Baryon currents in the C-broken phase of QCD
BBaryon currents in the C-broken phase of QCD
Biagio Lucini
Physics Department, Swansea University, Singleton Park, Swansea SA2 8PP, UKE-mail: [email protected]
Agostino Patella ∗ Scuola Normale Superiore, Piazza dei Cavalieri 27, 56126 Pisa, Italyand INFN Pisa, Largo B. Pontecorvo 3 Ed. C, 56127 Pisa, ItalyE-mail: [email protected]
Claudio Pica
Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USAE-mail: [email protected]
In a space with some sufficiently small compact dimension (with non-trivial cycles) and withperiodic boundary conditions for the fermions, the charge conjugation (C), spatial parity (P), timereversal (T) and CPT symmetries are spontaneously broken in QCD. We have investigated whatare the physical consequences of the breaking of these discrete symmetries, that is what localobservables can be used to detect it. We show that the breaking induces the generation of baryoncurrents, propagating along the compact dimensions.
The XXV International Symposium on Lattice Field TheoryJuly 30-4 August 2007Regensburg, Germany ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] O c t aryon currents Agostino Patella
1. Introduction
It is well known [1] that, in a space with some sufficiently small compact dimension (withnon-trivial cycles) and with periodic boundary conditions for the fermions, the charge conjuga-tion (C), spatial parity (P), time reversal (T) and CPT symmetries are spontaneously broken inQCD. Recently, some work was done on this subject, following studies about the orientifold planarequivalence.The orientifold planar equivalence [2] is the equivalence in the planar limit of two theories:QCD with one massless quark in the antisymmetric representation on one side; and the N =
2. Breaking of discrete symmetries
We consider a non-Abelian gauge theory with gauge group SU( N ) and N f families of funda-mental Dirac fermions. The manifold on which the theory is defined is R d × T n , where d + n = T n is a spatial n -dimensional torus. We assume that the torus has the same extension R in all thecompact dimensions. Moreover, spatial directions are closed with periodic boundary conditions forboth fermions and bosons.In what follows, ( x a ) a = ,..., d are the coordinates on R d , while ( z α ) α = ,..., n are the coordinatesof the compact dimensions.We compute the vacuum expectation value ( vev ) of the Wilson line in a compact direction [4] W ( A ) α ( x , z ) = Pexp (cid:18) i (cid:90) R A α ( x , z ) dz α (cid:19) . (2.1)The gauge can be fixed in such a way that W is diagonal W ( A ) α ( x , z ) = (cid:16) e iv ( A ) α ( x , z ) , . . . , e iv ( A ) N α ( x , z ) (cid:17) . (2.2)We focus on the effective potential for those eigenvalues : e iV d V ( v ,..., v N ) = (cid:90) e iS ∏ k δ (cid:18) v k − V d R n (cid:90) v ( A ) k ( x ) dxdz (cid:19) D A D ¯ ψ D ψ . (2.3) We drop the subscript α where this does not lead to ambiguities. aryon currents Agostino Patella
The absolute minima of V ( v , . . . , v N ) give the expectation values for the set of the eigenvalues. IfC-symmetry is broken, the set of eigenvalues is not invariant under the substitution v → − v . In theone-loop approximation, the effective potential for the Wilson line is given by [6] V ( (cid:126) v , . . . ,(cid:126) v N ) = (cid:34) N ∑ i , j = f ( ,(cid:126) v i − (cid:126) v j ) − N f N ∑ i = f ( m ,(cid:126) v i ) (cid:35) , (2.4)with f ( m ,(cid:126) v ) = R (cid:18) mR π (cid:19) ∑ (cid:126) k (cid:54) = K ( mRk ) k sin (cid:18) (cid:126) k · (cid:126) v (cid:19) , (2.5)where K is the modified Bessel function of the second kind of order 2 and the sum runs over n -index integer vectors (cid:126) k . The first term of eq. (2.4) gives rise to an attraction between the eigen-values. The second term produces an unconstrained absolute minimum at v i = π . When the SU ( N ) constraint is taken into account, the minima are: v ∗ = v ∗ = . . . = v ∗ N = (cid:40) ± N − N π for N odd π for N even . (2.6)The spontaneous symmetry breaking shows up as a non-zero imaginary part of the Wilsonline. Hence, if N is even, there is no symmetry breaking. If N is odd, then P, C, T, CPT are broken.There are 2 n vacua. This result is due to the spacial periodic boundary conditions for the fermions.The validity of the calculation in the non-perturbative regime has been checked on the lattice in [7],and the discrete symmetries are shown to be broken for R below the fermi scale.
3. Baryon currents
The Wilson line is an order parameter for the symmetry breaking. However, if the theory isnot invariant under P, C, T, we expect this to be reflected by some local observable. This observablemust be odd under the broken symmetries, but even under the combined action of two of them. Thespatial components of the baryon current j α = (cid:104) ∑ N f i = ¯ ψ i γ α ψ i (cid:105) satisfy this property.Why should we expect a non-zero baryon current? A non trivial vev of the Wilson line meansa non-zero value of the gauge field in that direction. Since the system is translationally invariantalong the compact direction, the value of the gauge field must be constant (note that in the presenceof toroidal topology a constant field cannot be gauged away). The background gauge field acts asa non-trivial source for the baryon current. Hence, we expect this current to be different from zero.In order to compute the expectation value of the baryonic current, we define the partitionfunction of the system in presence of a generalised "chemical potential" µ α : Z ( µ ) = (cid:90) exp (cid:40) iS G + i N f ∑ f = (cid:90) ¯ ψ f ( x ) (cid:0) i (cid:54) ∂ − (cid:14) A − µ α γ α − m (cid:1) ψ f ( x ) d x (cid:41) D A D ¯ ψ D ψ . (3.1)Since the insertion of the µ -term has the same effect as shifting A α ( x , x α ) → A α ( x , x α ) + µ α N (thegauge action is not affected by this shift), or as shifting the phases of the eigenvalues of the Wilsonlines v α k ( x ) → v α k ( x ) + R µ α , the partition function can be obtained as: Z ( µ ) = e iV d V ( v ∗ + R µ ,..., v ∗ N + R µ ) (3.2)3 aryon currents Agostino Patella
Figure 1:
The baryon current (one-loop approximation) vs. mR at different values of N . On the other side, as the chemical potential in 3.1 acts like a source for the current, which isobtained by deriving the partition function with respect to µ α : j α = iV d R n d log Zd µ α ( ) = − R n − ∑ k ∂ V ∂ v k ( v ∗ , . . . , v ∗ N ) . (3.3)Using the one-loop expression for the effective potential in 2.4, we obtain for the baryoncurrent: (cid:104) j α (cid:105) = − N f N ( mR ) R π ∑ (cid:126) k (cid:54) = K ( mkR ) k α sin ( (cid:126) k (cid:126) v ∗ ) k . (3.4) (cid:104) j α (cid:105) is zero when v ∗ α = v ∗ α = π (i.e. when the symmetry breaking does not occur in direction α ), is odd under v ∗ α → − v ∗ α , goes to zero when m → ∞ (fig. 1).
4. The lattice calculation
Details of the lattice action used in our simulation are provided in [5]. A 24 × lattice at β = . r [8]) to 0.125 fm. This means that L t = aN t = L s = aN s = . L t and L s below the fermi scale. The breaking of discretesymmetries can be checked by looking at the Wilson line wrapping around a spatial direction. Atypical behaviour is plotted in fig. 2, which shows that the Wilson loop magnetises along e i π , i.e. (cid:104) W (cid:105) acquires an imaginary part, as required by the symmetry breaking scenario.For the Euclidean rotated theory in the broken phase, we expect a non-zero value of the imag-inary part of the current. At the given lattice parameters we find | Im (cid:104) j α (cid:105)| = . ± .
002 (fig. 3),which should be compared with the perturbative prediction (cid:104) j α (cid:105) (cid:39) . ( ) : it is remarkablethat for compact dimensions of the order of 1 / Λ QCD the perturbative prediction still gives the cor-rect order of magnitude. 4 aryon currents
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Figure 2:
Plot of the Wilson loop. Clustering of the phase around 2 π / Figure 3:
Monte Carlo history of the imaginary part of the Wilson line and the current in a compact directionin the broken symmetry phase.
Since in simulations all the dimensions are kept finite, tunneling among the vacua are expected.In fig. 4 two consecutive tunneling events are shown. The correlation of the current with imaginarypart of the Wilson line is evident: both change their sign when tunneling occurs.
5. Conclusions and outlook
In the phase in which discrete symmetries are spontaneously broken, there is a persistent bary-onic current wrapping around the topologically non-trivial compact directions. This current is sim-ilar to the supercurrent observed in superconductors. However, there is a fundamental difference:unlike the case of superconductors, in QCD in compact not simply connected space the current5 aryon currents
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Figure 4:
Tunneling, due to the finite volume effects. is still conserved, since the U(1) baryon symmetry (which in the case of QCD is a global sym-metry) remains unbroken. The persistent flow is induced by the spontaneous breaking of discretesymmetries.In this case, pairs of quarks and antiquarks condense in the vacuum, while the total baryonicnumber is zero. The quark and the antiquark move with opposite momenta along the compactdimensions. The total momentum is zero, but there is a net baryonic current.The existence of this current is a clear physical signature of the symmetry breaking and couldbe used to determine the order of the symmetry restoring phase transition that happens at a criticalradius of the compact direction, which string-inspired calculations predict to be second order [9].Another open problem is the interplay between the aforementioned phase transition and the chiralsymmetry restoring phase transition at finite temperature.
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