Perturbation theory and Feynman rules

Interaction picture
The evolution of quantum system under time-dependent Hamiltonian can be studied in the
framework of interaction picture. If the solution of Schrödinger equation for the full time-dependent
Hamiltonian H(t) is either very difficult or practically impossible. In such cases, approximate
solution can be attempted if H(t) can be splitted in two parts,
H(t) = H0 + V (t) (1)
where H0 is the time independent component whose eigenstates can either be calculated or known.
The V (t) is the time dependent part which, when assumed small, enables us to use the eigenstates
of H0 to approximate the solution of the full Hamiltonian,
∂ ∂
i~ |ψ0 i = H0 |ψ0 i ⇒ i~ |ψ0 i ≈ (H0 + V (t)) |ψ0 i. (2)
∂t ∂t
If H0 is only present, the time evolution of |ψ0 i can be expressed in terms of the unitary time
evolution operator U0 (t, t0 ),
∂
|ψ0 (t)i = U0 (t, t0 ) |ψ0 (t0 )i ⇒
U0 (t, t0 ) = H0 U0 (t, t0 )
i~ (3)
∂t
The |ψ0 i are essentially states in Schrödinger picture. However, because of the time dependent
perturbation, the state |ψ0 i evolves differently. To describe this evolution, we defined interaction
states |ψI i and interaction Hamiltonian HI (t) as,
|ψI i = U0† (t, t0 ) |ψ0 i and HI (t) = U0† (t, t0 ) V (t) U0 (t, t0 ), U0 (t0 , t0 ) = 1 (4)
and the Schrödinger equation for the full Hamiltonian H(t) becomes,
∂ ∂
i~ |ψ0 i = (H0 + V (t)) |ψ0 i ⇒ i~ |ψI i = HI (t) |ψI i. (5)
∂t ∂t
The above equation can be interpreted as Schrödinger equation for |ψI i but with HI as the Hamil-
tonian. This description is known as interaction picture in quantum mechanics. In the spirit of (2),
the time evolution operator in the interaction picture can be defined as,
∂UI
|ψI (t)i = UI (t, t0 ) |ψI (t0 )i = HI UI .
satisfying i~ (6)
∂t
Therefore, U0 and UI are respectively the unitary time evolution operators of |ψ0 i and |ψI i cor-
responding to the Hamiltonian H0 and HI . If U (t, t0 ) be the time evolution operator of the full
Hamiltonian H(t), then it can be shown that
U (t, t0 ) = U0 (t, t0 ) UI (t, t0 ). (7)
The next step is to solve the differential equation (5) subjected to the initial condition UI (t0 , t0 ) = 1.
The differential equation together with initial condition is equivalent to the integral equation,
Z t Z t
d
i~ UI (t1 , t0 ) dt1 = HI (t1 ) UI (t1 , t0 ) dt1
t0 dt1 t0
Z t Z t
i~ d UI (t1 , t0 ) = HI (t1 ) UI (t1 , t0 ) dt1
t0 t0
Z t
t 1
UI (t1 , t0 )|t0 = HI (t1 ) UI (t1 , t0 ) dt1
i~ t0
i t
Z
U (t, t0 ) − UI (t0 , t0 ) = − HI (t1 ) UI (t1 , t0 ) dt1
~ t0
i t
Z
U (t, t0 ) = 1 − HI (t1 ) UI (t1 , t0 ) dt1 . (8)
~ t0

1
The unitary time evolution operator UI (t1 , t0 ) at time t1 can similarly be obtained by using the
above equation and integrating from t0 to t1 ,

i t1
Z
UI (t1 , t0 ) = 1 − HI (t2 ) UI (t2 , t0 ) dt2 (9)
~ t0

Substituting this expression in (8), an approximate solution to second order can be thus be obtained,

i t i t1
Z  Z 
UI (t, t0 ) = 1 − HI (t1 ) 1 − HI (t2 ) UI (t2 , t0 ) dt2 dt1
~ t0 ~ t0
 Z t  2 Z t Z t1
i i
= 1+ − dt1 HI (t1 ) + − dt1 dt2 HI (t1 ) HI (t2 ) UI (t2 , t0 ). (10)
~ t0 ~ t0 t0

Iterating this process, we eventually get

 Z t  2 Z t Z t1
i i
U (t, t0 ) = 1 + − dt1 HI (t1 ) + − dt1 dt2 HI (t1 ) HI (t2 ) +
~ t0 ~ t0 t0
 3 Z t Z t1 Z t2
i
− dt1 dt2 dt3 HI (t1 ) HI (t2 ) HI (t3 ) + · · ·
~ t0 t0 t0
 n X ∞ Z t Z t1 Z tn−1
i
= 1+ − dt1 dt2 · dtn HI (t1 ) HI (t2 ) . . . HI (tn ). (11)
~ n=1 t0 t0 t0

This series is known as Dyson series. One important thing to notice is that the various factors of
HI (t) stand in time order i.e. later on the left t1 > t2 > · · · > tn−1 . Here we note that

Z t Z t1 Z t Z t2
dt1 dt2 = dt2 dt1
t0 t0 t0 t0
Z t Z t1 Z t Z t
1
⇒ dt1 dt2 HI (t1 ) HI (t2 ) = dt1 dt2 T {HI (t1 ) HI (t2 )}. (12)
t0 t0 2! t0 t0

Hence the time ordered expression for UI is,

∞  n Z t
X 1 i
UI (t, t0 ) = 1 + − dt1 dt2 . . . dtn T {HI (t1 ) HI (t2 ) · · · HI (tn )}. (13)
n=1
n! ~ t0

The above series does not necessarily converges for arbitrary HI . But when it does, the Dyson series
can be written as,
i t 0
  Z 
UI (t, t0 ) = T exp − dt HI (t0 ) . (14)
~ t0
Converges or not, quantum field theory based on the above expression agrees remarkably with the
experiments. The connection with this and Mf i can be established through defining S-matrix,

i ∞
  Z 
S ≡ lim UI (t, t0 ) = T exp − dt HI (t) (15)
t→∞ ~ −∞
t →−∞ 0

2
The UI being unitary, the S-matrix is unitary too. Let |ii be the initial state at t = −∞ when
HI = 0 i.e. no HI and |f i be the final state at t = ∞ when again HI = 0. Obviously, both |ii and
|f i are the eigenstates of H0 . The state at time t evolving from |ii due to HI is,

|ψI (t)i = UI (t, t0 ) |ψI (t0 )i = UI (t, t0 → −∞) |ii (16)

The probability of finding the system in |f i of H0 which is different from |ii at time t under the
influence of HI is,
hf |ψI (t)i = hf |UI (t → ∞, t0 → −∞)|ii = hf |S|ii, (17)
where the S-matrix elements are over the vector space of free states. Using the expression (13), the
interacting part of the S-matrix can be identified, know as transition or T matrix,

hf |S|ii ≡ hf |1 + iT |ii = δf i + ihf |T |ii ≡ δf i + iTf i . (18)

The non-kinetic part is Mf i . The perturbative expansion of Tf i provides the picture of interaction
through exchange of particles. For details of how it arises and how Fermi’s Golden rule follows from
S-matrix see Sakurai, Halzen-Martin, Griffiths etc.
Wick’s theorem
In particle physics, for interacting theories we may need to calculate correlation functions such as,

hΩ|T {φ(x1 ) φ(x2 ) · · · φ(xn )|Ωi (19)

where, T {φ(x1 ) φ(x2 )} = φ(x1 ) φ(x2 ) t1 > t2
= φ(x2 ) φ(x1 ) t1 < t2
= θ(t1 − t2 ) φ(x1 ) φ(x2 ) + θ(t2 − t1 ) φ(x1 ) φ(x1 ) (20)

In quantum field theory, the fields are operators and admits normal mode expansion in terms of
creation and annihilation operators,
X 1
ap eipx + a†p e−ipx ≡ φ+ + φ−
 
φ(x) = p (21)
p
2Ep V
⇒ [φ(~x, t), Π(~x0 , t)] = i δ 3 (~x − ~x0 ) (22)
h i
⇒ ap , a†p0 = δ 3 (~ p − p~0 ).

Here φ+ and φ− are respectively the annihilation and creation operators, defined so based on their
action on the vacuum state |0i,

φ+ (x)|0i = 0 and h0|φ− (x) = 0. (23)

Consider the product of two fields for t1 > t2 ,

φ(x1 ) φ(x2 ) = [φ+ (x1 ) + φ− (x1 )] [φ+ (x2 ) + φ− (x2 )]

= φ+ (x1 )φ+ (x2 ) + φ+ (x1 )φ− (x2 ) + φ− (x1 )φ+ (x2 ) + φ− (x1 )φ− (x2 ) (24)
= φ+ (x1 )φ+ (x2 ) + φ− (x1 )φ+ (x2 ) + φ− (x1 )φ− (x2 ) +
φ− (x2 )φ+ (x1 ) + [φ+ (x1 ), φ− (x2 )] (25)

Rewriting (24) as (25) is known as normal ordering and has an interesting consequence, namely, the
annihilation operators are on the right of all the creation operators so that it annihilates vacuum
when acted on it. In short hand notation,

t1 > t2 φ(x1 )φ(x2 ) = : φ(x1 )φ(x2 ) : +[φ+ (x1 ), φ− (x2 )]

φ(x1 )φ(x2 ) = N {φ(x1 )φ(x2 )} + [φ+ (x1 ), φ− (x2 )]
t1 < t2 φ(x2 )φ(x1 ) = : φ(x2 )φ(x1 ) : +[φ+ (x2 ), φ− (x1 )] (26)

3
The normal ordering is denoted by the symbol : : around the field product and the two normal
products above in two cases are the same. We note that, if vacuum expectation value of the above
product (24) is considered then,
h0|φ(x1 ) φ(x2 )|0i = h0|φ+ (x1 )φ− (x2 )|0i ≡ h0|[φ+ (x1 ), φ− (x2 )]|0i. (27)
But the commutator [φ+ (x1 ), φ− (x2 )] is a number after all and, therefore, we get
h0|φ(x1 ) φ(x2 )|0i = h0|[φ+ (x1 ), φ− (x2 )]|0i = [φ+ (x1 ), φ− (x2 )]. (28)
Putting everything together, the expressions in (26) leads to,
t1 > t2 φ(x1 )φ(x2 ) = : φ(x1 )φ(x2 ) : +h0|φ(x1 ) φ(x2 )|0i
t1 < t2 φ(x2 )φ(x1 ) = : φ(x1 )φ(x2 ) : +h0|φ(x2 ) φ(x1 )|0i (29)
T {φ(x1 ) φ(x2 )} = : φ(x1 )φ(x2 ) : +h0|T {φ(x1 ) φ(x2 )}|0i (30)
This is essentially Wick’s theorem – connecting time-ordered product to normal-ordered product.
Actually, Wick’s theorem attempts to reduce arbitrary time-ordered products of creation and anni-
hilation operators to sums of products of pairs of these operators. Generalization can be achieved by
noting that on the r.h.s., we have vacuum expectation value of the time ordered product of two field
operators. This quantity is essentially the Feynman propagator (see Lahiri & Pal for instance),
h0|T {φ(x1 ) φ(x2 )}|0i = i ∆F (x1 − x2 ) ≡ φ(x1 ) φ(x2 ) (31)
For any arbitrary number of fields, using φn as shorthand for φ(xn ),
T {φ1 φ2 · · · φn } = N {φ1 φ2 φ3 · · · φn + φ1 φ2 φ3 · · · φn + perms
+ φ1 φ2 φ3 φ4 φ5 · · · φn + perms
+ φ1 φ2 φ3 φ4 φ5 φ6 φ7 · · · φn + perms
+ all remaining contractions & perms}, (32)
where, N {φ1 φ2 φ3 φ4 · · · φn } ≡ i∆F (x1 − x2 ) N {φ3 φ4 · · · φn }.
In the above expression N is used to employ normal ordering instead of : :. If vacuum expectation
value is calculated on T {φ1 φ2 · · · φn }, then any term in which there are uncontracted operators will
results in zero because of normal ordering; only fully contracted terms survive. An example with
four fields:
T {φ1 φ2 φ3 φ4 } = N {φ1 φ2 φ3 φ4 } +
N {φ1 φ2 φ3 φ4 } + N {φ1 φ3 φ2 φ4 } + N {φ1 φ4 φ2 φ3 } +
N {φ1 φ2 φ3 φ4 } + N {φ1 φ2 φ4 φ3 } + N {φ1 φ2 φ3 φ4 } +
N {φ1 φ2 φ3 φ4 } + N {φ1 φ3 φ2 φ4 } + N {φ1 φ4 φ2 φ3 } (33)
When the vacuum expectation value is calculated all the normal ordered uncontracted terms i.e. the
first three lines will result in zero. Only non-zero contribution will come from the fully contracted
last line above,
h0|T {φ1 φ2 φ3 φ4 }|0i = i∆F (x1 − x2 ) i∆F (x3 − x4 ) +
i∆F (x1 − x3 ) i∆F (x2 − x4 ) +
i∆F (x1 − x4 ) i∆F (x2 − x3 ). (34)
Although here the Wick theorem is discussed in terms of scalar fields, everything that are discussed
here hold equally well for fermions except that,
T {ψ(x1 ) ψ(x2 )} = ψα (x1 ) ψ β (x2 ) t1 > t2
= −ψ β (x2 ) ψα (x1 ) t1 < t2 (35)
i SF αβ (x1 − x2 ) = ψα (x1 )ψ β (x2 ) = −ψ β (x2 )ψα (x1 ). (36)

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Wick’s theorem, S-matrix & Feynman diagram: But what Wick’s theorem has got to
do with interaction picture? The interaction Hamiltonian HI , in the definition of S-matrix in
(15), is basically product of the interacting fields. For instance, consider the following interaction
Hamiltonian densities,

Yukawa interaction : HI = g N {ψ ψ φ} (37)

φ4 − interaction : HI = λ N {φ4 } (38)
a toy interaction : HI = g N {Φ† Φ ϕ} (39)
In the above Yukawa interaction Hamiltonian, if φ is replace with 4-potential Aµ then we have gauge
interaction and we will be looking at QED. Consider the first two order of S-matrix,
i ∞
  Z 
S = T exp − dt HI (t)
~ −∞
 Z 
(1) i 4
S = T − d x HI (x) (40)
~
"  2 Z #
1 i
S (2) = T − d4 x1 d4 x2 HI (x1 ) HI (x2 ) (41)
2! ~

As a warm-up, consider the toy model. Suppose s± are quanta of the complex scalar fields Φ and
Φ† and B is that of real scalar field ϕ. The corresponding expansion of Φ and ϕ in terms of creation
and annihilation operators are similar to φ in (21)
1 h i
ak e−ikx + a†k eikx ≡ ϕ+ + ϕ−
X
ϕ(x) = √ (42)
k
2ωk V
X 1  −ipx
+ b†p eipx ≡ Φ+ + Φ−

Φ(x) = p ap e (43)
p
2Ep V
1
+ a†p eipx ≡ Φ†+ + Φ†−
X
Φ† (x) =
 −ipx 
p bp e (44)
p
2Ep V

For ready reference, ϕ+ annihilates B and ϕ− creates B; Φ+ annihilates s− and Φ− creates s+ ; Φ†+
annihilates s+ and Φ†− creates s− , i.e.

ϕ− |0i = |Bi, Φ†− |0i = |s− i, Φ− |0i = |s+ i

ϕ+ |Bi = |0i, Φ+ |s− i = |0i, Φ†+ |s+ i = |0i. (45)
For this toy model, the first order S-matrix is (putting ~ = 1),
 Z  Z
S = −ig d x T {Φ (x)Φ(x)ϕ(x)} = −ig d4 x N {Φ† Φϕ} + Φ† Φ N {ϕ} .
(1) 4 †
 
(46)

Because of uncontracted ϕ, the vacuum expectation value h0|T {Φ† Φϕ}|0i = 0 and so is S (1) = 0.
But if we are considering the decay B → s− s+ , assuming mass M of B is greater than twice the
mass m of s± and is allowed by conservation laws, then the process can be represented as,
hs+ s− |T {Φ† Φϕ}|Bi = hs+ s− |N {Φ†− Φ− ϕ+ }|Bi =
6 0. (47)

5
The second term in (46) will go to zero since ϕ± acting on initial state |Bi will generate states that
are always orthogonal to the final state hs+ s− |. The matrix element of S (1) for the initial |ii = |Bi
state and final |f i = |s± i state is,
Z
(1) − †
Sf i = −ig d4 xhs+ p sq |N {Φ− (x)Φ− (x)ϕ+ (x)}|Bk i

e−ikx eipx eiqx

Z
= −ig d4 x √ p p hs+ s− |s+ s− i
2ωk V 2Ep V 2Eq V
Z
1
= −ig √ p p d4 x e−i(k−p−q)x
2ωk V 2Ep V 2Eq V
1
= −ig √ p p (2π)4 δ 4 (k − p − q)
2ωk V 2Ep V 2Eq V
1 1 1
≡ (2π)4 δ 4 (k − p − q) √ p p (iMf i ) . (48)
2ωk V 2Ep V 2Eq V
⇒ i Mf i = −ig

Mf i is the Lorentz invariant amplitude of the process at hand. Recall that kinematics of 2-body
decay 1 → 2 + 3 gives the decay rate Wf i in CM frame as,

p?
Z
1 1/2 2 2 2
Wf i = |Mf i |2 dΩ, where p? = λ (mi , m1 , m2 ). (49)
32π 2 m2i 2mi

For the problem at hand we have,

mi = M, m1 = m2 = m and |Mf i |2 = g 2
1
q
p? = [m2i − (m1 − m2 )2 ] [m2i − (m1 + m2 )2 ]
2mi
√
1 p 2 2 2
M 2 − 4m2
= M (M − 4m ) =
2M 2r
√ 2
1 M 2 − 4m2 g 4m2
Wf i = 2 2
4πg 2 = 1− . (50)
32π 2M 16πM M2
Therefore, the life-time of B is ≈ 1/Wf i ∝ g −2 implying smaller the coupling with s± longer B can
live. But this decay rate is only at tree-level i.e. calculated considering the first order in S-matrix.
Next consider the second order S (2) ,
Z
1
S (2) = (−ig)2 d4 x1 d4 x2 T {Φ† (x1 )Φ(x1 )ϕ(x1 ) Φ† (x2 )Φ(x2 )ϕ(x2 )}. (51)
2!
Using Wick’s theorem to the above expression gives 14 non-zero normal ordered term, but none of
them contributes to the process B → s+ s− as is evident from the list below. There are no diagrams
with B in the initial state and s± in the final state. Even though none of them corresponds to
B → s± decay, some of the diagrams corresponds to other physical processes. For instance, the
scattering processes like s+ s− → s+ s− or s± B → s± B are represented in diagrams 2, 3 and 4.
The other diagrams of physical interest are the so-called self-energy of s± or vacuum polarization
of B corresponding to the diagrams 9, 10 and 12. In these diagrams, the contractions appear
as internal lines meaning that they do not represent particles in the initial or final state. These
are called 1-loop diagrams and the momenta going around in the loop are not determined by the
external momenta but completely arbitrary. Therefore, this momenta must be integrated over the
entire range [−∞, +∞]. These are the potential sources of divergences in QFT. A huge theoretical
machinery of regularization and renormalization has been developed to tame these infinities and
make sense of the outcome.

6
 There are few other diagrams that do not correspond to any physical processes. The first diagram
are just two disconnected decays of B at space-time positions x1 and x2 . The loops in diagrams 5,
6, 7, 8, 11 and 14 are not kinematically allowed, while diagram 13 corresponding to non-interacting
or spectator B and s± .

Consider the elastic scattering s+ s− → s+ s− mediated by B as shown in the diagram 2 above. It

is evident that only ϕ ϕ contraction gives s± → s± scattering,
Z
(2) − † † + −
Sf i = (−ig)2 d4 x1 d4 x2 hs+
p0 sq 0 |N {Φ (x1 )Φ(x1 ) ϕ(x1 )ϕ(x2 ) Φ (x2 )Φ(x2 )|sp sq i (52)

Here the factor 2! is dropped since x1 and x2 are basically dummy indices and can be interchanged

7
to give the same situation. Now, unlike in S (1) , the creation and annihilation operators Φ± or Φ†±
are not explicitly shown because there can be two possibilities. (i) Φ† (x2 ) can be made to annihilate
−
|s+
p i at x2 or we may choose it to create |sq 0 i at x2 . Therefore, we can have two scattering scenario:
s-channel and t-channel scattering,

Using (31), ϕ(x1 )ϕ(x2 ) = i ∆F (x1 − x2 ) (see any QFT book, for instance Peskin, Lahiri-Pal etc.)
and following set of creation and annihilation operators in the integrand above (52,

d4 k e−ik(x1 −x2 ) d4 k −ik(x1 −x2 )

Z Z
∆F (x1 − x2 ) = ≡ e ∆F (k) (53)
(2π)4 k 2 − M 2 + i (2π)4
− † † + −
s − channel : i ∆F (x1 − x2 ) hs+
p0 sq 0 |Φ− (x1 )Φ− (x1 ) Φ+ (x2 )Φ+ (x2 )|sp sq i (54)
†
t − channel : i ∆F (x1 − x2 ) hs+
p0 s−
q 0 |Φ+ (x1 )Φ− (x1 ) Φ†− (x2 )Φ+ (x2 )|s+ −
p sq i (55)

Therefore, the matrix element of S (2) for initial state |ii = |s+ s− i and final state |f i = |s+ s− i is,
Z
(2) 1
Sf i = (−ig)2 d4 x1 d4 x2 p p p p ×
2Ep V 2Eq V 2Ep0 V 2Eq0
h 0 0 0 0
i
i∆F (x1 − x2 ) eip x1 eiq x1 e−ipx2 e−iqx2 + eip x1 e−ipx1 eiq x2 e−iqx2
i(−ig)2 d4 k
Z
= p d4 x1 d4 x2 ×
(2π)4
p p p
2Ep V 2Eq V 2Ep0 V 2Eq0
h 0 0 0 0
i
e−ik(x1 −x2 ) ∆F (k) ei(p +q )x1 e−i(p+q)x2 + ei(p −p)x1 ei(q −q)x2
i(−ig)2 d4 k
Z
= p ∆F (k) ×
(2π)4
p p p
2Ep V 2Eq V 2Ep0 V 2Eq0
Z h i
0 0 0 0
d4 x1 d4 x2 e−i(k−p −q )x1 e−i(p+q−k)x2 + e−i(k−p +p)x1 e−i(q −q−k)x2

i(−ig)2 d4 k
Z
= p ∆F (k) ×
(2π)4
p p p
2Ep V 2Eq V 2Ep0 V 2Eq0
(2π)4 δ 4 (k − p0 − q 0 ) (2π)4 δ 4 (p + q − k) + (2π)4 δ 4 (k − p0 + p) (2π)4 δ 4 (q 0 − q − k)
 

i(−ig)2
= p p p p ×
2Ep V 2Eq V 2Ep0 V 2Eq0
(2π)4 δ 4 (p + q − p0 − q 0 ) ∆F (p + q) + (2π)4 δ 4 (p − p0 + q − q 0 ) ∆F (p − p0 )
 
Y 1 Y 1
≡ (2π)4 δ 4 (p + q − p0 − q 0 ) √ p (iMf i ) (56)
i
2Ei V f 2Ef V
⇒ iMf i = i(−ig)2 [∆F (p + q) + ∆F (p − p0 )] (57)

This invariant amplitude Mf i can be used to calculate 2 + 2 body differential scattering cross-section
say in the CM frame,
1 p?f
Z
σ= |Mf i |2 dΩ? (58)
64π 2 s p?i

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Here Mf i depends on angle θ coming from the t-channel diagram. In CM frame p?i = p?f ,

1 1
∆F (p + q) = 2 2
= 2
(p + q) − M p + q + 2p · q − M 2
2

1 1
= = (59)
2m2 + 2E 2 + 2p?2 i − M 2 4m 2 − M 2 + 2p?2
i
1 1
∆F (p − p0 ) = = 2
(p − p0 )2 − M 2 p + p02 − 2p · p0 − M 2
1 1
= 2 2 ?2 2
= 2 ?2 (60)
2m − 2E + 2pi cos θ − M −M − 2pi (1 − cos θ)

Calculation of |Mf i |2 and subsequent use in (58) is basically tedious algebra.

Derive the invariant amplitude for inelastic scattering s± B → s± B, assuming it is
kinematically possible.
For contribution to B → s+ s− from higher order S-matrix, the next step to consider would be,
Z
1
S (3) = (−ig)3 d4 x1 d4 x2 d4 x3 T {Φ† (x1 )Φ(x1 )ϕ(x1 ) Φ† (x2 )Φ(x2 )ϕ(x2 ) Φ† (x3 )Φ(x3 )ϕ(x3 )}
3!
(61)
The terms that contribute to B → s+ s− are uncontracted Φ† Φϕ(x3 ) and they are,

There is, however, another diagram similar to (c) above where the loop is in the lower s+ arm but
its calculation will be exactly the same as (b). Actually, none of the diagrams (b) and (c) contribute
to the Mf i of the decay Proceeding exactly as before, the matrix element S (3a) corresponding to the

9
diagram (a) for initial state |Bi and final state |s+ s− i is,
Z
(3a)
Sf i = (−ig)3 d4 x1 d4 x2 d4 x3 i∆F (x1 − x2 ) iDF (x2 − x3 ) iDF (x1 − x3 ) ×
†
hs− +
p (x1 )sq (x2 )| Φ− (x1 ) Φ− (x2 ) ϕ+ (x3 ) |Bk (x3 )i
d4 k 0 d4 q1 d4 q2
Z Z
1 1 1
= (i)3 (−ig)3 d4 x1 d4 x2 d4 x3 √ ×
(2π)4 (2π)4 (2π)4
p p
2ωk V 2Ep V 2Eq V
0
e−ik (x1 −x2 ) ∆F (k 0 ) e−iq1 (x1 −x3 ) DF (q1 ) e−iq2 (x2 −x3 ) DF (q2 ) eipx1 eiqx2 e−ikx3
 

−i(−ig)3 d4 k 0 d4 q1 d4 q2
Z
= √ ∆F (k 0 ) DF (q1 ) DF (q2 ) ×
(2π)4 (2π)4 (2π)4
p p
2ωk V 2Ep V 2Eq V
Z h i
0 0
d4 x1 d4 x2 d4 x3 e−i(k +q1 −p)x1 e−i(q2 −k −q)x2 e−i(k−q1 −q2 )x3

−i(−ig)3 d4 k 0 d4 q1 d4 q2
Z
= √ ∆F (k 0 ) DF (q1 ) DF (q2 ) ×
(2π)4 (2π)4 (2π)4
p p
2ωk V 2Ep V 2Eq V
(2π)4 δ 4 (k 0 + q1 − p) (2π)4 δ 4 (q2 − k 0 − q) (2π)4 δ 4 (k − q1 − q2 )
 

−i(−ig)3
Z
= √ p p d4 k 0 ∆F (k 0 ) DF (p − k 0 ) DF (k 0 + q) δ 4 (k − p − q) (62)
2ωk V 2Ep V 2Eq V
1 Y 1
≡ (2π)4 δ 4 (k − p − q) √ p (iMf i ) .
2ωk V f 2Ef V

Hence the invariant amplitude for B → s− s+ in 1-loop order is,

d4 k 0
Z
⇒ iMf i = −i(−ig)3 ∆F (k 0 ) DF (p − k 0 ) DF (k 0 + q). (63)
(2π)4
The invariant amplitude Mf i involves integration over an unknown momentum k 0 . The k 0 is unknown
in the sense it is not determined by the external momenta k, p, q. This momentum can virtually
be anything ∈ [−∞, +∞] implying the particle carrying need not have to satisfy k 02 = M 2 . Such
particles in internal lines are known as off-shell or virtual particles. These loop integrals are source
of infinities or divergences in QFT, taming of which requires elaborate mechanism of regularization
and renormalization.
(1) (2) (3a)
Based on the explicit calculation of Sf i in (48), Sf i in (56) and Sf i in (62), we can construct the
following Feynman rules which can be used to write the invariant amplitudes Mf i or Sf i straight
away from the Feynman diagrams. The Feynman rules for this toy model are,

The internal s± line or propagator DF (p) is of the same form as that of B propagator ∆F (p) since
both arePscalar, Pexcept their masses are different. In addition to above, there will be a factor
(2π)4 δ 4 ( i pi − f pf ) to enforce overall momentum conservation. Also important is the fact that
the arrow indicating flow of momentum only makes sense in the external lines, but in the internal

10
lines they are just chosen as convenient. As an exercise, through explicit calculation, show that the
invariant amplitude corresponding to the diagram (c) is exactly equal to what we obtain using the
above Feynman rules,

d4 k 0
Z
(3c)
iMf i = −i(−ig) 3
∆F (k 0 ) DF (p) DF (p + k 0 ). (64)
(2π)4

Yukawa interaction: In field theory, the interaction among the scalar and fermion fields are
called Yukawa interaction and typically has the form,

HI = y ψ ψ ϕ. (65)

For the examples considered here, we will suppose electrons e± of mass m and B of mass M to be
the quanta of the fermionic field ψ and real scalar field ϕ respectively. The corresponding expansion
of ψ and ϕ in terms of creation and annihilation operators are,
1 h i
ak e−ikx + a†k eikx ≡ ϕ+ + ϕ−
X
ϕ(x) = √ (66)
k
2ωk V
X 1 Xh i
ψ(x) = p fs (p) us (p) e−ipx + fˆs† (p) vs (p) eipx ≡ ψ+ + ψ− (67)
p
2Ep V s
X 1 Xh i
ψ(x) = p fs† (p) us (p) eipx + fˆs (p) v s (p) e−ipx ≡ ψ − + ψ + (68)
p
2Ep V s

In the above expression, the us (p) and vs (p) are the Dirac spinors obeying,

Normalization : ur (p) us (p) = −v r (p) vs (p) = 2 m δrs (69)

X X
spin − sum : us (p) us (p) = p/ + m, vs (p) v s (p) = p/ − m. (70)
s s

For convenience, p is used for 3-momentum. For quick reference, the action of ϕ± and ψ± are
tabulated below,

ϕ− |0i = |Bi, ψ − |0i = |e− i, ψ− |0i = |e+ i

ϕ+ |Bi = |0i, ψ+ |e− i = |0i, ψ + |e+ i = |0i. (71)

Then the decay B → e− e+ corresponds to the matrix element,

Z
(1)
Sf i = −iy d4 x he− (p, s) e+ (p0 , s0 )|ψ − ψ− ϕ+ |B(k)i
0
e−i(k−p−p )x
Z
= −iy d4 x √ p p us (p) vs0 (p0 )
2ωk V 2Ep V 2Ep0 V
1
= (2π)4 δ 4 (k − p − p0 ) √ p p [−iyus (p) vs0 (p0 )]
2ωk V 2Ep V 2Ep0 V
⇒ iMf i = −iy us (p) vs0 (p0 ) (72)
2 2 0 2
or, |Mf i | = y |us (p) vs0 (p )| . (73)

If the final state electron and positron are in the spin state s and s0 then the square of the invariant
amplitude for the decay B → e± by Yukawa interaction is what given in (73). However, since the
final spin states are incoherent we could sum over all possible final spin states which gives
2
X
|Mf i |2 = y 2 |us (p) vs0 (p0 )| (74)
s,s0

11
To calculate the spin sum, we observe that
†
|us (p) vs0 (p0 )|2 = [us (p) vs0 (p0 )] [us (p) vs0 (p0 )]
†
vs†0 (p0 ) γ0 us (p) = v s0 (p0 ) us (p)
 †
us (p) γ0 vs0 (p0 ) = (75)
X X
⇒ |us (p) vs0 (p0 )|2 = [v s0 (p0 )α us (p)α ] [u(p)β vs0 (p0 )β ]
s,s0 s,s0
X X
= [us (p)α us (p)β ] [vs0 (p0 )β v s0 (p0 )α ]
s s0
" # " #
X X
0 0
= us (p) us (p) vs0 (p ) v s0 (p )
s αβ s0 βα
= [p/ + m]αβ [p/0 − m]βα
= Tr [(p/ + m) (p/0 − m)] (76)

Therefore, the sum over final state spin reduces to evaluating trace over Dirac indices. Expanding
the parenthesis above we get,

/ + m) (p/0 − m)] = Tr p
 0
/ + mp/0 − m2 = Tr γµ γν pµ p0ν − mγµ pµ + mγν p0ν − m2 1 (77)
  
Tr [(p /p− mp

Now, Tr 1 = 4 and the gamma matrices are traceless, so we have

Tr [m2 1] = 4m2 , Tr [m γµ pµ ] = m pµ Tr(γµ ) = 0 and similarly Tr [m γν p0ν ] = 0. (78)

The trace of the remaining term can be calculated as,

1 µ 0ν
Tr [γµ γν pµ p0ν ] = pµ p0ν Tr [γµ γν ] = p p Tr [γµ γν + γν γµ ]
2
1 µ 0ν
= p p Tr[2 gµν ] = gµν pµ p0ν Tr(1) = 4pν p0ν = 4p · p0 (79)
2
⇒ / + m) (p/0 − m)]
Tr [(p = 4 (p · p0 − m2 ) (80)

The invariant amplitude for the decay process (74) to first order in perturbation theory can thus be
expressed as,
|Mf i |2 = 4 y 2 p · p0 − m2 .


(81)
Let us evaluate the above expression in CM i.e. rest frame of B particle,

B: k = (M, ~0)
−
e : p = (E1 , p~? ) where E 2 = p?2 + m2
e+ : p0 = (E2 , −~
p? ) where E 2 = p?2 + m2 .
0
Hence k =p+p ⇒ k = p + p02 + 2p · p02 2

1
or M 2 = 2m2 + 2p · p0 ⇒ p · p0 = M 2 − 2m2


(82)
 2 
1
|Mf i |2 = 4 y 2 (M 2 − 2m2 ) − m2 = 2 y 2 M 2 − 4m2
 
⇒ (83)
2

The generic expression for 1 → 2 + 3 2-body decay rate is

p?
Z
2 1 1/2 2 2 2
Wf i = 2 2 |Mf i | dΩ, where p? = λ (mi , m1 , m2 ). (84)
32π mi 2mi

12
For the decay B → e− e+ , mi = M and m1 = m2 = m and, therefore,
√
1 M 2 − 4m2
q
? 2 2 2 2
p = [mi − (m1 − m2 ) ][mi − (m1 + m2 ) ] =
2m 2
√ i
M 2 − 4m2
Z
⇒ Wf i = 2 y 2 [M 2 − 4m2 ] dΩ
64π 2 M 2
y2  2 3/2
= 2 2
M − 4m2 4π
32π M
3/2
y2 M 4m2
 
= 1− . (85)
8π M2

• Find the decay rate of B → e± when HI = yψγ5 ψ ϕ.

• Find the decay rate of µ− → e− +ν̄e +νµ in Fermi’s original theory of β-decay using
4-fermion coupling. [Ref. Lahiri & Pal]
The final demonstration in this phase is studying the electron positron scattering in Yukawa
theory and eventual derivation of the Yukawa potential in the non-relativistic limit. For the process
e− e+ → e− e+ the S-matrix element is,

(−iy)2
Z
(2)
Sf i = d4 x1 d4 x2 he− (q, s) e+ (q 0 , s0 )|T {ψ(x1 ) ψ(x1 ) ϕ(x1 ) ψ(x2 ) ψ(x2 ) ϕ(x2 )}|e− (p, s) e+ (p0 , s0 )i.
2!
(86)
It is obvious that only ϕ ϕ contraction gives e± → e± scattering. The factor 2! will go away because
of x1 ↔ x2 interchange. Very much like all scalars theory, here too the reaction proceeds through s
and t channels as shown in the figure below.

For the above Feynman diagrams for s and t-channels we have,

s − channel : i∆F (x1 − x2 ) he− (q, s) e+ (q 0 , s0 )|ψ − (x1 ) ψ− (x1 ) ψ + (x2 ) ψ+ (x2 )|e− (p, s) e+ (p0 , s0 )i
= i∆F (x1 − x2 ) he− (q, s) e+ (q 0 , s0 )|us (q) vs0 (q 0 ) v s0 (p0 ) us (p)|e− (p, s) e+ (p0 , s0 )i ×
0 0
eiqx1 eiq x1 e−ip x2 e−ipx2 (87)
t − channel : i∆F (x1 − x2 ) he (q, s) e (q , s )|ψ − (x1 ) ψ+ (x1 ) ψ + (x2 ) ψ− (x2 )|e (p, s) e (p , s0 )i
− + 0 0 − + 0

= i∆F (x1 − x2 ) he− (q, s) e+ (q 0 , s0 )|us (q) us (p) v s0 (p0 ) vs0 (q 0 )|e− (p, s) e+ (p0 , s0 )i ×
0 0
eiqx1 e−ipx1 e−ip x2 eiq x2 (88)

13
Putting everything in the expression for S-matrix, we get
Z
(2) 1
Sf i = (−iy)2 d4 x1 d4 x2 p p p p i∆F (x1 − x2 ) ×
2Ep V 2Ep0 V 2Eq V 2Eq0 V
 0 0
[us (q) vs0 (q 0 ) v s0 (p0 ) us (p)] e−i(p+p )x1 ei(q+q )x2 +
0 0

[us (q) us (p) v s0 (p0 ) vs0 (q 0 )] e−i(p−q)x1 ei(p −q )x2
i(−iy)2 d4 k
Z
= p d4 x1 d4 x2 ∆F (k) e−ik(x1 −x2 ) ×
(2π)4
p p p
2Ep V 2Ep0 V 2Eq V 2Eq0 V
 0 0
[us (q) vs0 (q 0 ) v s0 (p0 ) us (p)] e−i(p+p )x2 ei(q+q )x1 +
0 0

[us (q) us (p) v s0 (p0 ) vs0 (q 0 )] e−i(p−q)x1 e−i(p −q )x2
i(−iy)2 d4 k
Z
= p ∆F (k) ×
(2π)4
p p p
2Ep V 2Ep0 V 2Eq V 2Eq0 V
 Z
0 0
[us (q) vs (q ) v s (p ) us (p)] d4 x1 d4 x2 e−i(k−q−q )x1 e−i(p+p −k)x2 +
0
0
0
0

Z 
0 0 4 4 −i(k+p−q)x1 −i(p0 −q 0 −k)x2
[us (q) us (p) v s (p ) vs (q )] d x1 d x2 e
0 0 e

i(−iy)2 d4 k
Z
= p ∆F (k) ×
(2π)4
p p p
2Ep V 2Ep0 V 2Eq V 2Eq0 V
[us (q) vs0 (q 0 ) v s0 (p0 ) us (p)] (2π)4 δ 4 (k − q − q 0 ) (2π)4 δ 4 (p + p0 − k) +
[us (q) us (p) v s0 (p0 ) vs0 (q 0 )] (2π)4 δ 4 (k + p − q) (2π)4 δ 4 (p0 − q 0 − k)

i(−iy)2
= p p p p ×
2Ep V 2Ep0 V 2Eq V 2Eq0 V
(2π)4 δ 4 (p + p0 − q − q 0 ) us (q) vs0 (q 0 ) ∆F (p + p0 ) v s0 (p0 ) us (p) −


(2π)4 δ 4 (p + p0 − q − q 0 ) us (q) vs0 (q 0 ) ∆F (p − q) v s0 (p0 ) us (p)

 
Y 1 Y 1
≡ (2π)4 δ 4 (p + p0 − q − q 0 ) √ p (iMf i ) (89)
i
2E i V f
2E fV

⇒ iMf i = i(−iy)2 [us (q) vs0 (q 0 ) ∆F (p + p0 ) v s0 (p0 ) us (p) − us (q) vs0 (q 0 ) ∆F (p − q) v s0 (p0 ) us (p)]
(90)

• Derive the invariant amplitude for e−(+) B → e−(+) B.

The e− e+ → e− e+ showed that interaction is mediated by a scalar field ϕ i.e. its corresponding
quanta B in Yukawa type interaction. In the non-relativistic or static limit, the invariant amplitude
reproduces the Yukawa potential. In this limit, E ≈ m. In CM frame, the 4-momentum of the initial
and final state particles are,

e− : p = (m, p~), q = (m, ~q), e+ : p0 = (m, −~

p), q 0 = (m, −~q), B: k = (M, ~k) (91)

Here, we need explicit form of the scalar propagator ∆F (k) which is,
1
i∆F (k) = . (92)
k2 − M 2 + 
In the static limit, the two scalar propagators in the above expression for Mf i (90) becomes,

(p + p0 )2 ≈ 4m2 ⇒ ∆F (p + p0 ) → constant.
1
(p − q)2 = −(~
p − ~q)2 ⇒ ∆F (p − q) → − .
p − ~q)2 + m2
(~

14
Making use of the above expression for scalar propagator and normalization criteria of the spinors
us (p) and vs (p) in (69), the t-channel invariant amplitude and S-matrix in static limit is,

i(−iy)2 i4m2 y 2
Mf i = − (2mδ ss ) (2mδ s 0 s0 ) =
p − ~q)2 + m2
(~ p − ~q)2 + m2
(~
3 3
i4m2 y 2
 
(2π) δ (~ pi − p~f )
Sf i = (2π)δ(Ei − Ef )
4m2 V 2 p − ~q)2 + m2
(~
2
y
→ i(2π)δ(Ei − Ef ) , where ~k = p~ − ~q. (93)
~
|k| + m2
2

In Born approximation to scattering amplitude in non-relativistic quantum mechanics, we get

hf |S|ii = −i(2π)δ(Ei − Ef ) V (~k) (94)

which when compared with (93) gives,

−y 2
V (~k) = . (95)
|~k|2 + m2

Assuming spherically symmetric potential,

d3 k −y 2
Z
~
V (x) = eik·~x
(2π) |~k|2 + m2
3
Z ∞Z π
−y 2 2 eikr cos θ
= 2πk sin θ dθ dk
(2π)3 0 0 k 2 + m2
∞ 1
−y 2 k 2 dk
Z Z
= d(cos θ) eikr cos θ
4π 0 k + m2 −1
2 2

−y 2 ∞ k 2 dk eikr − e−ikr
Z
=
4π 2 0 k 2 + m2 ikr
2 Z ∞ ikr
−y ke
= dk 2 (96)
i4π 2 r −∞ k + m2

The above complex integral can be carried out on the contour as shown in the figure below.

The contour integral gives us,

−y 2 k eikr y 2 e−mr
V (x) = 2
(2πi) =− . (97)
i4π r k + im k=im 4π r

This is the well known form of Yukawa potential. Yukawa worked this form out from static limit of
pion field with a nucleon source term, (∇2 − m2 )ϕ = −y 2 δ(~r). The next immediate step is to move
to more realistic calculations of QED!

15