On the particle spectrum and the conformal window

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Published for SISSA by Springer

Received: October 27, 2014

Accepted: December 8, 2014 Published: December 31, 2014

M.P. Lombardo,a K. Miura,b T.J. Nunes da Silvac and E. Pallantec,1

aINFN Laboratori Nazionali di Frascati,

I-00044, Frascati (RM), Italy

bKobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya University,464-8602, Nagoya, Japan

cVan Swinderen Institute, University of Groningen,

9747 AG, The Netherlands

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We study the SU(3) gauge theory with twelve avours of fermions in the fundamental representation as a prototype of non-Abelian gauge theories inside the conformal window. Guided by the pattern of underlying symmetries, chiral and conformal, we analyze the two-point functions theoretically and on the lattice, and determine the nite size scaling and the innite volume fermion mass dependence of the would-be hadron masses. We show that the spectrum in the Coulomb phase of the system can be described in the context of a universal scaling analysis and we provide the nonperturbative determination of the fermion mass anomalous dimension = 0.235(46) at the infrared xed point. We comment on the agreement with the four-loop perturbative prediction for this quantity and we provide a unied description of all existing lattice results for the spectrum of this system, them being in the Coulomb phase or the asymptotically free phase. Our results corroborate the view that the xed point we are studying is not associated to a physical singularity along the bare coupling line and estimates of physical observables can be attempted on either side of the xed point. Finally, we observe the restoration of the U(1) axial symmetry in the two-point functions.

Keywords: Lattice Gauge Field Theories, Conformal and W Symmetry, Beyond Standard Model, Renormalization Group

ArXiv ePrint: 1410.0298

1Corresponding author.

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP12(2014)183

Web End =10.1007/JHEP12(2014)183

On the particle spectrum and the conformal window

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Contents

1 Introduction 1

2 Theoretical premise 32.1 Two-point functions 42.1.1 Universal scaling laws at innite volume 62.1.2 Universal scaling laws at nite volume 82.1.3 New operators and corrections to universal scaling 92.2 The fermion mass anomalous dimension at the IRFP 112.3 The Edinburgh plot 12

3 Numerical setup 143.1 The action 143.2 Strategy for the spectrum measurements 16

4 Results 184.1 Two-point functions 184.2 Would-be hadrons in the QED-like region 204.3 The spectrum in a box 214.4 Extrapolation to innite volume 264.5 The spectrum at innite volume 284.6 Mass ratios and degeneracies 33

5 Conclusions 34

A Volume dependence and extrapolation to innite volume 36

1 Introduction

This work is devoted to the study of how the particle spectrum of non-Abelian gauge theories changes once they enter the conformal window. Strongly coupled scenarios beyond the standard model provide the phenomenological motivation to study these theories, in particular, the preconformal phase that precedes the conformal window. On the other hand, the properties of these theories inside the conformal window, such as the value of the anomalous dimension of the fermion mass operator along the infrared xed point (IRFP) line and the ordering of the would-be hadron states can shed light on the dynamics at and just below the lower endpoint of the conformal window. Our guiding principle, as in our previous studies of these theories, is the identication of symmetry patterns; conformal symmetry and chiral symmetry are the key ingredients in this case. Figure 1 guides us through the projection

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Figure 1. The phase diagram in the temperature (T ) and avours (Nf) plane for a non-Abelian gauge theory with Nf massless Dirac fermions in the fundamental representation. A chiral phase boundary (solid line) separates the hadronic phase from the quark-gluon plasma phase. The endpoint of the chiral phase boundary is where the conformal window opens.

on the plane temperature-avour of the phase diagram of the chosen prototype theory, an SU(3) gauge theory at zero chemical potential with Nf avours of massless Dirac fermions in the fundamental representation, in other words massless QCD with varying avour content. For small Nf and at zero temperature, QCD has spontaneously broken chiral symmetry and it is conning. At some nite temperature, the system enters the quark-gluon plasma (QGP) phase, where chiral symmetry is restored and the system deconnes. For increasing Nf the most favoured scenario, rst discussed in [1, 2]1 and supported by lattice studies [4], identies the endpoint of the chiral phase boundary with the opening of the conformal window at some critical value Ncf. The conformal window refers to a family of theories that develop an IRFP in the interval Ncf Nf Naff; at Naff asymptotic freedom is lost.

As long as the theory is renormalizable, scale invariance in the ultraviolet was long ago shown to imply conformal invariance if an energy-momentum tensor exists for which the scale current s satises s = x [5]. Recently, there has been progress in classifying the possible IR and UV asymptotics of eld theories in order to understand if four dimensional scale invariant unitary quantum eld theories are always conformally invariant [68]. QCD-like theories in the conformal window belong to the class of theories that ow between a (trivial) UV and a (non trivial) IR xed point, where scale invariance is expected to imply conformal invariance at least at the perturbative level and as long as nonrenormalizable operators (e.g., induced by a lattice discretization) do not play a role. To guarantee scale and conformal invariance, chiral symmetry has to be restored and the theory deconned in the usual sense. Everywhere in the parameter space of the theory, except at the xed point, the observables will only show remnants of conformality; these remnants and the realization of exact chiral symmetry determine features of the correlation functions and the spectrum that are quite distinct from those of QCD. The purpose of this study is to isolate the aforementioned features and identify the universal scaling properties of the IRFP. To this end, we take the Nf = 12 system as a prototype of theories inside the

1These works were preceded by the pioneering work in [3], where alternative scenarios with no phase transition at Ncf were considered.

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asymptotically free phase

QED-like phase

c h i r a l s y m m e t r y e x a c t b r o k e n

Figure 2. Phases of the many-avour SU(N) gauge theory formulated on the lattice inside the conformal window. As the bare lattice gauge coupling gL increases from left to right, the lattice theory encounters an asymptotically free phase (negative -function), it crosses the IRFP at some g L and enters a QED-like phase (positive -function); chiral symmetry is exact in all these phases.

At g L a phase transition occurs to a chirally broken phase. It may be preceded by an exotic phase (grey shaded area) due to the improvement of the lattice action [1012].

conformal window and study both the fermion mass and volume dependence of the would-be hadron masses in a lattice box, and the fermion mass dependence of the spectrum at innite volume. This analysis updates and largely extends our rst study of this system [4]. Importantly, this is the rst study of universal scaling for the complete would-be hadron spectrum pseudoscalar, scalar, vector, axial mesons and the nucleon. Inspired by [9], it also provides a unied description of all existing lattice spectrum results for this system.

The paper is organized as follows. Section 2 contains a theoretical premise, where we partly reformulate or adapt to this system existing knowledge for the scaling properties of two-point functions and the spectrum at, or close to, a xed point. Section 3 describes our lattice action and the strategy adopted to compute the spectrum of the Nf = 12 theory at a xed lattice gauge coupling. Section 4 is entirely devoted to results, in order, the universal scaling and violations of scaling for the would-be hadron spectrum at nite and innite volume, the fermion mass anomalous dimension at the IRFP, mass ratios, and the degeneracy of chiral and U(1)A partners. We conclude in section 5.

2 Theoretical premise

As always, symmetries guide our understanding of a physical phenomenon, and we should identify the observables sensitive to those symmetries. In this case, we are interested in the two-point functions inside the conformal window, and the relevant symmetries are conformal, chiral and, to a certain extent, the U(1) axial symmetry; the latter will be shortly discussed at the end of this work. Figure 2 describes the pattern of phases for a theory inside the conformal window formulated on the lattice, i.e. a gauge invariant formulation on a discretized spacetime. At small bare lattice coupling gL < g L the theory is in the asymptotically free phase, characterized by a negative -function. The lattice theory then crosses the IRFP at some lattice coupling g L and enters a Coulomb, or QED-like, phase characterized by a positive -function. For all gL > g L asymptotic freedom is lost and the lattice theory has no continuum limit the only exception being the possible

0 gL**

gL*

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appearance of a UVFP at stronger coupling. For all gL < g L in gure 2 chiral symmetry is exact. At g L the lattice theory exhibits a zero temperature, i.e. bulk, phase transition to a chirally broken phase. There are indications that the line of bulk phase transitions ends at a nite Nf Naff [13]. Finally, the grey shaded area in gure 2 indicates the

possible emergence of an exotic phase due to improvement of the lattice action [1012]; chiral symmetry is still exact in this phase [10, 12].

In our rst study [4] of the conformal window we outlined a strategy based on the physics of phase transitions, in order to characterize the di erent phases in gure 2. We highlighted the observables that are most suitable for probing the emergence of the conformal window and its properties: the chiral condensate, order parameter of chiral symmetry, the chiral susceptibilities and their ratios, and the would-be hadron masses. This goes beyond the direct probe of the IRFP, it probes the very existence of the conformal window and the nature of the symmetry breaking/restoring patterns. In fact, the existence of an IRFP might not necessarily imply the existence of a conformal window characterized by a phase transition at some Ncf. We also attempted for the rst time in the study of this theory a lattice determination of the fermion mass anomalous dimension, obtaining, with all due caveats and still sizable uncertainties, a value in good agreement with most of later results; ours were obtained in the QED-like region of gure 2, and we stress once again the importance of identifying the phase where observables are measured. In this work we concentrate on the would-be hadron spectrum in the QED-like phase of the Nf = 12

lattice system, as a specic probe of conformal and chiral symmetries. In line with recent work [14] and renormalization group (RG) theory la Wilson, gure 3 illustrates how the fermion mass term perturbs the RG ow of the continuum massless interacting theory inside the conformal window we have assumed that no other couplings are present besides the gauge coupling and the mass itself. The mass term is a relevant operator that drives the theory away from the IRFP, while the gauge coupling is irrelevant due to quantum corrections. As discussed in [14], mass deformed conformal eld theory (mCFT) can be used within the basin of attraction of the IRFP to uncover the universal scaling properties of the correlation functions at innite and nite volume. Away from the IRFP violations of universal scaling will gradually appear, while correlation functions still satisfy all constraints implied by the non spontaneously broken chiral symmetry. A UVFP at strong coupling may emerge, however no indications of it have been found in preliminary lattice studies of the Nf = 12 theory [15]. We also note that, at strong coupling, new operators may be promoted from irrelevant to marginal or relevant; if so, the xed point structure of the theory needs to be reanalyzed in an enlarged space of couplings, possibly complicating the lattice search for new UV/IR xed points.

2.1 Two-point functions

We are interested in the two-point functions made of currents of the type JM q Mq, with M = 1, 5, , 5 for the scalar (S), pseudoscalar (PS), vector (V) and pseudovector (PV) mesons, respectively, and the nucleon correlation function with current JN qqq.

In other words, we identify the would-be hadrons of QCD in order to allow for a direct comparison with the spectrum of theories inside the conformal window.

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Figure 3. Wilson RG ow and xed points of non-Abelian SU(N) gauge theories in d = 4 spacetime inside the conformal window, with fermion mass m and gauge coupling g. For the massless theory (m = 0), the trivial UVFP (g = 0, asymptotic freedom) becomes unstable towards g perturbations due to quantum corrections and the system ows toward the non-trivial IRFP at g , m = 0. The fermion mass operator is always relevant and the gauge coupling is irrelevant at the IRFP. Mass deformed conformal eld theory (mCFT) for g [similarequal] g and m ! 0 provides the universal scaling laws

for observables at the IRFP. The dashed line is a line of possible initial values (m, g) for the lattice system, and its critical point C ows into the IRFP. A UVFP at strong coupling may emerge.

At the IRFP the theory is massless and interacting, and its two-point functions satisfy universal scaling relations with nonzero anomalous dimensions

h0[notdef]T JH(x1)JH(x2)[notdef]0[angbracketright] (x1 x2) H H = M, N (2.1)

with H/2 the scaling dimension of the current JH. More specically, we are interested in the Euclidean two-point functions with zero total 3-momentum

CH(t) =

[integraldisplay]

= 0 dE K(E, t) (E) . (2.2)

This is the well-known power-law scaling of the correlations CH(t) at the IRFP. We have also introduced the representation of CH(t) in terms of the spectral function (E) and the kernel K(E, t). Spectral functions are a powerful probe, widely used in the study of the QCD phase diagram at nite temperature and chemical potential; for example, they are a direct probe of the gradual melting of bound states in the quark-gluon plasma close to the critical temperature.

In the presence of a nonzero fermion mass mCFT provides rigorous results inside the IRFP basin of attraction; these are derived in the next section. Away from the IRFP, known cases, such as QCD at zero temperature or QGP close to its critical temperature, provide a phenomenological insight. QCD is conning in the infrared, and free in the ultraviolet. Its complete spectral function (E) entails the low-energy resonances and the high-energy continuum. Schematically, it is made of a series of Dirac

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d3x [angbracketleft]0[notdef]T JH(t, [vector]x)JH(0, 0)[notdef]0[angbracketright] t H+3

[integraldisplay]

-functions (the propagator poles) and a continuum with a high-energy threshold (E)

E H4 AH (E2 m2H) + C (E E0)

, where for simplicity we assumed one resonance (pole) per channel H. If we take in eq. (2.2) the kernel K(E, t) = exp (Et), i.e. the

Fourier transform of a free boson propagator for innite temporal extent, the two-point function CH(t) can be exactly computed in terms of upper incomplete functions

CH(t) = CpoleH + CcontH

XHm H3HemHt +C

t H 3 ( H 3, E0t) . (2.3)

At large times t ! 1, using (s, x)/(xs1 exp (x)) ! 1 as x ! 1, one obtains for the

high-energy continuum contribution

CcontH(t)

E H40t eE0t , t ! 1 . (2.4)

The high-energy continuum generates an exponentially decaying contribution with time dependent coe cients, and leading 1/t behaviour at large times, di erently from the low-energy poles.

QGP close to its critical temperature is instead an example of a deconned, though strongly interacting, theory with restored chiral symmetry; it is almost analogous to a theory inside the conformal window, except that there is no IRFP. A realistic description of the QGP two-point functions is a rich subject of study that is beyond our analogy. It is here enough to observe that the system undergoes a gradual melting of the QCD bound states till their disappearance into a continuum. How gradual is the analogous transition inside the conformal window depends on various factors: the nature of the zero temperature phase transition that opens the conformal window at Ncf, the strength of the interactions at the IRFP, the quantum numbers of the would-be hadrons.

2.1.1 Universal scaling laws at innite volume

In order to understand the behaviour of the two-point functions CH(t) and the would-be hadron masses in the surroundings of the IRFP we summarize, partly reformulate and adapt to our case known aspects of the scaling theory at a conformal xed point. The scales of the system are the lattice spacing a (that can be thought as the inverse of an ultraviolet momentum cuto ), the characteristic length that will emerge in the scaling analysis, and the spatial length L of the lattice box; we shall consider specic ranges for a, and L. The couplings are the fermion mass m and the gauge coupling g. They have scaling dimensions

[m] = 1 + [g] = g , (2.5) where the scaling dimension is the sum of the canonical dimension and the anomalous dimension, and g, respectively. At the IRFP, g = g and m = 0, the anomalous dimen

sions have values and g, and we introduce the exponent = (1 + )1 for later use.

For the interacting theory, the coupling m is always relevant in the RG sense (0 < < 2), while g has g < 0 due to perturbative quantum corrections and it is thus irrelevant. We

rst consider the lattice system in the innite volume limit L ! 1 and in the continuum

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limit a ! 0.2 The invariance of the system under a rescaling of coordinates in the presence

of a small perturbation of the relevant coupling m at the IRFP provides the universal scaling relations for CH(t); they are universal in the sense that they do not depend on the microscopic details of the system. Under a rescaling of coordinates x[prime] = x/b where x indicates both space and time the form of the correlation is dictated by its scaling dimension sH

CH(t/b; b1/ m) = bsHCH(t; m) , (2.6)

where sH = H 3. Setting m = 0, eq. (2.6) yields CH(t/b) = bsHCH(t), hence

CH(t) tsH, the scaling form already introduced in eq. (2.2). Setting b = exp (l) gives

a better intuition of the approach to large distances. Since eq. (2.6) holds for any l, we can choose l = l so that m exp (l / ) = 1. We can also dene a characteristic length = exp (l ), so that we obtain

CH(t) = ZH sH F

[parenrightbigg]

. (2.7)

The coe cient ZH accounts for the microscopic details of the system and has zero scaling dimension. The second factor carries the scaling dimension sH of CH, while the third factor is the adimensional universal scaling function that only depends on the ratio t/, with zero scaling dimension. In the most general case, the scaling function F will depend upon all possible products with zero scaling dimension, made of a scale and a coordinate or a relevant coupling. The function F is universal in the sense that does not depend on the microscopic details of the system. It depends on the scaling dimension of the operators H = M, N, their spin and normalisation. For any nonzero m, one can think of as a nite characteristic length = m ; eq. (2.7) says that for a change of m, the coordinate t changes at the scale . In the limit ! 1, or equivalently t , the system ap

proaches the IRFP and eq. (2.7) should reproduce the form CH(t) tsH; this constrains the asymptotic behaviour of the scaling function F (t/). In the opposite limit t , the

two-point functions decay exponentially, so that

CH(t) tsH t

CH(t) sH f [parenleftbigg]

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[parenrightbigg]

et/ t . (2.8)

While the time dependence is dictated by the scaling dimension of CH, the numerator in the limit t depends on the spin and normalisation of the operators H = M, N. In the

opposite limit t , dimensional reasoning allows for all terms in f(t/) that reproduce

the correct scaling at the xed point, ! 1 or equivalently t . Specically, it allows

for all powers t , with 0 sH and including = 0. While the limit t is uniquely

determined by the conformal xed point, the details of the limiting behaviour for t

depend on the nature of the interactions in the quantum system. Our case is that of composite operators in a deconned and interacting non-Abelian gauge theory close to a non

2The latter limit is realized, in practice, whenever the characteristic length , dened later in eq. (2.7), is much larger than a.

trivial IRFP; we do not know a priori if the system is weakly or strongly coupled. Without a detailed knowledge of the quantum correlators, we can still infer their most general form close to the xed point. The presence of a mass threshold, i.e. nite, produces a pole and branch-cuts (i.e., a continuum of critical excitations) in the propagators of the quantum composite operators.3 Using for the sake of illustration the schematic form in eq. (2.3) for the continuum part, with threshold E0 = n/, for some n, one obtains

CH(t) sHet/ +

C tsH

sH, nt

[parenrightbigg]

, (2.9)

for any intermediate time and nite . The rst term is generated by the would-be hadron pole with residue sH. The second term is a continuum of excitations with energy threshold proportional to 1. The relative position of the pole and thresholds depends on the specic theory and interactions. It is always true, however, that all thresholds and the pole merge at the xed point, where the residue at the pole vanishes.

Note that the would-be hadron pole, the rst term in eq. (2.9), is proper of the deconned theory close to the IRFP and is not to be identied with the hadron poles of conned QCD. In fact, it vanishes at the xed point and the theory smoothly ows to pure Yang Mills at innite fermion mass, i.e. ! 0. This description of theories inside the conformal

window does not need, but does not exclude, the occurrence of a phase transition at a nite fermion mass between a deconned and a conned phase, a scenario proposed in [16]; note that connement is always realized in the limit m ! 1, where fermions decouple.

From the practical viewpoint of a lattice study of the system, it is su cient to observe that the two-point functions CH(t) in the presence of a fermion mass m = 1/ are dominated by a constant times an exponential, i.e. sH exp (t/), for t and [negationslash]= 0,

while the time-dependent power-law contributions to f(t/) in eq. (2.8) become increasingly important at smaller times and for decreasing masses. We also observe that the addition of a UV cuto , i.e. a nonzero lattice spacing a, or a nite temporal extent t T , do

not qualitatively change any of the properties discussed, nor a ect the extraction of the would-be hadron mass from the dominant pole contribution.

The comparison of eq. (2.7) with the large euclidean time behaviour CH(t) exp (mHt) for a would-be hadron of mass mH provides the universal scaling form

mH = cH m , (2.10)

at coupling g = g and with coe cient cH that depends on the spin of the operator H = M, N.

2.1.2 Universal scaling laws at nite volume

Keeping g = g , we now consider the system at nite volume L, with , L a, and we

trade the characteristic scale for the mass m, using 1 = m ; in this way, the system

3Analogous examples can be found in magnetic systems close to a quantum critical point, where a quasiparticle pole and multiparticle continuum thresholds are present in the system.

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has e ectively one relevant scale L, and we can study the scaling of CH under a change of L. In full analogy with eq. (2.7), the correlation is now4

CH(t; m, L) =H LsH F[parenleftbigg]

L, Lm [parenrightbigg]

. (2.11)

It is the product of a coe cientH, which accounts for the microscopic details, a scaling factor and the universal scaling function F with zero scaling dimension arguments, made of products of the scale L and a coordinate or a coupling. The leading scaling form of the would-be hadron mass in the channel H as a function of the fermion mass and L now reads

mH =H 1

L f(x) x = Lm , (2.12)

where the scaling function f(x) depends on the parameter x with zero scaling dimension. To recover the innite volume limit of eq. (2.10), one needs f(x) x as x ! 1 andH = cH.

One has also f(x) ! const as x ! 0. The function f(x) a priori depends on the channel

H = M, N through the spin of the corresponding operators. However, once the constant cH is factored out, its x ! 1 limit is H independent. Moreover, the study of section 4.3 and

gure 12 suggests that most of the H dependence is contained in cH for all x. Eqs. (2.10) and (2.12), important for the scaling analysis of the lattice results, were rst derived in [14]. Eq. (2.12) says that nding the universal curve followed by LmH/cH as a function of x, with 1/ = 1 + , is a way to determine the mass anomalous dimension at the IRFP.

2.1.3 New operators and corrections to universal scaling

The emergence of new marginal or relevant operators can occur in the system at su ciently strong coupling, e.g., the four fermion operator can turn from irrelevant to marginal, or relevant. In this case the xed point structure has to be reconsidered, together with the RG ow towards the xed point(s) in the enlarged parameter space. This is hardly the case for the IRFP studied here, but it may play a role in the possible emergence of a new ultraviolet xed point at stronger coupling.

On the lattice, one can isolate the universal scaling in eq. (2.12) by identifying the perturbative corrections in the volume L and mass m that produce a deviation from the scaling function f(x) and possible nonperturbative scaling violations. We discuss a few aspects below.

perturbative corrections in L and m: close to the xed point the irrelevant

couplings, i.e. g [negationslash]= g in our case, generate perturbative corrections made of products

of couplings and scales with zero scaling dimension.

nonperturbative corrections: they should be expected when the scale(s) of

the microscopic dynamics becomes comparable to or larger than the characteristic length of the system. Violations of universal scaling in this context can appear with nonuniversal functions that cannot be factored out of f(x). One example is when

4The large volume limit of the eld theory at the IRFP can be treated analogously to the low-temperature perturbation of a lattice spin system at a zero temperature critical point.

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the box size L becomes small compared to the Compton wavelength of the would-be hadron . Other examples are provided in this context by the occurrence of the bulk phase transition at strong coupling, which changes the underlying symmetries of the system, or the possible appearance of new phases that precede the bulk transition and are induced, e.g., by competing interactions in Symanzik-improved lattice fermion actions on coarse lattices [10].

The rst type of corrections, perturbative in L and m, deserves some discussion. They are induced by the irrelevant coupling g, whenever we take g [negationslash]= g . Consider a small

variation g; the smaller the scaling dimension g of g, the slower the system will ow back to the IRFP. The leading perturbative corrections to eqs. (2.7) and (2.11) can be of two types. Firstly, ZH (H) and m (the relevant coupling) should be redened, in a way that may generically be di cult to compute.5 Secondly, multiplicative corrections to the scaling function F appear. They can be written in terms of products with zero scaling dimension, made of the coupling g and the relevant scale, thus 1 + g g, or

equivalently 1 + gm g, in eqs. (2.7) and (2.11), and 1 + gL g in eq. (2.11). The rst corrections become increasingly unimportant as ! 1 (m ! 0), the second ones as

L ! 1. The same perturbative corrections for g [negationslash]= g will enter mH as redenitions of

cH, the fermion mass m itself (equivalently the scaling dimension ), and multiplicative corrections of the type 1 + g g and 1 + gL g.

The coe cient g, referred to as the tting parameter b in section 4, measures

the deviation of the gauge coupling from its value at the xed point, or, on the lattice, the deviation from g along the line of lattice parameters that ows to the

IRFP in the continuum limit. It is a nonuniversal parameter that vanishes at the IRFP, possibly changing its sign. In particular, when symmetries allow for a linear dependence on g g , we can expect g (or the parameter b) to change sign at

the boundary between the two phases of the lattice system, indicating if the latter is located on the strong coupling side of the IRFP, i.e. the QED-like phase on the lattice, or the weak coupling side of the xed point, i.e. the asymptotically free phase on the lattice. We also expect g to ow to zero in the continuum, i.e. L ! 1, where the lattice system reaches the IRFP.

The exponent g, entering the perturbative corrections to scaling, is universal and

it is given by the anomalous dimension of the fermion mass and the anomalous dimension of the gauge coupling at the xed point g; this tells that the universal

scaling function as well as its perturbative corrections contain information on .

We will encounter realizations of the perturbative corrections in m and nonperturbative corrections in L in section 4. As a nal note, the mixing of the couplings m and g under the RG ow cannot generate in this system a new xed point la Wilson-Fischer at m [negationslash]= 0 due to the chirality protection of the fermion mass term, i.e., dm/d[notdef] / m with renormalization

scale [notdef]. We study eq. (2.12) on the lattice in section 4.3 and eq. (2.10) in section 4.5.

5However, a perturbative treatment of the RG equations in the case of an IRFP at weak coupling may provide the leading contributions to these corrections.

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2.2 The fermion mass anomalous dimension at the IRFP

One aim of this study is to measure nonperturbatively on the lattice, by identifying universal scaling and the pattern of scaling violations in the would-be hadron spectrum. Numerous lattice studies have recently reported over a wide variety of evidences that the Nf =

12 theory is inside the conformal window, supporting the rst results in [17], based on the attening of the running gauge coupling, and [4] that probed the very existence of the conformal window. While the phenomenological interest mainly resides in theories just below the conformal window, it remains important to determine the pattern and sizes of observables inside the window, since they are directly, and in some cases smoothly, related to their value below the window lower endpoint; one such quantity is the fermion mass anomalous dimension along the IRFP line. Its value determines if the theory is strongly or weakly interacting at the IRFP. It is null in the free theory, at Nf = Naff, and it is bounded to be < 2 by the unitarity of the conformal theory [18, 19]. An appealing conjecture suggests = 1

at the lower endpoint of the conformal window, Nf = Ncf, so that chiral symmetry breaking is triggered [20] for = 1 the four-fermion operator becomes relevant. The theoretical question if = 1 holds exactly at Ncf is still open; it is however appealing to think that the exactness is realized [21], and some consequences of this scenario are discussed in section 4.5.

The value of can be determined nonperturbatively on the lattice6 and it has been computed in perturbation theory at two-loops [22], three and four loops [23, 24], see also [25, 26] for studies in the large-Nf limit, [27, 28] for analyses of renormalization scheme transformations and [29] for a recent xed point analysis of classes of gauge theories. It is thus mandatory to compare the genuinely nonperturbative lattice determination with perturbation theory, and in this spirit we analyze the lattice results in section 4. Here, we discuss a few relevant aspects that can be inferred from [23, 24] and we compute the gauge coupling anomalous dimension g to four loops. By inspection of the four-loop

-function and the mass anomalous dimension in [23, 24] one observes that:

The Nf = 12 IRFP coupling in the MS scheme moves from g2/(42) [similarequal] 0.24 at two

loops to g2/(42) [similarequal] 0.15 at four loops, and the mass anomalous dimension at the

IRFP a renormalization scheme independent quantity moves from [similarequal] 0.77 to [similarequal] 0.25. The latter value provides [similarequal] 0.8; this value will turn out to be in good

agreement with our lattice determination.

The IRFP coupling moves towards the origin when going from two to four loops,

while the lower endpoint of the conformal window stays around Nf = 8.

The Nf dependence of the coe cients i and i, i = 0, 1, 2, 3, of the four-loop -

function and , respectively, deserves some discussion. The four-loop coe cient 3

grows rapidly with Nf and it is responsible for the appearance of a new zero for

Nf 17, just above the conformal window. Analogously, the four-loop coe cient

3 grows rapidly with Nf and causes a change of sign of the running anomalous

6It is worth noting that other eld-theoretical techniques, such as conformal bootstrap and variations thereof, are becoming increasingly useful in constraining correlators and anomalous dimensions of operators in strongly coupled eld theories.

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dimension at some g for Nf 8; importantly, this happens at a coupling g g

for Nf = 12, suggesting that perturbation theory may still be reliable for Nf = 12 at g = g . This is no longer true for Nf 8, where g is larger and the change

of sign occurs for g < g . We do not further address the problem of the reliability of perturbation theory for Nf 8, with or without a truncation to a given order

in the loop expansion. We only note that this is the region where nonperturbative contributions can play a signicant role in the disappearance of the conformal window. An instructive comparison between perturbation theory and nonperturbative results can be carried out in the Veneziano limit, see [27] for studies in this limit, or with a large-Nf resummation as initially proposed in [25]. Recent progress in constraining by eld-theoretical techniques the correlators of large-N QCD and N = 1 SQCD [30, 31]

should help in the task of identifying the role of nonperturbative contributions.

Finally, we derived the value of the gauge coupling anomalous dimension at the

IRFP to two and four loops from [23] as g = @ (g)/@g[notdef]g=g

. As all other critical exponents, this is a renormalization group invariant quantity. A straightforward calculation gives g[notdef]2loop [similarequal] 0.360 and g[notdef]4loop [similarequal] 0.283, consistently with the fact

that the IRFP moves towards weaker coupling from two to four loops.

To summarize, four-loop perturbation theory predicts [similarequal] 0.25, i.e. [similarequal] 0.8, and the

universal exponent [similarequal] 0.23 for the Nf = 12 system at the IRFP. This prediction

misses the nonperturbative contribution, and some renormalization scheme dependence can be induced by the truncation of the perturbative expansion. A comparison with the nonperturbative lattice determination of is therefore instructive. Figure 4 collects recent lattice determinations of and the predictions of perturbation theory, anticipating the result of this work later discussed in section 4. The most salient feature of gure 4 is the agreement among lattice determinations, and their agreement with the four-loop perturbative prediction, once a universal scaling analysis is carried out. This is true for [9] and this work. Previous pioneering determinations of the mass anomalous dimension, the very rst one in [4] and the ones in [3236] were obtained through the analysis of specic H channels in the would-be hadron spectrum or the eigenvalues of the Dirac operator at some bare lattice coupling, without a systematic identication of the universal scaling contributions and violations thereof. Despite this, all determinations are contained in an interval that is well below = 1. This shows the stability of the prediction and the fact that the (lattice) system is not largely sensitive to deviations from the IRFP. Importantly, measurements can be done on both sides of the xed point, being it the asymptotically free side or the QED-like side. In section 4.5 we futher discuss our determination of

and the implications of the obtained value.

2.3 The Edinburgh plot

Besides conformal symmetry, one relevant ingredient characterizing the two-point functions in the conformal window is restored chiral symmetry. Everywhere in the asymptotically free and the QED-like phase, chiral Ward identities must be fullled by the renormalized

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present[0][1][2][3], b=3.7[3], b=4.0[4][5] perturb.

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4l-MS

3l-MS

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 g

Figure 4. Collection of recent lattice determinations of the mass anomalous dimension for the Nf = 12 system, and the perturbative prediction of its value at the IRFP to 2, 3 and 4 loops. From top to bottom, the value from this work, [0] from [4], [1] from [32], [2] from [33], [3] from [34], [4] from [9], [5] from [35], and the perturbative determinations from [23, 24].

correlation functions. There is no Goldstone boson, since chiral symmetry is not spontaneously broken, and all chiral partners, scalar and pseudoscalar, vector and axial, must be degenerate in the chiral limit.7

The Edinburgh plot, widely used in lattice QCD studies and suggested as a probe of the conformal window in [37], is constructed in terms of adimensional ratios of masses, and/or decay constants, and it traditionally o ers a powerful way to combine results of lattice calculations performed at di erent lattice spacings and with di erent lattice actions. In this case, it also provides a clear visualisation of di erent mass regimes and distinguishes between QCD and theories inside the conformal window; we use it here to illustrate the behaviour of the Nf = 12 system, adopted as a prototype of theories inside the conformal window. Figure 5 shows the Edinburgh plot for the Nf = 12 innite volume lattice results in table 1 for am > 0.025 and in table 4 for am 0.025. The physical point of QCD (left of

gure) corresponds to m/m [similarequal] 0.18 and mN/m [similarequal] 1.21. At the other side of the gure

a useful theoretical limit is the heavy quark mass limit, where all masses of the would-be hadrons are given by the sum of their valence quark masses so that m/m = 1 and mN/m = 3/2. A QCD scenario would correspond to a curve in gure 5 that extrapolates to the QCD physical point for decreasing quark masses, i.e., it would join the two red points in gure 5. Instead, we observe that the two mass ratios are stuck at a tiny corner of the plot, away from the heavy-quark limit and the QCD physical point for a wide range of bare fermion masses 0.01 am 0.07. This is to be expected inside the conformal

window; would-be hadron masses scale as mH = cH m at the IRFP, ideally producing one point in the Edinburgh plot. Moving away from the IRFP in mass, coupling(s) and

7Exact chiral symmetry implies the degeneracy of the complete renormalized two-point functions in the channels that are chiral partners, and the degeneracy of the corresponding renormalized chiral susceptibilities the integrals of the two-point functions.

Figure 5. The Nf = 12 Edinburgh plot for the innite volume lattice results in table 1 and 4 at L = 3.9: the ratio of the nucleon (N) and the vector () mass is shown as a function of the ratio of the pseudoscalar () and the vector mass. We show the scaling point (blue diamond) with coordinates (x, y) = (c/c, cN/c), with c,N from Fit I in table 5 and c in table 7 (This work).

The superimposed (blue solid) line y = (cN/c)x (error band not shown) entails the perturbative scaling violations derived in section 4. The coe cients cH, H = , , N are a priori L dependent, so that the solid (blue) line as well as the scaling point ow to their continuum value, as L ! 1,

cf. section 4.3. The QCD physical point (red star, leftmost) and the heavy quark limit (red star, rightmost) are also displayed.

volume produces some scattering of the data. A mild mass dependence of the innite volume mass ratios is induced by perturbative scaling violations for g [negationslash]= g and m [negationslash]= 0.

Anticipating the results of section 4, these are represented by points distributed along the solid line that passes through the scaling point in gure 5. All scaling violations still obey the constraints implied by the underlying restored chiral symmetry.

3 Numerical setup

3.1 The action

We have generated congurations of an SU(3) gauge theory with twelve degenerate avours Nf of staggered fermions in the fundamental representation using a tree level Symanzik improved gauge action

S =

where U(C) are the traces of the ordered product of link variables along the closed paths C divided by the number of colors. S0 and S1 contain all the 1 [notdef] 1 plaquettes and

1 [notdef] 2 and 2 [notdef] 1 rectangles, respectively. The SU(3) lattice coupling of the unimproved

action is given by = 6/g2L and the i are dened in terms of as 0 = (5/3) and 1 = (1/12) . According to the way lattice simulations are performed and reported, we

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4 Tr ln M(am, U) +

Xi=0,1 i(g2)

XC2Si

Re(1 U(C)) (3.1)

use L(= 0) = 10/g2L for the lattice results of this work, and = 6/g2L for some of the existing lattice results discussed in section 4.

Improvement is extended to the fermionic sector following the Naik prescription. The action of the fermionic sector can be written in terms of the one component staggered fermion eld [vector](x) as

SF = a4

Xx;

(x)

[vector](x) 1 2a

nc1

hU(x)[vector](x + [notdef]) U(x [notdef])[vector](x [notdef])[bracketrightBig] +c2 [U(x)U(x + [notdef])U(x + 2[notdef])[vector](x + 3[notdef])

U(x [notdef])U(x 2[notdef])U(x 3[notdef])[vector](x 3[notdef])[bracketrightBig][bracerightBig]

+a4m

[vector](x)[vector](x) (3.2)

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with the phase factor (x) = (1)(x0+x1...+x1). Order a2 accuracy at tree level is achieved by using the Naik choice c1 = 9/8 and c2 = 1/24.

This action is the same used in previous studies conducted by our group on SU(3) with Nf = 12 [4, 10, 12]. In particular, this action corresponds to the choice D of [10].

The theory under study exhibits a bulk transition separating a region at weak coupling where chiral symmetry is restored from a region at strong coupling where chiral symmetry is broken [4], as it is expected for all theories in the conformal window. For small enough bare fermion masses and with our choice of action, the competition induced by next-to-nearest neighbour interactions in eq. (3.2) causes the emergence of an intermediate phase at nite lattice spacing, just before chiral symmetry is broken, as one goes from weak to strong coupling [10], see also gure 2.

We generated congurations for a range of bare fermion masses going from am = 0.01 to am = 0.07 at xed coupling L = 10/g2L = 3.9. This choice guarantees that our simulations are carried in the chirally restored region, away from the bulk transition and the exotic phase induced by the improvement. For the heaviest bare masses am = 0.06 and am = 0.07 we simulated volumes 163 [notdef] 24 and 244. For bare masses am = 0.05, 0.025 and

0.020 we have simulated volumes 244 and 324. For bare masses am = 0.01, 0.04 we have simulated volumes 243 [notdef] 32 and 324. In addition, volume 163 [notdef] 32 was simulated for bare

masses am = 0.01, 0.02, 0.025 in order to make it possible for us to obtain innite volume estimates of the spectrum for these quark masses.

During the runs, the parameters for the acceptance rate have been tuned to yield a good acceptance while keeping a xed trajectory length l 0.4 for all ensembles.

Congurations were saved every ve simulated trajectories, so that saved congurations are separated by approximately two unit trajectory lengths. Measurements of observables such as the chiral condensate and the average plaquette were conducted on the y, while the measurements of the particle spectrum were conducted on the saved congurations following the strategy described in the next section.

3.2 Strategy for the spectrum measurements

We have measured the two-point functions with currents JM q Mq in the scalar, pseu

doscalar, vector and axial channels and the nucleon correlation function on the saved ensembles using corner-wall sources with xed Coulomb gauge. The gauge xing procedure in traditional QCD is known to reduce contamination from excited states and helps to better isolate the ground state of the system. In order to extract the lowest-lying masses we found it useful to construct the meson correlators from quark propagators with di erent combinations of temporal boundary conditions. This procedure was discussed in [38], it is extensively used in lattice QCD and, recently, it was explicitly implemented for SU(3) with Nf = 12 in [34]. Some caveats are in order inside the conformal window, where the two-point function has the form in eqs. (2.7) and (2.8). We rst summarize the strategy in the case of an exponentially decaying two-point function with constant coe cient, which is realized in all studied cases over a large time interval. Consider the quantity

C(t) = 12 [Cp.b.c.(t) + Ca.b.c.(t)] (3.3)

built from meson correlators with periodic (Cp.b.c.) and antiperiodic (Ca.b.c.) temporal boundary conditions on a lattice of temporal extent T . It can be shown that the resulting combined correlator C(t) has its lattice temporal extent e ectively doubled and it can also be written as the periodic correlator Cp.b.c.(t) with doubled period 2T [38]. In the case

of the pseudoscalar meson, the staggered two-point function does not contain a staggered parity partner state. Taking into consideration a possible constant oscillation term that might appear as a consequence of the wrapping of a quark line around the antiperiodic time boundary, we can then write

CPS(t) = A

emt + em(2Tt)

[parenrightBig]+ B(1)t (3.4)

The constant oscillation term can be removed by using the combination

PS(2t) = CP

S(2t)

2 +

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CPS(2t + 1)

4 +

CPS(2t 1)

4 (3.5)

so that the nal correlator is

PS(2t) = A

em2t + em(2T2t)

[parenrightBig]

. (3.6)

The e ective doubling of the temporal extent allows to better isolate the rst term in eq. (2.9) and enlarges the corresponding e ective mass plateau.

A similar combination of meson correlators with di erent boundary conditions in the time direction can be performed for the other mesons. We use the PV correlator to extract the masses of the would-be vector meson and the a1 axial meson. The averaged PV correlator can be written in this case as

CPV (t) = A

emt + em(2Tt)

[parenrightBig]+ Aa1(1)t [parenleftBig]

ema1 t + ema1 (2T t)

[parenrightBig]+ B(1)t , (3.7)

It is possible to proceed with a similar combination to that of eq. (3.5), so that

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Figure 6. Examples of antiperiodic (blue circles), periodic (red squares) and combined (black diamonds) meson correlators from eq. (3.3), obtained from congurations generated with bare quark mass am = 0.05 at volume V = 324. (Left) The pseudoscalar correlator, (right) the pseudovector correlator.

Figure 7. (Left) The periodic (red squares), antiperiodic (blue circles) and combined (black diamonds) scalar correlators, (right) the periodic (red squares) and antiperiodic (blue circles) nucleon correlator for a bare quark mass am = 0.05 and volume V = 324.

PV (2t) =[notdef]1

[notdef]2, and the resulting correlator is well approximated by a single exponential form with coe cient[notdef]1

PV (2t) [similarequal]

[notdef]1

em2t + em(2T2t)

[parenrightBig]+[notdef]2

ema12t + ema1(2T2t)

[parenrightBig]

. (3.8)

In fact, such an approach has been followed in [34], where it was noted that, for the range of volumes and quark masses studied by the authors,[notdef]1

em2t + em(2T2t)

[parenrightBig]

. (3.9)

We have noticed that while the approximation (3.9) holds true for our heavier quark masses, it starts to break down for our lightest quark masses. In addition to that, we are also interested in studying the behaviour of the mass of the axial meson ma1. For these reasons, we t the PV correlators obtained from our congurations to the complete functional form (3.7) in order to extract both m and ma1. Examples of the initial correlators and the quality of the nal combined correlators are shown in gure 6.

In similar fashion to eq. (3.7), we extract the mass m of the scalar meson from the averaged S correlator

CS(t) = A

emt + em(2Tt)

[parenrightBig]+ B(1)t [parenleftBig]

eMt + eM(2Tt)

[parenrightBig]+ B(1)t . (3.10)

Lastly, we built the nucleon correlators from quark and antiquark propagators with antiperiodic boundary conditions in the temporal direction. These were then tted with the usual expression for the lowest-lying states of a staggered baryon two-point function containing four parameters

CN(t) = AN

emNt (1)temN(Tt)[parenrightBig]+ BN(1)t [parenleftBig]

eMt (1)teM(Tt)[parenrightBig](3.11)

with parity partner of mass M. Examples of these correlation functions are shown in gure 7.

4 Results

Table 1 collects all lattice measurements of this work. Simulations have been done at inverse lattice coupling L = 3.9, located in the QED-like region of gure 2. The same coupling was also used in our rst study [4]. The masses of all would-be hadrons have been measured for a range of bare fermion masses between am = 0.01 and am = 0.07, and volumes between 163 [notdef]24 and 324. This section is organized as follows. After comparing the two-point func

tions with the ones of the free theory, and testing them against eqs. (2.7) and (2.8) in section 4.1, we provide in section 4.2 a tool to establish in which phase, QED-like or asymptotically free, the lattice system is. Section 4.3 is dedicated to the spectrum in a nite volume. It establishes the realization of universal scaling for the lattice results according to eq. (2.12) and provides a unied description of all available lattice data for the Nf = 12 system while identifying the pattern of scaling violations on both sides of the IRFP. Section 4.4 treats the extrapolation to innite volume, needed for the lightest masses. Section 4.5 is dedicated to the spectrum at innite volume, it established the realization of universal scaling for the lattice results according to eq. (2.10) and identies the pattern of scaling violations, in particular, for the spin-1 states of this work and those in [34]. This analysis leads to the determination of that consistently describes all available lattice data for the spectrum of the Nf = 12 system at nite and innite volume. Finally, section 4.6 is a brief discussion of mass ratios and degeneracies of chiral partners, probe of restored chiral symmetry.

4.1 Two-point functions

We have analyzed all two-point functions according to eq. (2.7) and its asymptotic forms in eq. (2.8). Our results are easily summarized. For the entire range of masses explored, the best ts to C(t) (with period 2T ) over the late time range are obtained for the form a exp (mt), symmetrized on 2T , with a constant and mass m. Time dependent

corrections will increasingly be present at small times for decreasing masses, rendering more di cult the determination of the would-be hadron masses. The corrected form a exp (mt) + b/t exp (nt), with b > 0 and n > m, ameliorates the ts at smaller times,

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am V olume am am amN am ama1 0.01 163 [notdef] 32 0.4421(17) 0.516(10) 0.867(26) 0.4388(15) 0.5086(36)

243 [notdef] 32 0.2701(44) 0.3002(61) 0.478(9) 0.2682(36) 0.305(30)

324 0.1942(21) 0.2114(52) 0.326(10) 0.2057(22) 0.2295(31) 0.02 163 [notdef] 32 0.4480(16) 0.499(25) 0.846(27) 0.446(8) 0.509(14)

244 0.3112(34) 0.3432(85) 0.528(14) 0.3297(35) 0.3903(91) 324 0.2624(13) 0.2857(11) 0.428(8) 0.3117(20) 0.3464(34) 0.025 163 [notdef] 32 0.4511(16) 0.5081(46) 0.892(31) 0.4598(48) 0.5102(32)

244 0.3447(12) 0.374(10) 0.591(21) 0.3916(82) 0.411(20) 324 0.3087(13) 0.3332(16) 0.530(13) 0.3786(49) 0.4048(55) 0.04 243 [notdef] 32 0.4236(57) 0.4890(60) 0.726(13) 0.543(24) 0.617(49)

324 0.4210(16) 0.4717(23) 0.709(2) 0.5359(51) 0.5783(83) 0.05 244 0.5020(23) 0.5652(76) 0.851(17) 0.6452(26) 0.703(41)

324 0.5031(21) 0.5689(17) 0.850(5) 0.6463(32) 0.7097(83) 0.06 163 [notdef] 24 0.5921(48) 0.678(12) 1.028(49) 0.747(13) 0.823(40)

244 0.5881(20) 0.6700(18) 1.003(8) 0.746(19) 0.831(10) 0.07 163 [notdef] 24 0.6600(19) 0.7663(48) 1.116(47) 0.831(14) 0.914(48)

244 0.6596(27) 0.7597(24) 1.111(6) 0.831(27) 0.918(34)

Table 1. Masses of the lowest-lying would-be hadrons, the pseudoscalar (), the vector (), the scalar () obtained from the quark-line connected part of the isoscalar correlator the axial (a1), and the nucleon (N) for bare quark masses am = 0.01 to 0.07 and lattice coupling L = 3.9.

The volumes span from 163 [notdef] 24 to 324.

as expected for our lightest mass am = 0.01, such corrections start to become relevant when considering times t < 10 and requires n [greaterorsimilar] m.

In gure 8 we compare the two-point functions at L = 3.9 with the corresponding ones obtained in the free case, with the same lattice staggered action and the same bare fermion masses; this comparison is useful to clarify how far the studied regime is from the free limit. In fact, if the theory is deconned one may expect a faster approach to the free limit than it is realised in the conned theory. One useful ingredient in the comparison is that two-point functions built with increasingly free quarks exhibit increasing sensitivity to the change of boundary conditions, both in the spatial and temporal direction. The zero-momentum free meson two-point functions reproduce the known analytical form for staggered correlators on even and odd temporal sites [39], and indeed gure 8 (left) shows the signicant di erence between the standard periodic meson two-point function built with periodic (P) and the one built with antiperiodic (A) temporal boundary conditions on the single free quark propagators. This di erence is absent in the two-point functions at L = 3.9 in gure 8 (right), for the same bare fermion mass. Note also that exact point-by-

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Figure 8. (Left) The pseudoscalar two-point function Cp.b.c. (logarithmic scale) in the free case with am = 0.05, made of two free quark propagators with antiperiodic boundary conditions (red squares) and periodic boundary conditions (black circles) in the temporal direction. (Right) The same two-point function at L = 3.9. Vertical axes are rescaled to match.

point degeneracy of the scalar and pseudoscalar free meson correlators in the chiral limit is only realized at zero lattice spacing, since they involve a sum over the momenta of the fermion and antifermion propagators. In accordance with their analytical form [39], we nd that pairs of chiral-partner free correlators are exactly degenerate at odd times and increasingly degenerate at even times towards the chiral limit. This comparison conrms that the two-point functions at L = 3.9 are signicantly away from the free limit, and well described by an exponential with a constant coe cient AH over a large time interval.

4.2 Would-be hadrons in the QED-like region

Figure 9 provides a tool to understand which phase of the Nf = 12 system we are looking at. It shows the mass ratio of the would-be pseudoscalar and vector mesons as a function of the bare fermion mass; these are the innite volume lattice results in table 1 for am > 0.025 and in table 4 for am 0.025. It also provides a direct comparison of our results with

those of [34], the latter obtained with a HISQ staggered action at two lattice couplings. In our initial study [4], where more than one lattice coupling including L = 3.9 and 4.0

was considered, we could conclude that our results were located in the QED-like region of the theory, i.e. on the strong coupling and non asymptotically free side of the IRFP, with a positive -function. The results in gure 9 update that study at lattice coupling L = 3.9;

data for the ratio at L = 4.0 would be located on a curve with similar slope, to the right of L = 3.9. The analogous study in [34] led the authors conclude that their results are instead located on the weak coupling and asymptotically free side of the IRFP. The same can be inferred from gure 9, where the crucial ingredients are the slopes and the ordering of curves. A line of constant physics would lead to a constant ratio m/m; a realization of such a line occurs at the IRFP, where the -function is zero and universal scaling holds with m , = c ,m . Away from the xed point, a family of curves at di erent lattice couplings as in gure 9 carries information about the sign of the -function. For our data, the crossing of a line of constant ratio with the curves at xed lattice coupling L = 3.9

and an ideal line for L = 4.0 at its right implies a positive sign of the -function, where

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Figure 9. The pseudoscalar () to vector () mass ratio as a function of the bare fermion mass. Data at L = 3.9 (left of gure) are at the largest volumes from table 1 (black circles) and the innite volume extrapolation from table 4 is shown for the three lightest points (red squares). Data at L = 4.0 would draw a line to the right of L = 3.9 [4]. Data from [34] (right of gure) are obtained with a HISQ staggered action, at = 6/g2L = 3.7 (green diamonds) and = 4.0 (blue triangles), am = 0.04 to 0.2.

to rst approximation we assume a constant physical mass between the intersections. The data of [34] have the opposite behaviour, and correspond instead to a negative sign of the -function. At rst sight, the reduced slope of the curves in the latter case would suggest that the data of [34] are less a ected by violations of scaling and plausibly closer to the IRFP. Another possibility, implied by the results in section 4.3 and in line with [9], is that di erent mass regimes are covered by the two sets of lattice measurements, both a ected in di erent ways and to di erent degree by violations of universal scaling. Summarizing, the combined set of data in gure 9 nicely covers the region on both sides of the IRFP. This illustrates the fact that observables in this system are actually sensitive to the change of sign of the -function and that some clever combination of these observables can be used to locate the IRFP; we are currently investigating a strategy along this line.

4.3 The spectrum in a box

Figure 10 illustrates the pseudoscalar and vector products LmH for each given L as a function of the bare fermion mass. It is clear that nite volume e ects are present at the largest spatial volume for the three lightest bare masses am = 0.01 , 0.02 and 0.025. At the same time, these data o er the interesting option of a nite size scaling study with the aim of identifying a universal scaling behaviour and dene the functional form appropriate to extrapolate these data to innite volume. Barring the emergence of new operators, we proceed to identify universal and nonuniversal behaviours in the space of couplings (g, m). A comparison with the superimposed best-t curves obtained at innite volume in section 4.5 helps locating the threshold where substantial deviations from a genuine power-law appear in the scaling of LmH, at xed L. These deviations can a priori contain

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Figure 10. LmH for the pseudoscalar (left) and vector (right) would-be hadrons as a function of am and for varying L. For su ciently large volumes and masses the points fall onto a curve (superimposed), which is the power-law best-t curve obtained at innite volume (table 5).

nonasymptotic contributions to the scaling function f(x) of eq. (2.12), as well as genuine scaling violations not described by f(x). The following analysis is devised to identify these contributions at small and large x.

Anticipating the results of section 4.5, we note that the innite volume best t to eq. (2.10) for the pseudoscalar, scalar and nucleon masses gives a critical exponent = 0.81, while the vector and axial states favour a slightly larger exponent = 0.86, see table 5. While the di erence in the values of for di erent H channels is in itself an indication of scaling violations, we note that universality appears to be realized in all but the vector channels and = 0.81 gives a value of = 1/ 1 in agreement,

within uncertainties, with the best t reported in [9] and, noticeably, with the four-loop perturbative prediction [23, 24]; it is thus tempting to conclude that our data are in the universal scaling regime and provide a measure of the mass anomalous dimension at the IRFP. The study that follows supports this conclusion.

In gure 11 we vary the scaling variable x about the best-t value for (central gures) and on the range = [0.5, 1] to study the x dependence of the ratio Lm/c (left) and

Lm/c (right). For x [greaterorsimilar] 1, the data in the central gures align on a common curve. They increasingly scatter and deviate from it when is moved away from its best-t value, over the range 0.5 to 1. Figure 12 reports all states for the reference value = 0.81; we observe the universal behaviour of the pseudoscalar, scalar and nucleon states at x [greaterorsimilar] 1 and the displacement and slight change of slope of the vector and axial states. For x [lessorsimilar] 1, the asymptotic behaviour of the universal scaling function f(x) ! const as x ! 0, is corrected

by nonperturbative L-dependent scaling violations. These are discussed in the next section.

Inspired by recent work [9], we now attempt a unied description of the nite volume results of this work and the results obtained for the same system with other lattice actions, at a priori di erent bare lattice couplings and fermion masses. In particular, we consider the results at = 2.2 in [40] and the results at = 3.7 and = 4.0 in [34]. We limit this analysis to the pseudoscalar channel, studied in all works, while later in section 4.5 we compare our results and those in [34] for the vector state. Figure 13 summarises this study,

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Figure 11. Ratios LmH/cH, cH from table 5, for the pseudoscalar state (left) and the vector state (right) as a function of the scaling variable x = Lm for varying on [0.5, 1] and L = 16, 24, 32. The central gures display the data alignment for the best-t values of in table 5.

where we show the collapse on a common universal curve of data obtained with di erent lattice actions and lattice couplings, once perturbative corrections to the universal scaling are divided out. The general conclusions of this analysis are in good agreement with the study in [9], while our analysis di ers from [9] in some details and interpretation of the parameters. We briey highlight the relevant ingredients and results.

Perturbative corrections to universal scaling are present whenever the system is close

to, but not at the xed point. As pointed out in [9], perturbative corrections due to g [negationslash]= g and a nite fermion mass can explain the deviations from universal scaling

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Figure 12. LmH/cH, cH from table 5, as a function of x = Lm with = 0.81, excluding the vector and axial states (top) and including them (bottom).

R b

This work 1 0

LH 1 0

LatKMI 3.7 1.054 0.5435

LatKMI 4.0 1.193 0.4926

Table 2. Values of R and b used in gure 13.

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Figure 13. Collapse of curves for the rescaled pseudoscalar product LmR/(1 + bm!), with ! = 0.23 from 4-loop perturbation theory and R, b in table 2, as a function of the universal scaling variable x = Lm with = 0.81 from this work. Data are from [40] at = 2.2 (LH, green crosses), [34] at = 4.0 (LatKMI, magenta hexagons) and = 3.7 (LatKMI, blue triangles), and from this work for L = 24, 32 (red squares) and L = 16 (red circles).

of many results for the Nf = 12 system. As explained in section 2.1.2, these contributions can be parameterized to leading order as multiplicative corrections 1 + b m!,

with universal exponent ! = g.

We nd that our best-ts favour values of ! lower than ! = 0.41 of [9], though we

cannot perform a fully unconstrained t. It is therefore appealing to consider the value ! = 0.23 given by = 0.81 our central value in good agreement with the 4-loop prediction within uncertainties and the 4-loop prediction g [similarequal] 0.283.

The values of b in table 2 and the analysis of the vector state in table 7 corroborate

the interpretation of g in section 2. We observe that all data are su ciently close to universal scaling. Our data are located in the QED-like phase of the system and do not show corrections to scaling at these light masses, thus b = 0. The data from [40] are close to our data, and again they seem not to be sensitive to corrections to scaling at the light masses they consider; the only di erence is that their data are located away from the asymptotic linear form of f(x) and we cannot clearly discriminate to which phase they pertain. The results from [34] are located on the other i.e. asymptotically free side of the xed point, thus showing b < 0. We expect the data of [9] to be located between ours and the data from [34] on the asymptotically free side, with b < 0, in agreement with their analysis. In section 4.5 we provide an example of a positive b for our system in the QED-like phase, needed to successfully describe perturbative corrections to scaling for the vector and axial states. Thus, the parameter b changes sign at the boundary between the two phases of the lattice system and we expect it to ow to zero in the continuum, i.e. L ! 1, where the

lattice system reaches the IRFP.

An overall rescaling is the usual procedure to bring together sets of data that follow

universal scaling. We perform a rescaling of Lm by the factor R in table 2. R is given by the ratio of the coe cients cH entering the innite volume functional form mH = cHm (1 + bHm!) for each given data set and channel H. Such a rescaling is asymptotically equivalent to a rescaling of the variable x, as used in [9], provided it does not enter the corrections to the universal scaling function. The factor R for LatKMI is derived in section 4.5. R shows a monotonic dependence on the lattice bare coupling L, converging to its universal value as L ! 1. This statement, as the previous

one for b, assumes that the lattice system is in the basin of attraction of the IRFP.

Once rescaled by R and once the perturbative mass corrections are divided out, the

product Lm for all the lattice data in gure 13 is described by a universal curve f(x) for all x values, except for the presence of nonperturbative violations of scaling for L [lessorsimilar] for some data of this work.

4.4 Extrapolation to innite volume

For am = 0.04 to 0.07, no residual nite volume dependence is left within the estimated uncertainties; we thus take the result at the largest available volume as the innite volume value for am = 0.04 to 0.07. For the three lightest bare fermion masses, am = 0.01, 0.02, 0.025 we have instead performed an extrapolation to innite volume. Lschers formula [41, 42] for particles in a box does in principle apply to any interacting quantum eld theory, provided the scattering amplitude of the particles involved is known. The latter problem is perturbatively solved by chiral perturbation theory ([vector]PT) for QCD-like theories in the chirally broken phase, predicting the leading order behaviour of the Goldstone boson mass to be [43, 44] m(L) = m + c exp (mL)/(mL)3/2 for mL 1.

The functional form to be used in our case is what describes the scaling violations at small x in gures 11 and 12.

The small x behaviour of the pseudoscalar would-be hadron is analyzed in gure 14, for = 0.81. At the smallest values of x, Lm shows an L-dependent deviation from a common curve. L dependent deviations from scaling can be expected whenever the box size L becomes comparable to or smaller than the would-be hadron Compton wavelength

. The entire range of x in gure 14 can be described in terms of the universal scaling

function f(x), with asymptotics f(x) x as x ! 1 and f(x) ! const as x ! 0, and

a nonperturbative L-dependent violation of scaling at small x note that perturbative corrections in L of the type 1 + gL g would instead multiply the entire scaling function f(x) and modify its behaviour at all x. Hence, F (x, L) = ax + g(L) ~f(x) should describe gure 14, except for the presence of nonlinear universal contributions to f(x) at intermediate x. The coe cient a is nothing but cH, H = of table 5, the function g(L) increases for decreasing L according to gure 14, and ~f(x) ! const as x ! 0. Figure 14 also displays

the best-t curves for the simplied ansatz F (x, L) = ax + c exp (kx), with best-t

values of a, c, k in table 3. Rather than aiming at the optimal [vector]2/d.o.f, the purpose of this example is to illustrate the trend of small volume corrections through e ective parameters c and k. The latter is quite stable for varying L, while, as expected, the parameter c

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Figure 14. Lm as a function of the scaling variable x = Lm with = 0.81, for L = 16, 24, 32. The curves are the best ts to the functional form F (x, L) = ax+c exp (kx), with a, c, k in table 3.

The analogous gures for H = S, V, P V, N are in gure 22 in the appendix.

L = 16 L = 24 L = 32 a 5.21(20) 5.59(10) 5.63(20) c 8.07(70) 7.1(2.0) 4.7(3.2) k 1.20(20) 1.31(30) 1.22(70) [vector]2/dof 8.5 10 9

Table 3. Best-t values of the parameters a, c, k and [vector]2/dof for the ts of Lm to the functional form F (x, L) = ax + c exp (kx), with x = Lm and = 0.81.

increases with decreasing L; a polynomial g(L) (1 bL) perfectly describes the data and

provides a volume dependence milder than the universal scaling form 1/L exp (L/) for

m(L) at small x. At the same time, the shift in the paramater a at L = 16, as compared to the larger volumes in table 3, should be attributed to the intermediate x contributions to f(x) that are not captured by the simple ansatz for F (x, L).

A volume dependence milder than QCD and milder than the universal scaling form can be traced back to the Coulomb dynamics in the QED-like phase and the absence of a conning potential. It is favoured by the combination of data in gure 14, gure 15 and the innite volume study of the heavier masses am > 0.025. Having only three volumes for each bare fermion mass, we have performed the extrapolation of the lightest would-be hadron masses to innite volume with the simplest ansatz

mH(L) = mH + c e~kmHL H=, , , a1, N (4.1)

with parameters c, ~k and the innite volume mass mH for the channel H. The results of the extrapolation are summarized in gure 15, table 4 and in the appendix. In order to account for the uncertainty induced by the lack of a complete knowledge of the function

Figure 15. Spatial L dependence of the pseudoscalar mass (left) and the vector mass (right) and extrapolation to innite volume according to eq. (4.1). From top to bottom, am = 0.025 (blue diamonds), am = 0.02 (red squares) and am = 0.01 (black circles). The extrapolated value and its uncertainty is indicated by the horizontal bands and reported in table 4. The channels H = S, P V, N are in gure 21 in the appendix.

am am am amN am ama10.01 0.1343(58)(+599321) 0.1496(49)(+618277) 0.228(16)(+8017) 0.1696(51)(+361149) 0.1842(64)(+453176)

0.02 0.2353(26)(+27177) 0.2522(79)(+335136) 0.382(11)(+573) 0.3084(27)(+320) 0.3205(40)(+2590) 0.025 0.2903(27)(+18460) 0.3155(24)(+17755) 0.515(18)(+50) 0.3755(76)(+300 ) 0.4043(66)(+50)

Table 4. Values of the would-be hadron masses extrapolated to innite volume for am = 0.01, 0.02, 0.025. The rst uncertainty is given by the best-t to eq. (4.1). The second uncertainty accounts for the lack of complete knowledge of F (x, L), see text.

F (x, L) we add a second uncertainty to each extrapolated mass obtained as follows. The simple parameterization g(L) = c(1bL) provides an explicit expression for F (x, L), which

in turn gives the volume dependence

mH(L) = mH + c [parenleftbigg]

Here, we used x = Lm and the innite volume mass relation mH = cHm inside F (x, L), where cH is nothing but the parameter a in the scaling study of gure 14 and table 3. For each channel H, the second asymmetric uncertainty on the innite volume mass in table 4 has the L = 32 mass value as upper bound, and as a lower bound we take the innite volume mass given by a t to eq. (4.2), with free parameters c, b and mH and xed k/cH equal to its L = 24 value in table 3 and table 8 in the appendix.

4.5 The spectrum at innite volume

Figure 16 shows the bare fermion mass dependence of the would-be hadron masses at innite volume, taken from tables 1 and 4. The results of a single power-law t on the heavier mass range (Fit I), the full mass range (Fit II), and a linear t with free intercept (Fit III) are summarized in table 5 and reproduced in gure 16. The linear t turns out

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L b

[parenrightbigg]

kcH mHL . (4.2)

Ch. Fit I Fit II Fit III = 0.81(3), c = 5.7(5) = 0.81(1), c = 5.8(2) m0 = 0.07(3), c = 8.6(3)

= 0.86(3), c = 7.5(6) = 0.86(2), c = 7.4(2) m0 = 0.06(2), c = 10.0(2) = 0.80(6), c = 7(1) = 0.81(1), c = 7.2(2) m0 = 0.08(1), c = 11.4(4) a1 = 0.87(7), ca1 = 10(2) = 0.85(2), ca1 = 9.0(5) m0 = 0.06(2), ca1 = 12.9(4) N = 0.81(4), cN = 10(1) = 0.81(2), cN = 9.7(6) m0 = 0.14(3), cN = 14.1(6)

Table 5. Best-t results for the fermion mass dependence of the would-be hadrons at L = 1.

Fit I is a power-law cH m on the range am = 0.04 to 0.07, Fit II is a power-law on the range am = 0.01 to 0.07 that includes only the rst uncertainty for the three lightest masses, and Fit III is a linear t with free intercept m0 + cHm on the range am = 0.01 to 0.07. Values of the [vector]2/d.o.f.

are reported in the gures.

Ch. Fit IIb

= 0.81(1) c = 5.71(19) [vector]2/dof = 0.95 = 0.86(1) c = 7.47(23) [vector]2/dof = 0.82 = 0.80(1) c = 7.08(24) [vector]2/dof = 0.43 a1 = 0.83(2) ca1 = 8.43(57) [vector]2/dof = 0.64 N = 0.81(2) cN = 9.58(46) [vector]2/dof = 1.72

Table 6. Fit IIb is Fit II of table 5 where the second (symmetrized) uncertainty for am = 0.01, 0.02 and 0.025 is added in quadrature to the rst one in table 4.

to be signicantly worse than the power-law ts in all cases. This conrms, once again, that chiral symmetry is restored. Fit IIb, reported in table 6, is a single power-law t on the full mass range where the symmetrized second uncertainty in table 4 has been added in quadrature to the rst uncertainty. In almost all cases in table 6 we obtain a [vector]2/dof [lessorsimilar] 1, likely indicating that the uncertainties on the lightest points are in this case slightly overestimated. What is most interesting is the value of the exponent and its dependence, or lack thereof, on the di erent quantum numbers H. A value [negationslash]= 1/2 for the

pseudoscalar state says that it is not a Goldstone boson and chiral symmetry is exact. A value < 1 says that we are away from the heavy quark limit where mH m. We observe

a common = 0.81 within errors in the channels H = P S, S, N a sign of universality and a slightly larger value = 0.86 within errors for the vector states H = V, P V ; this could be attributed to the di erent pattern of spin-spin interactions for spin-1 and spin-0 or 1/2 states. Noticeably, the value = 0.81 agrees with the four-loop prediction [23, 24] at the IRFP and it agrees with the best-t result of [9]. We conclude that the lattice results for the pseudoscalar, scalar and the nucleon states are in the universal scaling regime, i.e., at masses su ciently light to be insensitive to perturbative mass corrections to universal scaling arising for g [negationslash]= g . For this reason, we take = 0.81 and these results for the

pseudoscalar (scalar and nucleon) state as reference in the combined analysis with other lattice results, i.e., R = 1 and b = 0 in table 2.

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Figure 16. Would-be hadron masses at L = 1 in the pseudoscalar (top left), scalar (top right),

vector (centre left), axial (centre right) and the nucleon (bottom) channels as a function of the bare fermion masses. Three ts are shown: t I (solid black) is a power-law on the range am = 0.04 to 0.07, Fit II (dashed red) is a power-law on the range am = 0.01 to 0.07, and Fit III (solid blue) is a linear t with free intercept on the range am = 0.01 to 0.07. The total uncertainty used in Fit IIb is shown (red bar) for am = 0.01, 0.02, 0.025. Largest volume data are also shown for the same points (green diamonds). Fit results are in table 5 and 6.

c b c b

This work 5.7(5) 0 5.13(26) 0.52(12) LatKMI 3.7 5.408(86) 0.544(16) 6.899(68) 0.594(10)

LatKMI 4.0 4.778(64) 0.493(14) 5.96(11) 0.560(18)

Table 7. Best-t values for two power laws eq. (4.3) for the innite volume pseudoscalar and vector masses from this work and [34]. The exponents are xed to = 0.81 and ! = 0.23. The value of c from this work is from Fit I in table 5.

Figure 17. Collapse of the rescaled innite volume masses amHR/(1+bH(am)!), ! = 0.23, for the pseudoscalar () and vector () states in this work (red squares), and in [34] at = 4.0 (orange circles) and = 3.7 (green crosses). The values of R = cH/cKMIH and bH, H = , , are from table 7.

Instead, = 0.86 for the vector states suggests the presence of perturbative corrections to scaling. The results of a t with two power laws, according to the parameterization of the perturbative corrections to scaling discussed in section 2.1.3

mH = cH m (1 + bHm!) (4.3)

with = 0.81 and ! = 0.23 for the vector state are in table 7 (This work). Note that b > 0, as expected, consistently with the fact that our system is on the strong coupling side of the IRFP. To further corroborate this statement we combine the data for the vector and the pseudoscalar with those of [34], all at innite volume within uncertainties. The best-t values for eq. (4.3) are in table 7. While bH > 0 on the strong coupling side of the IRFP (This work), bH < 0 on the weak coupling side of the IRFP (LatKMI), and we expect bH ! 0 for L ! 1. Finally, gure 17 shows the collapse of the innite volume

pseudoscalar and vector states of this work and [34], after rescaling. The rescaling factor R = cH/cKMIH is the ratio of the leading power-law coe cients for the channel H in table 7.

This analysis leads to the determination of the mass anomalous dimension at the IRFP. We quote the value obtained from Fit I in the pseudoscalar channel from table 5

= 0.81(3) = 0.235(46) (4.4)

This value is in agreement with the perturbative four-loop prediction, with the best-t result of [9] and not far from the rst lattice determination of the fermion mass anomalous

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Figure 18. Ordering of the would-be hadrons in the QED-like phase of the Nf = 12 system. From bottom to top, the pseudoscalar (), the vector (), the scalar (), the axial (a1) and the nucleon N.

dimension for the Nf = 12 system in [4], though the latter was a ected by rather large uncertainties. The value of in eq. (4.4) suggests a rather weakly coupled Nf = 12 system at the IRFP, so that perturbation theory should be expected to hold. Conversely, four-loop perturbation theory seems to fail for Nf 8, where it predicts an IRFP at rather strong

coupling g and, even worse, a change of sign of the mass anomalous dimension for g < g , see end of section 2.2.8 This reinforces the idea that nonperturbative dynamics, known to be chiral dynamics in this case, has to play a role at the opening of the conformal window, for 8 [lessorsimilar] Nf [lessorsimilar] 12. Also, if = 1 has to be realized at the lower endpoint of the IRFP line, where the conformal window disappears, a rapid variation of the mass anomalous dimension for Ncf [lessorsimilar] Nf [lessorsimilar] 12 should be expected in a lattice (or any nonperturbative) determination of the IRFP line, where nonperturbative dynamics is fully encompassed. The present study also corroborates the view that the IRFP of these theories is not associated to a physical singularity, no discontinuity happens there and estimates of physical observables including the anomalous dimensions can be attempted on either side of the xed point.

We conclude this section with showing the ordering of the would-be hadrons in gure 18. To summarize, a universal power law with exponent = 0.81 describes all would-be hadrons, with additional perturbative mass corrections of the type 1+ gm g in the vector

and axial channels. The pseudoscalar is the lightest state, but it is not a Goldstone boson. The vector, the scalar, the axial, and nally the nucleon follow. It is worth noting that the scalar state9 is heavier than the vector state. Their ordering becomes phenomenologically relevant when the theory is just below the conformal window it remains, however, di cult to identify a broad scalar resonance, such as f0(500) of QCD,10 on the lattice.

8This e ect is not encountered at two loops.

9We remind the reader that of this work is the state extracted from the connected scalar two-point function.

10This state could in addition be an admixture of ordinary qq states and tetraquarks.

Figure 19. The ratio of the vector and axial masses (left) and the ratio of the pseudoscalar and scalar masses (right) as a function of the bare fermion mass.

4.6 Mass ratios and degeneracies

We conclude this work with some remarks on the interplay of conformal and chiral symmetry inside the conformal window. Ratios and degeneracies of would-be hadron masses are a combined probe of both symmetries, and, as shortly discussed below, the U(1) axial symmetry. At the IRFP conformal symmetry implies exact chiral symmetry. Away from the IRFP, inside the conformal window, restored chiral symmetry implies the degeneracy of chiral partners in the chiral limit. Figure 19 shows the mass ratios pseudoscalar-scalar and vector-axial. The two ratios are essentially constant 0.8 over the explored mass range,

as it can be deduced from the best-t values for the power-law exponent . Due to the presence of perturbative corrections to universal scaling for the vector states, deviations from a constant ratio will instead be observed in all cases that mix the vector (or axial) channel with the other ones, one example is gure 9. Before discussing the degeneracy patterns, it is important to recall that the scalar studied here is extracted from the quark-line connected piece of the scalar-isoscalar two-point function; for clarity, we then call this state c in the following discussion and with we refer to the lowest-lying state of the complete (connected plus disconnected) scalar-isoscalar correlator. The pseudoscalar and the scalar , not c, belong to the same chiral multiplet of SU(Nf)L [notdef] SU(Nf)R,

and they must be degenerate in the chiral limit m ! 0 when chiral symmetry is not spon

taneously broken. The vector and the axial a1 are also chiral partners and show the same degeneracy pattern. In other words, the mass degeneracy of the chiral partners and a1, and , can be used as an indicator of the restoration of chiral symmetry. What about the degeneracy of and c? For degenerate fermion masses, as in our case, the connected contribution to the scalar-isoscalar, c, equals the connected contribution to the scalar-isovector the latter has no disconnected contributions. The is the U(1)A partner of . We should thus conclude that the degeneracy of and c is probing the e ective restoration of U(1)A, at least at the level of the two-point functions.11 Figure 20

11A complete probe of U(1) axial and chiral restoration obviously includes the direct observation of the disconnected contributions and the degeneracy patterns of all states, including [prime] and .

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Figure 20. Degeneracy pattern and best-t curve of the pseudoscalar and the scalar c (connected) (top left) states. The ratio of the mass di erence to the mass average (top right). Analogous gures for the vector and the axial a1 states (bottom).

shows the mass di erence and the ratio (mi mj)/(mi + mj) for the spin-0 U(1)A part

ners c and the spin-1 chiral partners a1. The degeneracies in the chiral limit are

evident from the best-t curves of the mass di erence on the left of gure 20, thus conrming once again exact chiral symmetry and the e ective restoration of U(1)A. This also implies that the disconnected contributions to the scalar-isoscalar correlator are at least O(m). We defer to future work the question to which degree U(1)A is exact inside the conformal window beyond the two-point functions and how its restoration pattern compares with high-temperature QCD. The approximately constant ratios on the right of gure 20 highlight, once again, the realization of the scaling form cH m , modulo small perturbative corrections (1 + bHm!) in the spin-1 case, with universal exponents and ! and mass independent nonuniversal coe cients cH and bH.

5 Conclusions

We have studied the SU(3) gauge theory with twelve fundamental fermions as a prototype of theories inside the conformal window, with emphasis on the two-point functions and their properties when the IRFP is perturbed by a fermion mass. In order to disentangle the

imprint of the IRFP in the dynamics of the system, we have analyzed the complete would-be hadron spectrum, the would-be mesons and the nucleon, and performed a universal scaling study at nite and innite volume. The identication of universal contributions dictated by the conformal invariance at the xed point and deviations from universal scaling induced in the surroundings of the xed point has allowed for the nonperturbative determination of the fermion mass anomalous dimension = 0.235(46) and a unied description of all lattice results for the would-be hadron spectrum of the Nf = 12 theory.

This analysis shows that the lattice system retains all signatures of the underlying conformal symmetry of the xed point, in addition to the restored chiral symmetry, that the pattern of symmetries can be followed across the IRFP and the critical exponents or any other physical observable can be determined on either side of the xed point, be it the asymptotically free phase of the lattice system or the QED-like phase. In other words, one should conclude that no singularity is associated to such a xed point.

The obtained nonperturbative value of hints at a rather weakly coupled Nf = 12 system at the IRFP. It is thus amusing, and perhaps not unexpected, to observe the agreement with the four-loop perturbative prediction at the xed point. Based on this agreement, it is tempting to infer that the perturbative expansion is working well in this range of Nf,

and that the missing nonperturbative contributions and the e ects of a truncation of the perturbative series amount to a negligible correction for this system.

It also reinforces the idea that nonperturbative dynamics, known to be chiral dynamics in this case, has to play a role at the opening of the conformal window, for 8 [lessorsimilar] Nf [lessorsimilar] 12.

Should = 1 be realized at the lower endpoint of the IRFP line, where the conformal window disappears, a rapid variation of the mass anomalous dimension on the interval Ncf [lessorsimilar] Nf [lessorsimilar] 12 should be expected. Plausibly, nonperturbative contributions would become increasingly important towards Ncf, making mandatory the use of a nonperturbative strategy, lattice or else, in the study of the infrared dynamics of systems close to the lower endpoint of the conformal window.

As a byproduct, we have conrmed the restoration of chiral symmetry through the degeneracy of chiral partners and the e ective restoration of the U(1) axial symmetry at the level of the two-point functions.

Acknowledgments

We thank Dries Coone for his participation in the early stage of this work, and Slava Rychkov for an interesting discussion on conformal invariance. EP and MpL acknowledge the hospitality of the Aspen Center for Physics, which is supported by the National Science Foundation Grant No. PHY-1066293, as well as the many interesting discussions with the participants during their visit. KM and MpL thank Yasumichi Aoki and Hiroshi Ohki for fruitful discussions. This work was in part based on the MILC collaborations public lattice gauge theory code, see http://physics.utah.edu/~detar/milc/index.html

Web End =http://physics.utah.edu/ http://physics.utah.edu/~detar/milc/index.html

Web End =detar/milc/index.html . Simulations were

performed on the IBM BG/P at the University of Groningen, the Huygens IBM Power6+ system at Stichting Academisch Rekencentrum Amsterdam (SARA) and the IBM BG/Q

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Figure 21. Spatial volume dependence and extrapolation to innite volume with functional form eq. (4.1) for the masses of the scalar (top left), axial (top right) and the nucleon (bottom) states, and bare fermion masses am = 0.01, 0.02 and 0.025. The extrapolated value and its uncertainty is indicated by horizontal lines and can be read from table 4.

Fermi at CINECA. This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientic Research (NWO).

A Volume dependence and extrapolation to innite volume

This appendix completes the study in sections 4.3, 4.4 and 4.5 for some of the would-be hadrons. The extrapolation to innite volume for the scalar, axial and the nucleon are analogous to the ones presented in section 4.4 and are reported in gure 21. The nonperturbative L dependent violations of scaling at small x have been studied for all states. We report in gure 22 and table 8 the analogous of gure 14 and table 3 for the channels H = S, V, P V, N. The best-t values of a, c, k for the simplied ansatz F (x, L) = ax + c exp (kx) are reported in table 8.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Figure 22. LmH/(1+bHm!) for H = , N, , a1 as a function of the scaling variable x = Lm with = 0.81, for L = 16, 24, 32. The coe cient bH = 0 for H = S, N, b = 0.52(12) and ba1 = 0.45(24).

The curves are best ts to the functional form F (x, L) = ax + c exp (kx), see table 8.

H = H = N

L = 16 L = 24 L = 32 L = 16 L = 24 L = 32a 7.10(17) 7.25(10) 7.31(9) 7.8(1.4) 9.54(10) 9.56(19)

c 10.12(80) 12.2(3.4) 10.84(19.58) 14.83(2.26) 14.84(3.20) 13.49(21.02) k 2.23(23) 2.93(48) 3.15(2.35) 0.82(41) 1.57(35) 1.93(1.84)

H = H = a1L = 16 L = 24 L = 32 L = 16 L = 24 L = 32 a 4.77(17) 5.01(06) 5.08(06) 6.24(30) 6.53(12) 6.45(16) c 9.077(76) 7.52(70) 6.1(1.1) 10.4(1.7) 13.3(5.0) 3.8(1.9) k 1.46(20) 1.47(15) 1.57(25) 2.08(40) 2.84(66) 1.30(70)

Table 8. Best-t values of the parameters a, c, k for the ts of LmH, H = , N, , a1 to the functional form F (x, L) = ax + c exp (kx), with x = Lm and = 0.81.

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Abstract

We study the SU(3) gauge theory with twelve flavours of fermions in the fundamental representation as a prototype of non-Abelian gauge theories inside the conformal window. Guided by the pattern of underlying symmetries, chiral and conformal, we analyze the two-point functions theoretically and on the lattice, and determine the finite size scaling and the infinite volume fermion mass dependence of the would-be hadron masses. We show that the spectrum in the Coulomb phase of the system can be described in the context of a universal scaling analysis and we provide the nonperturbative determination of the fermion mass anomalous dimension γ^sup ^ = 0.235(46) at the infrared fixed point. We comment on the agreement with the four-loop perturbative prediction for this quantity and we provide a unified description of all existing lattice results for the spectrum of this system, them being in the Coulomb phase or the asymptotically free phase. Our results corroborate the view that the fixed point we are studying is not associated to a physical singularity along the bare coupling line and estimates of physical observables can be attempted on either side of the fixed point. Finally, we observe the restoration of the U(1) axial symmetry in the two-point functions.

Details

Title

On the particle spectrum and the conformal window

Author

Lombardo, M P; Miura, K; Nunes Da Silva, T J; Pallante, E

Pages

1-41

Publication year

2014

Publication date

Dec 2014

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP12(2014)183

ProQuest document ID

1708025262

SISSA, Trieste, Italy 2014

On the particle spectrum and the conformal window

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