Causal Amplitudes in the Schwinger Model at Finite Temperature

Causal amplitudes in the Schwinger model at ﬁnite temperature

Ashok Das,a,b R. R. Franciscoc and J. Frenkelc∗ a Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627-0171, USA b Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700064, India and c Instituto de F´ısica, Universidade de S˜aoPaulo, 05508-090, S˜aoPaulo, SP, BRAZIL

We show, in the imaginary time formalism, that the temperature dependent parts of all the retarded (advanced) amplitudes vanish in the Schwinger model. We trace this behavior to the CPT invariance of the theory and give a physical interpretation of this result in terms of forward scattering amplitudes of on-shell thermal particles.

PACS numbers: 11.10.Wx

In two earlier papers [1, 2] we derived the complete into a right handed and a left handed sector where thermal effective action for 0 + 1 dimensional QED as 1 1 1 1 well as for 1 + 1 dimensional Schwinger model (massless ψR = ( + γ )ψ, ψL = ( γ ), 2 5 2 − 5 QED) in the real time (closed time path) formalism [3, 4]. 0 1 ± x x The structure of the effective action in terms of the dou- x = ± , ∂± = ∂0 ∂1,A± = A0 A1. (3) bled thermal degrees of freedom allowed us to prove al- 2 ± ± gebraically that all the temperature dependent retarded At zero temperature the quantum corrections couple the (advanced) amplitudes vanish in these theories. While two sectors through the chiral anomaly. However, at fi- the algebraic proof in the real time formalism is indeed nite temperature there is no ultraviolet divergence and, quite simple and straightforward, it does not shed any therefore, there is no coupling between the two sectors so light on the physical origin of such an interesting result. that we can study each sector independently. Parentheti- On the other hand, the imaginary time formalism [4– cally we remark that in 1+1 dimensions, the Dirac matri- µ 8] is the natural framework in which retarded (advanced) ces satisfy special identities [12, 13] such as (A/ = γ Aµ, amplitudes can be derived conveniently without the need A± are the light-cone components as defined above and for doubling the degrees of freedom. Therefore, in this uµ = (1, 1) are the light-cone vectors) ± ± formalism the physical meaning of such a result may µ1 µ2 µ3 µ1 µ2 µ3 be better understood within the context of the original Tr (γ A/γ B/γ C/ ) = A+B+C+ u− u− u− ··· µ1···µ2 µ3 ··· quantum field theory. In this brief report we undertake + A−B−C− u u u , (4) ··· + + + ··· such an analysis for the 1 + 1 dimensional Schwinger so that even if we work in the complete theory, the loop model in the imaginary time formalism which leads to integrals would naturally separate into the two sectors consistent physical interpretations for the vanishing of because of such special identities. retarded (advanced) amplitudes in this model. The 0 + 1 Let us look at the two point amplitude, Fig. 1, in the dimensional model has already been studied extensively right handed sector in the imaginary time formalism in in the past in the imaginary time formalism [9, 10] and it some detail before generalizing the result to the 2n-point is known that the Ward identities require any amplitude amplitude (the odd point amplitudes vanish by charge to vanish when the external energies are nonvanishing. conjugation symmetry). In the imaginary time formal- It follows from this that the retarded (advanced) amplitudes must vanish in this theory since there is no analytic continuation available to obtain a nontrivial amplitude. k Therefore, we concentrate only on the 1 + 1 dimensional arXiv:1206.5677v2 [hep-th] 13 Jul 2012 Schwinger model in this work. pp The fermion sector of the Schwinger model [11] is de- scribed by the 1 + 1 dimensional Lagrangian density (see k + p [1, 2] for our notations) = ψγ¯ µ(i∂ eA )ψ. (1) L µ − µ FIG. 1: Photon self-energy in the right handed sector. Since the fermions are massless, the Lagrangian density naturally decomposes as ism, the energy becomes discrete, even multiples of πT for bosons and odd multiples of πT for fermions, with = R + L L L L T denoting the temperature. Correspondingly, the inte- † † = ψR(i∂+ eA+)ψR + ψL(i∂− eA−)ψL, (2) gral over loop energies becomes a sum over these discrete − − energies. As a result, the self-energy can be written as Z ∞ 2 dk X 1 1 ΠR = e T , (5) ∗ 2π iω + k i(ω + Ω ) + k + p e-mail: [email protected], [email protected] n=−∞ n n ` 2 where ωn = (2n + 1)πT, Ω` = 2`πT . integrand correspond to the two diagrams in the for- The sum over n in (5) can be evaluated using the ward scattering amplitude of an on-shell thermal particle. method of contours [6, 7] and allows us to separate the pp pp temperature dependent part of the amplitude to be

+ 2 Z Z k k + p k k p k k (T ) e dk 1 1 − − − ΠR = dk0 −2πi 2π C k0 + k k0 + iΩ` + k + p + k0 k0 nF (k0), (6) FIG. 3: Forward scattering description of the retarded self- → − energy. The external fermion is an on-shell thermal particle. 1 where nF (k ) = denotes the fermion distribution 0 ek0/T +1 function and the contour is closed (clockwise) on the pos- itive half of the real k0 axis (which does not enclose the However, when the fermion is massless, changing the momentum, changes the helicity taking a particle to its anti- poles of nF (k0) along the imaginary axis) as shown in Fig. 2. The contributions from the two poles in the denomi- particle. Therefore, in this case, the two diagrams correspond to the two contributions coming from the particle and anti-particle scattering. Cancellation between the Im k0 two terms in (7) simply corresponds to the anti-particle contribution exactly canceling the particle contribution leading to the conclusion that in 1 + 1 dimensions, an on-shell massless thermal fermion cannot scatter in the forward direction. This is not true in general, for exam- ple, if the fermion is massive this will not be the case Re k0 and we will understand this result later from symmetry arguments. A similar calculation can be carried out for the 2n- point amplitude, Fig. 4, in the imaginary time formal- C ism. After evaluating the sum over discrete energies us-

p2 FIG. 2: Contour C in the complex k0 plane along which the p integral needs to be evaluated. 1 nators can be easily evaluated. To analytically continue k to the retarded amplitude, one has to use the periodic- p ity condition nF (iΩ` + k + p) = nF (k + p) before letting 2n iΩ` p0. This leads to the temperature dependent part of the→ retarded self-energy to be Z (T ) 2 dk sgn(k)nF ( k ) ΠR (p0, p) = e | | + k k . − 2π p0 + p → − FIG. 4: 2n-point amplitude in the right handed sector. All (7) the external photon momenta are assumed to be incoming. It is clear that the two terms in the integrand cancel each other leading to a vanishing temperature dependent ing the contour method and using the periodicity of the retarded self-energy. fermion distribution function in the external (bosonic) However, to obtain a physical interpretation of this energies, we obtain the temperature dependent part of result, let us introduce an auxiliary variable of integration the retarded amplitude to have the form and rewrite (7) as Z dk h sgn(k) Z 2 (T ) 2n (T ) 2 d k 2πδ(k0 + k)sgn(k0 k) Γ2n = e + perm. Π (p , p) = e nF ( k ) 2n−1 R 0 2 − − 2π P (2π) | | p0 + p + k0 + k p1+(p1+ + p2+) ( pi+) ··· i=1 + k k . (8) i → − + k k nF ( k ), (9) → − | | This way of writing the self-energy makes contact with the forward scattering description of retarded amplitudes where the integrand again vanishes by anti-symmetry [14–16] as shown in Fig. 3. The two terms in the (pi+ denote the light-cone components of the momenta). 3

p1 p2 p2n p1 p2 p2n

··· ··· + + perm. k k + p1 k k p1 k k ··· − − ··· −

FIG. 5: Forward scattering description of the retarded 2n-point function. The external fermion is an on-shell thermal particle and we assume p1 + p2 + ··· + p2n = 0.

Introducing the auxiliary variable of integration as in (8) Consistency with the invariance under CPT, (12), then we can again give this the thermal forward scattering requires that the retarded amplitude must vanish and representation as shown in Fig. 5. The vanishing of the this is also consistent with the cancellation of the par- retarded amplitude can now be understood as arising due ticle and anti-particle contributions in the forward scat- to the exact cancellation between the particle and the tering description. It is worth emphasizing that if the anti-particle contributions to the process. fermion had a mass, each of the denominators would in- This vanishing can also be seen directly from the basic volve a mass term and, consequently, the denominators symmetries of the theory. Let us look at the thermal ef- would no longer be completely anti-symmetric and will fective action (in momentum space) at the 2n-point level have a symmetric part. As a result, the amplitudes will in the right handed sector, not have to vanish because of the requirement of CPT invariance. Similarly, if we are looking at a Feynman − ! 2n 1Z 2 amplitude, one or more factors may involve delta func- (2n)(T ) 2n Y d pi Γ = e A (p ) A (p − )A (p ) 2π + 1 ··· + 2n 1 + 2n tions of the external momentum which may make the i=1 amplitude symmetric so that it does not have to vanish. (T ) Γ (p1, , p2n−1), (10) This anti-symmetry, therefore, is a very special feature × 2n ··· of the retarded (advanced) amplitudes in this massless where p2n = p1 p2 p2n−1. CPT is an invariance theory which is why these amplitudes vanish. of the underlying− − quantum−· · ·− field theory and, therefore, the effective action must be invariant under CPT. Under This discussion generalizes in a straightforward man- a CPT transformation, ner to the left handed sector (as well as to advanced amplitudes) and shows not only that the temperature depen- CPT dent retarded (advanced) amplitudes in the Schwinger A±(pi) A±( pi), (11) −−−→− − model vanish, but also traces the physical reason for this so that the invariance of the effective action (10) would to be the CPT invariance of a massless theory in 1 + 1 require that the 2n-point amplitude should be an even dimensions. This also translates into an exact cancella- function of the external momenta, tion of the particle and anti-particle contributions in the forward scattering amplitudes. (T ) (T ) Γ ( p , , p − ) = Γ (p , , p − ). (12) 2n − 1 ··· − 2n 1 2n 1 ··· 2n 1 On the other hand, we see from the explicit evaluation of the retarded amplitudes in (7) and (9) that the depen- dence on the variable of integration completely drops out Acknowledgments of each of the (2n 1) denominators so that the amplitude is completely anti-symmetric− This work was supported in part by CNPq and (T ) (T ) Γ ( p , , p − ) = Γ (p , , p − ). (13) FAPESP (Brazil). 2n − 1 ··· − 2n 1 − 2n 1 ··· 2n 1

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