number of H and H † fields, in contradiction to the MCSD requirement.Therefore we conclude that the operator O belongs to ( Γ L , Γ R ) , where Γ L and Γ R must appear respectively in ( Adj L ) D / and ( Adj R ) D / . Proposition 2:
Suppose that the CDS of O is D and the MCDS fixed-charge operator corresponds to a tracelesscharge configuration. Let us also suppose O belongs to some irreducible representation ( Γ L , Γ R ) of SU ( N ) L × SU ( M ) R in the U (1) A -neutral sector. Then ( Γ L , Γ R ) must appear in the U (1) A -neutral sector of the decomposition of the D -indextraceless fully symmetric tensor of O (2 NM ) under the branching O (2 NM ) ⊃ SU ( NM ) × U (1) A ⊃ SU ( N ) L × SU ( M ) R × U (1) A (81) Proof of proposition 2: H is a complex N × M matrix field with 2 NM real components. As O is constructed withMCSD, it cannot contain derivatives and therefore if its CSD is D , it must live in the carrier space of a D -index fullysymmetric tensor of O (2 NM ). On the other hand the real symmetry of the theory is SU ( N ) L × SU ( M ) R × U (1) A ⊂ SU ( NM ) × U (1) A ⊂ O (2 NM ), therefore ( Γ L , Γ R ) must appear in the decomposition of a D -index fully symmetric tensorof O (2 NM ) under the branching in Eq. (81). In fact the D -index fully symmetric tensor must be traceless, becausewe are considering operators constructed with MCSD. If the tensor contains a trace part, then it would be possibleto factor out the trace and build a new operator with the same symmetry properties but with a smaller CSD, incontradiction to the MCSD assumption.
2. The correspondence between weight and charge configuration
In the above propositions, no explicit reference is made yet about the charge configuration matrix Q which wewill consider now. Let us start with the explicit form of Q and determine the precise correspondence between thecharge configuration matrix and the weight of an irreducible representation. The matrix Q belongs to the speciallinear algebra sl( N ; C ), which is the space of all N × N complex matrices X for which Tr X =
0. sl( N ; C ) is exactlythe complexification of the real Lie algebra of the SU ( N ) group [62]. The Cartan subalgebra h of sl( N ; C ) can becharacterized by h = λ ... λ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ j ∈ C , λ + ... + λ N = (82)A weight is a linear functional on h . Nevertheless, it is convenient to identify linear functionals on h with elements of h itself, by virtue of an inner product on h . Suppose K and K (cid:48) are two elements of h , we define their inner product by (cid:104) K , K (cid:48) (cid:105) = Tr( K ∗ K (cid:48) ) (83)If φ is a linear functional on h , there is a unique element λ in h such that φ ( K ) = (cid:104) λ, K (cid:105) (84)for all K ∈ h . Therefore, a weight µ can be thought of as an element in h , by virtue of the inner product defined inEq. (83).8The charge configuration matrix Q should be proportional to some weight µ of a representation of sl( N ; C ). Letus now determine the precise correspondence, assuming Q = Q L is normalized as in Eq. (71). Suppose Q can bedecomposed as Q = N − (cid:88) j = x j ˆ h j (85)with ˆ h j being a set of ortho-normal basis elements of h , with the orthogonality defined by virtue of the inner productin Eq. (83), and the normalization condition beingTr(ˆ h j ) = , j = , , ..., N − h =
12 ( E − E ) (87)where E ij denotes a N × N matrix with a ”1” in the ( i , j ) entry and ”0” elsewhere. The normalization of basis elementsis required as in Eq. (86) because, for example, one can compute the commutation relation[ˆ h , E ] = E (88)which implies a raising operator constructed with a single E will carry charge + h . One may wishto make this argument more precisely by rewriting Eq. (88) as a commutation relation between the correspondingcharge operator and the corresponding fixed-charge operator, computed with the help of canonical commutationrelations of fundamental fields.Then x j in Eq. (85) gives the Cartan charge associated with ˆ h j , and can be computed as x j = Q ˆ h j ) (89)On the other hand, for the weight µ , the Cartan charge associated with ˆ h j is given by µ (ˆ h j ) = (cid:104) µ, ˆ h j (cid:105) = Tr( µ ∗ ˆ h j ) (90)Here we use the same symbol µ for the weight as a linear functional and as an element in h . We will only be concernedwith the case of real µ and therefore the Cartan charge reads µ (ˆ h j ) = Tr( µ ˆ h j ) (91)Comparing Eq. (89) and Eq. (91) we conclude the correspondence between Q and µ is Q = µ (92)This leads to the following proposition. Proposition 3:
Suppose O is a fixed-charge operator that corresponds to a traceless charge configuration Q withMCSD, with the CSD of O being D . Let us also suppose O belongs to some irreducible representation ( Γ L , Γ R ) of SU ( N ) L × SU ( M ) R in the U (1) A -neutral sector. Then 2 Q must be a weight of Γ L , and − Q must be a weight of Γ R .Since we know that the weights of Lie algebra representations sit on a discrete weight lattice, we then deduce fromEq. (92) that the charge configuration Q is also quantized.Because we want to consider fixed-charge operators corresponding to a traceless charge configuration Q withMCSD, according to Proposition 1, the weights of our interest should belong to ( Adj L ) D / , when we consider the SU ( N ) L factor. The nonzero weights of Adj L are nonzero roots of sl( N ; C ), which are given by [62] α jk = e j − e k , j (cid:44) k , j , k = , , ..., N (93)9where e j ’s denote the standard basis elements of C N , that is e j = { , ..., (cid:124)(cid:123)(cid:122)(cid:125) j − , , , ..., (cid:124)(cid:123)(cid:122)(cid:125) N − j } (94)for j = , , ..., N . Note that in this representation of roots we have identified h with the subspace of C N consisting ofvectors whose components sum to zero [62]. Because all the weights of a tensor product representation are given bythe sum of weights of the component representations [61], we conclude that if µ is a weight of ( Adj L ) D / , then µ mustbe able to be expressed as µ = D / (cid:88) p = s p α j p k p , with s p = p = , , ..., D / α j p k p = e j p − e k p with j p (cid:44) k p is one of the weights in Eq. (93). From Eq. (95) we deduce that ifwe write µ = ( µ , µ , ..., µ N ) , then µ i ∈ Z , ∀ i = , , ..., N , and then Eq. (92) tells us the diagonal entries of Q must beintegers or half-integers.Now for any vector ν ≡ ( ν , ν , ..., ν N ) ∈ C N , define the A-length of A [ ν ] of ν as A [ ν ] ≡ N (cid:88) i = | ν i | (96)Suppose ν, ρ ∈ C N , then the following triangle inequality holds A [ ν + ρ ] ≤ A [ ν ] + A [ ρ ] (97)This can be proved easily: Suppose ν ≡ ( ν , ν , ..., ν N ) , ρ = ( ρ , ρ , ..., ρ N ), then A [ ν + ρ ] = N (cid:88) i = | ν i + ρ i | ≤ N (cid:88) i = | ν i | + N (cid:88) i = | ρ i | = A [ ν ] + A [ ρ ] (98)It is also obvious that the A-length has a linearity property with respect to multiplication by a c-number A [ c ν ] = | c | A [ ν ] , ∀ c ∈ C (99)Then by using the triangle inequality and linearity property of the A-length, from Eq. (95) we can deduce A [ µ ] ≤ D / (cid:88) p = A [ s p α j p k p ] = D / (cid:88) p = | s p | A [ α j p k p ] (100)Now let us note that | s p | ≤ , A [ α j p k p ] = A [ µ ] ≤ × D / = D (102)On the other hand, we see the charge configuration Q corresponding to µ satisfies Q = µ . To make a comparison weshould also map Q into C N in the obvious manner, i.e. Q = diag { Q , Q , ..., Q N } → ( Q , Q , ..., Q N ) ∈ C N (103)Then we can write A [ Q ] = A [ µ ] (104)0Combining Eq. (102) and Eq. (104) we see immediately that D ≥ A [ Q ] (105)This leads to the following proposition. Proposition 4
Suppose O is a fixed-charge operator that corresponds to a traceless charge configuration Q = diag { Q , Q , ..., Q N } . Let us denote the CSD of O by D . Then D satisfies the inequality D ≥ i = N (cid:88) i = | Q i | (106)
3. Scaling dimension and operator construction
All the conclusions achieved up to now are deduced without the need of explicitly constructing the fixed-chargeoperators. On the other hand, one can show that the equality sign in Eq. (105) and Eq. (106) can always be achievedby constructing an operator corresponding to a given charge configuration. To this end, we first consider buildingblocks that have simple definite transformation properties under SU ( N ) L × SU ( M ) R × U (1) A and are U (1) A -neutral.For example, we may consider Tr( τ L H τ † R H † ) (107)where τ L is an N × N matrix in some root subspace of the sl( N ; C ) Lie algebra, and τ R is an M × M matrix related to τ L in the following manner τ R = (cid:32) τ L N × ( M − N ) ( M − N ) × N ( M − N ) × ( M − N ) (cid:33) (108)Obviously, the building block in Eq. (107) lives in the bi-adjoint representation of SU ( N ) L × SU ( M ) R , i.e. ( Adj L , Adj R ).It is constructed in such a manner that Q L + Q R = Q L corresponding to a weight of Adj L .The explicit form of τ L is given by τ L = E pq (109)for some p , q = , , ..., N and p (cid:44) q . This is because we have the commutation relation[ˆ h j , E pq ] =
12 ( δ jp − δ jq − δ j + , p + δ j + , q ) E pq (110)Here j = , , ..., N − h j is defined by ˆ h j ≡
12 ( E j , j − E j + , j + ) (111)which satisfy the normalization condition Tr(ˆ h j ) = .Let us first identify the charge configuration associated with Eq. (107). Define a set of N linear functionals ε p , p = , ..., N acting on h as follows ε p (ˆ h j ) =
12 ( δ jp − δ j + , p ) (112)Then Eq. (110) can be written as [ˆ h j , E pq ] = { ε p (ˆ h j ) − ε q (ˆ h j ) } E pq (113)which means E pq corresponds to the root ε p − ε q , which when mapped into h using the inner product Eq. (83) gives α pq defined in Eq. (93) . This α pq just corresponds to the weight of Adj L associated with Eq. (107) and the correspondingcharge configuration is simply α pq , according to Eq. (92). This can be deduced from the results in Appendix G of the textbook by J. F. Cornwell [61]. (cid:104) Π j ( τ Lj H τ † Rj H † ) y j (cid:105) . (114)Here y j > τ Lj is an N × N matrix with the explicit form given by τ Lj = E p ( j ) q ( j ) for some p , q = , , ..., N that depend on j . The way that Eq. (114) is constructed implies that its charge configuration Q is justthe appropriate linear combination of the charge configuration Q j of its corresponding building blocks Q = (cid:88) j y j Q j (115)where Q j = α p ( j ) q ( j ) (116)We can now reverse the logic and ask for a given Q how one may choose τ Lj and y j in order to construct a MCSDoperator in the form of Eq. (114). To this end, we may rewrite Eq. (115) as2 Q = (cid:88) j y j α p ( j ) q ( j ) (117)Note 2 Q ∈ C N , with all entries being integers and the sum of all entries is zero. We also have α p ( j ) q ( j ) ∈ C N , which fora given j there exists only two nonzero entries, filled by + − (cid:80) j y j for a given Q . We can rewrite Eq. (117) as 2 Q − (cid:88) j y j α p ( j ) q ( j ) = α p ( j ) q ( j ) ’s from the given C N vector 2 Q . Supposeeach time we are only allowed to subtract one α p ( j ) q ( j ) , which we call an elementary subtraction. (For a given j wetherefore eventually subtract it y j times). The sum (cid:80) j y j therefore equals the total number of times we need toperform such elementary subtractions to make the resulting C N vector vanish. To minimize (cid:80) j y j it is then obviousthat during the subtraction process each entry of the C N vector should change in a monotonic manner (or remainunchanged for some steps). As a concrete example, suppose 2 Q = (2 , − , − , − , − → (1 , , − → (0 , ,
0) (119)while the following subtraction is not monotonic(2 , − , − → (1 , , − → (1 , − , → (0 , ,
0) (120)It can be seen manifestly in this simple example that non-monotonic subtraction leads to an increase of the totalnumber of times we need to subtract the vector to zero, and this obviously generalize to general cases. Monotonicsubtraction can always be realized, by subtracting from the positive entry with the largest absolute value andnegative entry with the largest absolute value each time. In such a case, the total number of times we need to performelementary subtractions simply equals A [2 Q ] / = A [ Q ], that is A [ Q ] = (cid:88) j y j (121)On the other hand, from it is obvious that the CSD D of the operator in Eq. (114) is D = (cid:88) j y j (122)2Therefore we conclude the MCSD can be achieved, with the relation D = A [ Q ] (123)which is compatible with our previous finding Eq. (106) without explicit construction of the fixed-charge operator.Therefore we are led to the following proposition Proposition 5
The equality sign in Eq. (106) can always be achieved.We emphasize that the method does not guarantee the unicity of the MCSD operator. In fact, one may choose toredistribute the trace operation (i.e. splitting one single trace to multiple traces), change the order of matrix productsfor di ff erent τ Lj H τ † Rj H † factors, or change the root basis, to obtain more operators associated with the same chargeconfiguration. Even if we impose the MCSD requirement there can be multiple solutions. Algebraically they maylead to di ff erent or identical results. It is also not known whether the above method based on the τ Lj H τ † Rj H † buildingblocks with appropriate application of the trace operation covers all fixed-charge MCSD operators. Nevertheless,for a special type of charge configuration matrix, there is a unique answer and we know the above way of explicitconstruction must lead to the unique answer. This charge configuration is Q L , J = diag {− J , J , , · · · , } (124)with J being an integer or half-integer. This charge configuration corresponds to the highest weight in the tensorproduct of Adj L , which is in turn the sum of the highest weight of Adj L . The uniqueness results from the fact that thehighest weight of a representation is always simple. The irreducible representation associated with such a highestweight then has the Dynkin label (2 J , , ..., , J ).The above five propositions we proved pave the way for a general identification of irreducible representations fora given charge configuration prescribed in a fixed-charge semiclassical computation. With the MCSD assumptionthe MCSD can be determined by virtue of Proposition 4 and 5 from the given charge configuration Q . Then thecandidate irreducible representations must satisfy the requirements of Proposition 1-3. IV. SEMICLASSICS AND ANOMALOUS DIMENSIONS IN THE U ( N ) × U ( M ) MODEL
In this section, we start the exploration of U ( N ) × U ( M ) model in 4 − (cid:15) dimensions with the fixed-charge semiclassicalmethod. The necessary group-theoretic results (especially the 5 propositions in section III) will be used, and we referthe readers who are interested in the detailed proofs to the previous section. In Euclidean spacetime, the Lagrangianof the theory reads L = Tr( ∂ µ H † ∂ µ H ) + u Tr( H † H ) + v (Tr H † H ) , (125)where H is a N × M complex matrix. For N = M and v >
0, it describes the finite-temperature phase transition inmassless quantum chromodynamics [64] with H the order parameter. We work in the MS scheme. The couplings arerenormalized as u M − (cid:15) = u + ∞ (cid:88) n = a ( n ) u ( u , v ) (cid:15) n , v M − (cid:15) = v + ∞ (cid:88) n = a ( n ) v ( u , v ) (cid:15) n , (126)The beta functions of the couplings are given by β u ≡ dud log M | (cid:15) = = − (cid:15) u + u ∂ a (1) u ∂ u + v ∂ a (1) u ∂ v − a (1) u , β v ≡ dvd log M | (cid:15) = = − (cid:15) v + u ∂ a (1) v ∂ u + v ∂ a (1) v ∂ v − a (1) v . (127)and, at 1-loop, read [65] β u ( u , v ) = − (cid:15) u + π (cid:16) uv + ( N + M ) u (cid:17) , (128) β v ( u , v ) = − (cid:15) v + π (cid:16) ( NM + v + N + M ) uv + u (cid:17) . (129)3At the 1-loop level there are always a Gaussian FP ( u ∗ = v ∗ =
0) and an O (2 NM ) one ( u ∗ = u ∗± = π A MN ∓ √ R MN D MN (cid:15) , v ∗± = π B MN ± ( M + N ) √ R MN D MN (cid:15) , (130)where A MN = NM + MN − N − M , B MN = − ( M + N ) , R MN = + M + N − MN , D MN = ( MN − M + N ) + . (131)The beta functions to five loops have been derived in [63], where the authors concluded that no stable FP exists for N = M and d =
3, suggesting that the chiral phase transition in light QCD at finite temperature is first-order. When R MN < walking type [66, 67].Since the u coupling breaks O (2 NM ) symmetry to SU ( M ) × SU ( N ) subgroup, it is convenient to think aboutrepresentations of this model as a decomposition of the O (2 NM ) multiplets with defining (vector) and the 2-indextraceless symmetric representations of O (2 NM ) as O (2 NM ) = = [ N , ¯ M ] ⊕ [ ¯ N , M ] . (132) O (2 NM ) = (1 , Adj) ⊕ (Adj , ⊕ (Adj , Adj) ⊕ (cid:104) ( , ∗ ) ⊕ ( , ∗ ) ⊕ c . c . (cid:105) . M + NM − = N − ⊕ M − ⊕ (cid:16) N − (cid:17) (cid:16) M − (cid:17) ⊕ (cid:32) N ( N + ) (cid:33) (cid:32) M ( M + ) (cid:33) ⊕ (cid:32) N ( N − ) (cid:33) (cid:32) M ( M − ) (cid:33) , (133)where in the last line we explicitly show the dimension of the representations appearing in the decomposition. A. Charging the system
In this section, we analyze the symmetry breaking pattern induced by charge fixing and set up the semiclassicalcomputation. After a Weyl map to the cylinder, our starting points are the cylinder action (68) and the spatiallyhomogeneous ground state ansatz given by Eq.(69). The Noether charges Q L and Q R associated with the U ( N ) × U ( M )global symmetry are given by Eq.(71) and satisfy the constraint (74) Q L + Q R =
0. The Euler-Lagrange equations read( m = (cid:16) d − R (cid:17) ) ∂ H + ∇ H + u H (cid:16) H † H (cid:17) + v Tr (cid:16) H † H (cid:17) H + m H = . (134)and for our homogeneous ansatz Eq. (69), they reduce to:2 M E B = − u B † B − v Tr (cid:16) B † B (cid:17) B − m B . (135)We label the entries on the diagonal of M E , ii with M E , ii = − i µ i . In this subsection µ i is a chemical potential andshould not be mistaken with the group theoretical weight matrix µ used elsewhere in the paper. For B ii we have B ii = b i , we can now rewrite the EOM as 2 µ i = u b i + v N (cid:88) k = b k + m , (136)while the corresponding ”charges” read J i ≡ ( Q L ) ii = − Vb i µ i . (137)4The classical energy E is given by evaluating the cylinder Lagrangian L cyl in (68) with an appropriate boundaryterm − (cid:80) Ni = µ i ∂ L cyl ∂µ i , which implements the charge fixing. We obtain EV = L cyl − N (cid:88) i = µ i ∂ L cyl ∂µ i = N (cid:88) i = b i µ i + u N (cid:88) i = b i + v N (cid:88) i = b i + m N (cid:88) i = b i . (138)We proceed by considering a 2-parameters family of charge configurations Q L , J , s = diag (cid:16) J , J , . . . (cid:124)(cid:123)(cid:122)(cid:125) s , − J , − J , . . . (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) s , , , . . . (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) N − s (cid:17) . (139)For N = M and varying s , this charge configuration interpolates between the ones considered in [16] ( s =
1; given inEq.(124)) and [12] ( s = N / s and J . This is the first time that thefixed-charge semiclassical methods are used to access the scaling dimension of operators with the same CSD anddi ff erent irreducible representation by varying the charge configuration.The classical energy for this charge configuration can be easily computed along the lines of Sec.II. By parameterizingthe M E and B matrices as follow µ i = µ i = , . . . , s , − µ i = s + , . . . , s , i = s + , . . . , N , b i = b i = , . . . , s , i = s + , . . . , N , (140)the charge condition and EOM become J = V µ b , µ = ( u + sv ) b + m . (141)From Eq.(123) we have that the CSD in four dimensions is ¯ Q = sJ . Since J ≥ /
2, this implies that the resultsobtained in [12] for the case M = N , s = N / Q ≥ N and not for arbitrary values of ¯ Q and N .It is useful to define rescaled (renormalized) ’t Hooft couplings as A h = J uN (4 π ) , A v = J svN (4 π ) , (142)Then the above equations imply2 µ m = + x x , x = N ( A h + A v ) + (cid:115) − + (cid:18) N ( A h + A v ) (cid:19) , (143)and our semiclassical expansion takes the form ∆ O J , s = (cid:88) k = − J k ∆ k ( A ∗ h , A ∗ v ) . (144)The leading order in the semiclassical expansion follows straightforwardly from the results above by setting d = A h , v = A ∗ h , v , where the star denotes the value of the couplings at the FP. We have ∆ − ( A ∗ h , A ∗ v ) = sN A ∗ h + A ∗ V ) 1 x ∗ (cid:16) √ x ∗ / − x ∗ / + √ x ∗ / + / x ∗ + / (cid:17) . (145)The expansion for small A ∗ h , v reads J ∆ − ( A ∗ h , A ∗ v ) = ¯ Q + (cid:32) A ∗ h + A ∗ v N (cid:33) − (cid:32) A ∗ h + A ∗ v N (cid:33) + (cid:32) A ∗ h + A ∗ v N (cid:33) + O (cid:32) A ∗ h + A ∗ v N (cid:33) . (146)5Notice that the leading order depends neither on M nor N when rewritten in terms of the original couplings u ∗ and v ∗ . This is because at the classical level only the fields which take a non-zero vev contribute and whose numberdepends on s .Before proceeding with the computation of ∆ , it is useful to study the induced symmetry breaking pattern. Theexplicit breaking can in general be deduced by adding the charge-fixing boundary term to the Lagrangian and checkwhich symmetries it preserves. We obtain SU ( N ) L ⊗ SU ( M ) R ⊗ U (1) A = ⇒ explicit C ( R ) L ⊗ SU ( M ) R ⊗ U (1) A , (147)where C ( R ) L is the SU ( N ) L subgroup that commutes with P = diag (cid:16) , , . . . (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) s , − , − , . . . (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) s , , , . . . (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) N − s (cid:17) and it is explictilygiven by C ( R ) L = SU ( s ) Lu ⊗ SU ( s ) Ld ⊗ SU ( N − s ) Ld ⊗ U (1) L ⊗ U (1) L , (148)where SU ( s ) Lu and SU ( s ) Ld are rotations in the first and second upper s × s blocks of SU ( N ) L while SU ( N − s ) Ld rotates thelower N − s × N − s block. Finally U (1) L and U (1) L are generated, respectively, by P and P = diag (cid:16) , , . . . (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) s , , , . . . (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) N − s (cid:17) and act on the left factor.The spontaneous symmetry breaking is determined by the vacuum configuration, which is proportional to the P matrix defined above. We have C ( R ) L ⊗ SU ( M ) R ⊗ U (1) A = ⇒ SSB SU ( s ) Lu ⊗ SU ( s ) Ld ⊗ SU ( N − s ) Ld ⊗ U (1) D ⊗ U (1) D ⊗ SU ( M − s ) Rd ⊗ U (1) A . (149)Here, U (1) D , are the diagonal subgroup of U (1) L , ⊗ U (1) R , where U (1) R , are the counterparts of U (1) L , actingon the right factor. Finally, SU ( M − s ) Rd is defined as the SU ( M − s ) rotation in the lower M − s × M − s block of SU ( M ) R . To define U (1) A , we first introduce U (1) L which is generated by diag (cid:16) , , . . . (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) s , , , . . . (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) N − s (cid:17) and acts on the left,and U (1) R which is generated by diag (cid:16) , , . . . (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) s , , , . . . (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) M − s (cid:17) and acts on the right. U (1) A is then defined as the axial partof U (1) L and U (1) R . Note the diagonal part of U (1) L and U (1) R is not independent from U (1) D and is thus notcounted.Altogether, the number of broken generators is M − − (cid:104) ( M − s ) − (cid:105) = s ( M − s ) . (150)We parametrize the fluctuations as H ( τ, x ) = e iM E τ ( B + Φ ( τ, x )) , (151)where Φ ( τ, x ) is a N × M matrix. We have L quad = N (cid:88) i = M (cid:88) j = ∂ µ Φ ij ∂ µ Φ ∗ ij − µ s (cid:88) i = M (cid:88) j = (cid:16) ( ∂ Φ ij ) Φ ∗ ij − Φ ij ∂ Φ ∗ ij (cid:17) − s (cid:88) i = s + M (cid:88) j = (cid:16) ( ∂ Φ ij ) Φ ∗ ij − Φ ij ∂ Φ ∗ ij (cid:17) + u b N (cid:88) i = s (cid:88) j = Φ ∗ ij Φ ij + u b s (cid:88) i = s (cid:88) j = (cid:16) Φ ij Φ ji + Φ ∗ ij Φ ∗ ji (cid:17) + v b s (cid:88) i = ( Φ ii + Φ ∗ ii ) + (cid:16) sv b + m (cid:17) N (cid:88) i = s + M (cid:88) j = Φ ij Φ ∗ ij . (152)It is useful to write Φ in block form as Φ = Φ (11)2 s × s Φ (12)2 s × ( M − s ) Φ (21)( N − s ) × s Φ (22)( N − s ) × ( M − s ) . (153)6The blocks decouple and we can decompose L quad as L quad = L (11) quad + L (12) quad + L (21) quad + L (22) quad . The dispersion relationsof the fluctuations read ω = (cid:113) J (cid:96) + µ s ( N − s ) d . o . f .ω = (cid:113) J (cid:96) + m N − s ) ( M − s ) d . o . f ω , = (cid:113) J (cid:96) + µ ∓ µ s (2 M − s ) d . o . f .ω , = (cid:113) J (cid:96) + µ + m ± µ s d . o . f .ω , = √ (cid:115) J (cid:96) + m + µ ± (cid:114)(cid:16) J (cid:96) + m + µ (cid:17) − J (cid:96) (cid:16) J (cid:96) + m (cid:17) s − . o . f .ω , = √ (cid:115) J (cid:96) + m + µ ± (cid:114)(cid:16) J (cid:96) + m + µ (cid:17) − J (cid:96) (cid:16) J (cid:96) + m (cid:17) . o . f . , (154)where m = µ − m , m = (cid:16) µ − m (cid:17) u u + sv , m = sv b + m . (155) ω describes Type II Goldstone bosons while ω and ω correspond to relativistic Type I Goldstone bosons. Theremaining dispersion relations describe gapped modes. It’s easy to check that the number of real d.o.f. sums to 2 NM while the counting of Goldstone modes with respect to the number of broken generators is2 × s (2 M − s ) + s − + = s ( M − s ) , (156)which agrees with Eq.(150), saturating the Nielsen-Chadha bound. The NLO in the semiclassical expansion is givenby the general formula Eq.(46). Regularization and renormalization are performed as explained in Sec.II, yielding ∆ ( A ∗ h , A ∗ v ) = ρ ( x ∗ , M , N , s , A ∗ h , A ∗ v ) + ∞ (cid:88) (cid:96) = R (1 + (cid:96) ) (cid:88) i g i ( M , N , s ) ω i ( (cid:96), x ∗ , A ∗ h , A ∗ v ) d = + σ ( (cid:96), x ∗ , M , N , s , A ∗ h , A ∗ v ) . (157)where x ∗ has been defined in Eq.(143) while ρ ( x ∗ , M , N , s , A ∗ h , A ∗ v ) and σ ( (cid:96), x ∗ , M , N , s , A ∗ h , A ∗ v ) are given in App.A.We checked the cancellation of the divergent terms between ∆ − and ∆ as explained above Eq.(20). For A h = O (2 NM ) model. Finally, for N = M and s = ∆ ( A ∗ h , A ∗ v ) = − N (cid:16) A ∗ h + A ∗ v (cid:17) (cid:104) s A ∗ h ( M + N + s ) + A ∗ h A ∗ v (2 s (2( M + N ) + s ) + + ( MN + A ∗ v (cid:105) − N (cid:104) s A ∗ h ( M + N − s ) + A ∗ h A ∗ v ( s ( M + N − s ) − + ( MN − A ∗ v (cid:105) + N (cid:104) A ∗ h A ∗ v (cid:16) ζ (3)( s ( M + N + s ) + + Ms + Ns − s − (cid:17) + A ∗ h A ∗ v (cid:16) ζ (3)( s ( M + N ) + + MN + Ms + Ns − s − (cid:17) + s A ∗ h (2 ζ (3)( M + N + s ) + M + N − s ) + A ∗ v (2 ζ (3)( MN + + MN − (cid:105) + O (cid:16) A ∗ v , h (cid:17) . (158)From the above results, we can also extract the full 1-loop scaling dimension, which we rewrite as a power series inthe couplings ∆ − loop Q J , s = ¯ Q (cid:18) − (cid:15) (cid:19) + N (cid:16) A ∗ h (cid:16) ¯ Q − s (cid:17) + ( ¯ Q − A ∗ v (cid:17) = ¯ Q (cid:18) − (cid:15) (cid:19) + ¯ Q (cid:16) ¯ Q − s (cid:17) (4 π ) s u ∗ + Q ( ¯ Q − π ) v ∗ , (159)and depends neither on N nor M when rewritten in terms of the original couplings u and v . We conclude that for theconsidered family of charge configurations, there is no scaling dimension degeneracy in the perturbative regime andthus the corresponding operators transform in di ff erent irreducible representations. These are accessed by varying7 s at fixed CSD ¯ Q . In the next section, we will study few concrete examples by setting ¯ Q = , ,
8. To this end, it isuseful to consider one more charge configuration given by Q L = diag {− , J , J , , · · · , } . (160)The M E and B matrices can be parametrized as M E = − i diag (cid:8) µ , µ , µ , , · · · , (cid:9) , B = diag { b , b , b , , · · · , } . (161)The EOM and the charge conditions read J = V µ b , µ = u b + v (cid:16) b + b (cid:17) + m , J = − V µ b , µ = u b + v (cid:16) b + b (cid:17) + m . (162)The physical solution is µ = J µ m v C , b = (cid:115) C π µ v , b = (cid:115) − Jm π µ , (163)where C = Jm ( u + v ) + π µ − π µ m and µ solves the following equation − J µ m v C + Jm (cid:18) − Jm v C + u + v (cid:19) π µ + Jm v C = . (164)We choose the solution of the above equation such that it reproduces the O (2 NM ) limit when u =
0. The perturbativeexpansion of this solution reads µ = − m − Jm ( u + v )4 π + J m (cid:16) u + u v + v (cid:17) π + J (cid:16) − mu − mu v − mu v − mv (cid:17) π + O (cid:16) ( u J ) , ( v J ) (cid:17) . (165)The range of validity of this solution is determined by the constraint C <
0, which we found out to be always satisfiedin the perturbative regime. The leading contribution in the semiclassical approximation is given by the classicalenergy (138) evaluated on this solution: J ∆ − = ¯ Q + ¯ Q (3 u ∗ + v ∗ )64 π − ¯ Q (cid:16) u ∗ + u ∗ v ∗ + v ∗ (cid:17) π + ¯ Q (cid:16) u ∗ + u ∗ v ∗ + u ∗ v ∗ + v ∗ (cid:17) π + O (cid:16) ( u ∗ ¯ Q ) , ( v ∗ ¯ Q ) (cid:17) . (166)where we set the couplings to their FP values and ¯ Q = J is the classical scaling dimension, as expected from Eq.(123).We checked that we recover the O (2 NM ) case when u =
0. To compute the fluctuation spectrum, we expand around8the classical trajectory as in Eq.(151). The result for the dispersion relations reads˜ ω = (cid:113) J (cid:96) + µ N −
3) d . o . f . ˜ ω = (cid:113) J (cid:96) + µ N −
3) d . o . f . ˜ ω , = (cid:113) J (cid:96) + µ ± µ × ( M −
3) d . o . f . ˜ ω , = (cid:113) J (cid:96) + µ ± µ × M −
3) d . o . f . ˜ ω = (cid:113) J (cid:96) + m N − M −
3) d . o . f . ˜ ω , = (cid:115) J (cid:96) + ub + µ ± (cid:114)(cid:16) J (cid:96) + ub + µ (cid:17) − J (cid:96) (cid:16) J (cid:96) + ub (cid:17) × . o . f . ˜ ω , , , : (cid:16) J (cid:96) − ω ± ω ( µ − µ ) (cid:17) (cid:104) J (cid:96) − ω ± ω ( µ − µ ) + u (cid:16) b + b (cid:17)(cid:105) ± u ω (cid:16) b − b (cid:17) ( µ + µ ) − ω ( µ + µ ) = × . o . f . ˜ ω , , , :det D A (cid:16) ω, J (cid:96) (cid:17) = . o . f . , (167)where m = v ( b + b ) + m and D A (cid:16) ω, J (cid:96) (cid:17) = ω − J (cid:96) + z − i ( µ + µ ) ω z − √ i ( µ − µ ) ω i ( µ + µ ) ω ω − J (cid:96) √ i ( µ − µ ) ω z − √ i ( µ − µ ) ω ω − J (cid:96) + z − i (2 µ + µ ) ω √ i ( µ − µ ) ω i (2 µ + µ ) ω ω − J (cid:96) , (168) z = − (cid:104) ( u + v ) b + v b b + u + v ) b (cid:105) , z = − √ b − b ) [( u + v ) b + ( u + v ) b ] , z = − (cid:104) u + v ) b − v b b + u + v ) b (cid:105) . (169)Although not obvious, for u = O (2 NM ) model discussed in Sec.II B when k = M charges have been fixed. In particular one of the last four d.o.f. reduces to the U (1) conformal mode.We weren’t able to find an analytical expression for ∆ in this case. Instead, we computed it numerically at fixedvalues of parameters and as a function of (cid:15) . The results are given in the next section. B. On how to identify the fixed-charge operators
In this section, we focus on identifying the fixed charge operators associated with a certain charge configuration.In particular, we propose a practical identification procedure which we outline by performing detailed examples inthe U ( N ) × U ( M ) model. Since the MSCD assumption can be violated at large coupling, the procedure is valid onlywhen the anomalous dimensions of the operators involved are much smaller than one, i.e. in the perturbative regime.Since in weakly coupled theories it is easy to find the explicit form of the MSCD operator once the irrep in which ittransforms is known, we focus on identifying the latter. We first list the conclusions below before conducting a moredetailed discussion. • The representation of the fixed charge operators can be uniquely determined when a charge configuration ischosen. This conclusion is based on the three conditions summarized in Propositions 1 − • To determine the representation of the fixed charge operators, both group theory and the actual semiclassicalcomputations of the scaling dimensions should be implemented. Group theory alone is not su ffi cient.9We start our analysis by considering operators with ¯ Q =
2; Eq.(123) implies that we can build only one N × N charge matrix Q L , / = diag {− / , / , , · · · , } . (170)This charge configuration is of the special type considered in Eq.(124). Then we can immediately identify theirreducible representation in which the corresponding fixed charge operator sits as the bi-adjoint representation( Adj , Adj ) = ( N − , M − Q = (cid:104) T a HT b H † (cid:105) . In [16], the anomalous dimension of this operator has been computed in the semiclassical expansionfor N = M and the result has been validated via a diagrammatic calculation at the 1-loop level. The result for general N , M has been given in the previous section, being a special case ( s = J = /
2) of the charge configuration (139).The simplest nontrivial example is obtained by fixing N = M = Q = Γ L , Γ R ) of SU ( N ) L × SU ( N ) R where Γ L = Γ R = (cid:0) Adj (cid:1) Q / . Thus, in the case at hand, the operators livein the decomposition of the tensor product ⊗ , which reads ⊗ = ⊕ ) ⊕ ⊕ ⊕ . (171)To construct all the relevant charge configurations, it needs to satisfy the following three requirements:1. The matrix of charge configuration is diagonal and traceless i.e. tr Q = Q / = ff erent charge matrices: Q (4)3 A = − , Q (4)3 B = − / − / . (172)It was shown in the previous section that the weight and the charge satisfy: µ = Q . The nonzero roots of SU (3) are α = − , α = − . α = α + α , (173)and − α , − α , − α . We, therefore, obtain µ (4)3 A = α and µ (4)3 B = α + α . By using the Cartan matrix of the SU (3)algebra A SU (3) = (cid:32) − − (cid:33) , (174)we obtain α = w − w , α = − w + w (175)where w and w are the fundamental weights of SU (3). Then we can decompose the weights µ (4)3 A , µ (4)3 B as µ (4)3 A = w − w = (4 , − ,µ (4)3 B = w = (3 , . (176)The next step is to determine the representations containing the above weights. By analyzing the weight diagramsof the irreducible representation appearing in the RHS of (171), we see that (4 , −
2) only appears in while (3 , and . Thus, we can set the following correspondence Q (4)3 A : ( , ) , Q (4)3 B : (cid:40) ( , ) (cid:16) , (cid:17) , (177)0 _ Q A ( ) . Q B ( ) ϵ Re [ Δ Q ] FIG. 1. The results for the real part of the scaling dimension at the fixed point for the U (3) × U (3) operators with CSD ¯ Q = Q (4)3 A (black line) and Q (4)3 B (red dots) as a function of (cid:15) . The error bars encode the numerical error in evaluating ∆ for Q (4)3 B . where we have already excluded asymmetric representations such as (cid:16) , (cid:17) since they do not appear in the decom-position of the four indices traceless symmetric O (2 NM ) tensor (Proposition 2). Notice that Q (4)3 A is again of the type(124) and, indeed, it can be directly uniquely associated with ( , ).Using group theory only, (177) is the best we can achieve. To further disentangle the representations, we need toemploy the fixed-charge semiclassical method, computing at least the first two orders in the semiclassical expansionfor both charge matrices. Then, if the corresponding scaling dimensions at the fixed point are di ff erent functions of (cid:15) we have that Q (4)3 B corresponds to (cid:16) , (cid:17) . The scaling dimension of Q (4)3 A at NLO in the semiclassical expansion hasbeen computed analytically in the previous section ( s = J = N = M = Q (4)3 B corresponds tothe J = / N = M = Q (4)3 A and Q (4)3 B . The error bar on the red dots takes into account all the potential numerical errors . Clearly,the two scaling dimensions are di ff erent functions of (cid:15) , and thus we can unequivocally associate representations andcharges as Q (4)3 A : ( , ) , Q (4)3 B : (cid:16) , (cid:17) . (178)Notice that the scaling dimension corresponding to Q (4)3 B is smaller than the one associated with Q (4)3 A , consistentlywith the minimal scaling dimension criteria selecting the fixed-charge operators. In the small ’t Hooft coupling regime we also expect that the scaling dimension corresponds to Q (4)3 B is smaller than that of Q (4)3 A . Since the fixedpoint values are complex, bigger / smaller refers to the real part of the scaling dimensions. The main source of error comes from performing a numerical Taylor expansion in (cid:15) of the one-loop functional determinant during therenormalization procedure. We estimate the numerical error as the di ff erence of the scaling dimensions of Q (4)3 B with the one of Q (4)3 A in thelimiting case of u →
0. As we know, in this case, the scaling dimensions for Q (4)3 B and Q (4)3 A should be equal and coincide with the O (18) result. Q = N = M =
4. We can now build three independent charge matrices Q (4)4 A = − , Q (4)4 B = − / − / , Q (4)4 C = / − / / − / . (179)To connect this example with our semiclassical calculations, we note that Eq.(139) encompasses Q (4)4 A ( s = J = N = M =
4) and Q (4)4 C ( s = J = / N = M =
4) while Eq.(160) reduces to Q (4)4 B when J = / N = M =
4. Bydenoting the three fundamental weights of SU (4) as W , W , and W , we can express the weights associated with theabove charge matrices as µ (4)4 A = W − W = (4 , − , ,µ (4)4 B = W − W = (3 , , − ,µ (4)4 C = W − W + W = (2 , − , . (180)The tensor product of two adjoint representations of SU (4) decomposes as ⊗ = ⊕ ) ⊕ (cid:48) ⊕ ⊕ ⊕ . (181)Inspecting the weight diagram of the above representations, we obtain the correspondence summarized below: Q (4)4 A : ( , ) , Q (4)4 B : (cid:40) ( , ) (cid:16) , (cid:17) , Q (4)4 C : ( , ) (cid:16) , (cid:17) ( (cid:48) , (cid:48) ) . (182)Again there is one charge matrix, Q (4)4 A , which is of the type considered in (124) and thus corresponds to a uniquerepresentation, ( , ). Then one can proceed by computing the scaling dimensions associated with Q (4)4 A and Q (4)4 B inthe semiclassical expansion. If they are di ff erent then Q (4)4 B corresponds to (cid:16) , (cid:17) and we can proceed by computingthe scaling dimension corresponding to Q (4)4 C . If the latter is also di ff erent from the previous ones, then we can furtherconclude that the operator with scaling dimension ∆ Q (4)4 C is in the ( (cid:48) , (cid:48) ) representation. This is actually the case,as can be seen from the results for the real and imaginary part of scaling dimensions at NLO, which are shownin Figs. 2 and 3, respectively. Due to the numerical error, in this case it is necessary to look also at the imaginarypart to disentangle the results. If the results for ∆ Q (4)4 A were the same, we wouldn’t have been able to identify therepresentation associated with Q (4)4 B , and would have been necessary to compute higher orders in the semiclassicalexpansion to check whether they broke the degeneracy or not. However, we could always deduce the irrep relatedto Q (4)4 C as ( (cid:48) , (cid:48) ) .In conclusion, we have Q (4)4 A : ( , ) , Q (4)4 B : (cid:16) , (cid:17) , Q (4)4 C : ( (cid:48) , (cid:48) ) . (183) From Fig.2, one could deduce that Re (cid:20) ∆ Q (4)4 B (cid:21) > Re (cid:20) ∆ Q (4)4 A (cid:21) , in violation of the minimal scaling dimension criteria. This apparent puzzle can besolved by taking into account the numerical errors. In addition, the fact that the comparison of the magnitude of the scaling dimensions atsmall (cid:15) should in principle be performed in the expansion of conventional perturbation theory rather than in the semiclassical expansion. Thedi ff erence is of order O (cid:16) ¯ Q (cid:15) (cid:17) , but prefactors may magnify it and invert the scaling dimension hierarchy. By making use of the full 2-loop resultof [16, 68], we estimated a di ff erence between 0 .
04% and 0 .
5% for ∆ Q (4)4 A at (cid:15) = .
1. However, the di ff erence in ∆ Q (4)4 B for the same value of epsilonmay be much larger. If Re (cid:20) ∆ Q (4)4 A (cid:21) = Re (cid:20) ∆ Q (4)4 B (cid:21) , it follows that Re (cid:20) ∆ ( , ) (cid:21) ≥ Re (cid:104) ∆ ( , ) (cid:105) , but since Re (cid:34) ∆ Q (4)4 C (cid:35) < Re (cid:104) ∆ ( , ) (cid:105) , then Re (cid:34) ∆ Q (4)4 C (cid:35) = Re (cid:104) ∆ ( (cid:48) , (cid:48) ) (cid:105) _ Q A ( ) . Q B ( ) _ Q C ( ) ϵ Re [ Δ Q ] FIG. 2. The results for the real part of scaling dimension for the U (4) × U (4) operators with CSD ¯ Q =
4, carrying the charges Q (4)4 A (black line), Q (4)4 B (red dots) and Q (4)4 C (blue lines) as a function of (cid:15) . The error bars encode the numerical error in evaluating ∆ for Q (4)4 B . _ Q A ( ) . Q B ( ) _ Q C ( ) - - ϵ Im [ Δ Q ] FIG. 3. The imaginary part of the scaling dimension for the U (4) × U (4) operators with CSD ¯ Q =
4, carrying the charges Q (4)4 A (blackline), Q (4)4 B (red dots) and Q (4)4 C (blue lines )at the fixed point values as a function of (cid:15) . The error bars encode the numerical error inevaluating ∆ for Q (4)4 B . As the last example, we keep N = M = Q =
8. The fourth tensor power of the adjoint representationdecomposes as ⊗ = ) ⊕ ) ⊕ (cid:48) ) ⊕ ) ⊕ ) ⊕ ) ⊕ ) ⊕ ) ⊕ ) ⊕ ) ⊕ ) ⊕ ) ⊕ ) ⊕ ) ⊕ ) ⊕ ) ⊕ (cid:48) ) ⊕ (cid:48) ) ⊕ (cid:48) ) ⊕ ) ⊕ ( ) ⊕ ) ⊕ ) . (184)We can build seven charge matrices Q (8)4 A = diag { , − , , , · · · , } , Q (8)4B = diag {− , , , , · · · , } , Q (8)4 C = diag { , − , , − , , · · · , } , Q (8)4D = diag { , − / , − / , , · · · , } , Q (8)4 E = diag { , − , − / , − / , , · · · , } , Q (8)4F = diag { / , / , − / , − / , , · · · , } , Q (8)4 G = diag { / , / , − , − , , · · · , } . (185)3The corresponding weights, expressed in the fundamental weight basis, read µ (8)4 A = (8 , − , , µ (8)4 B = (6 , , − , µ (8)4 C = (4 , − , , µ (8)4 D = (7 , , − ,µ (8)4 E = (6 , − , , µ (8)4 F = (2 , , − , µ (8)4 G = (2 , , . (186)By analyzing the weight diagrams and checking the decomposition of the O (2 NM ) traceless symmetric tensor of rank Q =
8, we obtain the correspondence below Q (8)4 A : ( , ) , Q (8)4 D : (cid:40) ( , ) (cid:16) , (cid:17) , Q (8)4 B : ( , ) (cid:16) , (cid:17)(cid:16) (cid:48) , (cid:48) (cid:17) , Q (8)4 F : ( , ) (cid:16) , (cid:17) ( , ) , Q (8)4 E : ( , ) (cid:16) , (cid:17)(cid:16) (cid:48) , (cid:48) (cid:17)(cid:16) , (cid:17) Q (8)4 G : ( , ) (cid:16) , (cid:17) ( , ) (cid:16) (cid:48) , (cid:48) (cid:17)(cid:16) , (cid:17) , Q (8)4 C : ( , ) (cid:16) , (cid:17) ( , ) (cid:16) (cid:48) , (cid:48) (cid:17)(cid:16) , (cid:17) ( , ) . (187)If all the corresponding semiclassical computations give di ff erent functions of (cid:15) as result, then we can uniquelyidentify all the irreps where the fixed charge operators sit. These are the representations outlined in red above. To bemore precise, to allow a complete identification it is not needed that all the scaling dimensions di ff er; for instance,the results for Q (4)4 B and Q (4)4 F or for Q (4)4 E and Q (4)4 F can be equal.To summarize, a general identification procedure consists in inspecting the weight diagrams of the various repre-sentations in order to obtain correspondences as Eq.(187) and then looking if the corresponding scaling dimensionsare degenerate or not, following a sort of ”identification chain” that starts from ”highest weight” charge matricesof the form (124) for which there is only one candidate for the corresponding irrep. In the second step, the choicewill be between this irrep plus one new candidate, and so on. We analyzed many other examples and our findingsstrongly suggest that this identification procedure can always be implemented and fails only in presence of particulardegeneracies in the semiclassical results. V. CONCLUSIONS
We introduced a general strategy apt at determining the relation between a given large charge configuration and theassociated operators. In fact, we demonstrated how, varying charge configurations, we could determine the specificanomalous dimensions of distinct operators transforming according to a variety of irreducible representations of thenon-abelian symmetry group going beyond traditional diagrammatical computations.To demonstrate the usefulness of our methodology, which fuses semiclassical methods with group theoreticalconsiderations, we determined the anomalous dimensions of several composite operators to the next-to-leadingorder in the semiclassical expansion of the U ( N ) × U ( M ) model in 4 − (cid:15) dimensions.Our work brings us one step closer to investigating the dynamics of theories similar in structure to the standardmodel of particle interactions which, in many respects, can be seen as a slight deformation of a CFT [69, 70].Additionally one can envision computing processes involving large number of SM Higgses useful for the nextgeneration of colliders [71–73] that have attracted past [74, 75] and recent attention [76].In fact, critical phenomena play an important role also for applications to social and health sciences. For example,it has been recently shown that (near) fixed points are a useful way to organise the dynamics and di ff usion ofinfectious diseases. The approach, known as the epidemiological Renormalization Group approach [77] has beenshown to emerge in [78] from either stochastic (percolation models, random walks, di ff usion models) or deterministic(compartmental-type) models that themselves can be viewed as mean field theories at criticality [79].4 ACKNOWLEDGEMENTS
The work of O.A. and J.B. is partially supported by the Croatian Science Foundation project number 4418 aswell as European Union through the European Regional Development Fund - the Competitiveness and CohesionOperational Programme (KK.01.1.1.06). F.S. and Z.W. acknowledge the partial support by Danish National ResearchFoundation grant DNRF:90. We thank Johan Henriksson for his very helpful comments. For the group theoreticalanalyses, we made use of the Mathematica package LieART [80, 81].
Appendix A: The functions ρ ( x ∗ , M , N , s , A h ∗ , A ∗ v ) and σ ( (cid:96), x ∗ , M , N , s , A h ∗ , A ∗ v ) In this appendix, we provide explicit expression for the functions appearing in Eq.(157). Recalling that x ∗ = N ( A ∗ h + A ∗ v ) + (cid:113) − + (cid:16) N ( A ∗ h + A ∗ v ) (cid:17) , we have ρ ( x ∗ , M , N , s , A h ∗ , A ∗ v ) = x ∗ / ( A h + A ∗ v ) (cid:104) − A ∗ h A ∗ v (cid:16) M (cid:16) Nx ∗ / + √ Nx ∗ / + / Nx ∗ + s (cid:16) √ x ∗ / + x ∗ / + √ x ∗ / +
72 3 / x ∗ +
48 3 / (cid:17)(cid:17) + Ns (cid:16) √ x ∗ / + x ∗ / + √ x ∗ / +
72 3 / x ∗ +
48 3 / (cid:17) + √ x ∗ / + x ∗ / + √ x ∗ / +
184 3 / x ∗ +
144 3 / (cid:17) + A ∗ h (cid:16) − M (cid:16) Nx ∗ / + s (cid:16) √ x ∗ / + x ∗ / + √ x ∗ / +
88 3 / x ∗ +
48 3 / (cid:17)(cid:17) − s (cid:16) N (cid:16) √ x ∗ / + x ∗ / + √ x ∗ / +
88 3 / x ∗ +
48 3 / (cid:17) + s (cid:16) √ x ∗ / + x ∗ / + √ x ∗ / +
56 3 / x ∗ +
48 3 / (cid:17)(cid:17)(cid:17) −A ∗ v (cid:16) MN (cid:16) √ x ∗ / + x ∗ / + √ x ∗ / +
88 3 / x ∗ +
48 3 / (cid:17) + (cid:16) √ x ∗ / + x ∗ / + √ x ∗ / +
64 3 / x ∗ +
48 3 / (cid:17)(cid:17)(cid:105) + ( M − s )( N − s ) (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) A ∗ h + (cid:16) x ∗ / + √ (cid:17) A ∗ v √ x ∗ / A ∗ h + A ∗ v + s (cid:115) x ∗ / (cid:16) A ∗ h + A ∗ v (cid:17) (cid:16) (cid:16) / x ∗ / + x ∗ / + √ (cid:17) A ∗ h + (cid:16) / x ∗ / + x ∗ / + √ (cid:17) A ∗ v (cid:17) / + s − − (cid:16) / x ∗ / + x ∗ / + √ (cid:17) A ∗ h + (cid:16) / x ∗ / + x ∗ / + √ (cid:17) A ∗ v x ∗ / (cid:16) A ∗ h + A ∗ v (cid:17) + (cid:32) (cid:16) x ∗ / + √ (cid:17) √ x ∗ / − (cid:33) A ∗ h A ∗ h + A ∗ v + (cid:16) x ∗ / + √ (cid:17) √ x ∗ / / + s − (cid:16) / x ∗ / + x ∗ / + √ (cid:17) A ∗ h + (cid:16) / x ∗ / + x ∗ / + √ (cid:17) A ∗ v x ∗ / (cid:16) A ∗ h + A ∗ v (cid:17) + (cid:32) (cid:16) x ∗ / + √ (cid:17) √ x ∗ / − (cid:33) A ∗ h A ∗ h + A ∗ v + (cid:16) x ∗ / + √ (cid:17) √ x ∗ / / + s (cid:16) x ∗ / + √ (cid:17) ( M − N )3 / √ x ∗ + s (cid:16) x ∗ / + √ (cid:17) ( N − s )3 / √ x ∗ + s (cid:16) x ∗ / + √ (cid:17) / √ x ∗ (A1)and σ ( (cid:96), x ∗ , M , N , s , A ∗ h , A ∗ v ) = − (cid:96) MN − (cid:96) MN + (cid:16) / x ∗ / + x ∗ / + √ (cid:17) (cid:96) x ∗ / (cid:16) A ∗ h + A ∗ v (cid:17) (cid:16) s A ∗ h ( M + N + s ) + A ∗ h A ∗ v (2 Ms + Ns + + ( MN + A ∗ v (cid:17) + (cid:96) x ∗ / ( A ∗ h + A ∗ v ) (cid:104) A ∗ h (cid:16) MNx ∗ / + Ms (cid:16) / x ∗ / + x ∗ / + √ (cid:17) + Ns (cid:16) / x ∗ / + x ∗ / + √ (cid:17)(cid:17) + A ∗ v (cid:16) MN (cid:16) / x ∗ / + x ∗ / + √ (cid:17) + / x ∗ / + x ∗ / + √ (cid:17)(cid:105) − x ∗ / ( A ∗ h + A ∗ v ) (cid:104) − A ∗ h (cid:16) MNx ∗ / + Ms (cid:16) / x ∗ / + x ∗ / + √ (cid:17) + Ns (cid:16) / x ∗ / + x ∗ / + √ (cid:17)(cid:17) − A ∗ v (cid:16) MN (cid:16) / x ∗ / + x ∗ / + √ (cid:17) + / x ∗ / + x ∗ / + √ (cid:17)(cid:105) (A2)5 [1] S. Hellerman, D. Orlando, S. Re ff ert and M. Watanabe, JHEP (2015), 071 doi:10.1007 / JHEP12(2015)071 [arXiv:1505.01537[hep-th]].[2] L. Alvarez-Gaume, O. Loukas, D. Orlando and S. Re ff ert, JHEP , 059 (2017) doi:10.1007 / JHEP04(2017)059[arXiv:1610.04495 [hep-th]].[3] D. Ja ff eris, B. Mukhametzhanov and A. Zhiboedov, JHEP (2018), 043 doi:10.1007 / JHEP05(2018)043 [arXiv:1710.11161[hep-th]].[4] S. Hellerman and S. Maeda, JHEP (2017), 135 doi:10.1007 / JHEP12(2017)135 [arXiv:1710.07336 [hep-th]].[5] L. `A. Gaum´e, D. Orlando and S. Re ff ert, [arXiv:2008.03308 [hep-th]].[6] D. Orlando, S. Re ff ert and F. Sannino, Phys. Rev. D (2020) no.6, 065018 doi:10.1103 / PhysRevD.101.065018 [arXiv:1909.08642[hep-th]].[7] D. Orlando, S. Re ff ert and F. Sannino, [arXiv:2003.08396 [hep-th]].[8] F. Sannino, Acta Phys. Polon. B (2009), 3533-3743 [arXiv:0911.0931 [hep-ph]].[9] G. Cacciapaglia, C. Pica and F. Sannino, Phys. Rept. (2020), 1-70 doi:10.1016 / j.physrep.2020.07.002 [arXiv:2002.04914[hep-ph]].[10] D. F. Litim and F. Sannino, JHEP (2014), 178 doi:10.1007 / JHEP12(2014)178 [arXiv:1406.2337 [hep-th]].[11] T. Banks and A. Zaks, Nucl. Phys. B (1982), 189-204 doi:10.1016 / ff ert and F. Sannino, JHEP , 164 (2019) doi:10.1007 / JHEP08(2019)164 [arXiv:1905.00026 [hep-th]].[13] G. Badel, G. Cuomo, A. Monin and R. Rattazzi, JHEP (2019), 110 doi:10.1007 / JHEP11(2019)110 [arXiv:1909.01269 [hep-th]].[14] G. Arias-Tamargo, D. Rodriguez-Gomez and J. Russo, JHEP (2019), 201 doi:10.1007 / JHEP10(2019)201 [arXiv:1908.11347[hep-th]].[15] O. Antipin, J. Bersini, F. Sannino, Z. W. Wang and C. Zhang, Phys. Rev. D (2020) no.4, 045011doi:10.1103 / PhysRevD.102.045011 [arXiv:2003.13121 [hep-th]].[16] O. Antipin, J. Bersini, F. Sannino, Z. W. Wang and C. Zhang, Phys. Rev. D (2020) no.12, 125033doi:10.1103 / PhysRevD.102.125033 [arXiv:2006.10078 [hep-th]].[17] I. Jack and D. R. T. Jones, Phys. Rev. D (2020) no.8, 085012 doi:10.1103 / PhysRevD.102.085012 [arXiv:2007.07190 [hep-th]].[18] I. Jack and D. R. T. Jones, [arXiv:2101.09820 [hep-th]].[19] S. Giombi and J. Hyman, [arXiv:2011.11622 [hep-th]].[20] G. Arias-Tamargo, D. Rodriguez-Gomez and J. G. Russo, [arXiv:2003.13772 [hep-th]].[21] A. Codello, K. Langæble, D. F. Litim and F. Sannino, JHEP (2016), 118 doi:10.1007 / JHEP07(2016)118 [arXiv:1603.03462[hep-th]].[22] M. Watanabe, [arXiv:1909.01337 [hep-th]].[23] G. Arias-Tamargo, D. Rodriguez-Gomez and J. G. Russo, JHEP (2020), 171 doi:10.1007 / JHEP01(2020)171 [arXiv:1912.01623[hep-th]].[24] G. Badel, G. Cuomo, A. Monin and R. Rattazzi, Phys. Lett. B (2020), 135202 doi:10.1016 / j.physletb.2020.135202[arXiv:1911.08505 [hep-th]].[25] K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972).[26] S. Rychkov, doi:10.1007 / ffi n, doi:10.1142 / (2017), 170 doi:10.1007 / JHEP10(2017)170 [arXiv:1702.07079 [hep-th]].[29] J. L. Cardy, J. Phys. A (1984), L385-L387[30] J. Cardy, J. Phys. A (1985) no.13, L757-L760 doi:10.1088 / / / / (1980), 215 doi:10.1016 / (1984), 277 doi:10.1007 / BF01212398[33] J. A. Gracey, Phys. Lett. B (1996), 178-184 [arXiv:hep-ph / (2011), 74-79 doi:10.1016 / j.physletb.2010.09.037 [arXiv:1006.2119 [hep-ph]].[35] O. Antipin, N. A. Dondi, F. Sannino, A. E. Thomsen and Z. W. Wang, Phys. Rev. D (2018) no.1, 016003 [arXiv:1803.09770[hep-ph]].[36] R. Mann, J. Me ff e, F. Sannino, T. Steele, Z. W. Wang and C. Zhang, Phys. Rev. Lett. (2017) no.26, 261802 [arXiv:1707.02942[hep-ph]].[37] G. M. Pelaggi, A. D. Plascencia, A. Salvio, F. Sannino, J. Smirnov and A. Strumia, Phys. Rev. D (2018) no.9, 095013[arXiv:1708.00437 [hep-ph]].[38] K. Kowalska and E. M. Sessolo, JHEP (2018), 027 doi:10.1007 / JHEP04(2018)027 [arXiv:1712.06859 [hep-ph]].[39] E. Molinaro, F. Sannino and Z. W. Wang, Phys. Rev. D (2018) no.11, 115007 [arXiv:1807.03669 [hep-ph]].[40] T. Alanne and S. Blasi, Phys. Rev. D (2018) no.11, 116004 doi:10.1103 / PhysRevD.98.116004 [arXiv:1808.03252 [hep-ph]].[41] Z. W. Wang, A. Al Balushi, R. Mann and H. M. Jiang, Phys. Rev. D (2019) no.11, 115017 [arXiv:1812.11085 [hep-ph]]. [42] F. Sannino, J. Smirnov and Z. W. Wang, Phys. Rev. D (2019) no.7, 075009 [arXiv:1902.05958 [hep-ph]].[43] N. A. Dondi, G. V. Dunne, M. Reichert and F. Sannino, Phys. Rev. D (2020) no.3, 035005 doi:10.1103 / PhysRevD.102.035005[arXiv:2003.08397 [hep-th]].[44] W. C. Huang, F. Sannino and Z. W. Wang, Phys. Rev. D (2020) no.9, 095025 [arXiv:2004.02332 [hep-ph]].[45] G. Cacciapaglia and S. Vatani, [arXiv:2005.07540 [hep-ph]].[46] D. Banerjee, S. Chandrasekharan and D. Orlando, Phys. Rev. Lett. (2018) no.6, 061603 doi:10.1103 / PhysRevLett.120.061603[arXiv:1707.00711 [hep-lat]].[47] A. Monin, D. Pirtskhalava, R. Rattazzi and F. K. Seibold, JHEP (2017), 011 doi:10.1007 / JHEP06(2017)011 [arXiv:1611.02912[hep-th]].[48] G. Cuomo, Phys. Lett. B (2021), 136014 doi:10.1016 / j.physletb.2020.136014 [arXiv:2010.00407 [hep-th]].[49] D. Banerjee, S. Chandrasekharan, D. Orlando and S. Re ff ert, Phys. Rev. Lett. (2019) no.5, 051603doi:10.1103 / PhysRevLett.123.051603 [arXiv:1902.09542 [hep-lat]].[50] S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, JHEP (2019), 038 doi:10.1007 / JHEP10(2019)038 [arXiv:1705.05825[hep-th]].[51] S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, [arXiv:1804.06495 [hep-th]].[52] H. B. Nielsen and S. Chadha, Nucl. Phys. B (1976) 445. doi:10.1016 / (1995) 777 doi:10.1016 / / (1991), 39-44 [erratum: Phys. Lett.B (1993), 545] doi:10.1016 / / (2013), 2544 doi:10.1140 / epjc / s10052-013-2544-1 [arXiv:1306.5644 [hep-th]].[56] M. Kompaniets and K. J. Wiese, Phys. Rev. E (2020) no.1, 012104 doi:10.1103 / PhysRevE.101.012104 [arXiv:1908.07502[cond-mat.stat-mech]].[57] P. Calabrese, A. Pelissetto and E. Vicari, Phys. Rev. B (2003), 054505 doi:10.1103 / PhysRevB.67.054505 [arXiv:cond-mat / / / / / (2002), 046115 doi:10.1103 / PhysRevE.65.046115 [arXiv:cond-mat / (2004), 018 doi:10.1088 / / / /
018 [arXiv:hep-ph / (2003), 029 doi:10.1088 / / / /
029 [arXiv:hep-ph / (1981), 3549-3552 doi:10.1103 / PhysRevB.23.3549[66] D. B. Kaplan, J. W. Lee, D. T. Son and M. A. Stephanov, Phys. Rev. D (2009), 125005 doi:10.1103 / PhysRevD.80.125005[arXiv:0905.4752 [hep-th]].[67] V. Gorbenko, S. Rychkov and B. Zan, JHEP (2018), 108 doi:10.1007 / JHEP10(2018)108 [arXiv:1807.11512 [hep-th]].[68] O. Antipin, E. Mølgaard and F. Sannino, JHEP (2015), 030 doi:10.1007 / JHEP06(2015)030 [arXiv:1406.6166 [hep-th]].[69] O. Antipin, M. Gillioz, J. Krog, E. Mølgaard and F. Sannino, JHEP (2013), 034 doi:10.1007 / JHEP08(2013)034 [arXiv:1306.3234[hep-ph]].[70] M. Shaposhnikov and C. Wetterich, Phys. Lett. B (2010), 196-200 doi:10.1016 / j.physletb.2009.12.022 [arXiv:0912.0208[hep-th]].[71] A. Abada et al. [FCC], Eur. Phys. J. ST (2019) no.4, 755-1107 doi:10.1140 / epjst / e2019-900087-0[72] [CEPC Study Group], [arXiv:1809.00285 [physics.acc-ph]].[73] J. B. Guimar˜aes da Costa et al. [CEPC Study Group], [arXiv:1811.10545 [hep-ex]].[74] V. A. Rubakov, “Nonperturbative aspects of multiparticle production,” [arXiv:hep-ph / (1996), 378-406 doi:10.1016 / / (2019), 1-52 doi:10.1016 / j.physrep.2019.06.004 [arXiv:1810.01722 [hep-ph]]. LaTeX(EU)[77] M. Della Morte, D. Orlando and F. Sannino, Front. in Phys. (2020), 144 doi:10.3389 / fphy.2020.00144[78] G. Cacciapaglia, C. Cot, M. Della Morte, S. Hohenegger, F. Sannino and S. Vatani, q-bio, PE, arXiv:2101.11399https: // arxiv.org / abs / / / / / (2015), 166-195 doi:10.1016 / j.cpc.2014.12.023 [arXiv:1206.6379[math-ph]].[81] R. Feger, T. W. Kephart and R. J. Saskowski, Comput. Phys. Commun. (2020), 107490 doi:10.1016 //