Gap in the Dirac spectrum and quark propagator symmetries in lattice QCD
aa r X i v : . [ h e p - l a t ] F e b Gap in the Dirac spectrum and quark propagator symmetries in lattice QCD
Marco Catillo ∗ Institute for Theoretical Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland (Dated: February 23, 2021)Recent studies on lattice QCD have shown the emergence of large symmetries at high temperature.This includes not only the restoration SU ( n F ) L × SU ( n F ) R , but also the effective emergence ofan unexpected symmetry group, namely SU (2) CS , which contains U (1) A as subgroup. At thesame time, at high T , a gap in Dirac spectrum appears. As it is argued in several works of L.Glozman et al. , there should be a connection between a gap in the Dirac spectrum and the presenceof SU (2) CS . In this paper, we analyze whether the quark propagator can be invariant under SU ( n F ) L × SU ( n F ) R and SU (2) CS transformations, in case of a gap in the Dirac spectrum, andconsequently the invariance of hadron correlators, giving the condition for a quark propagator to beinvariant under SU (2) CS . I. INTRODUCTION
In QCD, chiral and axial symmetry are symmetriesof the Lagrangian. However chiral symmetry is brokenspontaneously at low temperature, namely when
T < T c and axial symmetry is violated by the anomaly. Never-theless in lattice studies seem to be evident that chiralsymmetry get restored at high temperature and in manyworks (see e.g. Refs. [1, 2]) also the axial symmetryseems to emerge at high temperature (however we alsorefer to [3] for a recent study on this issue). However,this is not the whole story. In a range of temperatures,approximately T c − T c , also a further larger symmetryappears and this is what has been found in Refs. [4–6]. This symmetry includes the U (1) A and it is called SU (2) CS , see Refs. [4–10]. However, the full SU (2) CS isnot a symmetry of QCD Lagrangian, but still emerges athigh temperature in the calculation of hadron correlators.The important feature of QCD at high temperature isthat, the lowest eigenmodes as well as zero modes of theDirac operator become naturally suppressed. This canexplain why an effective emergence of U (1) A appears athigh temperature and, from the Banks-Casher relation[11], why SU ( n F ) L × SU ( n F ) R is restored.Regarding SU (2) CS , this fact is more fascinating, sincenot only it becomes evident at high temperature [4–6],when the lowest eigenmodes are suppressed, but its emer-gence is very explicit when the lowest eigenmodes areremoved manually from the quark propagator in the cal-culation of several hadron correlators, see Refs. [7–9].In this paper, we generalize the results of Refs. [12, 13]and we consider precisely the case where there is a gapin the Dirac spectrum. For simplicity, we will not con-sider possible anomaly terms in the action, which willarise the U (1) A breaking. In this situation, we study thesymmetries of the quark propagator starting from its for-mulation on the lattice. The reason is that, if the quarkpropagator has a symmetry, then also observables, that ∗ [email protected] can be written as only function of it (e.g. hadron corre-lators), contain such symmetry. We will show that, whenthere is a gap in the Dirac spectrum, the quark propaga-tor becomes SU ( n F ) L × SU ( n F ) R and U (1) A invariant.Driving by this fact, we impose also that the quark prop-agator is invariant under SU (2) CS , and see which condi-tions have to satisfy, in order to have the emergence of SU (2) CS . II. SOME PRELIMINARIES
We consider the euclidean formulation of QCD on thelattice with n F degenerate quark flavors. However wewill not examine the case where interaction terms amongquark with different flavors are present in the action (e.g.the presence of a ’t Hooft term) and we will not con-sider the presence of zero modes in the theory. In thiscase, the fermionic action can be split as S F = P n F i =1 S i ,where S i is the action of a single quark flavor, namely S i = a P x,y ¯ ψ ( x ) D ( m )lat ( x, y ) ψ ( y ), where a is the latticespacing and m is the mass of a single quark flavor. Thefull action is, then, given by S = S G + P n F i =1 S i , where S G is the gauge field action. D ( m )lat is the Dirac operator on the lattice for a given quarkflavor with mass m . Taking, for example, the Wilsondiscretization of such operator, it satisfies the relation D ( m )lat = ωD lat + m , with ω = 1 − am/ D lat isthe massless Dirac operator (see Refs. [14, 15]). D lat also satisfies the Wilson-Ginsparg equation and the γ -hermiticity, i.e. { D lat , γ } = a D lat γ D lat , D † lat = γ D lat γ . (1)The γ -hermiticity implies that the non-zero eigenvaluesof D lat come in pairs ( λ n , λ ∗ n ), since D lat v (+) n = λ n v (+) n , D lat v ( − ) n = λ ∗ n v ( − ) n , (2)where v ( − ) n = γ v (+) n and we have the following nor-malization, ( v ( ± ) n , v ( ± ) m ) = (1 /V ) δ nm and ( v ( ∓ ) n , v ( ± ) m ) =0, with V the total number of eigenvalues. Insteadthe Ginsparg-Wilson equation (together with the γ -hermiticity) tells us that the eigenvalues lie on a cir-cle with equation: λ n + λ ∗ n = a | λ n | . Denoting with η n = | λ n | , then such eigenvalues can be written as λ n = aη n / η n p − ( aη n / for η n ∈ (0 , /a ].The inverse of D ( m )lat can be written as D ( m ) − = X n h ( ωλ n + m ) − v (+) n v (+) † n +( ωλ ∗ n + m ) − v ( − ) n v ( − ) † n i . (3) D ( m ) − is the quark propagator and it is the main quan-tity for our interest. The reason is that, D ( m ) − is usedto compute hadron correlators and it is also equal to themean (cid:10) ψ i ¯ ψ i (cid:11) F = D ( m ) − , for a given quark flavor i . Thesubscript F stands for the average over the fermionicfields, which is weighted by the factor exp( − S F ). Thequantity (cid:10) ψ i ¯ ψ i (cid:11) F is connected with the chiral condensateΣ, which is the order parameter for the chiral symmetrybreaking. The thing that we want to show is how D ( m ) − transforms under chiral, axial and SU (2) CS transforma-tions and which are the criteria for having the emergenceof such symmetries in observables and hadron correla-tors. For doing so, we need to rewrite D ( m ) − in amore suitable way, separating different kind of contri-butions. At first we see that, using the expression of λ n in terms of η n , we can write the coefficients in Eq. (3) as,( ωλ n + m ) − = g ( m, η n ) + h r ( m, η n ) + i h i ( m, η n ), where g ( m, η ) = mk η + m h r ( m, η ) = 12 ω a η k η + m h i ( m, η ) = ω ηk η + m r − (cid:16) a η (cid:17) (4)with k = 1 − ( am/ and use that ( ωλ ∗ n + m ) − =( ωλ n + m ) − ∗ . The functions in Eq. (4) are basicallythe same introduced in Refs. [12, 13], generalized onthe lattice, but with the addition of a further function, h r ( m, η ), which comes from the lattice discretization.Now we can decompose D ( m ) − , in Eq. (3), as D ( m ) − = D ( m ) − (cid:12)(cid:12)(cid:12) h r = 0 h i = 0 + D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h i = 0 + D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h r = 0 , (5)where D ( m ) − (cid:12)(cid:12)(cid:12) h r = 0 h i = 0 = X n g ( m, η n ) h v (+) n v (+) † n + v ( − ) n v ( − ) † n i ,D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h i = 0 = X n h r ( m, η n ) h v (+) n v (+) † n + v ( − ) n v ( − ) † n i ,D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h r = 0 = X n i h i ( m, η n ) h v (+) n v (+) † n − v ( − ) n v ( − ) † n i . (6)The three functions, g ( m.η ), h r ( m.η ) and h i ( m.η ), havedifferent behavior for small and large values of η . There-fore they are important for us to analyze which partof the quark propagator D ( m ) − is dominant throughdifferent distributions of the Dirac eigenvalues. Naivelyspeaking, we can observe that for η → ∞ , we have that g ( m, η ) ∼ /η , h r ( m, η ) ∼ ωa/ k and h i ( m, η ) ∼ /η .Hence g ( m, η ) tends to be more suppressed for large η ,with respect h r ( m, η ) and h i ( m, η ). Vice versa for small η , then h r ( m, η ) ∼ h i ( m, η ) ∼
0, while g ( m, η )blows up. Furthermore, if we take the continuum limit( a →
0) and afterwards the massless limit ( m → g ( m, η ) becomes a δ -function centered in η = 0.Therefore g ( m, η ) tends to select the smallest eigenval-ues. This point is crucial for the Banks-Casher relation[11] and the chiral symmetry breaking, as we will clearin the next section. III. GAP IN THE DIRAC SPECTRUM
In this section, we want to show, from the Bank Casherrelation [11], how in presence of a gap in the Dirac spec-trum, the only relevant contribution in the quark prop-agator, in Eq. (5), is given by D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h r = 0 , while theothers, in Eq. (6), disappear in the following limit orderlim χ ≡ lim m → lim a → lim V →∞ , (7)where, for convenience, we define lim χ , as the chiral limit.At first we consider the distribution of the variable η as, ρ a ( m, V, η ) = 2 V X n : η n =2 /a δ ( η − η n ) , (8)where we excluded the eigenvalues η n = 2 /a , becausethese eigenvalues give some divergence problems in thecontinuum limit, as pointed out in Ref. [15]. ρ a ( m, V, η )is positive for all η, m, V and a , and its normalization isgiven by R ∞ dη ρ a ( m, V, η ) = 1 − ε . We need also to ε = (2 l/V ), where l is the number of eigenvalues 2 /a in thespectrum. By definition 0 ≤ l ≤ ( V/ − ε ≤ point out that all η n are strictly positive in the sum inEq. (8), which justifies the normalization in front of theequation, since there are exactly V / η n .Using Eq. (8), we can write the Banks-Casher rela-tion as lim χ h Σ( m, a, V ) i G = ( π/ h ρ (0 , i G , where wedefined ρ a ( m, η ) = lim V →∞ ρ a ( m, V, η ), and the factor1 / ρ a ( m, V, η )in Eq. (8). The average h·i G is over the gauge fields andweighted by the factor exp( − S G ). Σ( m, a, V ) is insteadthe chiral condensate on the lattice, which is given by[15] Σ( m, a, V ) = −h ¯ ψ i (cid:16) − a D lat (cid:17) ψ i i F = 1 ω Tr (cid:16) D ( m ) − − a V (cid:17) , (9)where i indicates a given flavor (no sum is understood).In the second line of Eq. (9), we are basically remov-ing by hand the contribution of 2 /a eigenvalues from thetrace. The Banks-Casher relation is also valid for a givengauge configuration, in the sense that lim χ Σ( m, a, V ) =( π/ ρ (0 , m, a, V )as Σ( m, a, V ) = 1 ω (cid:16) Tr( D ( m ) − ) − a (cid:17) = 2 ω Z ∞ dη ρ a ( m, V, η )( g ( m, η ) + h r ( m, η ))+ a lωV − a ω , (10)where, we observe that h i ( m, η ) does not contribute andthe further terms proportional to a will vanish in thelimits lim a → lim V →∞ . Hence, from Eq. (B7), we getΣ( m, a, V ) = 1 ω Tr( D ( m ) − (cid:12)(cid:12)(cid:12) h r = 0 h i = 0 )+Tr( D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h i = 0 ) − a . (11)Now we can take lim χ in both sides in (11) and useEqs. (B9) and (B10), and obtain that lim χ Σ( m, a, V ) =( π/ ρ (0 , g ( m, η ). Instead the h i ( m, η )-terms (see Eq. (5)) are traceless and h r ( m, η )terms are zero in the continuum limit.Now we go back to the Eq. (B9), and we assume thatthere is a gap in the Dirac spectrum. Namely ρ a ( m, V, η )is given by ρ a ( m, V, η ) = (cid:26) = 0 for η > Λ= 0 for η ≤ Λ (12)where Λ = Λ( m, V, a ) ≥ χ Λ >
0, then the Banks-Casher relationimplies that ρ (0 ,
0) = 0. This has a consequence in thestructure of lim χ D ( m ) − . In fact, we can use the decom-position in Eq. (5) and the result of Appendix B, in Eq.(B11). In this case, we getlim χ D ( m ) − = gap lim χ D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h r = 0 , (13)because the inequalities in Eq. (B11) tells us thatlim χ D ( m ) − ( x, y ) αa,βb (cid:12)(cid:12)(cid:12) h r = 0 h i = 0 = gap , lim χ D ( m ) − ( x, y ) αa,βb (cid:12)(cid:12)(cid:12) g = 0 h i = 0 = 0 , (14)where the first equation makes use of the gap assumptionand the second one is a consequence of the limits in (7).Hence only the h i -term becomes relevant and from Eq.(13), we can define the following continuum quark prop-agator: D − = gap lim χ D ( m ) − , (15)which is defined only in case of a gap in the distributionof the Dirac eigenvalues, namely ρ (0 ,
0) = 0.We can also give, explicitly, the matrix structure of D − .Take, in fact, the Dirac structure of the Dirac eigenvec-tors v ( ± ) n : v ( ± ) n = (cid:18) ± L n R n (cid:19) , (16)where we used γ defined in (A1). Then D − can beexpressed as D − = lim χ X η n > Λ i h i ( m, η n ) (cid:18) L n R † n R n L † n (cid:19) , (17)where in the sum we explicitly imposed that, there is agap in the Dirac spectrum of size Λ (according to Eq.(12)). Moreover, directly from the matrix structure inEq. (17), we have that D − satisfies the relations { γ , D − } = 0 , Tr( D − ) = 0 , (18)which are pretty trivial, since we are in the case wherechiral symmetry is restored and the massless limit istaken. IV. AXIAL AND CHIRAL SYMMETRY
Once we have made the hypothesis of a gap in the Diracspectrum (see Eq. (12)), we want to show how the prop-agator D − transforms under U (1) A and SU ( n F ) L × SU ( n F ) R groups. As we said in section II, we considerthe simple case in which no zero modes are present in thetheory and no ’t Hooft term is present in the action.At first we notice that, the quark propagator D − canbe also written as D − = gap lim χ h ψ i ¯ ψ i i F , (19)where we basically used Eq. (15) and the fact that D ( m ) − = h ψ i ¯ ψ i i F , with i a fixed flavor index. Now,looking how the fields ψ i and ¯ ψ i transform under U (1) A and SU ( n F ) L × SU ( n F ) R groups, we can see how D − ,in Eq. (19), transforms.Starting from U (1) A transformations, they are imple-mented as U (1) A : ψ i → exp (i αγ ) ψ i , ¯ ψ i → ¯ ψ i exp (i αγ ) , (20)with γ given in Eq. (A1). Typically on the lattice the γ matrix is replaced by ˆ γ = γ ( − ( a/ D − ), whichincludes a correction of finite lattice spacing. Howeverfor simplicity, we consider the standard γ , since what-ever correction we apply, should vanish anyhow in thecontinuum limit.Inserting the transformations (20) in Eq. (19), we get: U (1) A : D − → D − U (1) A gap = cos α D − + i sin α cos α ( γ D − + D − γ ) − sin α γ D − γ . (21)Using now the anticommutation relation in Eq. (18) for D − , we get that D − is invariant under U (1) A trans-formations, namely D − U (1) A gap = D − . This result isquite expected, if you consider that we are not taking inconsideration the eventual transformation of the measure D ¯ ψψ in h ψ i ¯ ψ i i F (see Eq. (19)) under U (1) A .Now we look the transformation of ( D − ) ij =lim χ h ψ i ¯ ψ j i (where i, j are generic flavor indices) underthe chiral group SU ( n F ) L × SU ( n F ) R .At first we need to remind again our assumptions.As we already said in section II, we suppose that thefermionic action can be split into the sum of the actionof different quark flavors, i.e. S F = P n F i =1 S i , then themean value h ψ i ¯ ψ j i turns to be zero for i = j , because h ψ i ¯ ψ j i = h ψ i ih ¯ ψ j i = 0, since h ¯ ψ j i and h ψ i i are null forall i, j . In this case we can write that ( D − ) ij = δ ij D − ,with D − = lim χ h ψ i ¯ ψ i i for some given flavor i .We are now ready for our calculations. Defining thequark fields ψ L/R = P L/R ψ and ¯ ψ L/R = ¯ ψ L/R P R/L , with P L = ( − γ ) / P R = ( + γ ) /
2, then the chiraltransformations are implemented as ψ L → exp(i α aL T a ) ψ L , ¯ ψ L → ¯ ψ L exp( − i α aL T a ) ,ψ R → exp(i α aR T a ) ψ R , ¯ ψ R → ¯ ψ R exp( − i α aR T a ) (22)where T a are the generators of the group SU ( n F ) . Usingthat ψ = ψ L + ψ R and ¯ ψ = ¯ ψ L + ¯ ψ R , we have that ( D − ) ij can be decomposed as (see Appendix C)( D − ) ij = ( D − LL ) ij + ( D − RR ) ij (23)where( D − LL ) ij = lim χ h ψ L,i ¯ ψ L,j i , ( D − RR ) ij = lim χ h ψ R,i ¯ ψ R,j i . (24)The Eq. (23) tells us that in presence of a gap onlythe LL and RR components survive in the quark prop-agator. Other possible terms like lim χ h ψ L,i ¯ ψ R,j i andlim χ h ψ R,i ¯ ψ L,j i are zero, because of Eq. (18), as it isshown in Appendix C.It is now straightforward proving that ( D − LL ) ij and( D − RR ) ij are invariant under the transformations in (22).Indeed, from the previous discussion, we can write that,( D − XX ) ij = δ ij D − XX for X = L or R , with D − XX =lim χ h ψ X,i ¯ ψ X,i i for a given index flavor i . Therefore un-der SU ( n F ) L × SU ( n F ) R transformations we have( D − XX ) ij → ( D − XX ) SU ( n F ) X ij = (exp(i α aX T a )) im ( D − XX ) mn (exp( − i α aX T a )) nj = (exp(i α aX T a )) im δ mn ( D − XX )(exp( − i α aX T a )) nj = δ ij ( D − LL ) = ( D − XX ) ij . (25)Hence from Eq. (23) we get( D − ) ij → ( D − ) SU ( n F ) L × SU ( n F ) R ij = ( D − RR ) SU ( n F ) R ij + ( D − LL ) SU ( n F ) L ij = ( D − ) ij , (26)where we used Eq. (25). Therefore D − is invariantunder SU ( n F ) L × SU ( n F ) R group, when a gap open inthe Dirac spectrum.These results are expected, because we assumed aneigenvalue distribution as in Eq. (12), with some gap Λ.The reason is that, we have basically looked the Banks-Casher relation from the other way around. If ρ (0 ,
0) =0 and no zero modes are present in the theory, then U (1) A and SU ( n F ) L × SU ( n F ) R are restored in the masslesslimit. V. SYMMETRIES IN CORRELATORS
The invariance of D − under U (1) A and SU ( n F ) L × SU ( n F ) R implies also the invariance of the hadron cor-relators. This is what we want to show in this section.A general observable, which is a function of a set ofvariables { ψ } and { ¯ ψ } , in different indices, can be writtenas C ( ψ, ¯ ψ ) = X k X I ,...,I k Γ x ...x k I ...I k k Y l =1 ψ ( x l ) I l ¯ ψ ( x k + l ) I k + l , (27) where the Γs represent different coefficients, I l = { i l , α l , a l } , for l = 1 , ..., k , represents a multi-index,which contains flavor, Dirac and color indices respec-tively. In the sense that, for a given l , then i l and i k + l gofrom 1 to n F (number of flavors), α l and α k + l go from 1to 4 (number of Dirac indices), and a l and a k + l go from1 to N c (number of colors). In Eq. (27), we have put thesame number of ψ and ¯ ψ variables, because otherwise hC ( ψ, ¯ ψ ) i F would be zero, for the Wick theorem and thefact that h ψ i F = h ¯ ψ i F = 0. Taking lim χ hC ( ψ, ¯ ψ ) i F , wecan write it in terms of the quark propagator. As it isshown in Appendix D, under the assumption of a gap inthe Dirac spectrum (12), C ( ψ, ¯ ψ ) can be written aslim χ hC ( ψ, ¯ ψ ) i F = gap X k X I ...I k X p s ( p ) Γ x ...x k I ...I k k Y l =1 δ p ( i l ) i k + l D − ( x p ( l ) , x k + l ) p ( α l ) p ( a l ) ,α k + l a k + l , (28)where we have put D − , given in Eq. (17). In Eq. (28),¯Γ = lim χ Γ, instead p ( l ) (as well as p ( α l ) and p ( a )) is thelabel obtained after the application of p transpositionsof l (respectively α and a ), and we are summing overall possible transpositions p ; s ( p ) is the sign which eachpermutation gives, coming from the exchange of I l and p ( I l ). The Kronecker delta δ p ( i l ) j l indicates that for dif- ferent quark flavors the contribution is zero, since we areassuming that no interaction terms among quark flavorsis present in the fermionic action.The Eq. (28) tells us that a generic correlator can bewritten as a linear combination of the quark propagator D − . Therefore under transformations ψ → ψ G and ¯ ψ → ¯ ψ G , with G = U (1) A or SU ( n f ) L × SU ( n f ) R , we havethatlim χ hC ( ψ G , ¯ ψ G ) i F = gap X k X I ...I k X p s ( p ) Γ x ...x k I ...I k k Y l =1 δ p ( i l ) i k + l D − ( x p ( l ) , x k + l ) G p ( α l ) p ( a l ) ,α k + l a k + l = X k X I ...I k X p s ( p ) Γ x ...x k I ...I k k Y l =1 δ p ( i l ) i k + l D − ( x p ( l ) , x k + l ) p ( α l ) p ( a l ) ,α k + l a k + l = lim χ hC ( ψ, ¯ ψ ) i F , (29)where in second line, we used the results of section IV,which tell us that D − G gap = D − . Now if the distribu-tion of the Dirac eigenvalues has a gap for all gaugeconfigurations, then the symmetry in Eq. (29) is heldeven after we average over the gauge fields h·i G , i.e. h lim χ hC ( ψ G , ¯ ψ G ) i F i G = gap h lim χ hC ( ψ, ¯ ψ ) i F i G .Therefore a symmetry in the quark propagator D − induces the invariance a generic observable and there-fore an invariance of hadron correlators, which are specialcases of Eq. (27). VI. CHIRALSPIN SYMMETRY ANDEIGENVECTORS
Chiralspin symmetry, seems to emerge in the hadronspectrum when the lowest eigenmodes of the Dirac op-erator are suppressed. This can happen surgically, byremoving them from the quark propagator see Refs. [7–9], or at high temperature (
T > . T c ) as shown in Refs.[4–6], where a gap in the Dirac spectrum as in Eq. (12)appears naturally, see Refs. [1, 2, 16] for that. There-fore, it seems that the necessary condition for having the SU (2) CS symmetry in the hadron spectrum is that, thereis a gap in the eigenvalue distribution, whenever if thishappens manually or naturally at high temperature.Such symmetry appears basically as a degenerationof hadron masses and, more in general, in hadron cor-relators (see Eq. (27)) summed over the space indicesand then correctly normalized with the space volume.However doing this sum analytically, in the most gen-eral case, can be difficult. Therefore we adopt a differ-ent procedure. Suppose to have two generic correlatorsof some hadrons connected via SU (2) CS group, namely C ( x, y ) = hO ( x ) ¯ O ( y ) i and C ( x, y ) = hO ( x ) ¯ O ( y ) i ,summed over x and y , i.e. C i ( t ) = P x , y C i ( x, y ), for i = 1 ,
2. In this case for large t = | x − y | , the dom-inant part is given by the exponential of the mass, i.e. C i ( t ) ∼ exp( − m i t ), with m i with i = 1 ,
2, the massesof the two given hadrons. Therefore in such limit, thedegeneration of the masses m i corresponds to a degener-ation of C i ( t ) and vice versa. We can take now the fol-lowing temporal correlators C i ( x , y ) = hO i ( x ) ¯ O i ( y ) i for i = 1 ,
2, where we have set x and y to zero. Then,also in this case for large t = | x − y | , we can writethat C i ( x , y ) ∼ exp( − m i t ). Hence even in this case, adegeneration of the masses m i correspond to a degenera-tion of C i ( x , y ) and vice versa, when t is large enough.Now, the hadron correlators C i ( x , y ) are basically equalto the ones given in Eq. (27), but with a proper choiceof the coefficients Γ and averaged over the fermionic andthe gluon action, i.e. h·i = hh·i F i G . Such correlators canbe always expressed in terms of the quark propagator aswe have seen in section V. In particular C i ( x , y ), uponthe limits lim m → lim a → lim V →∞ and the gap hypothe-sis (12), can be expressed in terms of the quark propa-gator D − ( x , y ). Therefore a symmetry of the quarkpropagator D − ( x , y ) induces a symmetry of C i ( x , y )and consequently a degeneration of the hadron masses.Hence, here we want to conjecture that the symmetry SU (2) CS arises from an invariance of D − , similar onwhat we have observed in the case of axial and chiralsymmetry in section IV. In particular we want to im-pose the SU (2) CS symmetry in D − ( x , y ), which isthe quark propagator in the time direction.In order to impose such symmetry, we observe that D − ( x , y ) = gap lim χ h ψ ( x ) i ¯ ψ ( y ) i i F , (30)for a given flavor i . Hence for the calculation of D − ( x , y ) we just need to consider the quark fieldsin the points x = ( x , ) and y = ( y , ).Before to proceed in our calculations, we remind whatthe SU (2) CS group is. The SU (2) CS group is defined bythe set of transformations of the quark fields generatedby Σ = { γ , i γ γ , − γ } , which forms an su(2) algebra.Taking two generic quark fields ψ ( x ) i and ¯ ψ ( x ) i for agiven flavor index i , in the point x = ( x , ), the SU (2) CS transformations are given by ψ ( x ) i → exp(i α a Σ a ) ψ ( x ) i , ¯ ψ ( x ) i → ¯ ψ ( x ) i γ exp( − i α a Σ a ) γ (31) where we used the definition of chiralspin transformationsas in Ref. [10]. We can now plug the transformations (31)in Eq. (30) and look how D − ( x , y ) transforms. This isdone in Appendix E. We found that D − ( x , y ) is notinvariant under SU (2) CS , but only under its subgroup U (1) A ⊂ SU (2) CS . Hence in Appendix E, we also givethe condition that D − ( x , y ) needs to satisfy in orderto be chiralspin symmetric. Defining as D − , CS the chi-ralspin symmetric quark propagator, then the conditionof chiralspin invariance of the quark propagator is γ D − , CS ( x , y ) γ = D − , CS ( x , y ) , (32)which is shown in Appendix E. Using now the expressionof the eigenvectors in (16) and the expression (17), weobtain that the sufficient condition for satisfying Eq. (32)is that L n ( x ) R † n ( y ) = CS R n ( x ) L † n ( y ) . (33)From this equation, we can give the following ansatz onthe eigenvector structure v ( ± ) n ( x ) = CS (cid:18) ± χ n ( x ) τ n χ n ( x ) (cid:19) , with τ ∗ n = τ n (34)where χ n ( x ) is some generic 2-component field and τ n isa real operator. Finally the quark propagator looks like D − , CS ( x , y ) = lim χ X η n > Λ i h i ( m, η n ) τ n χ n ( x ) χ † n ( y ) γ , (35)which is our SU (2) CS -invariant quark propagator in thepoints x = ( x , ) and y = ( y , ).Now the reason why chiralspin symmetry emergeswhen there is a gap in the Dirac spectrum is difficultto understand. It may be given by dynamical reasonsas pointed out in Ref. [13] and connected by a stringyfluid matter structure, as explained in Ref. [17]. Here wehave just imposed the chiralspin symmetry in the tem-poral quark propagator in order to get the structure ofthe Dirac eigenvectors (34) at x = ( x , ). If such quarkpropagator is implemented in the calculation of the tem-poral correlators C ( x , y ), it will induce the SU (2) CS symmetry in the hadron spectrum. However, we con-clude saying that the gap in the Dirac spectrum is just anecessary condition for having such symmetry, but stillnot sufficient and other input are needed. This is alsoremarked by the fact that at T & T c , the chiralspinsymmetry seems to disappear, as pointed out in recentworks [5, 6]. In such regime the gap in the Dirac spec-trum persists, but chiralspin symmetry does not. VII. CONCLUSIONS
We can summarize, now, our main results. We havestarted from a lattice formulation of the quark propa-gator and we have seen that when there is a gap inthe distribution of the Dirac eigenvalues and we arenot in presence of anomaly, then, under the limitslim m → lim a → lim V →∞ , we have that the quark prop-agator simplifies as in Eq. (17) and it becomes invariantunder SU ( n F ) L × SU ( n F ) R and U (1) A transformations.This induces an invariance of whatever observable whichis a function of the quark fields, especially the hadron cor-relators, bringing to a degeneration of the hadron massesconnected through SU ( n F ) L × SU ( n F ) R and U (1) A .Moreover, upon the lattice evidence [4–9] of a newsymmetry group (namely SU (2) CS ) when the low-lyingDirac eigenmodes are suppressed (by hand or going athigh temperature, T > . T c ), we have studied if thequark propagator, in Eq. (17), is also invariant under SU (2) CS . We found that the only gap is not sufficientfor the evidence of such symmetry. However we haveimposed which condition the quark propagator, in thetime direction, needs to satisfy in order to be SU (2) CS -invariant. This is given in Eq. (32) and we give also thestructure of such quark propagator in Eq. (35), arguingthat such kind of quark propagator can lead to a degen-eration of the hadron masses connected via SU (2) CS . ACKNOWLEDGMENTS
I am thankful to L. Glozman and C. B. Lang for intro-ducing me on this topic. I also thank Marina Marinkovi´cfor the support. This work is supported by the Institutefor Theoretical Physics, ETH Zurich.
Appendix A: Conventions
Here we present the main conventions and notationsused in this paper.The gamma matrices in euclidean space-time are takenin the following representation: γ µ = (cid:18) σ µ σ µ (cid:19) , γ = (cid:18) − (cid:19) (A1)where σ µ = ( , i σ ), ¯ σ µ = ( , − i σ ), for µ = 1 , , , σ ≡ σ , , are the Pauli matrices. The matrices inEq. (A1) satisfy the properties: { γ µ , γ ν } = 2 δ µν and { γ µ , γ } = 0, for all µ, ν .In this paper, vectors are indicated without indices,however you need to keep in mind that their indices struc-ture is given by v ≡ v ( x ) αa , where v is a generic given vec-tor and α = 1 , ..., a = 1 , ..., N c is thecolor index, and finally x is the space-time position. Thesame notation is also applied for matrices, namely given a generic matrix A , we have A ≡ A ( x, y ) αa,βb , with α, β Dirac indices, a, b color indices and x, y are two space-time points.The scalar product is defined as( v, Aw ) = X x,y,α,β,a,b v ( x ) † αa A ( x, y ) αa,βb w ( y ) βb . (A2)In the special case where the matrix A is A = h z † ,where h and z are two generic vectors, we have thatTr( A ) = X x,α,a h ( x ) αa z ( x ) † αa = ( z, h ) . (A3)The relation (A3) is often used in section III for thequark propagator. Appendix B: Traces of quark propagator
In this appendix we derive a few relations which relatethe trace of the quark propagator and its parts, with thedistribution of the Dirac eigenvalues.At first we observe that the orthogonality relationof the Dirac eigenvectors, defined in Eq. (2), namely( v ( ± ) n , v ( ± ) m ) = (1 /V ) δ nm , where the scalar product is de-fined as in Eqs. (A2) and (A3), implies thatTr( v ( ± ) n v ( ± ) † n ) = ( v ( ± ) n , v ( ± ) n ) = X x,α,a | v ( ± ) n ( x ) αa | = 1 /V. (B1)Therefore taking the trace in left-hand side of Eq. (6),we get Tr( D ( m ) − (cid:12)(cid:12)(cid:12) h r = 0 h i = 0 ) = 2 V X n g ( m, η n ) , Tr( D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h i = 0 ) = 2 V X n h r ( m, η n ) , Tr( D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h r = 0 ) = 0 , (B2)and consequently from Eq. (5),Tr( D ( m ) − ) = 2 V X n ( g ( m, η n ) + h r ( m, η n )) . (B3)Taking again the Eq. (B1), we have that each terminside the sum, satisfies the inequality | v ( ± ) n ( x ) αa | ≤ /V . Hence | v (+) n ( x ) αa v (+) n ( y ) † βb ± v ( − ) n ( x ) αa v ( − ) n ( y ) † βb | ≤| v (+) n ( x ) αa || v (+) n ( y ) † βb | + | v ( − ) n ( x ) αa || v ( − ) n ( y ) † βb | ≤ /V .This fact, together with Eq. (B2), gives the followinginequalities: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D ( m ) − ( x, y ) αa,βb (cid:12)(cid:12)(cid:12) h r = 0 h i = 0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Tr( D ( m ) − (cid:12)(cid:12)(cid:12) h r = 0 h i = 0 ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D ( m ) − ( x, y ) αa,βb (cid:12)(cid:12)(cid:12) g = 0 h i = 0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Tr( D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h i = 0 ) , (B4)for generic indices x, y, α, β, a, b . The inequalities in Eq.(B4) are obtained from Eq. (6).Now we want to use the definition of the eigenvaluedistribution in Eq. (8), where the eigenvalues η n = 2 /a has been removed by hand, and see the relation with theabove traces of the quark propagator. At first we needto consider that2 V X n : η n =2 /a g ( m, η n ) = Tr( D ( m ) − (cid:12)(cid:12)(cid:12) h r = 0 h i = 0 ) − a lV , V X n : η n =2 /a h r ( m, η n ) = Tr( D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h i = 0 ) − a lV , (B5)where l is the multiplicity of the eigenvalues 2 /a . Theterms ( a l/ V ) becomes irrelevant in the thermodynamic( V → ∞ ) and continuum ( a →
0) limit. Hence fora generic function f ( m, η ) (which can be g ( m, η ) or h r ( m, η )), we have,2 V X n : η n =2 /a f ( m, η n ) = 2 V Z ∞ dη X n : η n =2 /a f ( m, η ) δ ( η − η n )= Z ∞ ρ a ( m, V, η ) f ( m, η ) , (B6)where we used that R ∞ δ ( η − η n ) = 1 and the definitionin Eq. (B5). Therefore from Eq. (B6) and using Eq.(B5), we obtainTr( D ( m ) − (cid:12)(cid:12)(cid:12) h r = 0 h i = 0 ) = Z ∞ dη ρ a ( m, V, η ) g ( m, η ) + a lV , Tr( D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h i = 0 ) = Z ∞ dη ρ a ( m, V, η ) h r ( m, η ) + a lV . (B7)Hence the trace of Eq. (3) becomesTr( D ( m ) − )= a lV + Z ∞ dη ρ a ( m, V, η )( g ( m, η ) + h r ( m, η )) . (B8)Now we take the thermodynamic, continuum and mass-less limit of both sides in Eq. (B7). For the first equationin (B7) we have lim χ Tr( D ( m ) − (cid:12)(cid:12)(cid:12) h r = 0 h i = 0 ) = lim χ Z ∞ dη ρ a ( m, V, η ) g ( m, η )= lim m → lim a → Z ∞ dη ρ a ( m, η ) g ( m, η )= lim m → Z ∞ dη ρ a ( m, η )˜ g ( m, η ) = π ρ (0 , , (B9)where in the first line we have canceled the term a l/V ,since in the limits a → V → ∞ , it goes to zero,supposing that the multiplicity l doesn’t grow faster than V /a on those limits, otherwise we would have an accumu-lation of eigenvalues at infinity. In the second line of (B9)we have passed the limit V → ∞ inside the integral andin the third line we defined ˜ g ( m, η ) = m/ ( m + η ). Inthe last step, we used that ˜ g ( m, η ) is a Cauchy functionwhich becomes a δ -function in the limit m → χ Tr( D ( m ) − (cid:12)(cid:12)(cid:12) g = 0 h i = 0 ) = lim χ Z ∞ dη ρ a ( m, V, η ) h r ( m, η )= lim χ ω ak Z ∞ dη ρ a ( k m ′ , V, η ) η η + m ′ = 0 · lim χ Z ∞ dη ρ a ( k m ′ , V, η ) η η + m ′ = 0 , (B10)where we used the expression of h r ( m, η ) in Eq. (4)and the introduction of the variable m ′ = m/k . Inthe third line we split the two limits and we used thatthe integral R ∞ dη ρ a ( k m ′ , V, η ) η η + m ′ is finite. In fact | R ∞ dη ρ a ( k m ′ , V, η ) η η + m ′ | ≤ R ∞ dη ρ a ( k m ′ , V, η ) ≤ χ on both sidesof Eq. (B4) and use the two results in Eqs. (B9) and(B10), i.e.lim χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D ( m ) − ( x, y ) αa,βb (cid:12)(cid:12)(cid:12) h r = 0 h i = 0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π ρ (0 , , lim χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D ( m ) − ( x, y ) αa,βb (cid:12)(cid:12)(cid:12) g = 0 h i = 0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (B11)valid for all x, y, α, β, a, b . Those inequalities relates thequark propagator with the eigenvalue distribution in thesector of small η , when the limits lim m → lim a → lim V →∞ are taken in such order. Appendix C: Left and right components of thequark propagator
In this appendix we prove Eq. (23).At first we use that ψ = ψ L + ψ R and ¯ ψ = ¯ ψ L + ¯ ψ R ,then we can decompose ( D − ) ij as,( D − ) ij = lim χ h ( ψ L,i + ψ R,i )( ¯ ψ L,j + ¯ ψ R,j ) i = lim χ h ψ L,i ¯ ψ L,j i + lim χ h ψ L,i ¯ ψ R,j i + lim χ h ψ R,i ¯ ψ L,j i + lim χ h ψ R,i ¯ ψ R,j i . (C1)However lim χ h ψ R,i ¯ ψ L,j i and lim χ h ψ L,i ¯ ψ R,j i are null.Indeed, we can expand lim χ h ψ R,i ¯ ψ L,j i ,lim χ h ψ R,i ¯ ψ L,j i = lim χ (cid:0) P R h ψ i ¯ ψ j i P L (cid:1) = P R (cid:18) lim χ h ψ i ¯ ψ j i (cid:19) P L = P R δ ij D − P L = δ ij + γ ) D − ( + γ )= δ ij D − + γ D − + D − γ + γ D − γ ) , (C2)where in the first line we used the definition of ψ R and ¯ ψ L ,in the second line we have exchanged the operators P L/R with the limits lim χ , and used that ( D − ) ij = δ ij D − ,as described in section IV. Now using Eq. (18) we havethat the last line of (C2) is zero, since γ D − γ = − D − ,hencelim χ h ψ R,i ¯ ψ L,j i = 0 and lim χ h ψ L,i ¯ ψ R,j i = 0 , (C3)where the second equation is obtained as in (C2) justexchanging R ↔ L . Finally using the definitions in Eq. (24), we have that Eq. (C1) can be rewritten as in Eq.(23). Appendix D: Gap and correlators
Here we want show how from Eq. (27) we can arrive tothe Eq. (28), under the limits lim χ and when there is agap in the eigenvalue distribution of the Dirac operator.At first we can take the fermionic average on both sideof (27), hC ( ψ, ¯ ψ ) i F = X k X I ...I k h Γ x ...x k I ...I k k Y l =1 ψ ( x l ) I l ¯ ψ ( x k + l ) I k + l i F , (D1)where we passed the average as h·i F inside the sum. Nowusing the Wick theorem we have h k Y l =1 ψ ( x l ) I l ¯ ψ ( x k + l ) I k + l i F = X p s ( p ) k Y l =1 δ p ( i l ) i k + l h ψ ( x p ( l ) ) p ( I l ) ¯ ψ ( x k + l ) I k + l i F = X p s ( p ) k Y l =1 δ p ( i l ) i k + l D ( m ) − ( x p ( l ) , x k + l ) p ( α l ) p ( a l ) ,α k + l ,a k + l (D2)where, as explained in section V, we have written ex-plicitly the multi-index I l = { i l , α l , a l } , which indicatesflavor, Dirac and color indices respectively. The label p ( l )(as well as p ( α l ) and p ( a )) is obtained after the applica-tion of p transpositions of l (respectively α and a ), andwe are summing over all possible transpositions p ; s ( p ) isthe sign which each permutation gives, coming from theexchange of I l and p ( I l ).Now if we take the limits lim χ on both sides of Eq.(D1), then we getlim χ hC ( ψ, ¯ ψ ) i F = X k X I ...I k X p s ( p ) Γ x ...x k I ...I k k Y l =1 δ p ( i l ) i k + l (lim χ D ( m ) − ( x p ( l ) , x k + l ) p ( α l ) p ( a l ) ,α k + l a k + l ) (D3)where we defined ¯Γ = lim χ Γ. Then, under hypothesisof a gap in the Dirac spectrum, we have that D − = gap lim χ D ( m ) − , hence we obtain Eq. (28). Appendix E: Chiralspin symmetry and quarkpropagator
Here we want to show how from Eq. (30) we get thecondition in Eq. (32) and consequently the Eq. (33),just imposing the invariance of D − under the SU (2) CS transformations, which are given in Eq. (31).0At first we need to observe that SU (2) CS has three U (1) subgroups, one for each generator. We can see, infact, that for α a = { , , α } , the transformations (31)are the same of Eq. (20), just in the point x = ( x , ),hence the axial group is a subgroup of the chiralspingroup, namely U (1) A ⊂ SU (2) CS . Another U (1) sub-group is generated by γ which is defined, in x = ( x , ),as U (1) γ : ψ ( x ) i → exp(i α γ ) ψ ( x ) i , ¯ ψ ( x ) i → ¯ ψ ( x ) i exp( − i α γ ) , (E1)where we have just set α a = ( α , ,
0) in Eq. (31). Theother U (1) subgroup is generated by i γ γ , hence U (1) i γ γ : ψ ( x ) i → exp(i α (i γ γ )) ψ ( x ) i , ¯ ψ ( x ) i → ¯ ψ ( x ) i exp(i α (i γ γ )) , (E2)where we have just set α a = (0 , α ,
0) in Eq. (31).Since they are three distinct U (1) subgroups, in orderto impose the invariance of D − ( x , y ) under SU (2) CS ,is sufficient to impose the invariance under these three U (1) subgroup transformations. However, from sectionIV, we have shown that D − is already invariant under U (1) A , hence D − ( x , y ) will be also invariant. There-fore we just need to consider the other two subgroups.Regarding U (1) γ , we need to plug the transformations(E1), inside Eq. (30), then D − ( x , y ), transforms as D − ( x , y ) → D − ( x , y ) U (1) γ = cos α D − ( x , y )+ i sin α cos α ( γ D − ( x , y ) − D − ( x , y ) γ )+ sin α γ D − ( x , y ) γ . (E3) Instead if we plug (E2) inside Eq. (30) we get D − ( x , y ) → D − ( x , y ) U (1) i γ γ = cos α D − ( x , y )+ i sin α cos α (i γ γ D − ( x , y ) + D − ( x , y )i γ γ ) − sin α γ γ D − ( x , y ) γ γ . (E4)From Eqs. (E3) and (E4), we can see that, differentlyfrom SU ( n F ) L × SU ( n F ) R and U (1) A , D − ( x , y ) isnot invariant under SU (2) CS . However it would be in-variant if and only if γ D − ( x , y ) γ = D − ( x , y ).This condition can be rewritten in terms of the left andright components of the eigenvectors v ( ± ) n , given in Eq.(16). For doing this, we need to consider the expressionof D − ( x , y ) in Eq. (17), and passing γ inside lim χ and the sum over η n , then we get that the condition γ D − ( x , y ) γ = D − ( x , y ) is equivalent to X η n > Λ i h i ( m, η n ) γ (cid:18) L n ( x ) R † n ( y ) R n ( x ) L † n ( y ) 0 (cid:19) γ = CS X η n > Λ i h i ( m, η n ) (cid:18) L n ( x ) R † n ( y ) R n ( x ) L † n ( y ) 0 (cid:19) ⇒ X η n > Λ i h i ( m, η n )( L n ( x ) R † n ( y ) − R n ( x ) L † n ( y )) = CS , (E5)and assuming the last line of (E5) is zero if each terminside the sum is zero, then we get that L n ( x ) R † n ( y ) − R n ( x ) L † n ( y ) = 0, which is the condition in Eq. (33).Finally, calling D − ,CS ( x , y ) the quark propagator D − ( x , y ) satisfying the condition in Eq. (32), thenits invariance over the three U (1) subgroups of SU (2) CS ,which is generated by the matrices: { γ , i γ γ , − γ } , cor-responds to an invariance of D − ,CS ( x , y ) over the full SU (2) CS group. [1] A. Bazavov et al. The chiral transition and U (1) A symmetry restoration from lattice QCD using DomainWall Fermions. Phys. Rev. D , 86:094503, 2012. doi:10.1103/PhysRevD.86.094503.[2] A. Tomiya, G. Cossu, S. Aoki, H. Fukaya, S. Hashimoto,T. Kaneko, and J. Noaki. Evidence of effective ax-ial U(1) symmetry restoration at high temperatureQCD.
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