Eur. Phys. J. C (2020) 80:620 https://doi.org/10.1140/epjc/s10052-020-8174-5

Regular Article - Theoretical Physics

The QCD trace anomaly at strong coupling from M-theory

Aalok Misra1,a, Charles Gale2,b

1 Department of Physics, Indian Institute of Technology, Roorkee, Uttarakhand 247 667, India2 Department of Physics, McGill University, 3600 University St, Montreal, QC H3A 2T8, Canada

Received: 5 March 2020 / Accepted: 19 June 2020© The Author(s) 2020

Abstract Obtaining a lattice-consistent result for the tem-perature dependence of the QCD conformal anomaly from atop-down M-theory dual (valid) for all temperatures – both,T < Tc and T > Tc – of thermal QCD at intermediate gaugecoupling, has been missing in the literature. We fill this gapby addressing this issue from the M-theory uplift of the SYZtype IIA mirror at intermediate gauge/string coupling [bothobtained in Dhuria et al. (JHEP 1311:001, 2013)] of the UV-complete type IIB holographic dual of large-N thermal QCDof Mia et al. (Nucl Phys B 839:187, 2010), and comparingwith the very recent lattice results of Bazavov et al. (PhysRev D 97(1):014510, 2018). Estimates of the O(R4) higherderivative corrections in the D = 11 supergravity action rel-evant to considering the aforementioned M theory uplift inthe intermediate ’t Hooft coupling (in addition to gauge cou-pling) limit, are also presented. We also show that after a tun-ing of the (small) Ouyang embedding parameter and radiusof a blown-up S2 when expressed in terms of the horizonradius, a QCD deconfinement temperature Tc = 150 MeVfrom a Hawking–Page phase transition at vanishing baryonchemical potential consistent with lattice QCD in the heavy-quark limit, can be obtained.

1 Introduction

One of the breakthroughs of the relativistic heavy ion pro-gram has been the realization that the production of hadronicmatter in extreme conditions of temperature and density –as created during the high-energy collision of large nuclei –can be well modelled and understood using numerical simu-lations relying on relativistic fluid dynamics [4]. In that con-text, the QCD equation of state (EOS) is an essential quantity.Nonperturbative calculations based on lattice QCD have nowconfirmed the fact that at vanishing baryonic density the tran-

a e-mail: aalok.misra@ph.iitr.ac.in (corresponding author)b e-mail: gale@physics.mcgill.ca

sition between partonic degrees of freedom and those in theconfined sector is in fact a rapid crossover [3,5,6] occurringin the vicinity of T � 150 MeV. For higher baryon densi-ties and slightly lower temperatures, lattice calculations haveproven to be challenging because of the notorious sign prob-lem [7]. Some progress has nevertheless been made, througha variety of techniques [7]. At still lower temperatures andhigh densities, investigations of the hadronic EOS throughseveral different approaches suggest a first-order chiral sym-metry restoration phase transition [8–11]. These results andothers like them have fuelled much of the interest in thesearch for a critical end point (CEP) and the initiation of abeam energy scan (BES) at RHIC [12]. Thus, in parallel withtheoretical work, experimental explorations can be used touncover subtle but fundamental features of the EOS, such asthe existence of a possible critical point and of genuine ther-modynamic phase transitions [13]. Finally, the importanceof the hadronic EOS is not restricted to the field of relativis-tic heavy-ion collisions. The EOS is responsible for the bulkproperties of dense stellar objects such as neutron stars. Italso affects their cooling properties, which probes the parti-cle content and the state of the matter present in their core[14,15].

The EOS is an integral part of the energy–momentumtensor, Tμν . In a classical theory without any dimentionfulparameters, a scale transformation leaves the action invariant,and conversely leads to a traceless energy momentum tensor:Tμ

μ = 0. This is the case for classical, massless, Yang-Millstheory. However, quantum effects will spoil the conservationof the dilatation current, and make the theory scale-dependent[16], as clearly shown by the running of the coupling, g, viathe β-function: β(g) = μ∂g/∂μ. Then, the Yang-Mills Tμν

satisfies

Tμμ = β(g)

2gFμν a Fa

μν (1)

where a is a color index.This discussion brings us to the core of this paper, and

to its twofold intent. First, it is clear that calculations of the

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hadronic EOS clearly requires a treatment which goes beyondperturbation theory. In this context, the gauge–gravity duality[17,18] offers an appealing set of techniques which renderstrong coupling calculations analytically feasible. The origi-nal formulation of the duality was AdS/CFT: the field theorysector was conformal. More recently, extensions into fami-lies of theories which break conformal invariance have beenactively pursued. We now briefly describe the approach usedin this work and the path which lead to its development;details of the former are given in Sect. 2.

Gauge/gravity duality has proved to be very useful inunderstanding the properties of (thermal) QCD-like theories.The first top-down (type IIA) holographic dual of large-NQCD though catering only to the IR, was given in [19]. AUV-complete (type IIB) holographic dual of large-N thermalQCD, was constructed in [2]. It is believed that large-N ther-mal QCD laboratories like strongly coupled QGP (sQGP)require not only a large ’t Hooft coupling but also a inter-mediate gauge coupling [20]. Holographic models basedon this assumption, therefore necessarily require address-ing this limit from M theory. It is known that QCD pos-sesses a rapid crossover from a confining phase to a non-confining phase at T � Tc, and to explore the physics ofQCD at T ≈ Tc, we have to take a look at the strongly cou-pled regime of the theory. The holographic study of large-Nthermal QCD at intermediate coupling, was initiated in [1]which presented a M-theory uplift of the SYZ type IIA mirror(in the spirit of [21,22]) of a string theoretic dual of large-N thermal QCD-like theories at intermediate gauge/stringcoupling as part of the ‘MQGP’ limit of [1]. In this limit,the temperature dependence of a variety of transport coeffi-cients have been calculated in [23–25]. On the holographicphenomenology front, lattice/PDG-compatible glueball and(pseudo-)vector and (pseudo-)scalar meson masses as wellas (exotic scalar)glueball-to-meson decay widths were cal-culated in [26–28].

The QCD conformal anomaly and its temperature depen-dence are important quantities to be studied in the contextof, e.g., relativistic heavy ion collisions. In this paper, wewill describe how to evaluate the same and obtain, in partic-ular, the temperature dependence of the trace anomaly fromM-theory and compare our results with recent lattice results.Note, to the best of our knowledge, there is no precedence ofstudying the QCD conformal/trace anomaly from a top-downM theory dual (inclusive of higher derivative (O(R4)) correc-tions corresponding to considering the intermediate ’t-Hooftcoupling limit) at low and high temperatures consistent withrecent lattice results.1

The remainder of this paper is organized as follows. Sec-tion 2 is a brief review of (the UV complete) string/M-theory

1 See, e.g. [29] for earlier attempts at matching trace anomaly in bottom-up holographic models, with (older) lattice results; also see [30–33].

holographic dual of large-N thermal QCD as constructed in[2] (type IIB) and [1,34](type IIA and M-theory) to makethis paper self contained. Section 3 discusses obtaining alattice-compatible Tc from the type IIB holographic dual asconstructed in [2] from a Hawking–Page phase transition atzero chemical potential (improving upon a similar compu-tation done earlier in [23–25]). Section 4 has to do with aholographic computation of the QCD trace anomaly from Mtheory. This is partitioned into two Sect. 4.1 is on high tem-peratures, i.e., T > Tc and Sect.4.2 is on low temperatures,i.e., T < Tc. Section 5, apart from summarizing the results,discusses a very crucial point as regards compatibility of ourresults with lattice computations. The Hawking–Page phasetransition in our computation in Sect. 3 occurs at zero baryonchemical potential μC and one expects a smooth cross-overfor a non-zero μC above a critical value of μC which isexactly the opposite of what one generically expects from(lattice) QCD. We argue that our holographic gravity dualcomputation can still be justified in the heavy quark limit.There are two appendices. Appendix A discusses the eval-uation of the baryon chemical potential and the DBI actionon the flavor D7-branes. Appendix B summarizes the defi-nitions relevant to the O(R4) terms in the D = 11 SUGRAaction

2 String/M-theory dual of thermal QCD: a review of[1,2]

In this section, we provide a short review of a UV completetype IIB holographic dual (the only one we are aware of)of large-N thermal QCD constructed in [2], its Strominger–Yau–Zaslow (SYZ) type IIA mirror at intermediate stringcoupling and its subsequent M-theory uplift constructed in[1,34].

1. UV-complete holographic dual of large-N thermalQCD as constructed in [2]: The UV-complete holo-graphic dual of large-N thermal QCD as constructedin [2], subsumed the zero-temperature Klebanov–Wittenmodel [35], the non-conformal Klebanov–Tseytlin model[36], its IR completion as given in the Klebanov–Strasslermodel [37] and Ouyang’s [38] inclusion of flavor in thesame, as well as the non-zero temperature/non-extremalversion of [39] (but the non-extremality/black hole func-tion and the ten-dimensional warp factor vanished simul-taneously at the horizon radius), [40,41] (valid only atlarge temperatures) and [42,43] (addressing the IR), inthe absence of flavors. The following summarizes themain features of [2].

• Brane construct of [2]: The type IIB string dual of[2] consists of N D3-branes placed at the tip of six-

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Table 1 The Type IIB BraneConstruct of [2]

S. No. Branes World Volume

1. N D3 R1,3(t, x1,2,3) × {r = 0}

2. M D5 R1,3(t, x1,2,3) × {r = 0} × S2(θ1, φ1) × NPS2

a (θ2,φ2)

3. M D5 R1,3(t, x1,2,3) × {r = 0} × S2(θ1, φ1) × SPS2

a (θ2,φ2)

4. N f D7 R1,3(t, x1,2,3) × R+(r ∈ [|μOuyang| 2

3 , rUV]) × S3(θ1, φ1, ψ) × NPS2a (θ2,φ2)

5. N f D7 R1,3(t, x1,2,3) × R+(r ∈ [RD5/D5 − ε, rUV]) × S3(θ1, φ1, ψ) × SPS2

a (θ2,φ2)

dimensional conifold, M D5-branes wrapping thevanishing S2 and M D5-branes distributed along theresolved S2 placed at antipodal points relative to theM D5-branes. Let us denote the average D5/D5 sep-aration by RD5/D5. Roughly, r > RD5/D5, would bethe UV. The N f D7-branes, holomorphically embed-ded via Ouyang embedding [38] in the resolvedconifold geometry, “smeared”/delocalized along theangular directions θ1,2 as mentioned below (11), arepresent in the UV, the IR-UV interpolating regionand dip into the (confining) IR (but do not touch theD3-branes with the shortest D3 − D7 string corre-sponding to the lightest quark). In addition, N f D7-branes are present in the UV and the UV-IR interpo-lating region for the reason given below. The follow-ing Table 1 summarizes the aforementioned braneconstruct wherein S2(θ1, φ1) denotes the vanishingtwo-sphere and (NP/SP of) S2

a (θ2, φ2) is the (NorthPole/South Pole of the) resolved/blown-up two-sphere – a being the radius of the blown-up S2 – andrUV is the UV cut-off and ε(

RD5/D5−−|μOuyang|23

) < 1.

Also, μOuyang is the Ouyang embedding parameterthat is defined in (14) while describing the embed-ding of the flavor D7-branes in the resolved conifoldgeometry.

• In the UV, one has SU (N +M)× SU (N +M) colorgauge group and SU (N f ) × SU (N f ) flavor gaugegroup. There occurs a partial Higgsing of SU (N +M) × SU (N + M) to SU (N + M) × SU (N ) as onegoes from r > RD5/D5 to r < RD5/D5. This happensbecause in the IR, at low energies, i.e., at energies lessthanRD5/D5, the D5-branes are integrated out result-ing in the reduction of the rank of one of the productgauge groups (which is SU (N + number of D5 −branes) × SU (N + number of D5 − branes)). Bythe same token, the D5-branes are “integrated in” inthe UV, resulting in the conformal Klebanov–Witten-like SU (M +N )× SU (M +N ) product color gaugegroup [35].

• The pair of gauge couplings, gSU (N+M) and gSU (N ),were shown in [37] to flow oppositely; in fact theflux of the NS-NS B through the vanishing S2 is the

obstruction to obtaining conformality which is whyM D5-branes were included in [2] to cancel the netD5-brane charge in the UV. Further, as the N f flavorD7-branes enter the RG flow of the gauge couplingsvia the dilaton (see (13)), their contribution there-fore needs to be canceled by N f D7-branes which isthe reason for their inclusion in the UV in [2]. TheRG flow equations for the gauge coupling gSU (N+M)

– corresponding to the gauge group of a relativelyhigher rank – can be used to show that the same flowstowards strong coupling, and the SU (N ) gauge cou-pling flows towards weak coupling. One can showthat the strongly coupled SU (N +M) is Seiberg-likedual to weakly coupled SU (N − (M − N f )).2

• Obtaining Nc = 3: In the IR, at the end of a Seiberg-like duality cascade, the number of colors Nc getsidentified with M , which in the ‘MQGP limit’ tobe discussed below, can be tuned to equal 3. Thisis briefly explained now. One can identify Nc with

Neff(r) + Meff(r), (2)

where Neff(r) is defined via

F̃5 ≡ dC4 + B2 ∧ F3 = F5 + ∗F5, (3)

wherein F5 ≡ Neff Vol(Base of Resolved WarpedDeformed Conifold), and

Meff =∫S3

F̃3 (4)

(the S3 being dual to eψ ∧ (sin θ1dθ1 ∧ dφ1 −B1 sin θ2 ∧ dφ2), wherein B1 is an asymmetry factordefined in [2]; eψ ≡ dψ + cos θ1 dφ1 + cos θ2 dφ2)where [44]:

2 The Seiberg duality (cascade) is applicable for supersymmetric the-ories. For non-supersymmetric theories such as the holographic dualwe are working with, the same is effected via a radial rescaling:

r → e− 2π

3gs Meff r [38] under an RG flow from the UV to the IR.

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F̃3(≡ F3 − τH3) ∝ M(r)

≡ M1

1 + eα(r−RD5/D5), α 1. (5)

The effective number Neff of D3-branes variesbetween N 1 in the UV and 0 in the deep IR,and the effective number Meff of D5-branes variesbetween 0 in the UV and M in the deep IR. Hence,Nc varies between M in the deep IR and a large value[ in the MQGP limit of (9) for a large value of N ]in the UV. Hence, at very low energies, the numberof colors Nc can be approximated by M , which inthe MQGP limit is taken to be finite and can hencebe taken to be equal to three. Additionally, one canset N f = 2 + 1 for comparison with [3]. Hence,in the IR, this is somewhat like the Veneziano limitwherein

N fNc

is fixed (but, unlike [1,2], Nc, N f → ∞in the Veneziano limit in, e.g., s[29]) as (in the IR)N fNc

∼ N fM in [2]. Note, the low energy or the IR is

relative to the string scale. But these energies whichare much less than the string scale, can still be muchlarger than Tc. Therefore, as regards the energy scalesrelevant to QCD, the number of colors can be tunedto three.Thus, under a Seiberg-like duality cascade the N D3-branes are cascaded away and there is a finite M leftat the end corresponding to a strongly coupled IR-confining SU (M) gauge theory; the finite tempera-ture version of this SU (M) gauge theory is what wasconsidered in [2]. So, at the end of the Seiberg-likeduality cascade in the IR, the number of colors Nc isidentified with M , which in the ‘MQGP limit’ can betuned to equal 3.

• Color-Flavor Enhancement of Length Scale in theIR: In the IR in the MQGP limit, with the inclu-sion of terms higher order in gs N f in the RR andNS-NS three-form fluxes and the NLO terms in Nin the angular part of the metric, there occurs an IRcolor-flavor enhancement of the length scale as com-pared to a Planckian length scale in KS even for O(1)

M , thereby showing that quantum corrections will besuppressed. This was discussed in [34] and is sum-marized here. Using [2]:

Neff (r) = N

[1 + 3gsM2

eff

2πN

(log r + 3gs N eff

f

2π(log r)2

)],

Meff (r) = M + 3gs N f M

2πlog r +

∑m≥1

∑n≥1

Nmf M

n fmn(r),

N efff (r) = N f +

∑m≥1

∑n≥0

Nmf M

ngmn(r), (6)

wherein the type IIB axion C0 = N efff

(ψ−φ1−φ2)4π

,the ten-dimensional warp factor h, disregarding the

angular part, is given by:

h = 4πgsr4

[Neff(r) + 9gsM2

effgs Nefff

8π2 log r

]. (7)

At the end of a Seiberg-like duality cascade, Neff (r0 ∈IR) = 0 and writing h ∼ L4

r4 , the length scale L inthe IR will be given by:

L ∼ √MN

34f

√√√√√⎛⎝∑

m≥0

∑n≥0

Nmf M

n fmn(�)

⎞⎠

×⎛⎝∑

l≥0

∑p≥0

Nlf M

pglp(�)

⎞⎠

14

g14s

√α′

≡ N34f

√√√√√⎛⎝∑

m≥0

∑n≥0

Nmf M

n fmn(�)

⎞⎠

×⎛⎝∑

l≥0

∑p≥0

Nlf M

pglp(�)

⎞⎠

14

LKS

∣∣∣∣∣∣∣�:log �< 2π

3gs N f

,(8)

which implies that in the IR, relative to KS, there isa color-flavor enhancement of the length scale in theMQGP limit. Hence, in the IR, even for N IR

c = M =3 and N f = 2 + 1 (for comparison with [3]) uponinclusion of of n,m > 1 terms in Meff and N eff

f in(6), L LKS(∼ LPlanck) in the MQGP limit (9)involving gs

∼< 1, implying that the stringy correc-

tions are suppressed and one can trust supergravitycalculations. This is verified in Sect. 4.3 wherein it isexplicitly shown (at low temperatures, i.e., T < Tc;we expect a similar result though even for high tem-peratures, i.e., T > Tc) that the O(R4) correctionsare suppressed as compared to the LO terms in thesupergravity action.

• Gravity dual of the brane construct of [2]: Thefinite temperature on the gauge/brane side is effectedin the gravitational dual via a black hole in the latter.Turning on of the temperature (in addition to requir-ing a finite separation between the M D5-branes andM D5-branes to provide a natural scale above whichone is in the UV) corresponds in the gravitational dualto having a non-trivial resolution parameter of theconifold. IR confinement on the brane/gauge theoryside corresponds to having a non-trivial deformationof the conifold geometry in the gravitational dual.The gravity dual is hence given by a resolved warpeddeformed conifold wherein the D3-branes and theD5-branes are replaced by fluxes in the IR, and the

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back-reactions are included in the warp factor andfluxes.

Hence, the type IIB model of [2] make it an ideal holo-graphic dual of thermal QCD because: (i) it is UV con-formal (Landau poles are absent), (ii) it is IR confining,(iii) the quarks transform in the fundamental representa-tion of flavor and color groups, and (iv) it is defined forthe full range of temperature – both low and high.

2. TheMQGP limit, Type IIASYZmirror [21] of [2] andits M-theory uplift at intermediate gauge coupling:

• For constructing a holographic dual of thermal QCD-like theories, one would have to consider intermediategauge coupling (as well as finite number of colors) −dubbed as the ‘MQGP limit’ in [1]. From the perspec-tive of gauge–gravity duality, this necessitates look-ing at the strong-coupling/non-perturbative limit ofstring theory – M theory. The MQGP limit in [1] wasdefined as:

gs<∼ 1, M, N f ≡ O(1), N 1,

gsM2

N� 1. (9)

• The M-theory uplift of the type IIB holographicdual of [2] was constructed in [1] by working outthe SYZ type IIA mirror of [2] implemented viaa triple T duality along a local special Lagrangian(sLag) T 3 − which could be identified with the T 2-invariant sLag of [45] with a large base B(r, θ1, θ2)

(of a T 3(φ1, φ2, ψ)-fibration over B(r, θ1, θ2)) [23–25,34]3 Let us elucidate the basic idea. Let us con-sider the aforementioned N D3-branes oriented alongx0,1,2,3 at the tip of conifold. Further, assume theM D5-branes to be parallel to these D3-branes aswell as wrapping the vanishing S2(θ1, φ1). A singleT-dual along ψ yields N D4-branes wrapping the ψ

circle and M D4-branes straddling a pair of orthog-onal NS5-branes. These pair of NS5-branes corre-spond to the vanishing S2(θ1, φ1) and the blown-upS2(θ2, φ2) with a non-zero resolution parameter a –the radius of the blown-up S2(θ2, φ2). Two more T-dualities along φi and φ2, convert the aforementioned

3 Consider D5-branes wrapping the resolved S2 of a resolved coni-fold geometry [46], which one knows, globally, breaks SUSY (nicelyexplained in [47]). As in [48], to begin with, SYZ is implemented locallywherein the pair of S2s are replaced by a pair of T 2s in the delocalizedlimit, and the correct T-duality coordinates are identified. Then, whenuplifting the mirror to M theory, it is found that a G2-structure can bechosen that is in fact, free, of the delocalization. For the SYZ mirror ofthe resolved warped deformed conifold uplifted to M-theory with G2structure worked out in [1], the idea is precisely the same. Also note (aspointed out in Fig. 1), the type IIB/IIA SU (3) structure torsion classes(in the MQGP limit and in the UV/UV-IR interpolating region), satisfythe same relationships as satisfied by corresponding supersymmetricconifold geometries [49].

pairt of orthogonal NS5-branes into two orthogonalTaub-NUT spaces, the N D4-branes into N color D6-branes and the M straddling D4-branes also to D6-branes. Similarly, in the presence of the aforemen-tioned N f flavor D7-branes (embedded holomorphi-cally via the Ouyang embedding), oriented parallel tothe D3-branes and “wrapping” a non-compact four-cycle (4)(r, ψ, θ1, φ1), upon T-dualization yield N f

D6-branes “wrapping” a non-compact three-cycle (3)(r, θ1, φ2). An uplift to M-theory of the SYZtype IIA mirror, will convert the D6-branes to KKmonopoles, which are variants of Taub-NUT spaces.Therefore, all the branes are converted to geometryand fluxes, and one ends up with M-theory on a G2-structure manifold. Similarly, one may perform iden-tical three T-dualities on the gravity dual on the typeIIB side, which is a resolved warped-deformed coni-fold with fluxes, to obtain another G2 structure man-ifold, giving us the MQGP model of [1,34].

The type IIB brane construct, its type IIA mirror as wellas the type IIB gravity dual, its SYZ IIA mirror gravitydual along with the M-theory uplift of the type IIA gravitydual are summarized in Fig. 1. The SU (3)/G2 structure tor-sion classes (which measure the deviation of a six/sevenfoldfrom having SU (3)/G2 holonomy) are denoted respectivelyby Wi=1,2,3,4,5/Wi=1,2,3,4 (with superscripts in the G2-structure torsion classes denoting the respective dimension-alities) therein.

3 Lattice-compatible Tc

In this section after obtaining a lattice-compatible confine-ment-deconfinement phase transition temperature Tc as aHawking–Page phase transition at zero chemical poten-tial, we obtain the temperature variation of the QCD traceanomaly from M theory, both, at large temperatures T > Tcas well as low temperatures T < Tc.

The temperature at the horizon rh is given as under:

T (r = rh) = ∂rGM00

4π

√GM

00 GMrr

, (10)

which in the MQGP limit and utilizing the IR-valued warpfactor h(r, θ1, θ2):

h(r, θ1, θ2) = L4

r4

[1 + 3gsM2

eff2πN

logr

{1 +

3gs N efff

2π

(logr + 1

2

)

+gs N eff

f

4πlog

(sin

θ1

2sin

θ2

2

)}],

L ≡ (4πgs N )14 , (11)

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620 Page 6 of 16 Eur. Phys. J. C (2020) 80:620

Fig. 1 Dasgupta et al’s [2] Type IIB brane construct and the gravitydual of large-N QCD at high temperature, and their (SYZ) mirrors

will when evaluated near θ1 ∼ N− 15 , θ2 ∼ N− 3

10 [the spe-cific values of small θ1,2 facilitated in [34] construction ofexplicit SU (3)/G2 structures respectively for the string/Mtheory duals; the EH and GHY terms receive their most dom-inant contributions near very small values of θ1,2; the samealong with ψ = 0, 2π, 4π has the advantage of the decou-pling of M5(x0,1,2,3, r) and M̃5(θ1,2, φ1,2, ψ)], can be writ-ten out [23–25] in terms of N , M and N f . Now, we will take(as in [23–25]), the following form of the resolution parame-ter (the radius of the blown-up S2 of the non-Kähler warpedresolved conifold):

a(rh) =(

α + γgsM2

N+ β

gsM2

Nlog rh

)rh . (12)

We will now see how to obtain a lattice-compatible Tc.We will implement the idea that in the absence of a chemicalpotential the confinement-to-deconfinement transition in thegravitational dual side, can be understood as a Hawking–Page first order phase transition from a thermal (T < Tc)gravity dual to the one consisting of a black hole (T > Tc)[50]. Inspired by [2,23–25], the following type IIB dilaton(�) profile will be assumed:

(a)T (rh �= 0) :

e−� = 1

gs− N eff

f

8πlog(r6 + a2r4)

− N efff

2πlog

(sin

θ1

2sin

θ2

2

), r < RD5/D5,

e−� = 1

gs, r > RD5/D5;

(b)T (rh = 0) :

e−� = 1

gs− 3N eff

f

4πlog r

− N efff

2πlog

(sin

θ1

2sin

θ2

2

), r <

∣∣μOuyang∣∣ 2

3 ,

e−� = 1

gs, r >

∣∣μOuyang∣∣ 2

3 , (13)

wherein N efff is the effective number of D7-branes (or the

effective axionic charge) the Ouyang embedding parameteris defined via:

(r6 + 9a2r4

) 14e

i2 (ψ−φ1−φ2) sin

(θ1

2

)sin

(θ2

2

)= μOuyang.

(14)

Hence, setting the Newtonian constant to unity, performinga large-N expansion and then a large RUV-expansion, forthe thermal background (rh = 0) for which r ∈ [r0, RUV]where r0 and RUV are respectively the IR and UV cut-offs,the potential:

V1

�5= −1

2

∫ RUV

r=r0

d5x√−ge−2� (R − 2�)

−∫r=RUV

d4x√−he−2�K , (15)

where �5 ≡ ∫d5y

√g(θ1,2, φ1,2, ψ)δ

(θ1 − αθ1

N15

)δ(

θ2 − αθ2

N3

10

)δ (ψ − 2nπ) (n = 0/1/2, αθ1,2 being O(1)

numbers). Similarly for the black hole background, for whichr ∈ [rh, RUV] the potential:

V2

�5= −1

2

∫ RUV

r=rhd5x

√−ge−2� (R − 2�)

−∫r=RUV

√−hd4xe−2�K (16)

was worked out in [23–25]. Counter terms involving∫RUV

√−hThermal/BH need to be subtracted from V1,2 to ren-der them UV-finite and it was shown in [23–25] that assuming

RD5/D5 = √3a, |μOuyang| 2

3 = δ(≡ O(1)) × r0 and assum-

ing an IR-valued rh, r0, (V2 − V1)UV−finite = 0 yields:

123

Eur. Phys. J. C (2020) 80:620 Page 7 of 16 620

r0 = rh4

√√√√∣∣∣∣∣

9α4 − 1

2(δ83 − 1)

∣∣∣∣∣ + O(

1

log N

). (17)

Now, as we was shown in [28], the lightest 0++ scalar glue-ball mass is given by:

m0++ ≈ 4r0

L2 . (18)

Now, lattice calculations for 0++ scalar glueball mass [51],yield the lightest mass to be around 1, 700 MeV. Hence, tomake contact with lattice results, using (18), r0

L2 is replaced

by 1,7004 (in units of MeV) to yield:

Tc =mglueball

(1 + 3α2

2

)

274 π 4

√∣∣∣∣ 9α4−1

2(δ83 −1)

∣∣∣∣

∣∣∣∣∣∣∣∣∣∣α=0.6,δ=1.008

= 150 MeV, (19)

which is what is expected from lattice QCD in the heavy-quark-mass limit. Let us elaborate more. One should note thatin our gravity dual as proposed in [50], the Hawking–Pagephase transition occurs at zero baryon chemical potential μC

and one expects a smooth cross-over for a non-zero μC abovea critical value. However, it is exactly the opposite of whatone generically expects from (lattice) QCD. But, as explainedin [52], our holographic gravity dual computation can still bejustified in the heavy quark limit wherein the first order phasetransition at μC = 0 becomes a cross-over for μC �= 0. Letus explain how the heavy quark-mass limit is implied in ourcalculations and hence ensure compatibility with the latticeresults of [52]. We assume that in the type IIB dual (whoseuplift via the type IIA SYZ mirror is the M theory dual we areworking with in Sect. 2), all D7-branes have been identicallyembedded; in other words, in the type IIB brane picture, thequarks corresponding to the D3 − D7 strings, are either alllight or are all heavy – this will be determined by the modulusof the Ouyang embedding parameter. The reason is that the(modulus of the) Ouyang embedding parameter μOuyang has

the physical interpretation that |μOuyang| 23 gives essentially

the mass of the fundamental quarks arising from the D3−D7strings in the type IIB string theory dual as constructed in [2].Now, in [34] and the first reference in [23–25], it was shownthat : |μOuyang| ∼ r−α

h , α > 0. Further, the horizon radius rhwas estimated in the third reference in [23–25] to be:

rh ∼ exp

⎡⎢⎢⎣− 1

3(6π)13(gs N f

) 23(gsM2

N

) 13

⎤⎥⎥⎦ , (20)

implying a very small rh and hence a large |μOuyang| inthe large-N ‘MQGP limit’ of [1]. So, the quark mass indi-rectly enters our M theory computations via rh , which in theMQGP limit automatically implies considering the heavy-quark-mass limit.

The lattice calculations of [52] we have compared with inthis paper, have N f = 2(u, d) + 1(s) wherein mu/d = ms

20 .So, we can safely consider u/d quarks to be light and s quarkto be heavy. The trace anomaly and other thermodynamicquantities are seen in [52] to be insensitive to mu/d for T >

300 MeV; hence, at least for high temperatures it is acceptableif one assumes that only the heavy/strange quark contributesto the trace anomaly.

From the evaluation of the baryon chemical potential μC

in Appendix A we see that the |μOuyang| 1-limit corre-sponding to the heavy-quark limit of μC or |μOuyang| � 1-limit corresponding to the light-quark limit of μC whichwould imply that all N f flavors are respectively equallyheavy or light, yields: μC → 0. If one assumes that allquarks are s-like, in other words, “heavy”, we can also obtainat least a qualitative agreement between (A2) and (A3) andthe

(μCT

)2-vs-msT cross-section of Fig. 16 of [53].

Similarly, from the evaluation of the DBI action on theflavor D7-branes in Appendix A, one notices that in the light-quark-mass limit, effected by |μOuyang| � 1-limit, the UV-finite part of the DBI action (i.e. the part that remains finitein the large-UV-cutoff limit) vanishes. In the heavy-quark-mass limit effected by |μOuyang| 1-limit, using (20), onesees that there is no large-N -finite contribution that survivesfrom the UV-finite part of the DBI action. Therefore, in thelight- or heavy-quark mass limit wherein μC = 0, the UV-large-N finite contribution effectively arises only from thesupergravity action alone and not the DBI action; as shownabove, the former yields a first order Hawking–Page phasetransition. Hence, like the famous “Columbia plot” of phasetransition/cross-over in N f = 2 + 1 QCD, we have a phasetransition in the light/heavy quark-mass limit correspondingto vanishing baryon chemical potential.

4 QCD trace anomaly from M theory

In this section, we will compute the QCD trace anomalyhologarphically from M theory. This computation is dividedinto two Sect. 4.1 addresses the large temperature regime,i.e., T > Tc, and Sect. 4.2 addresses the low temperatureregime, i.e., T < Tc.

The UV-finite part of the D = 11 supergravity action isgiven as under:

SD=11 = 1

2κ211

[∫M11

√GMR +

∫∂M11

√hK

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620 Page 8 of 16 Eur. Phys. J. C (2020) 80:620

−1

2

∫M11

√GMG2

4 − 1

6

∫M11

C3 ∧ G4 ∧ G4

+(4πκ2

11

) 23

(2π)432.213

(∫M

d11x√GM

(J0 − 1

2E8

)

+∫

C3 ∧ X8 +∫

t8t8G2R3 + ·

)]− Sct, (21)

where GM is the determinant of the D = 11 metric, h is thesame restricted to r fixed at the UV cut-off, R is the D = 11Ricci scalar, K is the extrinsic curvature with

√hK being the

Gibbons–Hawking–York (GHY) surface term, G4 = dC3,C3 being the D = 11 three-form potential, κ11 being theD = 11 Newtonian constant, and J0, E8, X8, t8 and G2R4

are defined in Appendix B; the ellipsis in (21) denoting termsin [54] other than the one explicitly mentioned in (B1) – andthe counter-term Sct is added such that the Euclidean actionSD=11 is finite.

To evaluate the boundary trace anomaly consider the fol-lowing infinitesimal Weyl transformation [55,56]:

δr = −2rδσ (x), δhmn(x) = 2δσ (x)hmn(x), (22)

where δσ (x) is a local infinitesimal Weyl transformationparameter, and (m, n) �= r, r ∼ rUV rh , the UV cut-off.The trace anomaly is then given by:

T mm = − 1√

h

δ (Son−shell + Sct)

δσ (x). (23)

Let us now discuss the computation of the conformal traceanomaly via the application of (22), (23), separately for T >

Tc (Sect. 4.1) and T < Tc (Sect.4.2).

4.1 High temperatures (T > Tc)

In this subsection, we will evaluate the trace anomaly forlarge temperatures, i.e., T > Tc, and compare our resultswith the lattice results in [3]4. The upshot of this subsec-tion is that it is only the counter-term used to cancel theUV divergence generated from the GHY boundary term thatcontributes to the trace of the energy momentum tensor. Thevariation of the aforemetioned counter term, with respect tothe scalar appearing in the Weyl scaling of the radial coor-dinate and the metric along the other non-radial directions,generates the same temperature-dependent contribution asfrom the extrinsic curvature itself.

Now, on-shell:

RMN − gMN

2R − 1

12

(GMPQRG

PQRN − gMN

8G2

)

4 For previous bottom-up holographic computation-compatible latticeresults, see, e.g., [57,58]; see [59] for previous large-N lattice resultsfor the trace anomaly.

= −β1√−g

δ

δgMN

[√−g

(J0 − E8

2

)], (24)

β defined just above (35). As will be shown in 4.3, the RHSis sub-dominant as compared to the LHS of (24). Hence,RD=11 ∼ G2 > 0, and one can write:

∫ √GMR ∼

�∫ √

GM where the flux-generated cosmological constant

� is given by: �∫ √

GM ∼ ∫G4 ∧ ∗G4. One can show

[28]:

∫(G4 ∧ ∗G4 ≈)

√GM

∣∣∣H IIA3 ∧ AIIA

∣∣∣2

∼∫ √

GM∣∣∣H IIA

3 ∧ AIIA∣∣∣2∣∣∣∣Ouyang embeddingUV

∼∫ √

GM∣∣∣H IIA

3 ∧ AIIA∣∣∣2 δ

(θ1− αθ1

N15

)δ

(θ2− αθ2

N3

10

)δ (ψ − 2nπ)

∼a4gUV

s13/4

M4UVN 3/4NUV

f5/3 log

(R4

UVe

)

α11θ1

αθ2 log103 (N )

√1 − rh 4

R4UV

∫r=RUV

d4x√gM4(x0,1,2,3), (25)

where NUVf /MUV are average values of Meff(r), N eff

f (r) in

the UV, H IIA3 being the SYZ type IIA mirror/tripe T-dual NS-

NS three-form and A is the type IIA RR one-form generatedfrom the SYZ/triple T-dual of F IIB

1,3,5 of [2]. In the UV (to

ensure conformality), MUV, NUVf ≈ 0. The contribution to

the action from∫G4 ∧ ∗G4 is further suppressed by the

(small) resolution parameter-dependent factor a4.

As (the extrinsic curvature) K =√GM rr

2∂rdet hmndet hmn

,

(m, n) �= r , where hmn = GMmn(r = RUV = UV cut-off),

one can show that

K∂M5(x0,1,2,3;r=RUV) ∼ K∂M11(x0,1,2,3,θ1,2,φ1,2,ψ,x10;r=RUV)

∼⎛⎝ 1

L 3√N f

UV log N

⎞⎠

√1 − r4

h

R4UV

. (26)

Therefore, effectively we have dimensionally reduced the M-theory conformal anomaly to a D = 5 holographic conformal

anomaly. Near (θ1, θ2) ∼(N− 1

5 , N− 310

), one can show:

Son−shell + Sct

= Sflux � + SGHY

−αct

gUVs

7/4MUVN 19/20NUVf

5/3 log53 (N )

√1 − rh 4

R4UV

log(RUV)

α3θ1

α2θ2

×∫r=RUV

d4x√gM4(x0,1,2,3)(r = RUV), (27)

αct being an appropriate constant. As Sflux � + SGHY isinvariant under (22), therefore, from (23) it is only the RUV-dependent factors in the counter term required to cancel theUV-divergent contribution arising from the GHY boundaryterm (very similar to the example in [section 23.11.2 of] [56])

123

Eur. Phys. J. C (2020) 80:620 Page 9 of 16 620

that contributes to the trace of the energy momentum tensorand yields:

(T μ

μ

)grav ∼ −αct

√1 − rh4

R4UV

+ 2rh4 log(RUV)

R4UV

√1− rh

4

R4UV

L 3√NUV

f3√

log(N ) log(RUV)

∼ −αct1

L 3√N f

UV 3√

log(N )

[1

log RUV+ 2

(rhRUV

)4

+O((

rhRUV

)8)]

. (28)

Now, if MUV, NUVf , αθ1,2 are chosen such that the contribu-

tion from Sflux � exactly cancels off αct

L 3√

N fUV 3

√log(N ) log(RUV)

,

i.e.:

a =20√Nα2

θ14√

αct log54 (N )

O(1)gUVs

3/8M3/4UV

12√NUV

f4√

αθ2

. (29)

The temperature dependence on the right hand side of (29),

using (14), appears via αθ1,2 as R32UV

αθ1 αθ2√N

∼ |μOuyang|[given the temperature dependence of the Ouyang embeddingparameterμOuyang (see [34] and the first reference in [23–

25])]. Alternatively, one could instead consider(T̃ μ

μ

)grav

≡(T μ

μ

)grav+ αct

L 3√

N fUV 3

√log(N ) log(RUV)

for comparison with [3].

We thus see that(T μ

μ

)grav is given by the temperature/rh-

dependent contribution of ∼ K∂M5(x0,1,2,3;r=RUV). One henceconcludes that using dimensional consideration and notingthat string/M theory uses a mostly positive Minkowskiansignature whereas field theory uses a mostly negativeMinkowskian signature,

(T μ

μ

)grav → − (

T μμ

)FT ≡ −T μ

μ

(this notation will also be used in 4.2):

T μμ

T 4c

(T > Tc) ∼ 1

L 3√N f

UV 3√

log(N )

(rhRUV

)4

. (30)

Matching L(RUVR

)43√N f

UV 3√

log(N )T μ

μ

T 4 ≡ Tμμ

T 4 = r4h

R4 t̃4

(R being a scaling factor to match the lattice results of [3],and t̃ ≡ T

Tc) with the data points of [3] yields:

Tμμ

T 4 = γ

(PL[ωT ])4 , (31)

where γ and ω are numerical constants, and PL(z) is the so-called Product log function, also known as the Lambert W

250 500 750 1000 1250 1500 1750 2000

T (MeV)

0

1

2

3

4

5

6

Tµ µ/T

4

Nτ = 4Nτ = 6Nτ = 8Nτ = 10Nτ = 12

Fig. 2 The scaled trace anomaly, Tμμ

T 4 , plotted as a function of temper-ature for the high temperature region. The points represent lattice QCDresults (Fig. 2 of Reference [3]). The different Nτ values refer to thenumber of lattice cells in the imaginary time direction. The full curveis the fit obtained with the approach described in 2.1

function [60].5 A very good global Chi-squared minimiza-tion fit which includes the high temperature lattice results isshown on Fig. 2, and produces

γ = 3.08 ± 0.33, ω = 0.0100 ± 0.0006, (32)

where the uncertainties are obtained from the diagonal ele-ments of the covariance matrix.

5 As an example, requiring from [3], Tμμ

T 4 = 4.73291 at T = 199.184

MeV, and Tμμ

T 4 = 0.120615 at T = 1998.47 MeV implies that forgs = 0.3, N = 100, M = N f = 3:

R = 70501.1[(∑2n=0 αnβn

) (∑2m=0 γmβm

)] 14

PL (D)

;

β = 43.5944,

where α2 = γ2 = 1 and αn=0,1 and γm=0,1 are known numericalconstants, and:

D ≡12449

(520589

39898748

)N13N2 5N3

D;

N1 ≡2602633

10073826406 − 471991β8537539578

D,

N2 ≡ −N1 − 1,

N3 ≡ 4N1 − 5,

D ≡ 891009β

136759244780− 87826

69424325905.

This yields the following M-theory result for the anomaly:

Tμμ

T 4 = 3.449

(PL[0.0117T ])4 .

123

620 Page 10 of 16 Eur. Phys. J. C (2020) 80:620

4.2 Examining the low temperature region T < Tc

For low temperatures, i.e. T < Tc, it is the thermal back-ground (rh = 0) that is energetically preferred over theblack-hole gravitational dual. Therein, the GHY action toobecomes UV divergent in this limit. Further, from (30), wesee that one does not obtain any temperature-dependent con-tribution to the trace anomaly from the extrinsic curvature Kupon setting rh = 0. Now, unlike for temperatures T > Tc– corresponding to a gravitational dual with a black hole –wherein the temperature is constrained to be given in termsof rh , for low temperatures (T < Tc), the temperature isa free parameter. For the thermal case, we will continue touse a resolved warped conifold in the type IIA mirror of [1],assuming the existence of an rh-independent bare resolutionparameter a0 in the resolution parameter a guaranteeing theseparation of the M D5 − D5 branes RD5/D5 ≡ √

3a (as

for resolved conifolds radial distances exceeding√

3a, aretaken to be large). This on the supergravity side, provides anatural scale which will be the boundary common to the IR-UV interpolating region and the UV. The horizon radius rh isthe IR cut-off for high temperatures (T > Tc); for low tem-peratures (T < Tc) the IR cut-off is denoted by r0 (see above(15)). From (17), one sees that r0 and rh are proportional toeach other; from (20) one notes that rh , in the MQGP limit,is very small. Hence, for a0 �= 0, it is possible to arrange:RD5/D5 r0.

Consider the flux term:∫G4 ∧ ∗G4 in the rh = 0 limit.

Using the results of [1,28], one sees that:∫

G4 ∧ ∗G4 =∫

IRG4 ∧ ∗G4 +

∫UV

G4 ∧ ∗G4, (33)

where:

IR :∫

G4 ∧ ∗G4 ∼ g94s M4

IRN3f IR

N14

∫ √3a

r0

dr a6 d

dr

(r2 log r

),

UV :∫

G4 ∧ ∗G4 ∼ g94s M4

UVN 3f UV

N14

∫ RUV

√3a

dr a4 d

dr

(r4 log r

).

(34)

The fact, that on-shell, the only contribution from the fluxterms in the MQGP limit arises from

∫∂M11

C3 ∧∗G4, whichis what (34) in fact is, can be justified as follows. Defining

βl6p ≡(4πκ2

11

) 23

(2π)432.213 , the C3 EOM is:

d ∗ G4 = 1

2G4 ∧ G4 + βl6p X8. (35)

The flux-dependent terms in (21), disregarding the t8t8G2R3

term can be rewritten as:

1

2

(∫∂M11

C3 ∧ ∗G4 +∫M11

C3 ∧ d ∗ G4

)

+1

6

∫M11

G4 ∧ G4 ∧ C3 − βl6p

∫M11

C3 ∧ X8. (36)

On-shell, using (35), one obtains:

1

2

∫∂M11

C3 ∧ ∗G4 + 5

12

∫M11

G4 ∧ G4 ∧ C3

−βl6p2

∫M11

C3 ∧ X8. (37)

Now, it was shown in [1] that∫G4∧G4∧C3 = ∫

C3∧X8 =0 implying the assertion.

For rh = 0, one can arrange (∫R ∗ 1 ∼)

∫UV G4 ∧ ∗G4

to cancel off∫ √

hK (r = RUV). This will be effected via:

a = 1

O(1)

N1

20 (log N )54 α2

θ1(gUVs

) 38 M

34

UVα14θ2

, (38)

with the understanding that N ∼ O(102).Thus, effectively, it is not the boundary in the UV at

r = RUV but the boundary common to the IR-UV inter-polating region and the UV given by r = RD5/D5 = √

3athat acts as the effective boundary beyond which one doesnot generate a temperature-dependent UV-divergent counterterm. For RD5/D5 r0, the trace anomaly will be gener-ated from the infinitesimal Weyl transformation (22) of theRD5/D5( r0)-dependence in the counter term used for can-

celing the abovementioned “divergent” R2D5/D5

logRD5/D5

term (arising from∫

IR G4 ∧ ∗G4 ).For the purpose of applying (23) to calculate Tμ

μ, we willbe evaluating the same at the boundary r = RD5/D5 =√

3a(t̃) assuming that (23) is to be used with the same Weylscalar σ(x) (because x �= r ) at the boundary r = RD5/D5, as

the flux integral in the UV r >√

3a(t̃) is assumed to give anegligible contribution as compared to the IR. One can showthat:

∫IR

G4 ∧ ∗G4 ∼ a6gIRs

13/4N34 M4

IRNIRf

53 log(RD5/D5)

R2D5/D5

α11θ1

αθ2 log103 (N )

×∫r∈IR

d4x√gM4(x

0,1,2,3), (39)

and thus, using (23), implying the following conformalanomaly:

T μμ

T 4 (T < Tc) ∼ gIRs

54 M3

IRαθ2

N9

20 N IRf

13 α8

θ1(log N )

163 log(RUV)

(a(t̃)

T

)4 (a(t̃)

RUV

)4

log(√

3a(t̃))

, (40)

123

Eur. Phys. J. C (2020) 80:620 Page 11 of 16 620

120 130 140 150 160 170 180 190

T (MeV)

1

2

3

4

5T

µ µ/T

4

Nτ = 10Nτ = 12

Fig. 3 The trace anomaly, Tμμ

T 4 , plotted as a function of temperature forthe low temperature region. The points represent lattice QCD results(Fig. 2 of Reference [3]). The different Nτ values refer to the numberof lattice cells in the imaginary time direction. The full curve is the chi-square minimization fit obtained with the approach described in 4.2.The full curve represents the M theory result, Eqs. (40) and (43)

which using (38) yields:

T μμ

T 4 (T < Tc) ∼ gIRs

54 M3

IR

gUVs

32 N

14 MUV

3N IRf

13 NUV

f13 (log N )

13 log(RUV)R4

UV

×(a(t̃)

T

)4

log(√

3a(t̃))

. (41)

Further, assuming a(t̃)RUV

∼ t̃54 (inspired by the fractional

temperature dependence of the Ouyang embedding parame-ter arising from holographic computation of electrical con-ductivity in [23–25]), and definingTμ

μ ≡ log(RUV) (TcRUV)4

T μμ one obtains as the leading low-temperature contribution

to the QCD trace anomaly:

Tμμ

T 4 ∼ gIRs

54 M3

IR

gUVs

32 N

14 MUV

3N IRf

13 NUV

f13 (log N )

13

t̃ log(√

3αa(t̃) t̃)

× ≡ β t̃ log(√

3αa(t̃) t̃)

. (42)

The aforementioned behavior of a = a(t̃) was to ensurein the simplest way that Tμ

μ(T = 0) = 0 (as Tμμ(T ) =

GμνGμν(T = 0) − GμνGμν(T ) [61]). We may now com-pare with results of lattice QCD calculations, in the “lowtemperature” sector, as reported in [3]. Combining the latticedata sets which span 0.120 < T < 0.19 GeV (Nτ = 10, 12),one obtains the agreement shown in Fig. 3. A good globalchi-squared minimization fit is reached with:

β = 7.04 ± 0.5, αa0 = 0.78 ± 0.02. (43)

4.3 O(l6p R4) Terms (rh = 0)

Let us weigh in the higher derivative terms in the action (B1)up to O(R4) (that are of O(l6p)) in the rh = 0 limit.

1. First off, let us consider J0. One can show that in theMQGP limit [62]:

J0 ∼ Rφ2rθ1r Rφ1rθ1r Rφ2rφ1r Rθ1rθ1r

+ 1

2Rφ2rθ1θ2 Rrψθ1r R

rφ1rφ2

Rψrφ1r

, (44)

and for rh = 0:

∫UV

√GM J0 ∼ a4

MUV 2N 2f UV log N

N1310

(log a(t̃)

)2,

∫IR

√GM J0 ∼ a4

M IR 2N 2f IR log N

(log a(t̃)

)2

N1310

,

(45)

implying l6p∫

UV

√GM J0 � l6p

∫IR

√GM J0 � SGHY.

We thus notice that this contribution, in the MQGP limitof [1], is extremely suppressed relative to the one fromthe flux contribution (40), and hence would be discarded.

2. One can show that [62] (keeping track of only powers ofN and r -dependent terms), as a sample term:

E8 � Rtx1

t x1 Rx2x3

x2x3 Rrθ1rθ1

Rψx10

ψx10 , (46)

and hence:

∫IR

√GME8 � 10−6

MIRN IRf (log N )

143 R4

D5/D5

N3120

(logRD5/D5

) 203

,

∫UV

√GME8 � 10−6

MUVNUVf (log N )

143 R4

UV

N3120 (log RUV)

203

.

(47)

Hence, from (23) one sees that the contribution from theE8 term in the supergravity action would be extremelysuppressed relative to the flux contribution and thereforewill be discarded.

3. Finally, let us look at the t8t8G2R3 term in (B1). One

can show [62] that near θ1,2 ∼ N− 15 , N− 3

10 , the same is

O(

1

N11120

), and hence is extremely suppressed as com-

pared to the flux contribution, and will also be discarded.

5 Summary and discussion

In this paper, we showed that for Tc = 150 MeV, N f = 3,our M theory results for the variation of the QCD conformalanomaly with temperature can be made to be consistent with

123

620 Page 12 of 16 Eur. Phys. J. C (2020) 80:620

recent lattice results of [3], for both, high temperatures (T >

Tc) and low temperatures (T < Tc).The following points are noteworthy:

1. As explained in the first reference in [26,27] in the con-text of obtaining meson spectroscopy consistent withhadronic phenomenology, even though obtaining the typeIIA mirror of [2] and its M-theory uplift in [1] required alot of work, but once obtained, we are able to obtain thetrace/conformal anomaly from M theory which is veryclose to recent lattice results, already at O

(M0

)(which

is the non-conformal parameter) for T > Tc.6 There aretwo major reasons why this happens.

• As noted explicitly in the first reference in [26,27], thetype IIA SYZ (Strominger–Yau–Zaslow) mirror of[2] on account of the mixing of the type IIB metric andthe NS-NS BIIB under triple T duality, picks up sub-dominant terms in N of O

( 1Nκ

), 0 < κ < 1 which

are also of O(M0) in BIIA (the non-conformality intype IIB in [2] is because of BIIB which is accom-panied by M , the number of fractional D3 branes)

which are therefore bigger than the O(gsM2

N ) contri-butions, that were missed, e.g. in [63] in the contextof top-down meson spectroscopy.

• The type IIB gravity dual of [2] involved a resolvedwarped deformed conifold with a black hole andD5, D7 branes and D7 branes (plus fluxes). Thetype IIA mirror yields a non-Kähler warped resolvedconifold with a black hole and D6, D6 branes (plusfluxes). Now, warped resolved conifolds are moreeasier to deal with computationally than resolvedwarped deformed conifolds.7 It is the latter that areuplifted to M theory involving seven-folds of G2

structure (see [34]; the D6 branes get uplifted to KKmonopoles).

2. Strominger–Yau–Zaslow mirror construction is an entirelynew technique used for studying QCD, holographically,from a top-down string/M-theory dual.

3. The main Physics lesson that one learns in this work is thefollowing. It turns out that for high temperatures (T >

Tc) corresponding to a black-hole in the M theory dual, itis the counter term used to cancel the UV-divergent con-tribution of the GHY surface term (via reparametrizationof the UV boundary) that contributes to the temperaturedependence of the trace anomaly and guarantees vanish-

6 There is explicit dependence on gs , M, N f , N of the conformalanomaly for T < Tc; when written in terms of the temperature, thereis also an implicit dependence of the conformal anomaly on the afore-mentioned parameters via rh = rH (T ; gs , M, N f , N ) for T > Tc.7 One of us [AM] thanks K.Dasgupta for a short discussion on thispoint.

ing of Tμμ

T 4 at asymptotically large temperatures. On theother hand, for low temperatures (T < Tc) correspond-ing to a thermal background (no black hole), it is thecounter term used to cancel the divergent contributionarising from the flux term (at the boundary common tothe IR-UV interpolating region and the UV [assumingthe D5 − D5 separation to be much greater than the IRcut-off of the thermal background]) that contributes to thetemperature dependence of the conformal anomaly, and

guarantees increase of Tμμ

T 4 up to around Tc with increasein temperature – just like lattice calculations [3].

4. One should make note of the fact that was also statedearlier in Sect. 1, the type IIB string theory dual of ther-mal QCD as constructed in [2], unlike its earlier type IIAcousin – the Sakai–Sugimoto that catered only to the IR– is UV complete. So is hence the type IIA SYZ mirrorconstructed in [1]. Further, in the MQGP limit, the con-tributions of the higher order derivative corrections tothe trace anomaly, will be severely large-N suppressed.These two together ensure that the results of this paper onthe temperature variation of the trace anomaly consistentwith very recent lattice results, obtained from a top-downapproach unique to our work, can be completely trusted.

We have obtained the trace of the energy–momentum ten-sor in a top-down non-conformal M theory holographic dualwhich has a temperature behaviour consistent with that ofQCD. In the approach outlined in this paper, the low andhigh temperature regions correspond to two different limitsof the same theory, as is the case in the lattice QCD counter-part.

To conclude There have been several papers on a holo-graphic computation of the trace anomaly, but all bottom-up (e.g. [29,30], etc.). To the best of our knowledge, [1] isthe only (top-down) holographic M-theory dual (of thermalQCD) that is able to yield (as shown in this paper):

• (after a tuning of the (small) Ouyang embedding param-eter and radius of a blown-up S2 when expressed interms of the horizon radius) a deconfinement tempera-ture Tc from a Hawking–Page phase transition at vanish-ing baryon chemical potential consistent with the veryrecent lattice QCD results in the heavy quark limit

• a conformal anomaly variation with temperature compat-ible with the very recent lattice results at high (T > Tc)and low (T < Tc) temperatures – the latter missing, e.g.,in the bottom-up [29] (apart from the fact that the authorscompared with much older lattice results)

as well as (shown in earlier papers in the past few years):

123

Eur. Phys. J. C (2020) 80:620 Page 13 of 16 620

• Condensed Matter Physics: inclusive of the non-conformalcorrections, to obtain:

1. a lattice-compatible shear-viscosity-to-entropy-densityratio (first reference in [23–25])

2. temperature variation of a variety of transport coeffi-cients including the bulk-viscosity-to-shear-viscosityratio, diffusion coefficient, speed of sound (the lastreference in [23–25]), electrical and thermal conduc-tivity and the Wiedemann-Franz law (first referencein [23–25]);

• Particle Phenomenology: obtaining:

1. lattice compatible glueball spectroscopy [28]2. meson spectroscopy (first reference of [26,27])3. glueball-to-meson decay widths (second reference of

[26,27])

• Mathematics: using the beautiful concept of (SYZ) Mir-ror Symmetry in algebraic geometry, and the machineryof G-structures to provide, for the first time, an SU (3)-structure (for type IIB (second reference of [23–25])/IIA[34] holographic dual) and G2-structure [34] torsionclasses of the six- and seven-folds relevant to top-downholographic duals of thermal QCD.

The results of this note do demonstrate the potential ofthe methods outlined here to treat strongly-coupled problemsanalytically, whether conformal symmetry is manifest or not.Other applications to hadronic physics will include studies ofthe possible critical point in the hadronic phase diagram andof the high-density/low temperature color superconductingphase.

Acknowledgements This work was supported in part by the NaturalSciences and Engineering Research Council of Canada. We thank K.Dasgupta for many useful discussions, and P. Petreczky for pointing outthe location of the lattice data shown in Ref. [3], AM would like to thankthe McGill high energy theory group (K. Dasgupta in particular) for thewonderful hospitality during his visits to the same, at various stagesof this work, and M. Dhuria for help with some plots. AM was par-tially supported by a grant from the Council of Scientific and IndustrialResearch, Government of India, grant number CSR-1477-PHY.

Data Availability Statement This manuscript has associated data ina data repository. [Authors’ comment: The data used to plot the latticeresults in Figs. 2 and 3, is already available with the authors of reference[3] and was borrowed from one of its authors.]

Open Access This article is licensed under a Creative Commons Attri-bution 4.0 International License, which permits use, sharing, adaptation,distribution and reproduction in any medium or format, as long as yougive appropriate credit to the original author(s) and the source, pro-vide a link to the Creative Commons licence, and indicate if changeswere made. The images or other third party material in this articleare included in the article’s Creative Commons licence, unless indi-cated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permit-

ted use, you will need to obtain permission directly from the copy-right holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.Funded by SCOAP3.

A the baryon chemical potential µC and the DBI actionon the flavor D7-branes

In this appendix we discuss the evaluation of the baryonchemical potential μC and the DBI action on the world vol-ume of the flavor D7-branes. This is relevant to the discussionof showing lattice compatibility of our supergravity compu-tation of Tc in Sect. 3.

It was shown in [34] (by turning on a world-volume fluxFrt = ∂r At (r)) that the baryon chemical potential μC isgiven as under (with the simplifying assumption that using

the Ouyang embedding (14): e−φ = 1gs

− N IRf

2πlog |μOuyang|

for r ∈ [rh,√

3a] and e−φ = 1gs

− NUVf

2πlog |μOuyang| for

r ∈ (√

3a, RUV ]):

μC =∫ ∞

rhFrt dr

= −∫ ∞

rhdr

C√C2 + r9/2

(N f (r) log |μOuyang|

2π− 1

gs

)2

∼ −∫ RUV

rhdr

d

dr⎡⎢⎣r 2F1

⎛⎜⎝2

9,

1

2; 11

9;−

r9/2(N f (r) log |μOuyang|

2π− 1

gs

)2

C2

⎞⎟⎠⎤⎥⎦ ,

(A1)

which for |μOuyang| 1 yields:

−24/9C4/9�

(5

18

)�

( 119

) ( 1

NUVf

4/9 − 1

N IRf

4/9

)

(log |μOuyang|

) 49 18

√π

(A2)

if μOuyang does not depend on rh , and

−24/9C4/9�

(518

)�

( 119

)18√

πNUVf

4/9log

49 (|μOuyang|

− N f2rh11/2 log2(μ)

44π2C2 + rh

(A3)

if |μOuyang| ∼ r−αh , α > 0 [23–25].

Similarly, when |μOuyang| � 1, one obtains:

μC = −24/9C4/9�

(5

18

)�

( 119

) ( 1

NUVf

4/9 − 1

N IRf

4/9

)

(− log |μOuyang|) 4

9 18√

π

(A4)

123

620 Page 14 of 16 Eur. Phys. J. C (2020) 80:620

if μOuyang does not depend on rh , and:

μC = −24/9C4/9�

(518

)�

( 119

)18√

πNUVf

4/9 (− log |μOuyang|) 4

9

+N IRf

2rh11/2 log2 |μOuyang|44π2C2 − rh (A5)

if |μOuyang| ∼ r−αh , α > 0 [23–25].

Further, one notes that the D7-brane DBI action will begiven by:

SUV−finiteDBI ∼ TD7

∫ RUV

r=RD5/D5

√μr9/4

(1

gs− N f (r) log |μOuyang|

2π

)2√√√√√ r9/2

C2 + r9/2(

1gs

− N f (r) log |μOuyang|2π

)2

∼ TD7

∫ RUV

√3a

drd

dr

[2√

μr√

4π2(C2gs2 + r9/2

) + gs2N f (r)2r9/2 log2 |μOuyang| − 4πgs N f (r)r9/2 log |μOuyang|13πgs2

×(gs − 2πCgs 2F1

(2

9,

1

2; 11

9;−r9/2(gs N f (r) log |μOuyang| − 2π)2

4C2gs2π2

))]. (A6)

The “UV-finite part” of (A6), i.e., the action that remainsfinite in the large RUV -limit will be given by:

SLarge RUV−finiteDBI ∼ −

222/9√μ�(

518

)�

( 119

) ( CNUV

f

)4/9

13 18√

π log |μOuyang| 49

−311/4b11/2N f2rh

(11−α)2 log2(rh)

22π2C, (A7)

for |μOuyang| 1 (assuming |μOuyang| ∼ r−αh , 0 < α < 11)

and:

SLarge RUV−finiteDBI ∼ −

222/9C13/9√μ�(

518

)�

( 119

)13 18

√πNUV

f4/9|μOuyang| 4

9

−243 × 4

√3a11/2

√1C2

√μN f

2 log2(μ)

22π2 , (A8)

for |μOuyang| � 1.

B Definitions of symbols in the D = 11 supergravityaction (B1)

In (B1), (J0, E8, X8) are quartic polynomials in the curvaturetensor in 11-dimensional space and defined as:

J0 = 3.28(RMI J N RP I J Q R RSP

M RQRSN

+1

2RMN I J RPQI J R

RSPM RQ

RSN

),

E8 = εPQRM1N1...M4N4εPQRM ′1N

′1...M ′

4N′4RM ′

1N′1M1N1

...RM ′

4N′4M4N4

,

X8 = 1

192 · (2π2)4

[tr(R4) − 1

4(tr R2)2

], (B1)

where M, N , P, ... are D = 11 indices; the t8-symbol isdefined as (see [64]):

t N1...N88 = 1

16

( − 2(GN1N3GN2N4GN5N7GN6N8

+GN1N5GN2N6GN3N7GN4N8 + GN1N7GN2N8GN3N5GN4N6)

+8(GN2N3GN4N5GN6N7GN8N1

+GN2N5GN6N3GN4N7GN8N1

+GN2N5GN6N7GN8N3GN4N1)

−(N1 ↔ N2) − (N3 ↔ N4)

−(N5 ↔ N6) − (N7 ↔ N8)). (B2)

Also, from [54]:

t8t8G2R3 = t M1...M8

8 t8N1...N8

GM1N1PQGM2

N2PQ R N3N4

M3M4R N5N6M5M6

R N7N8M7M8

. (B3)

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