arXiv:0704.0559v1  [hep-ph]  4 Apr 2007
Signal for space-time noncommutativity: the
Z → γγ decay in the renormalizable gauge sector
of the θ-expanded NCSM ∗
Josip Trampetić†
Rudjer Bošković Institute, Zagreb, Croatia
Abstract
We propose the Z → γγ decay, a process strictly forbidden in the standard
model, as a signal suitable for the search of noncommutativity of coordinates at
very short distances. We compute the Z → γγ partial widthin the framework of
the recently proposed renormalizable gauge sector of the noncommutative standard
model. The one-loop renormalizability is obtained for the model containing the
usual six representations of matter fields of the first generation. Even more, the
noncommutative part is finite or free of divergences, showing that perhaps new
interaction symmetry exists in the noncommutative gauge sector of the model.
Discovery of such symmetry would be of tremendous importance in further search
for the violation of the Lorentz invariance at very high energies. Experimental
possibilities of Z → γγ decay are analyzed and a firm bound to the scale of the
noncommutativity parameter is set around 1 TeV.
∗ Based on presentation given at the IV Summer School in Modern Mathematical
Physics, Belgrad, Serbia, September 3-14, 2006 and LHC Days in Split, Croatia, October
2-7, 2006. Work supported by the Croatian Ministry of Science, Education and Sport
project 098-0982930-2900.
† e-mail address: josipt@rex.irb.hr
http://arxiv.org/abs/0704.0559v1
The title 2
Gauge theories can be extended to a noncommutative (NC) setting in
different ways. In our model, the classical action is obtained via a two-step
procedure. First, the noncommutative Yang-Mills (NCYM) is equipped
with a star product carrying information about the underlying noncommu-
tative manifold, and, second, the ⋆-product and noncommutative fields are
expanded in the noncommutative parameter θ using the Seiberg-Witten
(SW) map [1]. In this approach, noncommutativity is treated perturba-
tively. The major advantage is that models with any gauge group and any
particle content can be constructed [2, 3, 4, 5, 6, 7], so we can construct
the standard model (SM). Commutative gauge symmetry is the underlying
symmetry of the theory and is present in each order of the θ-expansion.
Noncommutative (NC) symmetry, on the other hand, exists only in the full
theory, i.e. after summation.
There are a number of versions of the noncommutative standard model
(NCSM) in the θ-expanded approach, [3, 4, 5, 6]. The action is gauge in-
variant; furthermore, it has been proved that the action is anomaly free
whenever its commutative counterpart is also anomaly free [8]. The ar-
gument of renormalizability was previously included in the construction
of field theories on noncommutative Minkowski space producing not only
the one-loop renormalizable model [9], but the model containing one-loop
quantum corrections free of divergences [10], contrary to previous results
[11, 12].
In [10] we analyzed the gauge theory based on the U(1)Y × SU(2)L ×
SU(3)C group: we succeeded in constructing a model which had the renor-
malizable gauge sector to θ-linear order. The condition of the gauge sector
renormalizability determines the additional θ-linear interactions between
gauge bosons.
Experimental evidence for noncommutativity coming from the gauge
sector should be searched for in the process of the Z → γγ decay, kinemati-
cally allowed for on-shell particles [10, 7]. As it is forbidden in the SM by an-
gular momentum conservation and Bose statistics (Landau-Pomeranchuk-
Yang Theorem), it would serve as a clear signal for the existence of space-
time noncommutativity. Signatures of noncommutativity were discussed
previously within particle physics in [7, 13, 14].
The noncommutative space which we consider is the flat Minkowski
space, generated by four hermitian coordinates x̂µ which satisfy the com-
mutation rule
[x̂µ, x̂ν ] = iθµν = const. (1)
The algebra of the functions φ̂(x̂), χ̂(x̂) on this space can be represented
by the algebra of the functions φ̂(x), χ̂(x) on the commutative R4 with the
Moyal-Weyl multiplication:
φ̂(x) ⋆ χ̂(x) = e
θµν ∂
∂yν φ̂(x)χ̂(y)|y→x . (2)
It is possible to represent the action of an arbitrary Lie group G (with
the generators denoted by T a) on noncommutative space. In analogy to
the ordinary case, one introduces the gauge parameter Λ̂(x) and the vector
The title 3
potential V̂µ(x). The main difference is that the noncommutative Λ̂ and V̂µ
cannot take values in the Lie algebra G of the group G: they are enveloping
algebra-valued. The noncommutative gauge field strength F̂µν is
F̂µν = ∂µV̂ν − ∂ν V̂µ − i(V̂µ ⋆ V̂ν − V̂ν ⋆ V̂µ). (3)
There is, however, a relation between the noncommutative gauge symmetry
and the commutative one: it is given by the Seiberg-Witten (SW) mapping
[1]. Namely, the matter fields φ̂, the gauge fields V̂µ, F̂µν and the gauge
parameter Λ̂ can be expanded in the noncommutative θµν and in the com-
mutative Vµ and Fµν . This expansion coincides with the expansion in the
generators of the enveloping algebra of G, {T a, : T aT b :, : T aT bT c :};
here : : denotes the symmetrized product. The SW map is obtained as a
solution to the gauge-closing condition of infinitesimal (noncommutative)
transformations. The expansions of the NC vector potential and of the field
strength, up to first order in θ, read
V̂ρ(x) = Vρ(x)−
θµν {Vµ(x), ∂νVρ(x) + Fνρ(x)}+ . . . , (4)
F̂ρσ = Fρσ +
2{Fµρ, Fνσ} − {Vµ, (∂ν +Dν)Fρσ}
+ . . . , (5)
where Dν = ∂ν − i[Vν , ] is the commutative covariant derivative.
The solution for the SW map given above is not unique and along with
(5) all expressions V̂ ′µ, F̂
µν of the form
V̂ ′µ = V̂µ +Xµ, F̂
µν = F̂µν +DµXν −DνXµ (6)
are solutions to the closing condition to linear order, if Xµ is a gauge
covariant expression linear in θ, otherwise arbitrary. One can think of this
transformation as of a redefinition of the fields Vµ and Fµν .
Taking the action of the noncommutative gauge theory, analogous to
that of the ordinary Yang-Mills theory with the commutative field strengths
replaced by the noncommutative ones,
S = −
d4x F̂µν ⋆ F̂
µν , (7)
and expanding the fields as in (4-5) and the ⋆-product in θ, we obtain the
expression
S = −
d4xFµνF
µν + θµν Tr
FµνFρσ − FµρFνσ
F ρσ , (8)
which is the starting point for the analysis of θ-expanded noncommutative
gauge models. Due to the renormalizability condition, we add term, includ-
ing NC freedom parameter 1
(a− 1), to the original Lagrangian, producing
the following general form of the noncommutative gauge field action:
S = −
d4xFµνF
µν + θµν Tr
d4x (
FµνFρσ − FµρFνσ)F
ρσ. (9)
The title 4
The most general form of the NC action, invariant under the NC gauge
transformation, is given in [3, 5, 6, 4],
Sgauge = −
R(F̂µν) ⋆R(F̂
. (10)
The sum in (10) is, in principle, taken over all irreducible representations
R of GSM with arbitrary weights CR. Obviously, gauge models are rep-
resentation dependent in the NC case: the choice of representations has a
strong influence on the theory, on both the form of interactions and the
renormalizability properties.
Expanding the NC gauge action (10) to first order in the noncommuta-
tivity parameter θ, we obtain
Sgauge = −
d4xR(Fµν)R(F
µν) (11)
R(Fµν)R(Fρσ)−R(Fµρ)R(Fνσ)
R(F ρσ).
The arbitrariness in the gauge action, introduced through the coefficient
a, reflects in part also the nonuniqueness of the SW map. As we have
already mentioned, renormalizability points out the value a = 3 as physical;
however, we keep the value of a arbitrary in calculations and use a = 3 at
the end.
Note that by generalizing the expression (5) to equivalent form
F̂µν(a) = Fµν +
2{Fµρ, Fντ} − a{Vρ, (∂τ +Dτ )Fµν}
, (12)
one could also obtain the actions (9,11) directly from (7,10).1 The im-
portant question, if the freedom parameter a is eventually comming from
different class of SW maps and/or some other new interaction symmetry
extends the purpose of this presentation and, consequentlly, shall be dis-
cussed elsewhere.
The noncommutative correction, that is the θ-linear part of the La-
grangian, reads
Lθi = g
′3κ1θ
fµνfρσf
ρσ − fµρfνσf
+ g3κ
BiµνB
ρσk −BiµρB
+ g3Sκ
GaµνG
ρσc −GaµρG
1This is in part due to the properties of the integral over the two-function ⋆-product,
i.e. the Stokes theorem.
The title 5
+ g′g2κ2θ
ρσi − fµρB
ρσi + c.p.
+ g′g2Sκ3θ
ρσa − fµρG
ρσa + c.p.
, (13)
where the c.p. in (13) denotes the addition of the terms obtained by a
cyclic permutation of fields without changing the positions of indices. Here,
fµν , B
µν , and G
µν are the physical field strengths which correspond to
U(1)Y, SU(2)L, and SU(3)C, respectively. The couplings κi, (i = 1, ..., 5),
as functions of the weights CR, that is of the Ci(= 1/g
i ), i = 1, ..., 6, are
parameters of the model. The couplings in (13) are defined as follows:
CRd(R2)d(R3)R1(Y )
3, (14)
CRd(R3)R1(Y )Tr (R2(T
L)R2(T
L)), (15)
CRd(R2)R1(Y )Tr (R3(T
S )R3(T
S)), (16)
CRd(R3)Tr ({R2(T
L),R2(T
L)}R2(T
L)), (17)
κabc5 =
CRd(R2)Tr ({R3(T
S ),R3(T
S)}R3(T
S)). (18)
The κ1, . . . , κ5 depend on the representations of matter fields through the
dependence on the coefficients CR. For the first generation of the standard
model there are six such representations, summarized in Table 1 of [4]; they
produce six independent constants CR
2. However, one can immediately
verify that κ
4 = 0. This follows from the fact that the symmetric coeffi-
cients dijk of SU(2) vanish for all irreducible representations. In addition,
we take that κabc5 = 0. The argument for this assumption is related to the
invariance of the color sector of the SM under charge conjugation. Although
apparently in Table 1 from [4] one has only the fundamental representa-
tion 3 of SU(3)C, there are in fact both 3 and 3̄ representations with the
same weights, C3 = C3̄. In the Lagrangian this corresponds to writing each
minimally-coupled quark term as a half of the sum of the original and the
charge-conjugated terms. Since the symmetric coefficients for the 3 and 3̄
representations satisfy dabc
= −dabc
, we obtain
κabc5 = C3d
= 0. (19)
2We assume that CR > 0; therefore the six CR’s were denoted by
, i = 1, ..., 6, in
[3, 6].
The title 6
-0.3 -0.2 -0.1
0 0.1
ΓΓΓ -0.2
-0.3 -0.2 -0.1 0 0.1
Figure 1: (a) The three-dimensional simplex that bounds possi-
ble values for the coupling constants Kγγγ , KZγγ and KZgg at the
MZ scale. The vertices of the simplex are: (−0.184,−0.333, 0.054),
(−0.027,−0.340,−0.108), (0.129,−0.254, 0.217), (−0.576, 0.010,−0.108),
(−0.497,−0.133, 0.054), and (−0.419, 0.095, 0.217). (b) The allowed region
for KZγγ and Kγγγ at theMZ scale, projected from the simplex given in Fig
(a). The vertices of the polygon are: (−0.333, −0.184), (−0.340, −0.027),
(−0.254, 0.129), (0.095, −0.419), (0.0095, −0.576), and (−0.133, −0.497).
We are left only with three nonvanishing couplings, κ1, κ2, and κ3, depend-
ing on six constants C1, . . . , C6:
κ1 = −C1 −
κ2 = −
C6 ; κ3 =
C5 . (20)
There are three relations among Ci’s:
= 2C1 + C2 +
C5 + C6 ,
= C2 + 3C5 + C6 ;
= C3 + C4 + 2C5 , (21)
in effect representing three consistency conditions imposed on (8) in a way
to match the SM action at zeroth order in θ. See detailes in [6].
Fig.(1) shows the three-dimensional simplex that bounds allowed values
for the dimensionless coupling constants Kγγγ , KZγγ and KZgg. For any
choosen point within the simplex in Fig.(1) the remaining coupling con-
stants KZZγ, KZZZ, KWWγ, KWWZ and Kγgg are uniquely fixed by the
NCSM [6, 4]. This is true for any combination of three coupling constants.
The title 7
Our total classical action reads
Scl = SSM +
Sθi = g
′3κ1θ
fµνfρσf
ρσ − fµρfνσf
+ g′g2κ2θ
ρσi − fµρB
ρσi + c.p.
+ g′g2Sκ3θ
ρσa − fµρG
ρσa + c.p.
. (22)
The term Sθ1 in (22) is one-loop renormalizable to linear order in θ [9] since
the one-loop correction to the Sθ1 is of the second order in θ. We need to
investigate only the renormalizability of the remaining Sθ2 and S
3 parts of
the action (22).
To realize the one-loop renormalization of the gauge part action (22), we
apply, as before [9, 10], the background field method [15, 16]. As we have
already explained the details of the method in [12], here we only discuss
the points needed for this computation. The main contribution to the func-
tional integral is given by the Gaussian integral. However, technically, this
is achieved by splitting the vector potential into the classical-background
plus the quantum-fluctuation parts, that is, φV → φV +ΦV , and by comput-
ing the terms quadratic in the quantum fields. In this way we determine
the second functional derivative of the classical action, which is possible
since our interactions (22) are of the polynomial type. The quantization
is performed by the functional integration over the quantum vector field
ΦV in the saddle-point approximation around the classical (background)
configuration φV .
First, an advantage of the background field method is the guarantee of
covariance, because by doing the path integral the local symmetry of the
quantum field ΦV is fixed, while the gauge symmetry of the background
field φV is manifestly preserved.
Since we are dealing with gauge symmetry, our Lagrangian (22) is sin-
gular owing to its invariance under the gauge group. Therefore, a proper
quantization of (22) requires the presence of the gauge fixing term Sgf [φ],
i.e. the Feynman-Fadeev-Popov ghost appears in the effective action
Γ[φ] = Scl[φ] + Sgf [φ] + Γ
(1)[φ], Sgf [φ] = −
d4x(DµΦ
)2 . (23)
The one-loop effective part Γ(1)[φ] is given by
Γ(1)[φ] =
log detS(2)[φ] =
Tr logS(2)[φ]. (24)
In (24), the S(2)[φ] is the 2nd-functional derivative of the classical action,
with the following structure:
S2 = ✷+N1 +N2 + T2 + T3 + T4 . (25)
The title 8
Here N1, N2 are commutative vertices, while T2, T3, T4 are noncommutative
ones. The indices denote the number of classical fields. The one-loop
effective action computed by using the background field method is
θ,2 =
Tr log
I +✷−1(N1 +N2 + T2 + T3 + T4)
(−1)n+1
−1N1 +✷
−1N2 +✷
−1T2 +✷
−1T3 +✷
As the conventions and the notation are the same as in [10], we only en-
counter and discuss the final results.
The divergent one-loop vertex correction to (22) as a function of the
SW freedom parameter a is [10]
Γdiv =
3(4π)2ǫ
BiµνB
µνi +
GaµνG
3(4π)2ǫ
g′g2κ2(3− a)θ
ρσ − fµρB
3(4π)2ǫ
g′g2Sκ3(3− a)θ
ρσ − fµρG
ρσa .
From (27) it is clear that the expanded gauge action (22) is renormalizable
only for the value a = 3 and, its noncommutative part is finite or free of di-
vergencies, so the noncommutativity parameter θ need not be renormalized.
The results for the bare fields and couplings, are given in [10].
Note that we have also analized the renormalizability properties of the
pure NC SU(N) gauge sector, for vector fields in the adjoint representation
[17]. We have found that this model is also renormalizable for a = 3.
However, to obtain renormalizability, we had to pay a price by necessity
for the renormalization of the noncommutative deformation parameter h.
In this way the parameter h and/or the scale of noncommutativity ΛNC
become running quantities, dependent on energy [17].
In addition, it was shown that the one-loop contributions to the U(1)
gauge-field part of the noncommutative gauge theories in the enveloping-
algebra formalism are renormalizable at first order in θ even if the scalar
matter, with and without spontaneous symmetry breaking, contributions
are taken into account [18]. There is reasonable hope that the same con-
clusion should hold for SU(N), but the computations are expected to be
extremely involving. Nevertheless, the results [18] further strengthen the
philosophy which is embraced in our latest papers [10, 17].
From the action (22) we extract the triple-gauge boson terms which are
not present in the commutative SM Lagrangian. In terms of the physical
fields A, W±, Z, and G they are
Lθγγγ =
sin 2θW Kγγγθ
ρτAµν (aAµνAρτ − 4AµρAντ ) ,
Kγγγ =
gg′(κ1 + 3κ2); (28)
The title 9
LθZγγ =
sin 2θW KZγγ θ
× [2Zµν (2AµρAντ − aAµνAρτ ) + 8ZµρA
µνAντ − aZρτAµνA
µν ] ,
KZγγ =
− 2g2
; (29)
where Aµν ≡ ∂µAν − ∂νAµ, etc. The structure of the other interactions
such as ZZγ, WWZ, ZZZ, Zgg, and γgg is given in [4, 6].
Next we focus on the branching ratio of the Z → γγ decay in the renor-
malizable model. Note that each term from the θ-expanded action (22),
(28) and (29) is manifestly invariant under the ordinary gauge transforma-
tions. The gauge-invariant amplitude AθZ→γγ for the Z(k1) → γ(k2) γ(k3)
decay in the momentum space reads
AθZ→γγ = −2e sin 2θWKZγγΘ
3 (a; k1,−k2,−k3)ǫµ(k1)ǫν(k2)ǫρ(k3). (30)
The tensor Θ
3 (a; k1, k2, k3) is given by
3 (a; k1, k2, k3) = − (k1θk2) (31)
× [(k1 − k2)
ρgµν + (k2 − k3)
µgνρ + (k3 − k1)
νgρµ]
− θµν [k
1 (k2k3)− k
2 (k1k3)]
− θνρ [k
2 (k3k1)− k
3 (k2k1)]
− θρµ [kν3 (k1k2)− k
1 (k3k2)]
+ (θk2)
gνρ k23 − k
+ (θk3)
gνρ k22 − k
+ (θk3)
gµρ k21 − k
+ (θk1)
gµρ k23 − k
+ (θk1)
gµν k22 − k
+ (θk2)
gµν k21 − k
+ θµα(ak1 + k2 + k3)α [g
νρ (k3k2)− k
+ θνα(k1 + ak2 + k3)α [g
µρ (k3k1)− k
+ θρα(k1 + k2 + ak3)α [g
µν (k2k1)− k
1 ] ,
where the 4-momenta k1, k2, k3 are taken to be incoming, satisfying the
momentum conservation (k1+ k2+ k3 = 0). In (31) the freedom parameter
a appears symmetric in physical gauge bosons which enter the interaction
point, as one would expect. The amplitude (30), for a = 3, with the Z
boson at rest gives the total rate for the Z → γγ decay:
ΓZ→γγ =
sin2 2θWK
~E2θ +
~B2θ ), (32)
where ~Eθ = {θ
01, θ02, θ03} and ~Bθ = {θ
23, θ31, θ12} are dimensionless coef-
ficients of order one, representing the time-space and space-space noncom-
mutativity, respectively. For the Z boson at rest, polarized in the direction
The title 10
of the third axis, we obtain the following polarized partial width:
Γ3Z→γγ =
sin2 2θWK
~E2θ +
~B2θ + 42
(θ03)2 + (θ12)2
. (33)
In order to estimate the scale of noncommutativity ΛNC from ΓZ→γγ,we
consider new experimental possibilities at LHC. According to the CMS
Physics Technical Design Report [19], around 107 Z → e+e− events are
expected to be recorded with 10 fb−1 of the data. From this one can
estimate the expected number of Z → γγ events per 10 fb−1. Assuming
that BR(Z → γγ) ∼ 10−8 and using BR(Z → e+e−) = 3 × 10−2, we may
expect to have ∼ 3 events of Z → γγ with 10 fb−1. Now the question
is: What would be the background from Z → e+e− when the electron
radiates a very high-energy bremsstrahlung photon in the beam pipe or
in the first layer(s) of the Pixel Detector and is thus lost for the tracker
reconstruction? In that case, the electron would not be reconstructed and
would be misidentified as a photon. The probability of such an event should
be evaluated from the full detector simulation. According to the CMS
note [20] which studies the Z → e+e− background for Higgs → γγ, the
probability to misidentify the electron as a photon is huge (see Fig. 3 in [20])
but the situation can be improved by applying more stringent selections to
the photon candidate when searching for Z → γγ events [21]. However,
the irreducible di-photon background (Fig. 3 in [20]) might also kill the
signal. In that case, one can only set the upper limits to the scale of
noncommutativity from the Z → γγ rate.
In accord with the analysis of the LHC experimental expectations [19,
20, 21] it is bona fide reasonable to assume that the lower bound for the
branching ratio is BR(Z → γγ)
∼ 10−8. Next, choosing the lower central
value of |KZγγ | = 0.05, from the figures and the Table in [6], we find that
the upper bound to the scale of noncommutativity is ΛNC
∼ 1.0 TeV for
~E2θ +
~B2θ ≃ 1. The obtained bound is strongly supported in [18].
Clearly, the measurement of the Z → γγ decay branching ratio would
fix the quantity |KZγγ/Λ
NC|, while the inclusion of other triple gauge boson
interactions through 2 → 2 scattering experiments [14] would sufficiently
reduce the available parameter space of our model by more precisely de-
termining the relations among the couplings Kγγγ , KZγγ , KZZγ, KZZZ ,
KWWγ, and KWWZ. Next, we summarize our results and compare with
those obtained previously.
The first Z → γγ calculation [22] was performed within a different
model which has different symmetries in comparison with ours and, because
of the absence of the SW map, the model does not possess the commutative
gauge invariance. Also, the Z → γγ rate obtained in [22] by imposing the
unitarity of the theory in the usual manner, θ0i = 0, [23, 24], vanishes 3.
The partial width for the same process was obtained in [6] in the frame-
work of similar theories, which, however, were not renormalizable. The
3The condition of unitarity can be covariantly generalized to θµνθ
µν = 2( ~B2θ −
~E2θ) > 0
[25].
The title 11
present results for the partial widths ΓZ→γγ and Γ
Z→γγ are about three
times larger than those in [6] and consistently symmetric with respect to
time-space and space-space noncommutativity. In the polarized rate (33)
the third components ((θ03)2 + (θ12)2) are enhanced relative to the other
two components by a large factor, as expected. Also, the rate (33) is en-
hanced by a factor of ∼ 3 with respect to the total rate (32). The upper
limit to the scale of noncommutativity ΛNC
∼ 1 TeV is significantly higher
than in [6]. This bound is now firmer owing to the regular behavior of
the triple gauge boson interactions (28-29) with respect to the one-loop
renormalizability of the NCSM gauge sector.
After 10 years of the LHC running the integrated luminosity is expected
to reach ∼ 1000 fb−1, [20]. This means that for the assumed BR(Z →
γγ) ∼ 10−8 we should have ∼ 300 events of Z → γγ, that is we should
be well above the background. On the other hand, this result can also be
understood as ∼ 3 events with the BR(Z → γγ) ∼ 10−10, which lifts the
scale of noncommutativity up by a factor of ∼ 3. Therefore, with a more
stringent selection of photon candidates and if the irreducible di-photon
contamination becomes controllable, the Z → γγ decay will become a clean
signature of space-time noncommutativity in LHC experiments.
Finally, the results of [17,18], while strongly supporting this computa-
tions, might also hint at the existence of new interaction symmetry of the
noncommutative gauge sector. Such new symmetry could be a responsible
for the renormalizability of the noncommutative matter sector including
fermions and, next, for the main goal, i.e. in general, the physical realiza-
tion of the Lorentz invariance breaking at very high energies, respectively.
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  We propose the Z -> gamma gamma decay, a process strictly forbidden in the
standard model, as a signal suitable for the search of noncommutativity of
coordinates at very short distances. We compute the Z -> gamma gamma partial
widthin the framework of the recently proposed renormalizable gauge sector of
the noncommutative standard model. The one-loop renormalizability is obtained
for the model containing the usual six representations of matter fields of the
first generation. Even more, the noncommutative part is finite or free of
divergences, showing that perhaps new interaction symmetry exists in the
noncommutative gauge sector of the model. Discovery of such symmetry would be
of tremendous importance in further search for the violation of the Lorentz
invariance at very high energies. Experimental possibilities of Z -> gamma
gamma decay are analyzed and a firm bound to the scale of the noncommutativity
parameter is set around 1 TeV.

arXiv:0704.0559v1  [hep-ph]  4 Apr 2007
Signal for space-time noncommutativity: the
Z → γγ decay in the renormalizable gauge sector
of the θ-expanded NCSM ∗
Josip Trampetić†
Rudjer Bošković Institute, Zagreb, Croatia
Abstract
We propose the Z → γγ decay, a process strictly forbidden in the standard
model, as a signal suitable for the search of noncommutativity of coordinates at
very short distances. We compute the Z → γγ partial widthin the framework of
the recently proposed renormalizable gauge sector of the noncommutative standard
model. The one-loop renormalizability is obtained for the model containing the
usual six representations of matter fields of the first generation. Even more, the
noncommutative part is finite or free of divergences, showing that perhaps new
interaction symmetry exists in the noncommutative gauge sector of the model.
Discovery of such symmetry would be of tremendous importance in further search
for the violation of the Lorentz invariance at very high energies. Experimental
possibilities of Z → γγ decay are analyzed and a firm bound to the scale of the
noncommutativity parameter is set around 1 TeV.
∗ Based on presentation given at the IV Summer School in Modern Mathematical
Physics, Belgrad, Serbia, September 3-14, 2006 and LHC Days in Split, Croatia, October
2-7, 2006. Work supported by the Croatian Ministry of Science, Education and Sport
project 098-0982930-2900.
† e-mail address: josipt@rex.irb.hr
http://arxiv.org/abs/0704.0559v1
The title 2
Gauge theories can be extended to a noncommutative (NC) setting in
different ways. In our model, the classical action is obtained via a two-step
procedure. First, the noncommutative Yang-Mills (NCYM) is equipped
with a star product carrying information about the underlying noncommu-
tative manifold, and, second, the ⋆-product and noncommutative fields are
expanded in the noncommutative parameter θ using the Seiberg-Witten
(SW) map [1]. In this approach, noncommutativity is treated perturba-
tively. The major advantage is that models with any gauge group and any
particle content can be constructed [2, 3, 4, 5, 6, 7], so we can construct
the standard model (SM). Commutative gauge symmetry is the underlying
symmetry of the theory and is present in each order of the θ-expansion.
Noncommutative (NC) symmetry, on the other hand, exists only in the full
theory, i.e. after summation.
There are a number of versions of the noncommutative standard model
(NCSM) in the θ-expanded approach, [3, 4, 5, 6]. The action is gauge in-
variant; furthermore, it has been proved that the action is anomaly free
whenever its commutative counterpart is also anomaly free [8]. The ar-
gument of renormalizability was previously included in the construction
of field theories on noncommutative Minkowski space producing not only
the one-loop renormalizable model [9], but the model containing one-loop
quantum corrections free of divergences [10], contrary to previous results
[11, 12].
In [10] we analyzed the gauge theory based on the U(1)Y × SU(2)L ×
SU(3)C group: we succeeded in constructing a model which had the renor-
malizable gauge sector to θ-linear order. The condition of the gauge sector
renormalizability determines the additional θ-linear interactions between
gauge bosons.
Experimental evidence for noncommutativity coming from the gauge
sector should be searched for in the process of the Z → γγ decay, kinemati-
cally allowed for on-shell particles [10, 7]. As it is forbidden in the SM by an-
gular momentum conservation and Bose statistics (Landau-Pomeranchuk-
Yang Theorem), it would serve as a clear signal for the existence of space-
time noncommutativity. Signatures of noncommutativity were discussed
previously within particle physics in [7, 13, 14].
The noncommutative space which we consider is the flat Minkowski
space, generated by four hermitian coordinates x̂µ which satisfy the com-
mutation rule
[x̂µ, x̂ν ] = iθµν = const. (1)
The algebra of the functions φ̂(x̂), χ̂(x̂) on this space can be represented
by the algebra of the functions φ̂(x), χ̂(x) on the commutative R4 with the
Moyal-Weyl multiplication:
φ̂(x) ⋆ χ̂(x) = e
θµν ∂
∂yν φ̂(x)χ̂(y)|y→x . (2)
It is possible to represent the action of an arbitrary Lie group G (with
the generators denoted by T a) on noncommutative space. In analogy to
the ordinary case, one introduces the gauge parameter Λ̂(x) and the vector
The title 3
potential V̂µ(x). The main difference is that the noncommutative Λ̂ and V̂µ
cannot take values in the Lie algebra G of the group G: they are enveloping
algebra-valued. The noncommutative gauge field strength F̂µν is
F̂µν = ∂µV̂ν − ∂ν V̂µ − i(V̂µ ⋆ V̂ν − V̂ν ⋆ V̂µ). (3)
There is, however, a relation between the noncommutative gauge symmetry
and the commutative one: it is given by the Seiberg-Witten (SW) mapping
[1]. Namely, the matter fields φ̂, the gauge fields V̂µ, F̂µν and the gauge
parameter Λ̂ can be expanded in the noncommutative θµν and in the com-
mutative Vµ and Fµν . This expansion coincides with the expansion in the
generators of the enveloping algebra of G, {T a, : T aT b :, : T aT bT c :};
here : : denotes the symmetrized product. The SW map is obtained as a
solution to the gauge-closing condition of infinitesimal (noncommutative)
transformations. The expansions of the NC vector potential and of the field
strength, up to first order in θ, read
V̂ρ(x) = Vρ(x)−
θµν {Vµ(x), ∂νVρ(x) + Fνρ(x)}+ . . . , (4)
F̂ρσ = Fρσ +
2{Fµρ, Fνσ} − {Vµ, (∂ν +Dν)Fρσ}
+ . . . , (5)
where Dν = ∂ν − i[Vν , ] is the commutative covariant derivative.
The solution for the SW map given above is not unique and along with
(5) all expressions V̂ ′µ, F̂
µν of the form
V̂ ′µ = V̂µ +Xµ, F̂
µν = F̂µν +DµXν −DνXµ (6)
are solutions to the closing condition to linear order, if Xµ is a gauge
covariant expression linear in θ, otherwise arbitrary. One can think of this
transformation as of a redefinition of the fields Vµ and Fµν .
Taking the action of the noncommutative gauge theory, analogous to
that of the ordinary Yang-Mills theory with the commutative field strengths
replaced by the noncommutative ones,
S = −
d4x F̂µν ⋆ F̂
µν , (7)
and expanding the fields as in (4-5) and the ⋆-product in θ, we obtain the
expression
S = −
d4xFµνF
µν + θµν Tr
FµνFρσ − FµρFνσ
F ρσ , (8)
which is the starting point for the analysis of θ-expanded noncommutative
gauge models. Due to the renormalizability condition, we add term, includ-
ing NC freedom parameter 1
(a− 1), to the original Lagrangian, producing
the following general form of the noncommutative gauge field action:
S = −
d4xFµνF
µν + θµν Tr
d4x (
FµνFρσ − FµρFνσ)F
ρσ. (9)
The title 4
The most general form of the NC action, invariant under the NC gauge
transformation, is given in [3, 5, 6, 4],
Sgauge = −
R(F̂µν) ⋆R(F̂
. (10)
The sum in (10) is, in principle, taken over all irreducible representations
R of GSM with arbitrary weights CR. Obviously, gauge models are rep-
resentation dependent in the NC case: the choice of representations has a
strong influence on the theory, on both the form of interactions and the
renormalizability properties.
Expanding the NC gauge action (10) to first order in the noncommuta-
tivity parameter θ, we obtain
Sgauge = −
d4xR(Fµν)R(F
µν) (11)
R(Fµν)R(Fρσ)−R(Fµρ)R(Fνσ)
R(F ρσ).
The arbitrariness in the gauge action, introduced through the coefficient
a, reflects in part also the nonuniqueness of the SW map. As we have
already mentioned, renormalizability points out the value a = 3 as physical;
however, we keep the value of a arbitrary in calculations and use a = 3 at
the end.
Note that by generalizing the expression (5) to equivalent form
F̂µν(a) = Fµν +
2{Fµρ, Fντ} − a{Vρ, (∂τ +Dτ )Fµν}
, (12)
one could also obtain the actions (9,11) directly from (7,10).1 The im-
portant question, if the freedom parameter a is eventually comming from
different class of SW maps and/or some other new interaction symmetry
extends the purpose of this presentation and, consequentlly, shall be dis-
cussed elsewhere.
The noncommutative correction, that is the θ-linear part of the La-
grangian, reads
Lθi = g
′3κ1θ
fµνfρσf
ρσ − fµρfνσf
+ g3κ
BiµνB
ρσk −BiµρB
+ g3Sκ
GaµνG
ρσc −GaµρG
1This is in part due to the properties of the integral over the two-function ⋆-product,
i.e. the Stokes theorem.
The title 5
+ g′g2κ2θ
ρσi − fµρB
ρσi + c.p.
+ g′g2Sκ3θ
ρσa − fµρG
ρσa + c.p.
, (13)
where the c.p. in (13) denotes the addition of the terms obtained by a
cyclic permutation of fields without changing the positions of indices. Here,
fµν , B
µν , and G
µν are the physical field strengths which correspond to
U(1)Y, SU(2)L, and SU(3)C, respectively. The couplings κi, (i = 1, ..., 5),
as functions of the weights CR, that is of the Ci(= 1/g
i ), i = 1, ..., 6, are
parameters of the model. The couplings in (13) are defined as follows:
CRd(R2)d(R3)R1(Y )
3, (14)
CRd(R3)R1(Y )Tr (R2(T
L)R2(T
L)), (15)
CRd(R2)R1(Y )Tr (R3(T
S )R3(T
S)), (16)
CRd(R3)Tr ({R2(T
L),R2(T
L)}R2(T
L)), (17)
κabc5 =
CRd(R2)Tr ({R3(T
S ),R3(T
S)}R3(T
S)). (18)
The κ1, . . . , κ5 depend on the representations of matter fields through the
dependence on the coefficients CR. For the first generation of the standard
model there are six such representations, summarized in Table 1 of [4]; they
produce six independent constants CR
2. However, one can immediately
verify that κ
4 = 0. This follows from the fact that the symmetric coeffi-
cients dijk of SU(2) vanish for all irreducible representations. In addition,
we take that κabc5 = 0. The argument for this assumption is related to the
invariance of the color sector of the SM under charge conjugation. Although
apparently in Table 1 from [4] one has only the fundamental representa-
tion 3 of SU(3)C, there are in fact both 3 and 3̄ representations with the
same weights, C3 = C3̄. In the Lagrangian this corresponds to writing each
minimally-coupled quark term as a half of the sum of the original and the
charge-conjugated terms. Since the symmetric coefficients for the 3 and 3̄
representations satisfy dabc
= −dabc
, we obtain
κabc5 = C3d
= 0. (19)
2We assume that CR > 0; therefore the six CR’s were denoted by
, i = 1, ..., 6, in
[3, 6].
The title 6
-0.3 -0.2 -0.1
0 0.1
ΓΓΓ -0.2
-0.3 -0.2 -0.1 0 0.1
Figure 1: (a) The three-dimensional simplex that bounds possi-
ble values for the coupling constants Kγγγ , KZγγ and KZgg at the
MZ scale. The vertices of the simplex are: (−0.184,−0.333, 0.054),
(−0.027,−0.340,−0.108), (0.129,−0.254, 0.217), (−0.576, 0.010,−0.108),
(−0.497,−0.133, 0.054), and (−0.419, 0.095, 0.217). (b) The allowed region
for KZγγ and Kγγγ at theMZ scale, projected from the simplex given in Fig
(a). The vertices of the polygon are: (−0.333, −0.184), (−0.340, −0.027),
(−0.254, 0.129), (0.095, −0.419), (0.0095, −0.576), and (−0.133, −0.497).
We are left only with three nonvanishing couplings, κ1, κ2, and κ3, depend-
ing on six constants C1, . . . , C6:
κ1 = −C1 −
κ2 = −
C6 ; κ3 =
C5 . (20)
There are three relations among Ci’s:
= 2C1 + C2 +
C5 + C6 ,
= C2 + 3C5 + C6 ;
= C3 + C4 + 2C5 , (21)
in effect representing three consistency conditions imposed on (8) in a way
to match the SM action at zeroth order in θ. See detailes in [6].
Fig.(1) shows the three-dimensional simplex that bounds allowed values
for the dimensionless coupling constants Kγγγ , KZγγ and KZgg. For any
choosen point within the simplex in Fig.(1) the remaining coupling con-
stants KZZγ, KZZZ, KWWγ, KWWZ and Kγgg are uniquely fixed by the
NCSM [6, 4]. This is true for any combination of three coupling constants.
The title 7
Our total classical action reads
Scl = SSM +
Sθi = g
′3κ1θ
fµνfρσf
ρσ − fµρfνσf
+ g′g2κ2θ
ρσi − fµρB
ρσi + c.p.
+ g′g2Sκ3θ
ρσa − fµρG
ρσa + c.p.
. (22)
The term Sθ1 in (22) is one-loop renormalizable to linear order in θ [9] since
the one-loop correction to the Sθ1 is of the second order in θ. We need to
investigate only the renormalizability of the remaining Sθ2 and S
3 parts of
the action (22).
To realize the one-loop renormalization of the gauge part action (22), we
apply, as before [9, 10], the background field method [15, 16]. As we have
already explained the details of the method in [12], here we only discuss
the points needed for this computation. The main contribution to the func-
tional integral is given by the Gaussian integral. However, technically, this
is achieved by splitting the vector potential into the classical-background
plus the quantum-fluctuation parts, that is, φV → φV +ΦV , and by comput-
ing the terms quadratic in the quantum fields. In this way we determine
the second functional derivative of the classical action, which is possible
since our interactions (22) are of the polynomial type. The quantization
is performed by the functional integration over the quantum vector field
ΦV in the saddle-point approximation around the classical (background)
configuration φV .
First, an advantage of the background field method is the guarantee of
covariance, because by doing the path integral the local symmetry of the
quantum field ΦV is fixed, while the gauge symmetry of the background
field φV is manifestly preserved.
Since we are dealing with gauge symmetry, our Lagrangian (22) is sin-
gular owing to its invariance under the gauge group. Therefore, a proper
quantization of (22) requires the presence of the gauge fixing term Sgf [φ],
i.e. the Feynman-Fadeev-Popov ghost appears in the effective action
Γ[φ] = Scl[φ] + Sgf [φ] + Γ
(1)[φ], Sgf [φ] = −
d4x(DµΦ
)2 . (23)
The one-loop effective part Γ(1)[φ] is given by
Γ(1)[φ] =
log detS(2)[φ] =
Tr logS(2)[φ]. (24)
In (24), the S(2)[φ] is the 2nd-functional derivative of the classical action,
with the following structure:
S2 = ✷+N1 +N2 + T2 + T3 + T4 . (25)
The title 8
Here N1, N2 are commutative vertices, while T2, T3, T4 are noncommutative
ones. The indices denote the number of classical fields. The one-loop
effective action computed by using the background field method is
θ,2 =
Tr log
I +✷−1(N1 +N2 + T2 + T3 + T4)
(−1)n+1
−1N1 +✷
−1N2 +✷
−1T2 +✷
−1T3 +✷
As the conventions and the notation are the same as in [10], we only en-
counter and discuss the final results.
The divergent one-loop vertex correction to (22) as a function of the
SW freedom parameter a is [10]
Γdiv =
3(4π)2ǫ
BiµνB
µνi +
GaµνG
3(4π)2ǫ
g′g2κ2(3− a)θ
ρσ − fµρB
3(4π)2ǫ
g′g2Sκ3(3− a)θ
ρσ − fµρG
ρσa .
From (27) it is clear that the expanded gauge action (22) is renormalizable
only for the value a = 3 and, its noncommutative part is finite or free of di-
vergencies, so the noncommutativity parameter θ need not be renormalized.
The results for the bare fields and couplings, are given in [10].
Note that we have also analized the renormalizability properties of the
pure NC SU(N) gauge sector, for vector fields in the adjoint representation
[17]. We have found that this model is also renormalizable for a = 3.
However, to obtain renormalizability, we had to pay a price by necessity
for the renormalization of the noncommutative deformation parameter h.
In this way the parameter h and/or the scale of noncommutativity ΛNC
become running quantities, dependent on energy [17].
In addition, it was shown that the one-loop contributions to the U(1)
gauge-field part of the noncommutative gauge theories in the enveloping-
algebra formalism are renormalizable at first order in θ even if the scalar
matter, with and without spontaneous symmetry breaking, contributions
are taken into account [18]. There is reasonable hope that the same con-
clusion should hold for SU(N), but the computations are expected to be
extremely involving. Nevertheless, the results [18] further strengthen the
philosophy which is embraced in our latest papers [10, 17].
From the action (22) we extract the triple-gauge boson terms which are
not present in the commutative SM Lagrangian. In terms of the physical
fields A, W±, Z, and G they are
Lθγγγ =
sin 2θW Kγγγθ
ρτAµν (aAµνAρτ − 4AµρAντ ) ,
Kγγγ =
gg′(κ1 + 3κ2); (28)
The title 9
LθZγγ =
sin 2θW KZγγ θ
× [2Zµν (2AµρAντ − aAµνAρτ ) + 8ZµρA
µνAντ − aZρτAµνA
µν ] ,
KZγγ =
− 2g2
; (29)
where Aµν ≡ ∂µAν − ∂νAµ, etc. The structure of the other interactions
such as ZZγ, WWZ, ZZZ, Zgg, and γgg is given in [4, 6].
Next we focus on the branching ratio of the Z → γγ decay in the renor-
malizable model. Note that each term from the θ-expanded action (22),
(28) and (29) is manifestly invariant under the ordinary gauge transforma-
tions. The gauge-invariant amplitude AθZ→γγ for the Z(k1) → γ(k2) γ(k3)
decay in the momentum space reads
AθZ→γγ = −2e sin 2θWKZγγΘ
3 (a; k1,−k2,−k3)ǫµ(k1)ǫν(k2)ǫρ(k3). (30)
The tensor Θ
3 (a; k1, k2, k3) is given by
3 (a; k1, k2, k3) = − (k1θk2) (31)
× [(k1 − k2)
ρgµν + (k2 − k3)
µgνρ + (k3 − k1)
νgρµ]
− θµν [k
1 (k2k3)− k
2 (k1k3)]
− θνρ [k
2 (k3k1)− k
3 (k2k1)]
− θρµ [kν3 (k1k2)− k
1 (k3k2)]
+ (θk2)
gνρ k23 − k
+ (θk3)
gνρ k22 − k
+ (θk3)
gµρ k21 − k
+ (θk1)
gµρ k23 − k
+ (θk1)
gµν k22 − k
+ (θk2)
gµν k21 − k
+ θµα(ak1 + k2 + k3)α [g
νρ (k3k2)− k
+ θνα(k1 + ak2 + k3)α [g
µρ (k3k1)− k
+ θρα(k1 + k2 + ak3)α [g
µν (k2k1)− k
1 ] ,
where the 4-momenta k1, k2, k3 are taken to be incoming, satisfying the
momentum conservation (k1+ k2+ k3 = 0). In (31) the freedom parameter
a appears symmetric in physical gauge bosons which enter the interaction
point, as one would expect. The amplitude (30), for a = 3, with the Z
boson at rest gives the total rate for the Z → γγ decay:
ΓZ→γγ =
sin2 2θWK
~E2θ +
~B2θ ), (32)
where ~Eθ = {θ
01, θ02, θ03} and ~Bθ = {θ
23, θ31, θ12} are dimensionless coef-
ficients of order one, representing the time-space and space-space noncom-
mutativity, respectively. For the Z boson at rest, polarized in the direction
The title 10
of the third axis, we obtain the following polarized partial width:
Γ3Z→γγ =
sin2 2θWK
~E2θ +
~B2θ + 42
(θ03)2 + (θ12)2
. (33)
In order to estimate the scale of noncommutativity ΛNC from ΓZ→γγ,we
consider new experimental possibilities at LHC. According to the CMS
Physics Technical Design Report [19], around 107 Z → e+e− events are
expected to be recorded with 10 fb−1 of the data. From this one can
estimate the expected number of Z → γγ events per 10 fb−1. Assuming
that BR(Z → γγ) ∼ 10−8 and using BR(Z → e+e−) = 3 × 10−2, we may
expect to have ∼ 3 events of Z → γγ with 10 fb−1. Now the question
is: What would be the background from Z → e+e− when the electron
radiates a very high-energy bremsstrahlung photon in the beam pipe or
in the first layer(s) of the Pixel Detector and is thus lost for the tracker
reconstruction? In that case, the electron would not be reconstructed and
would be misidentified as a photon. The probability of such an event should
be evaluated from the full detector simulation. According to the CMS
note [20] which studies the Z → e+e− background for Higgs → γγ, the
probability to misidentify the electron as a photon is huge (see Fig. 3 in [20])
but the situation can be improved by applying more stringent selections to
the photon candidate when searching for Z → γγ events [21]. However,
the irreducible di-photon background (Fig. 3 in [20]) might also kill the
signal. In that case, one can only set the upper limits to the scale of
noncommutativity from the Z → γγ rate.
In accord with the analysis of the LHC experimental expectations [19,
20, 21] it is bona fide reasonable to assume that the lower bound for the
branching ratio is BR(Z → γγ)
∼ 10−8. Next, choosing the lower central
value of |KZγγ | = 0.05, from the figures and the Table in [6], we find that
the upper bound to the scale of noncommutativity is ΛNC
∼ 1.0 TeV for
~E2θ +
~B2θ ≃ 1. The obtained bound is strongly supported in [18].
Clearly, the measurement of the Z → γγ decay branching ratio would
fix the quantity |KZγγ/Λ
NC|, while the inclusion of other triple gauge boson
interactions through 2 → 2 scattering experiments [14] would sufficiently
reduce the available parameter space of our model by more precisely de-
termining the relations among the couplings Kγγγ , KZγγ , KZZγ, KZZZ ,
KWWγ, and KWWZ. Next, we summarize our results and compare with
those obtained previously.
The first Z → γγ calculation [22] was performed within a different
model which has different symmetries in comparison with ours and, because
of the absence of the SW map, the model does not possess the commutative
gauge invariance. Also, the Z → γγ rate obtained in [22] by imposing the
unitarity of the theory in the usual manner, θ0i = 0, [23, 24], vanishes 3.
The partial width for the same process was obtained in [6] in the frame-
work of similar theories, which, however, were not renormalizable. The
3The condition of unitarity can be covariantly generalized to θµνθ
µν = 2( ~B2θ −
~E2θ) > 0
[25].
The title 11
present results for the partial widths ΓZ→γγ and Γ
Z→γγ are about three
times larger than those in [6] and consistently symmetric with respect to
time-space and space-space noncommutativity. In the polarized rate (33)
the third components ((θ03)2 + (θ12)2) are enhanced relative to the other
two components by a large factor, as expected. Also, the rate (33) is en-
hanced by a factor of ∼ 3 with respect to the total rate (32). The upper
limit to the scale of noncommutativity ΛNC
∼ 1 TeV is significantly higher
than in [6]. This bound is now firmer owing to the regular behavior of
the triple gauge boson interactions (28-29) with respect to the one-loop
renormalizability of the NCSM gauge sector.
After 10 years of the LHC running the integrated luminosity is expected
to reach ∼ 1000 fb−1, [20]. This means that for the assumed BR(Z →
γγ) ∼ 10−8 we should have ∼ 300 events of Z → γγ, that is we should
be well above the background. On the other hand, this result can also be
understood as ∼ 3 events with the BR(Z → γγ) ∼ 10−10, which lifts the
scale of noncommutativity up by a factor of ∼ 3. Therefore, with a more
stringent selection of photon candidates and if the irreducible di-photon
contamination becomes controllable, the Z → γγ decay will become a clean
signature of space-time noncommutativity in LHC experiments.
Finally, the results of [17,18], while strongly supporting this computa-
tions, might also hint at the existence of new interaction symmetry of the
noncommutative gauge sector. Such new symmetry could be a responsible
for the renormalizability of the noncommutative matter sector including
fermions and, next, for the main goal, i.e. in general, the physical realiza-
tion of the Lorentz invariance breaking at very high energies, respectively.
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