AAnalytic pseudo-Goldstone bosons
Riccardo Argurio , Andrea Marzolla ,Andrea Mezzalira , and Daniele Musso Physique Th´eorique et Math´ematique and International Solvay InstitutesUniversit´e Libre de Bruxelles, C.P. 231, 1050 Brussels, Belgium Institute Lorentz for Theoretical Physics, Leiden UniversityP.O. Box 9506, Leiden 2300RA, The Netherlands School of Mathematics, Trinity College, Dublin 2, Ireland Abdus Salam International Centre for Theoretical Physics (ICTP)Strada Costiera 11, I 34151 Trieste, Italy
Abstract
We consider the interplay between explicit and spontaneous symmetry breaking in stronglycoupled field theories. Some well-known statements, such as the Gell-Mann-Oakes-Rennerrelation, descend directly from the Ward identities and have thus a general relevance.Such Ward identities are recovered in gauge/gravity dual setups through holographicrenormalization. In a simple paradigmatic three dimensional toy-model, we find analyticexpressions for the two-point correlators which match all the quantum field theoreticalexpectations. Moreover, we have access to the full spectrum, which is reminiscent of linearconfinement. a r X i v : . [ h e p - t h ] F e b ontents d and generic ∆
16B Higher order corrections to GMOR 17
Gaining systematic understanding of symmetry breaking at strong coupling constitutesone of the main open fronts in contemporary Quantum Field Theory investigations. Con-ceptual questions are side by side to diverse applications in motivating this field of re-search. Supersymmetry breaking in strongly coupled hidden sectors and high- T c super-conductors are just two important instances of such a wide range of applications.The purpose of the paper is to delve into the theoretical aspects of symmetry breakingat strong coupling to provide neat and general descriptions of the dynamics in genericcases featuring concomitant explicit and spontaneous breaking. The key theoretical tools,not surprisingly, are the correlators of the gauge invariant quantities and the relationsamong them descending from the original symmetry and its breaking, namely the Wardidentities.We first rely on a purely field theoretical setting illustrating how the Ward identitystructure, supplemented with appropriate and generically valid consistency requirements,can alone provide a description of correlators whose low energy and momentum behavioris characterized by pseudo-Goldstone bosons. As the Goldstone theorem guarantees thepresence of Goldstone modes whenever the breaking of a global internal symmetry ispurely spontaneous [1], similarly the QFT framework is capable of grasping precisely theessential dynamics of pseudo-Goldstone modes whenever the breaking is both explicit andspontaneous; specifically, when the explicit component is parametrically small [2, 3].The standard theoretical difficulties of strong coupling manifest themselves in thecontest of symmetry breaking since both the fields responsible for and arising from thesymmetry breaking dynamics are in general composite and impossible to treat in a per-turbative fashion. A fruitful way of approaching the problem is in gauge invariant, “op-eratorial” terms, where one describes the structure of correlators and expectation values1f observables.The above setting is particularly suitable to be treated in a gauge/gravity context [4–6].The holographic renormalization of a gauge/gravity model precisely implements the Wardidentities satisfied by correlators of the dual strongly coupled field theory [7, 8]. In thisframework we describe one of the simplest holographic models for a prototypical U (1)symmetry breaking allowing for a complete analytic treatment. In this model we are ableto determine exactly and analytically the correlators of the operators in the symmetrybreaking sector. From them we can extract the complete spectrum of composite boundstates. Of particular interest is the lightest mode, i.e. the pseudo-Goldstone boson. Therelevant input scales of the problem are actually the parameters controlling the explicitand spontaneous components of the symmetry breaking, respectively. The output datais represented by the quantities characterizing the spectrum, namely the masses andthe residues of the poles. For the pseudo-Goldstone bosons, these quantities organizeaccording to the celebrated Gell-Mann-Oakes-Renner relation [2] in the limit of smallexplicit breaking. The toy-model has the additional interesting feature of having, in thepurely spontaneous case, a spectrum reproducing linear confinement, i.e. massive boundstates equally spaced in the squared masses.Though the analytic power of the approach is somehow restricted to our specific toy-model, we believe that the qualitative picture generalizes to a full class of holographicbottom-up models encompassing different space-time dimensions and diverse operatorcontent. In particular, the physical understanding emerging from the present analysismakes contact with numerous previous studies in the holographic literature, both in thetop-down [9–13] and in the bottom-up spirit [14–17].The paper is organized as follows. In section 2 we derive purely in field theory andin full generality some properties that the correlators have to satisfy as a consequenceof the Ward identities in the presence of symmetry breaking. One such property is theGell-Mann-Oakes-Renner (GMOR) relation for the pseudo-Goldstone boson (PGB) in thelimit of small explicit breaking. In section 3 we present our three-dimensional toy-modeland show how holographic renormalization reproduces exactly the Ward identities. Westress that only near-boundary analysis is utilized to reach these conclusions. In section4 we solve the bulk equations for the fluctuations and find exact analytical expressionsfor the correlators for any values of the explicit and spontaneous symmetry breakingparameters. We then explore several limits, extracting the spectrum, and showing thatthe correlators satisfy non-trivially all the consistency checks. In the two appendices wepresent respectively the generalization of the holographic model to arbitrary dimensions,and the departure from the GMOR relation when the explicit breaking is no longer small. Consider a field theory which is invariant under a symmetry, which for simplicity we taketo be a U (1). It has a conserved current J µ . We can add to the action a term which2xplicitly breaks the symmetry S tot = S inv + (cid:90) d d x m O φ + c.c. , (1)where O φ is a scalar operator of dimension ∆ which is charged under the U (1). Again forsimplicity, we take its charge to be unity and the explicit breaking parameter m to be real.The current is no longer conserved, rather one has the operator identity ∂ µ J µ = m Im O φ .Furthermore, if in the general case the operator develops a VEV (cid:104) O φ (cid:105) = v , which we taketo be real, the Ward identity (cid:104) ∂ µ J µ ( x )Im O φ (0) (cid:105) = m (cid:104) Im O φ ( x )Im O φ (0) (cid:105) + i (cid:104) Re O φ (cid:105) δ d ( x ) (2)implies that the following two-point functions all depend on a single non-trivial function f ( (cid:50) ), namely (cid:104) Im O φ Im O φ (cid:105) = − if ( (cid:50) ) , (3) (cid:104) ∂ µ J µ Im O φ (cid:105) = − imf ( (cid:50) ) + iv , (4) (cid:104) ∂ µ J µ ∂ ν J ν (cid:105) = − im f ( (cid:50) ) + imv , (5)where we have kept the delta function implicit. The last correlator is just a consequenceof the operator identity.When v = 0 (pure explicit breaking case), then also the second correlator is a trivialconsequence of the operator identity. On the other hand, when m = 0 (pure sponta-neous breaking case), the second correlator is a constant directly determined by the Wardidentity. It implies the presence of a massless pole in (cid:104) J µ Im O φ (cid:105) . The same massless ex-citation has however to appear in (3), though its description requires an analysis of the(IR) dynamics.When both v (cid:54) = 0 and m (cid:54) = 0, we see that (4) has both features, a term related to (3)and a constant term. On the other hand, since the symmetry is broken explicitly, we donot expect a massless mode in the spectrum contributing to this set of correlators. As wewill see, requiring continuity in the m → In momentum space we write (cid:104) Im O φ Im O φ (cid:105) = − if ( k ) , ik µ (cid:104) J µ Im O φ (cid:105) = − imf ( k ) + iv . (6)Note that the dimensions are [ f ] = 2∆ − d , [ v ] = ∆ and [ m ] = d − ∆.Using Lorentz invariance of the vacuum, which imposes (cid:104) J µ Im O φ (cid:105) = k µ g ( k ), thesecond relation leads to (cid:104) J µ Im O φ (cid:105) = − k µ k ( mf ( k ) − v ) . (7) Standard derivations of such a relation can be found, e.g., in [3, 18]. Though close in spirit, thederivation we present in this section is original, to our knowledge. Its starting point is precisely theoutcome of the holographic analysis of the next section. See also, e.g., [11] for the equally standardderivation based on the effective action. Even though (cid:50) corresponds to − k , with a venial abuse of language, we keep denoting the function f with the same symbol also in momentum space.
3e immediately see that there cannot be a massless excitation in the (cid:104) Im O φ Im O φ (cid:105) chan-nel, otherwise there would be a double pole in (cid:104) J µ Im O φ (cid:105) . Moreover, the massless pole inthe above correlator should be spurious, which means that f ( k ) has to satisfy mf (0) − v = 0 . (8)In general, we could wonder whether there exist local finite counter-terms that modify theconstant part of f ( k ) in order to impose the above condition through a scheme choice.This is a question that depends on the specifics of the model, in particular d and ∆. Wewill discuss an example where there are no gauge-invariant, local, finite counter-terms and(8) has to emerge directly and unambiguously from the computations. Note that whatwe have determined until now is true for any values of m and v .For m and k both small with respect to √ v , we can approximate f by a pole corre-sponding to the PGB of mass Mf ( k ) (cid:39) µk + M − µM + vm , (9)where we have implemented the condition (8), and the residue µ is a dynamical quantityof dimension 2∆ − d + 2. We now require that in the m → f ( k ) goes oversmoothly to what we expect in the pure spontaneously broken case. Namely, we expect µ to be (roughly) constant in the limit, as of course v , while M →
0, so that f ( k ) → µk , (10)up to possibly an additive finite constant. From (9) we see that this is possible only ifthere is a relation between all the constants such that M = µv m . (11)This is the generalization of the GMOR relation [2], which indeed states that the squaredmass of the PGB scales linearly with the small parameter which breaks explicitly thesymmetry. The two other constants entering the expression are both of the order of thedynamical scale generating the VEV, i.e. the spontaneous breaking of the symmetry.Note that since µ has to be positive because of unitarity, then the signs (and moregenerally the phases) of m and v have to be correlated in order to avoid tachyonic PGBs.This can be understood by the fact that the small explicit breaking removes the degeneracyof the vacua, and thus the phase of the VEV v is no longer arbitrary but has to be alignedwith the true vacuum selected by m .A last remark is that the usual way in which the GMOR relation is stated is in termsof the residue of the (cid:104) J µ J ν (cid:105) correlator (cid:104) J µ J ν (cid:105) = − iµ J k + M k µ k ν + . . . , (12)where µ J is related to the square of the “PGB decay constant”.4ote that implementing (11) in the correlators we get f ( k ) = µk + M , mf ( k ) − v = − vk k + M , (13)so that, at k = − M , we have µ J M = mvM , (14)which leads to M = m vµ J , (15)namely the usual GMOR relation, which is thus completely equivalent to (11).Above we have kept both d and ∆ arbitrary, and the relation is valid in all generality.In the following we will discuss a specific model where d = 3 and ∆ = 2, so that m hasindeed the dimension of a mass. To illustrate the interplay between explicit and spontaneous symmetry breaking, we useas a toy-model a simplified, Abelian version of the model used in [19, 20], which coincideswith the model of the very first holographic superconductor [21, 22]. The Ward identitystructure emerges through the precise holographic renormalization procedure [7, 8] whichtherefore constitutes our first task.We start considering the action S = (cid:90) d x √− g (cid:26) − F MN F MN − D M φ ∗ D M φ + 2 φ ∗ φ (cid:27) , (16)where D M φ = ∂ M φ − iA M φ , the metric is AdS and we choose the most general profile for φ , namely ds = 1 z ( dz + dx µ dx µ ) , φ b = mz + vz . (17)We keep Lorentz invariance unbroken, and hence we have a vanishing background for A µ . Moreover we have chosen the squared mass of the scalar to be − φ = √ ( φ b + ρ + iπ ), where φ b is assumed to be real for simplicity (and, as we havealready remarked, also for consistency). The rescaling prefactor √ A z = 0 gauge the regularized action up to quadratic order can be written as thefollowing boundary term S reg = − (cid:90) d x (cid:26) − z ∂ z φ b ρ − A µ ∂ z A µ − z ( ρ∂ z ρ + π∂ z π ) (cid:27) . (18)It is possible to rewrite the above expression using the equation of motion coming fromthe variation of (16) with respect to A z , which in the “radial” A z = 0 gauge reads − z ∂ z ∂ µ A µ + iz φ∂ z φ ∗ − iz φ ∗ ∂ z φ = 0 . (19)Linearizing and taking A µ = A tµ + ∂ µ A l (namely splitting the longitudinal and transversepart), it rewrites z ∂ z (cid:50) A l − φ b ∂ z π + ∂ z φ b π = 0 . (20)Noting that the second term of (18) has a longitudinal part that can be rewritten, afterpartial integration, as A l ∂ z (cid:50) A l , the regularized action for the longitudinal part and thescalars becomes S reg = − (cid:90) d x (cid:26) − z ∂ z φ b ρ − z A l ( ∂ z φ b π − φ b ∂ z π ) − z ( ρ∂ z ρ + π∂ z π ) (cid:27) . (21)We now expand the fluctuations as A l = A + A z + . . . , ρ = ρ z + ρ z + . . . , π = π z + π z + . . . . (22)Eq. (21) then becomes S reg = − (cid:90) d x (cid:26) − z mρ − vρ − mρ (23) − z ( ρ + π ) −
32 ( ρ ρ + π π ) + 12 A ( mπ − vπ ) (cid:27) . The counter-term needed to cancel the divergencies is S ct = − (cid:90) d x √− γφφ ∗ = − (cid:90) d x (cid:26) z φ b ρ + 12 z ( ρ + π ) (cid:27) . (24)Note that we neglect the constant term as it would only be relevant with dynamicalgravity. After adding the counter-term (24) to (23), we obtain the renormalized action S ren = − (cid:90) d x (cid:26) − vρ − ρ ρ − π ( π − mA ) − vA π (cid:27) . (25)Let us now discuss gauge invariance. Under a gauge transformation that preserves A z = 0 we have δA l = α, δφ = iαφ . (26)6he first transformation above tells that α should be considered of the same order as thefluctuations A l and ρ, π . It then implies that the gauge variations of ρ, π have actuallyterms of first and second order δρ = − απ , δπ = αφ b + αρ . (27)On the coefficients of the near-boundary expansions (22), the transformations read δA = α , δA = 0 , δρ = − απ , δρ = − απ ,δπ = αm + αρ , δπ = αv + αρ . (28)With the transformations given above, one can check that all the actions S reg , S ct and S ren are gauge invariant. We note that gauge invariance requires the cancellation between thevariations of the linear and quadratic parts of the actions, and we have of course neglectedorders higher than quadratic (i.e. in the variations of the quadratic part of the action,only the terms of first order in the variations of ρ, π are considered).In the renormalized action (25) it is manifest that the ρ sector decouples from the A l , π sector. We will not be concerned with the former, except for the obvious fact thatthe linear, ρ -dependent term gives the VEV of the operator.In order to solve for π in terms of the sources π and A , one should be careful thatthe deep bulk (IR) boundary conditions will impose relations between gauge invariantquantities. At linear order, the gauge invariant combinations are π − mA and π − vA .As a consequence, one can express the subleading mode of π in terms of the sources as π = vA + f ( (cid:50) )( π − mA ) . (29)The renormalized action for this sector can be rewritten accordingly S ren = − (cid:90) d x (cid:104) −
12 ( π − mA ) f ( (cid:50) )( π − mA ) − vA π + 12 mvA A (cid:105) . (30)We observe that we have a term linear in m , which encodes the operator identities that arepresent when the symmetry is explicitly broken. Then we have a term linear in v , whichembodies the Ward identities when the symmetry is spontaneously broken. Eventuallywe have a term which is linear both in m and v and is necessary in order to recover theproper Ward identities in the case of concomitant spontaneous and explicit breaking.We now derive the holographic correlators, assuming that the terms coupling thesources to the operators are (cid:90) d x ( ρ Re O φ + π Im O φ − A ∂ µ J µ ) , (31) It is possible also to parametrize the complex scalar in terms of its modulus and phase as in [24];the latter parametrization, being well-adapted to gauge transformations (which consist in shifts of thephase), features manifest gauge invariance without mixing among different orders in the fluctuations. O φ ,we have (cid:104) Re O φ (cid:105) = δiS ren δiρ = v . (32)For the two-point correlators in the longitudinal sector, we have: (cid:104) Im O φ Im O φ (cid:105) = δ iS ren δiπ δiπ = − if ( (cid:50) ) , (33) (cid:104) ∂ µ J µ Im O φ (cid:105) = − δ iS ren δiA δiπ = − imf ( (cid:50) ) + iv , (34) (cid:104) ∂ µ J µ ∂ ν J ν (cid:105) = δ iS ren δiA δiA = − im f ( (cid:50) ) + imv . (35)These are exactly the equations (3)–(5) obtained from QFT arguments, that we usedto derive the GMOR relations. Now we proceed to compute holographically the non-trivial function f ( k ) and show that it reproduces the physics that one expects on generalgrounds. In this section we study the bulk equations of motion for the fluctuations, in order toextract exact expressions for the correlators. We will thus be able to verify explicitly thatthey satisfy the non-trivial conditions discussed in section 2.We start again from the action (16). Allowing for the moment for a possible backre-action of the scalar’s background profile on the metric, the latter is now defined by ds = g MN dx M dx N = 1 z (cid:0) dz + F ( z ) η µν dx µ dx ν (cid:1) , (36)where the warp factor F ( z ) is such that F (0) = 1 (i.e. asymptotically AdS ) while itdecreases monotonically for increasing z . We assume that φ has a profile along z , thatproduces a non-trivial F through backreaction. In this way we can express the fields interms of fluctuations over the background A M ( z, x ) dx M = A z ( z, x ) dz + A µ ( z, x ) dx µ , (37) φ ( z, x ) = 1 √ φ b ( z ) + ϕ ( z, x )) . (38)We furthermore gauge-fix A z = 0, and we eventually derive the following equations of8otion for the fluctuations z F / ∂ z ( F / ∂ z A µ ) + z F ( (cid:50) A µ − ∂ µ ∂ ν A ν ) − i φ b ∂ µ ( ϕ − ϕ ∗ ) − φ b A µ = 0 ,z F ∂ z ∂ µ A µ + i φ b ∂ z ( ϕ − ϕ ∗ ) − i ∂ z φ b ( ϕ − ϕ ∗ ) = 0 , (39) z F / ∂ z (cid:18) F / z ∂ z ϕ (cid:19) + z F (cid:50) ϕ − i z F φ b ∂ µ A µ + 2 ϕ = 0 . We then split the gauge field in its transverse and longitudinal parts as follows A µ = A tµ + ∂ µ A l , ∂ µ A tµ = 0 , (40)and we define ϕ = ρ + iπ , so that eqs. (39) split into five equations: two equations for A tµ and ρ , respectively, decoupled from each other and from the rest, which we will notconsider further; and a set of three coupled equations for A l and π , namely z F / ∂ z ( F / ∂ z A l ) + φ b π − φ b A l = 0 , (41) z F (cid:50) ∂ z A l − φ b ∂ z π + ∂ z φ b π = 0 , (42) z F / ∂ z (cid:18) F / z ∂ z π (cid:19) + 2 π + z F (cid:50) π − z F φ b (cid:50) A l = 0 . (43)We can extract π from the first equation π = φ b A l − z F / φ b ∂ z ( F / ∂ z A l ) . (44)Note that gauge transformations, at linear order, are given by δA l = α and δπ = φ b α ,where α does not depend on z because of the gauge fixing condition A z = 0. We then seethat both A (cid:48) l and π − φ b A l are gauge invariant quantities.We then plug (44) into (42), and we find a second order differential equation for F / A (cid:48) l ≡ B (as expected from gauge invariance) z B (cid:48)(cid:48) + 2 zB (cid:48) − z F (cid:48) F B (cid:48) − z φ (cid:48) b φ b B (cid:48) + z F (cid:50) B − φ b B = 0 . (45)The system of three equations is therefore reduced to a single second order ODE.We have included backreaction to show that it does not change substantially theequations for the fluctuations. Its effects are subdominant, as we will argue below. Hence,let us consider from now on the case without backreaction, i.e. F = 1. In this case thescalar profile is φ b = mz + vz , where the leading term encodes explicit symmetry breaking,whereas the sub-leading one corresponds to spontaneous symmetry breaking. Eventuallyequation (45) simplifies to B (cid:48)(cid:48) − vm + vz B (cid:48) − k B − ( m + vz ) B = 0 , (46)9nd, by the simple change of variable y = z + mv , we obtain B (cid:48)(cid:48) − y B (cid:48) − k B − v y B = 0 , (47)which can be recast as a general confluent hypergeometric equation. Its solutions are givenin terms of the Tricomi’s confluent hypergeometric function U [ a, b ; x ] and the generalizedLaguerre polynomial L [ a, b ; x ] B ( y ) = exp (cid:20) − vy (cid:21) (cid:18) C U (cid:104) k − v v , −
12 ; vy (cid:105) + C L (cid:104) v − k v , −
32 ; vy (cid:105)(cid:19) . (48)In the deep bulk ( y → ∞ ), we have e − vy U ∼ e − vy whereas e − vy L ∼ e + vy . Since ∂ z A l isgauge-invariant, we are allowed to impose IR boundary conditions on it, and we choosebulk normalizability of the solution setting C ≡
0. We thus obtain B ( y ) = C e − v y U (cid:104) k − v v , −
12 ; vy (cid:105) . (49)Note that this solution has a very fast decrease towards the interior of the bulk, confirmingthat backreaction will only affect mildly the correlators that we will extract from it.In this way we have obtained an exact analytical solution for the derivative of thegauge field, but we still have to derive a solution for π , in order to compute the scalarcorrelator. If we consider the near-boundary expansion for the fluctuations π = z π + z π + . . . , (50) A l = A + z A + z A + z A + . . . , (51)then we need to know the expressions for π and π in order to compute the scalarcorrelator. Indeed, from (29) and (33), we see that (cid:104) Im O φ Im O φ (cid:105) = − i δπ δπ = − if ( k ) . (52)In other words, the correlator is essentially extracted from (29), that we rewrite here as π − v A = f ( k ) ( π − mA ) . (53)From equation (44) with F = 1, we can express the gauge invariant combination appearingin eq. (53) in terms of A l π − φ b A l = − z φ b A (cid:48)(cid:48) l ( z ) . (54)Order by order near the boundary, through the expansions (50)–(51), we obtain π − mA = − m A ; (55) π − v A = m A + vm A − m A . (56)10e can then realize that A , A and A can be associated to B , B (cid:48) and B (cid:48)(cid:48) evaluated at z = 0, or equivalently at y = mv , in the following way B ( x/ √ v ) = A (cid:48) (0) = A ,B (cid:48) ( x/ √ v ) = A (cid:48)(cid:48) (0) = 2 A ,B (cid:48)(cid:48) ( x/ √ v ) = A (cid:48)(cid:48)(cid:48) (0) = 6 A , with x ≡ m √ v . Thus we can establish the expression for f in terms of B and its derivatives, f ( k ) = π − v A π − mA = − vm + B (cid:48)(cid:48) ( x/ √ v ) − vx B ( x/ √ v ) B (cid:48) ( x/ √ v ) . (57)We can then express the correlator in terms of Tricomi functions (cid:104) Im O φ Im O φ (cid:105) = i x ( k − v ) (cid:16) v U (cid:104) k +3 v v , ; x (cid:105) + ( k + 3 v ) U (cid:104) k +7 v v , ; x (cid:105)(cid:17) √ v (cid:16) v U (cid:104) k − v v , – ; x (cid:105) + ( k − v ) U (cid:104) k +3 v v , ; x (cid:105)(cid:17) . (58)Let us show now how this expression reproduces all the physical features required bythe field theory analysis. First of all, in the limit of zero momenta, f ( k ) as given in (57)satisfies relation (8), i.e. f (0) = vm . This can be easily seen by using (47) in order toobtain f ( k ) = vm + k B ( x/ √ v ) B (cid:48) ( x/ √ v ) . (59)Moreover, we can graphically find the poles of the propagator by plotting the correlatorfor specific values of the ratio x = m √ v . For instance, with x = 0 .
01, that is spontaneousbreaking dominating on explicit breaking, we find a first pole close to zero (see fig. 1),and then a complete spectrum of higher massive poles with a gap considerably biggerthan the mass of the first pole (see fig. 2). This is the hallmark of a pseudo-Goldstoneboson. Furthermore, the gapped spectrum presents an interesting feature that we willshow analytically for the purely spontaneous case in the next section: the poles areseparated by a regular gap in squared mass (except for the first higher pole after the PGB,which exhibits a slightly bigger gap from the rest of the spectrum). This is reminiscentof linear confinement. In addition, we are able to analytically find the GMOR linear relation (11). Indeed,one finds that the numerator of expression (58) is just a constant in the limits k → x → x and afterwards to the first order in k , it vanishes for k = − √ v Γ[ ]Γ[ ] m , (60) Indeed a phenomenological model like [16], that is designed in order to achieve linear confinement,also ends up having confluent hypergeometric equations for the bulk fluctuations. .01 (cid:45) k i (cid:60) Im O Im O (cid:62) Figure 1:
The lightest pole (PGB) in (cid:104) Im O φ Im O φ (cid:105) , for v = 1 and x = 0 .
01, which is ofthe order of x . (cid:45) k i (cid:60) Im O Im O (cid:62) Figure 2:
The first poles of the spectrum in (cid:104) Im O φ Im O φ (cid:105) , for v = 1 and x = 0 .
01; theyexhibit a gap of the order of 5 v with respect to the first pole (PGB). where Γ is the Euler function. So we have found the explicit value for the residue µ appearing in (11) for the specific model at hand, namely µ = vM m = 2 v / Γ[ ]Γ[ ] . (61)We are also able to find the deviations from the linear GMOR behavior to the desiredorder in m √ v , as we show in appendix B.Let us underscore that expression (57) is valid not only for small m . We can thentake x (cid:29) m and it12s pushed towards the rest of the spectrum, as can be seen in fig. 3. Actually, the ratiobetween the first gap and the subsequent ones increases with x , so that if one keeps thefirst pole fixed, the other poles will be increasingly dense just after it. This is the signalthat a cut is emerging in the x → ∞ limit, i.e. in the purely explicit case. (cid:45) k i (cid:60) Im O Im O (cid:62) Figure 3:
The low | k | portion of the spectrum of (cid:104) Im O φ Im O φ (cid:105) , for v = 0 . x = 10;the first pole is of the order of m = 100 v = 10. In the next subsections we make further comments on the sub-cases of purely sponta-neous or purely explicit breaking.
For purely spontaneous breaking, i.e. m = 0, equation (46) becomes B (cid:48)(cid:48) − z B (cid:48) − k B − v z B = 0 . (62)This is the same as (47), but directly in the z variable. Its solution is given in (49), wherenow B is a function of z B ( z ) = C e − v z U (cid:104) k − v v , −
12 ; vz (cid:105) . (63)In this case π is gauge invariant by itself, so f ( k ) = π − vA π , (64)and by using the equations of motion (41) (for F = 1 and φ b = vz ), we obtain (cid:104) Im O φ Im O φ (cid:105) = − i f ( k ) = − i B (cid:48)(cid:48)(cid:48) (0)2 B (cid:48)(cid:48) (0) = 8 i v k Γ (cid:104) k +5 v v (cid:105) Γ (cid:104) k − v v (cid:105) , (65)13hich, in the limit of small momenta, actually exhibits a massless pole, signature of theexpected Goldstone boson, (cid:104) Im O φ Im O φ (cid:105) ∼ − i v Γ[ ]Γ[ ] 1 k . As a double check, we can recover the same result of eq. (65) by taking the limit m → m n = (5 + 4 n ) v , with n being a non-negative integer. As anticipated in the previous section (see fig. 2),this spectrum presents the feature of equally gapped poles, except for the first massiveone, whose gap from zero is bigger than the others by one unit in v . For v = 0 the equation reduces to B (cid:48)(cid:48) − k B − m B = 0 . (66)Note that the limit of vanishing scalar profile is trivially achieved putting m = 0 in theabove equation, and in its solutions. The solutions are B = C + e √ k + m z + C − e −√ k + m z . (67)Bulk normalizability imposes C + = 0. The gauge invariant combination is π − mzA l = − zm B (cid:48) = C − √ k + m m (cid:16) z − z √ k + m + . . . (cid:17) . (68)From this we read π − mA = C − √ k + m m , π − mA = − C − k + m m . (69)We can extract A directly as the constant term of B , so that A = C − . This gives π = − C − k m = − k √ k + m ( π − mA ) . (70)Finally the correlator is given by (cid:104) Im O φ Im O φ (cid:105) = − i δπ δπ = i k √ k + m . (71)It presents a cut starting after a gap given by m . This is what was expected from the m/ √ v → ∞ limit of the correlator in the general case. Note that when m = 0 we obtainthe conformal result (cid:104) Im O φ Im O φ (cid:105) = ik , with a cut as well, but without any gap.It is important also to notice that (71) goes as k for small k , which is necessary toensure that the correlator (cid:104) J µ Im O φ (cid:105) does not have a spurious massless pole.14 Conclusion
The present paper systematizes knowledge that was in part already present in the litera-ture but in a scattered fashion. Our considerations naturally split in three blocks. Firstwe rely on purely quantum field theoretical arguments to determine the Ward identitystructure expected on general grounds in the presence of a U (1) symmetry breaking. Theanalysis encompasses the generic case where the breaking can be explicit, spontaneousor concomitantly explicit and spontaneous. Consistency arguments pinpoint the Wardidentity structure independently of the strength of the coupling, encoding the symmetrybreaking pattern at the operatorial level. In particular, neither the explicit knowledge ofthe QFT Lagrangian nor that of the actual microscopic degrees of freedom are needed.This approach is able to encompass the generically composite nature of the fields respon-sible and emerging from the symmetry breaking at strong coupling. Furthermore it allowsfor both a qualitative and quantitative control on the Goldstone modes and their pseudorelatives. In fact, their masses and residues are constrained by the Ward identities andwe show the validity in full generality of Gell-Mann-Oakes-Renner type of relations whichrelate the (pseudo)-Goldstone pole structure to the parameters of the symmetry breaking.We then shift to holography to show how the Ward identity structure and symmetrybreaking pattern can be neatly embodied in a simple and paradigmatic toy-model. Theprecise relations among the correlators are realized by the holographic renormalization ofthe gauge/gravity model and rely on just an asymptotic near-boundary analysis. Thismeans that, in order to describe the Ward identities, only UV knowledge is necessary.The analysis can therefore be performed before actually solving the model and discussingits IR properties.In turn, in order to access quantitative data such as masses and residues, the IRproperties are crucial and hence solving for bulk fluctuations becomes necessary. Wethus explicitly study the toy-model which allows for complete analytic control of its so-lutions and the dual correlators. Holography, already in one of its simplest realizations,is therefore able to reproduce general quantum field theoretical expectations and allowsfor explicit quantitative computations. We expect that the results of this analytic studyremain qualitatively true in general for the entire class of toy-models in different space-time dimensions and featuring U (1) breaking by means of charged operators of differentscaling dimension.The present study of pseudo-Goldstone modes has potentially far reaching future per-spectives when applied to other kinds of symmetries. For instance, it would be interestingto consider supersymmetry breaking in holography along the lines of [25–27], and theemergence of a putative pseudo-Goldstino. One should also consider non-relativistic set-ups, such as in the presence of temperature and/or chemical potentials. There is also theappealing possibility of considering directly the breaking of space-time symmetries whichcommute with the Hamiltonian. Regarding this latter possibility, it is very interestingto study translation symmetry breaking. This corresponds, when explicit, to a dual Of particular interest in this context is the line of holographic studies initiated in [28] featuringspecific mass terms for the bulk graviton which break spatial diffeomorphisms. Later similar models were
Acknowledgements
We would like to thank Lasma Alberte, Andrea Amoretti, Daniel Arean, Matteo Baggioli,Matteo Bertolini, Francesco Bigazzi, Aldo Cotrone, Johanna Erdmenger, Manuela Kulax-izi, Nicodemo Magnoli, Andrei Parnachev, Oriol Pujolas, Diego Redigolo, Javier Tarrioand Giovanni Villadoro for useful and enjoyable conversations and exchanges on the topicsof the paper. RA and AMa would like to thank the Galileo Galilei Institute for hospital-ity during completion of this work. This research is supported in part by IISN-Belgium(convention 4.4503.15), by the “Communaut´e Fran¸caise de Belgique” through the ARCprogram and by a “Mandat d’Impulsion Scientifique” of the F.R.S.-FNRS. RA is a SeniorResearch Associate of the Fonds de la Recherche Scientifique–F.N.R.S. (Belgium). Thework of AMe ( £ ) is supported by the NWO Vidi grant. A Generic d and generic ∆ In this appendix we consider the analogues of eqs. (41)–(43) in the case of generic d andgeneric ∆, given by m φ = ∆(∆ − d ). The analysis shows that when we move away fromthe d = 3, ∆ = 2 case studied in the main text, an analytic treatment of the equations ofmotion, whenever possible, is more involved. The generalized equations of motion read z d − F d/ − ∂ z (cid:0) z − d F d/ − ∂ z A l (cid:1) + φ b π − φ b A l = 0 , (72) z F (cid:50) ∂ z A l − φ b ∂ z π + ∂ z φ b π = 0 , (73) ∂ z (cid:20) F d/ z d − ∂ z π (cid:21) − F d/ z d +1 m φ π + F d/ − z d − ( (cid:50) π − φ b (cid:50) A l ) = 0 . (74)By defining B ≡ z − d F d/ − ∂ z A l , (75) realized by means of neutral scalars through a Stueckelberg mechanism which allows for a Ward identityprecisely accounting for the translation breaking [29]. Along these lines, different further analyses havetackled or commented the possibility of having phonons in holography, see for instance [30–32].
16e can derive π from eq. (72) and plug it into eq. (73), obtaining z B (cid:48)(cid:48) + z (cid:20) − z φ (cid:48) b φ b + d − (cid:18) − d (cid:19) F (cid:48) F z (cid:21) B (cid:48) + (cid:16) z (cid:50) F − φ b (cid:17) B = 0 . (76)Therefore, by setting F = 1 and by writing explicitly the scalar background φ b = mz d − ∆ + vz ∆ , (77)we have z φ b B (cid:48)(cid:48) + z (cid:2) m (2∆ − d − z d − ∆ + v ( d − − z ∆ (cid:3) B (cid:48) + (cid:0) z φ b (cid:50) − φ b (cid:1) B = 0 . (78)It is easy to see that the manipulations that have been used to recast (46) into a confluenthypergeometric form depend very much on d = 3 and ∆ = 2. Thus the generic case,namely in the presence of concomitant explicit and spontaneous breaking, will not havesimple analytic solutions such as the ones of our toy-model. B Higher order corrections to GMOR
In this appendix we discuss the corrections to the GMOR relation. First, we derivethe corrections in the small m expansion of the GMOR relation given in (60). Usingan iterative procedure, it is possible to obtain the analytic form of the first pole of thecorrelator (58) up to the desired order in the small m expansion. To do so, we expand k as follows k = N (cid:88) i =1 Ξ i m i , (79)and we solve order by order the equation (cid:104) Im O φ Im O φ (cid:105) − = 0 . (80)At first order we obtain the GMOR relation (60) as expected, while by pushing furtherthe analysis we obtainΞ = 1 − π Γ (cid:2) (cid:3) Γ (cid:2) (cid:3) , Ξ = 2 √ π Γ (cid:2) − (cid:3) + 8 ( −
32 + 16 c + 3 π ) Γ (cid:2) (cid:3) √ v Γ (cid:2) − (cid:3) , (81)Ξ = 16 π ( −
56 + 24 c + 3 π ) Γ (cid:2) − (cid:3) − π ( −
96 + 48 c + 5 π ) Γ (cid:2) (cid:3) v Γ (cid:2) − (cid:3) , where c (cid:39) . m/ √ v regime.17 (cid:45) k Figure 4:
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