Yukawa and Gauge couplings uniﬁcation with the variation ... · Yukawa and Gauge couplings uniﬁcation with the variation of SUSY breaking scale in the light of LHC data XXI DAE-BRNS

Yukawa and Gauge couplings unification with the

variation of SUSY breaking scale in the light of LHC dataXXI DAE-BRNS HIGH ENERGY PHYSICS SYMPOSIUM 2014, IITG

k Sashikanta Singh

Department of PhysicsGauhati University

December 10, 2014

k Sashikanta Singh (Department of Physics Gauhati University)Yukawa and Gauge couplings unification with the variation of SUSY breaking scale in the lightDecember 10, 2014 1 / 41

Introduction Idea of Unification

Idea of Unification

Excluding Gravity we have three types of interactions

Electromagnetic.

Weak Interaction.

Strong Interaction.



Unification condition with 1 loop threshold correction in

MSSM

1

1

αi (µ)=

1

αi (mz)−

bi2π

ln

(

µ

mz

)

+∆i

where∆i = δi (2) + δi (light) + δi (GUT )

i = 1, 2, 3.δi (2)=threshold correction from 2-loop contribution of the couplingsbetween the energy scales.δi (light)=threshold correction from all superpartners (SUSY sectorparticles).δi (GUT )=threshold correction from full SUSY GUT particles.



How to??

We study the nature of variation of the coupling constant with the help ofRenormalization Group Equation (RGE)


Introduction Definition of RGE

Definition of RGE

RGEs are a set of differential equations of first order in the parameter µ.Its generic form for one coupling constant (g) is

µ∂g

∂µ= β(g)

where µ is the renormalization point or the extraction point. β(g) is a non-linear function that can be calculated as a series of powers of the couplingconstants according to the perturbation theory.


Introduction Input data

Input data

Table : Initial data from Standard Model

mass in GeV coupling constant

mz = 91.1876 ± 0.0021GeV αe.m(mz ) = 127.944 ± 0.014

mt(mt) = 173.5 ± 0.6 αs(mz) = 0.1184 ± 0.007

mb(mb) = 4.18 ± 0.03 α1(mz) =

mτ (mτ ) = 1.77682 ± 0.0016 α2(mz) =

Weinberg mixing angles = sin2θW (mz) = 0.23116 ± 0.00012



1 Matching condition

1

αem(mz )=

3

5

1

α1(mz)+

1

α2(mz)




1

αem(mz )=

3

5

1

α1(mz)+

1

α2(mz)

2 Weinberg mixing angle

sin2 θw (mz ) =αem(mz)

α2(mz )




1

αem(mz )=

3

5

1

α1(mz)+

1

α2(mz)

2 Weinberg mixing angle

sin2 θw (mz ) =αem(mz)

α2(mz )

3

α1(mz) = 1.7100+0.00012−0.00017 × 10−2

andα2(mz) = 3.3753−0.00215

+0.02150 × 10−2



The bottom quark mass mb and tau lepton mass mτ in Table 1 are atz-mass scale their respective values at top quark mass scale usingQCD-QED rescaling factor η are

mb(mt) =mb(mb)

ηb, ηb = 1.53

mτ (mt) =mτ (mτ )

ητ, ητ = 1.015



Masses at various energy scales

Table : Masses at various energy scales scale in SM

Lower limit Best value Upper limit

mz(mz) 91.855 91.1876 91.1897

mt(mt) 172.9 173.5 174.1

mb(mb) 4.1500001 4.1799998 4.2100000

mτ (mτ ) 1.776660 1.77682 1.77698

mτ (mt) 1.7504039 1.505616 1.7507193



What is DR scheme and Why we need it?

1 DR scheme ⇒ Dimensional Regularization by dimensional reduction



What is DR scheme and Why we need it?

1 DR scheme ⇒ Dimensional Regularization by dimensional reduction

2 Radiative correction is achievedmost commonly used for an invariant regularization of the

supersymmetric theories.



mb and α3/αs in DR scheme

mDRb (µ) = mMS

b (µ)

(

1−1

3παs(µ)−

29

72παs(µ)

2

)

and for strong coupling constant αs it is defined as

1

αs(µ)DR=

1

αs(µ)MS−

1

4



The values of mb at various scales both in the MS and DR scheme areshown in Table 3.

Table : mb in MS and DR schemes

at Lower limit Best value Upper limit

mb(mb) 4.1500001 4.1799998 4.2100000

MS mb(mz) 2.7605150 2.8618159 2.9618413

mb(mt) 2.6922452 2.7860069 2.8781033

mb(mb) 4.0431719 4.0713134 4.0994658

DR mb(mz) 2.7264879 2.8242269 2.9205322

mb(mt) 2.6605568 2.7511761 2.8400190



3rd generation Yukawa couplings at mt scale

1

ht(mt) =mt(mt)

174 sin β=

mt(mt)√

1 + tan2 β

174 tan β




1

ht(mt) =mt(mt)

174 sin β=

mt(mt)√

1 + tan2 β

174 tan β

2

hb(mt) =mb(mb)

174 ηb cos β=

mb(mt)√

1 + tan2 β

174




1

ht(mt) =mt(mt)

174 sin β=

mt(mt)√

1 + tan2 β

174 tan β

2

hb(mt) =mb(mb)

174 ηb cos β=

mb(mt)√

1 + tan2 β

174

3

hτ (mt) =mτ (mτ )

174 ητ cos β=

mτ (mt)√

1 + tan2 β

174



The three gauge couplings constants at mt scale

Table : α′

i s or g′

s at top quark mass (mt) scale in SM

Lower limit Best value Upper limit

α1 1.70986541 × 10−2 1.71003081 × 10−2 1.71019584 × 10−2

α2 3.7748118 × 10−2 3.37533131 × 10−2 3.37318331 × 10−2

αDR3 0.10948129 0.11620533 0.12290786

g1 0.46353859 0.46356103 0.46358338

g2 0.65148050 0.65127313 0.65106583

gDR3 1.17293750 1.20842020 1.24278150


Unification for ms = mt Unification of gauge couplings for ms = mt

2-loop RGE for gauge couplings for ms = mt

2-loops RGEs for gauge couplings2

dgidt

=bi

16π2g3i +

(

1

16π2

)

⎡

⎣

3∑

j=1

bij g3i g2

j −∑

j=t,b,τ

aij g3i h2j

⎤

⎦

wheret = lnµ and bi , bij , aij are β function coefficients in MSSM,

bi=

(

6.6, 1.0,−3.0

)

, bij =

⎛

⎜

⎜

⎜

⎜

⎝

7.96 5.40 17.60

1.80 25.00 24.00

2.20 9.00 14.00

⎞

⎟

⎟

⎟

⎟

⎠

, aij =

⎛

⎜

⎜

⎜

⎜

⎝

5.2 2.8 3.6

6.0 6.0 2.0

4.0 4.0 0.0

⎞

⎟

⎟

⎟

⎟

⎠


Unification for ms = mt Unification of Yukawa couplings for ms = mt

2-loop RGE Yukawa couplings for mt = ms :

dht

dt=

ht

16π2

⎛

⎝6h2t + h2b −

3∑

i=1

ci g2i

⎞

⎠ +

ht

16π2

⎡

⎣

∑

i=1

(

ci bi +c2i

2

)

g4i + g

21 g

22 +

136

45g21 g

23 + 8g22 g

23 +

(

6

5g21 + 6g22 + 16g23

)

h2t +

2

5g21 h

2b − 22h4t − 5h4b − 5h2t h

2b − h

2bh

2τ

]

dhb

dt=

hb

16π2

⎛

⎝6h2b + h2τ

+ h2t −

3∑

i=1

c′

i g2i

⎞

⎠ +

hb

16π2

⎡

⎣

∑

i=1

⎛

⎝c′

i bi +c′2i

2

⎞

⎠ g4i + g

21 g

22 +

8

9g21 g

23 + 8g22 g

23 +

(

2

5g21 + 6g22 + 16g23

)

h2b

+4

5g21 h

2t +

6

5g21 h

2τ

− 22h4b − 3h4τ

− 5h4t − 5h2bh2t − 3h2bh

2τ

]



dhτ

dt=

hτ

16π2

⎛

⎝4h2τ

+ 3h2b −

3∑

i=1

c′′

i g2i

⎞

⎠

+hτ

16π2

⎡

⎣

∑

i=1

⎛

⎝c′′

i bi +c′′2i

2

⎞

⎠ g4i +

9

5g21 g

22 +

(

6

5g21 + 6g22

)

h2τ

+

(

−2

5g21 + 16g23

)

h2b + 9h4b − 10h4

τ− 3h2bh

2t − 9h2bh

2τ

]

where

ci =

(

1315 , 3,

1613

)

, c′

i =

(

715 , 3,

163

)

, c′′

i =

(

95 , 3, 0

)



At tanβ g3 Unification points (in GeV)

Experimental Gauge Yukawa

Ug1,g2,g3 Uht ,hb,hτ

Central value 59.9905 1.2084 ∼ 2.5899 ×1016 1.9974 ×1012

Table : Approximate unification points for gauge couplings and Yukawa couplingsfor gDR

3 = 1.2084 and ms = mt .



At tanβ g3 Unification points (Energy in GeV)

experimental Gauge Yukawa


Central Value 60.1380 1.2240 2.9515 × 1016 3.8828 × 1011

Table : Exact unification points for gauge couplings and Yukawa couplings forinput values of gDR

3 in the range 1.2084+0.0344−0.0355 and ms = mt .



0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

5 10 15 20 25 30 35 40

Yuka

wa

coup

lings

Energy (in ln scale)

evolution of Yukawa couplings with energy

(3.8828E11, 0.8637)

h_th_b

h_tau

Figure : Unification points for Yukawa couplings at ms = mt = 173.5GeV



0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

5 10 15 20 25 30 35 40

Gau

ge c

oupl

ings

Energy (in ln scale)

evolution of gauge couplings with energy

(2.9518E16, 0.7284)

g1g2g3

Figure : Unification points for Gauge couplings at ms = mt = 173.5GeV


Unification for ms > mt (ms = 500GeV , 1TeV , 3TeV , 5TeV , 7TeV )gauge couplings RGEs for mt < µ ≤ ms

2-loop RGE for gauge couplings for mt < µ < ms

2-loops RGEs for gauge couplings2

dgidt

=bi

16π2g3i +

(

1

16π2

)

⎡

⎣

3∑

j=1

bij g3i g2

j −∑

j=t,b,τ

aij g3i h2j

⎤

⎦

where

bi=

(

4.100, −3.167,−7.000

)

, gij =

⎛

⎜

⎜

⎜

⎜

⎝

3.98 2.70 8.8

0.90 5.83 12.0

1.10 4.50 −26.0

⎞

⎟

⎟

⎟

⎟

⎠

, aij =

⎛

⎜

⎜

⎜

⎜

⎝

0.85 0.5 0.5

1.50 1.5 0.5

2.00 2.0 0.0

⎞

⎟

⎟

⎟

⎟

⎠


Unification for ms > mt (ms = 500GeV , 1TeV , 3TeV , 5TeV , 7TeV )Yukawa couplings RGEs for mt < µ ≤ ms

2-loop RGE Yukawa couplings for mt < µ < ms :

dht

dt=

ht

16π2

⎛

⎝

3

2h2t −

3

2h2b + Y2(S) −

3∑

i=1

ci g2i

⎞

⎠ +

ht

(16π2)2

[

1187

600g41 −

23

4g42 − 108g43 −

9

20g21 g

22 +

19

15g21 g

23 + 9g22 g

23

+

(

223

80g21 +

135

16g22 + 16g23

)

h2t −

(

43

80g21 −

9

16g22 + 16g23

)

h2b

+5

2Y4(S) − 2λ

(

3h2t + h2b

)

+3

2h4t −

5

4h2t h

2b +

11

4h4b

+ Y2(S)

(

5

4h2b −

9

4h2t

)

− χ4(S) +3

2λ2]




dhb

dt=

hb

16π2

⎛

⎝

3

2h2b −

3

2h2t + Y2(S) −

3∑

i=1

c′

i g2i

⎞

⎠ +

hb

(16π2)2

[

−127

600g41 −

23

4g42 − 108g43 −

27

20g21 g

22 +

31

15g21 g

23 + 9g22 g

23

−

(

79

80g21 −

9

16g22 + 16g23

)

h2t +

(

187

80g21 +

135

16g22 + 16g23

)

h2b

+5

2Y4(S) − 2λ

(

h2t + 3h2b

)

+3

2h4b −

5

4h2t h

2b +

11

4h4t

+Y2(S)

(

5

4h2t −

9

4h2b

)

− χ4(S) +3

2λ2]




dhτ

dt=

hτ

16π2

⎛

⎝

3

2h2τ

+ Y2(S) −3∑

i=1

c′′

i g2i

⎞

⎠ +

hτ

(16π2)2

[

1371

200g41 −

23

4g42 −

27

20g21 g

22 +

(

387

80g21 +

135

16g22

)

h2τ

+5

2Y4(S) − 6λh2t +

3

2h4τ

−9

4Y2(S)h

2τ

− χ4(S) +3

2λ2]




dλ

dt=

1

16π2

[

9

4

(

3

25g41 +

2

5g21 g

22 + g

42

)

−

(

9

5g21 + 9g22

)

λ + 4Y2(S)λ − 4H(S) + 12λ2]

+

1

(16π2)2

[

−78λ3 + 18

(

3

5g21 + 3g22

)

λ2 +

(

−73

8g42 +

117

20g21 g

22 +

1887

200g41

)

λ

+305

8g62 −

867

120g21 g

42 −

1677

200g41 g

22 −

3411

1000g61 − 64g23

(

h4t + h

4b

)

−8

5g21

(

2h4t − h4b + 3h4

τ

)

−3

2g42 Y2(S) + 10λY4(S) +

3

5g21

(

−57

10g21 + 21g22

)

h2t

+

(

3

2g21 + 9g22

)

h2b +

(

−15

2g21 + 11g22

)

h2τ

− 24λ2Y2(S) − λH(S) + 6λh2t h

2b

+20(

3h6t + 3h6b + h6τ

)

− 12(

h4t h

2b + h

2t h

4b

)]




Y2(S) = 3h2t + 3h2b + h2τ

Y4(S) =1

3

[

3∑

ci g2i h

2t + 3

∑

c′

i g2i h

2b + 3

∑

c′′

i g2i h

2τ

]

χ4(S) =9

4

[

3h4t + 3h4b + h4τ

−2

3h2t h

2b

]

H(S) = 3h4t + 3h4t + h4τ

λ =m2

h

V 2, is the Higgs self coupling (mh = Higgs mass). (1)

ci =

(

0.85, 2.25, 8.00

)

, c′

i =

(

0.25, 2.25, 8.00

)

, c′′

i =

(

2.25, 2.25, 0.00

)


Unification for ms > mt (ms = 500GeV , 1TeV , 3TeV , 5TeV , 7TeV )Unification of gauge couplings for µ ≥ ms

2-loop RGEs for gauge couplings for µ ≥ ms

2

dgidt

=bi

16π2g3i +

(

1

16π2

)

⎡

⎣

3∑

j=1

bij g3i g2

j −∑

j=t,b,τ

aij g3i h2j

⎤

⎦

where, bi , bij and aij are the coefficients of β function.For Minimal Supersymmetric Standard Model

bi=

⎛

⎜

⎜

⎜

⎜

⎝

6.6

1.0

−3.0

⎞

⎟

⎟

⎟

⎟

⎠

, bij =

⎛

⎜

⎜

⎜

⎜

⎝

7.96 5.4 17.60

1.8 25.0 24.0

2.2 9.0 14.0

⎞

⎟

⎟

⎟

⎟

⎠

, aij =

⎛

⎜

⎜

⎜

⎜

⎝

5.2 2.8 3.6

6.0 6.0 2.0

4.0 4.0 0.0

⎞

⎟

⎟

⎟

⎟

⎠


Unification for ms > mt (ms = 500GeV , 1TeV , 3TeV , 5TeV , 7TeV )Unification of Yukawa couplings for µ ≥ ms

2-loop RGEs for Yukawa couplings for µ ≥ ms

dhtdt

=ht

16π2

(

6h2t + h2b −

3∑

i=1

ci g2i

)

+

ht16π2

[

∑

i=1

(

cibi +c2i2

)

g4i + g2

1 g22 +

136

45g21 g

23 + 8g2

2 g23 +

(

6

5g21 + 6g2

2 + 16g23

)

h2t +2

5g21h

2b − 22h4t − 5h4b − 5h2t h

2b − h2bh

2τ

]




dhbdt

=hb

16π2

(

6h2b + h2τ + h2t −

3∑

i=1

c′

i g2i

)

+hb

16π2

[

∑

i=1

(

c ′i bi +c′2i

2

)

g4i + g2

1 g22 +

8

9g21 g

23 + 8g2

2 g23+ h2b

+

(

2

5g21 + 6g2

2 + 16g23

)

+4

5g21 h

2t +

6

5g21 h

2τ − 22h4b

−3h4τ− 5h4t − 5h2bh

2t − 3h2bh

2τ

]




dhτdt

=hτ

16π2

(

4h2τ+ 3h2b −

3∑

i=1

c′′

i g2i

)

+hτ

16π2

[

∑

i=1

(

c ′′i bi +c′′2i

2

)

g4i +

9

5g21 g

22 +

(

6

5g21 + 6g2

2

)

h2τ

+

(

−2

5g21 + 16g2

3

)

h2b + 9h4b − 10h4τ − 3h2bh2t − 9h2bh

2τ

]



For MSSM the coefficients are

ci =

⎛

⎜

⎜

⎜

⎜

⎝

0.866

3

5.333

⎞

⎟

⎟

⎟

⎟

⎠

, c′

i =

⎛

⎜

⎜

⎜

⎜

⎝

0.4666

3

5.333

⎞

⎟

⎟

⎟

⎟

⎠

, c′′

i =

⎛

⎜

⎜

⎜

⎜

⎝

4.5

3

0

⎞

⎟

⎟

⎟

⎟

⎠



0

0.2

0.4

0.6

0.8

1

1.2

1.4

5 10 15 20 25 30 35 40 45

Yuka

wa

coup

lings

energy (in ln scale)

evolution of Yukawa couplings with energy

(5.0175E9, 0.7968)

h_th_b

h_tau

(a)

Figure : Unification of third generation Yukawa couplings at ms = 7TeV .



0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

5 10 15 20 25 30 35 40 45

gaug

e co

uplin

gs

energy (in ln scale)

evolution of gauge couplings with energy

(5.417E16, 0.7024)

g1g2g3

(a)

Figure : Unification of gauge couplings at ms = 7TeV .



SUSY breaking tanβ Unification points (Energy in GeV)

scale (ms) Gauge Yukawa


500GeV 60.9070 3.7447 × 1016 1.9315 × 1011

1 TeV 61.4656 4.1134 × 1016 8.6171 × 1010

3 TeV 62.4180 4.8372 × 1016 2.5719 × 1010

5 TeV 62.8150 5.1843 × 1016 1.6339 × 1010

7 TeV 63.0523 5.4012 × 1016 1.2611 × 1010

Table : Approximate gauge unification points and Yukawa unification points forcentral value of gDR

3 = 1.2084



SUSY breaking tanβ gDR3 Unification points (Energy in GeV)

scale (ms) Gauge Yukawa


175.5 GeV 60.1633 1.2248 2.9762 × 1016 1.7118 × 1011

500GeV 61.0600 1.2151 3.7526 × 1016 1.1423 × 1011

1 TeV 61.6230 1.2091 4.1142 × 1016 8.2059 × 1010

3 TeV 62.5450 1.1992 4.8237 × 1016 4.6046 × 1010

5 TeV 62.9750 1.1952 5.1569 × 1016 3.6490 × 1010

7 TeV 63.2500 1.1928 5.3722 × 1016 3.1866 × 1010

Table : Exact Unification points for gauge couplings and Yukawa couplings forinput values of gDR

3 in the range 1.2084+0.0344−0.0355.



22

22.5

23

23.5

24

24.5

25

25.5

26

26.5

27

0 1000 2000 3000 4000 5000 6000 7000

unifi

catio

n po

ints

energy in GeV

Yukawa couplings unification points at various energies

central value

(a)

Figure : Nature of variation of unification points for Yukawa couplings with thevariation in SUSY breaking scale ms



2.5

3

3.5

4

4.5

5

5.5

0 1000 2000 3000 4000 5000 6000 7000

unifi

catio

n po

ints

x 1

0E16

energy in GeV

gauge couplings unification points at various energies

central value

(a)

Figure : Nature of variation of unification points for gauge couplings with thevariation in SUSY breaking scale ms


Discussion

Discussion

1 We use 2-loop RGEs to check the idea of unification for both gaugecouplings and Yukawa couplings.

2 We use FORTRAN program for this process

3 The technique we used is the 4th order Runge Kutta method

4 We use DR scheme only for the mb and αs to convert from MS toDR scheme.

5 Here in our analysis we make an oversimplified assumptions by takinga single SUSY breaking scale.

6 We haven’t included the 1-loop threshold correction also.


Discussion

Conclusion

gauge couplings unify at an energy scale ∼ 1016GeVwhereas for Yukawa couplings unification is achieved at energy scale∼ 1010 − 1011GeV.The nature of variation in unification points of the two are in reverse trend.


Discussion


Documents

Yukawa and Gauge couplings uniﬁcation with the variation ... · Yukawa and Gauge couplings uniﬁcation with the variation of SUSY breaking scale in the light of LHC data XXI DAE-BRNS