Table Of ContentOff-Shell Tachyons
Yi-Lei Tang
Institute of Theoretical Physics, Chinese Academy of Sciences,
and State Key Laboratory of Theoretical Physics,
P. O. Box 2735, Beijing 100190, China∗
5 (Dated: February 2, 2015)
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Abstract
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The idea that the new particles invented in some models beyond the standard model can appear
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only inside the loops is attractive. In this paper, we fill these loops with off-shell tachyons, leading
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to a solution of the zero results of the loop diagrams involving the off-shell non-tachyonic particles.
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h We also calculate the Passarino-Veltman Ao and Bo of the off-shell tachyons.
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Electronic address: [email protected]
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I. INTRODUCTION
Recently, an interesting and attractive idea that all the supersymmetric particles could
only appear inside the loops has been introduced [1, 2]. By modifying the quantization
techniques of the supersymmetric particles, they cannot appear in the out-legs of any Feyn-
mann diagrams, just like the Faddeev-Popov ghosts. Thus, it means that we can only detect
the existence of these off-shell particles by measuring the radiative loop effects rather then
finding these particles directly on colliders.
This idea can be generalized to other models. The new particles invented in these models
can be hidden inside the loop in order to escape the detections. In general, these new
particles should be assigned with charges of some unbroken symmetries in order for them to
form closed loops, without any channels decaying into pure standard model (SM) particles.
e.g., in Ref. [2], it is the R-parity of the supersymmetric particles to play this role.
In this paper, we invent off-shell tachyons [3–6] to be quantized in the unconventional
way. This lead to a solution to the zero result of the loop-diagrams involving off-shell
non-tachyonic particles when we apply the half-retarded and half-advanced propagators
introduced in Ref. [2, 7, 8]. Thus, We could not see an “on-shell” tachyon so that we
need not worry about observing something moving faster than the light, and these off-shell
tachyons contribute to the loop-diagrams, leaving us some observable effects.
II. ORDINARY OFF-SHELL NON-TACHYONIC PARTICLES SHOULD FORM
A CLOSED LOOP
Without loss of generality, we introduce an unbroken Z symmetry in this paper. Usually,
2
the non-tachyonic Z -odd particles can decay into the lightest Z -odd particle. If some of
2 2
these Z -odd particles are quantized through the unconventional way described in Ref. [2],
2
and the other Z -odd particles are quantized through the normal way, inconsistencies will
2
be the case.
Suppose AandB aretwo Z -oddparticles, andm < m . If bothparticles are quantized
2 A B
through the normal way, the decay B → A+{CP-even particles} can usually happen. The
self-energy diagrams B → A+{CP-even particles} → B also contain imaginary parts which
contribute to the width of the B’s Breit-Wigner propagator i , where Γ is the
p2−m2B+imBΓB B
2
FIG. 1: The probable t-channel diagram in which A might be near the shell. X and Y might be
the SM-particles or other particles that do not carry the charges of the A and B.
decay-width.
However, if A is quantized through the normal way, and B is quantized through the
unconventional way, the diagram B → A+{CP-even particles} can still move the pole of
the B’s propagator by a quantity of im Γ , which destroys the structure of the propagator
B B
1 1 1 1
P = + , (1)
(cid:18)p2 −m2(cid:19) 2 (cid:18)p2 −m2 +iǫ p2 −m2 −iǫ(cid:19)
or
1 1
P = ±iπδ(p2 −m2) (2)
(cid:18)p2 −m2(cid:19) p2 −m2 ±iǫ
invented in Ref. [2]. These are the principal values of the propagator, and should be applied
when a particle is quantized through the unconventional way.
If A is quantized through the unconventional way, and B is quantized through the normal
way, the discussions are a little bit complicated. Without loss of generality, let A be the
lightest Z -odd particle. It should be noted that the in-lined A can still be near the shell
2
through the t-channel diagrams, e.g. B +X → B + Y in Fig. 1. The X and Y are some
Z -even particles and m > m +m , m > m +m .
2 B X A B Y A
The t-channel near-shell stable particles are rarely discussed in the literature, but this
case does exist. The integral over the phase space is actually infinite due to the divergence
2
of the propagator 1 . If A and B are both quantized through the normal way, this
(cid:12)t−m2A±iǫ(cid:12)
non-physical infinit(cid:12)e can be(cid:12) subtracted by eliminating the on-shell C-effects
(cid:12) (cid:12)
1 d3~p d3p~ d3p~ 1
σ = Y B2 A
OS 2E ·2E |~v −~v | Z (2π)3 (2π)3 (2π)32E ·2E ·2E
B X B X Y B2 A
1
|M |2|M |2 (2π)4δ(p −p −p )(2π)4δ(p −p −p ). (3)
pB1→pY,pA pX,pA→pB 2ǫ B1 A Y A X B2
3
In our appendix, we will derive (3) and will show that how the on-shell effects be subtracted.
However, in our case that A is quantized through the unconventional way, A cannot be on-
shell and thus (3) is absent, leaving us an infinite result of the diagram in Fig. 1.
In a word, A and B should be both quantized through the normal way, or the unconven-
tional way. In the latter case, these particles can only form a closed loop.
III. ZERO RESULT OF THE NON-TACHYONIC OFF-SHELL PARTICLES’
LOOP DIAGRAM
Let’s start from calculating this integral
+∞ 1
I = dz, (4)
± Z z2 −a2 ±iǫ
−∞
whereǫisaninfinitesimal positivenumber introducedinordertoavoidthetwopolesz = ±a.
As the integrand fades out as ∼ 1 , one can close the contour upwards or downward to pick
z2
up the different residues as shown in Fig. 2-3, resulting in the similar consequence
iπ
I = ∓ , (5)
±
a
Hence,
I +I
+ − = 0. (6)
2
Now we are going to calculate this integral in another way. Notice that Re(I ) = 0, and
±
theIm(I )only comes fromthearea near thetwo poleswhen thecontour is bypassing them.
±
Suppose there is a pole z = a located on the real axis with is residue to be Res(z = a),
when the contour is going above this pole, it contributes a iπRes(z = a), and when it is
going beneath this pole, it becomes −iπRes(z = a). Then for (4), Res(z = −a) = − 1 and
2a
Res(z = a) = 1 , so
2a
iπ iπ iπ
I = ±( + ) = ± , (7)
± 2a 2a a
which is compatible with the (5).
Generalize this method to calculate
∞ 1
I(z ,z ,...,z ,P ,P ,...,P ) = dz, (8)
1 2 n 1 2 n Z (z −z1+iP ǫ)(z −z2+iP ǫ)...(z −z2+iP ǫ)
∞ 1 2 n
4
Im(z) Im(z)
I+,closingthecontourdownwards.
z=−a z=a Re(z) z= −a z=a Re(z)
I+,closingthecontourupwards.
FIG. 2: I , different contour closing path.
+
Im(z) Im(z)
I ,closingthecontourdownwards.
−
z=−a z=a Re(z) z= −a z=a Re(z)
I−,closingthecontourupwards.
FIG. 3: I , different contour closing path.
−
where z , z , ..., z are real numbers which define the positions of the poles, and P , P ,
1 2 n 1 2
...P can be +1 or −1 which decide how the contour bypasses the poles. If P = +1, it
n i
means that the contour bypasses z = z upwards, and if P = −1, it means that the contour
i i
bypasses z = z downwards. Then we can immediately write down
i
n
I(z ,z ,...,z ,P ,P ,...,P ) = P Res(z = z )πi. (9)
1 2 n 1 2 n i i
Xi=1
Then we are prepared to calculate the loop diagrams involving the non-tachyon off-shell
particles. Any of this diagrams should contain at least one subloop, each line formed by a
non-tachyonic off-shell particle. To calculate this subloop, we need to calculate
∞ d4q i i i
M = P P ...P (1.0)
sub Z (2π)4 (cid:18)[(q −p )2 −m2](cid:19) (cid:18)[(q −p )2 −m2](cid:19) (cid:18)[(q −p )2 −m2])(cid:19)
∞ 1 1 2 2 n n
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If we adopt (1) to calculate (9), and integrate out dp at first,
0
+∞ dq0 i i i
P P ...P
Z (2π)4 (cid:18)[(q −p )2 −m2](cid:19) (cid:18)[(q −p )2 −m2](cid:19) (cid:18)[(q −p )2 −m2])(cid:19)
−∞ 1 1 2 2 n n
= I(P ,P ,...,P ), (11)
1 2 n
X
{P1,P2,...,Pn}
where P = ±1, i = 1-n, and means to enumerate all the combinations of
i
{P1,PP2,...,Pn}
{P ,P ,...,P } and then sum over them. The definition of I(P ,P ,...,P ) is
1 2 n 1 2 n
I(P ,P ,...,P )
1 2 n
+∞ dq0 in 1
= (12.)
Z (2π)42n[(q −p )2 −m2 +iP ǫ][(q −p )2 −m2 +iP ǫ]...[(q −p )2 −m2 +iP ǫ]
−∞ 1 1 1 2 2 2 n n n
We are not going to talk about the massless particles, so m2 > 0, and then (~q −
i
p~)2 + m2 > 0 always holds. Hence, all the poles are located in the q ’s real-axis. No-
i i 0
tice that I(P ,P ,...,P ) bypasses the poles in a totally opposite manner compared with
1 2 n
I(−P ,−P ,...,−P ), e.g. Fig. 4, thus
1 3 n
I(P ,P ,...,P ) = −I(−P ,−P ,...,−P ), (13)
1 2 n 1 2 n
so
+∞ dp0 i i i
P P ...P
Z (2π)4 (cid:18)[(q −p )2 −m2](cid:19) (cid:18)[(q −p )2 −m2](cid:19) (cid:18)[(q −p )2 −m2])(cid:19)
−∞ 1 1 2 2 n n
= I(P ,P ,...,P )
1 2 n
X
{P1,P2,...,Pn}
1
= I(P ,P ,...,P )− I(−P ,−P ,...,−P )
1 2 n 1 2 n
2
X X
{P1,P2,...,Pn} {P1,P2,...,Pn}
1
= I(P ,P ,...,P )− I(−P ,−P ,...,−P )
1 2 n 1 2 n
2
X X
{P1,P2,...,Pn} {−P1,−P2,...,−Pn}
1
= I(P ,P ,...,P )− I(P ,P ,...,P )
1 2 n 1 2 n
2
X X
{P1,P2,...,Pn} {P1,P2,...,Pn}
= 0. (14)
Thus,
M ≡ 0. (15)
sub
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FIG. 4: An example of comparing the contour path of I(P ,P ,...,P ) and I(−P ,−P ,...,−P ).
1 2 n 1 2 n
Notice that the they bypass the poles in totally opposite manners.
IV. TACHYONIC CASES
(13) holds only when m2 > 0 and all the poles are located in the q0’s real-axis. If the
i
particles appeared in the loops are tachyons, things could be different.
Tachyons are the hypothesised particles which satisfy E2 = p2 −m2 , and m2 > 0. In
Tac Tac
order to calculate the propagators of the off-shell tachyons, we should quantize the tachyonic
fields in the unconventional way [2]. Unlike the normal particles, there are two different
momentum areas, which are the unstable region p2 < m2 and the stable region p2 > m2 .
Tac Tac
These should be treated differently.
A. Scalar Tachyons
The Lagrangian of the scalar tachyons is
L = ∂µφ∂ φ+m2 φ2 (16)
µ Tac
The unstable region p2 < m2 is quantized according to Ref. [4], and the stable region
Tac
p2 > m2 is treated similar to [2]. Then the propagator is [7]
Tac
d4p i
h0|T{φ(x )φ(x )}|0i = P , (17)
1 2 Z (4π)4 (cid:18)p2 +m2 (cid:19)
Tac
where
1 1
P = ±iπδ(p2 +m2 ). (18)
(cid:18)p2 +m2 (cid:19) p2 +m2 ±iǫ Tac
Tac Tac
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B. Spinor Tachyons
The Lagrangian of the spinor tachyons is [5]
L = iψ¯γ5γµ∂ ψ −m ψ¯ψ. (19)
µ Tac
The quantization of the tachyonic spinors is a little bit complicated. We follow Ref. [6],
d3~p 1
ψ(x) = aσuσ(p)e−ip·x +bσ†vσ(p)eip·x , (20)
Z (2π)3 2Ep~ Xσ (cid:16) p~ p~ (cid:17)
p
where σ = ±1 is the helicity of the plane wave solutions. The uσ and vσ are normalized
according to
u¯σ1(p)uσ2(p) = σ2m δσ1σ2
Tac
v¯σ1(p)vσ2(p) = −σ2m δσ1σ2
Tac
uσ1†(p)uσ2(p) = vσ1†(p)vσ2(p) = 2E δσ1σ2. (21)
p~
The commutators of the operators are
{aσ1,aσ2†} = {bσ1,bσ2†} = (−σ)(2π)3δ3(p~ −~p ). (22)
p~1 p~2 p~1 p~2 1 2
The Hamiltonian operator becomes
d3p~
H = σ aσ†aσ +bσ†bσ E . (23)
Z (2π)3 p~ p~ p~ p~ p~
Xσ (cid:16) (cid:17)
For |p~| < m and |p~| > m , we respectively proceed the Ref. [6] and [2], again we acquire
Tac Tac
d4p i
h0|T{ψ(x )ψ¯x }|0i = (iγ5γµ∂ −m ) P . (24)
1 2 µ Tac Z (4π)4 (cid:18)p2 +m2 (cid:19)
Tac
V. CALCULATION OF Ao AND Bo FUNCTIONS
0 0
All the loop diagrams can be reduced into A , B , C , ... Passarino-Veltman functions
0 0 0
[9], and only A and B contribute to the divergences in the usual cases, so we are going to
0 0
calculate the Ao and Bo functions, which are the corresponding version of the A and B
0 0 0 0
functions in the off-shell tachyonic case,
1 ∞ 1
Ao(m ) = d4qP ,
0 Tac iπ2 Z (cid:18)q2 +m2 (cid:19)
−∞ Tac
1 ∞ 1 1
Bo(p2;m ,m ) = d4qP P . (25)
0 Tac1 Tac2 iπ2 Z (cid:18)q2 +m2 (cid:19) (cid:18)(q +p)2 +m2 (cid:19)
−∞ Tac1 Tac2
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A. Calculation of the Ao Function
0
Let’s integrate out q0 at first. Notice that only in the unstable area m2 > ~q2 can the pole
q = ±i m2 −~q2 be located on the imaginary axis, avoiding the situation of (13).
0
p
1 mTac +∞ dq0
Ao(m ) = d3~q
0 Tac iπ2 Z Z q0 2 −~q2 +m2
0 −∞
1 mTac 1 +∞ dq0
= d3~q
iπ2 Z m2 −~q2 Z 1+ q0 2
0 −∞ m2−~q2
1 mTac 1
= π d3~q
iπ2 Z m2 −~q2
0 Tac
1 mTacp 1
= π ·4π q2dq = −im2 π. (26)
iπ2 Z m2 −q2 Tac
0
p
We can see that there is no divergence in the result. In fact, the traditional counting of the
“divergence degree” is applied after the Wick’s rotation, which is impossible in our cases.
B. Calculation of the Bo Function
0
To calculate Bo, we adopt a different form of the propagator
0
1 1
P = +iπδ(p2 +m2 ), (27)
(cid:18)q2 +m2 (cid:19) q2 +m2 +iǫ Tac
Tac Tac
then
Bo(p2;m ,m )
0 Tac1 Tac2
= B (p2;im ,im )+B (p2;m ,m )
0 Tac1 Tac2 δ Tac1 Tac2
+ B (p2;m ,m )+B (p2;m ,m ), (28)
δ Tac2 Tac1 δδ Tac1 Tac2
where
µ4−D 1
B (p2;im ,im ) = dDq (29)
0 Tac1 Tac2 iπ2 Z (q2 +m2 +iǫ)[(q +p)2 +m2 +iǫ]
Tac1 Tac2
is the usual B function with the traditional Feymann propagators, and
0
µ4−D 1
B (p2;im ,im ) = dDq iπδ(q2 +m2 ),
δ Tac1 Tac2 iπ2 Z (q+p)2 +m2 +iǫ Tac1
Tac2
1
B (p2;m ,m ) = d4qiπδ(q2 +m2 )iπδ((q +p)2 +m2 ). (30)
δδ Tac1 Tac2 iπ2 Z Tac1 Tac2
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To calculate the B (p2;im ,im ), the traditional tricks involving Feynmann integral
0 Tac1 Tac2
and Wick’s rotation are applied,
B (p2;im ,im )
0 Tac1 Tac2
2 1 µ2
= −γ +ln4π + dxln , (31)
ε Z (cid:18)p2(x2 −x)−m2 x−m2 (1−x)−iǫ(cid:19)
0 Tac1 Tac2
where ε = 4−D.
B (p2;m ,m ) is just cancelling all the imaginary part of B (p2;im ,im )
δδ Tac1 Tac2 0 Tac1 Tac2
according to the optical theorem.
To calculate B (p2;m ,m ), we work in the p = (p0,~0) reference frame, then
δ Tac2 Tac1
B (p2;m ,m )
δ Tac2 Tac1
µ4−D 2(π)D2−1 +∞ 1
= qD−2dq
π Γ D−1 Z (cid:20)(2E )(p2 +2E p −m2 +m2 +iǫ)
2 mTac1 ~q 0 ~q 0 Tac1 Tac2
(cid:0) (cid:1) 1
+
(2E )(p2 −2E p −m2 +m2 +iǫ)(cid:21)
~q 0 ~q 0 Tac1 Tac2
µ4−D 2(π)D2−1 +∞ (E~q2 +m2Tac1)D2−3 1
= dE
π Γ D−1 Z 2 ~q(cid:20)p2 +2E p −m2 +m2 +iǫ
2 0 0 ~q 0 Tac1 Tac2
(cid:0) (cid:1) 1
+ ,
p2 −2E p −m2 +m2 +iǫ(cid:21)
0 ~q 0 Tac1 Tac2
(32)
where E = ~q2 −m2 .
~q Tac1
p
The full calculations of (32) are too complicated to be discussed in this paper. We
+∞
only note that in the complex-E plane, the integral can be divided into =
~q 0
±imTac1+ ∞ . The ±imTac1 part is finite and the ∞ part contributes to R
0 ±imTac1 0 ±imTac1
R R R R
2(m2 −m2 +p2)
B (p2;m ,m ) = − Tac2 Tac1 0 , (33)
δ Tac2 Tac1 div 2p2ε
0
which means
2
B (p2;m ,m ) +B (p2;m ,m ) = − , (34)
δ Tac2 Tac1 div δ Tac1 Tac2 div ε
that cancels the divergent part of (31) accurately.
VI. SUMMARY
We have proved that the loop contributions from the half-retarded and half-advanced
propagators of the off-shell particles are always zero unless these particles are tachyons. We
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