Table Of Content[hep-th]
The study of the photon’s pole structure in the noncommutative Schwinger model
M. Ghasemkhani
∗
Institute for Research in Fundamental Sciences (IPM)
School of Particles and Accelerators, P.O. Box 19395-5531, Tehran-Iran
The photon self energy of the noncommutative Schwinger model at three-loop order is
analyzed. It is shown that the mass spectrum of the model does not receive any correction
fromnoncommutativityparameter(θ)atthisorder. Alsoitremainsunchangedtoallorders.
The exact one-loop effective action for the photon is also calculated.
3
1
0 PACS numbers:
2
n
a
J
7 I. INTRODUCTION
]
h The idea of the noncommutative quantum field theory originates from 1940’s when it was applied
t
- to cure the divergencies in quantum field theory before the renormalization approach was born [1].
p
e Nevertheless, it was demonstrated that the divergencies were not removed [2]. Later on, it was shown
h
[ in [3] that the noncommutative quantum field theory describes effectively the low energy limit of the
1 string theory on a noncommutative manifold. In the simplest case, the description of the noncom-
v mutative space-time is given by a constant parameter, θµν, that its space-space (-time) components
1
3 correspond to the magnetic (electric) field. The space-time noncommutative field theories suffer from
2
1 the unitarity violation of the S-matrix [4] while the space-space noncommutative field theories face
.
1 another obstacle, mixing of ultraviolet and infrared singularities [5]. The problem of the non-unitary
0
S-matrix was studied in [6] but includes some inconsistencies.
3
1 In fact, space-time noncommutativity leads to the higher orders of time derivatives of the fields in the
:
v Lagrangian which make the quantization procedure of the theory different from that of the commu-
i
X tative counterpart. For example in [7], the perturbative quantization of the noncommutative QED in
ar 1+1 dimensions has been analyzed up to O(θ3).
Inthepresentwork,thenoncommutativetwodimensionalQEDwithmasslessfermionsinEuclidean
space (x ≡ it) is considered. The purposeof this paper is to concentrate on the mass spectrum of the
2
theory at higher loops. The commutative counterpart of this model, Schwinger model, was studied in
[8] where it was shown that the photon in two dimensions acquires dynamical mass, arising from the
loop effect, without gauge symmetry breaking. The mass spectrum of the Schwinger model contains
a free boson with a mass proportional to the dimensionful coupling constant. Fermions disappear
from the physical states due to the linearity of the potential that is similar to the quark confinement
potential in the quantum chromodynamics (QCD). Hence, Schwinger model can be a toy model to
understand the quark confinement.
∗Electronic address: [email protected]
The extension of the Schwinger model to the noncommutative version from different aspects has been
addressedin[7,9–14]. Here, wefocusonthedynamicalmassgeneration inthenoncommutative space.
The noncommutative version of this phenomenon at one-loop has been discussed in [12], where it was
found that the Schwinger mass receives no noncommutative correction at this order. Then, the work
[13] suggests that the Schwinger mass acquires noncommutative correction at three-loop order.
However, our analysis shows that the Schwinger mass remains unaffected by noncommutativity to all
orders.
This paper is organized as follows: in Sec. II, we introduce the noncommutative Schwinger model
in the light-cone coordinates in order to simplify our calculations. In Sec. III, to obtain the mass
spectrum of the theory at three-loop order, the photon self energy is studied at this level. Using
the explicit representation of the Dirac γ-matrices provides a straightforward method to compute
the trace of the complicated fermionic loops. Then, it is shown that the noncommutativity does not
affect the Schwinger mass in this order. Extension of the three-loop computations to all orders is
discussed in Sec. IV where the exact mass spectrum is also obtained. In Sec. V, we demonstrate that
the noncommutative one-loop effective action for the photon is exactly the same as the commutative
counterpart. Finally, Sec. VI is devoted to the concluding remarks.
II. NONCOMMUTATIVE SCHWINGER MODEL IN LIGHT-CONE COORDINATES
TheLagrangian of the noncommutative Schwinger model can beobtained from its commutative coun-
terpart by replacing the ordinary product with the star-product which is defined as follows
iθ ∂ ∂
µν
f(x)⋆g(x) ≡ exp f(x+a)g(x+b) , (II.1)
2 ∂a ∂b
µ ν (cid:12)
(cid:18) (cid:19) (cid:12)a=b=0
(cid:12)
(cid:12)
where θ is an antisymmetric constant matrix related to the noncommutative structure of the space-
µν (cid:12)
time. In two-dimensional space-time, θ can be written as the antisymmetric tensor ǫ which
µν µν
preserves the Lorentz symmetry, namely
[x ,x ]= θǫ . (II.2)
µ ν µν
To avoid the unitarity problem in noncommutative space-time field theories, we use the Euclidean
signature throughout this paper. The Lagrangian of the two-dimensional noncommutative massless
QED is given by
1 1
L = −iψ¯⋆γ ∂µψ+eψ¯⋆γ Aµ⋆ψ+ F ⋆Fµν + (∂ Aµ)⋆(∂ Aν)−∂ c¯⋆(∂µc−ie[A ,c] ),
µ µ µν µ ν µ µ ⋆
4 2
(II.3)
where F is defined as
µν
F = ∂ A −∂ A +ie[A ,A ] , (II.4)
µν µ ν ν µ µ ν ⋆
with [A ,A ] = A ⋆A −A ⋆A .
µ ν ⋆ µ ν ν µ
One of the useful properties of the two dimensional space is that our calculations in the light-cone
2
coordinates, x =x ±ix , are simplified significantly. The Lagrangian (II.3) in the light-cone gauge,
1 2
±
A = 0, has the following form
−
i e 1
L = ψ¯⋆(γ ∂ +γ ∂ )ψ+ ψ¯⋆γ A ⋆ψ− (∂ A )⋆(∂ A ), (II.5)
+ + + + +
2 − − 2 − 2 − −
where γ = γ ±iγ and A = A ±iA .
1 2 1 2
± ±
In this particular gauge, the non-linear term in the field strength tensor is removed. Therefore, the
photon self interaction parts, three and four photon interaction vertices, are eliminated and the ghost
fields are decoupled from the theory. The resulting Feynman rules are shown in Fig. 1.
q
-1/q2
-
p
i/p
q
eγe2ipθq
-
FIG. 1: Feynman rules for noncommutative Schwinger model in the light-cone gauge.
Note that only γ appears in the fermion-photon vertex.
−
III. THREE-LOOP NON-PLANAR CORRECTION TO THE SCHWINGER MASS
As it was mentioned before, Schwinger showed that the photon in two dimensions acquires dynamical
mass, µ = e . This mass generation originates from the presence of a special singularity in the scalar
√π
vacuum polarization at one loop order. Using the non-perturbative method shows that the obtained
mass does not receive any correction from loops at higher orders [8, 15].
The noncommutative extension of this kind of mass generation at one-loop level was discussed in
[12] where it was proved that the Schwinger mass gets no noncommutative correction in this order.
Higher loop contributions, e.g. two-loop and three-loop, have been pointed out in [13] withoutexplicit
computation of the loop integrals.
At two-loop order, there is only one non-planar diagram but three-loop order includes two non-planar
graphs. In the present section, we analyze the calculation of three-loop non planar graphs to find
the noncommutative corrections to the Schwinger mass. The general structure of the exact photon
propagator1 in two dimensional noncommutative space is the same as its commutative counterpart
1 Here, we work in Feynman gauge.
3
[12], namely
δµν
Dµν(q)= − , (III.1)
q2[1+Π(q2)]
where the scalar vacuum polarization, Π(q2), is related to its tensor form via the following
Πµν = (q2δµν −qµqν)Π(q2), (III.2)
here,Π(q2)includestheplanarandnon-planarparts. Thepolestructureisobtainedfromthefollowing
limit
lim q2Π(q2,e2,θ)= lim q2Π (q2,e2)+ lim q2Π (q2,e2,θ), (III.3)
planar non planar
q2 0 q2 0 q2 0 −
→ → →
with fixed θ. The first term yields the exact commutative Schwinger mass with Π(q2,e2) = e2 and
πq2
the second term gives the noncommutative corrections to it. We concentrate on the analysis of the
second term at three loops (Fig. 2).
q k q
l
p
q q
l k
p
FIG. 2: Three-loop non-planar diagrams.
The contributions related to the above graphs are given by
d2p d2k d2ℓ 1 1
Π(3)| = e6
µν n.p (2π)2 (2π)2(2π)2 k2ℓ2
Z
1 1 1 1 1 1
× e i(kθℓ+kθq+ℓθq)tr γ γρ γλ γ γ γ
− µ ν ρ λ
(6q+6p) (6q+6p+6ℓ) (6q+6p+6ℓ+6k) (p6 +6ℓ+6k) (p6 +6k) 6p
(cid:26) (cid:18) (cid:19)
1 1 1 1 1 1
+ e i(kθq+ℓθq)tr γ γρ γλ γ γ γ .
− µ ν λ ρ
(6q+6p) (6q+6p+6ℓ) (6q+6p+6ℓ+6k) (6p+6ℓ+6k) (p6 +6ℓ) 6p
(cid:18) (cid:19)(cid:27)
(III.4)
Rewriting (III.4) in the light-cone coordinates, we obtain
dp dp dk dk dℓ dℓ g+ g+
Π(−3−)|n.p = e6 (2−π)2+ (2+π)2− (2+π)2−k2ℓ2(q+p)2(q+p+ℓ)2(q−+p−+ℓ+k)2(p+ℓ+k)2p2
Z
− −
N e i(kθℓ+kθq+ℓθq) N e i(kθq+ℓθq)
1 − 2 −
× + ,
(p+k)2 (p+ℓ)2
(cid:20) (cid:21)
(III.5)
4
where
N = tr γ (q6 +6p)γ (6q+6p+6ℓ)γ (6q+6p+6ℓ+6k)γ (6p+6ℓ+6k)γ (p6 +6k)γ 6p
1
− − − − − −
(cid:18) (cid:19)
N = tr γ (q6 +6p)γ (6q+6p+6ℓ)γ (6q+6p+6ℓ+6k)γ (6p+6ℓ+6k)γ (p6 +6ℓ)γ 6p , (III.6)
2
− − − − − −
(cid:18) (cid:19)
and kθℓ= θ k ℓ = iθ(k ℓ −k ℓ ). Using the explicit matrix form of γ is useful to find the trace
µν µ ν 2 + +
− − −
of the fermionic loop in a simple way (see APP. A for more details). Therefore, the quantities N and
1
N are given by
2
N =26 (p+q) (p+q+ℓ) (p+q+ℓ+k) (p+ℓ+k) (p+k) p ,
1
− − − − − −
N =26 (p+q) (p+q+ℓ) (p+q+ℓ+k) (p+ℓ+k) (p+ℓ) p . (III.7)
2
− − − − − −
Plugging them in (III.5), we have
dp dp dk dk dℓ dℓ (p+q) (p+q+ℓ) (p+q+ℓ+k) (p+ℓ+k) p
Π(−3−)|n.p = 16e6 (2−π)2+ (2+π)2− (2+π)2−k2ℓ2(q+−p)2(q+p+−ℓ)2(q+p+ℓ+−k)2(p+ℓ+−k)2−p2
Z
− −
e i(kθℓ+kθq+ℓθq)(p+k) e i(kθq+ℓθq)(p+ℓ)
− −
× − + − . (III.8)
(p+k)2 (p+ℓ)2
(cid:20) (cid:21)
Since, the non-planar phase factor is independent of the fermionic loop momentum, p, the related
integral can be evaluated separately. Consider the first term of (III.8)
dp dk dk dℓ dℓ 1
Π(3,1)|n.p = 16e6 (2π−)2 (2+π)2− (2+π)2− k2ℓ2 e−i(kθℓ+kθq+ℓθq)F, (III.9)
−−
Z
− −
where
1
F = dp . (III.10)
+
(p+q) (p+q+ℓ) (p+q+ℓ+k) (p+ℓ+k) (p+k) p
+ + + + + +
Z
For simplicity, + indices are dropped. After a long algebraic manipulations, (III.10) is reduced to the
following
1 1 1 1 1 1 1 1 1 1 1 1
F = dp − + −
kqk+ℓq+ℓq+ℓ+k p p+q+k+ℓ ℓqk+ℓk−qℓ+k−q p+k+ℓ p+q
Z (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)
1 1 1 1 1 1
+ − (III.11)
kℓq+ℓk−qk−q−ℓ p+q+ℓ p+k
(cid:18) (cid:19)(cid:21)
(3,1)
Having changed variables, we arrive at F = 0 that leads to Π = 0. Similarly, the second term,
(3,2) −−
Π , vanishes. As a consequence
−−
(3)
Π | = 0. (III.12)
n.p
−−
According to (III.3), the commutative Schwinger mass remains free from noncommutative correction
(2)
at three-loop order. Similarly, at two loop order, it is proved that Π = 0. In what follows, this
−−
calculation will be extended to all orders of the quantum corrections.
5
IV. ALL-LOOP NON-PLANAR CORRECTION TO THE SCHWINGER MASS
In this section, we generalize three-loop computation to all orders to obtain the exact mass spectrum.
At n−looplevel, thereareseveral non-planardiagrams contributingtothevacuumpolarization tensor
that one of them may be found in Fig. 3 for which n is an odd number. The general Feynman form
for the non-planar diagram related to the photon’s vacuum polarization at n−loop (n 6=1) is written
as
dp dp dk dk dk dk dk dk 1
Π(n,i) = (e2)n + − 1+ 1− 2+ 2− ··· (n−1)+ (n−1)−
−− Z (2π)2 (2π)2 (2π)2 (2π)2 k12 k22 ···k(2n 1)
− − − −
n−1
n 1 2 n 1
− −
× exp i qθ k + k θ k g+ ···g+
r r s − −
(cid:20) (cid:18) Xr=1 Xr=1 s=Xn+21 (cid:19)(cid:21) n−1
n 1 n 1
| {z− } −
tr γ (q6 +6p)γ (6q+6p+6k )···γ (6q+6p+ 6k )γ (6p+ 6k )···γ 6p
1 i i
− − − − −
× (cid:18) Xi=1 Xi=1 (cid:19), (IV.13)
n 1 n 1
(q+p)2(q+p+k )2···(q+p+ − k )2(p+ − k )2···p2
1 i i
i=1 i=1
P P
(n,i)
whereΠ showsthei-th non-planarcontribution tothetotal self energy atn−looplevel. Analogous
−−
to Sec. III, the numerator can be easily computed as
dp dp dk dk dk dk dk dk 1
Π(n,i) = 21−n(e2)n (2+π)2− (12+π)21− (22+π)22− ··· (n−1()2+π)2(n−1)− k2 k2 ···k2
−− Z 1 2 (n 1)
− − − −
n−1
n 1 2 n 1
− −
× exp i qθ k + k θ k
r r s
(cid:20) (cid:18) Xr=1 Xr=1 s=Xn+21 (cid:19)(cid:21)
n 1 n 1
22n (q+p) (q+p+k ) ···(q+p+ − k ) (p+ − k ) ···p
1 i i
× − − i=1 − i=1 − −.
n P1 n 1P
(q+p)2(q+p+k )2···(q+p+ − k )2(p+ − k )2···p2
1 i i
i=1 i=1
P P (IV.14)
(n,i)
Then Π is reduced to
−−
dp dk dk dk dk dk dk 1
Π(n,i) = 2n+1e2n − 1+ 1− 2+ 2− ··· (n−1)+ (n−1)−
−− Z (2π)2 (2π)2 (2π)2 (2π)2 k12 k22 ···k(2n 1)
− − − −
n−1
n 1 2 n 1
− −
× exp i qθ k + k θ k G, (IV.15)
r r s
(cid:20) (cid:18) Xr=1 Xr=1 s=Xn+21 (cid:19)(cid:21)
and G is defined as
1
G = dp .
+
n 1 n 1
Z (q+p) (q+p+k ) (p+q+k +k ) ···(q+p+ − k ) (p+ − k ) ···p
+ 1 + 1 2 + i + i + +
i=1 i=1
P P (IV.16)
6
Here the non-planar phase factor is independent of p . It is proved that for a fixed n, similar to the
+
previous section, the value of G is zero, i.e.
(n,i)
Π = 0. (IV.17)
−−
(n,i)
The obtained result is correct for any non-planar graph. Therefore, we conclude that Π = 0.
i −−
Accordingly, the noncommutativity does not affect the Schwinger mass up to all orders.P
k
n+1
2
q . . .kn2-1 ... q
k1 kn-1
p
FIG. 3: N-loop non-planar diagram.
V. NONCOMMUTATIVE ONE-LOOP EFFECTIVE ACTION
Thecomputation methodusedintwo previoussections willbeusefultosimplifythephoton’sone-loop
effective action in noncommutative space. The one-loop effective action is given by integrating out
the fermionic degrees of freedom
Γ[A] ≡ Dψ¯Dψexp i d4x ψ¯ i6Dψ , (V.18)
Z (cid:20) Z (cid:21)
where D = ∂ −ieA and A is an external abelian gauge field. The quantity Γ[A] is equivalent to
µ µ µ µ
the following functional determinant from Fig. 4
det(∂6 −ie6A) ∞ 1 1
Γ[A] ≡ −ln = Tr[ (ie6A)]n. (V.19)
det(6∂) n 6∂
n=1
X
Using the non-perturbative approach in two dimensions, the expression Γ[A] is exactly determined.
In other words, (V.19) has non-zero value only for n = 2 which is equal to
e2 d2k kµkν
Γ[A]= − A (k)(δµν − )A (−k), (V.20)
µ ν
2π (2π)2 k2
Z
Therefore, it shows that the photon has received the mass from the one-loop quantum correction.
Thenoncommutative version of Γ[A] inthree dimensionsfor non-abelian gauge fields has beenalready
7
. . . .
.
k
3
k
2
p
k
1
k
n-1
k
n
FIG. 4: Relevant graph for the n-th term of one-loop effective action
discussed in [16]. In what follows, we determine the one-loop effective action for the noncommutative
Schwinger model.
According to (V.18), we can define
Γ [A] ≡ Dψ¯Dψexp i d4x ψ¯⋆ i6Dψ , (V.21)
nc
Z (cid:20) Z (cid:21)
where D = ∂ −ieA ⋆. Similar to the commutative part, Γ [A] can be represented as
µ µ µ nc
det(6∂ −ie6A⋆) ∞ 1 1
Γ [A] = −ln = Tr[ (ie6A⋆)]n, (V.22)
nc
det(6∂) n 6∂
n=1
X
which is equivalent to the following expression
∞
Γ [A] = d2z ···d2z Aµ1(z )···Aµn(z ) Γnc (z ,··· ,z ). (V.23)
nc 1 n 1 n µ1 µn 1 n
···
n=1Z
X
The quantity Γnc (z ,··· ,z ) is given by
µ1 µn 1 n
···
Γnc (z ,··· ,z ) = (−1)n n d2kj (2π)2δ( n k ) eijP=n1kjzje2i jP<nℓkjθkℓΓnc (k ,··· ,k ),
µ1 µn 1 n n (2π)2 j µ1 µn 1 n
··· ···
Z j=1 j=1
Y X
e (V.24)
with,
tr γ (6p+6k )γ (6p+6k +6k )γ (6p+6k +6k +6k )···γ 6p
d2p µ1 1 µ2 1 2 µ3 1 2 3 µn
Γnc (k ,··· ,k ) = (cid:18) (cid:19).
µ1 µn 1 n (2π)2 (p+k )2(p+k +k )2(p+k +k +k )2···p2
··· 1 1 2 1 2 3
Z
(V.25)
e
The non-zero leading term starts from n = 2 with phase factor equal to 1 which leads to its com-
mutative value. For n > 3, similar to the Sec. III and IV, the phase factor is independent of the
momentum p, so the integral over it vanishes. Thus, the noncommutativity has no effect on one-loop
effective action and its exact commutative form is preserved.
8
VI. CONCLUSION
In this paper, we have concentrated on the mass spectrum of the noncommutative Schwinger model
with Euclidean signature at higher loops. It is demonstrated that the Schwinger mass receives no
noncommutative corrections to all orders.
To prove this in a perturbative method, we have used the light-cone gauge to simplify the Lagrangian
form. In this gauge, only the fermion-photon vertex remains and consequently the fermionic loops
contribute to our calculations. Having fixed the gauge, the study of the non-planar sectors of the
photon self energy at three-loop and all-loop order has been performed.
At three-loop, the non-planar parts of the photon self energy were analyzed. Since the non-planar
phase factor, appeared in Feynman integrals, is independent of the fermionic loop momentum, the
corresponding loop integral is easily evaluated. This analysis showed that the contributions from the
non-planar graphs are zero. Hence, the commutative mass spectrum does not change at this order.
Then, the three-loop calculation was extended to all orders. Analogously to three-loop level, the
non-planar phase factor is independent of the fermionic loop momentum and the resulting integral
vanishes. This proves that the Schwinger mass remains intact up to all orders in the noncommutative
space.
The technique applied for computing the trace of the fermionic loops inspired us to study the
relevant one-loop effective action. As a consequence, the exact one-loop effective action in the light-
cone gauge with no noncommutative corrections was obtained.
VII. ACKNOWLEDGMENTS
I would like to express my special gratitude to F. Ardalan for his inspiration, illuminating discussions
and reading the manuscript. I appreciate all the comments from A. Armoni and N. Sadooghi. Last
but not least, I would like to especially thank M. Mohammadi Najafabadi for helping building up
figures and A. A. Varshovi for fruitful discussions.
9
Appendix A
In this appendix, we present more details on computation of the trace expressions appeared in the
numerator of the relation (III.5).
N = tr γ (q6 +6p)γ (6q+6p+6ℓ)γ (6q+6p+6ℓ+6k)γ (6p+6ℓ+6k)γ (p6 +6k)γ 6p
1
− − − − − −
(cid:18) (cid:19)
N = tr γ (q6 +6p)γ (6q+6p+6ℓ)γ (6q+6p+6ℓ+6k)γ (6p+6ℓ+6k)γ (p6 +6ℓ)γ 6p . (A.1)
2
− − − − − −
(cid:18) (cid:19)
To calculate these, we start from representation of the gamma matrices in Euclidean space
0 −i 0 −1
γ = , γ = , (A.2)
1 2
i 0 ! −1 0 !
which in the light-cone coordinates are defined as
0 −2i 0 0
γ = γ +iγ = , γ =γ −iγ = , (A.3)
+ 1 2 1 2
0 0 ! − 2i 0!
and the light-cone metric by using gµν = ∂xµ∂xνδρσ is obtained
∂xρ ∂xσ
g++ g+ 0 1
gµν = − = 2 . (A.4)
g−+ g−− ! 12 0 !
The terms such as 6p appeared in (A.1) can be revised as the following
1 0 −ip
6p = (p+γ +p γ+)= − , (A.5)
2 − − ip+ 0 !
consequently
0 0 0 −ip 0 0
γ 6p = − = . (A.6)
− 2i 0! ip+ 0 ! 0 2p !
−
Plugging the above in (A.1)
0 0 0 0 0 0
N = tr
1
0 2(p +q )! 0 2(p+q+ℓ) ! 0 2(p+q+ℓ+k) !
(cid:20)
− − − −
0 0 0 0 0 0
0 2(p+ℓ+k) ! 0 2(p+k) ! 0 2p !
(cid:21)
− − −
0 0
= tr
0 64(p+q) (p+q+ℓ) (p+q+ℓ+k) (p+ℓ+k) (p+k) p !
− − − − − −
= 64 (p+q) (p+q+ℓ) (p+q+ℓ+k) (p+ℓ+k) (p+k) p , (A.7)
− − − − − −
ultimately, the first denominator of e.q (III.5) is given by
(q+p)2(q+p+ℓ)2(q+p+ℓ+k)2(p+ℓ+k)2(p+k)2p2 = (q+p) (q+p) (q+p+ℓ) (q+p+ℓ)
+ +
− −
(q+p+ℓ+k) (q+p+ℓ+k) (p+ℓ+k) (p+ℓ+k) (p+k) (p+k) p p . (A.8)
+ + + +
− − − −
10
