Renormalizability of noncommutative quantum electrodynamics at θ2 order

Physics Letters B 678 (2009) 250–253

Contents lists available at ScienceDirect

Physics Letters B www.elsevier.com/locate/physletb

Renormalizability of noncommutative quantum electrodynamics at θ 2 order Jia-Hui Huang, Zheng-Mao Sheng ∗ Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, PR China

a r t i c l e

i n f o

a b s t r a c t In θ -expanded approach, we expand the noncommutative quantum electrodynamics action to θ 2 order and calculate the one loop divergent corrections to gauge field propagators. It is shown that the gauge field propagators are one loop renormalizable at θ 2 -order with a massive or massless fermion field. © 2009 Elsevier B.V. All rights reserved.

Article history: Received 1 May 2009 Accepted 16 June 2009 Available online 18 June 2009 Editor: A. Ringwald PACS: 11.10.Nx 11.10.Gh 11.15.-q Keywords: Renormalization Noncommutative gauge field Background field method

1. Introduction Field theories on noncommutative space were suggested long time ago by Snyder [1] to find a natural cutoff for the loop integrals in quantum field theories. Recently, due to the development of string theories, noncommutative field theories have been studied extensively [2–5]. The noncommutative space is defined as follows:

 μ ν xˆ , xˆ = i θ μν ,

(1)

where xˆ μ are generators of the noncommutative space and θ μν are real antisymmetric constants. From results of noncommutative geometry, the algebra (1) can be represented by the algebra of functions on ordinary space with deformed multiplication, Moyal– Weyl product [6], which is defined as



i μν ∂ ∂ f (x)  g (x) = exp θ 2 ∂ xμ ∂ y ν



  f (x) g ( y )

.

(2)

y −>x

Field theories on noncommutative space can be obtained by replacing ordinary product between fields with the Moyal–Weyl product. Then the noncommutative effects can be explored by the usual perturbative method. One special noncommutative effect is the nonplanar graph which leading the phenomena of UV/IR

*

Corresponding author. E-mail address: [email protected] (Z.-M. Sheng).

0370-2693/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2009.06.034

mixing [7–9]. Noncommutative U(1) and U( N ) gauge theories can be discussed in this approach. But for general noncommutative gauge theories, the Moyal–Weyl product makes the noncommutative gauge fields cannot close in gauge algebras. The noncommutative gauge fields should be enveloping-algebra valued [10]. The expansion of noncommutative gauge fields in enveloping algebra can be determined by Seiberg–Witten map [5]. Seiberg and Witten have argued that there is a mapping between noncommutative gauge theories and ordinary gauge theories based on their research in string theory. The noncommutative fields can be expanded by ordinary fields in powers of θ μν . From the Seiberg–Witten map, one can obtain a θ -expanded action and consider the noncommutative effects order by order in θ . In this θ -expanded approach, all gauge theories can be discussed. Obviously, one can study the renormalizability of noncommutative gauge theories in this approach. Renormalizability of noncommutative gauge theories in θ expanded approach have been discussed in many papers [11–14]. It is shown that the pure noncommutative U(1) gauge propagator is one loop renormalizable to all orders in θ [11]. For pure noncommutative SU( N ) theory, the gauge propagators are renormalizable at the first order in θ [14]. But when matter fields appear, the renormalizability of the gauge theories become complicated problems, which are still not completely solved. In noncommutative U(1) Higgs–Kibble model, the gauge sector is one loop renormalizable at first order in θ no matter whether the gauge symmetry is spontaneously broken or not [16], but the whole θ -order renor-

J.-H. Huang, Z.-M. Sheng / Physics Letters B 678 (2009) 250–253

malizability is spoiled by the matter sector. There is a similar result for fermion fields. In noncommutative quantum electrodynamics (QED), the gauge sector is one loop renormalizable at first order in θ [12,15], and a four fermions term can not be renormalizable by field redefinitions. As it is argued in [16], the renormalizability of gauge sector is very surprising when matter fields appear. If the gauge sector can be renormalized to all order in θ , we may expect that there are new symmetries in these noncommutative gauge theories, which make the gauge sector be renormalizable. Then we can modify the matter sector according these new symmetries so that the whole theory becomes renormalizable. But in order to achieve these, we should first prove that the gauge sector is renormalizable to all orders in θ when matter fields exist. In this direction, the oneloop UV-divergent matter contributions to gauge sectors have been computed in Ref. [17] using path integral method for noncommutative fields. In this Letter, using the θ -expanded approach, we expand the noncommutative QED action to θ 2 -order by Seiberg– Witten map, and calculate all the divergent fermionic contributions to gauge field propagators up to θ 2 -order. It is shown that these divergent terms can be renormalized by field redefinitions. When one only considers contributions from action of θ -order, there are non-renormalizable gauge field divergent terms involving fermion mass [12]. Because the divergent gauge field contributions to the gauge field propagators are renormalizable [11], we can conclude that the gauge field propagators are one loop renormalizable to θ 2 -order in noncommutative QED. We have more confidence in the renormalizability of the gauge propagators to all orders in θ in noncommutative QED. In Section 2, we give the θ 2 order noncommutative QED action using the Seiberg–Witten map. In Section 3, we calculate the divergent contributions to the gauge field propagator and Section 4 is for the conclusion. 2

2. θ -order action The QED action on noncommutative space is

S =−

1



4

d4 x Fˆ μν  Fˆ μν +



ˆ d4 x ψ¯ˆ  (iD /ˆ − m)ψ,

(3)

251

Then plugging the above results (6) and (7) into (3), one can get the action to θ 2 order as follows:

S = S 0 + S 1 + S 2,a + S 2,ψ , where

(8)



 1 α μν ¯ ¯ , S 0 = d x ψ i γ D α ψ − mψψ − f μν f 4

 1 S 1 = − θ μν d4 x f αμ ψ¯ i γ α D ν ψ + f αμ f β ν f α β 2   1 1 ¯ − f αβ f αβ , + f μν ψ¯ i γ α D α ψ − mψψ 2 2

 1 β S 2,a = θ μν θ κ λ d4 x 2 f αμ f βλ f νκ f α β − f αμ f β ν f κα f λ 

4

1

S 2,ψ

+ f κ λ f αμ f β ν f α β + f μκ f ν λ f α β f α β 4  1 − f μν f κ λ f α β f α β , (11) 8

1 μν κ λ 1 = θ θ f μλ f νκ ψ¯ i γ α D α ψ + f μν f κ λ ψ¯ i γ α D α ψ 16

2

+ 2 f μν f ακ ψ¯ i γ α D λ ψ + 6 f αμ f νκ ψ¯ i γ α D λ ψ + 2∂μ f ακ ψ¯ γ α D λ ( D ν ψ) + ∂μ f κ λ ψ¯ γ α D ν ( D α ψ) 

−m

1 2



3. One-loop divergent terms In this section, we calculate the divergent contributions to the U(1) gauge field propagator at one-loop level. These contributions come from two sectors, the gauge sector and fermionic field sector. Because the gauge sector is renormalizable, we just consider the fermionic contributions. To carry out the calculation, we consider terms in the action (8) quadratic in ψ . These terms can be written as



ˆ μ ψˆ = ∂μ ψˆ − i Aˆ μ  ψ, ˆ D

(4)

where B (m,n) are terms containing m factors of

(5)

of aμ . From the action (8), we can obtain

According to Seiberg–Witten map, one can express the NC fields

given by [18]

ˆ α = aα + A 1

+ θ μν θ κ λ ( f κμ f αν aλ + aμ aκ ∂λ f να − aλ ∂κ aμ ∂ν aα ),

(2 )

+ B (2,1) + B (2,2) + B (2,3) ψ,

(13)

θ and n factors

B (0,0) = i∂/ − m, B

(0,1)

= a/,

(14)





1 i m B (1,1) = − θ μν i γ α f αμ ∂ν + f μν ∂/ − f μν , 2 2 2

1 μν θ (aμ f αν − aμ ∂ν aα ) 2



¯ + m∂μ f κ λ ψ¯ i D ν ψ . (12) f μν f κ λ − f μκ f ν λ ψψ

S ψ = ψ¯ B (0,0) + B (0,1) + B (1,1) + B (1,2)

ˆ Aˆ μ using ordinary QED fields ψ, aμ . In order to get the θ 2 -order ψ, action, one should express the NC fields to θ 2 -order, which are

(10)

4

where  is Moyal–Weyl star product. The NC covariant derivative ˆ μ ψˆ and the noncommutative U(1) gauge field strength Fˆ μν are D defined as

ˆ ν − ∂ν Aˆ μ − i [ Aˆ μ ,  Aˆ ν ]. Fˆ μν = ∂μ A

(9)

(16)



(6)

2 1 ψˆ = ψ − θ μν aμ ∂ν ψ 2 1 μν κ λ  + θ θ ∂μ aλ ∂κ aν ψ − 2i ∂μaκ ∂ν ∂λ ψ + 4aμ ∂ν aκ ∂λ ψ 16 + 2aμ aκ ∂λ ( D ν ψ) + 2aμ f νκ ∂λ ψ , (7) where f μν = ∂μ aν − ∂ν aμ is the usual U(1) gauge field strength and D ν ψ = ∂ν ψ − iaν ψ is the usual covariant derivative of fermion field ψ .

 1 1 B (1,2) = − θ μν γ α f αμ aν + f μν a/ ,

(15)

2

B (2,1) =

1 16

2

 θ μν θ κ λ 2γ α ∂μ f ακ ∂λ ∂ν

(17)

+ 2∂μ ∂κ aλ ∂ν ∂/ + im∂μ f κ λ ∂ν , (18)

1 μν κ λ i θ θ i f μλ f νκ ∂/ + f μν f κ λ∂/ + 2i f μν f ακ γ α ∂λ B (2,2) = 16

2

− 2i ∂μ f ακ γ α (∂λ aν + aν ∂λ + aλ ∂ν )   1 f μν f κ λ − f μκ f ν λ −m 2

252

J.-H. Huang, Z.-M. Sheng / Physics Letters B 678 (2009) 250–253

+ 6i f αμ f νκ γ α ∂λ − 2i ∂μ ∂κ aλ γ α (∂ν aα + aα ∂ν + aν ∂α )  + m ∂μ f κ λ a ν , (19) 

1

B (2,3) =

θ μν θ κ λ f μλ f νκ a/ +

16

1 2

f μν f κ λa/ + 2 f μν f ακ γ α aλ

+ 6 f αμ f νκ γ α aλ − 2∂μ f ακ γ α aλ aν  − 2∂μ ∂κ aλ aν a/ .

e

 =

= det

(20)



i Γ (1) = ln det B (0,0) + B (0,1) + B (1,1) + B (1,2)

+ B (2,1) + B (2,2) + B (2,3)  = Tr ln B (0,0) + B (0,1) + B (1,1) + B (1,2) + B (2,1) + B (2,2) + B (2,3)   = Tr ln I − im2−1∂/ + 2−1 B (0,1) + B (1,1) + B (1,2) + B (2,1) + B (2,2) + B (2,3) i∂/ + Tr ln 2 − Tr ln(i∂/ ), (22)

where 2 = −∂ . The last two terms are infinite constants and can be ignored in the effective action. The contribution term to the effective action is the first term in the final equation. We can calculate it by power expansion 2

+ B (2,1) + B (2,2) + B (2,3) i∂/

∞ n  (−1)n+1  −1 −1 ( j ,k) 2 B i∂/ 2 − im∂/ + . = Tr

(23)

Using the above equation, we can calculate the divergent contributions to the gauge field propagators. For n = 1, Tr(2−1 B (2,2) i∂/ ) = 0. The nonvanishing terms from the n = 2 case are:

 i 2 d4 k 4 a˜ μ (−k)˜aν (k) k2 g μν − kμ kν , (24) = 2 4 3 (4π )  (2π )  Tr 2−1 B (1,1) i∂/ 2−1 B (1,1) i∂/    i 2 d4 k μν κ λ 1 2 2 ˜ θ θ a (− k )˜ a ( k ) − k k k m = ν λ μ κ , 2 (4π )2  (2π )4

=



(25)

 μν θ κ λ a˜ (−k)˜ θ aβ (k) k2 g α β − kα kβ α 2 4 (4π )  (2π )   1 1 4 × g ν λ k2 kμ kκ − a˜ ν (−k)˜aλ (k) k kμ kκ . (26) i

24

2



d4 k

60

For n = 3, the divergent terms are as follows:

B

4

2

d k

(4π )2 

(2π )



−1 (1,1)

i∂/ 2

B

i∂/

(27)

 A.





3 θ μν θ κ λ a˜ ν (−k)˜aλ (k) − m2k2kμ kκ , 4 2

3 



2−1 B (2,2) i∂/



i

2

A.

d4 k

(29)

(4π )2 

12

+ a˜ α (−k)˜aρ (k)   1 1 2 2 2 × − g αρ g ν λm k kμkκ + g ν λm kα kρ kμkκ , 4



Tr −im2−1∂/

=

i

2

2  

(30)

6

  2−1 B (2,1) i∂/ 2−1 B (0,1) i∂/ A . d4 k

θ μν θ κ λ (4π  (2π )4

1 × a˜ α (−k)˜aκ (k) m2k2 ( g ν λkμkα + g αν kμ kλ ) )2

12

+ a˜ α (−k)˜aρ (k)   1 1 × − g αρ g ν λm2k2 kμkκ + g ν λm2kα kρ kμkκ 4 6   1 2 2 + a˜ ν (−k)˜aλ (k) m k kμkκ , 



6





i

2





Tr −im2−1∂/

=

i

2

2  

(31)

 A.

d4 k

θ μν θ κ λ (4π )2  (2π )4 

 1 × a˜ α (−k)˜aκ (k) − g ν λm2k2kμ kα 6   1 + a˜ α (−k)˜aρ (k) g αρ g ν λm2 k2 kμ kκ 6   1 2 2 − a˜ ν (−k)˜aλ (k) m k kμkκ ,

=





−1 (1,1)

Tr −im2−1∂/ 2−1 B (2,1) i∂/ −im2−1∂/ 2−1 B (0,1) i∂/

j ,k

Tr 2−1 B (0,1) i∂/ 2−1 B (2,1) i∂/ + 2−1 B (2,1) i∂/ 2−1 B (0,1) i∂/



 θ μν θ κ λ a˜ ν (−k)˜aλ (k) 2m4kμ kκ , (2π )4  2    Tr −im2−1∂/ 2−1 B (0,1) i∂/ 2−1 B (2,1) i∂/ A .  i 2 d4 k μν κ λ = θ θ (4π )2  (2π )4

1 × a˜ α (−k)˜aκ (k) m2k2 ( g ν λkμkα − g αν kμ kλ )





A.

  1 2 2 μν κ λ˜ θ θ aν (−k)˜aλ (k) − m k kμ kκ , 4 (2π )4 d4 k

∂/ 2 



Tr ln I − im2−1∂/ + 2−1 B (0,1) + B (1,1) + B (1,2)

Tr 2−1 B (0,1) i∂/ 2−1 B (0,1) i∂/

i



where [ A BC ] A . means the sum of all inequivalent arrangements of A BC . For n = 4 case, the divergent contributions are

=

Then the effective action can be written as

n =1



(4π )2 

Tr −im2−1∂/

(21)

n



(28)

m,n



2

−1

=

m,n

 B (m,n) .

=

i

Tr −im2

   4 ¯ (m,n) ¯ Dψ Dψ exp i d x ψ B ψ 





¯ ψ , we can obtain the contributions to the efIntegrating over ψ, ( 1) fective action Γ , i Γ (1)



Tr −im2−1∂/ 2−1 B (0,1) i∂/ 2−1 B (2,1) i∂/

6

2−1 B (1,1) i∂/ d4 k

(32)

2  A.

θ μν θ κ λ

(4π )2  (2π )4

 1 × a˜ α (−k)˜aρ (k) k2 g αρ − kα kρ g ν λm2kμ kκ 3

 − a˜ ν (−k)˜aλ (k) m2k2kμ kκ 3   4 ˜ + aν (−k)˜aλ (k) 6m kμ kκ . 4

(33)

J.-H. Huang, Z.-M. Sheng / Physics Letters B 678 (2009) 250–253

For n = 5, we can obtain the nonvanishing contributions as follows:



Tr −im2−1∂/

4 

2−1 B (2,2) i∂/

 (34)

Tr −im2−1∂/

4 

2−1 B (1,1) i∂/

(35)

=

2



A.

d4 k

i

2

(4π

)2





d4 k

)4



a˜ μ (−k)˜aν (k)

(37)

4 2 k g μν − kμ kν 3

60

+

24

 a˜ α (−k)˜aβ (k) k2 g α β − kα kβ g ν λ k2 kμ kκ

 .

(38)

We can rewrite i Γ (1) in coordinate space

i Γ (1 ) =

i

2

(4π +

)2

1 48







d4 x

∂μ ∂κ f α β

2 3

 f μν f μν − θ μν θ κ λ

2 f αβ

 .

1 240

f μν 22 f κ λ (39)

As we have already mentioned, we want to investigate the oneloop renormalizability of noncommutative QED at θ 2 -order. One main point in this investigation is to make use of the freedom in Seiberg–Witten map, which has been discussed in [11]. The freeˆ (μn) is a dom of Seiberg–Witten map allows field redefinitions. If A n solution of Seiberg–Witten map at θ -order. Then, a gauge covari(n) ˆ (μn) to form ant term Aˆ μ of exactly θ n -order can be added to A another solution 

ˆ (μn) = Aˆ (μn) + Aˆ (μn) . A

(40)

These field redefinitions yield additional terms  S a to the original action which can be used to absorb divergences.  S a is of the form



 Sa =

d4 x ∂μ f μν Aˆ ν . (n)

d4 k

(2π )4

 θ α β θα β aμ k2 g μν − kμkν k4 aν ,

(42)

θ α β θ μν aμkν k4 kβ aα ,

(43)

d4 k 

α β 



aμ k4 kα kβ aμ − aμ kμ k2 kα kβ kν aν ,

(44)

α β  α β θ 2 aα k6 aβ + θ 2 aμ kμ k2 kα kβ kν aν (2π )4  α β  − 2 θ 2 aα kβ k4 kμ aμ .

(45)

(2π )4

θ2

d4 k 

4. Conclusion

(2π  1 + θ μν θ κ λ − a˜ ν (−k)˜aλ (k)k4 kμ kκ 1

(2π

)4

(36)

For n = 7, 8, . . . , there is no divergent contribution. Sum up all the divergent contributions (24)–(37), we can obtain

i Γ (1 ) =

d4 k

It is easy to check that the θ 2 -order divergent contributions in (39) are of the forms (43) and (44). So the gauge field propagators are one-loop renormalizable at θ 2 -order in noncommutative QED with massive matter fields.

2 

  9 4 μν θ κ λ a˜ (−k)˜ , θ a ( k ) k k m ν λ μ κ 2 (4π )2  (2π )4  4    Tr −im2−1∂/ 2−1 B (0,1) i∂/ 2−1 B (2,1) i∂/ A .    i 2 d4 k μν κ λ 3 4 ˜ = θ θ a (− k )˜ a ( k ) k k m ν λ μ κ . 2 (4π )2  (2π )4 i

 

For n = 6, the divergent contributions are





A.

  i 2 d4 k μν κ λ 5 4 = θ θ a˜ ν (−k)˜aλ (k) m kμ kκ , 2 (4π )2  (2π )4     3 Tr −im2−1∂/ 2−1 B (0,1) i∂/ 2−1 B (2,1) i∂/ A .    i 2 d4 k μν κ λ 25 4 ˜ = θ θ a (− k )˜ a ( k ) k k m ν λ μ κ . 2 (4π )2  (2π )4 



253

(41)

At θ 2 -order, there are four additional terms to the original action. In momentum space, these additional terms can be expressed as

In this Letter, we investigate the renormalizability of gauge field propagators at θ 2 order in noncommutative QED. We expand the noncommutative QED action to θ 2 order. Using the background field method, we calculate the one loop divergent contributions of the fermion fields to the U(1) gauge field propagators. We have proved that the gauge propagators are renormalizable at θ 2 order no matter whether the fermions are massive or massless. If we only consider contributions from action of θ -order as that in Ref. [12], there are non-renormalizable gauge field divergent terms of θ 2 -order when the fermion is massive. From this result, we have more confidence in extrapolating that the gauge propagators are one loop renormalizable to all order in θ in noncommutative QED. In order to prove renormalizability at θ n order, we should expand the action to θ n and calculate effective action to θ n order, of course, this is not an easy work. Acknowledgement This work is supported in part by the funds from NSFC under Grant No. 90303003. References [1] H. Snyder, Phys. Rev. 71 (1947) 38; H. Snyder, Phys. Rev. 72 (1947) 68. [2] A. Connes, M.R. Douglas, A. Schwarz, JHEP 9802 (1998) 003. [3] Y.-K.E. Cheung, M. Krogh, Nucl. Phys. B 528 (1998) 185. [4] C.S. Chu, P.M. Ho, Nucl. Phys. B 550 (1999) 151. [5] N. Seiberg, E. Witten, JHEP 9909 (1999) 032. [6] R.J. Szabo, Phys. Rep. 378 (2003) 207. [7] C.P. Martin, D. Sanchez-Ruiz, Phys. Rev. Lett. 83 (1999) 476, hep-th/9903077. [8] I.F. Riad, M.M. Sheikh-Jabbari, JHEP 0008 (2000) 045, hep-th/0008132. [9] M. Hayakawa, Phys. Lett. B 478 (2000) 394, hep-th/9912094. [10] B. Jurco, S. Schraml, P. Schupp, J. Wess, Eur. Phys. J. C 17 (2000) 521, hepth/0006246. [11] A. Bichl, et al., JHEP 0106 (2001) 013, hep-th/0104097. [12] M. Buric, V. Radovanovic, JHEP 0210 (2002) 074, hep-th/0208204. [13] M. Buric, V. Radovanovic, JHEP 0402 (2004) 040, hep-th/0401103. [14] M. Buric, D. Latas, V. Radovanovic, JHEP 0602 (2006) 046, hep-th/0510133. [15] R. Wulkenhaar, JHEP 0203 (2002) 024. [16] C.P. Martin, D. Sanchez-Ruiz, C. Tamarit, JHEP 0702 (2007) 065. [17] C.P. Martin, C. Tamarit, Phys. Lett. B 658 (2008) 170. [18] S. Fidanza, JHEP 0206 (2002) 016.