Abstract
We consider the neutral pion and nucleon fields interacting via the pseudoscalar
Yukawa-type coupling. The method of unitary clothing transformations is used to
handle the so-called clothed particle representation, where the total field Hamiltonian H and the three boost operators in the instant form of relativistic dynamics
take on the same sparse structure in the Hilbert space of hadronic states. In this
approach the mass counterterms are cancelled (at least, partly) by commutators
of the generators of clothing transformations and the field interaction operator.
This allows the pion and nucleon mass shifts to be expressed through the corresponding three-dimensional integrals whose integrands depend on certain covariant
combinations of the relevant three-momenta. The property provides the momentum
independence of mass renormalization. The present results prove to be equivalent
to the results obtained by Feynman techniques.
1
CLOTHED PARTICLE REPRESENTATION IN QUANTUM
FIELD THEORY (QFT): MASS RENORMALIZATION
V.Yu. Korda and A.V. Shebeko
NSC Kharkov Institute of Physics & Technology, Kharkov, Ukraine
Contents
i) Some Recollections
ii) Underlying Formalism: Clothed Particles in QFT
iii) The Clothing Procedure in Action: Elimination
of Mass Counterterms
iv) Comparison with an Explicitly Covariant Calculation: Elimination of Divergences in the S-matrix
v) Some General Links: Introduction of Cutoff Functions in Momentum Space
vi) Concluding Remarks
Partly, what follows is based upon our paper published in Phys. Rev. D
70 , 2004
2
1
Recollections
In: Progr. Part. Nucl. Phys. 44 (2000) 75; Phys. Part. Nucl. 32 (2001) 31
by Shebeko, Shirokov this method has been employed for an approximate treatment of
simplest eigenstates of total field Hamiltonian H:1
– physical vacuum Ω (lowest-energy H eigenstate)
– observable one-particle states
√ 2 |pi2 with the momentum p,
|pi ∈ H eigenvalue Ep = p + m ,
where m mass of a free particle (e.g., fermion), i.e., m here via relativistic
dispersion law vs. m as a pole of full particle propagator.
Miscellaneous self-energy contributions give rise to undesirable divergences. Their removal
requires considerable intellectual efforts associated with a consequent regularization of
the divergent integrals involved.
In this context, note Dubna Report P2-2000-277 by Shirokov, where one-particle energies have been calculated for a nonlocal model of interacting charged and neutral mesons
using stationary perturbation theory and suggested a fresh look at regularization problem
within noncovariant approach.
Korchin, Shebeko, Phys.At.Nucl. 56 (93) and Krüger, Glöckle, Phys.Rev.
C60 (99) applied Okubo’s idea to blockdiagonalize H, viz., for π, ρ,
ω and σ mesons coupled with nucleons via Yukawa-type interactions, as in
the former, and for scalar ”nucleons” and mesons with a simpler coupling, as in
the latter.
This enabled not only to derive ...
Krüger , Glöckle have shown that their expression for second-order nucleon mass shift
coincides with the corresponding expression found by Feynman technique. In particular,
this shift is independent of nucleon momentum.
We continue that study by Shebeko, Shirokov (2001) of the mass renormalization problem in clothed particle representation (CPR) for the operator H.
2
Underlying Formalism
Clothing procedure in question
is aimed at expressing a total Hamiltonian
H ≡ H(a) = H0 (a) + HI (a)
= K0 (ac ) + KI (ac ) ≡ K(ac )
(1)
through ”clothed” particle creation (destruction) operators a†c (ac ) such that
ac (k) Ω = 0, Ha†c (k) Ω = k0 a†c (k) Ω, ∀ k = (k0 , k)
(2)
They obey the same algebra as ”bare” operators a† (a) do.
1
Note also our attempts (see my talks at ISHEPP’02 (Dubna), FB17’03 (Durham) ) to express
S–matrix through interactions between clothed particles, these quasiparticles within approach under
consideration
3
Clothing itself is implemented via
a (k) = W (ac ) ac (k) W † (ac ) , ∀ k
(3)
where unitary transformation (UT)
W (ac ) = W (a) = expR
(4)
with R† = −R removes ’bad’ terms from H. In the context, K0 (ac ) 6= H0 (a) but coincides
with H0 (ac ), viz.,
Z
K0 (ac ) = H0 (ac ) =
k0 a†c (k) ac (k) dk + ...
(5)
√
where k0 = k2 + µ2 with phys. mass µ!
Operator KI (ac ) consists of interactions between clothed particles, responsible for
processes with phys. particles. At last,
K(ac ) = W H(ac )W †
= exp [R(ac )] H(ac )exp [−R(ac )]
(6)
is a constructive tool to evaluate
KI (ac ) = K(ac ) − H0 (ac ) = HI (ac ) + [R, H0 ]
1
+ [R, HI ] + [R, [R, H0 ]] + ...
2
(7)
For instance, at first stage by requiring
{HI }bad = [H0 , R]
(8)
to eliminate ”bad” terms of the g 1 -order from
HI = V + Mren
(9)
V primary interactions between bare particles, Mren = O (g 2 ) are necessary mass counterterms.
After eliminating these bad terms (strictly speaking together with their Hermitian
counterparts to provide unitarity at every stage of our procedure) via W1 = expR1 ,
K(ac ) = K0 (ac ) + Mren (ac ) +
1
[R1 , V ] + [R1 , Mren ]
2
1
+ [R1 , [R1 , V ]] + ...
3
2
Four-operator (g -order) interactions between clothed particles stem from
(10)
1
2
[R1 , V ] (see,
e.g., Korda’s talk at ISHEPP’02 and our contr. by Korda, Canton and me to FB17). Two(2)
(ac ) bringing definition
operator contributions to it can be compensated by species Mren
of the latter.
4
3
Clothing procedure in action. Elimination of mass
counterparts
In the following model, where a spinor (fermion) field ψ interacts with a neutral pseudoscalar meson field φ by means of the Yukawa
bare´
³
´ coupling,³H can be
´ expressed through
³
destruction (creation) operators a (k) a† (k) , b (p, r) b† (p, r) and d (p, r) d† (p, r)
of the meson, the fermion and the antifermion, respectively. Here k and p momenta, r
spin index.2 In fact, we have its free part
Z
Z
†
HF ≡ H0 (a) =
dkωk a (k) a (k)+
dpEp
Xh
i
b† (p, r) b (p, r) + d† (p, r) d (p, r) , (11)
r
and primary interaction
Z
V (a) =
dp0 dpdk
X
Z
F † (p 0 ,r0 ) V k (p 0 ,r0 ; p ,r) F (p,r) a(k)+H.c. ≡
dkF † V k F a(k)+H.c.,
rr0
(12)
where
operator column
F and row F of bare nucleon and antinucleon operators (e.g., F † (p,r) ≡
h
i
b† (p,r) , d (−p,r) ), and we have introduced c-number matrices ,
†
"
V k (p 0 ,r0 ; p ,r) =
V11k (p 0 ,r0 ; p ,r) V12k (p 0 ,r0 ; p ,r)
V21k (p 0 ,r0 ; p ,r) V22k (p 0 ,r0 ; p ,r)
"
×
#
=
ig
3/2
(2π)
q
ū (p 0 ,r0 ) γ5 u (p,r)
ū (p 0 ,r0 ) γ5 v (−p ,r)
0 0
v̄ (−p ,r ) γ5 u (p,r) v̄ (−p 0 ,r0 ) γ5 v (−p ,r)
m
2ωk E Ep
δ (p + k − p0 )
p0
#
.
(13)
Corresponding generator R(ac ) = R − R† that meets
V + [R, HF ] = 0
(14)
repeats operator structure of V linear in field φ , viz.,
Z
dkFc† Rk Fc ac (k),
R=
(15)
where c-number matrices Rk can be written as
h
i
k
k
Ri,j
(p 0 ,r0 ; p ,r) = Vi,j
(p 0 ,r0 ; p ,r) / (−1)i−1 Ep0 − (−1)j−1 Ep − ωk , (i, j = 1, 2) . (16)
Our interest in terms bilinear in meson and fermion operators from
Z
Z
+
Z
+
2
i
h
dk1 dk2 Fc† Rk1 , V k2 Fc ac (k1 )ac (k2 )
[R, V ] =
h
i
dk1 dk2 Fc† Rk1 , V −k2 Fc a†c (k2 )ac (k1 )
dkFc† Rk Fc · Fc† V −k Fc + H.c. .
For brevity, sets a and ac
5
(17)
As mentioned, they may be cancelled by respective counterparts from the operator Mren (ac ) =
Mmes (ac ) + Mf erm (ac ) with
Mmes
and
Mmes (ac ) =
1 2Z
= δµ
dxφ2 (x) .
2
(18)
i
δµ2 Z dk h †
2ac (k) ac (k) + ac (k) ac (−k) + a†c (k) a†c (−k) ,
4
ωk
(19)
Z
Mf erm = δm
and
dxψ̄(x)ψ(x)
(20)
n
Mf erm (ac ) = mδmFc† M Fc = mδm b†c M11 bc + b†c M12 d†c + dc M21 bc + dc M22 d†c
o
(21)
where the matrix M is given by
"
M=
M11 M12
M21 M22
#
δ (p0 − p)
=
Ep
"
δr 0 r
ū (p 0 ,r0 ) v (−p ,r)
v̄ (−p 0 ,r0 ) u (p ,r) −δr0 r
#
(22)
We used standard δm = m0 − m and δµ2 = µ20 − µ2 .
As shown by Shebeko, Shirokov (2001) meson mass shift of the g 2 -order is
(
)
(
)
2g 2 Z dp
p− k
pk
2g 2 Z dp
µ4
δµ =
−
=
1+
, (23)
(2π)3 Ep µ2 + 2p− k µ2 − 2pk
(2π)3 Ep
4 (pk)2 − µ4
2
i.e., it is independent of the meson momentum k. Here we have introduced the 4-vectors
p = (Ep , p), p− = (Ep , −p)and k = (ωk , k). Note a misprint in signs in Eq. (A.20) from
Shebeko, Shirokov (2001).
Now, we will show how the second-order contributions to the fermion mass counterterm
can be cancelled by certain terms of the commutator [R, V ]. In this connection, let us
separate out in Eq. (17) fermionic two-operator contribution,
Z
Z
n
o
1
†
k
k
k †
k
k †
[R, V ]2f erm = dkFc X Fc = dk b†c X11
bc + b†c X12
dc + dc X21
bc + dc X22
dc . (24)
2
With aid of normal ordering of fermion operators one can obtain formulae for evaluation
of c-number matrix elements Xijk
k
k
k
2X11
= R11
V11−k − V12−k R21
+ H.c.,
³
k†
k
k
k
k
k
= R11
V12−k − V12−k R22
+ R21
V11−k − V22−k R21
2X12
= 2X21
´†
,
k
k
k
2X22
= R21
V12−k − V22−k R22
+ H.c..
Now, we are interested in cancellation of the b†c bc and dc d†c terms responsible for the
transitions one fermion → one fermion to get a prescription in determining the fermion
(nucleon) mass renormalization (of course, in the g 2 -order). Doing so, we assume that
Z
mδm(2) M11 +
Z
(2)
k
dkX11
= 0,
k
= 0,
dkX22
mδm M22 +
6
(25)
or in the spinor space,
Z
(p0 − p)
k
δr0 r = − dkX11
(p 0 ,r0 ; p ,r) ,
mδm
Ep
Z
0
(2) δ (p − p)
k
mδm
δr 0 r =
dkX22
(p 0 ,r0 ; p ,r) .
Ep
(2) δ
(26)
After this all we need is to prove that each of these integrals depend on fermion momentum and spin as Cδ (p0 − p) δr0 r /Ep and find constant C. As shown in that ref. to
PRD,
Z
g 2 δ(p0 − p)
k
dkX11
(p0 , r0 ; p, r) = −
δr0 r I(p),
(27)
4(2π)3
Ep
where
Z
I(p) =
dk
Ep−k ωk
(
)
m2 − Ep Ep−k + p(p − k)
m2 + Ep Ep−k + p(p − k)
−
,
Ep − Ep−k − ωk
Ep + Ep−k + ωk
(28)
and using specific transformations,
I(p) = I1 (p) + I2 (p),
with
Z
(
)
dk
1
1
I1 (p) =
pk
−
,
ωk
µ2 − 2pk
µ2 + 2pk
(
)
Z
dq
m2 − pq
m2 + pq
I2 (p) =
+
.
Eq 2 [m2 − pq] − µ2
2 [m2 + pq] − µ2
Thus, mass shift of interest is
δm(2) =
g2
g2
I(p)
=
[I1 (m, 0, 0, 0) + I2 (m, 0, 0, 0)] .
4m(2π)3
4m(2π)3
(29)
k
k
The second relation (26) leads to the same result since X22
= −X11
. The integrals involved
in Eq. (29) can be reduced to the elementary ones.
Remaining crossed b†c d†c and dc bc terms in Eq. (24) are bad having nonvanishing matrix
elements between the vacuum Ω and two-fermion states. It turns out that they are not
covariant (see Appendix) and, unlike meson mass renormalization, are not cancelled with
respective terms of the operator Mf erm (ac ).
4
Comparison with an explicitly covariant calculation. Elimination of divergences in the S-matrix
The considered procedure enables us to remove from Hamiltonian in CPR not only ”bad”
terms. Simultaneously, ”good” two-particle terms are eliminated too being compensated
with corresponding mass counterterms. Along the guideline some ultraviolet divergences
inherent in the conventional form of H cannot appear in the S-matrix. In the context,
let us recall Dyson expansion for S operator,
7
Z∞
S =1−i
−∞
∞
Z∞
1 Z
dt1 HI (t1 ) + (−i)
dt1
dt2 P [HI (t1 ) HI (t2 )] + ...,
2!
2
−∞
(30)
−∞
where, as usually, HI (t) = exp [iHF (a) t] HI (a) exp [−iHF (a) t] is an interaction in Dirac
picture. To be definite, we consider the interacting neutral pion and nucleon fields with the
operator HI (a) = V (a) + Mren (a) (see Eqs.(12) and (21)) and matrix elements hf | S (2) |ii
of the S operator in g 2 -order, sandwiched between initial and final π 0 N states,
|ii = a† (k) b† (p, r) Ω0 , |f i = a† (k0 ) b† (p0 , r0 ) Ω0 .
(31)
We are interested in competition between fermion mass renormalization contribution to
S (2) and the so-called fermion self-energy diagram contribution:
(2)
hf | SSE |ii = −
g2
δ(p0 − p)
2
m
I
(p)
δ (k0 − k) δ (Ep0 + ωk0 − Ep − ωk ) δr0 r ,
F
(2π)3
Ep
Z
IF (p) =
(
d4 q
p (p − q)
1−
2
2
q − µ + i0
m2
or
Z∞
Z
IF (p) =
dq
−∞
×
)
(32)
1
,
(p − q) − m2 + i0
2
(
1
p0 (p0 − q0 ) − p (p − q)
1−
dq0 2
2
q0 − ωq + i0
m2
)
1
.
2
(p0 − q0 ) − Ep−q
+ i0
2
It is pertinent to stress that, unlike the notation p − k = (Ep−k , p − k) adopted in
Appendix, the vector p − q here is the difference of the two four-vectors p and q, i.e.,
p − q = (p0 − q0 , p − q).
The ”forward-scattering” process associated with this diagram would be responsible
for appearance of certain infinity in the πN scattering amplitude hf | T |ii. Following a
(2)
common practice, the divergence should be compensated by the hf | Mf erm (a) |ii piece,
viz., it is required that
(2)
(2)
2πi hf | Mf erm (a) |ii δ (Ef − Ei ) = hf | SSE |ii .
(33)
At this point, one should emphasize that similar well-known steps become unnecessary if
from the beginning we operate with clothed particle representation K(ac ) of Hamiltonian
H(a). This new form of H does not contain ultraviolet divergences and, being constructed
via the sequential unitary transformations, gives new unitarily equivalent forms of the S
operator (see my talks at ISHEPP’02 and FB17). It is important that the approach
enables us to evaluate one and the same S matrix with nonperturbative methods.
In addition, we would like to note that the crossed (nondiagonal) terms in (21) do not
(2)
contribute to hf | Mf erm (a) |ii. In fact,
(2)
(2)
hf | Mf erm (a) |ii = hΩ0 | a (k0 ) b (p0 , r0 ) Mf erm a† (k) b† (p, r) |Ω0 i
(2)
= hΩ0 | b (p0 , r0 ) Mf erm b† (p, r) |Ω0 i δ (k0 − k)
= mδm(2) hΩ0 | b (p0 , r0 ) b† M11 bb† (p, r) |Ω0 i δ (k0 − k)
δ (p0 − p)
= mδm(2)
δr0 r δ (k0 − k) .
Ep
8
Obviously, this observation is related to all g 2n -orders (n = 1, 2, ...).
Now, by taking into account the pole disposition for the propagators involved and
carrying out the q0 -integration, one can get,
(2)
hf | SSE |ii =
πi g 2 δ(p0 − p)
δ (k0 − k) δ (Ep0 + ωk0 − Ep − ωk ) δr0 r
2 (2π)3
Ep
(
)
Z
dq
m2 − Ep Ep−q + p(p − q)
m2 + Ep Ep−q + p(p − q)
×
−
(34).
Ep−q ωq
Ep − Ep−q − ωq
Ep + Ep−q + ωq
The three-dimensional integral in (34) coincides with the integral I (p) defined by Eq.
(28). Hence, one can write
(2)
hf | SSE
πi g 2
δ(p0 − p)
|ii =
I (p)
δ (k0 − k) δ (Ep0 + ωk0 − Ep − ωk ) δr0 r .
3
2 (2π)
Ep
(35)
It follows from (32) and (35) that
IF (p) = −
πi
I (p) ,
2m2
(36)
i.e., we have found another proof of the p-independence of I (p) since IF (p) is the explicitly
covariant quantity. Besides, we have expressed the Feynman one-loop integral IF (p)
through other covariant integrals I1 (p) and I2 (p).
5
Some general links
Let us consider the momentum independence in question from a general point of view,
viz., one-particle matrix elements
h
i
hk0 | S |ki = hk0 | 1 + S (1) + S (2) + ... |ki
to be definite with spinless states |ki = a† (k) |Ω0 i as in case of pion, where, e.g.,
hk0 | S (2) |ki = −2πiδ (ωk0 − ωk ) hk0 | T (2) (ωk ) |ki
with second order T -operator
T (2) (ωk ) = HI (ωk + i0 − HF )−1 HI
To set links with previous discussions it is sufficient to note that on certain conditions
1 0
hk | [R1 , V ] |ki = hk0 | V (ωk + i0 − HF )−1 V |ki
2
(37)
It holds within model discussed if pion mass µ < 2m. In particular, it means that
propagator with intermediate nucleon-antinucleon states in Eq. (37) is not singular for
ωk > µ. Then, according to Shebeko, Shirokov (2001), generator R1 is
Z∞
dtVD (t) e−εt ,
R1 = −i lim
ε−→0+
0
9
and proof of Eq. (37) is trivial. R
Further, let us assume V = dxV (x) with interaction density in D-picture to be
Lorentz scalar
U (Λ) V (x) U −1 (Λ) = V (Λx) .
(38)
By the way, such introduction of interaction density may be inherent not only in a local
field theory!
Exploiting Eq. (38) and representation
−1
(ωk + i0 − HF )
Z∞
dtei(ωk +iε−HF )t
= −i lim
ε−→0+
0
one can show,
hk0 | V (ωk + i0 − HF )−1 V |ki
µ
∞
¶
µ
¶
Z
Z
δ (k0 − k)
1
1
= −i (2π)
lim
dte−εt dρ hΩ0 | a (k) VD
ρ VD − ρ a† (k) |Ω0 i
ε−→0+
ωk
2
2
3
0
Here, as in Shebeko, Shirokov (2001), we are addressing operators a (k) =
meet the covariant commutation rules
h
√
ωk a (k) that
i
a (k) , a† (k) = ωk δ (k0 − k) .
It results in appearance of a typical combination
δ (ωk0 − ωk ) δ (k0 − k)
C (k)
ωk
in the correspondent S-matrix element. It is time to remember relativistic invariance
property
hk 0 | S |ki
hΛk 0 | S |Λki
= √
√
ωk0 ωk
ωk0Λ ωkΛ
Doing so, we arrive to a possible (probably, general) way when finding the momentum
independence of mass shifts within this three-dimensional formalism.
6
Summary
i) We have demonstrated here how the mass shifts in the system of interacting pion
and nucleon fields can be calculated by the use of the clothed particle representation.
The respective mass counterterms are compensated and determined directly ! in the
Hamiltonian.
ii) The procedure described above has an important feature, viz., the mass renormalization is made simultaneously with the construction of a new family of quasipotentials
(Hermitian and energy independent) between the physical particles (the quasiparticles
of the method). Explicit expressions for the quasipotentials can be found in Shebeko,
Shirokov (2001) and Korda’s talk at ISHEPP’02.
10
iii) By using a comparatively simple analytical means, we could show that the threedimensional integrals, which determine the pion and nucleon renormalizations in the second order in the coupling constant g, can be written in terms of the Lorentz invariants
composed of the particle three-momenta. In other words, these integrals are independent
of the particle momentum.
iv) The experience acquired has allowed us, on the one hand, to reproduce the manifestly covariant result by Feynman techniques and, on the other hand, to derive a new
representation for the Feynman integral that corresponds to the fermion self-energy diagram. Of course, here we are dealing with the coincidence of the two divergent quantities:
one of them is determined by the nucleon mass renormalization one-loop integral, while
the other stems from the commutator [R, V ]. We are trying to overcome this drawback
by means of the introduction of the cutoff functions in momentum space. Such functions
have certain properties to do the theory to be satisfied the basic symmetry requirements.
We would like to thank Drs. A Korchin and L. Levchuk for useful discussions.
11