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JHEP05(2015)104 Springer e May 20, 2015 April 29, 2015 : ation Group : quantum field February 20, 2015 : Published Accepted , 10.1007/JHEP05(2015)104 s formalism also leads us Received doi: them. In the case of nearby n on the space of fields. The se distributions and quantifies malization group flow from the metric on the space of couplings. el Hill, e field theories generated by the ces, ia, d International Solvay Institutes, and Alexander Maloney d Published for SISSA by [email protected] , Jonathan J. Heckman a,b,c . 3 1410.6809 The Authors. c Conformal and W Symmetry, Statistical Methods, Renormaliz

We study the question of how reliably one can distinguish two , vijay@.upenn.edu Department of Physics, McGillMontreal, University, Canada E-mail: New York, U.S.A. Theoretische Natuurkunde, Vrije Universiteit Brussel,Pleinlaan an 2, B-1050 Brussels, Belgium Department of Physics, UniversityChapel of Hill, North U.S.A. Carolina at Chap David Rittenhouse Laboratories, University ofPhiladelphia, Pennsylvan U.S.A. CUNY Graduate Center, Initiative for the Theoretical Scien [email protected] b c e d a ArXiv ePrint: UV to the IR andto leads a us criterion to for astring distinguishability quantification theory of of landscape. low fine-tuning. energy Thi effectiv Keywords: theories (QFTs). Each QFTrelative defines a provides probability a distributio notionthe of number proximity of between measurements the requiredconformal field to theories, distinguish this reduces between toOur the formulation Zamolodchikov quantifies the information lost under renor Relative entropy and proximity oftheories quantum field Open Access Article funded by SCOAP Abstract: Vijay Balasubramanian, JHEP05(2015)104 3 3 6 7 8 9 9 1 2 5 8 10 10 11 , how reliably can we q tor content, correlation oximity of QFTs in any ny UV completions of a in statistical inference and dels of particle physics and and ld theories. Additionally, it ions, we shall convert well- p uantify the distinguishability oblem is challenging, especially n probability distributions into uation) of a quantum field theory – 1 – theory 4 φ 4.3.1 This formalism gives a concrete method for evaluating the pr Broadly speaking, theory determination is a basic question Here we ask: given two competing theories of the world 4.4 Large field range inflation 3.2 Metric proximity 4.1 Flux vacua 4.2 2D CFTs 4.3 Quantifying fine-tuning 2.1 Master theories 2.2 Perturbative calculability 3.1 group flows analogous measures of proximity between QFTs. . By interpreting theas (Euclidean contin a probability distributionstudied information on theoretic the notions space of of proximity field betwee configurat distinguish them given a finite number of measurements? A central aimfunctions, of and high coupling constants energy ofin physics the real the is world. to context Thisgiven pr determine of low the string energy effective opera theory, fieldcosmology. theory because such there as the are Standard a Mo priori ma 1 Introduction 5 Discussion 4 Landscapes 3 Conformal field theories Contents 1 Introduction 2 Proximity in quantum field theory UV complete theory with aprovides landscape a of way low energy to effective coarse-grainof fie different for effective specific field features, theories. and to q JHEP05(2015)104 ) ). z z ( not ( q (2.3) (2.4) (2.2) (2.1) (2.5) p drawn . The } } ) = ) } N x ) and ∗ Consider a ( ]. The special z ξ φ 5 ( 1 { |{ p z ( , . . . , e q ). In the context 1 . different distribu- analytic continua- , e z ]) ( ξ { (1) = 0 [ ]. φ p f [ 4 nce. Nevertheless, in p = – ribution is a Euclidean S 2 erated by a Euclidean nt of information which E − . almost surely. stributions iven spatial configuration ) q q qual time configurations in . N ) ) exp( = ]) over Euclidean trajectories log ed the Kullback-Leibler (KL) z z . ∂ξ ( ( istribution on the space of field φ p ∂ φ [ q p )) q q , see e.g. [ S D || M − p oximity obtained by taking the expectation Z ( log ) log ∂ξ z ∂ N = ( KL ) p p z δξ ( Z ) q M ND z a convex function such that ( δξ ) − independent events f z ( dµ on a fixed Euclidean time surface. We will – 2 – N ) to model the distribution ) such that for some value of 0 z exp( } dµ Z ) for ( Fisher MN φ ] depending on field configurations ξ , the probability that these draws could have been q ≃ φ Z G |{ ]) with [ p/q N ) z ( ) = φ S ( [ ≡ ≃ f q E p q | || ) ) is a choice of measure on the space of outcomes. The of S p p q z ( ( q || − p Fisher MN 0 and vanishes if and only if Pr( ( dµ KL G ≥ D exp( KL ) p ). At large are nearby, the KL divergence reduces to a metric. q D 1 z Z || ( q ). is p q ( x ) around this point yields the Fisher information metric: } ( q ] = ) φ and || KL φ x ) is: [ ( p z p D corresponds to the KL divergence. p ( ( φ p { u ]: KL 1 log D u ]). This distribution defines the quantum field theory via its is the partition function. A draw from this probability dist φ is the outcome, and [ ) = p z S u Z ( ) is generated by summing the distribution exp( − Note that in quantum physics we are often also interested in a In this note we will consider probability distributions gen In information theory, the KL divergence quantifies the amou As it is not symmetric, the KL divergence is not really a dista f 0 This local metric also arises from a more general notion of pr φ 1 Expanding Ψ( tion — i.e. thethe square Lorentzian of the theory. ground The state wavefunction ground over state e wavefunction for a g obtained from from the distribution of statistical inference, one can consider is lost when one uses the distribution For additional discussion and references to the literature case the limit where parametric family of distributions relative entropy value with respect to the distribution field configuration One way to quantify the proximity between two probability di 2 Proximity in quantum field theory Here, Here that approach the boundary condition quantum field theory with action on a probability spacedivergence) is [ via the relative entropy (also call exp( action of the Euclidean field theory defines an (unnormalized tion to Lorentzian signature.configurations The normalized probability d JHEP05(2015)104 − p 2 ρ . (2.8) (2.7) (2.6) p log . Hence, p ρ master ) in Euclidean p s of field con- x ( φ ories with actions . This point will be is the density matrix ]. , tant, although we will j ). (2.9) 6 i x ous low energy effective  ρ ( ork for comparing QFTs q p λ and Z Z Θ r string compactification. . ired, however, is that there ld theories generated by the i ) erturbation theory. Consider ce in stress energies between that of theory µν ibution for x g ( λ d theory rather than Lorentzian , where = q Θ ) + log ories, see e.g. [ p in the ground state of theory ) . )) , which is also a quantity of physi- x ) S p 2 ( x ) for all | x ( g and j Z ) ( − λ i q 0 / ) q p p q || φ Θ x O i S µν x ( ) Z q ( ) the quantum relative entropy between two x D ( x Ψ( δg | (  ( d i g p λ KL ]. This master theory could be either a lattice p S KL Z √ – 3 – ’s descend from the same master theory, we can ( D − Z ≡ D q are constant. We then get a family of probability δ ) + log e ) p ) i ] = φ x x S λ master ( φ ( [ 2 D g ], which we perturb by a linear combination of local φ λ p − [ φ √ S Θ [ q Z p S − S = . Nor is ] i ) master ) = φ p p [ ( q µν q || T S p ( − ) is not the same thing as the quantum relative entropy, Tr( . Since all of the q KL ( µν M ] defined by the deformation: D T KL } λ D | { : , . . . , q φ q ]: 1 [ q q φ [ q and S ) specifies a source, i.e. a position dependent coupling cons x p ), between the ground states of theories ( q i ρ λ ] and Given two quantum field theories which depend on the same clas From this master theory we can consider deformations to vari Note that this For a wholly different discussion on the distance between the φ 2 log [ p p signature — is mostquantum directly field interpreted theory. in statistical fiel cal interest. Our object of interest — the probability distr formulation of a field theory, a continuum CFT, orfield a theories particula ρ S figurations, we can now study the KL proximity between two the for the ground state of theory continue to label the field and operator content according to This is the expectation value of ( Each with the same field/operatoris content. some “master All UV that theory” is really requ density matrices of a given theory. 2.1 Master theories At first glance, our notion of proximity would appear to only w here be studying the associated distribution we can still speak of the KL divergence This deformation is proportionaltheories to the trace of the differen The notion of proximity wea have introduced Euclidean is theory calculable with in action p especially important when we turnstring to landscape. the study of effective fie 2.2 Perturbative calculability sources for operators: distributions primarily focus on the case where the JHEP05(2015)104 , ) ) q q || ( UV ive, p ( (2.10) (2.13) (2.11) (2.12) KL and then ) treated and Λ D x ) as defining ( p ) ( UV λ p q . This defines ( UV Λ ) p . . Operationally, < ( UV q  | Λ he system at finite and p k / i | with Θ . ) j toffs Λ p and can be packaged ..., q . This finite piece is p and ( UV < O i the partition functions p + h ) p f divergences — an IR ) UV p y q f i correlation functions. In must be adjusted to hold ( ( UV ming from contact terms i = Λ dependent. If a particular nce comes from such UV λ ormation contained in the f couplings is ] to quadratic order in the cheme unambiguously. As µ conn O b eory ) Θ φ [ h uplings is reflected in the fact ce, in the usual way: one first q f the finite piece of p − h . Operationally, we introduce i x, y p ) µ ( i x λ j ( , where G λ O . ) i ) y Θ ( q O k/b h g || µ  p − h p ( ) → with momenta Λ p are related by flow; y y i ( log k p ) D q g KL ∂ y d ( ) p λ x ∂D – 4 – y ( and . We can regard theories and g D ) )Θ ≡ d p p bx x p ( UV ) ( is calculable in since the term x x Λ ( λ in theory → KL D g q Θ k β d x < φ p x Z ≡ h , the values of the couplings at ) ) is independent of the UV cutoff Λ and q D 2 1 ( UV q d UV p || p conn Z ( ) ) = 2 1 q KL || x, y and can a priori be positive, zero, or negative. p = ( D ( λ , but can depend on a choice of scheme, and can a priori be posit UV is the connected two point function G KL UV D Fisher ij G conn coincide. The IR divergence is easily regulated by putting t ) y x, y ( λ and are the generating functions of the correlation functions o depends only on the expected value of the deformation, while G x , p q S An important special case is when The finite part of Integrated two-point functions of this sort have two types o We now study the leading order behavior of the KL divergence, Z , − p q S Z is independent of Λ We conclude that the Fisher information metric on the space o where divergence proportional to thewhere volume and a UV divergence co in terms of the values of the couplings at an RG scale volume. The UV divergentfinite contact terms UV are completion generally isone scheme known, might this expect, fixesdivergent the pieces. the largest regularization contribution s to the KL diverge The KL proximity between a regularization scheme, along with some counterterms in th independent of Λ we do not eventhis need sense an the action KL forcorrelation divergence either functions also of theory, a but quantifies theory. only the their amount of inf respectively. We will take Λ they describe a given quantum field theory with UV momentum cu as a small perturbation to the original theory. Expanding zero, or negative. This makes the physical interpretation o perturbation, the KL divergence is rescales positions and momenta as more subtle. As we change Λ fixed the long distance behavior.that This the active beta tuning function: of the co different probability measures on theintegrates same out configuration field configurations spa JHEP05(2015)104 ] i D IR by has UV ℓ δλ ), a p Λ (3.2) (3.1) (3.3) lo , = p ectively 2.2 can be IR sity, as × i V [Λ hi δλ p ) is the mutual master p UV,IR is a conformal field ( I q tice of volume ) = lost as one coarse grains lo onventions, the two-point ribed above is a particular p , l fail to distinguish between × molodchikov metric! D ich remains in the continuum is 2 s the lattice cutoff is taken to commutative field theories, and in hi 1 | The small variation p . or theory x || D | ]. Then, since the CFT one-point d by renormalization group flow.  p D 16 ) at. The product distribution ting with a master theory 1 D UV V π master ℓ p f the relations between renormalization group ( (2 ij (Zam)  G KL × ]. For further discussion on information geometry j D 2 is another CFT obtained by perturbing 13 (Zam) ij δλ , i q δλ G 12 – 5 – δλ , and ) is proportional to the length of the vector ]. The second operation is somewhat awkward in local ≃ = ) q p UV q . i Λ 2 ]. 2.10 || = master p p δλ µ, lattice sites, each draw from the Euclidean probability 15 (0) ( [ j , − δλ lo e O p 14 KL D UV ) ], as well as [ , marginalize out either high or low momentum shells, to resp x D ( 11 i UV – V/ℓ 7 O ], and . Then h = is a CFT and that , µ UV and Λ p K IR ℓ IR [Λ pieces of data about the couplings of the theory. Following ( hi p 3 K . in the space of couplings can then be interpreted as the KL den ) q ( UV δλ is the Zamolodchikov metric of the CFT [ to Λ ) (Zam) ij p ( UV G To make this more explicit, consider theories defined on a lat First, suppose that From the perspective of a continuum theory, the lattice desc Another quantity of interest is the mutual information. Star 3 with lattice-separation support on the same momentum modes as quantum field theory,theories but which makes have a sense gravity dual both with in a finite the length context AdS thro of non- Thus the Fisher information metric is proportional to the Za information between the UV andflow IR. and For related information discussions theory o see [ produce distributions with respect to the Zamolodchikov metric: with IR and UV cutoffs Λ where in the context of AdS / CFT, see e.g. [ measurement of the fieldthe configuration two at theories with one probability lattice site wil functions vanish, the KL divergence ( The distance distribution gives theory (CFT). The KL divergence thenfrom provides Λ a measure of the information 3 Conformal field theories We now consider the special limit where either theory two distributions with different coupling constants relate follows. Since there are regularization scheme, which willzero contain size. UV divergences It a is also of interest to extract the finite piece wh a linear combination of exactly marginal scalar operators. viewed as a vectorfunction in for the a space marginal of primary marginal scalar couplings. of dimension In ∆ our = c JHEP05(2015)104 ) ]. p ese 18 (3.4) (3.5) (Zam) ij 2.10 G 1 leaves n ( ) depends − 2 3.4 2) all lead to ) is proportional > D/ q is large. Irrelevant is, up to a factor of || > D/ p IR ( ℓ D ∆ ) is dominated by the KL − q i KL ange the Zamolodchikov D || aw from the distribution an RG flow from the UV . In the special case where p D 2∆ rdinate transformations on ( teresting to understand how D > ory. Relevant perturbations  because there are long range orrespond to different choices . e space of couplings: KL < D cise coefficient in ( q IR es in various dimensions, see e.g. [ UV V tor when D ℓ ℓ al. Nearly marginal perturbations that is suppressed by the infrared  nd hence measurements at different ). ) is nearby. Similar statements apply, i from 3 . ∆ p KL 2 p is not a CFT, but is related to theory δλ − . We will work in a basis where i ( D q δλ D O O 2( IR ∝ ℓ are small the above derivation can be easily ) 2 i ) +  q i p || – 6 – δλ || p ), the finite piece of δλ q ( ( 4 3.5 i KL c KL D D i X for dimensional reasons. We can interpret this as follows: ) = ≃ q with volume, but the unitarity bound ∆ ) in a way that is almost insensitive to volume because nearly || q /V ]). The scheme-independent piece of p || . The resulting flow and subsequent form of the KL divergence ( is an IR fixed point, and KL p 17 q D UV ; the Zamalodchikov metric transforms covariantly under th KL ( q ℓ i D KL is not proportional to the volume D λ KL D has disappeared, since the regulated integrals appearing i . As expected, the contribution to D ) KL i D UV D ) make a finite contribution to ∆ − D V/ℓ > D 2( IR are numerical constants of order one. ℓ i 2 c  i to the IR theory ) contribute to δλ p D For each summand in equation ( Let us now consider the case when the theory Conversely, suppose i ∼ For some examples of such computations for CFTs on round spher c 4 The factor of the finite piece of correlations in aessentially conformal gives one field piece theory, of data and about the hence theory. a The pre given dr by the addition of some non-marginal operators to is diagonal. Then when the perturbations conformal theories have long-range spatial correlationslocations a do not givewith independent dimensions information above about the the Breitenlohner-Freedman the bound ( where the on the nature of the IR regularization. must be proportional to generalized. In the lattice regularization order one, proportional to the Zamolodchikov distance in th of contact terms inmetric. the However, OPE. changes Naively, of thesethe scheme might space can appear of be to couplings interpreted ch as coo limit. In a CFT, different choices of regularization scheme c (∆ scale because the low-energy effective theories are identic diffeomorphisms (see e.g. [ contribution of the lowest dimension (most relevant) opera couplings (∆ sub-extensive scaling of will be dominated by the operator(s) of lowest dimension ∆ a narrow window with super-extensivethis scaling. arises in It terms would of be measurements in distinguishing 3.1 Renormalization groupLet flows us now considertheory the case where the deformation initiates since to leading order JHEP05(2015)104 , ) ], µν } KL 3.4 T µν (3.6) (3.8) D g |{ φ n for [ p -dimensional ry along the 1, this flow is 0. In two and D in the positions, ≥ ≪ ng an RG flow, ) (3.7) , ) δ y ) q ( = 2 supersymmetric x || s. Along these lines, , ( even, it is determined p ρσ ) D for ( t N 2 D ral charge close to one, ( δg δ is proportional to x 0 in a reflection positive ∂t ) larization) precisely ( KL Recall that in flat space, µν,ρσ x ∂c . Specializing to the case harp information-theoretic − T rge is closely related to the I ( s. Perturbing about a fixed D > µ C T 12 anomalies of a CFT measure D µν T C ombination of central charges − ion C δg field theory on a ≡ is calculable. The KL divergence p = i = log δ ) D j ded in local degrees of freedom. t y R ( i ∂t ∂λ defines a family of theories i ρσ (0) T ∂t ]). Here, ∂λ ) µν ρσ g x 19 T ( ) x µν ( (Zam) ij – 7 – G hT µν ) , and in three-dimensional T y ∝ c ]). h ( t , i.e. the normalization constant for the R-symmetry g 20 = = is the stress energy with the one-point function q p t RR p D

y τ i R ) ) D i q x t d ( ) , and consider two UV CFTs which differ only in the choice || x (0) t ( µν ( 2 g UV ρσ T ∂t T p KL ) x − h D , the line element for the information metric is: conformally equivalent to flat space, the two-point functio x ) 2 D ( odd this one-point function vanishes, and for x d ∂ δg D is proportional to ( µν D M + T Z µν hT C g 2 1 T ) is a specific dimensionless combination of terms quadratic → . Varying the background metric x ) = ) is the c-function of a two-dimensional conformal field theo g ( D x ) as a function of the RG flow parameter ( RG q M t t µν ( µν,ρσ ]. || c p T I t 16 ( We can also extend this calculation to cover the case of RG flow Evaluating on We can also consider the KL divergence between two points alo This is a satisfying result. It tells us that the quantity KL of a 2D CFT, weinformation learn lost that in the moving initial from change in the the UV central to cha the IR. We have flow [ directly quantifying the level of distinguishability enco we introduce a UV cutoff Λ is closely related towhich the counts the evaluation of localwe a degrees have: particular of linear freedom c in the field theory. where D where subtracted off. For by the conformal anomaly. and it is natural tobackground consider the proximity of two such member its degrees of freedom. Weinterpretation. now show Consider that a this Euclidean statement signature has a conformal s 3.2 Metric proximity More generally, there is a deep intuition that the conformal described above. theory, which agrees with the information theoretic condit where the operator is marginally relevant, i.e. has dimension ∆ = short. In manyand situations various 4D such supersymmetric as quantum 2D field theories, minimal models with cent four dimensions, as dictated by conformal invariance (see e.g. [ manifold in this case can again be computed and we get (in a lattice regu current two-point function (see e.g. [ field theories it is proportional to JHEP05(2015)104 ) T C −→ M || (4.2) (4.3) (4.1) N −→ minima ( l . Suppose KL µν D δg cations of the g the massive vacua we can evaluate ] for some fields l + 5 N −→ , µν . g n ! generates an effective 2 ) he IR, ) ng of our background metric. N −→ k h of the ( see that if the two theories minima of the theory with , . . . , ϕ as nothing to do with the oser in the IR. φ 1 isolated vacua: tence of a large landscape of ϕ l − [ supersymmetric models. , φ S ) ( how how to deploy our formalism j ( l =1 such that the form of the effective with Y k δφ 2 i ) 4 φ i −→ M ( m − l 1 2 × δφ Λ ij l ij 2 . So in other words, the statement that δ − G , we can compute the proximity 2 IR T ) × ≃ – 8 – −→ M ) V ∂φ ( > C −→ M ≃ 2 1 || and ): UV ij T }

N −→ is the mass-squared of the real scalar expanded around ) ( l N C G −→ ( 2 x ) 4 ) KL i d ( D φ Z is the regulated volume of the spacetime. on the lattice of fluxes. − , . . . , φ ) , and with it an effective action ] for a review), the flux quantum numbers define an integrally k V and a small perturbation to another metric (1) ] = ( −→ M N φ −→ φ 21 N ( −→ µν i |{ g φ 6= and φ, ( [ k p S Q N −→ 4 − l 2 2 . This means the KL divergence can be evaluated by just varyin Λ is the information metric from varying with respect to the lo N −→ . Given two flux vectors = n ij 2 i critical point, and G m th Suppose now that we have another flux vector To illustrate, consider a toy model in which our flux vector i In a CFT with a UV cutoff, we can perform an RG flow by a Weyl rescali 5 , . . . , ϕ 1 potential for this fluxflux vector vector has minima which are nearby the yields the approximation: minima. Working in a saddle-point approximation around eac where in the distribution where quantized lattice vector the KL proximity for theseare two closer background metrics. together in Hence, the we IR, then the We thank H. Verlinde for this comment. of background metric This form readily generalizes to complex scalars, as well as 4.1 Flux vacua In flux vacua (see e.g. [ 4 Landscapes One clear lesson fromself-consistent recent low work energy in effective field string theories. theoryin We is the now s the study of exis the landscape. typically decreases under RG flow means CFTs typically get cl we perturb this UV CFT by a relevant operator. Upon flowing to t ϕ between the two effectivedistance theories. between Note that a priori, this h action for a single canonically normalized real scalar JHEP05(2015)104 - + n 2 | 2 , 4 and (4.4) ween master − 3 p χ | ,..., + ] centered 2 | ]. Following 1 = 1 22 , + 3 r 22 χ | + Consider the two 2 terms. The latter is a | ice spacing, and as 2 , 120 for − 2 / − r,s cua in (perturbative) χ h restricted knowledge χ | 1) . 6 ons, which respectively e this, observe that the 2 + − FT. The KL divergence which upon further flow T 2 2 fine a notion of distance | th descend from the same . ) ese theories [ i 1 V master theory, we might s , rect comparison of the two q M is infinite in both directions. ne our master theory 5 + 2 . tate . Although χ | − nt, the parent has non-singlet since it involves integrated ]. Though the untwisted sector ditional states mean that some ! s whether such UV completions re is a collection of intermediate r s untwisted sector. Clearly, our KL + 23 ons. 2 D | , . . . , p 1 diag diag sing full , 1 = ((6 + 1 p Z Z χ | such RG trajectories, we can therefore r,s

h = e modulus of the M by omitting all of the full -point functions. Though the details differ, Z diag sing , n ) = log Z ± r,s – 9 – χ diag full p || ]. A formal notion of distance given in [ . We get diag sing 2 22 | p 4 . Given a set of , ( − 4 M χ | KL + D 2 | orbifold of the diagonal theory [ 3 , , . . . , q + 4 2 1 symmetry, it is also natural to compute the KL divergence bet χ q Z | 2 + 5 model has primaries with weights Z 5 models viewed as effective field theories 2 / | / 2 , − 4 = 4 ) is still infinite, but in the other direction, we get: χ | = 4 c c + sing 2 | p 1 , || . In terms of the Virasoro characters + 4 χ | 5 theories, with diagonal and off-diagonal partition functi full 5 CFTs, but provides a way of telling us whether a theorist wit / p + / ( , . . . , r 2 Let us illustrate in more detail for 2D CFTs defined on a torus. The value of the KL divergence strongly depends on the UV latt Since we have a | The diagonal 3 6 , = 4 KL + 3 = 4 = 1 χ UV spin system and so there ought to be a “distance” between th correspond to the tetra-criticalthey cannot Ising be model connected and by the an three operator s deformation, they bo 4.3 Quantifying fine-tuning To a low energy effective fieldlead theorist, what to really novel matters constraints i onimagine that IR upon physics. an appropriate operator Startingvalues deformation, from the of some the U couplings, and corresponding theories Similar considerations apply forconsiderations the extend orbifold to theory more and general it orbifold constructi c respectively descend to of the orbifold coincidesstates, with and the the singlet orbifold has sectorstates of twisted of each sector the theory states. are pare These not present ad in the other, and so our general considerations, we take the UV spin system to defi on defining a metric on the values of local function on the space Teichm¨uller for the complex structur evaluating the KL divergence intuitively agrees with this, point functions. with the two 2D CFTs are anotherstring machine theory. for generating Thesince a it Zamolodchikov vast cannot metric number connect is all of insufficient CFTs va [ to de 4.2 2D CFTs | s the singlet sector andc the full theory. This is not really a di about their CFTD could ascertain information about the full C off-diagonal theory is a we now argue, diverges as we pass to the continuum limit. To se JHEP05(2015)104 ) = φ (4.7) (4.6) (4.8) (4.5) ( V 2 potential / 2 φ 4! and address ): 2 / 4 the UV boundary m 4.6 1, then we say that λφ , # # ≫ 8 t since the inflaton is 16 / llows from dimensional ij / 2 . 2 UV y. Here, we quantify the 3 F 2 4 L Λ L λ 2 f order  off dependent contributions: 4 he simple n the IR parameters of the evaluated by a similar token. theory, viewing the reduced m λ/ m oximity of two theories in the scheme, the leading log contri- PL − in equation ( of the inflaton potential would 4 m eal scalar with potential as a soft IR cutoff. Using our φ M 2 UV 4 UV 8 3 λλ 6= 0. / Λ Λ  1. When MS 2 m G / 4 λ etball diagram entering in the computation L 1 λ/ × / 2 ≪ 2 " L ), and the corresponding ratios: j m λ ij . q × ) ) F − || q p ∼ 4 ( i ( ij ij " q )  ( D D } × IR UV ≡ = 0 versus KL V – 10 – Λ 6= 0 Λ λ D ij = λ  F = ∼ ) }||{ q MS the regulated volume of the spacetime. ( ij # as bare parameters of a UV theory, we can evaluate the # λ D λ λ V 2 = 0 2 λλ λλ m λ m G G { G G ( and 2 ) and GeV as a UV cutoff, and 2 , and 2 j 2 2 m m KL m p 18 2 π 2 λm || D λm m i m 10 G 16 p G G ( G / ) " ∼ " 2 KL D /m PL 2 4!. This theory is fine-tuned because small perturbations in = µ M / ) ]. 4 p theory ( ij 24 λφ 7 4 D φ = log( 2+ , see [ / L 2 Along these lines, we return to our calculation for λλ For further details on the evaluation of the three-loop bask 2 information metric in this case to find the leading order cut φ 7 G 2 × methodology, we get that the KL divergence scales as Planck scale the distinguishability of the theory with 2 m of large field range inflation. We consider a correction term o As an illustrative example, consider the theory of a single r conditions of theeffective coupling theory. constants Treating lead to large changes i the theory has no fine-tuning. Intermediate cases can also be 4.3.1 with 4.4 Large field rangeA inflation common claimsensitive in to the Planck scale studyprovide physics, of learning a the wealth string exact of compactificationsamount shape information of is on information tha the obtained UV from structure the of first a correction theor to t of where each entry isanalysis considerations. multiplied by As anIR, expected, there “order if is one we a evaluate power number”, law the as divergent pr contribution. fo In the evaluate We say that a pair of theories is fine-tuned when bution is: JHEP05(2015)104 2 ) φ (4.9) ∆ ommons m/ ( λ < heoretical Physics nt DE-FG02-05ER- ant. Freytsis, A. Hebecker, . te center for hospitality of the KL divergence in ergy effective field theo- 4 eory group for hospitality − xtra sector. It would also boration at an early stage ories. This dovetails with or helpful discussions and ability distributions on the aws of field configurations) on content over the entire Penn, and the organizers of ]. 10 eed to assume redited. energy observer could recon- een how to track information ospitality at Schloss Ringberg 26 sses, where it is quite common ginal deformations, we recover ∼ ximity in the space of QFTs. In , 4 25  φ PL (for CFTs). ∆ M T  C < ], and also to apply our formalism in various ) } 27 – 11 – 6= 0 λ }||{ , we learn that the KL divergence is bounded above: ), which permits any use, distribution and reproduction in = 0 PL λ { M ( 10 KL ∼ D φ CC-BY 4.0 This article is distributed under the terms of the Creative C Using this setup, we can coarse-grain any landscape of low en In future work, it would be interesting to study the behavior during part of this41367, work. and the The work work of of AM is VB supported isOpen by supported an Access. by NSERC DOE Discovery gr gra any medium, provided the original author(s) and source are c correspondence. JJH thanks the highthe energy workshop “Frontiers theory in group String at Phenomenology”during U for kind part h of this work.during AM his thanks sabbatical the leave. Harvard JJHduring high and energy part AM th of thank the this CUNY work. gradua AM and VB thank the Aspen Center for T Acknowledgments JJH and AM thank T.of Hartman this for work. helpful discussions WeS. and also colla Hellerman, thank A.N. R. Blumenhagen, Schellekens, T. J. Dumitrescu, Sonner, M. and H. Verlinde f various covariant regulator schemes such asscenarios [ where operators ofbe the exciting Standard to Model consider more mixto general with encounter systems an such statistical e as ensembles spin of gla coupling constants. space of fields, the relative entropythe leads special to a case measure of of conformal pro the field familiar theories case connected of by the mar loss Zamolodchikov metric. both in We have terms also of s RG flows and the value of volume of the spacetime. 5 Discussion Viewing quantum field theory as a machine for generating prob This upper bound is rather charitable, as it is the informati Attribution License ( On the other hand, to not spoil slow roll in the first place, we n so for a field range ∆ ries. We simply askmust how be many idealized performed measurements beforerecent (i.e. investigations we dr aimed can at reliably understandingstruct how distinguish — well even two a in the low principle — different UV completions [ JHEP05(2015)104 ]. s , , . (1951) up acting eory , . SPIRE (2012) 62 22 IN n a 2D , ][ ]. t Terms (2012) 045014 anics on the A 391 ical Decision ]. , SPIRE D 86 arXiv:1310.3188 IN ]. , , ][ Physica SPIRE , IN Momentum-space , ][ , Wiley, New York U.S.A. Ann. Math. Statist. SPIRE , IN cond-mat/9601030 Phys. Rev. ][ ]. , ]. ]. ]. SPIRE SPIRE SPIRE hep-th/9307010 IN [ IN IN (1997) 349 [ ][ Sphere Partition Functions and the ][ ][ 9 SPIRE arXiv:1405.7271 IN [ [ ]. hep-th/0507082 [ Instantons, the information metric and the – 12 – ]. (1994) 311 SPIRE IN [ 231 (2014) 001 Neur. Comp. SPIRE , hep-th/9709175 Elements of Information Theory [ IN hep-th/9511090 (2006) 051 11 [ On Information and Sufficiency [ theory entropy and the renormalization group (1986) 730. Translations of Mathematical Monographs. Vol. 191: Method Implications of conformal invariance in field theories for ]. ]. , American Mathematical Society, Providence U.S.A. (1982) cond-mat/0610375 Information geometric approach to the renormalisation gro Renormalisation as an inference problem [ ]. 08 hep-th/0108122 Distance between physical theories based on information th A c-theorem for the entanglement entropy 43 , On the symmetry of real space renormalization Black hole as emergent holographic geometry of weakly inter , American Mathematical Society, Providence U.S.A. (2000) (2011) 319 JHEP SPIRE SPIRE , Statistical inference, occam’s razor and statistical mech “Irreversibility” of the Flux of the Renormalization Group i SPIRE IN IN Annals Phys. JHEP (1998) 588 IN (1989) 153 , [ , ][ ][ (1996) 5163 A 26 Translations of Mathematical Monographs. Vol. 53: Statist (2007) 7031 Information theory and renormalization group flows JETP Lett. Geometry on the Space of Conformal Field Theories and Contac , B 522 B 220 D 54 A 40 ]. SPIRE IN arXiv:0910.2097 arXiv:1108.3568 Rules and Optimal Inference (1991). AdS/CFT correspondence Phys. Lett. Zamolodchikov Metric general dimensions hot Yang-Mills gas arXiv:1206.7004 [ [ Phys. Rev. J. Phys. Nucl. Phys. Mod. Phys. Lett. entanglement and renormalization in[ quantum field theory of Information Geometry 79. space of probability distributions Field Theory [6] J. Calmet and X. Calmet, [4] T.M. Cover and J.A. Thomas, [5] N.N. Chentsov, [8] H. Casini and M. Huerta, [9] D.C. Brody and A. Ritz, [7] V. Balasubramanian, M.B. McDermott and M. Van Raamsdonk [3] S.-I. Amari and H. Nagaoka, [1] S. Kullback and R.A. Leibler, [2] V. Balasubramanian, [14] M. Blau, K.S. Narain and G. Thompson, [18] E. Gerchkovitz, J. Gomis and Z. Komargodski, [19] H. Osborn and A.C. Petkou, [15] S.-J. Rey and Y. Hikida, [16] A.B. Zamolodchikov, [13] C. and B´eny T.J. Osborne, [10] S.M. Apenko, [11] J.C. Gaite and D. O’Connor, [12] C. Beny and T.J. Osborne, [17] D. Kutasov, References JHEP05(2015)104 , ]. SPIRE (2013) 012011 IN [ (2007) 733 462 79 The Exact (2005) 210 , (2013) 125025 B 730 ]. ]. D 88 ]. d J. Wright, arXiv:1305.3621 SPIRE , SPIRE Rev. Mod. Phys. IN SPIRE [ IN , J. Phys. Conf. Ser. IN ][ , Nucl. Phys. ][ , Phys. Rev. The Massive thermal basketball diagram , (1989) 673 – 13 – Extended Chiral Algebras and Modular Invariant B 327 hep-ph/0002048 [ arXiv:1401.1810 Flux compactification [ Covariant non-commutative space-time ]. ]. ]. ]. SPIRE SPIRE SPIRE SPIRE Nucl. Phys. IN IN (2015) 58 , IN IN ][ ][ (2000) 045004 ][ ][ Spaces of Quantum Field Theories Statistical Inference and AdS/CFT for accelerator physics B 894 D 62 arXiv:1305.6311 hep-th/0507137 hep-th/0610102 arXiv:1005.2779 [ Nucl. Phys. Partition Functions Phys. Rev. superconformal R-symmetry minimizes tau(RR) [ [ [ [25] J.J. Heckman, [26] A. Hebecker, [27] J. Heckman and H. Verlinde, [23] A.N. Schellekens and S. Yankielowicz, [24] J.O. Andersen, E. Braaten and M. Strickland, [21] M.R. Douglas and S. Kachru, [22] M.R. Douglas, [20] E. Barnes, E. Gorbatov, K.A. Intriligator, M. Sudano an