Higgs-Higgs Interaction. The One-Loop Amplitude in the Standard Model

0.66 MB·

by  Valeriy V. Dvoeglazov

Higgs-Higgs Interaction.
The One-Loop Amplitude in the Standard
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Model ∗
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n Valeriy V. Dvoeglazov
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J Universidad de Zacatecas

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A.P. 636, Suc. 3 Cruces, Zacatecas 98068 Zac. M´exico
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E-mail: valeri@fisica.uaz.edu.mx
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Received on: August 20, 2015
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1 Abstract
v
3 The amplitude of Higgs-Higgs interaction is calculated in the
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Standard Model in the framework of the Sirlin’s renormalization
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5 scheme in the unitary gauge. The one-loop corrections for λ, the
0 constant of 4χ interaction are compared with the previous results of
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0 L. Durand et al. obtained on using the technique of the equivalence
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theorem, and in the different gauges.
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∗The talk presented at the Meeting of the DPF of the APS, May 24-28, 2002, The
College W&M, Williamsburg, VA, USA, May 26, 2002. Originally, it was on the web-
site http://www.dpf2002.org/abstract display.cfm?abstractid=26, which was promised
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to be permanently active.
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1 Introduction.
The Higgs sector of the electroweak theory attracts much attention because
of its connection with the cornerstones of the theory. The search for Higgs
scalars is included in most of the experimental programs of the newcoming
and acting accelerators [1]. The Higgs particles are suggested to be found
in the decays of different particles (Z0 bosons, heavy quarkoniums etc.) as
well as in photon-photon interactions and gluon-gluon fusion. In connection
with that, let us mention non-so-long-ago attempts to explain anomalous
events seen at SpS as the manifestation of the bound state of two Higgs
bosons, i. e. of Higgsonium [2]-[4] or that of the bound state of vector bo-
son [5] in accordance with the Veltman paper, Ref. [6]. Nowadays, even
after clarifying the experimental situation with these anomalous events, the
interest in Higgsonium has still the rights for existence at least from the
viewpoint of preparedness to new unexpected news from the experiment.
The previous investigations of the problem of existence of two-Higgs bound
states were based on the Born approximation of their interaction ampli-
tude [2]-[4]. In the present paper we present the results of our calculation
of the amplitude of Higgs-Higgs interaction up to the fourth order in the
framework of the Standard Model (SM) of Weinberg, Salam and Glashow
with one Higgs doublet. This problem is also of present interest since there
is some relations with the idea that gauge vector bosons could originate
from a strong interacting scalar sector of the electroweak theory [7]. The
amplitude obtained in this paper could also be useful for the consideration
of the problem of the unitarity limit (e. g., [8]). Moreover, information on
the behaviour of the Higgs coupling constants at mass scale M would be
also of interest.
These are the reasons why we start a more complete study of the prob-
lem of Higgs-Higgs interaction. To reduce the volume of the article we
shall use the standard notation used in [9], dimensional regularization and
the renormalization scheme on the mass shell, which is analogous to that
suggested in [9, 10]. We also choose the unitary gauge (ξ → ∞, to avoid
ghosts) and the parameters recommended by the Trieste conference [11],
namely (e ,M ,M ,M ,m ).
0 W0 Z0 H0 f0
The Higgs sector of the Lagrangian of the SM with one Higgs field has
2
the following form, e. g. [12]:1
1 1 e
L = − (∂ χ)2 − M2χ2 − (cid:88)m f¯fχ−
2 µ 2 χ 2M (1−R)1/2 f
W f
eM eM e2
− W W+W−χ− Z Z2χ− W+W−χ2 −
(1−R)1/2 µ µ 2R1/2(1−R)1/2 µ 4(1−R) µ µ
e2 eM2 e2M2
− Z2χ2 − χ χ3 − χ χ4, (1.1)
8R(1−R) µ 4M (1−R)1/2 32M2 (1−R)
W W
where e is the electron charge, M is the Higgs mass, M and M are the
χ W Z
masses of the vector bosons, m are the fermion masses, R = M2 /M2.
f W Z
Thepaperisorganizedasfollows. InSection2wepresenttheexpressions
for the self-energy and vertex parts (see also calculations in details in [13]).
The results of the calculation of the total Higgs-Higgs amplitude will be
presented in Section 3. The Appendix contains the definitions of some
integralsmetincalculations. Theirconnectionswiththeintegralscalculated
in [14, 15] is given.
1It is possible to add the pseudoscalar fermionic interaction, ∼(1+b γ ).
i 5
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Table I. Coupling constants in the case of the Standard Model.
(cid:113)
fW − √1 g fWWZA ig2 R(1−R)
2 2
fZ −1gMZ fWχ −igM
4 MW W
fA eQ fZχ −igMZ2
i MW
fWWZ −gMW f2W2χ −ig2
MZ 2
fWWA e f2Z2χ −ig2MZ2
2M2
W
f2W2Z −ig2MW2 fχ −i gmi
MZ2 2MW
f4W ig2 f3χ −i3gMχ2
2MW
f2W2A −ie2 f4χ −i3g2Mχ2
4M2
W
The Kobayashi-Maskawa K matrix, g = e/sinθ are used in the full
ij W
Lagrangian of the SM, 2P = −1+γ+ln(M2 /4πµ2) are used in the dimen-
(cid:15) W
sional regularization, γ is the Euler constant.
4
(cid:31)
p
q q
(cid:31) (cid:31)
1 2
Figure 1:
2 Self-energy, vertex and box diagrams for
the scalar boson.
2.1 Self-energy diagrams.
Here they are:
if4χ (cid:34) M2 (cid:35)
Π = M2 2P −1+log χ . (2.2)
16π2 χ M2
W
if2V2χ (cid:34) M2 (cid:35)
Π = M2 6P −1+3log V . (2.3)
16π2 V M2
W
if3χf3χ (cid:104) (cid:105)
Π(q2) = 1 2 −2P −I (q2,M2,M2) . (2.4)
16π2 0 χ χ(cid:48)
ifχfχ (cid:110)(cid:104) (cid:16) (cid:17) (cid:105)
Π(q2) = 1 2 (1−b b ) q2 +2m2 +2m2 +(1+b b )2m m P+
4π2 1 2 1 2 1 2 1 2
(cid:20)1 (cid:16) (cid:17) (cid:21)
+ (1−b b ) q2 +m2 +m2 +(1+b b )m m I (q2,m2,m2)+(2.5)
2 1 2 1 2 1 2 1 2 0 1 2
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V
p
q q
(cid:31) (cid:31)
1 2
Figure 2:
(cid:31)
p
q q
1 2
p + q
(cid:31) (cid:31)
1 2
0
(cid:31)
Figure 3:
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m
1
q 1 p 2 q
p + q
m
2
Figure 4:
1 (cid:32) m2 m2 q2(cid:33) 1 (cid:41)
+ (1−b b ) m2log 1 +m2log 2 + + (1+b b )m m .
2 1 2 1 M2 2 M2 2 2 1 2 1 2
W W
ifVχfVχ (cid:40)(cid:34) q4 q2 (cid:35)
Π(q2) = 1 2 − −3 −6 P− (2.6)
16π2 2M4 M2
V V
(cid:32) q4 q2 (cid:33) q2 M2 q2 (cid:41)
− + +3 I (q2,M2,M2)− log V + −2 .
4M4 M2 0 V V 2M2 M2 2M2
V V V W V
In the framework of the SM with one Higgs doublet only we obtain
ig2 (cid:40)(cid:34) 3 q4 q2 3 q2 q2
Πχ(q2) = M2 − −3 − + Trm2−
16π2 χ 4M2 M2 M2 2RM2 M2 M2 i
W χ χ χ W χ
9 6 (cid:35) 3q4
− 3r −9r−1 − r−1 + Trm4 P +tadpoles+ +
W W 2R Z M2 M2 i 4M2 M2
W χ W χ
(cid:18)5 5 (cid:19) q2 21 9 9 3 (cid:32) q4
+ + + r + r−1 + r−1 − r log r + +
2 4R M2 8 W 2 W 4R Z 2 W W 8M4
χ W
3 q2 9 (cid:33) 3q2
+ + r−1log R− Trm2+
4RM2 4R2 W 4M2 M2 i
W W χ
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V
p
q
1 2 q
p + q
(cid:31) (cid:31)
1 2
Figure 5:
q2 m2 7 3 m2
+ Trm2log i − Trm4 + Trm4log i −
2M2 M2 i M2 2M2 M2 i M2 M2 i M2
W χ W W χ W χ W
(cid:32) q2 1 3M2 (cid:33) 1
− + + W L(q2,M2 ,M2 )−
8M2 2 2q2 M2 W W
W χ
(cid:32) q2 1 3M2(cid:33) 1 1
− + + Z L(q2,M2,M2)− (2.7)
16M2 4 4q2 RM2 Z Z
Z χ
9 1 1
− r L(q2,M2,M2)− Trm2L(q2,m2,m2)+
16 Wq2 χ χ 4M2 M2 i i i
W χ
(cid:41)
1 1
+ Trm4L(q2,m2,m2) .
M2 M2q2 i i i
W χ
The corresponding counterterms are
δM2 ig2 (cid:40)(cid:34) 3 15 9 1
χ = Z −Z = 3+ − r −9r−1 − r−1 − Trm2+
M2 Mχ χ 16π2 2R 4 W W 2R Z M2 i
χ W
6 (cid:35) Πχ(tadpoles) 5 5 27 9 9
+ Trm4 P + − − + r + r−1 + r−1−
M2 M2 i M2 2 4R 8 W 2 W 4R Z
W χ χ
3 (cid:18)1 3 9 (cid:19) 3
− r log r + r − + r−1 log R+ Trm2−
2 W W 8 W 4R 4R Z 4M2 i
W
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1 m2 7 3 m2
− Trm2log i − Trm4 + Trm4log i +
2M2 i M2 2M2 M2 i M2 M2 i M2
W W W χ W χ W
(cid:18)r 1 3 (cid:19) 1
+ W − + r−1 L(−M2,M2 ,M2 )+
8 2 2 W M2 χ W W
χ
(cid:18)r 1 3 (cid:19) 1 1
+ Z − + r−1 L(−M2,M2,M2)+ (2.8)
16 4 4 Z RM2 χ Z Z
χ
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+ L(−M2,M2,M2)+
16M2 χ χ χ
W
(cid:41)
1 1
+ Trm2L(−M2,m2,m2)− Trm4L(−M2,m2,m2)
4M2 M2 i χ i i M2 M4 i χ i i
W χ W χ
and
ig2 (cid:40)(cid:34) 3 3 1 (cid:35) 3 3
Z −1 = −3− + r + Trm2 P + + +3r−1+
χ 16π2 2R 2 W M2 i 2 4R W
W
3 (cid:18) 3 1 (cid:19) 1 1 m2
+ r−1 + − r log R− Trm2 + Trm2log i −
2R Z 4R 4 W 4M2 i 2M2 i M2
W W W
(cid:32) (cid:33)
2 1 1 3 1
− Trm4 + − r − L(−M2,M2 ,M2 )+
M2 M2 i 4 4 W r (r −4) M2 χ W W
W χ W W χ
(cid:32) (cid:33)
1 1 3 1 1
+ − r − L(−M2,M2,M2)+
8 8 Z 2r (r −4) RM2 χ Z Z
Z Z χ
3 1
+ L(−M2,M2,M2)− (2.9)
8M2 χ χ χ
W
(cid:41)
1 1
− Trm2L(−M2,m2,m2)− Trm4L(−M2,m2,m2)
4M2 M2 i χ i i 2M2 M4 i χ i i
W χ W χ
Consequently,
ig2
Πren(q2) = Πχ(q2)−δM2 −(Z −1)(q2 +M2) = M2×
χ χ χ 16π2 χ
(cid:40)(cid:34) 3 q4 3 3 q2 (cid:35) 3q4 q4
× − − r − P + + log R+
4M2 M2 4 W 2M2 4M2 M2 8M2 M2
W χ W W χ W χ
q2 (cid:18) 1 3 (cid:19) q2 1
+ 1+ −3r−1 − r−1 + log R+ r log R+
M2 2R W 2R Z 4M2 8 W
χ W
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1 3 3 (cid:32) q2 (cid:33) 1
+ 1+ − r −3r−1 − r−1 − +1 Trm2+
2R 4 W W 2R Z M2 2M2 i
χ W
(cid:32) q2 (cid:33) 2 (cid:32) q2 1 3M2 (cid:33)
+ +1 Trm4 − + + W ×
M2 M2 M2 i 8M2 2 2q2
χ W χ W
1 (cid:32) q2 1 3M2(cid:33) 1 1
× L(q2,M2 ,M2 )− + + Z L(q2,M2,M2)−
M2 W W 16M2 4 4q2 RM2 Z Z
χ Z χ
9 1
− r L(q2,M2,M2)+ Trm2L(q2,m2,m2)+
16q2 W χ χ 4M2 M2 i i i
W χ
1 1 (cid:32) q2 q2 3q2
+ Trm4L(q2,m2,m2)+ − + − +
M2 M2q2 i i i 4M2 4M2 M2r (r −4)
W χ χ W χ W W
(cid:33)
1 1 3 3 1
+ + r − r−1 + L(−M2,M2 ,M2 )+
4 8 W 2 W r (r −4) M2 χ W W
W W χ
(cid:32) q2 q2 3q2 1 1 3
+ − + + + + r − r−1+
8M2 8M2 M2r (r −4) 8 16 Z 4 Z
χ Z χ Z Z
(cid:33)
3 1 1
+ L(−M2,M2,M2)−
2r (r −4) RM2 χ Z Z
Z Z χ
(cid:32) 3q2 15 (cid:33) 1
− + r L(−M2,M2,M2)+
8M2 16 W M2 χ χ χ
W χ
q2 1 (cid:32) q2 (cid:33)
+ Trm2L(−M2,m2,m2)+ 3+ ×
4M2 M4 i χ i i 2M2 M4 M2
W χ W χ χ
(cid:111)
× Trm4L(−M2,m2,m2) (2.10)
i χ i i
In this Section and in what follows r = M2/M2 , r = M2/M2, b are
W χ W Z χ Z 1,2
constants defined by the strength of the Higgs -fermion pseudoscalar inter-
action. 2 The form of the integral I (q2,M2,M2) is given in Appendix A.
0 1 2
2In the Standard Model they are equal to zero.
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