Light-front zero-mode contribution to the Ward Identity

Nuclear Physics B (Proc. Suppl.) 199 (2010) 211–214 www.elsevier.com/locate/npbps

Light-front zero-mode contribution to the Ward Identity. J. H. O. Salesa , A. T. Suzukib a Universidade Estadual de Santa Cruz, Departamento de Matem´ atica km 16 Rodovia Ilh´eus-Itabuna - 45662-000 - Ilh´eus, BA, Brazil. b Instituto de F´ısica Te´orica-Univ. Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II - 01140-070 S˜ ao Paulo, SP, Brazil In a covariant gauge we implicitly assume that the Green’s function propagates information from one point of the space-time to another, so that the Green’s function is responsible for the dynamics of the relativistic particle. In the light front form one would naively expect that this feature would be preserved. In this manner, the fermionic field propagator can be split into a propagating piece and a non-propagating (“contact”) term. Since the latter (“contact”) one does not propagate information, and therefore, supposedly can be discarded with no harm to the field dynamics we wanted to know what would be the impact of dropping it off. To do that, we investigated its role in the Ward identity in the light front. Here we use the terminology Ward identity to identify the limiting case of photon’s zero momentum transfer in the vertex from the more general Ward-Takahashi identity with nonzero momentum transfer.

1. Introduction One of the most important concepts in quantum field theories is the question of renormalizability. In QED (Quantum Electrodynamics) specifically, the electric charge renormalization is guaranteed solely by the renormalization of the photon propagator. This result is a consequence of the so-called Ward-Takahashi identity, demonstrated by J.C.Ward in 1950 [1–3]. The importance of this result can be seen and emphasized in the fact that without the validity of such an identity, there would be no guarantee that the renormalized charge of different fermions (electrons, muons, etc.) would be the same. In other words, without such identity, charges of different particles must have different renormalization constants, a feature not so gratifying nor elegant. Moreover, without the Ward-Takahashi identity, renormalizability would have to be laboriously checked order by order in perturbation theory. Over half a century ago, P.A.M.Dirac [4] discussed the dynamical forms for relativistic particles. In his work, he demonstrated that the front-form (light-cone, light-front) of particle dynamics may have some subtle advantages over the instantaneous form. Since then, many studies and applications in quantum field theory have 0920-5632/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2010.02.031

been considered by many authors. In particular, S.J.Chang and S.K.Ma [5] and S.Brodsky et al [6], who have respectively considered the Feynman rules for quantum electrodynamics (QED) and renormalization of QED in the light-front. In the former work, it has been pointed out that QED presents “an outstanding feature in the new rules” (i.e., Feynman rules) which is the constrained integration variables in mometum space. This constraint clearly limits the integration variables within definite limits or ranges. In the latter work, it is shown that the renormalization program for QED requires the inclusion of the socalled z-graphs. Another situation where the extra instantaneous term can appear is in gauge field theories constrained in the light front gauge. Here the absence or presence of this extra instantaneous term is an open problem depending on the process of gauge fixing [7,8]. Current literature on the subject is rather sparse and as far as we know this question has not been addressed with consistency and conclusively. Moreover we know that light-front dynamics is plagued with singularities of all sorts and because of this the connection between the covariant quantities and light-front quantities cannot be so easily established. If we want to describe

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our theory in terms of the light-front coordinates or variables, we must take care of the boundary conditions that fields must obey. Thus, a simple projection from the covariant quantities to lightfront quantities via coordinate transformations is bound to be troublesome. This can be easily seen in our checking of the QED Ward identity in the light-front. The terminology Ward identity here specifies a particular case of zero momentum transfer by the photon in the vertex function to distinguish it from the more general WardTakahashi identity wherein momentum transfer is not zero. In the light-front coordinates, the fermionic propagator does bear an additional term proportional to γ + (p+ )−1 oftentimes called “contact term” in the literature, which, of course, is conspicuously absent in the covariant propagator. For the external fermion fields, this contact term can be discarded if we project the free propagator onto γ + component because (γ + )2 = 0. This kind of projection facilitates the study of the BetheSalpeter equation in the light-front [9,10]. However, there is no guarantee that such projection is legitimate, preserving all the symmetries of the theory. This term, as we will see, is crucial to the Ward identity in the light-front. The covariant propagating term solely projected onto the lightfront coordinates therefore violates Ward identity, and therefore breaks gauge invariance. Such result is obviously wrong and unwarranted. 2. Light-front Coordinates

=

x⊥

=


1 √ x0 ± x3 2 1ˆ x i + x2ˆj,

so, the scalar product is given by
aµ bµ = a+ b− + a− b+ − a⊥ · b⊥ .

The Dirac’s γ matrices in the light-front are given by: 1 (5) γ ± = √ (γ 0 ± γ 3 ), 2 (6) γ i = γ 1,2 . 3. The Ward Identity

There are several ways to write down the Ward identity for fermions, and one of them is inferred from manipulations of their propagator, namely, S(p). Multiplying by its inverse, we get the identity: S(p)S −1 (p)

= I,

(7) µ

which upon differentiation with respect to p on both sides we get: ∂S −1 (p) ∂S(p) −1 S (p) + S(p) ∂pµ ∂pµ

=

which leads to ∂S −1 (p) ∂S(p) −1 S (p) = −S(p) µ ∂p ∂pµ

0,

(8)

(9)

Finally, multiplying both sides from the right by the propagator itself yields ∂S −1 (p) ∂S(p) = −S(p) S(p) ∂pµ ∂pµ

(10) −1

The light-front is characterized by the nullplane x+ = t + z = 0, x+ is known as the “time” coordinate. All of the coordinates are set regarding this plane, and one has new definitions of the scalar product, for example. The basic relations on the light-front are x±

Using (3), one can write the product p / on the light-front:

(4) p / = p µ γ µ = γ + p − + γ − p + − γ ⊥ · p⊥ .

(1) (2)

(3)

Now, using S(p) = i (p / − m) it follows that / − m). Differentiantits inverse is S −1 (p) = −i (p ing this last expression with respect to pµ we get ∂S −1 (p) = −iγµ , ∂pµ

(11)

which inserted into (10) leads to the differential form of the Ward Identity, namely, ∂S(p) = iS(p)γµ S(p). ∂pµ

(12)

Here it is important to stress that if the propagator were raised to a power as in the analytic regularization scheme, i.e. if it had the form −σ S(p) = i (p / − m) , with σ 6= 1, the identity (12) would not be fulfilled.

J.H.O. Sales, A.T. Suzuki / Nuclear Physics B (Proc. Suppl.) 199 (2010) 211–214

4. Fermion propagator in the Light-Front The covariant propagator for the cle is

spin- 12

S(xµ ) = (iγ µ ∂µ + m) Sb (xµ ),

parti(13)

where Sb (xµ ) stands for the scalar boson propagator: Z i d4 p −ipµ xµ . (14) Sb (xµ ) = 4 p2 − m2 + iε e (2π) Using Eq.(14) in Eq.(13), we have: Z i(/ p + m) −ipµ xµ d4 p e , S(xµ ) = 4 (2π) p2 − m2 + iε

(15)

An important difference between Eq.(14) and Eq.(15) is the presence of the term (/ p + m) in the numerator. This will yield, after integration in p− an instantaneous term in the light-front time x+ . The quantum propagator in the light-front propagates the wave function from x+ = 0 to x+ > 0. It corresponds to the definition of the time ordering in the coordinate x+ . We make this instantaneous piece explicit by adding and subtracting the on-shell mass condition in the numerator of the propagator, so that: p2

+

/+m p − m2 + iε

=

/pon + m γ
+ + , (16) − − 2p p − pon

2p+

(p⊥ )2 + m2 , is the on-shell energy 2p+ in the light-front. The propagator Eq.(15) in the light-front time is therefore: Z dp+ d2 p⊥ dp− S(p− ) S(x+ ) = 4 (2π) where p− on =

×

−i(p− x+ +p+ x− −p⊥ ·x⊥ )

e

.

(17)

Comparing this result with the scalar propagator Eq.(14), the fermion propagator has the additional term known as the instantaneous piece. Taking the Fourier transform of Eq.(17), we get: S(p− ) = i

/pon + m iγ +
. + 2p+ 2p+ p− − p− on

(18)

General Ward-Takahashi identity relates the quantum corrections to this propagator with the

213

vertex function corrections to all orders of perturbation. However, for our purposes, manipulating Eq.(18) will be enough to infer the Ward identity, that is, to relate it to the vertex function with intermediate photon with zero momentum. 5. The Ward Identity on the Light-Front There are two manners to test whether the propagator (18) on the light-front satisfy the Ward identity (12). The simplest and most direct ∂S −1 (p) for each one is to do the differentiation ∂pµ component and put them in Eq.(10). The second manner to test the identity on the light-front is to work out explicitly both sides of Eq.(12) component by component. Whatever manner is employed the results are: ∂S(p) ∂p+ ∂S(p) ∂p− ∂S(p) ∂p1,2

=

iS(p)γ − S(p),

(19)

=

iS(p)γ + S(p),

(20)

=

iS(p)γ 1,2 S(p),

(21)

where p1,2 = p⊥ and γ 1,2 = γ ⊥ are the transversal or perpendicular components. So, we verify that the Ward identity is satisfied. 6. The Ward Identity for the propagator without contact term Here we work on the Ward Identity for the simplified propagator S(p) =

i (p /on + m)
. + 2p p− − p− on

(22)

Differentianting the above equation with respect to p− yields, −i (p /on + m) ∂S(p) =
2 ; + ∂p− 2p p− − p− on

(23)

while the term iS(p)γ + S(p) yields: iS(p)γ + S(p) =

−i (p /on + m)
2 , + 2p p− − p− on

(24)

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J.H.O. Sales, A.T. Suzuki / Nuclear Physics B (Proc. Suppl.) 199 (2010) 211–214

since (γ + )2 = 0. So, for the negative component, the Ward identity is satisfied: ∂S(p) = iS(p)γ + S(p). ∂p−

(25)

For the plus component and the transversal components, however, one does not have the equality between the two sides of Eq.(20) and Eq. (21), so that ∂S(p) ∂p+ ∂S(p) ∂p1,2

6=

iS(p)γ − S(p),

(26)

6=

iS(p)γ 1,2 S(p),

(27)

and the Ward Identity is not fulfilled. 7. Conclusions We have shown here that the Ward identity for the fermionic field in the light-front is preserved to guarantee that the charge renomalization constant depends solely on the photon renormalization constant, as it is expected. However, one important point emerges in our computation, and that is that the Ward identity in the light-front is valid provided the fermionic field propagator bears the relevant “contact” term piece, which is absent in the propagating piece of the covariant propagator and its straightforward projection into light-front variables. We have shown here that the contact term present in the free fermion propagtor may not carry any information, but is nonetheless important in ensuring the correct original symmetries present in the theory. Our computation has demonstrated once again the significance of the light-front zero-mode contribution that the so-called “contact” term bears in it, without which the Ward identity would be violated. Although the zero-mode term does not carry physical information, its non-vanishing contribution nonetheless is crucial to the validity of the Ward identity in the light-front formalism. In other words, “contact” term may not carry information from one space-time point to another in the light front, but contains relevant physical information needed to ensure the Ward identity,

and therefore, for the correct charge renormalization. Of course, our calculation was restricted to the case of photon zero momentum transfer, but it is significant that even in this simple approximation, the instantaneous term plays a prominent role in preserving the Ward identity. Acknowledgments: A.T.S. wishes to thank Capes for travel grant; J.H.O. Sales thanks FAPESP and FAPEMIG for partial financial support and the hospitality of the Institute for Theoretical Physics, UNESP, where part of this work has been performed. REFERENCES 1. J.C.Ward, Physical Review, 77, (1950) 293, 2. J.C.Ward, Physical Review, 78, (1950) 182 3. Y. Takahashi, Nuovo Cimento, 6, ser. 10, (1957) 370 4. P.A.M.Dirac, Rev.Mod.Phys. 21 (1949) 392 5. S.-J.Chang and S-K.Ma, Phys Rev. 180 (1969) 1506 6. S.J.Brodsky et al, Phys.Rev. D8 (1973) 4574 7. A.T.Suzuki and J.H.O.Sales, Nucl. Phys. A 725 (2003) 139 8. A.T.Suzuki and J.H.O.Sales, Mod. Phys. Lett. A 19, 38 (2004) 2831. 9. J. H. O. Sales, T. Frederico, B. V. Carlson, and P. U. Sauer, Phys.Rev. C 63, 064003 (2001) 10. J.H.O.Sales, T. Frederico, B.V. Carlson and P.U. Sauer, Phys. Rev. C C61 (2000) 044003.