JHEP03(2019)107 max b . Along Springer ∞ ) model at March 3, 2019 N March 19, 2019 : ( : January 23, 2019 O : is taken to , Accepted Published max Received and Lorenzo G. Vitale b using conformal truncation, i.e. with Published for SISSA by f https://doi.org/10.1007/JHEP03(2019)107 N [email protected] , Emanuel Katz b [email protected] , . In both the Chern-Simons theory, and in the . A. Liam Fitzpatrick, 3 a fermions in the limit of large max f 1811.10612 N The Authors. 1/N Expansion, Chern-Simons Theories, Field Theories in Higher Dimen- c

f We set up and analyze the lightcone Hamiltonian for an abelian Chern-Simons , [email protected] , we compute the current spectral functions analytically as a function of ∆ N N Department of Physics, StanfordStanford, University, CA 94305, U.S.A. Department of Physics, BostonCommonwealth University, Avenue, Boston, MA 02215,E-mail: U.S.A. [email protected] b a Open Access Article funded by SCOAP Keywords: sions, Gauge Symmetry ArXiv ePrint: infinite and reproduce previous results inthe the way, limit we that the determinediscrete truncation how ∆ basis to for the preserve momenta gauge of invariance states and in how the truncation to space. choose an optimal Abstract: field coupled to a truncated space ofmaximum states cutoff corresponding ∆ to primary operators with dimension below a Luca V. Delacr´etaz, Conformal truncation of chern-simonslarge theory at JHEP03(2019)107 7 4 31 18 9 33 39 28 41 5 34 29 – 1 – 14 27 10 11 25 17 31 42 f N 2 i 18 9 20 TT h 8 = 0 in primary state basis 23 f 35 ) 15 m f 37 13 N N ( CS f CS at interaction matrix elements NO N f f 27 N N ) warm-up N ( C.1 Large C.2 Large A.3 Inner productsA.4 and matrix elements Projection toA.5 Dirichlet basis Spin even sector and A.1 Hamiltonian andA.2 mode expansion Higher spin currents and 2-particle states 4.3 Interaction subspace notation 5.1 Hamiltonian eigenstates 5.2 Computing correlators 5.3 Convergence rate 3.3 Diagonalization of3.4 the mass term Massive free fermionO correlators 4.1 Hamiltonian and4.2 matrix elements Optimal basis 2.2 Simplifications at infinite 3.1 Massless primary3.2 basis Massive theory and IR divergences as projectors: the Dirichlet basis 2.1 Construction of on-shell hamiltonian E Large F Wavefunction normalizations C Connection to covariant formulation D Large B Dirichlet basis from dimensional regularization 6 Discussion and futureA directions Algebra 5 CS at infinite 4 3 Free theory warm-up Contents 1 Introduction and summary 2 Setup JHEP03(2019)107 ]. 12 , ] vs a 8 11 – 5 ]. In general, 4 ]. An important , but are avoided = 3, where again 3 f – d 1 N = 2 [ d limit significantly simplifies quantum Hall state [ f 1 2 N = ν ] and the ]. Various evidence has been put forward to – 2 – 10 15 – 13 ]. In the present case, the most important reason we take LC 9 is not locally gauge invariant. Nevertheless, conformal truncation µ J µ of fermion flavors is infinite. The large A f . N f N We will use the lightcone (LC) conformal truncation framework of [ Our motivation for starting a Hamiltonian Truncation study of CS gauge theories Rather than considering CS gauge theories in general, we will restrict to a limit where One of the major obstacles to applying conformal truncation widely is that there are non-perturbative check. stems from several aspects.such as They deconfined are quantum relevantFurthermore, CS to criticality theories describe [ with matter different havewhich physical been phenomena, generalize conjectured to rank-level undergo aconfirm duality web these of [ dualities, dualities, and we believe the Hamiltonian truncation could provide a strong there are tradeoffs betweenLC adopting quantization an scheme equal-time [ quantization (ET) is quantization that scheme the [ dimensionless interaction interactions is in dimensionless, ET and Hamiltonian it truncation. remains unclear how to treat the number the theory, and will allowcorrelators us to to their perform known solution allare from calculations resumming important analytically Feynman conceptual as diagrams. well issues Inat as that addition, infinite to there remain compare to be resolved at finite be integrated out ofto the consider action Chern-Simons directly. (CS)the From gauge gauge this theories field point coupled isconformal of nondynamical to truncation view, and matter to a can in such be natural theories will integrated next be out. step our is Understanding main how goal to in apply this paper. free theories are athat problematic interacting starting gauge point theorieslocal for are studying not operator; gauge really deformations theories.has of been The their successfully reason free applied is theorysimplification to limit in nonabelian this by case gauge a is theories that in 2d gauge fields have no local degrees of freedom, and can methods generally as Conformalof Truncation QFT methods; that these takes are CFTs an rather appealing than formulation Lagrangians asfew the CFTs starting where point. the CFTto ‘data’ high — precision. OPE coefficients Free and theories scaling are dimensions — an are obvious known case where this data is known. However, where it can be exactlylarge diagonalized, class usually of numerically. QFTs A thatdeformed special are by case points is along a to the relevant considerof RG operator. the flow the of primary In a operators Conformal thisgiven Field of case, by Theory the if the (CFT) original one CFT CFT, OPE fixes then coefficients the the and truncation Hamiltonian its basis matrix spectrum in elements of terms are operators. We will refer to such Compared to the perturbative regime,ories the (QFTs) strong remains coupling poorly regimeperturbatively of is understood. through Quantum their Field The Lattice The- integral definition, standard front and which method center. puts the of Antion alternative Lagrangian defining to methods, and the QFTs the which Lattice path involve non- description truncating are the Hamiltonian trunca- Hamiltonian of the theory to a finite subspace 1 Introduction and summary JHEP03(2019)107 i of − 4 J (1.2) (1.3) (1.1) . We − H J h max ` for the , ) −− in this basis are IR q ρ ( 2 , κ limit, fermion particle H  f ) Im 1 2 +1 N λ ) − 1 2 q , which is just the Hamilto- α +1 ( max 2 π − ` π 2 α H µνλ ( max  ` π i 2 ) 2 f sec ], and contains 2-, 4-, and 6-fermion . m cos 2 f 6 16 , we recover the Lagrangian result, m H , the interaction terms mix states with +1 4 2 1 . sgn( + H − max 4 − ` → ∞ 2 max ) H q ` q ( + r – 3 – max τ , and the eigenstates become nontrivial functions ` 2 2 +1 H Re µ 2 =1 X α max = µν ` η H 2 2 − 4 q q | q | − ν ) = q q µ ( q Our first task will be to construct the lightcone Hamiltonian in a −− ) = πρ q acting on it analytically, for any value of the truncation level ( 2 µν , the only interaction terms that survive are in the four-body piece H ] took such an approach, where the general basis was a particular fixed are known functions. f πρ 4 N κ and τ . One way to deal with this mixing is simply to choose a priori a general basis, evaluate At large In this basis, diagonalizing the quadratic Hamiltonian 2 µ the Hamiltonian. Unlike thedifferent invariant free momentum-squared Hamiltonian of the matrix elements inThe this basis, work and in numerically [ diagonalize the resulting Hamiltonian. inator is positive. Moreover, in the limit where where terms are only included in the sum if the argument of the square root in the denom- and diagonalize can then construct the spectralthe functions resulting of eigenstates. local operators Forcurrent-current as instance, two-point sums function we over in find the the overlaps the with free spectral theory function is nian for the freethe massive goal is theory, to is performinvolves already dealing the a diagonalization with analytically, the nontrivial but fact problem.divergent. even that numerical To some This diagonalization deal of is with thethat clearly matrix this, has true elements finite we of energy if impose asas an the the IR IR “Dirichlet” regulator regulator, subspace. is and Fortunately, taken it project to turns onto zero. out the We to refer subspace be to possible states to in find this space this subspace number is conserved andprimary we operators are can therefore restrictbasis fermion to of bilinears, such the and operators up our two-particle toto truncation sector. a keeping scheme maximum operators scaling is The up dimension, to which to corresponding keep in a a this maximum case spin is equivalent terms, In lightcone quantization, these interactions can2, change fermion and particle 4 number by particles up at to 0, a time, respectively. However, in the large basis of primary operators ofwe the will free, massless allow theory. a Withcancel mass the off counterterm benefit the for of UV some the contributionsproportional hindsight, due gauge to to boson, the a which non-gauge-invariant gauge regulator. willfields boson eventually Aside has mass, from be appeared the a in tuned resulting term the to Hamiltonian literature in previously, e.g. terms [ of the fermions Summary of results. JHEP03(2019)107 , . f 5 N N = 0, (1.4) − . a  ) 1 2 +1 − α ( max , we construct . In section ` π 3 2 N , we warm up for -dependence of the sec 4 µ ) model at large +1 , where we compute 2 1 f N m ( max N ` 4 two-point function takes O 2 − φ 2 q to infinity as well, reproducing  , we set up the LC Hamiltonian  2 max r ` to the interactions of the particular ) model at infinite Chern-Simons theory, and find they µ 2 =1 N max analytically. We show that in the limit X α ( ` above a cut-off Λ. However, the more f O N µ . ≡ max ) ` q max ( `  – 4 – P , ) q ( 2 + , this spectral function reduces to the Lagrangian result P 2 16 λ max ` ) + q ]), though with some modifications in order to preserve gauge in- 2 ( ) model, the spectral function for the ), vanishing sharply at )) + 16 implicitly includes only terms where the argument of the square q µ N P ( ( ( k correlators. α − O , we conclude with a discussion of potential future directions. g P f 6 4 λ N (1 + . In the where the truncation level is taken to infinity, the correct results are reproduced. limit, we will see that we can actually choose a small, optimal basis, selected by ) = max f q ` ( N 2 → ∞ The rest of the paper proceeds as follows. In section In fact, essentially the same strategy works to simplify the φ πρ max 2 Setup In this section, we setDirac up fermions the coupled Lightcone to (LC) a Chern-Simons Hamiltonianprevious gauge work formulation field. (see of e.g. the Much [ of theoryvariance the of once construction we will pass follow to a truncation framework. After removing the non-dynamical fields we apply these methodsthe spectral to functions the of Chern-Simons the` currents theory at at finite infinite Finally, in section the CFT primary basisdeformation states, of and the showthe how UV to eigenstates CFT analyze of is the thecompute free the mass the theory, fermion spectral term where mass functions in thethe of term. the only interacting the CS truncated conserved case basis, The currents. with and analysis an In using involves analysis section these finding of eigenstates the to spectral functions ofreduce the to currents the Lagrangian in results the at large infinite for the abelian Chern-Simonsand gauge integrate field out coupled the to nondynamical fermions components in of lightcone the gauge fields. In section Again, the sumroot on is positive.from At summing infinite bubble diagrams. Similarly, we obtain explicit analytic expressions for the diagonalize the Hamiltonian, and so weat obtain explicit any expressions for the spectralthe functions form bra and ket state. Asin a fact consequence, we we will will see be thatthe we able known can to large take remove the the truncation cut-off limit analytically, and We first treat thisinvariance case or as a spin warm-up, indices. where we In do the not have interacting to theory, deal we with again issues find of gauge we can analytically closely one can tailortheory the at basis hand, of the eigenfunctions smallerlarge in the basis that willthe be structure needed of for the apossible interactions particular is themselves. accuracy. that the In The interaction the matrix feature elements that factorize into makes separate this optimal choice set of polynomials JHEP03(2019)107 µν η (2.4) (2.5) (2.6) (2.7) (2.1) (2.2) (2.3) Λ ∼ , limit produces a = 1 ) and two relevant → ∞ . −⊥ k f 2 is an integer. However + . /  ! m µ f i 0 − a N i µ 0 = a + − .... , a k

λ + m a 012 i ν = ) flavor symmetry and a global ν + f ∂ f j ⊥ µ λ µ N f a a j h ν ,  . ) can be thought of as a deformation ν ∂ µνλ SU( 2 ⊥ a λ µ  µ a 2 a , γ π ν a dx k , 2.13 Chern-Simons gauge field 4 ∂ Ψ ! limit, this distinction is unimportant here and we + − µνλ k µ  λ − γ µνλ π 2 0 a  k ¯ 0 0 4 Ψ ν )Ψ + dx √ π – 5 – → ∞ 1 ∂ f 2 + ≡ µ

m , k a µ f f dx = = Consequently, our original Lagrangian may need a = 0. This gauge has a number of advantages, the j )Ψ + + N µ f µνλ − a ) quantization with the following conventions j − 3  = 2 . m a +  D  π µ 2 k i x 4 a + ¯ a Ψ( is linearly divergent and may produce a contribution , γ  D i  ] = = i ν f a ! [ j This theory has a global L ¯ , ds 2 µ Ψ( f eff ) j 1 √ h 1 = L ). ). The theory for general values of these couplings will be studied x 0 0 0 a µ ), and the theory is therefore gauge invariant if L f

 ia m 0 m = + x ( µ + and sgn( 2 ∂ γ Dirac fermions coupled to a U(1) 1 ( f 2 f √ µ N f γ m N = + as a free parameter in the following. ≡ k k  a x → ].  D  k 17 Our choice of regulator will be such that integrating out the fermion in the Technically, the global symmetry is a more complicatedHigher if order one terms keeps are track UV of finite discrete by symmetries, Lorentz invariance see and power-counting. 2 3 1 shift since we will mostlywill be treat concerned with the e.g. [ The two-point function that generates a massgauge term boson mass for counter-term to cancel this contribution: we can write chief one being thateliminating the the gauge gauge field field also isunfortunately eliminates nondynamical the any and issues actual with can situation preserving beis is gauge integrated that invariance, not but out. if so we simple. Naively, then first One we integrate may way over generate to the a think fermion mass about term degrees the for of problem the freedom gauge with field. a Defining generic the regulator, “fermion current” and make the following choice of gamma matrices: We will work in light-cone gauge We will work in light-cone time ( with U(1) symmetry carried by the ‘topological’ current of a free fermioncouplings theory ( parameterized by aby diagonalizing dimensionless the coupling Hamiltonian ( in a truncated2.1 basis of free fermion Construction states. ofWe on-shell study hamiltonian in light-cone gauge, the theory as formulated in ( JHEP03(2019)107 ≡ ) in µ (2.8) (2.9) , the , one j (2.15) (2.11) (2.12) (2.14) (2.10) (2.13)
χ 2.2 ∗ ∗ χ ψ ¯ Ψ = , j ,
ψ . χ ψ ∗ j χ , ψ + ∗ f . ψ ∗ j χ 2  − ψ , 2 2 1 − j 1 ∂ − ∂ + f i 1 ψ √ j ∂ ∗ j + f ψ ) j ∗ − = i ψ 1 χ χ ∂ 2 ψ ∗ − − f 1 ⊥ 2 2  ∂ π ψ ∂ . After integrating out i k 2 π  ∗ f k . ψ a − 2 π ψ λ 2 k 2 m  ψ a √ + ∗ , j ν a  i ) ∂ χ , ψ a + 2 π m µ χ k ,...,N + 2 ⊥ 2( f ∗ m ⊥ A f j + ∂ j ψ √ ψ . ∗ − i − ∗ + 1 f 1 k = 1 χ i ∂ + 2 µνλ − j π ψ k  ψ 2
π − − = 2 ψ − f k 1 π ,i 1 2 1 i, j ∂ ∗ ) ∂ 2 ⊥ os m – 6 – χ ψ ⊥ − f ψ ) + ( χ j + i ∗ + − χ − = . The Lagrangian reduces to ∂ π ∗ ψ k L : ∂ ∗ ψ = 2 ⊥ 2 ψ ⊥ ,i ψ ψ a ⊥ a ⊥  f ∗ os ∂ + ] = + ) indices π  χ k f ) without non-dynamical fields is then given by A 2 ψ χ [ )Ψ + and − ∗ − 1 , j N f  − L ∂ i χ + ∂ a ( + f 2 m ψ 2.13 a ∗ f j 1 m + χ ψ , j √ 1 k + m i ∂ ∗ ∗ i is non-dynamical, the solution to its constraint equation is ∂  2 ( ψ = + + 2 i ψ = 2 χ √ i + ,i ¯ √ Ψ( 2 j os = √ = = χ i L + f L = j , or L =0 A ) to a background gauge field: | µ 2.7 /δA ] It will be useful to keep track of what has become of the global U(1) current ( The action can be further simplified because a component of the spinor is non- Next, we want to integrate out the nondynamical fields from the Lagrangian. First, acts as a Lagrangian multiplier enforcing A [ + The current inδS the theory ( This formulation of the theory will be our startingthe point process for of conformal integrating truncation. outtheory the ( non-dynamical fields. This can be done by coupling the where we reintroduced theobtains SU( the following Lagrangian for The spinor component dynamical in light-cone quantization.currents are Writing the fermions as Ψ = and the action for the components is which allows us to eliminate to encounter divergences that are cancelled by a massa term for the gauge boson. In particular, when we attempt to diagonalize the lightcone Hamiltonian, we should expect JHEP03(2019)107 by = 0 and ), if a µ (2.16) (2.17) n m j µ 2.13 ∂ , ) mixes i 6 , one can obtain 0 | H -channel exchange ( A y t 4 → 2 , and we can restrict , H 1 f = 0, breaking of gauge y  N − a ): 2 y . We will now restrict to the ( f , having to do with the proper ψ f ) N ) to be Lorentz invariant; this is , because the 1 N f y ( ). Two-particle states in this sector N ∗ . f 2.13 6 ψ ]. A similar analysis might be possible N H conserve total particle number because ): 19 , 2 + 2 4 i∂ 18 H H − , 1 + – 7 – . 2 i∂ 5 H − f ( is on-shell. This current conservation is in contrast = φ N f ψ  H iyP − ). Note that the current is still identically conserved: y e ⊥ 3 j d basis is simple enough to allow for a purely analytic treatment. ⊥ ∂ Z f + 4) particle states. + f in terms of the physical fermion modes from the above Lagrangian. N n φ + N j A H , including particle-number-mixing interactions is a tractable challenge, + p f ∂ ( = N , which has a number of simplifications. The first of these simplifications is − 1 i ∂ + 2 and ∞ − = φ, P is a normalization factor. The neutral states of the theory form a Generalized Free | = n, n f φ − N A j A second issue is that the one-loop fermion propagator from integrating out the gauge There are also conceptual issues that arise at finite At finite Following some straightforward manipulations outlined in appendix . Moreover, because we are using a Lorentz violating gauge a + 2 ( divergence cancels between multiple diagrams [ here. However, the issue isof avoided a entirely photon at vanishes infinite in this limit. field has a non-analytic contribution. The one-loop contribution (and moreover the full see that the infinite formulation of the Hamiltonianinterpreted itself. literally, leads The to firstthe IR issue t’ divergences is Hooft in that model. matrix theIR elements. In double divergence that pole by A case, analyzing similar in one Feynman issue ( diagrams can arises in derive in the the covariant correct formulation, prescription where for the removing IR the and Field Theory (GFF), andstandard they way. can be constructed from the two-particlebut eigenstates it in requires a numerically constructing as large a basis as possible. By contrast, we will take the form where here and in the following the summation over flavor indices will be kept implicit, Up to this point,case the construction has beenthat for particle a general number isour conserved analysis to by the the two-particleMoreover, interaction we states at will as focus infinite these on will the not singlet mix sector with of higher SU( particle numbers. In LC quantization, theparticles cannot matrix be elements pair of n produced from the vacuum. By2.2 contrast, Simplifications at infinite what will be done in practice in section the Hamiltonian The Hamiltonian has two-particle, four-particle, and six-particle terms: holds whether or notwith the gauge fermion invariance of them underlying theory, which holdsinvariance is only tied for to a that particularrequiring of value Lorentz correlators of invariance. of Therefore, the we purely can fermionic equivalently tune theory ( and JHEP03(2019)107 , kin (3.1) (3.2) (3.3) H (2.18) (2.19) in future f . N ) p − ( ψ ) p ( ∗ ψ to the LC energy of the = 0 region of the photon. 2 f − . + p − ) p m 2 q p : ) 2 − p ( 2 π ] in lightcone gauge and found . mass d ψ 4 A.1 a f ) 16 H p N m ( Z ∗ + . ] in the covariant approach. However, H, = = ψ − − p a 0 20 kin 2 f P + , keeping the following ratios fixed m H p mass ], only states that map to primaries of the − 2 4 p + = 2 = 2 p 2 µ 2 ⊥ c limit. → ∞ p P k H – 8 – N f a µ 2 p ,H 2 N P π ) d 4 , m p ) = f p , k, m ) was computed in [ limit provides a promising setting in which to analyze − k Z c f ( N c M ≡ Σ( ) on-shell, effectively setting ψ N N = N ) = 2 p ( λ H 2.18 ∗ ψ = in the following for simplicity. − 2 ⊥ a p H p 2 m 2 p 2 π d 4 Z = kin H Given these additional complications, we will address the problem at finite We will start by the describingand the then conformal we basis will that consider diagonalizes the the kinetic mass term term. A number of interestingin issues this arise section already wecurrent will in correlators therefore in the the set freemassless free up theory. “kinetic” theory. the piece For We conformal and will the a truncation separate mass sake basis the term, of and quadratic clarity, compute Hamiltonian into the a to consider only the free part of the mass matrix where the light-cone Hamiltonian is simply (see appendix 3 Free theory warm-up In the conformal truncation frameworkfree of fermion ref. CFT [ need to be considered. To set up this basis, it will therefore be sufficient work and focus here on the limit We drop the prime on corresponding individual parton.correspond On to the sums other overterms physical hand, in states the generally in action. nonanalytic thethis The terms issue theory infinite further, must and since the cannot theoryrainbow be is diagrams solvable absorbed there vanish [ into in local the infinite When we integrateterm, out and the normal-ordering photon thisrainbow in four-fermion diagram. our term LC involves However,rainbow Lagrangian, the it diagram same we ought is contractions generate to notwould in a be be clear the simply evaluated four-fermion how to in evaluate the the ( nonanalytic LC contribution Hamiltonian. from A the naive interpretation to be In terms of the loopcontribution comes integral entirely of from the the zero rainbow lightcone diagram momentum in Lorentzian space, this nonanalytic resummed propagator at infinite JHEP03(2019)107 − is 0. ¯ ΨΨ M µ , we (3.7) (3.8) (3.9) (3.4) (3.5) (3.6) → (3.10) =  . The B 0 ]. Ward − µ P J . 21 ) q 0 , ) , = `, ` x i 2 0 | A : − min( p has a spectrum that 2 f (1 − ` P + m C ψ ) ∗ p M x . , = 8 ψ 0 0 . 2 −  ss + 2 ) `` ): 0 δ µ 0 (1 for details. In appendix σ`/ M · `` x `, ` 0 δ x, σ ) ( ): p ) are diagonal and given by ss . 2 0 δ A.4 `s 0 ) µ f 3.3 `` x max( A.3 ) =  δ − 2 ) − = − with 2 2 X 0 √ σ denotes the sign under parity, and  µ x, σ µ (1 1 2 ( 1 ( σ/ from ( √ , which are known in closed form [ x )  − −  x πδ ` 0 p 2 2 s kin `` = ,... – 9 – − µ / ) ( 2 = 2 H 0 M s , 0 (1 ) = 2 f i ` · πδ δ x x 0 m ( + = 1 ss 8 0 p − ` δ = 2 ` 0 δ ( `, s, µ f π | , f 2 dx 0 , `s ) for 0 1 mod2 s x 0 0 = , µ 0 ` 0 kin ` 2 µ Z 0 `` δ − , s ··· `` f ) − 0 M 1 2 ` as expected. The parity odd sector however has IR divergences: ` µ 0 N M h A , which are parity even and odd respectively. These are shown in (1 J µ 2 f 1 , but has 4 eigenvalues that are strictly infinite in the limit p 2 f − m − −⊥ 1 ` = 2 ... m C ` µ − i ( are J πδ `s ) = 2 f `, s, µ and to lead to the states ): | = 2 x, σ 6= 0, the parity even sector has the following matrix elements for the mass term ( 3.3 −− A.2 + f denotes the spin of the state, ... s, `s is a sharp IR momentum cutoff, see appendix ` eigenvalue. We have also introduced the normalization factor 0 1, and the singlet is given by ` mass ` − f in (  m ` 2 J ≥ M P ` mass where discuss an IR regulator basedis instead also on bounded dimensional by regularization. 4 The The corresponding spectrum of states willdensities. thus It be is then projected convenient out to and eliminate these will states not from contribute the to start, by spectral projecting the Notice that spin isis conserved bounded mod below 2. by One 4 can easily check that 3.2 Massive theoryWhen and IR divergences asH projectors: the Dirichlet basis The matrix elements of the kinetic term These states are properly normalized (see appendix wavefunctions for Here, the The 2-body primary operators ofand the free higher fermion spin CFT currents are spannedidentities by reduce the singlet the numberto of be independent components toappendix 2 per current; we take these 3.1 Massless primary basis JHEP03(2019)107 (3.18) (3.15) (3.16) (3.17) (3.12) (3.13) (3.14) (3.11) . This − M ,
+ 1) states 1 2 is equivalent to + 1 to refer to parity 1) states − 2 max . i ` α − H 2 ( β max 1 2 | ` , ≥ π i max ` α ` by states of the form ≡ | ( . i 1 2 1 + 1 β α | − − ) with now − 2 β 2 , . , . . ) ) 0 i 0 3.4 i , φ + M ) for + −i α + −i , , `, ` `, ` x , , φ =  2 1 2 i 1 2 max max − − + 1 β ` ` by the following states: sec min( | | max( − j j i (1 2 2 f 2 j is free of IR divergences. | | 2 M| . Our goal will be to find the eigenbasis of α s ) ) | m  ) − − + β α 2 0 + ` α ( j ( j ), but now with + e e ,..., ,..., C α A.5 i M + 4 `, ` 8 1 M – 10 – 3.9 φ ) +1 − 2 + −i 2 , , 2 2 x = µ =1 =1 1  3 5 j j X X i max | | max min( − ` ` = − , , α 2 f i cos 2 . We will use Greek letters + (1 = = ` j m −i + α x M| i i , , √ 3 8 1) 3 1 β α mass p | | | M | − = σ ( H 0 − `` + 1 ) = M and leads to states of the form ( 2 x, σ max ) is block diagonal in four sectors labelled by parity and spin mod 2, ( ` − A.4 , where √ ` 3.9 f +1 = 2 ) + α max ( j ` e ,..., 2 denotes the parity even/odd conformal primary eigenstates of the kinetic term. , Let us start with the parity-even, spin odd sector, truncated to i = 1 `, In this sector, the Hamiltonian is diagonalized and with eigenvalues Next we consider the parity-odd, spin odd sector truncated to α The Hamiltonian is diagonalized term is diagonal,diagonalizing diagonalizing the the mass massive term even quadratic and odd Hamiltonian eigenstates, respectively,| of the (truncated) quadratic Hamiltonian, whereas 3.3 Diagonalization ofThe the mass mass matrix term ( each of which canand be treat diagonalized the independently. spinthe even We massive sector focus quadratic in here Hamiltonian appendix on in the the spin truncated odd conformal sector, basis. Because the kinetic The parity even sector is unchanged since is done in appendix The matrix elements still have the form ( basis obtained in the previous section to the kernel of the IR divergent part of JHEP03(2019)107 ). ), χ ∗ 3.7 ψ (3.24) (3.19) (3.20) (3.22) (3.23) (3.21) − . In the ψ ∗ , k a χ implicit (e.g. , but we will ( ) i , ~q , − β , i , m ( j − and a sum over . These currents i f 2 f e 2 + 2 m = and m j |O µ |O 2 ⊥ , ~q . . j µ + 4 , ~q 2 ’s small enough such ) 2 . γ µ i + 1 γ, µ 2 γ, q ( = πβ ih + h.c. . Since current is conserved, |O 2  ), with max  , ~q ψ ⊥ ` γ 2 ), we can perform the integral f i, q , j γ, µ ≡ ) depends on 3.18 M ih m − ih γ, µ − j − | 3.19 − ∂ γ β 1 i, q , ~q | 2.12 = ) min ⊥ 1 ( hO ∂ is an integral over ) O ) and ( ∗ χ hO 2 γ, q i →∞ ) ( ψ q 2 ) and ( , φ 2 2  q 3.14 1 − max − ` √ β γ − 3.16 γ, µ φ i | = – 11 – 2 1 M ( Ψ so that the charge is an integer, whereas the M includes both parity even and parity odd states. ⊥ δ µ ( . The sum is over all hO sec 2 δ γ γ  π 2 f )) ¯ γ Ψ and lim 2 i dµ m φ X 2 γ, q = Z ( + 4 only overlap with the spin odd, two-particle primaries, the ψ , j µ 2  sec ∗ 2 f ) = j 2 γ 2 µ , q 2 f ψ X µ 1 ( m 2 2 m − Θ( O ≡ O q 4 √ were defined in eqs. ( 1 + 4 γ − O i 2 = X − β 6= 0, we have ρ , ~q − 2 − f ) = 2 π 2 j M q q q > µ . We will not need the explicit form of the vectors m ( 1 2  ) and the sum over − γ, µ γ O | i ) = 2 0. 1 q O ) = , ~q max ( M ` ≥ . ρ 2 2 ) O 1 γ, q ( O α, µ α, β γ, q 2  ρ ,..., to obtain ( µ 2 = 2  2 ≡ | , γ µ µ We will take the operators to be the currents ” i = 1 α | the ones defined insince the the free CFT. solution to Thisfree the is case constraint the with only case equation in for particular for correlators involving these components entirely fixdiffer correlators from involving the currentshave used a to define natural the normalization higher basis spin currents in used two in the ways. constructionSecond, of the First, the physical the basis currents of were physical the normalized interacting according currents (or to massive) ( theory do not necessarily match where that where the states “ Given the simple formover of the eigenvalues ( primaries. If the operators spectral density takes the form As an application of thespectral diagonal basis densities constructed in the previous section we can compute in the free fermion theory. Here, the sum over with 3.4 Massive free fermion correlators need the explicit form of the eigenvalues: Note that the spectrum satisfies β JHEP03(2019)107 , (3.29) (3.27)   . Note (3.28c) (3.28a) (3.25a) (3.26a) (3.28b) (3.25b) (3.26b) π 1 4 ⊥⊥ ρ  · , − , − β ) β φ q φ ( 2 µν sin . ˜ sin ρ , 2 − is not in the truncated π s / i 1 δ β + 1 K − + 1 + 1 ) ) + 1) α J ). The spectral densities | q φ − x ( − max ⊥ , ( 2 ` β, q κ max + 1mod2 ` ( ` Θ( ` sec √ L 2 − δ α √ + Im ˜ √ + 1 µ − x α J − q µ λ , − φ f q µ q , so √ √ β ≡ max  π µq √ √ 2 X , ` im 2 −i cos 2 π 2 − µνλ α √ q √ 4 + K  = `, i | + x µ )  = − −  J q f ) · q + 2 f √ π s α f π ( j √ 2 m δ 2 + ) + 1) m e α β − , im ( j j φ  and so on — in the primary basis truncation ) one finds ( e +1 overlaps with parity-odd states: · + – 12 – 2 − j sgn( = i s 2 =1 1mod2 δ  + ) j X ⊥ ` ⊥ max p − 1 · α j α cos 3.23 δ ` ` J ( 1 ) | 1 φ 2 2 δ f e f − q , as discussed below. − / / − 2 ( µ 1 1 − µ − 2 =1 τ q µq im q µq − − im j X ⊥⊥ √ √ max K K sec ` into − ρ √ 4 √ − ) ) √ 4 √ 2 is rotated onto all ) into ( Re ˜ / i − i 1 = = = = q µ − α, q α, q µν − + i i i i J ( ( with the basis states. Using Wick contractions one finds that K √ η √ , | 3.26 ) 2 2 + 2 + 4 1 i | q | µ µ 0 α, q α, q | J J q | | = | = `, s, q `, s, q α, q − | | − ⊥ ( i α α i ν j j − ⊥ 2 + X X . The component h h q · − j j . Consequently, parity-odd states only appear in the cut for = 0) ) µ µ h h J β, q 1 2 J | q | +1 y +1 − ( + 1 + 1 + 1 α can rotate α πβ 1 1 1 X (0) max ( max ` ) = π `  ⊥ q µν max max max j ( ` ` ` h × L = = µν + − |Oi ≡ O α β ) = ) = ) = πρ φ q q q φ ( ( ( can be probed by studying Plugging the overlaps ( ⊥− ⊥⊥ −− ρ ρ ρ max where we introducedtherefore the take short-hand the form notation ` that Lorentz invariance isgenerators broken by truncationtherefore in keeps the or Dirichlet removesthe entire basis. Dirichlet Lorentz basis multiplets Indeed, without Lorentz Hilbert mutilating space any. and Lorentz However invariance is in broken. This breaking of Lorentz invariance at finite where where The overlaps with the parity-even Hamiltonian eigenstates are therefore interest the overlaps with the Dirichlet states are The first step in obtaining spectral densities is to compute the overlap of the states of JHEP03(2019)107 ) ⊥ ν δ ⊥ µ (3.31) (3.34) δ (3.33a) (3.30a) (3.32a) (3.33b) (3.30b) (3.32b) ∝ µν dependence ρ µ , , ) q λ ( . , q , 0 ) π τ + q 2 ( µνλ α = 0 0  φ ) κ 2 = 0 we have ˜ q = Re πx ( πx 0 ⊥ 2 2 cos . ] q 2 κ q 2 2 q ) = Im / / 22 f 1 1 cos 2 4 sec q 16 − − π . 4 + 2 m 2 K K -dependence, but in the remaining q 0 − + + − κ = 2 = α α − q ) q , and therefore the energy eigenstates φ φ 2 2 4 q 2 2 p πx µ q πx + sgn ( , 2 π 2 8 q 2 dx µν πx 4 sec 4 sec (4 + 1 = 1. q 2 η i 2 − 4 sec 2 1 4 sec − − − q | q − 2 2 acos − − q cos | 1 q q π – 13 – − 2 q 2 J J tanh 0 i q ν q π Z q q 1 4 α α p µ p . X X f dependence of the wavefunctions. This basis will in π − − q 4 ) m dx µ dx q = = q -dependence of the wavefunctions. Our first step will be 2 q 2 ( + 1 + 1 A.5 4 0 0 µ 0 / π τ τ κ ) = q acos acos q max max 1 π 1 ( = π ` ` 0 0 0 Z i Z ⊥⊥ . Once we turn on interactions, the Hamiltonian will mix states ) ˜ ρ 2 q 4 π q ) = ) = ( µ q q ν ( ( j →∞ τ κ ) = ) = µ q q lim j ( ( h max Re ˜ Im ˜ and τ ` κ limit, one recovers the Lagrangian result [ f s, so we will have to contend with the Hamiltonian in a basis both of | 1 2 f N Re ˜ µ Im ˜ m | → ∞ / ) is the Lorentz violating part (with our kinematics | →∞ →∞ q ) warm-up q ( | lim lim max max max N µν ` ` = ` ( ρ q In the above expressions, we have kept track of the O with different primary operators and a basisto of choose an optimalturn basis allow for us the to solveand for their the components eigenstates for analytically, the both CFT in primary terms operators. of their 4 Having seen how to diagonalize theHamiltonian. free The theory, free we Hamiltonian next was want diagonalwere to in also diagonalize eigenstates the of interacting One can similarly recoverstates stress — tensor this spectral is densities done which in instead appendix involve spinsections even we will use boost invariance to set and the Lorentz violating part vanishes Indeed, with In the and where where ˜ JHEP03(2019)107 ) N ( (4.7) (4.3) (4.4) (4.5) (4.6) (4.1) (4.2) O . 1 1 ): − − 2 2 ) . Finally, the ) 0 0 0 3.16 2 αα max `, ` `, ` ` δ α. ) . 2 0 ∀ ) j µ . min( max( φ ,...,
j − , parameterized by a spin 1 2 s i φ . 2 + 1 ) + 1 0 − µ )( = 1 i ( 0 1 ): ], to which we refer the reader α φ ,... `, ` max 4 , i α max πδ 1 6 `, s, µ ` µµ ` 2 φ | , π ( 3.12 µ √ √ 4 · , = 2 min( N ≡ µ λ 0 1) 1 i 4 2 0 α `` √ δ = 2 − m − ) , µ ` 2 8 3 0 2 i 0 0 label for conciseness. ` α ) = φ µ | i s = µ φ ( 0 , φ 1)( − λ 2 α `` = 1. 2 α α, µ − m h φ µ − ( 2 1 2 – 14 – 2 q ` , v ), ( ) πδ − ,M 0 . The interaction preserves parity, so we can focus sec 0 ) 2 p µ , 3.8 i 2 `` ( 2 µ i 0 φ = 2 m α M µ v = , ` · ∂ 0 ) j, µ are two-particle currents kin ` 0 , but similar in many ways: namely, the 3d scalar 2 + 4 µ )( | i ( f ) int `,` 2 N M φ α α N µ mod2 ( j v µ 0 M e ∂ 2 λ `` = ( δ 1 2 ) 2 α =1 = max 2 j X 0 ` 0 = µ M = − int α,α L i run over positive even integers, 2 . This warm-up will allow us to demonstrate how to choose a basis to M 0 µ ( N α, µ of the mass term. This change of basis can be written as | `, ` i πδ ; we will not need the exact form. Instead, we will just need the interaction ) α | α ( j , and momentum-squared = 2 e s denotes the eigenvector number, and runs over , ` α 0 mass ` ) singlet sector at infinite It is much easier to diagonalize the full Hamiltonian if we first change to the basis of The kinetic term is the same as ( In this section, we will go through these manipulations in a model that is simpler M N ( , parity for some matrix elements in the mass eigenbasis.than These the matrix change elements of turn out basis to itself: be much simpler eigenstates Here, interaction is The values of Consequently, the quadratic Hamiltonian eigenvalues are the same as in ( and the mass term in the Dirichlet basis is the same as ( for more details.Hamiltonian Here matrix we will elements briefly inO summarize this the basis. conformal As primary` in basis the states CS and theory,entirely on the the basis parity-even states states for and the drop the The Lagrangian of the theory is The LC truncation approach to this theory was analyzed in [ than Chern-Simons at large model at large efficiently deal with thegauge interactions, invariance. without In the having following to we also set 4.1 deal with issues Hamiltonian relating and to matrix elements JHEP03(2019)107 2 − = q ): ,S q 2 α ; q (4.9) C v µ in the (4.10) (4.11) (4.12) (4.13) . The ( and ≡ α α ψ 2 ) C ψ µ ) (4.8) q only through q, α ; . ( 0 ) but due to the 2 ψ µ 2 ( µ , where 4 / , 0 , 1 v dependence. in the mass eigen- α  , γ K . ) T 2 2 i ψ 2 φ x 2 2 v ) µ / φ µ 0 2 µ − 1 µ J K ) − is the eigenvalue of the ( ( ) 0 sec , so the wavefunctions were . µ ) 2 α 2 2 ) α ( 2 q q v µ α in terms of the parameters πv q, α ( m 2 v 0 ( q, α α π ( 2 S v 2 2 µ dµ + 4 , but not ≡ − µ 2 ∝ − 2 ) q J Z ) µ ( x ) q , 2 α ; − ) µ = ( q µ S P.V. 2 ( ; α S v 0 0 q v )Θ( 1 2 α µ x 2 λ ( ψ ) + 0 0 − 2 − α is the component of the eigenstate ) µ αα ψ √ 2 i ) = ) δ ( / 0 are also eigenstates of q − ) q 1 2 α µ ; ; K α 2 ) ( ) – 15 – µ µ 0 ψ φ πv | ( α 2 0 − v q, α α q, α 2 ( 2 solutions here, one for each parameter 0 ( ψ 2 sec α, µ = / 0 2 h implicitly depend on µ 2 dµ µ 2 ( ), and we have used the following identity: αα J m µ max δ ( x 4 ,S Z ) ` πδ 1 ) = α − α α 2 q v ) of definite − α ; φ . In the free theory, we did not need a separate )Θ( α C x ). 2 γ µ 2 C α C H ( µ x φ 3.14  α ( − is proportional to the outer product 2 sec α − ψ P.V. P ) = 2 α ) v eigenstate wavefunctions will have nontrivial 2 sec q X µ q 2 ; 2 m int 2 ( ( 2 4 λ µ q 2 m α 1) is just the wavefunction of the singlet operator ( M mass eigenstate. As in the previous section, 4 v ) should be thought of as an equation for α − 2 = i − ψ 2 α 2 | µ dµ S µ ,..., 4.12 ) denotes 1 ∞ − − γ , 0 2 K 2 . The parameters Z q q (1 x ( α ( J δ ( φ +1 α 2 ∝ v 1 ) max q ` sec ; Substituting this general form back into the Hamiltonian equation of motion, the More precisely, the states ( √ 2 µ , because there are actually 4 ( µ α α m 1 √ The equation ( C ψ where 4 problem is converted fromcoefficients: an integral equation to a simple linear equation for the One can therefore immediately write down a complete basis for the solutions to where “P.V.” denotes “principal value,” and as in the previous section so the Hamiltonian equation takes the form 4.2 Optimal basis The key features of thisthat integral the mass equation term are has that beenthe the diagonalized. definite interaction The integral term interaction term is depends factorizable on and in the mass basis, where direction of the mass-squared operator 2 label on the eigenstatesinteraction since term they the were related by basis. So, the equation for the eigenstates of the interacting Hamiltonian takes the form In other words, JHEP03(2019)107 = i (4.20) (4.17) (4.18) (4.19) (4.14) (4.15) (4.16) µ, ` | 2 function φ h δ . However, α , we need its φ i , . The spectral 2 φ , v i m q ) ] to be φ 2 q 4 sec ( 1 2 2 ≡ + − m P 2 4 2 φ 16 λ , coth > . q ) + 1  2 2 1 q ) q 2 πq ( 2 / , we have essentially already = )) + . 1 is a continuous parameter, so . The states are 2 = . q ) K i ) P 2 ) ( 2 ) 2 / 2 2 µ / ψ 0 − 1 / | 1 φ that is parallel to q . 1 K were computed in [ P ; the result is )) ) ) ) i q, α 4 λ 2 2 i − ( / ψ q F 2 πx | such that 1 . 2 ( K ) µ q, α q `, µ − 2 | ) J ( ( α µ ( 2 ) (1 + ( 2 α α µ πδ sec = 0, since it does not see the interaction dx v α, q v 2 πx = 2 dx  (  ( ψ J S 2 i 2 = ( m 2 µ N α 2 4 α i J φ X | v ( – 16 – sec q − = α 2  ; α 2 i 2 α, µ 0 m | ψ X q ), and the space perpendicular to this vector. The S q 2 ( C v 2 ih ; (4 1 4 / φ ) ≡ q 1 h = ) ψ ; m q K | ) + 2 ) m q q α q ψ ( 2 | 2 ; ( 5 1 ( C 2 1 − ψ  C functions simplify to φ q, α h − h P (  with the mass eigenstates when we computed the interaction cos 2 P ψ cos 1 2 1 µ = 1 − J N ) that we can separate this space of solutions into one direction φ 1 2 − π ( ψ 0 π α Z Z , the N v 4.12 ) = q ( we have one eigenstate for each ) = ) = 2 → ∞ q q φ q ( ( only has overlap with the eigenstate + − πρ i max P P 2 ` φ | , and from this one can perform the change of basis to obtain the overlap with the mass eigenstates. 1 − 2 1 ` , there is really only one state that is affected by the interaction. The upshot of this Lastly, to obtain the spectral function of the singlet operator We also need to compute the norm of the state More explicitly, the overlap with the Dirichlet states √ 5 µ 1 max √ In the limit where we have defined matrix elements. The overlap is Therefore, function reduces to the overlap with this state divided by its norm: overlap with the basiscomputed states. the overlap Since of the interaction is ( normalized: We discuss how to compute these norms in appendix ` discussion is that we can focus on the special case for any value of it is also clearparallel from to ( the vector perpendicular space has theterm. trivial In solution other words, we can solve the Hamiltonian equation of motion because for any reason there is a large degeneracy of eigenvalues is that JHEP03(2019)107 is 2 H (4.23) (4.24) (4.25) (4.28) (4.26) (4.27) (4.21) (4.22) , whose 2 φ 6 . 2 , where µ 1 − ) 2 − ) µ µ ) 2 ( , solution. ) H α − v q, α ) : m − ( 1 N 2 ψ 2 + µ q, α − q ˆ D. ( q 2 i 2 ( µ θ . P.V. − ) . 2 2 )  = ˜ µ D µ q P.V. λ ( i 8 2 α − ∼ =  = φ φ ) | 0 ) , , i + ˜ ) ) µ D q | αα 2 ( µ associated with the operator q, α + ; ( , ∗ α ˜ ( function piece of D i 2 µ i α 2 χ  ( 2 v δ ψ α, µ µ 2 1 H α | m m φ h ( π 2 | 2 ψ q 2 1 dµ 2 +2 − πδ i ≡ , − q q , 2 ) 2 ) 2 – 17 –  2 Z α φ i ≡ 2 q | µ µ C ψ functions or principal value poles on their diagonal, | α log = − X − δ ≡ α, µ ) ) ) h λ qπ α, µ 8 i . Finally, the spectral function reduces to (0) h i ≡ i for the purely q, α q, α m + φ ( ψ ( | 2 | 2 2 1 + 2 χ (0) µ  µ h i H 1 ( ( δ ψ α, µ q < 0 | − h will just be denoted πδ αα + ) = )2 q ψ q ( µ πδ are just the real and imaginary parts of ( 2 2( ( φ α ˆ D v = 2 πρ 0 ) model is simple enough that writing all the states out in components was i ≡ αα N and 2 ( ˆ D φ ˜ | D ] equation (2.34). 4 ˆ , and vanish for D NO | m 2 α, µ h The overlap between states is defined as We begin by defining the following state q > See e.g. [ 6 and we will also introduce Finally, the eigenstates the quadratic part of the Hamiltonian: as written above. More explicitly, Essentially, We also define the matrices to be diagonal and have, respectively, equations in Dirac notation, which willwe make are the essentially equations dealing more with compact and a the single-state fact system that morewavefunction transparent. is 4.3 Interaction subspaceThe notation large sufficiently concise that the resulting equationsHowever, were as still the reasonably size compact of andquickly the readable. becomes interaction more subspace of increases, a keeping distraction. track of In all this components subsection, we will rewrite the Hamiltonian which matches the known result from the standard covariant large for JHEP03(2019)107 . | 2 φ h (4.33) (4.32) (4.35) (4.29) (4.30) (4.31) as long (4.34a) (4.34b) ˜ D , 2  i 2 φ | ˆ D 4 | 2 is trivial to solve, and we φ | h . First, the matrix elements 2 λ . . Acting on this state, the : f i φ i   2 h . 2 . i N i φ i . 2 φ + | 2 | C,S . φ ψ 2 ˆ φ i | | ˆ . D (0) | ) D 2 2 ) i | i ˜ ˆ as long as it is orthogonal to D q 2 φ D 2 φ | ψ | | ( ∝ | 2 , φ φ 2 ih | ˜ ) − h whose components are all proportional D φ 2 (0) φ q + ˜ h P i h D S ( φ (0) | i | S 1 2 i ψ  (0) 2 , so we can multiply through by + | ψ 2 + λ i 2 ψ | φ ˜ P − + | D i h 2 ψ S orthogonal to | i 2 = 2 λ φ 1 4 = = 2 φ i | ih φ i i ) is – 18 – | − (0) 2 + 2 2 ψ ˆ 2 i D | ˆ φ φ φ 2 D ) | | | ψ (1 | H C 2 | 2 C ˜ 2 ˆ ˜ D D D H φ | | =  = 2 h λ − vanish, and only ‘planar’ contractions of the four-body 2 2 2 ), simplifies at infinite 2 i 2 φ φ C 6 = h h − q = i| ψ  | A.6 H function piece i ψ ψ ψ 1 | 2 λ 2 ψ function piece δ 2 N N | q φ ( δ = |h S f ). ) = N q becomes the following equation for ( i 2 4.18 φ ψ | )): ), the inverse of ( πρ α, q α, q ( ( can be purely its 2 2 µ µ ψ − 6= 2 2 µ µ ( δ 5 CS at infinite 5.1 Hamiltonian eigenstates The full Hamiltonian, givenof in the ( six-body interaction term which reproduces ( The spectral function is again obtained from taking the overlaps and dividing by the norm: The norm of the states is It is straightforward to verify that the inner products are given by The equation for Clearly, Therefore the subspace ofcan states restrict with ourinteraction attention manifestly to generates the only states state of with the form as we allow an extrato purely problem will be fairlycompact quick. form: The Hamiltonian equation can be written in the following For Now that we have introduced a formalism tailored to the problem, solving the Hamiltonian JHEP03(2019)107 ) (0) N i ( (5.3) (5.4) (5.2) ψ | O (5.1a) (5.1b) simply as i , ψ i . | ) in the parity ψ , function pieces . | (0) q ) for the ) ) i i ; δ q | q µ ψ ; ; . ) function on their − | 0 4.8 0 ( 2 j µ + µ f µ ( + ( j ih ) (+) α ) limit should reduce to , and a parity odd set i 2 − to denote the diagonal 0 − − 0 − − ψ 1 j f ( β ψ ( β ) ∂ | +1 | ˆ ψ a D ψ i N 2 + f ) | ) 0 j 0 m γ, q max − ` 2 ( 2 j 2 λ and  ). We will work with the two- ih µ, µ 2 µ µ, µ ( π ( ( − ˜ π k 0 0 D δ 2 6= 0. In the bra and ket notation j 8 ,..., | max integrated out, int int αβ ββ a f  ` or − µ a m M m M 1 a
= 1 2 2 2 | 0 m 0 represents the purely − , we work out the form of the matrix λ α − ) has components 2 j . Recall that the mass eigenstates came 2 + dµ dµ E 2 (0) π µ ih i ψ + 8 f are the states corresponding to the current H Z Z ⊥ j − ψ ) with i | j with ) ) − ) − q 1 q − ) in the parity-odd space. The Hamiltonian . i
∂ ; j ; 2.9 | q ) + ) + | α µ ; | q µ ⊥ q γ, q f – 19 – − | − | ( ( ( ; ; j j µ 3.4 0 0 ) ⊥ 2 ( 0 j π 0 ) µ ih µ − k µ (+) and α ( β ( 2 ( − ⊥ ih ( β ψ i ψ j 0 0 0 − 0 (we restrict to odd ψ ⊥ (+) (+) α α j | j 1 αα ββ | ψ ψ δ ) δ ) − | − | )Ψ + 0 0 ) ) are not affected by the interaction, and the eigenstates in , we will use the matrices 2 f ⊥ = 0 from the very beginning, so that we do not need to = 1 primary states; we perform the explicit analysis of this , we worked out the conformal primary states as well as the + − i j πiλ max α β m ` ` 2 − f µ, µ µ, µ ih 4.3 φ φ h ( ( J 3.3 + 0 0 − | m 2 2 j = | ∂  int int βα αα i i sec sec ,..., M M and 2 f 2 f ψ ¯ in the mass eigenbasis, so Ψ( | 2 2 0 0 πiλ i ) . m m = 1 2 ψ 2 . More generally, in appendix i = ⊥ 4 4 ) of the Hamiltonian equation makes it manifest that states in h dµ dµ β J ψ H | | L ˜ D − − 2 D , we can write the Hamiltonian equation for the eigenstates Z Z 5.4 2 2 of the two-particle mass eigenstates at − = µ µ = = 4.3 with i 1 int − − 2 i ψ | q 2 2 β ( M | q q survive. In section ( ( 4 components of the current operator. This expectation is correct. It is particularly H ⊥ The form ( From the structure of the Lagrangian ( operator in the absenceand of mass the eigenbasis were gauge given boson inof section mass the term; eigenstate their components in the Dirichlet perpendicular to Recall that, as in subsection matrices with the principal value polediagonal, ( respectively. The states elements of subsection or equivalently, and clear if we takeuse the the Dirichlet limit states.only In on the this two case, independent case the in interaction piece appendix of the Hamiltonian is nonzero one might expect that thesomething interaction Hamiltonian with in components the infinite only in a two-dimensional subspace, corresponding to the + equation for the eigenstatesmodel. takes Now, a we have similar form to what we saw in ( mass eigenstates of the quadraticin Hamiltonian two families, a parityof even states set of states particle eigenstates even space as well as components term JHEP03(2019)107 ). and (5.8) (5.9) (5.5) (5.6) (5.7) i as 2.15 (5.10) . The S i i i . j i | TT j h j | . ˆ D ! | i j , and therefore h j 0 πiλ S  1 ,j 0 − ij = 0 current . s (or vice versa) using ψ V i a S πiλ 0 2 j ! C 2 , m . in canceling divergences. 1 † ψ,i P −  S (0) a a . Substituting the general k i 4 1

= m S ψ , iλ i i | i = + k i 1 0 ψ j | j 1 + πiλm ,j | i | = 0 − i j 4 ˜ ˜ ψ ) h D D | ψ ∗ i − | C j ∗ S j ⊥−  X

h , the norm of the eigenstates is | † ψ,i ( V j + j C = F + in terms of the V i i  k ih i = j i ≡ | a s given in terms of C j | i = ˆ 0 i D k 0 ), we have i ˜ S , m D −⊥ j i | C – 20 – ij j V | ψ,ψ 5.7 ˆ ,X ⊥ V D i , | j N j ⊥ − , j ,V j X and the stress tensor two-point function | j = h − i ) model. We can write the Hamiltonian equation even ij and is independent of the interactions, so it is just a , S i  = X ) , and N ij X = ) and ( 2 ij a i,j 0 ( ) JJ V 1 q i h j O m A.5  5.5 − ψ 2 0 X − | j,k λ V = X 2 ( = − q i ( i = j ψ a i X | = πδ S 2 , m = 0 i j i | −− ψ,ψ ψ | ,V N a = , m i i ), and orthonormalizing using the above inner product. = 0 0 j h q ; 5.7 0 ) into the Hamiltonian equation gives the following relation between the ⊥⊥ ψ | V q 5.5 ; To compute the spectral densities for the currents, we also need to take into account the Following the general discussion in appendix ψ h coefficients: i Note that by using equations ( fact that the current is modifiedWe by can the gauge write boson the mass term modified according current to state equation ( to compute spectral functionsthe for local current operators. two-point function Twolatter natural is correlators computed to in considerfree are appendix theory computation. Soexplicitly in see this the subsection, role we will of compute the the gauge former, boson where mass we counterterm will equation ( 5.2 Computing correlators Next we want to use the expression from the previous subsection for the eigenstates in order The eigenstates are simply the states with form ( C more compactly as where without loss of generality we can parameterize them as similarly to what we did in the these directions are the free ones. There are therefore only two nontrivial eigenstates, and JHEP03(2019)107 , i ) ): 2 S / 1 = 1 5.7 − 3.28 ⊥ (5.17) (5.15) (5.16) (5.12) (5.14) (5.11) (5.13) µ ,S vanishes . 0 1 V = 0 − ij .  − 0 , which are of ), using the S ! i V j i i i j j | − − | , since j . The former are in 5.10 j ˜ | ˆ | i D ij . D | | j ˜ ρ ˜ i D | D . This growth leads to j  | j | h 2 ˜ h D ⊥ † / 1 − Λ . . Because equation ( | 0 j 1 . i j i j (recall we are using boosts V i  a ) and divide by the inverse µ h j
1 4 , of the form 2 ,C i h j i 1 4 i O | µ j 5.7 , m ⊥ s to parameterize the physical + S γ . j i j · j i h | + and ; | 1 2 − j  | + C ⊥ ˜ µ 0 i ˜ 1 2 ) 0 D − D j α | j | | K K ψ V i ih | φ . Because of ( h i x x j ˜ ⊥ i 0 ˆ D 2 h γ, q j j D | . | S − | − ( ) J J − γ, µ j always appear in the spectral function ˜ q 2  | ψ,ψ D j h cos 1 2 1 2 ( i | − h ) i µ s or the j 2 j 1 j i a h / − −h j + − S 1 free ij | m − − h ˜ = π Λ 2 N D K πρ | ( ) i 2 P.V. i − X

j 2 µ = h ψ  π | – 21 – 1 = i α, q 2 1 a in terms of − dµ j ( πiλ − 2 j i x i | 2 + 2 µ , m j Z C ˆ µ i | D µ J − | j − ˆ i h α D γ 0 j | P.V. ij ) is given by summing over a basis of the eigenstates: x X h j δ q 2 P h ( X ψ,ψ = πµ † = ij dµ 2 i  +1 P.V. ρ j 0 1 2 j ,kj ) = V | π Z 0 max q i 2 ` ˜ ( dµ V j D | | i . If we take a hard cut-off on Λ, then the divergent integrals take ij i Λ k i = ˜ j D j ≤ h | | ⊥ i πρ µ j j ˜ − | D ≤ j | 0 ˜ | , the spectral function will be independent of the specific basis we choose i D 0 − h | j Z ˆ = 0, and solve for D ⊥ | X j ψ,ψ ⊥ − − h h  j N h ij  ,S X = 1). This fact is consistent with the above equation for ) = = 1 q − = 1 in the free theory, where there is no gauge boson counterterm needed. ( q − ij S X Finally, note that the matrix elements So the only new matrix elements we need to compute are πρ term in Λ and keep the finite piece.in For now, the we combination will keep the linear divergence as above. the form By contrast, something like dimensional regularization would simply discard the linear and consequently they lead to convergent integrals in However, the overlaps with the parity-odddivergences eigenstates in grow like the form The overlaps of the currents with the parity-even Hamiltonian eigenstates decay like For instance, to set and To go farther, we needfact expressions just for the the free theory matrices spectral function components, and were already computed in ( and parameters is particularly convenient. AsGram long matrix as we impose ( for doing the sum. The result is The space of eigenstates isrelates parameterized them by to the each coefficients other, wetwo-dimensional can space. use either For the instance, we can choose our basis states to be Finally, the spectral function JHEP03(2019)107 , 7 f ), 0. m ) for 3.4 . We 3.29 → (5.20) (5.22) (5.21) (5.19) (5.18a) (5.18b) E f 5.12 m . 0, because they  ) and the matrix π . limit, all the sums Λ → ), we also need to − always come in the 16 5.12 f 1) a + 1) m 5.12 − → ∞   m . . 1  , max max ! 2 ` q − + 1 ` max . 1 ( ` q 0 − 16 16 q  16( f max max q q m = 1 ` ` 0 = 16 m    . One way to do this is to note that θ  all vanish at

1 4 a 1 4 = ij . In the i )  m f j 1    j  + | − +1 m = 0 from the beginning, because in the limiting 2 1 4 − α ˜ f q β q D f φ |  q 4 = 0. The only nonvanishing piece is the max max + m φ i ` ` 2 m 2 f i j 16 − 2 h 1 2 2 m β − 1
4  cos φ sin + (2 + 4 log 2) K – 22 – 2 16 q 1 2 πλ q q q . In this case, as we explicitly saw in section 2 Λ 3 f − q + q sin β at nonzero α 4 J 2 f  m and the gauge boson mass X X 16 i i ) must be equal for both Lorentz and gauge invariance m j π q 4 1 i = j (1 + → ∞ 8 ( + 1 | /π ⊥ + 1 + 1 1 2 j ˜   − Λ D | = 1 1 ⊥⊥ |  i ˜ − max max K ρ D = ij j ` | , which means that the mass term successfully removes the ` ) max max h ) = i a ` ` ⊥ q πρ ⊥ j β ( m h j 0 γ, q X | 2 ( Next, we want to compute the spectral function at nonzero ) and → 2 ˜ ) = ) = free ij D q f : = | q q µ lim ( i ( ( m . i ⊥ πρ − ⊥ Let us first see how this works in detail in the massless limit j ⊥ J j −− | j free −− free ⊥⊥ reduces to a diagonal matrix, with the following diagonal components: ρ | ˜ 2 − h 0 is slightly different from taking D ˜ | D q πρ πρ 0. | a → ∞ free ij ⊥ → ⊥ j to cancel this piece. Substituting these expressions into our result ( ρ can be approximated as integrals, which are evaluated in appendix m j h f → ) is sufficient to fix the counterterm: h a m β f max m E.8 ` m and α The finite pieces of the matrix elements The limit 7 elements ( case we are still working within the Dirichlet subspace of states. over must also choose the gauge bosonLorentz- and mass gauge-invariance counterterm constrain the correlatorand to in be particular of the formto hold. in equation This ( condition together with the form of the spectral function ( To obtain the interactingcompute the spectral matrix function elements from our expression ( Case 2: and take the truncationthe limit free theory spectral function reduces to We choose the full spectral function, we find are proportional to divergence in combination 2 UV divergences as promised! Case 1: The free piece Crucially, the component JHEP03(2019)107 , , 0 0 . as V , 1 G † A 0 1 ) − − i G (5.24) (5.25) (5.26) (5.27) (5.28) (5.29) (5.23) ! j  |  0 ˜ 0 D = 1. The | and G V j 0 ) i H 0 − V j q i | −h 2 G ˆ D X | − , j 1 h ij i 2  ! ) = ( H 0 . i ) . ) j i− 0 G | 1 ij τ 0 j 1 | ˜ − κ D ! − q of the free correlator ˜ | D Im(  ) j |  . 0 0 ) πi 0 j H 0 Re( 0 A 0 κ 2 h G V τ −h ( G 0 † 0 G q V − X ) G Im( i ( + 2 0 Re( i . 2 0 ) ). X τ is an important part of how useful q πi 1 0 1 H 0 V 2 −  κ − G 1 1 ) Im( 1 + 5.27  max 0  0− 0 ) ` =  can be written in terms of κ i Re( πi 0 H 0 G j 2 τ | ij 0 ! = G − ρ ) for the spectral function in the form ˆ V 0 Im( 1 D † 0 | i 2 κ − Re( 0

πi j 1 G π q 1 2 h 1  1 2 2 : 5.12 λ − qτ † 0 , and more generally the analysis is limited by ). Note that – 23 – − − A 0 1 V − )  + † 0 0 0 G

 0 max V 0 V ij G 0 3.32 κ ` † 0 ) τ δ is just the gauge boson propagator at = i 2 i π q 1 G 1 G 2 j 0 | = i 2 V = ˆ free ij

D 1 + | from ( ,kj G j  πρ 0 = h 0 V i 2 , we find 0 i = , τ a 2 k 0 + (1 + i G j ) = j | κ i− 1 m q j in the interacting theory can be solved by a standard resummation j ( ˜ | − | D ) | ij ˆ G i ˜ D 0 D ρ | j | i G j j 0 h h ( − h V i 2 − ij − X X limit is a rare case where conformal truncation can be evaluated analytically  (1 f

0 N G ) = = q ( H The Hermitian piece of this correlator is The full correlator Taking this choice of ij G πρ 5.3 Convergence rate The large as a function ofnumerical the resources truncation to level result some approaches finite the exact maximum result as value. a function The of rate at which the truncated which agrees with our conformal truncation result ( result is that of 1PI diagrams; in this picture, which also has an anti-Hermitian piece By inspection, we see that our spectral function This free theory spectral function is the Hermitian piece We also have, from the free theory results, that in terms of the free theory which allows us to rewrite our expression ( JHEP03(2019)107 . f + )” on x m f ) in max a with ` x m (5.31) (5.30) ( ρ = 3 2 (3 , at any φ ) q for some 2 ) that the φ −− . Plots for . We find ˜ I ρ α,q to ( δ 2 max happening to f max b max µ was shown in 3.28 1 ` ` . So, the finite − a` m q 2 − µ max . To quantify the e ` max so that there exist f √ and ` = 2 λ m q λ ∼ , which fits the overall . Empirically, we find | ∼ )), and correspondigly 2 2 2 1 / = 3 q φ 3 and the truncation result 2 q − max φ from 4.21 ` ≈ ρ i −− ∆˜ x I ∼ − | j | For simplicity, we will focus on . ˜ . D ) | ) . ). For any spectral density , µ ) = 0 and µ − ( ( j µ ρ h λ ( ρ max 5.30 x we have chosen for the plot. ` . We leave this interesting question for + 2 µν x ρ f iµ ρ − 2 2 e q m µ 2 2 i dµ − dµ 2 = 3 e q (by inspection of ( 2 2 0 q – 24 – ∞ m Z 4 λ 16 dµ Z ≡ R ) ≡ . q ∼ ) ( dependence of ˜ q for a couple of values of coupling x has to have an imaginary part in order to regulate the ( x µν 1 ˜ ρ I x . ). In the interacting case, we choose = 0) m ⊥ , we show the result for 5.31 are qualitatively similar. 2 , q − µν , however we haven’t investigated the analytical dependence of I , q 2 3 ) model, whose exact spectral density at finite + x N ∼ ( ( one can define its Fourier transform , analytic value. For comparison, we show b G O . In figure ∞ is shown the relative error between the numerical and exact ˜ m ). q are shown in figure to probe the spectral density in this window, 3 = , i.e. x , whereas the exact spectral functions are smooth functions of 4.18 − max q ` max ` max ` spectral functions do not converge pointwise. Rather, we must consider the spectral In figure Finally, we would like to comment on another quantity related to the spectral density, We can also read off the rate at which the integrated spectral function from truncation We must first discuss what quantities we want to compare at finite vs infinite . We imagine it could be possible to extract physical quantities, such as mass gaps and max that in bothnumerically cases that the convergencex is exponentially fast, anomalous dimensions, from the future work. the free and interactingUV cases. divergence in integrala ( strongly coupledwe region choose 2 Note that this is essentiallyat the finite Wightman functionthe as case a of functionequation the of ( lightcone time be better or worse at the particular point which we have found toa converge mass even gap faster at than ( fixed value of rate of convergence for thebetween interactions as the well, integrated we value also of lookand the at its the matrix difference element “ trend. There is some spread around this trendline, with some values of will be sufficient. Comparisonsat between finite the exact result for the other components approaches the exact result by looking at the difference as a function of ` functions integrated against sufficiently smoothspectral kernels. functions, In practice, taking the integrated corrections to the exact spectral functions at finite It is clear by inspectionspectral functions of themselves the have terms inverse-square-root in singularities, finite the sums of the spectral functions ( conformal truncation can be in practice. In this subsection, we will consider the size of the JHEP03(2019)107 1) − � = 0 and � � �� λ exact ��� ��� truncation interactions � � ��� ��� for comparison. 2 / ��� ���� � � ��� 3 − max ��� � ` ��� ��� relevant � � � � ���� ���� ���� ���� ��� -���� -���� -���� -���� -��� -��� -��� -��� -��� λ =�� =�� ��� =�� λ =��� � � ��� ��� � � � � �� 2 dimensions. We have focused . i is lifted to infinity. Instead, the − -� -� -� -� -� -� � � j | d > �� �� �� �� �� �� . The residuals (defined as ˜ D max ) | � ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� − max j ` -- -- h � � (�� --  δ� . The black solid line is 0.1 – 25 – � max �� � ` � ��� ��� � � ��� ��� ���� and truncation level ��� � � ��� between exact analytic result and truncation result at λ ��� � ��� ��� −− � � � � ���� ���� ���� ���� ���� ��� ��� ��� -���� -���� -��� -��� -��� λ =� =�� δI ��� =� λ =��� � � ��� ��� � � � � �� -� -� -� -� -� � � : difference �� �� �� �� �� ) � ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� Left as a function of truncation level . . Comparison with exact analytic result (red, dashed) and truncation result (black, solid) -- -- (�� � � f -- m : analogous plot, but for the integrated value of δ� = 3 theories. Unlike inworried 2d, a the priori interaction that inin conformal our order truncation case to methods is convergecomplications require as dimensionless, from strictly the and the truncation one increasedmanageable. parameter dimensionality might Specifically, of ∆ have the the interaction leads interaction to term UV divergences, turned which out must be to regulated. be 6 Discussion and future directions In this paper, weformal have attempted Truncation can to be make applied progresson to in 3d gauge Chern-Simons understanding theories theories how coupled in Lightconethe to Con- gauge matter field, since, such in theories terms are of a the natural degrees next of step freedom after in successful work in 2d Yang-Mills Figure 2 q Right Figure 1 for various values of the coupling are shown in the insets. JHEP03(2019)107 c is N ]. In limit. num 80 ρ 20 f is lifted = 24 and N and ˜ λ max ` for num ˜ 60 ρ ˜ ρ − ˜ ρ max ` 0.506528 ≡ ρ max 40 l versus integrated out. One could , must be resolved in order ρ χ 2.2 x 0.56851 - 0.422162 , where ∆˜ 20 max `

0

-1 -2 -3 -4 -5

Δρ Δ log ˜ ) versus log 6.55 – 26 – : dependence of log ∆˜ free . The linear best fit is shown above the plot. There ], where the authors attempted to derive the singlet channel ρ i 23 01 6.50 . Right log ∆˜ − 6. . 2 + 0 6.45 0 . ∼ = 0 b 6.40 max x l log CS theories from a certain kinematical configuration of the momentum space c with 6.35 , at N b max aL 0.4281433224 - x 0.6087645655 ` − 6.30 . These puzzles, described briefly in section e f : dependence of log ( We leave this important step for future work. N ∼ 8 6.25 ρ . . Exponential best fit is shown above the plot. i Left 2 . C . 0 e

3.35 3.50 3.45 3.40 3.55

16 free

Δρ Δ (- ) log

log . ˜ One shortcoming of our approach is that we use an IR regulator that breaks Lorentz From a condensed matter perspective, it would be interesting to try to incorporate Our analysis was restricted to the case of Abelian gauge theories in the large 2 S-matrix in large A related computation was performed in [ 8 = 0 → this problem. invariance, and Lorentz invariance is restored only when the truncation level 2 correlator. For technical reasons they did not evaluate it directly, but it would be interesting to return to field. These can in principlehelp be studied address within the the problem samevery framework, of which interesting could a to potentially Fermi try surfacetemperature to coupled quantum generalize to criticality. the critical Accessing methodsystems bosons. the to is real nonzero It notoriously time temperature difficult, would dynamics even also and of numerically. study be strongly finite correlated exactly, and one might hope towith gain the insight into gauge how boson to properlyalso and define nondynamical attempt the fermion LC to Hamiltonian component defineappendix the Hamiltonian indirectly via the Bethe-Salpeter equation,Lorentz as violating in deformations such as a chemical potential or a background magnetic arise at small to apply LC ConformalA Truncation relatively to simple the setting generallimit, where case where the some of exact of Chern-Simons solution is those matterthis also limit, problems theories. known the from can more rainbow standard be diagrams covariant that methods isolated renormalize [ is the fermion the propagator large can be computed can be subtracted off by including a corresponding counterterm in the originalThis Hamiltonian. restriction was partly fortruncation analysis simplicity can and be performed clarity analytically, —— and compared but in to mostly this the to known limit, exact avoid the answer some entire conceptual conformal puzzles about how to implement the method that is a trend ∆˜x Taking a hard cutoff onfinite the answers, Lorentz-invariant momentum-squared but as also we breaks did here gauge produces invariance and generates a gauge boson mass, which Figure 3 evaluated at finite JHEP03(2019)107 (A.5) (A.1) (A.2) (A.3) (A.4) .  ) − p 0. The creation Θ( < † p b − . p ) ) + . ⊥ q − ) ψ ∗ p ⊥ − ψ y − ⊥ 2 − 1 − p Θ( ∂ ( p ⊥ δ ψ − ) x ∗ ( a − ψ . δ  q ) 6 2 x · −  − H y ip k π − e +  p − 2 ( 4 ) a p δ − 2 π H 2 m x d ) ( (2 + π δ 2 2 + 2 – 27 – Z 1 H √ os 4 = (2 / χ 1 1 = = } − 2 † q ∂ } H ) , b ∗ os y p ( ) = χ b ∗ 2 p { ( √ , ψ ψ = i ) x · } x = † q ( ip e ψ , a H 2 { p ) p ) leads to the Hamiltonian density a 2 π { d (2 2.13 Z 4 / 1 1 2 ) = x ( Finally, although the interaction in our analysis is dimensionless, its effect is relatively ψ differently from the other spacetime directions. It would be more satisfying, and likely − The Hamiltonian has two-particle, 4-particle, and six-particle terms: The fields can be expanded into modes as The step functions areand necessary annihilation because operators there satisfy are no modes with with anti-commutation relation A Algebra A.1 Hamiltonian andThe mode Lagrangian expansion ( Number DE-SC-DE-SC0015845, and ALFFellowship. was ALF, supported EK, and in LGVGrant part were on by also the a supported Non-Perturbative Sloan inEnergy Bootstrap. Foundation part Office by of LD the Science was Simons under supported Collaboration Award by Number the DE-SC0018134. US Department of We would like toespecially thank Matthew Walters Jeremias for Aguilera-DamiaWalters many and discussions. and Max Xi We Zimet wouldwere Yin for also supported collaboration for like in at correspondence, to part an thank and by early Matthew the stage US of Department the of project. Energy ALF Office and of EK Science under Award test of the methodanomalous would dimensions be of the to deformedApplying consider theory, LC a or conformal even marginal to truncation deformationlikely induce shed to that log light such allows running on of examples, one the couplings. to even details dial of two-dimensional how ones, and would Acknowledgments whether the technique works more generally. with the fermion massx term, and ouruseful, regulator to develop effectively a treats regulator the that preserves lightcone Lorentz direction invariance. mild in that it does not change the scaling dimension of any operators. A more stringent to infinity. An IR regulator is needed in order to deal with the matrix elements associated JHEP03(2019)107 ) , ) 00 A.3 q (A.8) (A.9) (A.6) ( (A.7c) (A.12) (A.11) (A.7a) (A.7b) k (A.10a) (A.10b) ψ ) 0 . Current p ( . The parity j ··· , µ − ψ ν ) ) δ , z 0 , x i q µ 2 y 0 , = z | − → − : µ ··· 2 − 2 1 , z p q p 1 µν ( (1 y i 2 − `

A ψ P  C ) ): ψ ) π 00 ≡ 2 k ∗ p 2 q x y , , ˆ ψ A (which is still null to leading i i  2 − − Ψ( 0 0 a 0 | | µ ⊥ ): γ q ) ) (1 )ˆ m ( σ`/ y y δ ) by taking x 1 , ∗ k ( ( ) y x, σ + 0 ψ + p ( ) ψ , q A.9  0 −− −⊥ ¯ µ ( − ∗ Ψ( `s p j f δ ...... : f χ label denotes the charge under parity. ` ` ψ − − ) = m −  ) − =  J J + q q = q  2 2 + µ X ( σ ). Further using the mode expansion ( χ ˆ ˆ − ∂ ∂ z x, σ ∗ j − ∗ iyP iyP p ⊥ ( 1 2 ψ − − ψ p − + ) − ) ip ( A.9 ` x i 1 1 q y e y e ˆ ˆ ∂ ∂ ψ – 28 – + 3 3 − ), and the functions are given by ) − 0 d d  x f q p 1 . ¯ (1 ΨΨ = − m Z Z − − x 1 − ` − 0 − ( q = q ∗ i C q − , in momentum space these take the form p (1 − ( 1 0 ∗ ) part of ( − ψ ) 2 i ∝ i ∝ x ∗ j ⊥ )  q − J q 6 ) , f ` ( p dx ψ − , ) q ( h p ) ) ,P ,P 1 O 2 p − . We then define the corresponding states as p x ψ 2 f 0 00 ˆ 2 ∂ − + − 2 p 2 σµ p Z − , ( π m 2  p ( ) `, `, −⊥ − + 4 ( | | `s − d ∗ i = p k p π ) = i ... 1 + 0 2 A ψ p 2 ˆ `  ∂ − −  q ⊥ (1 2 ⊥ ( 4 π 2 J − 1 p k p π = 4 − h ( d ψ − ∗ ) 1 0 0 i ` 2 2 p p q = = = ψ ) = ( C ( π π 2 2 and and y 6 2 4 ∗ 4 4 d d ( h h h ` ψ − 2 2 q q ˆ J 2 `, s, P −− 2 2 π π | ) = 2 h d d /P 4 4 ... ` − − 2 2 2 p p p p J 2 2 2 π π π x, σ d d d 4 4 4 ( ), and picking up the =  + ` Z Z Z x f ] = = = 21 2 4 6 H H H where with the normalizations toThe be parity even fixed states later. canodd be states The can simply be obtained foundorder from for in ( example by taking one finds take to be as [ where hats denote contractionsconservation of implies tensors that with only a two components null of vector each current are independent, which we A.2 Higher spinThe currents two-particle and primaries 2-particle in states the free fermion theory are spanned by the singlet and the higher spin currents which can be expressed in terms of Gegenbauer polynomials with Using the notation JHEP03(2019)107 . . `s . ) is A A.1 )] p (A.20) (A.16) (A.17) (A.18) (A.19) (A.13) (A.14) (A.15) will be A.14 − `s ) gives P . A ( ) 2 p i h A.8 ), 0 | − − i ) 0 ): | A.13 p p y p, P − ( ( − → ), ( φ 2 2 , f h P 1 ) [ ( y p  . ψ − A.12 ) 0 . − P ): p ss 2 ) ( )2 δ y p ∗ 0 p ( ψ − `` p, P ψ ( δ ( − ) 0 ) ψ 1 ): ∗ . φ 2 ) 0 y p f ) p ( µ H ( ∗ x p, P − ⊥ ∗ − ( ψ − p 2 X φ ψ − P 2 /µ ) f ) √ ) ): p µ x − (1 p, P ( 2 ( p ( = 2 σ/ P x 2 − φ i∂ h − πδ µ f p p 2 p − (1 P 2 , π ⊥ – 29 – µ x = 1 d p ) which we will not write explicitly. The choices 4 = 2 X 0 p, P P φ ) = ). Doing the same with the singlet ( i ( p ~ i∂ 1 2 P 0 y x Z that with the choices ( A ∗ ) φ ( ( − π dx f − ( x − − m ` φ ). We now choose 0 1 P · ~ M ≡ ⊥ f C P `, s, µ − f x 0 ) A.3 | p (  0 2 X Z 1 0 m 2 − ) (1 ) is then orthonormal ) = 2 δ µ 0 , µ 2 x x 2 0 0 iyP 2 ) (1 → − µ − − p π x , s m 0 2 M A.13 ` − µ y e p (1 π h ( dx 3 2 µ lim x d ), ( µ 1 . ( πδ  0 p ≡ s Z Z ) πδ = π f f dx y A.12 = 2 ( 0 φ φ ⊥ , and to get the to the second line we used the mode expansion from 1 i N N ` and p 0 = 2 A A − C ` Z p p i . Using Wick contractions, the inner product of two states is given by /P φ, P φ × | = = − 2 i p φ, P | 0 ≡ |M 0 1. Here ,P φ, P x 0 | ≥ ,P φ 0 Matrix elements of the mass operator h ` φ h can be similarly computed. As an example, we can consider a generic two-body hamiltonian Using Wick contractions, one finds that its matrix elements in two-particle states ( Note that all inner productspreserving and matrix delta elements function will contain (2 anof overall basis spatial functions momentum ( The sum is over for all states where independent of A.3 Inner productsTwo-particle and states matrix in elements the singlet sector take the form The normalizations of theseHowever functions we is will arbitrary show for in now, section since we have not fixed for JHEP03(2019)107 (A.28) (A.25) (A.26) (A.27) (A.21) (A.22) (A.23) (A.24) , 4 ˜ h ) . p . in the primary , . i − .  2 0 ) mod2 .  s µ 0 0 `` ) · p `` 0 # + h.c. p, P 0 M δ ) `, s, µ x ( 0 ) | − · + ss 0 φ 0 4 0 x 0 δ − s f 1 0 ) x δ ss 0 ,P `, ` − `` δ 2 0 we have (1 − |M p δ ) 0 ) the quantity in brackets is 0 p s ) 2 4 (1 δ x − 0 2 does not contribute and the 0 − 0 1 ) − , µ µ 0 H 1 0 0 x min( µ ` 6 + p − δ A.7a 2 f − ,P , p , s p ) 1 H 0 0 − 0 P 0 ` − 2 ` x m p p δ 2 σ ( µ 0 0 0 µ − ( ) between two-particle states can be − µ 1 µ φ = 2 ( ( ), the matrix elements of the kinetic ≡ h iµ = 8 4 f 4 . πδ (1 h r πδ limit, ) 0 ⊥ A.6 ) − ,`s x 0 x p, P, P ,p M x s )  8 A.17 2 0 = 2 ⊥ X int ` 2 f ) + = 2 x iπλ − f p − i p x − i m − 1 4 − 2 P M s − → ∞ (1 ) ( δ 0 (1 have the form 4 ) x − (1 1 πλm p, x f + – 30 – h x s x 2 f `, s, µ − 1 0 | `, s, µ N + δ − − 1 − ` | − − 1 m p 0 2 0 a C ` kin ) µ (1 ) kin δ 0 p (1 m 1 x p, p x 2 x ` iµσ − 2 |M δ λ 0 |M " − 0 p 2 p 2 0 − P, ) π λ , µ x 0 P , µ − 2 πλ 8 are orthonormal ( 0 ( (1 µµ ). In the , π 4 0 1 , s − i + − 0 a , s √ = +) they are given by p h − 0 ` · 1 4 `  ` A.7 0 s m (1 − = ( C f x 2 4 − 4 − 1 ≡ h µµ ≡ h N ˜ h `, s, µ h P π p | dx 2 = 0 √ − , `s − + 1 ,`s 0 P 0 ) vanishes. In this limit one has 2 s 0 s 0 = f ,`s 0 0 π Z mass ` kin ` = s i N 0 2 f dxdx int ` 4 M A.26 M 1 ˜ m h 0 M φ, P Z | 4 f = 8 ) given by = 0, this leads to matrix elements is given in eq. ( 0 N + f `` |M 4 0 × h m A.12 M ,P 0 The matrix elements of the interaction terms ( φ h When basis ( second line in ( where with is discussed in the next section. similarly computed. For the quartic interactions In the parity-even sector ( The matrix elements of the mass term in the parity-odd sector have an IR divergence which The matrix elements of the mass term Since the basis states term are just For example for the Hamiltonian of a free massive particle ( JHEP03(2019)107 . 9 3.4 (A.33) (A.29) (A.30) (A.31) (A.32) (A.34c) (A.34a) (A.34b) . Doing so − ) M x ], which found − 24 , , (1 in general, except in ) x  . . x 2 p 2 p )) ) . ν x − 2  / 2 ) mod2 q 2 0 3  x (1 ↔ 0 `` q 2 − )] 2 + perm ` δ . x µ C −  (1 ) νβ  ) 2 − 0 atanh x + ( q P 2 − (1 0 2 2 λ atanh 2 2 ` (1 `, ` p . + νβ − x 4) C − 2 ` [ 16 P ) q q 2 C q − x 2 diverge (one in each spin even and odd µαλ (1 − 8 2 µα 2 ` are functions of ε 2 mod2 ) max( 4 0 − q P . These functions can be obtained in the  C g 0 x ( q ( − g 1 `` | − κ τ 0 π δ g M atanh π p g − τ dx − + − (1  | 1 − 1 κ 2 1  2 2 2 1 √ 192 (1 2 − ` − 0 q q – 31 – + − 2 4 = ` and )  x q Z 2 2 √ f C − 0 g ) 2 g ) 2 αβ 0 0 p τ / − τ / m 4 + 4 ) P for the massive free fermion theory was diagonalized π ) σ 0 dx 1   , 0 0 i ] `, ` 0 2 f 0 `, ` ` g µ µν ` 1 3 3 δ τ δ 0 P q q m `` P  22 + + ) = Z | µ i i 8 0 π π TT g 2 f 0 q 3 ` p ` + sgn( h τ δ | P min( δ max( ( m 1) 64 64 − 2 x, σ = = ( s − 1+( = = = ) i − . Here 0 = 2 2 ` 0 0 g g g ν 2 f 0 M f ` . τ τ p αβ κ − = `` `, ` µ m T 0 p 1)( M − there `` µν µν T 128 − − T min( M h 2 2 f ` µν 0, two of the eigenvalues of = 2 ( η m 2 p 8 3 → p here µν the matrix  T = = ≡ 0 3.3 − `` µν P M Note that 9 where a CFT where theyLagrangian are approach by constants computing and a fermion loop. This is done in ref. [ Ward identities to take the form [ A.5 Spin evenIn sector section and in the spin odd sector, whichIn allowed this us section to compute we current obtain spectralodd stress densities sector. tensor in spectral section The densities stress by tensor similarly two-point diagonalizing function the is spin constrained by Lorentz invariance and gives the orthonormal Dirichlet basis In this basis, the matrix elements of the mass term are finite and given by In the limit sectors), and the corresponding statesthem decouple by from hand the by theory. projecting It is the convenient basis to to remove the kernel of the divergent part of The matrix elements of the mass term in the parity-odd sector are given by and are all divergent. A sharp cutoff on the integral gives A.4 Projection to Dirichlet basis JHEP03(2019)107 = = ` ` , i (A.41) (A.38) (A.39) (A.40) , (A.37c) (A.35c) (A.35a) (A.37a) (A.35b) (A.36b) (A.37b) + −i `, | `, | , Ψ, the relevant + 1 ¯ ) αj ν π ∂ 2 ↔ max . ` µ ( # γ 2 sin ¯ ` Ψ j i δ + c.c. 3 1) ψ . 2 = . √ − f ( ). π m g − µν , , . − κ T 96 ¯ 1 ¯ −
β α + 1 . 0mod2 ∂ A.33 0 ` g 3 − i φ φ δ ⊥ τ − 10 2 p ) 2 2 ∂ + ) 2 f max ∗ 0mod2 f ` ` , j, ` by the states + 2 sec sec ψ ) (A.36a) δ 0 2 g √ m im 2 √ | g i f 2 f 2 − τ ψ ) `, τ " | ¯ − β ¯ ∂ β δ − | p m m ( ( j = by the states | − 2 ¯ β ∂ | e 3 − sgn( ) − 4 − ∗ i i 2 − µ p µ p √ α | p p µp p − ¯ + 4 + 4 (¯ j α ψ M 2 – 32 – 2 √ | √ =1 e max √ q 2 2 √ j ¯ X α ` = = = − = µ µ + 1 2 1 8 1 4 i i i i ψ = M ψ = = ¯ ∗ β = = = ∗ i −⊥ −⊥ −− ¯ ¯ β ψ = α i i ¯ ψ β T T T | − i ⊥ + −i + M| ∂ M M ¯ α with ∂ ( −⊥ −− −− `, `, `, 2 | | | i T T T i 2 h h h M| √ 3 = 0, we can extract these coefficients from (0) (0) (0) 2 √ , ⊥ i = = −− −⊥ −⊥ p T T T + h h h 2. The corresponding eigenvalues are j, −− −⊥ The parity-even, spin even sector spanned by the states / 2 T T | ) α max . (¯ j e +1 ¯ α and is diagonalized is diagonalized 2 =1 π max j X ` max ; the exact form will not be needed. The corresponding eigenvalues are , . . . , ` ` ¯ β j 2 max max = e , = i ¯ α ¯ α | = 1 φ The parity-odd, spin even sector is spanned by the Dirichlet states α We consider the improved stress tensor (as opposed to the Noether current that generates translations) , . . . , ` , . . . , ` 4 4 10 , , for some because only if it is symmetric will its correlator take the form ( 2 for ¯ with Diagonalization. 2 The overlaps with the Dirichlet states are given by The stress tensor incomponents are the fermion theory is given by In the kinematic regime JHEP03(2019)107 ). , 3.23 (A.45) (A.46) 4 (A.42c) (A.43c) (A.44a) (A.42a) (A.43a) (A.43d) (A.44b) (A.42b) (A.43b) 3 − q 2 q , π , 6 . 48 3 −  4 p − ) 3 − 4 3 f 2 − p π q q q f 8 2 , m . 96 q 64 ). im ! π , + 8  2 −  4 sgn( 2 −  p · i q 16 p = 6 4 . 2 f , | 4 − A.34 ¯ , α − 4 −  g 16 p p φ m | p · κ = ¯  β ¯ α ¯ ¯ + 1 . β

α + 1 · + 1 φ = φ · φ φ tan Im ¯ α 2 πx ¯ 16 α 4 ¯ α φ πx max 2 max 2 q φ πx max φ tan by computing matrix elements with 2 ` ` − cos 2 ` 2 2 sin 2 2 64 ], and projecting to the kernel of the 4 √ √ , tan √  q πx ) cos 4 sec − | − sin sin 2 tan A.4 ¯ µ 2 3 − β 2 p p µ − 4 sec 16 4 πx | ) ) ) p 2 µ − µp √ q ¯ ¯ ¯ 2 α 2 α 2 α √ 1 2 − √ 2 ¯ ¯ ¯ − − α α α q √ p µ µ µ f q Θ( sin 64 , 2 4 µ µ µ ¯ 1 4 1 4 [ β sin 2 2 q q p Θ( Θ( Θ( – 33 – µ → im − − ∈ p = ¯ ¯ ¯ ¯ β α α α dx = = = 0 −⊥ g X X X X − − T q 2 i i i τ dx p P ¯ ¯ ¯ β α α q 2 | | | −⊥ Re + 1 + 1 + 1 + 1 T acos = 1 1 1 1 1 π (0) (0) (0) acos 0 πρ x 1 π Z max max max max 0 −⊥ −⊥ −− ` ` ` ` , Z | T T T 3 − h h h | f 2 p = = f f m 4) ) = ) = 4 | 4 − m p p q | 8 p We are now ready to compute the spectral densities using eq. ( im ( ( − 64 2 16 −⊥ −− q odd cut even cut | T T ( | ) → − → ) limit, one finds p − p −− −− ( ) ) ( T T p p = ( ( −⊥ −⊥ g πρ πρ T T τ −⊥ −− → ∞ T T −⊥ −⊥ T Re T −− −− max T T ` πρ πρ πρ πρ B Dirichlet basis fromThe dimensional Dirichlet regularization basis wasa constructed in sharp appendix IR momentum cutoff The result from truncation is therefore These results exactly agree with the Lagrangian predictions ( Similarly adding the parity-even and odd cuts one finds In the Spectral densities. From the overlaps above, these are found to be The overlaps in the diagonal basis are given by JHEP03(2019)107 ) N max ( ` (B.3) (B.4) (B.1) (B.2) O ). In the . . ) B.3 . We perform  i − 0 | dp  0 . y  − i p → 0 | 2 , : dim. reg.) 1 = y p | − (IR cutoff − − p p 0 0 P ):   2 ψ y ∗ p 1+ 1+ ( . d d ψ ψ ). Removing the corresponding ( p in any inner product or matrix ) ψ 1 0 → ):  0 one finds y p B.4 ( )] − ipx ∗ x e − → ψ dp mod2  − 0 0 −  p `` ): 2 δ p, P 2 ) (1 p (  mod2 2 π x 0 ) i∂ f d 0 `` (2 ) and are all divergent. A sharp momentum − δ ⊥ , p `, ` X  1 Z – 34 – ) 0 0  4 i∂ A.30 2 / 1 1+ 1 − `, ` ( max( 2 )] φ x f −   ) = − max( 1 x √ ( iyP (1 −  − ) is evaluated with the dim. reg. prescription explained above, ψ x 0 [ 2  / ) y e π 0 2 dx 3 0 / 2 f ` A.30 d ) 1 δ 0 0 0 m + 2 f ` Z Z 8 0 δ ` m δ + ( = = 8 0 ` 2 i δ the spectra in both schemes agree, except that the infinite eigenvalues ( 2 = 0 2 and then analytically continue to φ, P / = − | `` ) are replaced by negative eigenvalues in ( 1 0 M − → ∞ `` > B.3 0 M  max in ( `  Chern-Simons theory. 1 √ f In this section we show howthe to relate correlators, the Bethe-Salpeter to equation, whichmodel the is first, used Hamiltonian to and equation. compute expressshow how the To to Hamiltonian warm derive the equation up, HamiltonianN in equation we from the will the conformal Bethe-Salpeter consider equation basis. in the the Next, large we will eigenvectors, one recovers the Dirichlet basisthe with spectra the and dim. corresponding reg. prescription. bases will At however finite differ in both schemes. C Connection to covariant formulation The effect oflimit dim. reg. is∼ simply to remove the power law divergence in ( If instead the integral in ( assume elements. Let us now lookof at primary the matrix states. elements of Thesecutoff the on are mass the term given in integral by the gave ( parity-odd sector This replacement will simply add a factor of [ Two particle states now take the form (ignoring overall factors) to states of higher particle numberthis by prescription cutting does off the not momentumregularization generalize of schemes to each that particle. generic can CFTs, However, theory be motivating using the more dimensional study readily regularization of generalized. (dim.this alternative In reg.) replacement this in section the we mode study expansion the divergent part of this matrix. For free CFTs, this regularization scheme can be generalized JHEP03(2019)107 , 0 + p 3 dp (C.7) (C.5) (C.6)  d (C.8) (C.3) (C.4) ) ) In this , r 0 i integral can + 12 . (C.1) 2 k, p + i )  ( m α ) r dp J kφ − | − 2 p ) model are physical 2 ⊥ d p 1 , k, r p ( 0 N − . ( p − S Z ≡ h  ( ) ) O ) and the 0 2 − λ p ) − p r 2 ( kφ r ( p ) − + 2 ( S 0 − p − d ) 2 ) + (2) p α 0 in the (2) α − m p , r i F − p Z 0 ) can be related to the con- ) F φ (C.2) + )( − − λ ) . The Bethe-Salpeter equation r , q 1 + 2 ⊥ 0 p ( p, p r C.3 − p , r ( . ( N − 0 2 1  ) + p − 0 δ  ) 2 ~p,~p kψ p + ( x − −  − p, p m d ) decays like p Next, we consider φ ( p P 2 r d − ) − ) r + 2 ψ x p p −   11 )(2( (1 ( 2 ⊥ 0 + ( Z − p x ) p − i − 2 − ( 1 λ πi p (2) − P δ dp α S + − p  2 ) (1 + F to obtain 2 1 2 p 0 − x − ( r − ) + − p   m 1 0 ) dp ` S 2  1 − ) ]. However, instead of considering directly the 13 p πir x C r − r )  – 35 – ( ) model at large Z pd x − 18 ` ⊥ − 2 ⊥ − − 2 , so − p P C p p N 1 − d p πi ) ( p ( − + (1 − ( − p ) = δ (1 θ O Z ) = ) = p ) x ) − S . p p r ) p ) , r x ( ( x 0 − p x = ( + − ⊥ ( − p ` ` θ − = ) explicitly, i − r S F F ) p ~p,~p (2 ( − ( ( ’t Hooft model which are confined. q θ + (1 p + ) θ φ S C.3 − ) = ) − D ) dp dp ( p x p ( ( , r α ( 0 just denote the spin and parity quantum numbers. Z θ θ Z J S )  α = = q ) p, p = 1 and ( ( ) = α r − N φ ) = J r ( ) = h −  , r 0 p , r ) 2 ( 0 x S m ) NO p, p p − ( p, p ’s are the wavefunctions of the bilinear states, ( + ( ψ S F φ (1 2 ⊥ + x p dp − Z 2 We will now show how the Bethe-Salpeter equation ( r The scalar propagators each decay like A physical motivation to consider the full correlator is thatExplicitly, the the states wavefunctions are proportional to Gegenbauer polynomials,  13 11 12 In the last line, we set be done by residues: states, unlike the quarks in the 2 formal truncation Hamiltonian equation,in and terms how of the the wavefunctionsintegral spectral of in density the the can Hamiltonian. first be To term show in computed that, ( we first perform the where the model, the parameters This correlator can berents, used which to compute do the notsymbolically two-point mix in functions terms with between of the higher a bilinear currents discrete cur- at set large of parameters, N. We can indentify the currents homogenous equation for the “blob”so as that in we ’t have Hooft, an we equation will for not integrate the entirely full over correlator. Which satisfies the equation We will begin as a warm-upin with this the case is We now follow steps similar to ’t Hooft [ C.1 Large JHEP03(2019)107 , ) i i x ( ` µ, ` f E, n | | (C.14) (C.15) . Here ) x − 1 (1 ∗ (C.9) x ) ) is equivalent  k √ ) 0 )) ( p ( , q (C.10) 0 kf 0 C.11 ) 2 α (C.13) ) d F 0 , r (C.12) 0 ~p,~p ) ~p 0 ) ( ( 0 Z ~p φ p ( ) p, ~p . Then, we can construct ( ∗ E,n x ,n n − | φ φ q ). Furthermore, the spectral | p 2 ) − )Re(  ( 0 . We define the action of the πi r φ ~p 0 2 ) p ( − (1 p ( δ x 0 2 x ) C.10 1] with measure /r α ) + (C.11) ) onto the conformal basis d ) E,n 0 x , − ) 2 ⊥ − F x p φ ~p ) p p − Z 1 ( (1 p denotes a countable set of quantum i − − ( x C.11 = (1 α p 2 n + E,n (1 (   x x F 2 2 ) φ 2 0 θ − ) 2 δ p p E ( 2 ) 2 x dE E ( α ) = x µ , where 0 − pd F i ~p  2 2 ) λθ − ( iπ d q ) = δ ~p – 36 – ) ( p µ, ` (1 n | x ∗ E,n ) + ,n Z ]( x E | ( p φ x ` q h Z | ) ( − = f p ~p E,n f 2 i ( n pφ φ ) 2 ) = [ d 2 X q x 2 i d ) = m ( E,n R = 1, and defined 0 p iπ − φ M α + Z 2 − `, µ J )]( Next, we project eq. ( | ) (1  2 ⊥ p , r q i 2 x p ) = , r h dE 0 − n

is general. 0 φ ) dµ a ` f e − ! ) f − )Ψ x i φ κ π ) R ! ) ν ( p p 0 ) ` x ( α γ x ( dxx j x ` ( F ) ( )( P 0 0 ¯ 0 f Ψ ) Ψ + ` α p Z ) ) = . Equation ( ) i , r x f i 0 r ) (see figure p F x γ 0 ) ( − ) ` − − 0 ¯ 0 x , r Ψ – 37 – X 0 p − 0 i 1 x k, p ( p 2 D E, n 1 (1 a ( − ( 3 | (1 i − x δ 1 m ae p, p bf = − ) x 1 (1 ( r ψ p i )Ψ (1 µ, ` x 0 0 ) be the fermion propagator. Then, the Bethe-Salpeter ) h ) can be expressed as a sum over these wavefunctions, p h ad bc ) + p )Ψ p p x − dx ψ p ( µ ( f i Tr dx p − p S ( ( √ ¯ m µ, ` 0 Ψ Z dx C.14 d k ν b ( h Z 3 a S + ≡ dx ` d ) ) Z

/ E,n p X ∂ p ) ( i λ ( Z φ Z

µ a c 2 2 + × ¯ a 0 Ψ( S − µ, ` µ ` h ( X

X E,n ) = E ) = | L φ × CS n , r 0 µ, ` . Similarly to the previous subsection, we consider the Bethe-Salpeter f ( N ) = p, p 2 ( E,n π/κ q φ ( ad bc 0 2 = ψ E αα λ ρ delta function factored out. Let equation can be written as where equation for the “transfer matrix” correlator in momentum space We will write the tree-level gauge boson propagator as C.2 Large In a similar way,Simons we can theory show can thatis, be the however, Bethe-Salpeter cast a equation as subtlety forlight-cone. the related the In Hamiltonian LC large to gauge, equation the our for presence Lagrangian the is of wavefunctions. unphysical degrees There of freedom on the where we specialized toIn the this parity-even model, sector, the butextract the spectral the correlators generalization density and is is spectral straightforward. proportional densities from to Finally, the spectral densities ( where we defined formal Hamiltonian truncation equation, leading to the matrix elements in section Taking into account the completeness relation JHEP03(2019)107 : shell (C.30) (C.26) (C.27) (C.25) − on ) + ) (C.29) p r I f − = ( m p + ( p d + CS theory u )¯ ⊥ f p γ ( N (C.28) ⊥ a ) p u 2 f )¯ (C.24) + p p, r ) m ( ( r c − is proportional to a projection u − γ ad bc (C.23) − ) ) ) 2 − r p p p OS , r ( reg ! + 0 − d S b φ γ + ) 2 p m S p' p − − ) 1 ( x p + p, p − ). m 1 b p p 2 ( γ ⊥ − ~p ( 2 u − √ ) + ( ( a + 2 ⊥ c ip ad bc 2 0 p u 1 S − ψ (1 2 ⊥ E f + p

p, r + ) + x as p 0 + 2 f ) 2 ( we need to evaluate the tensor p m − x m ( dp dp p − ad bc ad bc φ ) – 38 – − + φ is evaluate at the single particle pole, √ OS − = Z r S OS dp (1 + ) = 2 + ≡ i φ θ p ~p r ( = ) ) ) = ) ( Z 2 ) is the usual solution in 3d, with + x p ~p p p' u ~p I ( ( ≡ ( i ) 2 f ≡ ( f ≡ θ u p, r u ) u )¯ ) ( m + m 0 m ad bc p x 2 f ( ad bc φ + = p, r + − ) + u ( − m 0 2 iπ / p µ φ p ad bc γ (1 ( − ) 2 µ 0 x ,E 2 1 p φ p √ 2 ( − − − r r √ p ≡ . Symbolic Bethe-Salpeter equation for the large ) = ) ≡ p p , p p' ( ( ) = x φ ), so that ( S OS p, r 0 1 1 0 S ( ( Figure 4 ad bc ∗ ) u OS = φ ( , and an “on-shell” term, where u + p The right prescription, fromcalculation, comparison turns of out the to finalSalpeter be correlators equation. to with discard a This the Feynman prescriptiondegrees diagram “regular of intuitively term” corresponds freedom in on to thePerforming the discarding the derivation light the of integral cone, unphysical we the though obtain Bethe- we do not have an a priori derivation for it. and ¯ operator on a one-dimensional subspacecontributions of to the space of spinors. Consequently, we have two where the fermion wavefunction To derive the Bethe Salpeter equation for To simplify the analysis, we will splitin the propagator into a “regular term”, which is analytic We define the integrated transfer matrix JHEP03(2019)107 ) 0 p , the (D.1) i (C.33) (C.34) (C.36) (C.31) (C.32) − , µ p . (  2  , δ 1 -dependence ) )) | 0 r µ p , will just give ) (C.35) ( . = 0 0 − a p p i )) ( u 2 p )¯ r a b ( ) r ) u pd r − ρ 2 , − d γ p `, s, µ ∗ E,n 0 − ) | . ( ! φ R p p p u c d 0 ( ( ( ) )( ρ b u d ν p γ u ( u ) γ ) )(¯ d c )¯ µ/µ p r r ) )( ) p ( 0 ( ( r p u − c , r can be computed as 0 E,n ,νρ 8 u )(¯ p − πλ 0 ) ( φ r d c i ⊥ p ( d ( x ) D p, p ( J u ( d i i )¯ , − ,νρ /µ f e u a p 0 0 E,n 0 ad bc − = 0, which can be addressed using the ) )¯ ( + m φ µ J φ (1 c ν p D µµ 2 f ( 2 b a ( θ u γ e f λ 2 1 c ) ) √ p ) ) E 2 m 2 µ )( u x ν E π x 8 ) ( γ πλ = 1 in this subspace) γ − , r ( θ x i ) − 0 0 = − ` 0 2 )( ) iπ p − p r d c − k x 2 – 39 – ] ( 2 (1 ), we get k, p

− f ) x e − ( (1 pd πi p E,n p 2 2 θ dE = ( ae bf ( ) kf d φ (1 d 2 c n √ [ matrices, and integrating in 2 C.22 φ i x x 2 δ f ( E h d ) ) Z 2 γ θ 0 Z , µ just sets the momentum frame we are working with, we p M ) p , and for conciseness we will not write the k ) = ( Z √ 0 ( 2 2 x µ ) can be written as 2 | n 0 a p d r µ X × + ( = 0 in primary state basis E u − 1 2 int a b with )¯ iπ ) Z iπ f r ) = Re ≡ C.31 (1 = φ q |M ) − x ( − 0 m ∗ E,α p + 0 2 ) = ( φ µν p , µ d c , r √ ( ρ )( , r b )] ) 0 p . h u p ( ( (  = 1 d c 1 f 2 ). Since the interaction only acts on the states ) instead of the Dirichlet basis. The matrix elements of the interaction 0 p, p ) [ CS at ( 2 − × E 3.4 f ad bc E,α , r M φ A.28 φ N ( ) = 2 = 0 , r 0 ⊥ dE r p, p Z ( ad bc α X φ are given in ( Hamiltonian is already diagonal exceptit in is the given subspace by spanned (we by drop these the two states spin where index since D Large In this appendix weprimary study basis the massless ( theory from which the spectral densities for the currents The wavefunctions will be normalized so that Therefore, the solution of ( for the wavefunctions.neous Formally, the equation, solution can be found by first solvingwhere we the defined homoge- the squared mass operator Taking the contraction of the expression for the correlators. However, we will not do this, since we seek an equation will set explicitly. Integrating the equation ( Since the external momentum JHEP03(2019)107 5 6= 0 (D.6) (D.7) (D.8) (D.9) (D.2) (D.3) (D.4) (D.5) f (D.10) m ) model was , 2 N leads to
( . 2 O | 16 πλ = 0 (see section q . 1 − a i j ⊥ m ih . This reproduces the j ) which can be written | 1 + . − µ 2 16 j ˆ i D | µ µ | 2 1 , ψ , . | = ⊥ λ 16 − i i j µ
2 i (note that here the matrix i | h 2 4 where the − j π 2 = ⊥ q √ j | − a i | j j | ˆ i , D ˆ − = m D | 4.3 ˆ ih i πiλ j ⊥ | i D i | 8 , | j i j ⊥ j | − ˆ h j | j ⊥ D j ) with eigenvalue − ˆ | j , µ h D ˜ C D = 2  | h
i − i | 2 j j ⊥ | |− S | − | D.3 j i h j − = 1. It has the form πiλ | ⊥ h 2 j ⊥ S 2 j −⊥ ) 2 + | ˆ and j i − D h ih ) V | i 1 4 ⊥ ih j i C ⊥ µ πλ j j j 1 j − πλ | | h + 16 = j 1 ˜ | ˆ 2 ij D D + ( = 0, unlike in the general case with | , i – 40 – + ( i V = i | − | ∗ i C µ 1 ⊥− i C − j ⊥ | i − j 1 j − V j j − πiλ √ | i | X i −  j = 0 and 4 | ih j ˆ X ˆ − D | D . One then finds | ˆ = | j − ⊥ i D = 2 | ˆ j are = j | = i D − ⊥ X | C j ˆ i j i , − D i S h integrals, and we can therefore take h | j = ψ = V ψ − | h 2 − | , µ ) i = j µ 2 h πiλ + = ψ −i i | µ | i ψ ) for regularization using a sharp cutoff, where a counterterm − = 0. ψ = | ψ, j − + 2 h | f h satisfy ψ 2 only overlaps with this state, we can directly compute its 2 1 i q m 5.19 2 ( S ≡ h − µ −i| j ψ , where ). The norm of the state is ψ = ψ, i ) at N | i 5 N j − | is diagonal and M j ˆ 5.29 D |h i ij δ C = i P −− i ∝ j = j πρ | ˆ 6= 0 is needed). Looking for an eigenvector of ( D (0) | i a i j ψ where in the lastknown step result we ( used with norm Since thespectral current density Let us focus on the solution with where the non-zeroh entries of studied in section where the coefficients m This equation can be inverted| up to a term in the kernel of ( The rest of thediagonalized, diagonalization using will Dirac closely notation. followregularization section For to simplicity perform in the thisand section in we particular will eq. use ( dimensional the full mass matrix in this subspace can be written Using the overlaps of the current operators with the basis states JHEP03(2019)107 . ). In : (E.6) (E.3) (E.4) (E.5) (E.1) i + (E.2c) + 1 (E.2a) (E.2b) α ). This , ) φ `, s 3.27 | µ max 4 ( ` sec β A.25 . In the free s √ ⊥ |M 0 1 2 2 = ) and ( , j πλ − , s i 0 0 − µ ` ⊥ j f 1 2 j 3.26 | µµ m ≡ h √ that diagonalizes the . , . | , ≡ − ) β, µ ,`s 0 − h 0 ) + j s α 0 3.3 . . 0 µ int ` α 2 ( ih 1mod2 ) is simply given by − , 0 φ α µ ` ↔ ) − + h.c. β M δ j µ 0 φ , | 1 ) α − ( E.2 µ ` a µ 2 cos µ ) α δ , a µ ( m sin a + + + ( r 2 α 2 were given in ( 0 f α ) − c s + λ q, α i φ + − δ + α µ + 1 4 ( ) 2 0 β β im ( α 2 + iπλ φ α | π 2 s φ α φ µ φ 8 , 1 cos δ c q, α i − max ( cos ` − α − sin − 2 | cos . cos = 1 2 √ µ P.V. + 1 a c 0 ` i . 2 1 2 1 + 1 π m ⊥ √ µ + 1 1 µµ 2 j in two-particle states was given in ( 2 dµ | max 1 2 + h 1 P.V. λ – 41 – 4 ` √ 2 | max 4 2 ` Z π = ⊥ max M α, µ λ 0 j ` 1 1mod2 β i α h 0 ≡ − 2 ` α 0 ` ih µ µ δ − π ) X δ µµ j a πλ − − 1 | µ , µµ j ` s f r √ ( | = ˜ δ δ m D β √ | m 2 i + + 2 0 0 = 0 − s s iπλ πiλ − − j i δ δ | α, µ 1 , s − denote matrices whose diagonal components are the principal + 0 = = = h ˜ s ` D j = 2 | | 0 0 δ δ ˆ f a + D − + 1 CS αβ α m αα m αα j − + int h φ β, µ M h M M M = max and ` cos ˜ , ,`s D √ ) 0 interaction matrix elements s 1 2 µ 0 int ` function, respectively. For instance, ( − f α µ δ c M N 1 2 = i ≡ − − j ) | µ The matrices ( α α, µ c h and value pole or Using this notation, the interaction with matrix elements ( where we have defined the following vectors: We will particularly be interested intheory, their the spectral overlaps with densities the of massthe the current eigenstates interacting theory, the overlaps are given by It will be convenientmass to term. work in In the this basis basis, constructed the matrix in elements section read The general form of matrixleads elements of to the following matrix elements in the Dirichlet basis E Large JHEP03(2019)107 . → free ! ρ + (F.1) (E.8) (E.7) α max φ ` and . 2 function replaced ˆ D 1 δ !# sec −
1 2 = 0. q 2 − q K 1 ) , − log 2 2 α In the limit f α, q m ( q 2 m coth 2 14 , − . µ
π + 1 , 2 1 2 3 − α
µ J φ q 2 + 4 − 2 , − α 2 2 ) 1 2 X q 1 2 q / q − 2 f 1 − cos π K ) πx 4 1 2 ) 2 m − 2 q 4 − 1 K P.V. coth , and the connection between − ) 2 sin − ) α, q i 1)

/ , ( 2 − µ 1 free 2 f ( q β ) − ρ = ! πx µ α, q α 2 φ m π (
2 f q 2 − 2 2 2 q 2 πx / J + max 1 µ 2 − (4 + ` ) sin 1 α ) + −  2 J X 1 2 − 2 α f q dx (4 sec Λ( cos πx K ) . 2 α ) m m − 3 – 42 – q 2 − were obtained in section f X dx ( q q coth − dx 1 m + 1) 2 ˆ

D β, q πx
2 − ( 1 2 µ = (4 sec 2 + 1) cos max ) µ = 0. These states are eigenstates of a Hamiltonian q + 4 1 − ` q 2 + 1) 1 1 2 − − 2 ( 2 J µ 1 8( π 1 q max q (4 sec − ` Z ( f β ) max (2 + 4 log 2) δ ` q 1 8 2 X 8( cos α − ( im − f 1 1 8( C + − 2 1 2 − − 1 m 2 q Λ π = 2 = = Z cos

− − 2, the expressions are the same except with every coth 1 ) = i i ∗ f f  1 2 i q − π ⊥ − 1 < ; " π m j j ⊥ Λ 1 | | j µ 8 16 Z q | π πm ) for the free theory spectral function ( ˜ ˜ 1 D D i 8 8 4 ˜ − − α | | D < | ψ ⊥ − 3.28 j j − = = = = = h h j i i i h . Recall that we define − − ⊥ At 0 1 j j j = is regular away from | | | − i limit of the sum, and the approximation of the sum as an integral over the summand. Shamefully, ˜ ˜ ˜ 15 D D D α − | | | 2. f j | ⊥ ⊥ − ). j j j ˜ > h h h D → ∞ We will also use the following matrix elements: | See equation ( The log 2 term in the second-to-last line is a bit subtle because it is due to the difference between the q 5.13 , the above expressions simplify to ⊥ 14 15 j max h with a continuous set of eigenvalues, and thereforeis ( their norm receives only ` we discovered it numerically. It is easiest to derive analytically by evaluating the sum at In our analysis, we often encounter eigenstates with wavefunctions of the form where at by a tanh F Wavefunction normalizations The corresponding matrix elements for ∞ JHEP03(2019)107 i ) (F.5) (F.6) (F.7) (F.8) (F.2) (F.3) (F.4) 2 α m − ∞ 2 q to p ( . . Integrals of the 0∗ α function terms, as a f −∞ , . ) δ b 2 ∼ 2 α fixed  1 − x m  y π , but to see the relevant from x + 4  2 − . 2 y 2 2 δ q and  ) P.V. 2 a  δ p , a + ) ( ( b f 2 α 1 − ) . So we can restrict our attention f 0 simultaneously. More precisely, − b b = 2 x x π , − ) 6= ( 1 x → 0 2 x +  a  ( P.V. ) + finite ) → 0∗ α a 2 + x P.V.  C 2 ( − 2 ) and α + = ) ) δ b b C δ a ,  ( 2 2 dxg h ) 0 δ −  2 − − ) a → y – 43 – 2 α Z π x + y → x ) − . ( a ( 2 m ) a )(( a y x 2 ( x 2 0 (  − f q − , and also the limit of small and ) + 2 b 0 x + x function, so we have = − q a ( lim → 2 ∼ 2  δ I ≡ y 1 . By construction of the principal value part, integrals of − q Θ( ), which permits any use, distribution and reproduction in ( a b ( + dxf x 0 δ α dy ∼ lim → × X Z  , y x 0 Z P.V. above involves two kinds of integrals over principal values: ) ) ) = a x → I ≡ 0 α ( ( and ψ a f CC-BY 4.0 a µ, q − dxg ( . ) in the integrand, and we can integrate over This article is distributed under the terms of the Creative Commons ∼ b α b a I → Z ψ representation of the principal value part, we therefore have to evaluate x ( ≡ ) ∼  f q δ a ; µ → ( ) ∗ ) is regular. We are interested only in divergences that arise in the integral from α 0 limit of the r.h.s. is a x x ψ ( ( 2 f g → dµ  Using the Z as desired. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. The Then, We are interested in thebehavior region we have towe take take a the limit limit where the region near the first kind do notsecond kind produce are any clearly divergences equivalent in toto the the the first region region kind around if The normalization of where The contribution from the principalfollowing value representation parts of can the be principal derived value as part: follows. First, recall the well as from the overlap of two principal value terms: contributions. These contributions can come from the overlap of two JHEP03(2019)107 ]. X , 02 B 47 ]. (2016) SPIRE IN JHEP C 72 [ X 6 SPIRE , Phys. Rev. IN ]. . using a , ][ Phys. Rev. ]. , N QCD coupled to ]. ]. ]. (2015) 025005 dimensions and SPIRE N (2015) 031027 (1974) 461 IN SPIRE theory in two Phys. Rev. ][ SPIRE , SPIRE IN SPIRE 4 Eur. Phys. J. D 91 (2004) 1490 , IN ϕ 2 + 1 IN IN X 5 ][ B 75 ][ ][ ][ ]. ]. 303 QCD at finite ]. arXiv:1606.01989 D SPIRE SPIRE [ 2 Phys. Rev. bosonization IN IN , Phys. Rev. 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