divergence in noncommutative chiral models

Absence of the 4 divergence in noncommutative chiral models

Maja Buric,* Dusko Latas,† and Voja Radovanovic‡

Faculty of Physics, University of Belgrade, P.O. Box 368, RS-11001 Belgrade, Serbia

Josip Trampeticx

Rudjer Boskovic Institute, Theoretical Physics Division, P.O. Box 180 HR-10002 Zagreb, Croatia(Received 7 November 2007; published 27 February 2008)

In this paper we show that in the noncommutative chiral gauge theories the 4-fermion vertices are finite.The 4 vertices appear in linear order in quantization of the �-expanded noncommutative gauge theories;in all previously considered models, based on Dirac fermions, the 4 vertices were divergent andnonrenormalizable.

DOI: 10.1103/PhysRevD.77.045031 PACS numbers: 11.10.Nx, 11.10.Gh, 11.15.�q

I. INTRODUCTION

Although the issue of regularization of quantized fieldtheories was the original motivation to introduce the non-commutativity of coordinates in the 1950s [1], the questionof renormalizability of field theories on noncommutative(NC) Minkowski space is still far from being settled.

In a definition of ‘‘noncommutative theory’’ there areseveral steps which are not straightforward and need to bespecified. The first one is a definition of noncommutativespace. By noncommutative Minkowski space one usuallymeans the algebra of functions on commutative R4 whichamong themselves multiply with the Moyal-Weyl ? prod-uct. This product is associative but not commutative. The ?product of functions integrated in the usual sense has acyclic property, which is necessary to define the action andwith it the NC generalization of a classical field theoryderived from the action principle.

The second nonunique step is the very definition of afield theory. This is because, clearly, many theories canhave the same commutative limit. In the flat NC space amost straightforward way is to start with a commutativetheory and replace the commuting fields with the non-commuting ones and the ordinary products with the ?products. A noncommutative scalar �4 theory and theU�N� gauge theory were formulated in this manner initially[2,3]. The most prominent result was that the short and thelong distances were related: this was seen through themixing of ultraviolet and infrared divergencies. The UV/IR mixing also obstructs the renormalization. There areother variants of noncommutative scalar theories: as hasbeen shown recently [4], one can define a renormalizablenoncommutative �4 theory by modifying the originalcommutative action by a potential term x2�2. There aresimilar proposals for gauge theory too [5].

Generalization of the notion of gauge symmetry is alsonot unique. Initial proposals [3] contain UV/IR mixing asdoes the scalar field theory. In the most recent models, thesymmetry in the ordinary sense is not changed; the cop-roduct in the Hopf algebra is deformed [6]. These modelshave not yet been tested for renormalizability. We shallwork in the framework of the �-expanded gauge theory [7].The original idea [8] that, in addition to gauge symmetry,nonuniqueness of the Seiberg-Witten (SW) map can beused to establish renormalizability proved to be very usefulin a couple of models. A rough summary of the obtainedresults [8–11] is as follows. In general, divergencies re-lated to the gauge fields are weaker than those for thefermions. When fermions are included an immediate ob-stacle to renormalizability is found—the so-called ‘‘4 ’’divergence, which is of the form

D jdiv �1

�� 5 �: (1)

It appears independently of whether fermions are massiveor massless. This divergent vertex of the form of Fermiinteraction cannot be regularized in any well-defined orsystematic way.

Recently, some potentially interesting and encouragingresults on renormalizability of the �-expanded theorieshave been obtained [12,13]. It was shown that the pureSU�N� noncommutative gauge theory is one-loop renor-malizable in linear order in � [12]; also, it was possible todefine NC generalization of the standard model (SM)which has the gauge sector free of divergencies at oneloop in the same linear order [13]. These results point outthe importance of the choice of representation for therenormalizability properties. With this in mind and recall-ing that all previously considered models have only in-cluded Dirac fermions, we decided to redo the calculationof divergencies for the chiral fermions. As an indicativeand most important check we choose to do first the 4 divergence. The result was that for chiral fermions 4 divergence is absent. In this paper we present the calcu-

*[email protected]†[email protected]‡[email protected]@rex.irb.hr

PHYSICAL REVIEW D 77, 045031 (2008)

1550-7998=2008=77(4)=045031(7) 045031-1 © 2008 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.77.045031

lations for the noncommutative U(1) and SU(2) gaugetheories.

II. NONCOMMUTATIVE U(1) THEORY

A. Notation

We will work in the noncommutative Minkowski space,defined by the relation

�x� ;? x�� x� ? x� � x� ? x� � i��: (2)

The commutator in (2) is a ? commutator given by theMoyal-Weyl product,

f�x� ? g�x� � e�i=2��@=@x��@=@y��f�x�g�y�jy!x: (3)

The action for the left chiral fermion ’ interacting withthe U(1) gauge field A� is, in commutative theory, given by

SC �Zd4xLC

�Zd4x

�i �’ ��@� iA��’�

1

4F��F��

�: (4)

The noncommutative U(1) symmetry can be realized bythe same set of fields. We denote the noncommutativegauge potential by A�, the NC field strength by F��,F�� @�A� � @�A� � �A� ;? A��, and the NC Weylspinor by ’. The noncommutative U(1) symmetry is ofcourse non-Abelian; it can however be related to the usualAbelian U(1) symmetry by the SW map [14], which givesthe basic, i.e., noncommutative, fields as expansions intheir commutative approximations. The SW map to firstorder in �� reads [7,15]

A � � A� �14��fA�; @�A� F��g ; (5)

F�� F�� 12��fF��; F��g �

14��fA�; �@� D��F��g

; (6)

’ � ’� 12��A�@�’

i4��A�A�’ : (7)

The f; g denotes the anticommutator and D� is the commu-tative covariant derivative. The fields ’ and strengthsF�� @�A� � @�A� in (5)–(7) carry representations ofthe commutative U(1) symmetry.

The action for noncommutative chiral electrodynamicsis analogous to (4)

SNC �Zd4xLNC

�Zd4x

�i �’ ? ��@� iA�� ? ’�

1

4F�� ? F

��:

(8)

Expanding to first order in �� we obtain

L NC � L0 L1;A L1;’; (9)

where

L 0 � LC � i �’��@� iA��’�14F��F

��; (10)

L 1;A � �12��F��F��F��

14F��F��F

��; (11)

L 1;’ � �18��

��F �’ ��@� iA��’: (12)

The antisymmetric � is defined by �� .

For non-Abelian theories formulas (10) and (11) contain anadditional trace in the group generators, as we will see inthe special case of SU(2) later.

Obviously the parameter �� of dimension �length�2 issmall—of order of magnitude & �TeV��2 [16], and there-fore the expansion (9) is useful to compute the almost-classical effects of noncommutativity. Its relevance in con-siderations of renormalizability is not quite clear since thedivergent contributions can be nonperturbative in ��; in-deed this is what happens with the UV/IR mixing.Nevertheless, we shall work with the truncated expression(9) for two reasons: first, an expansion like this might be aviable or a correct way to define a renormalizable theory.Second, we presume that an additional structure given bynoncommutative Ward identities exists: it then relates then-point functions of different orders in the � expansion.Thus it is possible, in principle, to use NC Ward identitiesin order to lift renormalizability from � linear to higher-�orders. In any case, if the theory is not renormalizable inlinear order, it will not be renormalizable at all because ofits essentially iterative definition.

B. Quantization

We start with the action (9) for noncommutative chiralelectrodynamics and quantize it by using the path-integralmethod. We treat the �-dependent terms as interactions, theparameter �� as a coupling constant. The propagators forthe spinors and for the gauge fields are the same as in thecommutative theory. In order to compute the functionalintegral we need either to complexify the gauge potentialor to work with the Majorana spinors; we choose the latter.Using

�’�’ _

� �; (13)

we can rewrite the commutative part of the Lagrangian (4)as

L 0 �i2

� ��@� � i�5A�� 14F��F

��: (14)

To obtain (14) from (10) we use the �matrices in the chiralrepresentation; further details of the notation are given inthe Appendix. Written in terms of the Majorana spinors theU(1) symmetry becomes axial; this is no surprise as chiralLagrangian is not invariant under parity. For the �-linearspinor part of the Lagrangian we obtain

MAJA BURIC et al. PHYSICAL REVIEW D 77, 045031 (2008)

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L 1; � �1

16��

��F � ��@� � i�5A�� ; (15)

while the gauge self-interaction is given by (11).In order to preserve gauge covariance we integrate using

the background field method [10,11]. Briefly, in the firststep we expand fields around their classical configurations;we replace therefore in the classical action

A� ! A� A�; ! �; (16)

where A�; are the classical fields and A�;� are thequantum fluctuations. After the integration of the quantumfields, in the saddle point approximation we obtain the one-loop effective action

��A�; � � S�A�; � �1

2iS Tr�logB�A�; ��: (17)

The first term S�A�; � is the classical (gauge-fixed) actionand the second is the one-loop quantum correction �1.Finally, the gauge fixing and the ghost terms in (17) arethe same as in the commutative theory: the ghosts propa-gate only between �-independent vertices and thereforethey do not contribute to the one-loop �-linear corrections[10,11].

The operator B�A�; �, a result of Gaussian integration,is the second functional derivative of S�A�; �; it can beobtained in our case by expanding S�A� A�; �� tosecond order in A� and �

S�2� �Zd4x A�

��

BA

�

� �: (18)

We can divide B into its commutative part B0 and a�-linear contribution B1: B � B0 B1. After the gaugefixing, the matrix B0 is given by

B 0 �1

2g� � � ��5

� �5 i@6 A6 �5

� �: (19)

It contains the kinetic part

B kin �1

2g� � 0

0 i@6

� �; (20)

and the interaction. In order to expand the logarithm in (17)around identity we have to multiply B by C,

C � 2g� 00 �i@6

� �: (21)

Then we can write

BC � �I N1 T1 T2; (22)

with

I �g� 00 1

� �: (23)

The expression

�1 �i

2S Tr log�I ��1N1 ��1T1 ��1T2�

�i

2

X ��1�n1

nS Tr��1N1 ��1T1 ��1T2�

n

(24)

gives the perturbation expansion. The �1 can be identifiedwith the one-loop effective action because

S Tr�logB� � S Tr�log��1BC� � S Tr�C��1�: (25)

As the last term does not depend on the fields A� and , itcan be included in the (infinite) normalization.

In (22) we have divided the interaction term to threeparts in the following way. Operator N1 contains the com-mutative 3-vertices, i.e., the terms with one classical andtwo quantum fields. Analogously, operator T1 is a termlinear in �� containing one classical field, and T2 is linearin �� containing two classical fields. From (19) we seethat N1 equals to

N1 �0 �i � ��5@6

� �5 �5A6 @6

� �: (26)

The noncommutative vertices T1 and T2 require a bit morework. Using the Majorana spinor identities we obtain

T1 �1

8��

��V�� 2��@ � ��@�@6

�2i� ��@ �@� �F��@�@6

!; (27)

and

T2 �1

8��

��

�2

� ��5 @� @�� 5 �� 2i��@ � ��5A� � ��5F� � � ��5A@��@6�� 2��5A @� F��5 � iAF��5@6

!; (28)

with the bosonic part V�� given in [10]. As we are looking at just the four-fermion divergence, V�� will not contribute toour calculations as we shall explain shortly.

ABSENCE OF THE 4 DIVERGENCE IN . . . PHYSICAL REVIEW D 77, 045031 (2008)

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C. Divergencies

We have noted that the full perturbation expansion isgiven in (24). For example, the 4-point functions are con-tained in terms which have the sum of the operator indicesequal to 4. If in addition we look only for the contributionslinear in ��, the relevant expressions can have at most oneof the T1 or T2. There are two such terms,

D 1 � S Tr��1N1�3��1T1�� (29)

and

D 2 � S Tr��1N1�2��1T2��: (30)

As here we restrict to the 4-fermion vertex, we can furthersimplify the calculation by putting A� � 0. Under this

condition we also have V�� 0.In order to find the divergencies we write the traces in

the momentum representation and afterwards perform thedimensional regularization. The result which we obtain isthat the term D1 is finite due to the structure of themomentum integrals. For the divergent part of D2 we get

D 2jdiv �1

�4��2�

3i

8�� 5 �� 5 �:

(31)

Because of the antisymmetry of the Levi-Civita symbol theabove expression vanishes too. In fact, in retrospect, it iseasy to see that the 4 divergence has to be zero in thechiral case: because of antisymmetry of �� the onlypossible expression is (1). However, for the Majoranaspinors � �� 0 and therefore (1) vanishes identically.

III. NONCOMMUTATIVE SU�2� THEORY

A. Lagrangian

Analogous analysis for the chiral fermions in the funda-mental representation of SU(2) is in order. We start with adoublet of fermions and a vector potential,

’ �’1

’2

� �; A� � Aa�

�a2; (32)

and the following commutative Lagrangian

L 0 � L0; L0;A � i �’ ��@� iA��’�14 TrF��F��;

(33)

which we want to rewrite in terms of the Majorana spinors

1 �’1

�’1

� �; 2 �

’2

�’2

� �: (34)

As the fundamental representation of SU(2) is not real,when we write the Lagrangian in the Majorana spinors weapparently break the SU(2) symmetry; i.e., we have towrite each component of the vector potential separately.That is,

L 0; � i �’1 ��@�

i

2A3�

�’1 �

1

2�’1 ��A��’2

�1

2�’2 ��A�’1 i �’2 ��

�@� �

i

2A3�

�’2

�1

4� 1

� 2

� � 2i@6 A6 3�5 A6 1�5 iA6 2

A6 1�5 � iA6 2 2i@6 � A6 3�5

� � 1

2

� �:

(35)

We have denoted A�� A1� � iA2

�. Now, of course,Majorana � 1

2� is not a SU(2) doublet.

The �-linear bosonic part of the SU(2) Lagrangian,

L 1;A � �12�� Tr�F��F��

14F��F��F

��

� �12dabc��Fa��Fb��

14F

a��Fb��F��c; (36)

vanishes, because it is proportional to the symmetric co-efficients: dabc � Tr��af�b; �cg� � 0. In fact, dabc � 0for all irreducible representations of SU(2). On the otherhand, the fermionic linear part of the Lagrangian,

L 1; �1

32��

�� ’ ��2iFa�a@� A

aF

a�

� i�abcAaFb��c�’; (37)

in the Majorana representation is rewritten as

L 1; �1

64��

�� 1� 2

� � 2iF3�

�@� AaFa��5 2iF1

��@� � 2F2

��5@�

2iF1�

�@� 2F2�

��5@� �2iF3�

�@� AaFa��5

! 1

2

� �: (38)

B. Quantization and divergencies

As we have written the Lagrangian in an appropriateform, the procedure of quantization is straightforward andfollows closely that which was done for the U(1). It isinteresting that the results, as we shall see shortly, arecompletely analogous, though the intermediate calcula-tions are now considerably more involved. The part ofthe action which is of second order in quantum fields wewrite as

S�2� �Z

d4x A1� A2

� A3�

��1��2

� �B

A1� 0

A2�0

A3�0

�1

�2

0BBBBBB@

1CCCCCCA:

(39)

Matrices Bkin and C have the same, just enlarged, structureas in the U(1) case:


045031-4

The interactions of course differ: now we have the 4-bosonvertex in the commutative part for example. The one-loopeffective action has the form

�1 �i2S Tr log�I ��1�N1 N2 T1 T2 T3��:

(42)

As the formulas are cumbersome we shall here restrictimmediately to the subset of fermionic diagrams defined bythe condition Aa� � 0, which of course simplifies the cal-culations. With this restriction we also have N2 � 0 andT3 � 0. The remaining interaction vertex is

In �-linear order we have the following T1 matrix:

while the T2 matrix is given by


045031-5

where

E� � � 1��5 1 � 2��5 2; (46)

F�� i�@� � 1�� 1 i � 1��@� 1� i�@� � 2�� 2

� i � 2��@� 2�; (47)

G�� @� � 1��5 2 � � 1��5�@� 2�

� �@� � 2��5 1 � 2��5�@� 1�; (48)

H�� i�@� � 1��

� 2 i � 1��@� 2� � i�@� � 2��

� 1

i � 2��@� 1�: (49)

As before, the divergent contributions in principle comefrom D1 and D2. However, D1 is finite, while for thedivergent part of D2 we obtain

D 2jdiv � �1

�4��2�

9i

64�� 1��5 1

� 2��5 2�� 1��5 1 � 2��5 2�;

(50)

which identically vanishes, too.

IV. CONCLUSIONS

When one thinks about the quantization of the chiralmodels, the first question which naturally arises is the oneabout anomalies. The issue of chiral anomalies for the�-expanded models has been analyzed in detail in [17],and the result was that, for the compact gauge groups,anomalies are the same as in the commutative theory.This, for example, means that the noncommutative chiralelectrodynamics which we analyzed in Sec. II cannot bequantized consistently. On the other hand, it also meansthat we can build the particle physics models as in theordinary theory. For example, the noncommutative chiralU�1� � SU�2� gauge theory is consistent if lepton andquark multiplets are the same as in the SM. Our presentresult asserts that in addition such a noncommutativemodel has no four-fermion divergencies.

Construction of a consistent noncommutative standardmodel (NCSM) is the main motivation of our investigation.We have previously proposed a model which is renorma-lizable in the gauge sector [12,13,16] and the present resultopens a possibility to extend it. Of course the Higgs sectorshould also be investigated [18]. There are a number ofphenomenological predictions arising from the NCSMmodels [19–22]. They would, however, become morerobust if one could prove the one-loop renormalizabilityfor the complete NCSM.

Obviously, there is a long way to go to show the fullrenormalizability of the NC chiral gauge models: what wehave done here is just an initial step. As from the one-looprenormalizability no immediate conclusions can be madeabout the all-loop properties; likewise, from the renorma-lizability in �-linear order nothing automatically followsfor the full SW expansion. There are many steps to be done:some, as extension from linear to higher orders in ��, wejust see as viable possibilities. Others, like the analysis ofall one-loop divergent vertices in the � linear order, arestraightforward and require additional work, and this iswhat we plan to do in our following work. One shouldremember that in the �-expanded theories one has an addi-tional tool for renormalizability: the SW field redefinition.Note also that the renormalizability principle could help tominimize or even cancel most of the ambiguities of thehigher order SW maps [23].

If indeed the �-linear order of the chiral gauge modelsproves to be renormalizable, then it will really be importantto analyze the noncommutative Ward identities and theirimplications to renormalizability more systematically.

ACKNOWLEDGMENTS

The work of M. B., V. R., and D. L. is a done within theproject 141036 of the Serbian Ministry of Science. Thework of J. T. is supported by the project 098-0982930-2900of the Croatian Ministry of Science Education and Sports.Our collaboration was partly supported by the UNESCOproject 875.834.6 through the SEENET-MTP and by ESFin the framework of the Research Networking Programmeon ‘‘Quantum Geometry and Quantum Gravity.’’

APPENDIX: CONVENTIONS

The notation and the rules of chiral-spinor algebra fol-low basically [24]. We use the following chiral representa-tion of the � matrices:

�� 0 ��

�� 0

� �; �5 �

�1 00 1

� �; (A1)

with

�� 1; ~��; �� 1;� ~��: (A2)

This means, in particular,

�� _ � � _ _�� _: (A3)

The chiral ; � spinors multiply as

’� � �’; �’ �� ’; (A4)

�’ �� ’; �� ’�y � ’�� : (A5)

Those relations, as can be seen easily, give the usual


045031-6

identities for the Majorana spinors �; which we use:

�� ; ��5 � � �5�; (A6)

�� ; ��5 � � ��5�: (A7)

Majorana Lagrangians are obtained from the correspond-ing chiral ones using the identities (A4) and the fact thatLagrangians are real.

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divergence in noncommutative chiral models

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