Comments on the U(1) axial symmetry and the chiral transition in QCD

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Nuclear Physics B Proceedings Supplement 00 (2014) 1–7

Supplement

Comments on the U(1) axial symmetry and the chiral transition in QCD

Enrico Meggiolaro

Dipartimento di Fisica, Universit`a di Pisa, and INFN, Sezione di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy

Abstract

We analyze (using a chiral effective Lagrangian model) the scalar and pseudoscalar meson mass spectrum of QCD at finite temperature, above the chiral transition atTc, looking, in particular, for signatures of a possible breaking of theU(1) axial symmetry aboveT_c. A detailed comparison between the case with a number of light quark flavors N_f ≥3 and the (remarkably different) caseN_f =2 is performed.

Keywords: finite-temperature QCD, quark-gluon plasma, chiral symmetries, chiral Lagrangians

1. Introduction

The so-calledchiral condensate,hqqi ≡¯ P^Nf

l=1hq¯lqli, is known to be an order parameter for the S U(Nf)⊗ S U(Nf) chiral symmetry of the QCD Lagrangian with Nf massless quarks (chiral limit), the physically rele- vant cases beingNf =2 andNf =3. Lattice determi- nations ofhqq¯ i(see, e.g., Refs. [1]) show that there is a chiral phase transitionat a temperatureTc∼150÷170 MeV, which is practically equal to thedeconfinement temperature T_d, separating the confined(or hadronic) phase atT <T_d, from thedeconfinedphase (also known asquark-gluon plasma) atT >T_d. ForT <T_c∼T_d, the chiral condensatehqqi¯ is nonzero and the chiral symmetry is spontaneously broken down to the vectorial subgroupS U(Nf)V, and the N²_f −1 J^P = 0⁻ lightest mesons are just the (pseudo-)Goldstone bosons associated with this breaking. Instead, forT >Tc ∼ Td, the chiral condensatehqqi¯ vanishes and the chiral symmetry is restored. But this is not the whole story, since QCD withNfmassless quarks also has aU(1) axial symmetry [U(1)A], which is broken by an anomaly at the quantum level [2, 3]: this anomaly plays a fundamental role in explaining the large mass of theη⁰meson [4, 5].

Email address:[email protected](Enrico Meggiolaro)

Now, the question is: What is the role of the U(1) axial symmetry for the finite temperature phase structure of QCD? One expects that, at least forT T_c, where the density ofinstantons is strongly suppressed due to a Debye-type screening [6]), also theU(1) axial symmetry will be (effectively) restored. This question is surely of phenomenological relevance since the particle mass spectrum aboveTcdrastically depends on the presence or absence of theU(1) axial symmetry. From the theoretical point of view, this question can be inves- tigated by comparing (e.g., on the lattice) the behavior at nonzero temperatures of the two-point correlation functions hOf(x)O^†_f(0)i for the variousqq¯ meson channels (“f”). For example, forNf = 2 [7, 8], one can study the meson channels (traditionally calledσ,~δ,ηand~π) which are listed in Table 1, together with their corresponding interpolating operators and their isospin (I) and spin-parity (J^P) quantum numbers. UnderS U(2)A

Meson channel Interpolating operator I J^P

σ(or f0) Oσ=qq 0 0⁺

~δ(or~a0) O~δ=q^~τ₂q 1 0⁺

η Oη=iqγ₅q 0 0⁻

~π O~π=iqγ5~τ

2q 1 0⁻

Table 1:q¯qmeson channels (forNf =2) and their quantum numbers.

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andU(1)Atransformations, theqq¯ meson channels are mixed as follows:

σ ←→^U(1)^A η

S U(2)_Al lS U(2)_A

~π ←→^U(1)^A ~δ

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The restoration of theS U(2) chiral symmetry implies that theσand~πchannels become degenerate, with identical correlators and, therefore, with identical (screening) masses,Mσ= Mπ. The same happens also for the channels η and~δ. Instead, an effective restoration of theU(1) axial symmetry should imply thatσbecomes degenerate withη, and~π becomes degenerate with~δ.

(Clearly, if both chiral symmetries are restored, then all σ,~π,η, and~δcorrelators should become the same.)

In Ref. [9] the scalar and pseudoscalar meson mass spectrum, above the chiral transition atTc, has been ana- lyzed using, instead, a chiral effective Lagrangian model (which was originally proposed in Refs. [10, 11, 12] and elaborated on in Refs. [13, 14, 15]), which, in addition to the usual chiral condensatehqq¯ i, also includes a (possible) genuine U(1)_A-breaking condensate that (possi- bly) survives across the chiral transition atT_c, staying different from zero atT >T_c. The motivations for considering this Lagrangian (and a critical comparison with other effective Lagrangian models existing in the literature) are recalled in Sec. 2. The results for the mesonic mass spectrum forT > Tc are summarized in Sec. 3, for the caseNf ≥3, and in Sec. 4, for the caseNf =2.

Finally, in Sec. 5, we shall make some comments on (i) the remarkable difference between the caseNf ≥3 and the caseNf = 2, and (ii) the comparison between our results and the available lattice results for Nf = 2 (or Nf =2+1).

2. Chiral effective Lagrangians

Chiral symmetry restoration at nonzero temperature is often studied in the framework of the following effective Lagrangian [16, 17, 18, 19, 20], written in terms of the (quark-bilinear) mesonic effective field Ui j ∼ q_jRq_iL=q_j₁₊_γ₅

2

q_i,¹

L₁(U,U^†)=L₀(U,U^†)+ B_m 2

√ 2

Tr[MU+M^†U^†] +L_I(U,U^†), (2)

1We use the following notation for the left-handed and right- handed quark fields:qL,R≡¹₂(1±γ5)q, withγ5≡ −iγ⁰γ¹γ²γ³.

where M = diag(m1, . . . ,mNf) is the quark mass matrix andL₀(U,U^†) is a term describing a kind of linear sigma model,

L₀(U,U^†)=1

2Tr[∂µU∂^µU^†]−V0(U,U^†), V0(U,U^†)=1

4λ²_πTr[(UU^†−ρ_πI)²]+1

4λ⁰²_π[Tr(UU^†)]², (3) whileL_I(U,U^†) is an interaction term of the form:

L_I(U,U^†)=c_I[detU+detU^†]. (4) Since underU(N_f)_L⊗U(N_f)_Rchiral transformations the quark fields and the mesonic effective fieldUtransform as

U(Nf)L⊗U(Nf)R: q_L,R→V_L,Rq_L,R ⇒ U→VLUV_R^†, (5) whereVLandVRare arbitraryNf×Nf unitary matrices, we have thatL₀(U,U^†) is invariant under the entire chiral groupU(Nf)L⊗U(Nf)R, while the interaction term (4) [and so the entire effective Lagrangian (2) in the chiral limitM=0] is invariant underS U(Nf)L⊗S U(Nf)R⊗ U(1)Vbutnotunder aU(1) axial transformation:

U(1)A: qL,R→e^∓iαqL,R ⇒ U→e^−i2αU. (6) However, as was noticed by Witten [21], Di Vecchia, and Veneziano [22], this type ofanomalousterm does not correctly reproduce the U(1) axial anomaly of the fundamental theory, i.e., of the QCD (and, moreover, it is inconsistent with the 1/N_cexpansion). In fact, one should require that, under a U(1) axial transformation (6), the effective Lagrangian, in the chiral limitM =0, transforms as

U(1)_A : L^(M⁼⁰⁾

e f f → L^(M⁼⁰⁾

e f f +α2N_fQ, (7) whereQ(x)= _64π^g²2ε^µνρσF_µν^a(x)F^a_ρσ(x) is thetopological charge densityandL_{e f f} also containsQas an auxiliary field. The correct effective Lagrangian, satisfying the transformation property (7), was derived in Refs. [21, 22, 23, 24, 25] and is given by

L₂(U,U^†,Q)=L₀(U,U^†)+ B_m 2

√ 2

Tr[MU+M^†U^†] + i

2QTr[logU−logU^†]+ 1

2AQ², (8) where A = −iR

d⁴xhT Q(x)Q(0)i|Y M is the so-called topological susceptibilityin the pure Yang–Mills (YM)

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theory. After integrating out the variableQin the effective Lagrangian (8), we are left with

L₂(U,U^†)=L₀(U,U^†)+ Bm

2

√

2Tr[MU+M^†U^†] +1

8An

Tr[logU−logU^†]o2

, (9)

to be compared with Eqs. (2)–(4).

For studying the phase structure of the theory at finite temperatureT, all the parameters appearing in the effective Lagrangian must be considered as functions of T. In particular, the parameterρπ, appearing in the first term of the potentialV0(U,U^†) in Eq. (3), is responsible for the behavior of the theory across the chiral phase transition atT = Tc. Let us consider, for a moment, only the linear sigma modelL₀(U,U^†), i.e., let us ne- glect both the anomalous symmetry-breaking term and the mass term in Eq. (9). Ifρ_π(T <Tc)> 0, then the valueUfor which the potentialV₀is minimum (that is, in a mean-field approach, thevacuum expectation value of the mesonic fieldU) is different from zero and can be chosen to be

U|_ρ_π_>0=vI, v≡ F_π

√ 2 =

s ρπλ²_π

λ²_π+Nfλ⁰²_π , (10) which is invariant under the vectorialU(Nf)Vsubgroup;

the chiral symmetry is thus spontaneously broken down toU(Nf)V. Instead, ifρπ(T >Tc)<0, we have that

U|ρπ<0=0, (11)

and the chiral symmetry is realized `a laWigner–Weyl.

The critical temperatureT_cfor the chiral phase transition is thus, in this case, simply the temperature at which the parameterρ_πvanishes:ρ_π(T_c)=0.

However, the anomalous term in Eq. (9) makes sense only in the low-temperature phase (T < Tc), and it is singular for T > Tc, where the vacuum expectation value of the mesonic fieldUvanishes. On the contrary, the interaction term (4) behaves well both in the low- and high-temperature phases.

The above-mentioned problems can be overcome by considering amodifiedeffective Lagrangian (which was originally proposed in Refs. [10, 11, 12] and elaborated on in Refs. [13, 14, 15]), which, in a sense, is an “extension” of bothL₁andL₂, having (i) the correct transformation property (7) under the chiral group, and (ii) an interaction term containing the determinant of the mesonic fieldU, of the kind of that in Eq. (4), assuming that there is aU(1)_A-breaking condensate that (pos- sibly) survives across the chiral transition atTc, staying different from zero up to a temperatureTU(1) >Tc.

(Of course, it is also possible that TU(1) → ∞, as a limit case. Another possible limit case, i.e., TU(1) = T_c, will be discussed in the concluding comments in Sec. 5.) The newU(1) chiral condensate has the form C_U(1) = hO_U(1)i, where, for a theory with N_f light quark flavors, O_U(1) is a 2N_f-quark local operator that has the chiral transformation properties of [3, 26, 27]

O_U(1)∼det( ¯qsRqtL)+det( ¯qsLqtR), wheres,t=1, . . . ,Nf

are flavor indices. The color indices (not explicitly indi- cated) are arranged in such a way that (i)O_U(1)is a color singlet, and (ii)CU(1) =hO_U(1)iis agenuine2Nf-quark condensate, i.e., it has nodisconnectedpart proportional to some power of the quark-antiquark chiral condensate hqq¯ i; the explicit form of the condensate for the cases Nf = 2 and Nf = 3 is discussed in detail in the Ap- pendix A of Ref. [15] (see also Refs. [12, 28]).

The modified effective Lagrangian is written in terms of the topological charge density Q, the mesonic field U_{i j}∼q¯_jRq_iL, and the new field variableX∼det( ¯q_sRq_tL), associated with theU(1) axial condensate [10, 11, 12],

L(U,U^†,X,X^†,Q)= 1

2Tr[∂µU∂^µU^†]+1

2∂µX∂^µX^†

−V(U,U^†,X,X^†)+ i

2ω1QTr[logU−logU^†] + i

2(1−ω₁)Q[logX−logX^†]+ 1

2AQ², (12) where the potential termV(U,U^†,X,X^†) has the form

V(U,U^†,X,X^†)

=1

4λ²_πTr[(UU^†−ρπI)²]+1

4λ⁰²_π[Tr(UU^†)]² +1

4λ²_X[XX^†−ρX]²− Bm

2√

2Tr[MU+M^†U^†]

− c1

2

√

2[X^†detU+XdetU^†]. (13) Since under chiral U(Nf)L⊗U(Nf)R transformations [see Eq. (5)] the fieldXtransforms exactly as detU,

U(Nf)L⊗U(Nf)R: X→detVL(detVR)^∗X, (14) [i.e.,Xis invariant underS U(Nf)L⊗S U(Nf)R⊗U(1)V, while, under a U(1) axial transformation (6), X → e^−i2N^f^αX], we have that, in the chiral limit M = 0, the effective Lagrangian (12) is invariant underS U(N_f)_L⊗ S U(N_f)_R⊗U(1)_V, while under aU(1) axial transformation, it correctly transforms as in Eq. (7).

After integrating out the variable Q in the effective Lagrangian (12), we are left with

L(U,U^†,X,X^†)=1

2Tr[∂_µU∂^µU^†]+1

2∂_µX∂^µX^†

−V(U,˜ U^†,X,X^†), (15)

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where

V˜ =V−1

8A{ω₁Tr[logU−logU^†]

+(1−ω₁)[logX−logX^†]}². (16) As we have already said, all the parameters appearing in the effective Lagrangian must be considered as functions of the physical temperatureT. In particular, the parametersρπ andρX determine the expectation values hUi and hXi, and so they are responsible for the behavior of the theory across theS U(Nf)⊗S U(Nf) and theU(1) chiral phase transitions. We shall assume that the parametersρπ andρX, as functions of the temperature T, behave as reported in Table 2; Tρ_π is thus the temperature at which the parameterρπ vanishes, while TU(1) > Tρπ is the temperature at which the parameter ρ_X vanishes (with, as we have said above,TU(1) → ∞, i.e., ρ_X > 0 ∀T, as a possible limit case). We shall

T <Tρπ Tρπ<T <TU(1) T >TU(1)

ρπ>0 ρπ <0 ρπ<0

ρ_X >0 ρ_X >0 ρ_X <0

Table 2: Dependence of the parametersρπ,ρX on the temperatureT.

see in the next section that, in the case Nf ≥ 3, one hasTc=Tρ_π (exactly as in the case of the linear sigma model L₀ discussed above), while, as we shall see in Sec. 4, the situation in which Nf = 2 is more com- plicated, beingTρπ < Tc < TU(1) in that case (unless Tρπ =Tc=TU(1); this limit case will be discussed in the concluding comments in Sec. 5).

Concerning the parameter ω₁, in order to avoid a singular behavior of the anomalous term in Eq. (16) above the chiral transition temperature T_c, where the vacuum expectation value of the mesonic fieldUvan- ishes (in the chiral limit M = 0), we shall assume that ω1(T ≥Tc)=0.

Finally, let us observe that the interaction term between theUandXfields in Eq. (13), i.e.,

L_int = c1

2

√

2[X^†detU+XdetU^†], (17) is very similar to the interaction term (4) that we have discussed above for the effective LagrangianL₁. How- ever, the term (17) is not anomalous, being invariant under the chiral groupU(Nf)L⊗U(Nf)R, by virtue of Eqs. (5) and (14). Nevertheless, if the field X has a (real) nonzero vacuum expectation valueX [the U(1) axial condensate], then we can write

X=(X+hX)eⁱ^{S X}^X (with : hX=SX =0), (18)

and, after susbstituting this in Eq. (17) and expanding in powers of the excitationshXandSX, one recovers, at the leading order, an interaction term of the form (4):

L_int=c_I[detU+detU^†]+. . . , c_I ≡ c1X 2

√ 2

. (19)

In what follows (see Ref. [9] for more details) we shall analyze the effects of assuming a nonzero value of the U(1) axial condensateXon the scalar and pseudoscalar meson mass spectrumabovethe chiral transition temperature (T >Tc), both for the caseNf ≥3 (Sec. 3) and for the caseNf =2 (Sec. 4).

3. Mass spectrum forT>Tcin the caseNf ≥3 Let us suppose to be in the range of temperatures Tρπ<T <TU(1), where, according to Table 2,

ρπ≡ −1

2B²_π<0, ρX ≡1

2F²_X>0. (20) Since we expect that, due to the sign of the parameter ρXin the potential (13), theU(1) axial symmetry is broken by a nonzero vacuum expectation value of the field X(at least forλ²_X → ∞we should haveX^†X → ¹

2F²_X), we shall use for the fieldUa simple linear parametrization, while, for the field X, we shall use a nonlinear parametrization (in the form of apolar decomposition),

U_{i j} =a_{i j}+ib_{i j}, X=αe^iβ =(α+h_X)eⁱ

β+^{S X}α

, (21) whereX=αe^iβ(withα,0) is the vacuum expectation value of X anda_{i j},b_{i j},h_X, andS_X are real fields. In- serting Eq. (21) into the expressions (13) and (16), we find the expressions for the potential with and without the anomalous term (withω1=0),

V˜ =V−1

8A[logX−logX^†]²=V+1

2Aβ², (22) V =1

4λ²_πTr[(UU^†)(UU^†)]+1

4λ⁰²_π[Tr(UU^†)]² +1

4λ²_πB²_π(a²_{i j}+b²_{i j})+1

4λ²_X α²−1 2F²_X

!2

− Bm

√

2miaii− c1

2

√ 2

[αcosβ(detU+detU^†) +iαsinβ(detU−detU^†)]+N_f

16λ²_πB⁴_π.

At theminimumof the potential we find that, at the leading order inM=diag(m₁, . . . ,mNf):

U= 2B_m

√2λ²_πB²_πM+. . . , α= F_X

√

2+O(detM), β=0.

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In particular, in the chiral limitM=0, we find thatU= 0 andX =α= ^F^√^X₂, which means that, in this range of temperaturesTρπ<T <TU(1), theS U(Nf)L⊗S U(Nf)R

chiral symmetry is restored so that we can say that (at least forNf ≥3)Tc≡Tρπ, while theU(1) axial symmetry is broken by theU(1) axial condensateX. Concern- ing the mass spectrum of the effective Lagrangian, we have 2N²_f degenerate scalar and pseudoscalar mesonic excitations, described by the fieldsai j and bi j, plus a scalar (0⁺) singlet fieldhX =α−αand a pseudoscalar (0⁻) singlet fieldSX =αβ[see Eq. (21)], with squared masses given by

M_U² = 1

2λ²_πB²_π, M²_h

X =λ²_XF²_X, M_S²

X = A

X²

= 2A F²_X.

(24) While the mesonic excitations described by the field U are of the usualqq¯ type, the scalar singlet fieldhX

and the pseudoscalar singlet fieldSX describe instead twoexotic, 2Nf-quark excitations of the formhX (α)∼ det( ¯qsLqtR) + det( ¯qsRqtL) and SX ∼ i[det( ¯qsLqtR) − det( ¯qsRqtL)]. In particular, the physical interpretation of the pseudoscalar singlet excitationS_Xis rather obvious, and it was already discussed in Ref. [10]: it is nothing but thewould-beGoldstone particle coming from the breaking of theU(1) axial symmetry. In fact, ne- glecting the anomaly, it has zero mass in the chiral limit of zero quark masses. Yet, considering the anomaly, it acquires atopologicalsquared mass proportional to the topological susceptibilityAof the pure YM theory, as required by the Witten–Veneziano mechanism [4, 5].

4. Mass spectrum forT >Tcin the caseNf =2 As in the previous section, we start considering the range of temperaturesTρπ <T <TU(1), with the param- etersρ_π andρ_X given by Eq. (20) (see also Table 2).

We shall use for the fieldUa more convenient variant of the linear parametrization, while, for the fieldX, we shall use the usual nonlinear parametrization given in Eq. (21),

U= 1

√ 2

[(σ+iη)I+(~δ+i~π)·~τ], X=αeîβ, (25) whereτâ(a=1,2,3) are the three Pauli matrices [with the usual normalization Tr(τâτ^b) =2δ_ab] and the fields σ,η,~δ, and~πdescribe, precisely, theqq¯mesonic excitations which are listed in Table 1.

Inserting Eq. (25) and M = diag(m_u,m_d) into the expressions (13) and (16), we find the following expres- sion for the potential with and without the anomalous

term (withω₁=0), V˜ =V−1

8A[logX−logX^†]² =V+1

2Aβ², (26) V =1

4λ²_πTr[(UU^†)(UU^†)]+1

4λ⁰²_π[Tr(UU^†)]² +1

4λ²_πB²_π[σ²+η²+~δ²+~π²]+1

4λ²_X α²−1 2F²_X

!2

− Bm

2 [(mu+md)σ+(mu−md)δ3]

− c1

2√

2[αcosβ(σ²−η²−~δ²+~π²) +2αsinβ(ση−~δ·~π)]+1

8λ²_πB⁴_π.

When studying the equations for astationary point of the potential, one immediately finds thatη=π_a =β=0 (P-invariance requires thatU = U^†andX = X^†), and alsoδ₁=δ₂ =0, while for the other valuesα,σandδ≡ δ3one finds the following solution (at the first nontrivial order in the quark masses):

σ = Bm

λ²_πB²_π−c1FX

(mu+md)+. . . ,

δ = Bm

λ²_πB²_π+c1FX

(mu−md)+. . . ,

α = FX

√

2+O(m²), (27)

which, in the chiral limitmu=md=0, reduces to U=0, X=α= FX

√

2, (28)

signalling that theS U(2)_L⊗S U(2)_Rchiral symmetry is restored, while the U(1) axial symmetry is broken by theU(1) axial condensateX.

Studying the matrix of the second derivatives (Hes- sian) of the potential with respect to the fields at the stationary point, one finds that (in the chiral limit mu = md = 0) there are (as in the case Nf ≥ 3) two exotic 0^± singlet mesonic excitations, described by the fields hX = α−αandSX = αβ, with squared masses M²_h

X = λ²_XF_X², M_S²

X = ^A

X² = ^2A_F2 X

, and, moreover, two qq¯chiral multiplets appear in the mass spectrum of the effective Lagrangian, namely,

(σ, ~π) : M²_σ=M²_π= ¹₂(λ²_πB²_π−√ 2c1X), (η, ~δ) : M²_η=M_δ²=¹₂(λ²_πB²_π+√

2c1X), (29) signalling the restoration of theS U(2)L⊗S U(2)R chiral symmetry.² Instead, the squared masses of theqq¯

2From the results (29) we see that the stationary point (28) is a

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mesonic excitations belonging to a same U(1) chiral multiplet, such as (σ, η) and (~π, ~δ), remainsplitby the quantity

∆M_U(1)² ≡M²_η−M²_σ=M_δ²−M_π²= √

2c1X, (30) proportional to theU(1) axial condensateX= ^F^√^X₂. This result is to be contrasted with the corresponding result obtained in the previous section for theNf ≥ 3 case, see Eq. (24), in whichall(scalar and pseudoscalar)qq¯ mesonic excitations (described by the field U) turned out to be degenerate, with squared massesM_U² = ¹₂λ²_πB²_π. 5. Comments on the results and conclusions

The difference in the mass spectrum of the qq¯ mesonic excitations (described by the fieldU) forT >

Tc between the caseNf = 2 and the caseNf ≥ 3 is due to the different role of the interaction termL_int = c_I[detU+detU^†]+. . ., withc_I ≡ ^c¹^X

2√

2, in the two cases.

WhenNf =2, this term is (at the lowest order) quadratic in the fieldsUso that it contributes to the squared mass matrix. Instead, whenN_f ≥3, this term is (at the lowest order) an interaction term of orderN_f in the fieldsU (e.g., acubicinteraction term forN_f =3) so that, in the chiral limit, whenU = 0, it does not affect the masses of theqq¯mesonic excitations.

Alternatively, we can also explain the difference by using a “diagrammatic” approach, i.e., by considering, for example, the diagrams that contribute to the following quantityD_U(1), defined as the difference between the correlators for theδ⁺andπ⁺channels:

D_U(1)(x)≡ hT Oδ⁺(x)O^†_δ₊(0)i − hT Oπ⁺(x)O^†_π₊(0)i

=2h

hTu¯RdL(x) ¯dRuL(0)i+hTu¯LdR(x) ¯dLuR(0)ii . (31) What happens below and aboveTc? ForT <Tc, in the chiral limitm1=. . . mNf =0, the left-handed and right- handed components of a given light quark flavor can

minimum of the potential, provided thatλ²_πB²_π > c1FX; otherwise, the Hessian evaluated at the stationary point would not be positive definite. Remembering that, forTρ_π <T<T_U(1),ρπ≡ −¹₂B²_π<0, the condition for the stationary point (28) to be a minimum can be written asG_π≡c₁F_X+2λ²_πρπ=c₁F_X−λ²_πB²_π<0. In other words, assumingc1FX >0 and approximately constant (as a function of the temperatureT) aroundTρ_π, we have that the stationary point (28) is a solution, i.e., a minimum of the potential, not immediately above Tρ_π, where the parameterρπvanishes (see Table 2) andGπis positive, but (assuming thatλ²_πB²_πbecomes large enough increasingT, starting fromλ²_πB²_π=0 atT=Tρ_π) only for temperatures that are sufficiently higher thanTρ_π, so that the conditionGπ<0 is satisfied, i.e., only for T>T_c>Tρ_π, whereT_cis defined by the conditionGπ(T=T_c)=0, and it is just what we can call thechiral transition temperature.

be connected through the qq¯ chiral condensate, giving rise to a nonzero contribution to the quantity D_U(1)(x) in Eq. (31). But for T >T_c, theqq¯ chiral condensate is zero, and, therefore, also the quantityD_U(1)(x) should be zero forT >T_c,unlessthere is a nonzeroU(1) axial condensateX; in that case, one should also consider the diagram with the insertion of a 2Nf-quark effective vertex associated with theU(1) axial condensateX. For Nf = 2 (see Figure 1), all the left-handed and right- handed components of theupanddownquark fields in Eq. (31) can be connected through the four-quark effective vertex, giving rise to a nonzero contribution to the quantityD_U(1)(x). Instead, forNf = 3 (see Figure 2),

Figure 1: Diagram with the contribution toD_U(1)from the 2Nf-quark effective vertex in the caseN_f=2.

the six-quark effective vertex also generates a couple of right-handed and left-handedstrangequarks, which, for T >Tc, can only be connected through the mass operator−msq_sqs, so that (differently from the caseNf =2) this contribution to the quantityD_U(1)(x) should vanish in the chiral limit; this implies that, for Nf = 3 and T >T_c, the~δand~πcorrelators are identical, and, as a consequence, alsoM_δ=M_π. This argument can be eas-

Figure 2: Diagram with the contribution toD_U₍₁₎from the 2Nf-quark effective vertex in the caseN_f=3.

ily generalized to include also the other meson channels and to the caseNf >3.

Finally, let us see how our results for the mass spectrum compare with the available lattice results. Lat- tice results for the case Nf = 2 (and for the case Nf =2+1, withmu,d →0 andms∼100 MeV) exist in the literature, even if the situation is, at the moment, a bit controversial. In fact, almost all lattice results [29, 30, 31, 32, 33, 34, 35, 36] (usingstaggered fermions ordomain-wall fermionson the lattice) indicate thenon- restorationof theU(1) axial symmetry above the chiral

(7)

transition at Tc, in the form of a small (but nonzero) splitting between the~δand~πcorrelators aboveTc, up to

∼1.2T_c. In terms of our result (30), we would interpret this by saying that, forT >T_c, there is still a nonzero U(1) axial condensate, X > 0, so that cI = ₂^c¹^√^X₂ > 0 and the above-mentioned interaction term, containing the determinant of the mesonic fieldU, is still effective forT >Tc.

However, other lattice results obtained in Ref. [37]

(using the so-calledoverlap fermionson the lattice; see also Ref. [38]) do not show evidence of the above- mentioned splitting above T_c, so indicating an effec- tiverestoration of theU(1) axial symmetry aboveT_c, at least, at the level of theqq¯ mesonic mass spectrum.

In terms of our result (30), we would interpret this by saying that, for T > Tc, one has c1X = 0, so that cI =₂^c¹^√^X₂ =0 and the above-mentioned interaction term, containing the determinant of the mesonic fieldU, is not present forT >T_c. For example, it could be that also theU(1) axial condensateX (like the usual chiral condensatehqqi) vanishes at¯ T =Tc, i.e., using the notation introduced in Sec. 2 (see Table 2), thatTU(1) =Tc. (Or, even more drastically, it could be that there is simplyno genuineU(1) axial condensate . . . )

In conclusion, further work will be necessary, both from the analytical point of view but especially from the numerical point of view (i.e., by lattice calculations), in order to unveil the persistent mystery of the fate of the U(1) axial symmetry at finite temperature.

Also the question of the (possible) exotic pseudoscalar singlet fieldSX ∼ i[det( ¯qsLqtR)−det( ¯qsRqtL)]

for T > T_c, with squared mass (in the chiral limit) given byM_S²

X|_M₌₀ = ^A

X² = ^2A_F2 X

, should be further inves- tigated, both theoretically and experimentally. As we have already said, the excitationSX is nothing but the would-beGoldstone particle coming from the breaking of theU(1) axial symmetry, as required by the Witten–

Veneziano mechanism [4, 5]. So, it is precisely what we should call the “η⁰” forT >T_c: is there any chance to observe it? Lattice results seem to indicate thatA(T) has a sharp decrease forT >Tcand it vanishes at∼1.2Tc

[39]. (And, maybe,A(T >Tc)→ 0 forNc → ∞, as it was suggested in Ref. [40].) Could this explain the “η⁰” mass decrease, which, according to Ref. [41], has been observed inside the fireball in heavy-ion collisions?

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