A Construction of an Effective Hamiltonian from Feynman Diagrams: Application to the LightFront Yukawa Model
Yuki Yamamoto
0
1
0
Department of Physics, Kyushu University
, Fukuoka 8128581,
Japan
1
A Construction of an Effective Hamiltonian from Feynman Diagrams
We study a similarity transformation to construct an effective Hamiltonian systematically, which does not contain particlenumberchanging interactions, by means of the FukudaSawadaTaketaniOkubo method. We show that such a Hamiltonian can be constructed from Feynman diagrams and give rules for constructing it in the lightfront Yukawa model. We prove that it is renormalized by the familiar covariant perturbative renormalization procedure. It is very advantageous that the effective Hamiltonian can be obtained from our rules immediately. We also numerically diagonalize it to second order in the coupling constant as an exercise.

transformations. One is a transformation in momentum space, which is equivalent
to integrating out states that exchange energies greater than some energy cutoff,
proposed by G:lazek and Wilson, 3) and independently by Wegner. 4) It is
interesting that this transformation gives the nonperturbative low energy physics and a
logarithmic confining potential in LF QCD, 5) although it is difficult to obtain the
effective Hamiltonian, even at lowest order in the coupling constant. The details of
this method and applications are discussed in Ref. 6), and recent progress is reported
in Ref. 7).
The second similarity transformation is a transformation in the particle number
space, which reduces the Hamiltonian to one which has no particlenumberchanging
interactions, so that the transformed one can be solved easily. This transformation
was proposed by Harada and Okazaki. 8) It can avoid the problem of sectordependent
counterterms, because its origin is the general property of the Hamiltonian that it has
particlenumberchanging interactions and its eigenstates need an infinite number of
particles. However, applying it in actual calculations is complicated and tedious.
Although their method was considered in the LF field theory, it is not new in the ET
context and has been used for obtaining the TMO potential of nuclei. 9) We call it the
FukudaSawadaTaketaniOkubo (FSTO) method. 10), 11) This method provide an
easier method for constructing the effective Hamiltonian and seems to be promising
in the LF framework. However, it lacks manifest Lorentz covariance, and therefore it
is difficult to tell what sort of divergences the effective Hamiltonian has before doing
actual calculations. This makes the renormalization procedure more complicated
than that in usual covariant perturbation theory. It is important to determine how
the divergences emerge in the FSTO framework.
The purpose of this paper is to show that the effective Hamiltonian constructed
with the FSTO method can be immediately obtained from Feynman diagrams, as the
Smatrix element in covariant perturbation theory, and that one can use the usual
renormalization procedure to renormalize it. This makes the construction of the
effective Hamiltonian systematic and simpler. In particular, it allows us to perform
higherorder calculations. 12)
Our strategy is the following. First, we show that the FSTO effective
Hamiltonian is a sum of the auxiliary operator G and its products to fourth order in the
interaction part of the Hamiltonian. The advantages of using G are that it is
constructed from the same Feynman diagrams as the Smatrix elements, and that it has
no particlenumberchanging interaction, so that one can easily calculate the normal
ordering of their products. We give the rules for constructing G from Feynman
diagrams. They are slightly different from the familiar Feynman rules in the covariant
perturbation theory. This set of rules is one of the main results of the present paper.
Our method is more convenient and powerful for constructing the effective
Hamiltonian and investigating its renormalization than other similarity methods. Then,
we find that there are three types of UV divergence. One is the familiar loop
divergence, which can be renormalized by the usual renormalization procedure. The
second comes from the difference between our construction rules and those of
Feynman. We can make this second divergence harmless by using an ambiguity. These
two divergences emerge only in G. The last divergence emerges from the products of
the renormalized operators G. We show that this divergence works as a box
counterterm, 13) which is needed to cancel the cutoff dependence of the eigenvalue, in
diagonalizing the effective Hamiltonian.
This paper is organized as follows. In 2, we briefly review the FSTO method. In
3, we show that the effective Hamiltonian constructed from the FSTO method can
be written in terms of the auxiliary operators F or G. (F is written in terms of G.)
We consider the LF Yukawa model, 13) whose Lagrangian and Hamiltonian are given
in Appendix A, as a concrete example, and give the rules for the construction of G
from Feynman diagrams and the renormalization of it and the effective Hamiltonian.
In 4, we summarize and discuss the validity of our method. In Appendix B, as an
exercise, we calculate the eigenvalue of the ground states of the effective Hamiltonian
to second order in the coupling constant in the LF Yukawa model. In Appendix C,
we explain an ambiguity of the energy integrations.
2. Review of the FSTO method
This section briefly reviews the FSTO method, partly following Ref. 11). The
FSTO method is to reduce a Hamiltonian to blockdiagonal form using a similarity
transformation. We can obtain the effective Hamiltonian for the subspace of the
Fock space without loss of necessary information.
We wish to solve the Schrodinger equation
for the secondquantized Hamiltonian H, which consists of the free part H0 and the
interaction part H :
The eigenstate  can be expanded in terms of the complete set of eigenstates of
H0. To divide the Fock space into two, we introduce a projection operator that
commutes with H0:
H = H0 + H .
where both 1 and 2 are states written in terms of the eigenstates of H0. 1
is a state in the target Fock space, which is at our disposal. For our purpose, we
choose to be an operator that selects states with definite particle number. In
matrix notation, and  are written as
and we express an arbitrary operator O as
O =
= U
1 is completely decoupled from 2 , so that we can concentrate on only the
equation for Heff and 1 in the subspace selected by :
We assume a form for the similarity (unitary) transformation operator as
U =
It is convenient to use J ,
(1 + AA)1/2
A(1 + AA)1/2
A(1 + AA)1/2
(1 + AA)1/2
J =
instead of A. In order for the offdiagonal parts in Eq. (2.6) to become zero, J must
satisfy
(1 ) H J + [H0, J ] J H J
where the brackets are defined by
O O + (1 )O(1 ) =
Using Eqs. (2.6), (2.9), and (2.11), we formally obtain the effective Hamiltonian,
Heff = J J 1/2J HJ J J 1/2,
= (1 ) H (t)J (t) J (t)H (t)J (t) ,
where J (0) is identified with J . H (t) and J (t) are defined in this picture as
H (t) eiH0tH eiH0tet ,
J (t) eiH0tJ eiH0tet ,
where et is an adiabatic factor with 0+. We hereafter omit it in order to keep
equations simple. Provided that we set the initial condition as
J () = 1,
J can be found order by order. Alternatively, if we solve
= H (t)V (t) V (t)H (t)V (t),
under the initial condition
J is given by
V () = 1,
It is easily found that the solution of Eq. (2.17) is given by
V (t) = U (t)U (t) 1
where U (t) is the usual time evolution operator,
U111(t) and U221(t) are the inverse operators of U11(t) and U22(t), respectively. J is
explicitly obtained from Eqs. (2.19) and (2.20), and then we expand it in terms of
increasing order in H to obtain the effective Hamiltonian perturbatively:
J =
U21(t)U111(t)
U (t) = T exp
i
U21(t)U111(t)
= 1 +
where n indicates the order of H . The resultant effective Hamiltonian becomes
Heff =
H0 + H + H J1 + H J2
1
+H J3 + 4 [J1J1, H J1] + h.c. + O(H5 ),
to fourth order. It is known that this can produce the TMO potential in the
symmetrical pseudoscalar pion theory with pseudovector coupling, which describes the
properties of the deuteron very well.
Although Eq. (2.23) is general and can be calculated very straightforwardly, the
calculation is absurdly tedious, because J is not a timeordered operator but a
product of such operators, and possesses particlenumberchanging interactions. In order
to renormalize the theory perturbatively, we need to calculate the counterterms, but
it is not clear what sort of the divergences emerge in the effective Hamiltonian before
doing actual calculations. Neither is it obvious whether it is renormalizable even in
the renormalizable theory.
3. Construction and renormalization of the effective Hamiltonian
In this section, we describe the construction method of the FSTO effective
Hamiltonian from Feynman diagrams and discuss its renormalization. First, we introduce
the auxiliary operators F and G for convenience, and then show that the effective
Hamiltonian can be written in terms of them. G can be constructed from the same
Feynman diagrams as those for Smatrix elements in the covariant perturbation
theory. We give rules for its construction and show that it is renormalized by the usual
renormalization procedure. Lastly, we show that the effective Hamiltonian has
divergent terms even if G is renormalized. We discuss their role in diagonalizing the
effective Hamiltonian.
3.1. Definitions of F and G
Considering the first few terms in Eq. (2.23), we define the operator
We will show that the effective Hamiltonian can be rewritten in terms of H0 and F .
By using Eqs. (2.19) and (2.20), we write J as
= (1 ) + U (0)U (0) 1.
J =
U21(0)U111(0)
Therefore, F becomes
where we define the operator G as
H (0) exp i
dt H (t)
H (0) +
(i)n
tn1
As we will shortly see, introducing G is crucial for using the the diagrammatic rules.
An advantage of using G is that it is diagonal in the particle number space, so that
we can normalorder products of Gs more easily than those of J s and H . Note
that G is similar to the Smatrix operator, but there is one important difference: the
upper limit of the time integration is 0.
We find that U (t) can be written in terms of G by integrating t from
to t after sandwiching G between eiH0t and eiH0t :
i
= i
dt G(t )
i
dt H (t ) +
= F2(t) + F2(t),
= {H (t) + H (t)J1(t)} + {H (t) + J1(t)H (t)}
(3.10)
(i)n+1
tn1
(i)n+1
Next, we solve the differential equation
which is obtained from Eq. (2.14). It is easily found that
J1J1 = i
dt {F2(t) F2(t)},
By substituting U (0) in Eq. (3.5) into Eq. (3.3), we can write F as
It is important that F depends only on G. Here, we expand F and G in terms of
increasing order in H :
Then, it is convenient to define
G(t) = eiH0tGeiH0t.
= G
F =
G =
Fn =
Gn =
1
[J1J1, H J1] + h.c. = [J1J1, H + H J1] + [H + J1H , J1J1]
0
= i
dt [F2(t) F2(t), F2] + h.c.
dt [F2(t) F2(t), F2] + h.c. + O(H5 ). (3.13)
All interactions are written in terms of Gn through Fn to fourth order in H . To
summarize, the effective Hamiltonian can be easily constructed once Gn is obtained.
3.2. Rules for the construction of G
In this subsection, we give rules for the construction of G from Feynman
diagrams. We do not need to use the oldfashioned perturbation theory. Knowledge
gained from covariant perturbation theory helps us to find them.
From Eq. (3.5), we immediately find that
where S is the familiar Smatrix operator. Comparing order by order, we can relate
the nth order term of S to Gn as
dt Gn(t).
Gn =
(i)n1
(n 1)!
dtn1 T(H (0)H (t1) H (tn1)). (3.17)
It is important to make the difference between Sn and Gn clearer. For this purpose,
it is useful to rewrite them in terms of the Tproduct and compare them order by
order to find the correspondence with Feynman diagrams:
Wicks theorem tells us that Tproducts of the H (t) can be written as sums of
normalordered products of creation and annihilation operators with amplitudes,
which are given by the same Feynman diagrams, acting as the coefficients. The
difference between Sn and Gn arises from the time integrations. It is apparent
that the role of dt is to give the energyconserving delta function 2(Eout Ein)
to each vertex in the Feynman diagrams, where Eout is the outgoing energy from the
vertex and Ein is the incoming energy to it. Because the time integrations end at 0
in Gn, the energy denominators (i)(Eout Ein i )1 appear instead of the delta
functions. The sign of the infinitesimal constant must be taken to be positive to
ensure the convergence of the integrations at t = . The n factors of H (t) become
equivalent and cancel the factor 1/n! in Sn due to n time integrations, while there
are (n 1) time integrations and the factor 1/(n 1)! is canceled in Gn. However,
there is a H (0) which is not integrated, and therefore there are n terms for each
vertex which have (n 1) products of the energy denominators.
Let us consider the LF Yukawa model) as a concrete example. In this model,
the interaction part of the Hamiltonian has the form
H = gH(1) + g2H(2) .
The effect of H(2) , which is the instantaneous interaction, is absorbed into the
fermion propagator, and we hereafter omit it, because we assume that LF diagrams
are equivalent to the covariant one.) We define as the projection operator for the
twobody state that consists of a fermion and an antifermion. Because the power
of g is equal to the number of the scalar fields and ()n = 0 is satisfied for odd m,
the relation
[H (x1+)H (x2+) H (xn+)](m) = 0,
F2 = G2 + O(g4),
F4 = G4 + iG2
dx+G2(x+) + O(g6).
Also, Gn can be written as
where we define
Gn =
Mn b(p1)b(p2)(2)33(p1 p2)
Mn d(p1)d(p2)(2)33(p1 p2)
(2)33(p1 + l1 p2 l2),
p
3(p q) (p+ q+)2(p q).
) The Lagrangian and Hamiltonian in this model are given in Appendix A.
) This point is explained in the last paragraph of Appendix A.
We hereafter call p the energy and (p+, p) the threemomentum. The creation
and annihilation operators of the fermion, b(p) and b(p), and those of the
antifermion, d(p) and d(p), satisfy the anticommutation relations
i
i
i
i
is assigned to each vertex, where pout (pin) is the outgoing (incoming)
fourmomentum from (to) the vertex. The energy denominator corresponds to the
energyconserving delta function in the Feynman rules.
An independent energy p is assigned for each propagator and integrate it. We
do not impose the conservation of the energies.
A total energy difference (p1 + l1 p2 l) is assigned as an overall factor.
2
It is obvious that the first two differences come from the domain of the time
integrations. The origin of the last one is that there are n terms that have different products
of the energy denominators, as mentioned above. For example, in the case of the
diagram Fig. 1(a), we consider the energy denominators obtained from Eq. (3.17)
with factors of i:
(i)3i4
i
l1 + k2 q1
i
q2 k2 l2
i
k1 q2 p2
p1 + q1 k1
q2 k2 l2
k1 q2 p2
p1 + q1 k1
l1 + k2 q1
k1 q2 p2
p1 + q1 k1
l1 + k2 q1
q2 k2 l2
i
i
i
i
i
i
Finally, Vn becomes
Vn = (p1 + l1 p2 l2)
(iAn),
i
p1 k1 q1 i
l1 k2 + q2 i
i
dk1
dk2
dq1
dq2
vertices (ig5). In general, the total factor of i is (i)V 1iV , where V is the
number of vertices in the diagram. Since the sum in Eq. (3.27) is the combination
of removing one from four denominators, we can make it a product of all the energy
denominators:
= (p1 + l1 p2 l2)
i
p1 + q1 k1
i
q2 k2 l2
i
k1 q2 p2
l1 + k2 q1
i
This factor corresponds to the elimination of the total energyconserving delta
function.
where An is the usual amplitude constructed from the Feynman diagrams, but each
energy of the propagators is an independent variable, which is integrated outside of
An. Therefore, An is constructed from the propagators SF (k) and F (q) with each
independent energy, the vertex g5, the external fermion line u(p1, 1), u(p2, 2),
v(l1, 1), v(l2, 2), and an integration of the fourmomentum for each loop. The
threemomenta are conserved, but energies are not.
Although each propagator includes both the loop and independent energy, it is
easy to split them. For example, in the box diagram Fig. 1(a), the corresponding
energy denominators are
i
p1 + q1 k1 i
i
l1 + k2 q1 i
i
q2 k2 l2 i
i
k1 q2 p2 i
If we consider q2 as the loop momentum, by shifting the momentum as
k1 k1 + q2, q1 q1 + q2 and k2 k2 + q2,
the denominators become
i
p1 + q1 k1 i
i
k1 p2 i
i
l1 + k2 q1 i
i
k2 l2 i
Fig. 1. An example of the oneloop box diagrams. In (a), energies are assigned to each propagator
individually. In (b), energies are assigned to each propagator but the loop fourmomentum q2
is specified. All the threemomenta are conserved in both cases. A continuous line corresponds
to a fermion propagator, and a broken line correspond to a scalar propagator.
whose energies are assigned in Fig. 1(b). Because the loop momenta are not restricted
by the usual conservation law of fourmomenta in the covariant perturbation theory,
it is apparent that the energy denominators do not depend on them.
3.3. Renormalization of G
As mentioned in 3.2, An corresponds to the usual amplitude. What is different
from the amplitude is that the energies of the propagators in An are independent of
each other.
Even if the energies are not conserved, we can renormalize An with the usual
prescription in the covariant perturbation theory, because UV divergences from the
integrations of the loop momenta are local. They emerge as the coefficients of the
polynomial of the other momenta. If the zerothorder term in these expansions
includes a divergence, it is renormalized by shifting masses or a coupling constant.
The other divergences are logarithmic and depend on momenta. They must be
renormalized by the field renormalization.
After renormalizing An, we must consider the energy integrations. It is not clear
whether the energy integrations are finite, because the energies are not conserved.
First, we show that a new divergence arises when we integrate the energies of the
renormalized An with the energy denominators. We discuss the field renormalization
and show that an ambiguity which may cancel the abovementioned divergence comes
from the energy integrations.
3.3.1. Divergences in G
and the threemomentum conservation law
for the fermion line, we find that the oneparticle states conserve the energies:
p1+ = p2+,
p1 = p2
p1 =
p12 + m2
p22 + m2
= p2.
The same conservation law is applied to the antifermion line. It is obvious from
Eq. (3.14) that G multiplied by the total energyconserving delta function is S.
Therefore the oneparticle states of G are equivalent to those of S. If we renormalize
the selfenergy part under the physical renormalization condition, Mn and Mn in
Eq. (3.22) vanish. An example of order g2 is considered in Appendix B.
In multiparticle states, even though
the external lines satisfy the onshell p1 p2
condition and the threemomentum
conservation law, one cannot say that the
total energy is conserved. We will show l1 l2
that such divergences disappear even in
the multiparticle states, except for the
case that the outgoing external fermion Fig. 2. A diagram of the oneparticle state.
line has a selfenergy part. The blob is a connected diagram.
First, we consider the oneparticle
irreducible part (k1, k2, ) of the renormalized An, where k1, k2, are the
fourmomenta of the propagators, except for the loop momentum, and satisfy the
threemomentum conservation law. Since (k1, k2, ) is covariant, the analysis for large
fourmomentum ki , which is used in the operator product expansion, is valid when
we investigate the asymptotic behavior for large ki . 14) Therefore, after realizing
that the loop momenta associated with the ki are as large as ki , we regard ki as
larger than ki+ and ki. By power counting, we find that the leading contribution
comes from the selfenergy part. For the scalar selfenergy part, the leading behavior
of (k1, k2, ) is
(k1, k2, )scalar selfenergy
cij(ki+ kj + ki kj+ ),
and for the fermion selfenergy,
(k1, k2, )fermion selfenergy
where cij and ci may depend on logarithmic factor of ki. It is important that both
are proportional to ki . If we multiply (k1, k2, ) by the energy denominators and
Fig. 3. (a) includes a scalar selfenergy in an internal scalar line. (b) (d) include a fermion
selfenergy part in an internal, incomingexternal and outgoingexternal fermion lines, respectively.
In all cases, the threemomenta are conserved. The blobs are some oneparticle irreducible
graphs.
integrate over the energies, it seems that the energy integration is logarithmically
divergent. However, such a term cannot exist when the energies are conserved; such
a divergence is not possible in S, so it must be proportional to the total energy
difference. For the scalar selfenergy in the internal line, we regard the blob in
Fig. 3(a) as (k1, k2, )scalar selfenergy. The contribution from it is
dk2 i
2 q2 k1 k2 i (k1, k2, )scalar selfenergy
i
q2 q1 i {i(q1, q2) + iq+(q2 q1) log(q+ )
+(q2 q1)(finite terms)},
where (q1, q2) becomes the usual scalar selfenergy if we replace the energy
denominator with the energy delta function, and is the UV cutoff of the energy. The
second and third terms are proportional to (q2 q1) for the reason stated above. q+
and q+ are proper longitudinal momenta. The divergence only arises in the second
term, because Eq. (3.38) is regarded as a Taylor expansion in (q2 q1). The first
term is the ordinary renormalized finite term. Although the second term is
divergent, it is found that it is logarithmically divergent at most by Lorentz covariance
and power counting. The third term converges, because the second term is
logarithmically divergent, and differentiation by the energy decreases the power of the
divergence by 1.
Although Eq. (3.38) includes the divergence, it vanishes when it is inserted in the
internal scalar line. Let us consider scalar propagators and energy denominators in
vertices around the blob in Fig. 3(a). It is crucial that the second term in Eq. (3.38)
dq1
dq2
i
2 k4 k3 q2 i
i
F (q2)
i
But the threemomenta are conserved (q1+ = q2+, q1 = q2), and thus it is obvious
that Eq. (3.39) vanishes.
The fermion selfenergy is logarithmically divergent, like Eq. (3.38). For the
internal fermion line in Fig. 3(b), the contribution from the divergent part is
dk2
dk3
i
2 k4 + q2 k3 i
i
k3 k2 i
SF (k2) k2 k1 q1 i
i
where k+ is a proper longitudinal momentum. Unlike F (q), SF (k) has a term
which does not depend on the energy but is proportional to +:
SF (k) =
Fortunately, + has the property that (+)2 = 0, so that the first term has no effect
in Eq. (3.40). The second term behaves in the same way as in the scalar case, and
hence Eq. (3.40) vanishes.
Note that the above discussed mechanism only works for internal lines. The
diagrams which have selfenergy in the external fermion line may have the same
divergence. We consider the incoming external fermion line as in Fig. 3(c). The
contribution from the divergent part becomes
dk1
2 q1 + k2 k1 i
i
k1 p2 i
i
i+(k1 p2) log(k+) u(p2, 2)
= SF (q1 + k2)(p2+) log(k+)u(p2, 2),
) We have used
dq i
2 F (q) (q k) i =
dq i i
2 2q+(q q2q++ 2 + iq +) (q k) i
where k+ = q+ and k = q.
= u(p1, 1)SF (k2 + q1)(p1+) log(k+).
Since the outgoing line satisfies p1+ > 0, this does not vanish and is logarithmically
divergent.
3.3.2. Field renormalization
Fig. 4(a) =
dk
i
(1 Z3) F1(k)
i
k q1 i
(1 Z3) cq1+ +
i
q2 q1 i
F1(q1) ,
where c is an ambiguous constant that depends on the manner of taking the limit
k . More generally, the contribution from n counterterms connected by the
scalar propagators is
where all the threemomenta are conserved. Since the incoming line has positive
longitudinal momentum, p2+ > 0, this vanishes. In the case of the outgoing external
fermion line like in Fig. 3(d), the contribution from the divergent part becomes
i
p1 k1 i
Fig. 4(b) =
dk1
i
i
i
dk2
dkn
dq3
dqn
dq2
= (1 Z3)n sn(q1+)q1+ +
i
qn+1 q1 i
F1(q1) ,
where all the threemomenta are conserved. sn(q+) is defined as
sn(q+) = an(q+) + bn(q+),
Fig. 4. Diagrams that contribute to the renormalization of the scalar field. Here, X represents a
counterterm for the scalar field. (a) has a counterterm for the scalar field between scalar lines.
q1 and q2 are fourmomenta assigned to the external scalar lines, and k is the fourmomentum
for the counterterm. (b) has n counterterms connected by the scalar propagators. In (c), (b)
is inserted between two vertices. (d) is an infinite sum of the diagrams (c). In all cases, the
threemomenta are conserved.
where an and bn are ambiguous constants, because there is an ambiguity in the order
of the energy integrations.) If we insert Eq. (3.46) into an internal scalar line, the
contribution becomes
Fig. 4(c) =
dq1
dqn+1
) See Appendix C.
(1 Z3)n
sn(q1+)q1+ +
F (q1)
k4 k3 qn+1 i
i
i
qn+1 q1 i
F (qn+1)
F1(q1)
i
i
i
dq
(1 Z3)n F (q)
(1 Z3)n F (q)
i
i
q + k2 k1 i
This does not depend on sn(q+) and is equivalent to a free propagator multiplied by
the constant (1 Z3)n. The mechanism responsible for the vanishing of sn(q+) is
the same as that responsible for the vanishing of the logarithmic divergence in the
selfenergy. We take the sum of diagrams as in Fig. 4(d) and obtain the result
F (q)
dq i
Z3 2 k4 k3 q i
F (q)
(1 Z3)n
Fig. 5(a) =
dq1
i
kn+1 qn i
where the threemomenta are conserved. This property is the same as that of the
usual field renormalization.
Similarly, we can treat the fermion field. If we connect n counterterms for the
fermion field with fermion propagators, the contribution becomes
i
= (1 Z2)n fn(k1+)+ +
i
kn+1 k1 i
SF1(k1) ,
dqn
i
i
where all the threemomenta are conserved and Z2 is the field renormalization
constant for the fermion field. The function fn(k+) is defined as
fn(k+) = cn(k+) + dn(k+),
where cn and dn are ambiguous constants, due to a similar ambiguity.
When we insert Fig. 5(a) into the internal fermion line, its contribution is given
Fig. 5(b) =
dk1
dkn+1
i
SF (kn+1)
SF1(k1)
(1 Z2)n fn(k1+)+ +
i
kn+1 k1 i
i
dk
i
i
2 q2 + p2 k i
(1 Z2)nSF (k)
(1 Z2)nSF (k)
i
i
k q1 p1 i
which is a free propagator multiplied by the constant (1 Z2)n and the energy
denominators. The infinite sum of the diagrams becomes
dk
Z2 2 q2 + p2 k i
i
i
k q1 p1 i
i
q1 + p1 k1 i
where k12 = m2 and k1+ > 0. Of course, all the threemomenta are conserved. In this
case, fn(k+) does not affect the incoming external fermion line. The infinite sum
becomes
which is 1/Z2 times the tree external line. k is the momentum of the external
fermion.
i
q1 + p1 k1 i
For the outgoing external fermion line, the contribution is
i
kn+1 q1 p1 i
= (1 Z2)n(cn + 1)u(kn+1, )
+
where kn2+1 = m2 and kn+1 > 0. The infinite sum becomes
(1 Z2)n(cn + 1) u(k, )
i
i
kn+1 q1 p1 i
k q1 p1 i
3.3.3. Summary of the renormalization of G
We have shown that UV divergences from the loop integrations are renormalized
using the usual procedure in the covariant perturbation theory. After renormalizing
An, we carry out the energy integrations. As a result, the new UV divergence and
ambiguous constant cn arise only from the diagrams in which the fermion selfenergy
and its counterterm are inserted into the outgoing external line.
We fix cn so that it can remove this divergence. This is always possible, because
both arise in the outgoing lines and are closely related. Although this is not the only
way to fix cn, we believe that it is natural for these to cancel.
3.4. Divergences in Heff
In 3.3, we showed that G is not divergent after renormalizing it by the usual
prescription in the usual perturbation theory. In this subsection, we show that Heff
is divergent even if G is finite.
From Eqs. (3.13), (3.20), and (3.21), the effective Hamiltonian can be written
i 0
+ dx+[G2(x+) G2(x+), G2] + h.c. + O(g6), (3.58)
4
to order g4. We can renormalize G2 and G4, but new divergences arise from the
product of G2. One of the examples is the product of two onescalar exchange
diagrams in Fig. 6. The corresponding term in G2 is
G2ex = i
[ F (l2 l1)(l2+ l1+) + F (p1 p2)(p2+ p1+)]
where we have integrated the energy of F (q). The third term in the brackets of
Eq. (3.58) is
x(1 x)x (1 x )
+(finite terms),
where is the cutoff of the transverse momentum of the external fermion. Here,
x and x are defined as
x =
x =
Equation (3.60) is logarithmically
divergent as . A Feynman box dia p1
gram, which is finite, as seen from power
counting, consists of the sum of various G2ex =
timeordered diagrams which may be
divergent individually. The above product
is one such diagram, and it is logarith l1
mically divergent.
It is important to recognize that
Heff should be divergent. The reason
for this can be understood from its perturbative diagonalization:
Epert = (H0 + Heff ) +
Fig. 6. The onescalar exchange diagram that
contributes to G2.
where Heff = Heff H0 and  is an eigenstate of H0. Because the logarithmic
divergence in Eq. (3.60) comes from the integral of the threemomentum of the
intermediate state, the divergences of 2i 0 dx+G2(x+)G2 and of 2i G2 0 dx+G2(x+)
have opposite signs, and they cancel each other. In general, using the eigenstates of
H0, we obtain the matrix element
i
 2 G2
1 1
G2 G2 2 p p i +
in which the power of the intermediate energy p is decreased by 1, and thus the
power of the transverse momentum of the intermediate states is decreased by 2.
It is important that although these divergences must be canceled by adding
artificial counterterms in other similarity methods, they automatically arise in the
higherorder terms in our method. As shown in Appendix B, if we diagonalize
the effective Hamiltonian to order g2, the eigenvalue is divergent. In general, such
divergences are mainly related to the box diagrams and arise not only in this case but
also in the case of the TD approximation in (3 + 1) dimensions. 13) Equation (3.63)
shows that such divergences are canceled if we include g4 order terms. In this case,
the cutoff dependence in the eigenvalue is weakened. However, new divergences arise
in diagonalizing the effective Hamiltonian due to new interactions. We expect that
they are canceled by higherorder interactions, because the exact eigenvalue should
not depend on the cutoff, and the similarity transformation does not change the
eigenvalue.
4. Summary and discussion
In this paper, we have shown that the effective Hamiltonian, which is obtained
with the FSTO similarity transformation in the particle number space, can be written
in terms of only G (or F ) to fourth order in H .
Introducing G is crucial for constructing the effective Hamiltonian more easily
than in the traditional methods, especially at higher orders. G has the favorable
property that it is diagonal in the particle number space. Since it also has the same
form as the formula of the Smatrix operator, we can use the Feynman diagrams and
the rules mentioned in 3.2 in the LF Yukawa model. By using knowledge obtained
from covariant perturbation theory, we can avoid complexity in calculating the
effective Hamiltonian and make the renormalization procedure transparent. Note that
our construction rules are slightly different from the usual Feynman rules, because
energies are not conserved at each vertex.
The divergences due to the integrations of the loop momenta are renormalized
by a familiar prescription of covariant perturbation theory. The divergences due to
the energy integrations can be canceled by terms which come from the ambiguity of
the counterterms for the fields.
The mechanism of the cancellation of the extra divergences from the energy
integrations is valid only in the LF field theory. It will be applied not only to the
LF Yukawa model but also to the other LF models. Although we cannot use it in
the ET field theory, we expect that more precise integrations and a treatment of the
limit are necessary for the cancellation of the divergences.
Although G is constructed from Feynman diagrams and renormalized, the
effective Hamiltonian has divergent terms which are written in terms of products of
the renormalized G. It is very important that such terms do not need counterterms,
but act as those which cancel the divergences in diagonalizing the effective
Hamiltonian. Although the exact eigenvalue should not depend on the cutoff, our method
is perturbative, and then it is not possible to treat nonperturbative divergences in
diagonalization and to get the exact eigenvalue. It is important to find the
nonperturbative counterterms for them. When we find the low energy states, our method
is sufficiently useful in the small coupling region to allow us to ignore the cutoff
dependence if the cutoff is much larger than lower eigenvalues.
There is also the problem of a vanishing energy denominator. Here, we only
consider the LF Yukawa model without a massless particle, so that we may avoid
this problem by including the term i in denominators. If there is a massless
particle, as in QCD, we must consider it.
We proved here that the effective Hamiltonian can be written in terms of G to
fourth order by explicit calculation. However, the general proof for higher orders
is lacking. Although we do not know how a general proof might be constructed,
we think that it is likely that this feature persists to all orders. Recently, Hansper
proposed a nonperturbative approach for the FSTO method in LF field theory. 15)
Although he applied it only to the parton distributions, it is very useful to solve the
mesonic bound states in QCD if it is applicable to our method.
As reported in Appendix B, we have shown numerically that in the case that the
coupling constant is sufficiently large, the lowestenergy state has positive binding
energy, but we do not regard it as a bound state, because it depends on the transverse
cutoff . As mentioned above, our method is valid for small coupling, and we expect
that it will be improved in higherorder calculations. We are now extending the
present work to the next order to confirm the cancellation of the cutoff dependence
and to obtain bound states. 12)
It is interesting to note that this method might also be applied to the similarity
transformation in momentum space, because Eq. (3.13) does not depend on the
choice of . If this can be done, it will be easier to obtain the effective Hamiltonian
than in the traditional method.
The author is grateful to K. Harada for helpful discussions and encouragement,
and also thanks A. Okazaki for his useful work.
Acknowledgements
Appendix A
LightFront Yukawa Model
1
{(x), (y)}x+=y+ = (x y)2(x y)+,
2
where (x) is the dynamical part of the fermion field (x):
are the projection operators of the fermion field and are defined as
= 1 0. (A.7)
2
From Eq. (A.3), the familiar Legendre transformation gives the LF Hamiltonian,
where k+ = k = (k0 + k3)/2, k = k+ = (k0 k3)/2 and ki = (k1, k2).
Nonzero elements of the metric tensor are g+ = g+ = g+ = g+ = g11 = g22 = 1.
If k satisfies the onshell condition k2 = m2, k can be written
H0 =
H =
H = P = H0 + H ,
dxd2x 2 ()2 + 1 22 + 2 + m2 ,
1
2 2i
+ g22 () 21i () 21i (T )()
= 0, = 0.
In the interaction picture, they become
H0(x+) =
H (x+) =
dxd2x 2 ()2 + 1 22 + + 2 + m2 ,
1
2 2i
dxd2x ig5 + g22
2i () ( 2i )T()T
SF (k) =
d4x0T +((x)(0))0e ikx =
includes the noncovariant term. In Fig. 7, the contributions of the noncovariant term
in the first two diagrams are naively canceled by the vertex
2
k3+ + q2+ + k3+ q1+ ,
in the last diagram. This comes from the instantaneous interaction in the
Hamiltonian. For example, let us consider the scalar selfenergy in Fig. 8. Using (+)2 = 0,
we can see that the noncovariant contribution from the first diagram,
dq2
i
where all the field operators are defined in this picture. Note that (x) is not the
same as that in Eq. (A.3) but, rather, is a new field made from (x) in this picture,
and the solution of the free Dirac equation: 16)
i
+(g5) k/ + /q2 m + i (g5) 2ik++ ,
i
is canceled by the second one,
dk2
i
i
2 k3 k2 i k2 k1 i
is canceled by the second one,
i
k3 k1 i
Fig. 7. The first two diagrams are examples which include an internal fermion line. The propagators
include the noncovariant term. The last diagram is the contribution from the instantaneous
vertex included in the LF Hamiltonian.
Fig. 8. The scalar selfenergy. The first diagram is covariant, except for the fermion propagators in
the loop, which have a noncovariant term. The second one comes from the instantaneous vertex
in the Hamiltonian.
Fig. 9. The fermion selfenergy. The first diagram is covariant, except for the noncovariant term
in the fermion propagator. The second one comes from the instantaneous vertex.
Therefore, we regard the fermion propagator as the first term in Eq. (A.15) and
effectively covariant, and omit the second term in Eq. (A.15) and the vertex Eq. (A.16)
together. As a result, the scalar and fermion propagator are given by
F (q) =
SF (k) =
Appendix B
Explicit Calculations andNumerical Result
We calculated the ground state energy numerically to second order in g. Of
course, the calculations to this order are not so different from those using other
methods. 17) The advantage of the present formulation becomes apparent at higher
orders. 12) The reason we present the second order calculation here is to demonstrate
some features of our method and to clarify what would be expected at the next order.
From Eqs. (3.13) and (3.20), the LF effective Hamiltonian is
Fig. 10. Feynman diagrams that are necessary to construct the effective Hamiltonian to second
order in g. The first diagram is the free part. The next four diagrams are the selfenergy parts,
and X represents the counterterms for the fermion mass and field. The last two diagrams are
the onescalar exchange part and the fermion annihilation part.
to g2 order. Also, we add the flavor of fermions. It is easy to estimate Eq. (B.1)
from Feynman diagrams by using our rules.
Since possible graphs are the same as those needed in constructing the Smatrix,
we can immediately imagine those which contribute to Eq. (B.1). The Feynman
diagrams associated with the effective Hamiltonian to second order are shown in
Fig. 10.
Terms which should be renormalized are only the fermion selfenergy terms M2
and M2 at this order. Although we do not give the rules for these, they are similar
to those for Vn. The selfenergy terms correspond to the first and the third graphs
in the second line of Fig. 10 and are written
G2self =
(p1 p2)
dk
+ (1 Z2i)(2)SFi1(k) imi(2)
bi(p1)bi(p2)(2)3(p1 p2),
where the m(2) and (1 Z2i)(2) are the masses and field renormalization constants
i
of order g2, respectively, and the onshell condition (p21 = p22 = mi2) for the external
fermion lines is satisfied. Although there is no energyconserving delta function,
G2self =
1
p 2p+ ui(p, )i(2)(p)ui(p, ) bi(p)bi(p),
(p2 = mi2)
G2ex + G2ex = g2
(2)33(p1 + l1 p2 l2)
ui(p1, 1)5ui(p2, 2)vj(l2, 2)5vj(l1, 1)
1 1
(p1 p2)2 2 + (l2 l1)2 2
ui(p1, 1)5vi(l1, 1)vi(l2, 2)5ui(p2, 2)
1 1
(p1 + l1)2 2 + (p2 + l2)2 2
(p1 p2)
p1 k i
k p1 i
G2an + G2an = g2
(2)33(p1 + l1 p2 l2)
which are all second order interactions in the LF effective Hamiltonian. Of course,
since it does not have particlenumberchanging interactions, the eigenstate is a pure
twobody state.
ei(m1/22/2)ij(x, ; 1, 2, m)
]0.6
V
e
[G0.5
y
rg0.4
e
n
e0.3
g
idn0.2
n
iB0.1
]
eV0.25
G
[
yg 0.2
r
e
ne0.15
indg 0.1
n
iB0.05
0.5
]
V
[eG0.4
y
rg0.3
e
n
ge0.2
n
i
d
in0.1
B
Appendix C
Ambiguity in Integrations
In this appendix, we show that there is in general an ambiguity in integrations,
like that appearing in Eq. (3.46), by considering an explicit example.
Let us consider the following integration:
i
i
the residue theorem, I becomes
I =
k1 k2 i
where we keep the radii finite in the second term, because I has various values
depending on the manner in which the limits are taken. For example,
1. 1 before 2 ,
2. 2 before 1 ,
I =
k1 k2 i
I =
k1 k2 i
I =
k1 k2 i
The other limits may yield other constants. Note that such an ambiguity arises from
the terms in which the total dimension of the variables of the integrations is zero.