Banishing AdS ghosts with a UV cutoff
NNSF-KITP-11-267
Banishing AdS ghosts with a UV cutoff
Tom´as Andrade and Donald Marolf
Department of Physics, UCSB, Santa Barbara, CA 93106, USA
Thomas Faulkner
KITP, Santa Barbara, CA 93106, USA
Abstract
A recent attempt to make sense of scalars in AdS with “Neumann boundary conditions” outsideof the usual BF-window − ( d/ < m l < − ( d/ + 1 led to pathologies including (dependingon the precise context) either IR divergences or the appearance of ghosts. Here we argue thatsuch ghosts may be banished by imposing a UV cutoff. It is also possible to achieve this goal incertain UV completions. An example is the above AdS theory with a radial cutoff supplemented byparticular boundary conditions on the cutoff surface. In this case we explicitly identify a region ofparameter space for which the theory is ghost free. At low energies, this theory may be interpretedas the standard dual CFT (defined with “Dirichlet” boundary conditions) interacting with an extrascalar via an irrelevant interaction. We also discuss the relationship to recent works on holographicfermi surfaces and quantum criticality. a r X i v : . [ h e p - t h ] D ec . INTRODUCTION AdS/CFT relates a set of Conformal Field Theories to gravitational theories in AdS [1–3].Interesting field theory dynamics follows from simple relevant deformations of these CFTs.The inclusion of multi-trace deformations has lead to many results [4–8], and in particularto recent attempts to drive a theory across a quantum phase transition [9–12]. In addition,the role of multi-trace deformations in the holographic renormalization group has recentlybeen emphasized in [13, 14] (see also [15]). As a result, one would like to have as completean understanding as possible of which multi-trace deformations are allowed, and when theycan lead to useful dynamics.Linear scalars in AdS offer a good starting point for such analyses. Within the BFwindow − ( d/ < m l < − ( d/ +1 there are two possible boundary conditions preservingconformal invariance [16], often called the standard and alternate quantizations [6]. Thesefixed points are characterized by the existence of a single trace operator with dimensions d/ ν and d/ − ν respectively, where ν = m l + ( d/ . From the bulk perspective,it is natural to think of these as generalized Dirichlet and Neumann boundary conditionsrespectively. There are many other boundary conditions that do not preserve conformalinvariance but which correspond to multi-trace deformations of the aforementioned choices[4, 5]. For example, when it is relevant, the double trace deformation leads to an RG flowbetween the alternative and standard theories with the former being a UV fixed point andthe later an IR fixed point.The obstruction to playing these games for ν > d/ − ν < d/ − and the addition of double-trace operators. It is thus hard to make sense of this By taking it to be a cylinder [17], de Siter space, or anti-de Sitter space [18]. S (cid:48) = S ( std ) CF T + 12 (cid:90) d d x (cid:0) − κ ( ∂ Φ) − λ Φ + . . . (cid:1) + S int , S int = g (cid:90) d d x ˆ O Φ (1)where S CF T denotes the action of the dual CFT which contains an operator ˆ O of dimension∆ = d/ ν . Note that the BF window corresponds to 0 < ν <
1. The free operatordimension of Φ is ( d − / g is 1 − ν ; thus the interaction term is relevant for 0 < ν < g/λ ) ˆ O and upon substitution in the action one finds S (cid:48) = S ( std ) CF T + (cid:90) d d x g λ ˆ O , (2)which is just a double trace deformation of the CFT in standard quantization. Furthermore,sending λ → < ν < κ = 0) and sending λ →
0. The modeΦ, which is being integrated over, plays the role of an operator in S ( alt ) CF T with dimension d/ − ν (from power counting with respect to the coupling term after setting [ g ] = 0). Italso enacts the Legendre transformation which relates the two theories [6].Let us attempt to continue these arguments to ν >
1. It is no longer valid to integrateout Φ due to the importance of the kinetic terms. Indeed, since g → κ > λ should be allowed as λ < (cid:104) Φ (cid:105) (cid:54) = 0 without pathology, at least so long as appropriate higher order interactionterms (such as Φ ) are present . Note that this condensation will have a residual effect on the CFT through an irrelevant interaction. κ > N factorization (see for example[19]), is given by G Φ ( p ) = 1 − κp − λ − g G O ( p ) . (3)Here G O is the two point function of O in the interacting CFT. Conformal invariancefixes G O = c ν ( p ) ν (where p = − ω + (cid:126)p ) and the condition that the spectral densityIm G O ( ω + i(cid:15), (cid:126)p ) be positive for ω > c ν sin( πν ) >
0. Let us now examine G Φ for potential ghosts. For simplicity, we restrict to the case 1 < ν < c ν < ν . This case was studied explicitly in [17], whichshowed that ghosts arise for all values of λ and κ (though we discuss only κ > λ > p (cid:63) >
0. Expanding G Φ ( p ≈ p (cid:63) ) around thispole one can show that it has a negative residue. For λ < λ merge and move into the complex plane. In the real case one of thesetwo poles is a ghost while the other is a non-ghost tachyon. As usual, the complex casenecessarily contains a ghost.On the other hand, it is clear that no ghost is present for g = 0 . Studying the change inthe corresponding pole perturbatively in g would not have indicated the presence of ghosts.This suggests that the ghosts correspond to new poles that enter from p = ∞ and thus that,at least in some sense, they are a UV issue. Indeed, since the coupling between the CFTand Φ is governed by an irrelevant interaction we expect to run into problems at energyscales above: p > Λ g = ( g/κ / ) / (1 − ν ) . (4)One can show that the ghost found using (3) for κ > | p (cid:63) | > N ν Λ g where N ν = ( − νc ν ) − ν − is a number which depends only on ν . So it is natural to expect thatcutting off (or appropriately modifying) the theory at p > N ν Λ g will banish our ghosts.The purpose of the present paper is to construct examples in which this can be demon-strated precisely. But let us first comment on some related examples already known in theliterature. The low energy theory for the fermions analyzed in [20–22] and identified in [19]was given by an action similar to (1). The free fermion plays the role of our free scalar above, This explicit bound corresponds to the value of p/ Λ g which maximizes the expression κp + g G O ( p ),associated with the case λ = 0. AdS × R (or an interesting generalization thereof) . Although the details are different, there wereagain two interesting cases distinguished by conditions analogous to the cases 0 < ν < ν > AdS and AdS × R , with the transition happening at an energy scale µ set bythe chemical potential. This µ provides an effective UV cutoff on the AdS × R theory.The kinetic terms (analogous to κ ) and g were computed in [22] and one may check thatthey satisfy Λ g (cid:38) µ . As a result, the above prediction of ghosts (based on analyzing the lowenergy action) is not reliable and one must instead consider the full RG flow.In this way, the action (1) may generally be taken to model the IR regime of a domainwall flow between two different scale invariant fixed points. The low energy theory thennaturally comes with a cutoff Λ; the scale where the domain wall begins to deviate fromthe IR fixed point. So long as we start with a good theory in the UV, we expect the full theory to be ghost-free. But it is easy to engineer models in which the IR fixed point has afield satisfying ν > ν ) subject to an irrelevant double-tracedeformation. In this case our discussion above implies that the low-energy effective kineticterms and the low energy coupling will satisfy Λ g (cid:38) Λ.The problem of the existence of negative norm states can be studied systematically ona case by case basis. Here we take a much simpler approach and study the
AdS theorywith a radial cutoff. This problem is then a simple generalization of the analysis in [17]whose results will confirm the above intuition. This in turn increases one’s confidence in thetheories studied in [9, 11, 12, 19]. The generalization of fermions to scalars in the extremal charged black hole background was consideredin [9, 11, 12] and a similar discussion applies. radial cut-off. Whilethis is not equivalent to a UV cut-off (since arbitrarily high momenta along the boundaryare still allowed), it corresponds to a non-trivial (and non-local, see e.g. [14]) deformationof an appropriate dual CFT defined by removing the radial cut-off. This theory is easy tostudy and ghost-free, but it is ill-defined at the quantum level due to an IR divergence in thetwo-point function (of the sort seen in [23], [17]). Section III then studies a two-parameterfamily of (quadratic) deformations of our reference theory. It was shown in [17] that, withoutthe radial cut-off, these deformations remove the IR divergence but also introduce ghosts.Nevertheless, we show that (at least in a certain regime of parameter space) the ghostsmay be banished by imposing a suitably strong radial cut-off. We close with some finaldiscussion in section IV, which in particular shows that the models of section III suffice togive a ghost-free UV-modified version of all models studied in [17] for which a certain UVcoupling is positive.
II. REFERENCE SYSTEM WITH RADIAL CUTOFF
As stated above, the explicit model that we will study is that of a scalar field φ on(Poincar´e) AdS d +1 . We impose a radial cut-off at some r = r in coordinates associatedwith the metric ds = dr r + 1 r η ij dx i dx j . (5)In particular, we take r ∈ ( r , ∞ ) and note that r = ∞ is the Poincar´e horizon. We focus onthe mass range 1 < ν < r = 0 (no cut-off), we write the action in a form thatparallels the r = 0 action for Neumann boundary conditions (see [17]), I Ref = I + (cid:90) ∂M √ γ (cid:20) ρ µ ∂ µ φφ − ∆ − φ + 14( ν − γ ij ∂ i φ∂ j φ (cid:21) , (6)where I = − (cid:82) M √ g [ g µν ∂ µ φ∂ ν φ + m φ ], ∆ − = ( d/ − ν ), ∂M denotes the surface r = r and ρ µ is the unit normal to this surface (we denote the normal derivative by ∂ ρ below).6he boundary conditions must be chosen to make I Ref stationary. Varying (6) with respectto φ we obtain the boundary condition ∂ ρ φ = ∆ − φ + 12( ν − (cid:50) γ φ at r = r . (7)Noting that ∂ ρ φ = r∂ r φ , (cid:50) γ φ = r (cid:50) φ and that at small r the field φ has the asymptoticexpansion φ = r d/ − ν ( φ (0) + r φ (1) + r ν φ ( ν ) + . . . ) with φ (1) = 14( ν − (cid:50) φ (0) , (8)we can readily verify that (7) reduces to φ ( ν ) = 0 in the limit r →
0. Here (cid:50) is theD’Alembertian associated with the flat boundary metric, i.e. (cid:50) = η ij ∂ i ∂ j .Using the prescription of [24], we can read off the inner product associated with the action(6), including necessary contributions from the boundary kinetic terms on ∂M . We take thebulk Klein-Gordon current associated with a pair of solutions φ , φ to be j bulkµ = i φ ∗ ↔ ∂ µ φ , (9)and introduce a corresponding boundary current j bndyj = i φ ∗ ↔ ∂ j φ , (10)where A ↔ ∂ B = A∂B − B∂A and the index j ranges only over boundary directions. Therenormalized inner product is then simply( φ , φ ) = ( φ , φ ) bulk − ν −
1) ( φ , φ ) bndy , (11)where ( φ , φ ) bulk , ( φ , φ ) bndy are given by introducing some surface Σ with boundary ∂ Σ at r = r , contracting the currents (9), (10) with either the normal n µ to Σ or the normal n µ∂ to ∂ Σ within the surface r = r , and integrating over Σ or ∂ Σ using the volume measureinduced by (5).
A. Spectrum
In order to solve the wave equation, we shall use the mode decomposition φ = e ik · x ψ k ( r ) , (12) The explicit variation is of the form δI Ref = (cid:82) ∂M φδ b . c . so that the b.c. plays the role of a source in thedual theory. k i = ( ω, k ) and ψ k ( r ) is a radial profile that depends on the eigenvalue of (cid:50) , whichwe will denote as the “boundary mass”, i.e. m bndy := − k i k i . This eigenvalue may be used toclassify the modes as ( m bndy > m bndy = 0) and space-like or tachyonic ( m bndy < m bndy , and refer to the associated modesas “complex tachyons” below.Let us first consider the time-like solutions. In this case, a general mode can be written ψ = φ ( ν ) ψ + + φ (0) ψ − , (13)where φ (0) and φ ( ν ) are arbitrary constants and ψ + = C − ν r d/ J ν ( m bndy r ) ψ − = C ν r d/ J − ν ( m bndy r ) , (14)with C ν = 2 − ν Γ(1 − ν ) m νbndy . (15)Here J ν ( x ) are Bessel functions of the first kind. The radial profiles (14) oscillate rapidly nearthe Poincar´e horizon and it can be shown both solutions are plane-wave normalizable withrespect to the inner product (11), see e.g. [17]. Thus time-like modes form a continuum andexist for all values of r . The solution is completely specified by noting that the boundarycondition (7) imposes a r -dependent relation between φ (0) and φ ( ν ) , whose explicit form willnot be important for the moment. The norm of these modes follows from expression (11)and can be computed by the methods of [17] . This quantity is positive definite for all r and is given by ( φ , φ ) = (2 π ) d − δ ( d ) ( k i − k i ) | φ (0) k C ν,k + e iπν φ ( ν ) k C − ν,k | . (16)As stated above, the coefficients φ (0) and φ ( ν ) are related by the boundary conditions so that(16) is fixed up to a normalization constant. Since for r → φ ( ν ) →
0, the UV behavior of (16) is guaranteed to agree with the Neumann resultof [17]. Integrating by parts reduces the inner product to a sum of boundary terms at r = r and r = ∞ . But aself-adjointness argument requires the result to be proportional to a Dirac delta-function, which can comeonly from the region near the horizon where the modes are plane-wave normalizeable. It follows that onlythe asymptotics near r = ∞ are needed to compute the inner product.
8n the other hand, using the boundary condition to express φ (0) in terms of φ ( ν ) for small m bndy one finds( φ, φ ) | φ ( ν ) | ≈ ν Γ(1 + ν ) (2 π ) − d m − νbndy + O (1) , ( φ, φ ) | φ (0) | ≈ − ν Γ(1 − ν ) (2 π ) − d m νbndy + O (1) , (17)which coincide respectively with the Dirichlet and Neumann results for r = 0 to leadingorder in m bndy . As expected, the leading small momentum behavior is not modified by theradial cut-off at r . But the second expression in (17) means that our reference theory suffersfrom the same IR divergence in the bulk two-point function identified in [17] for r = 0 (thisdivergence also appeared in the pure CFT context in [23]). Thus the theory is ill-defined atthe quantum level.Let us nevertheless complete the mode analysis for this theory. We next consider thelight-like modes, i.e. m bndy = 0, whose general profile is ψ = Ar d/ − ν + Br d/ ν , (18)where A and B are arbitrary constants. The boundary condition (7) then implies B = 0.One can check that light-like modes (18) with B = 0 are normalizable for ν > m bndy := − p <
0. Byconvention, we restrict ourselves to Re p >
0. With this choice, the normalizable solutionat the Poincar´e horizon is ψ T = r d/ K ν ( pr ) , (19)where K ν ( x ) is the modified Bessel function of the second kind. The boundary condition(7) then yields K ν − ( pr ) = 0 which, provided Re p >
0, has no solutions anywhere in thecomplex plane [25]. It follows that there are no tachyonic solutions.To summarize, our reference theory is ill-defined at the quantum level due to an IRdivergence in the two-point function. This divergence is associated with the presence of nullstates (the light-like modes). However, the theory has no negative-norm states. One maytherefore hope that a suitable IR modification will render the theory well-defined withoutintroducing ghosts. We exhibit a two-parameter family of such deformations in section IIIbelow. 9
II. DEFORMED THEORY
We now deform the action (6) by considering I = I Ref + I def with I def = − ν (cid:90) ∂M √ γr ν (cid:20) κr ( ∂φ ) + λφ (cid:21) , (20)where all the quantities are taken to be tensors with respect to γ . This parametrization ofboundary couplings behaves smoothly in the limit r → r = 0) such deformations always give rise to ghosts. But below we will see that for any κ > r sufficiently large. Note that stationarity of the deformed action requires theboundary condition ∂ ρ φ − (∆ − + 2 νλr ν ) φ − (cid:20) ν − − νκr ν − (cid:21) (cid:50) γ φ = 0 at r = r . (21)It should be noted that the deformation term (20) contains a new boundary kinetic term,so that it modifies the boundary symplectic current. As a result, the total inner productreads ( φ , φ ) = ( φ , φ ) bulk − (cid:20) ν − − νr ν − κ (cid:21) ( φ , φ ) bndy . (22)Below, our main focus will be to find a region in the space of parameters ( λ, κ ) that isghost-free. To do so, we shall concentrate in the tachyonic modes, since, as shown in [17],time-like and light-like modes necessarily have non-negative norms for all κ > κ < κ > A. Existence of Tachyons
We now study the existence of tachyonic solutions as we vary r holding fixed λ and κ .As above, we define p = − m bndy < p > ψ = r d/ K ν ( pr ) . (23)10ntroducing q = pr , the boundary condition (21) implies K ν − ( q ) K ν ( q ) = ˆ κ + ˆ λ/q , (24)where κ c = ν ( ν − r ν − , ˆ κ = κ/κ c , and ˆ λ = λr /κ c .To analyze (24), it is useful to note the following facts. First, the asymptotic form of K µ ( q ) for fixed µ at large | q | is K µ ( q ) = (cid:114) π q e − q (cid:20) µ − q + O ( | q | − ) (cid:21) . (25)Hence, letting q = Re iθ we have for large RK ν − ( q ) K ν ( q ) ≈ − ν ) R (cos θ − i sin θ ) + O ( R − ) . (26)Second, for q ≈ µ >
0, we have K µ ( q ) ≈ Γ( µ )( q ) − µ . In order to use thisexpression for ν − <
0, we note that K − µ ( q ) = K µ ( q ). It follows that for small R we canwrite K ν − ( z ) K ν ( z ) ≈ − ν ) Γ(2 − ν )Γ( ν ) R ν − { cos[2( ν − θ ] + i sin[2( ν − θ ] } . (27)The behavior of the real and imaginary parts of the ratio of the two relevant Bessel functionsis plotted in figures 1(a) and 1(b). With these observations in mind, let us go back to (24). R e (cid:64) K Ν (cid:45) (cid:72) R e i Θ (cid:76)(cid:144) K Ν (cid:72) R e i Θ (cid:76) (cid:68) (a) I m (cid:64) K Ν (cid:45) (cid:72) R e i Θ (cid:76)(cid:144) K Ν (cid:72) R e i Θ (cid:76) (cid:68) (b) FIG. 1: On the left we plot Re K ν − ( q ) K ν ( q ) vs. R for ν = 1 . θ = 3 / π . This function is invariantunder θ → − θ . On the right we plot Im K ν − ( q ) K ν ( q ) vs. R for ν = 1 . θ = 7 / π . This functionchanges sign under θ → − θ . The peak is smaller for smaller values of ν Let us first show that there are no tachyons at complex momenta for λ >
0. To do so,we let q = Re iθ with | θ | < π/
2, so (24) readsRe K ν − ( q ) K ν ( q ) = ˆ κ + ˆ λR cos(2 θ ) , (28)11m K ν − ( q ) K ν ( q ) = − ˆ λR sin(2 θ ) . (29)Now, using (26) and the fact – justified by numerics – that Im K ν − ( q ) K ν ( q ) has no zeroes or polesfor Re q >
0, we conclude that Im K ν − ( q ) K ν ( q ) is bounded and positive definite for 0 < θ < π/ − π/ < θ <
0. For ˆ λ > θ (cid:54) = 0, the left and right hand sideof (29) have different signs for all R . Thus there are no complex solutions.Consider now q ∈ R . It is not hard to show that the left hand side of (24) rangesmonotonically over (0 ,
1) as q varies between (0 , ∞ ). Thus, for λ >
0, it follows that (24)has one and only one real solution if ˆ κ < κ < κ c ) and no solutionsotherwise. Recalling the definition of κ c , we conclude that for λ > κ > r is sufficiently large. Thus, at least in this regime, the resultingtheories are both well-defined and ghost-free.To make contact with the introduction note that the the condition for a ghost free spec-trum ˆ κ > r − (cid:46) κ / ν − ≡ Λ g (30)where we have appropriately set g = 1 in the expression (4) for Λ g . So as long as the cutoffenergy scale r − is smaller than Λ g the theory is ghost free. B. Complete Analysis
For completeness, we now analyze the case λ < λ ). Though our arguments above were largely analytic, we relyon simple numerics below to establish some general trends.We begin with the case λ <
0, ˆ κ >
1. For real q , it is easy to see that there is one realtachyon (at some positive q ). But numerical investigation shows that there are no complexsolutions; see figure 2(a). On the other hand, due to a new branch of solutions to (28) that This involves using the above expansions to evaluate the LHS of (24) at large and small real z > W ν ,ν = z ( K ν ∂ z K ν − K ν ∂ z K ν ) for ν < ν . To show positivity of W ν ,ν , one uses the Bessel equationto show that W ν ,ν is strictly decreasing for ν < ν and real z >
0. The argument is completed bynoting that (25) implies W ν ,ν > z . κ = 1, for ˆ κ < κ/ ˆ λ . See figure 3(a). (cid:45) (cid:45) (cid:45) Ω I m Ω (a) (cid:45) (cid:45) Ω I m Ω (b) FIG. 2: The case ˆ κ ≥ , ˆ λ <
0. We plot numerical solutions of (28) – dashed and dotted lines –and (29) – solid lines, including both the straight lines along the real axes and the rough circles.A simultaneous solution to both equations would requires these curves to intersect. Since theintersection at q = 0 corresponds to the light-like modes already studied (and is not a tachyon),there is a single real tachyon in each case shown. Figure (a) shows results for ˆ λ = − ν = 1 . κ . For (29) we show ˆ κ = 1 . κ = 1 (dottedcurve). Figure (b) shows results for ˆ λ = −
8, ˆ κ = 1 . ν = 1 .
4. The structure is similar for allˆ κ ≥ , ˆ λ < It remains to compute the norms of the tachyonic solutions for both λ > λ < p (cid:63) , p ∗ (cid:63) . The inner product ( ψ ( p (cid:63) ) , ψ ( p ∗ (cid:63) )) is non-zero, anddiagonalizing the the resulting symplectic structure gives one degree of freedom with positivenorm and a second with negative norm. Thus complex tachyons are necessarily associatedwith ghosts and it remains only to analyze real tachyons.Following [17] we find that for tachyonic solutions of real momentum, the inner product(22) simplifies to( φ , φ ) = 12 ( ω + ω )(2 π ) d − δ ( d − ( (cid:126)k − (cid:126)k ) e it ( ω − ω ) (cid:104) ψ , ψ (cid:105) SL . (31)13 (cid:45) (cid:45) Ω I m Ω (a) (cid:45) (cid:45) Ω I m Ω (b) FIG. 3: The case ˆ κ ≤ , ˆ λ <
0. We again plot numerical solutions of (28) – dashed and dottedlines – and (29) – solid lines, including both the straight lines along the real axes and the roughcircles. Simultaneous solutions occur at the intersections. Again, q = 0 corresponds to the light-likemodes already studied (and is not a tachyon). Figure (a) shows results for ˆ λ = − . ν = 1 . κ , while for (28) the dashed and dotted curves respectivelydescribe ˆ κ = 0 . , . κ through this range causes thedashed curve to pinch off and to separate into two pieces (as shown by the dotted curves). Furtherincreasing ˆ κ →
1, the rightmost dotted line moves off to infinity and we recover figures 2(a)) and2(b). Changing ˆ λ appears to simply change the overall scale of the figures as indicated by figure(b) which shows ˆ λ = − . ν = 1 .
4, and ˆ κ = 0 . , . Here (cid:104)· , ·(cid:105) SL is a Sturm-Liouville-like product with an explicit boundary contribution: (cid:104) ψ , ψ (cid:105) SL = (cid:104) ψ , ψ (cid:105) bulk + (cid:104) ψ , ψ (cid:105) bndy , (32)where (cid:104) ψ , ψ (cid:105) bulk = − r − d p − p ( ψ ψ (cid:48) − ψ ψ (cid:48) ) (cid:12)(cid:12) r = ∞ r = r , (33) (cid:104) ψ , ψ (cid:105) bndy = r − d (cid:20) − ν −
1) + 2 νr ν − κ (cid:21) ψ ψ . (34)Note that (33) is singular when evaluated in tachyonic solutions that satisfy the boundaryconditions since this fixes a particular value of p . In order to evaluate (33) for a mode withmomentum p which lies in the discrete part of the spectrum, we consider two solutions withmomenta p and p which do not satisfy the boundary conditions, take the limit p , p → p ,14nd impose the boundary condition that sets p = p at the end. Applying this procedure to(33) and taking into account the contribution (34) we obtain( φ , φ ) = (2 π ) d − ω δ ( d − ( (cid:126)k − (cid:126)k ) δ p ,p (cid:104) ψ , ψ (cid:105) SL , (35) (cid:104) ψ, ψ (cid:105) SL = A (cid:26) (ˆ κ −
1) + ( ν − (cid:20) K ν − ( q ) K ν +1 ( q ) K ν ( q ) − (cid:21)(cid:27) , (36)where q is given implicitly by (24) and A is the positive quantity A = 2 p ν r K ν ( q ) ν ( ν − ν ) . (37)Numerical results indicate that the second term in (36) (including the factor of ν −
1) ispositive for real q and decays monotonically from 1 to 0 as q ranges over (0 , ∞ ). TheKronecker delta in (35) reflects the facts that the tachyonic spectrum is discrete and thatthe SL product (36) vanishes when p (cid:54) = p . Noting that (31) also vanishes for (cid:126)k (cid:54) = (cid:126)k ,we conclude that the frequencies must also be equal in order for (36) to be non-zero. Thusthe time-dependent exponentials in (31) cancel, making manifest that the inner product isconserved. As a consistency check, we note that taking the limit r → (cid:104) ψ, ψ (cid:105) SL = − ν (cid:20) κ ( ν −
1) + λνp (cid:21) + O ( r ) . (38)We now study (36) for the tachyons found above: Case ˆ λ > , ˆ κ <
1: in this region we find one real tachyon. Since the second term in (36)decays monotonically, the maximum of the norm occurs when the value of q that solves (24)acquires its minimum. For any fixed ˆ κ <
1, the value of q (ˆ λ, ˆ κ ) defined by (24) decreasesmonotonically with ˆ λ , arriving at the minimum when ˆ λ = 0, see figure 1(a). Thus if thenorm (36) is negative for ˆ λ = 0 and all ˆ κ <
1, it is in fact negative everywhere in the regionbeing considered, i.e ˆ κ <
1, ˆ λ >
0. To help see that this is indeed the case, we solve (24)with ˆ λ = 0 for ˆ κ and insert the result into (36) to obtain: (cid:104) ψ, ψ (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) λ =0 = A (cid:8) K ν ( q ) − [ K ν − ( q ) K ν ( q ) + ( ν − K ν − ( q ) K ν +1 ( q )] − ν (cid:9) . (39)Plotting (39) for q > < ν < Case ˆ λ < , ˆ κ >
1: Here both terms in (36) are positive definite, so there are no ghosts.15 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) A (cid:45) (cid:60) Ψ , Ψ (cid:62) Λ (cid:61) FIG. 4: The left hand side of (39) is plotted as a function of q for ν = 1 . ν = 1 . ν = 1 . Case ˆ λ < , ˆ κ <
1: We found a pair of complex tachyons that can move to the real axisfor certain values of ˆ κ , ˆ λ , As mentioned above the complex ghosts constitute a ghost/non-ghost pair. In the region in which the tachyons are real, one may show that one (and onlyone) of the tachyons is a ghost by using the fact that the norm is given by the derivative of(24) up to multiplication by a positive definite function . The norms vanish at the criticalpoint where the tachyons leave the real axis. At this point we expect logarithmic modes toappear with the corresponding associated ghosts. IV. DISCUSSION
Our main point above is that the ghosts found in [17] may, at least in some cases, bebanished by either imposing a suitable low UV cut-off Λ g , or by appropriately modifyingthe theory on energy scales above Λ g . We argued that this is a general property of renor-malization group flows that approach the IR fixed points of [17] and which start from awell-defined UV theory, analogous to those analyzed in [9, 11, 12, 19–22].In addition, we exhibited a simple new class of examples in which the ghosts are banishedby imposing a radial cut-off on the AdS space. As discussed in [14], this corresponds to anon-local UV modification of the usual CFT dual to bulk AdS. We found a two-parameterfamily of such theories corresponding to further quadratic deformations which are ghost-free While this may be checked explicitly using Bessel identities, it also follows from the general relationbetween the norm and the residues of the 2-point function.
16n a certain regime of parameter space. In particular, gathering the results found in theprevious sections, leads to the phase diagram shown in figure 5. Here, regions I and II (i.e.ˆ κ >
1) constitute the ghost-free regime. More specifically, in region I there are only time-likeexcitations whereas in region II there is a non-ghost tachyon. On the dividing line λ = 0a light-like mode of zero norm is present with the associated IR divergence in the 2-pointfunction. The remaining regions contain ghosts: in region III there are two real momentumtachyons, one of which is a ghosts; in region IV there are two complex tachyons, whosepresence is tied to ghosts, as explained above; finally, in region V there is one real tachyonwith negative norm. Here the dotted line that marks the boundary between the regions withtwo real (III) and two complex tachyons (IV) is to be considered very approximate. We havenot investigated this boundary in detail, though the fact that K ν − ( q ) /K ν ( q ) is positive for q > q = 0 shows that it lies to the right of the λ -axis and terminates atthe origin. For small q we can send the cutoff r to zero and the boundary between regionIII and IV satisfies λ ∼ − κ ν/ ( ν − . IIIIIIIVV ΚΛ FIG. 5: Different regions in parameter space ( λ, κ ) It is natural to ask whether a similar simple radial cut-off can banish more general ghosts.Consider for example the addition of a new term to I def involving η ( (cid:50) γ φ ) (a p term withcoefficient η ). The higher order boundary condition will then give rise to additional ghosts.Our preliminary numerical investigations indicate that for η (cid:54) = 0 there are no values of κ, λ, η, r for which the theory is ghost-free, so that the ability to banish ghosts by using asimple radial cut-off is not generic. However, it is again likely that for at least some valuesof the parameters that a more complicated UV modification of the IR fixed point (such asthat associated with RG flow from a good UV theory) that renders the theory ghost-free.17e conclude by making explicit the sense in which the radial cut-off theories of sectionIII are UV modifications of a theory with no cut-off. This may be done by comparing thetwo point functions of the theories with finite and vanishing r in the deep IR, which we taketo mean m bndy = 0. This is in turn equivalent to studying expression (16) for the norms atsmall m bndy . We take the cut-off free theory to be given by the same action I = I Ref + I def with couplings ˜ κ, ˜ λ and r = 0. As noted in section III, our parametrization was chosen tobehave smoothly as r → λ = r λ/κ c and ˆ κ = κ/κ c as r -dependent functions of ˜ λ, ˜ κ throughˆ λ (˜ λ, ˜ κ ) = r κ c ˜ λ r ν ˜ λ ≈ ν ( ν − , (40)ˆ κ (˜ λ, ˜ κ ) = [ r ˜ λ (1 + ν + r ν ˜ λ ) + 2˜ κ ( ν − κ c ( ν − r ν ˜ λ ) ≈ νν + 1 , (41)where we have displayed the behavior for large r . Thus we see that given any ˜ κ and anypositive ˜ λ in the r = 0 theory, for large r the IR behavior is described by the universalvalues ˆ κ univ = νν +1 and ˆ λ univ = 4 ν ( ν − . Since our analysis holds for 2 > ν > κ univ > λ univ >
0. In this sense, subjecting such r = 0 theories to a radial cut-off atlarge r renders them both ghost- and tachyon-free.The expressions (40) and (41) can be interpreted as RG flows for the couplings ˆ λ and ˆ κ asa function of the cutoff r . Indeed they are solutions to the RG equations of [13, 14] wherethe multi-trace couplings (or in the language of [14] the boundary action S B ) are truncatedto second order in boundary derivatives. The constant ˜ κ and ˜ λ are integration constants.Since the Wilsonian RG equations of [13, 14] are exact, and since the spectrum does notchange under exact RG, the full solutions that include all higher derivative couplings (butwhich continue to fix all other couplings to zero in the r = 0 theory) would necessarilydescribe radial cut-off theories with ghosts. In this case it is the truncation that leads to awell-defined ghost-free theory. 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