Appendix A Baryon and Lepton Number Anomalies in the Standard Model

A.1 Baryon Number Anomalies

The introduction of a gauged baryon number leads to the inclusion of quantum anomalies in the theory, refer to Fig. 1.2. The anomalies, for the baryonic current, are given by the following, 2 For SU(3) U(1)B , ⎛ ⎞ 3   A (SU(3)2U(1) ) = Tr[λaλb B]=3 × ⎝ B − B ⎠ = 0. (A.1) 1 B 2 i i lef t right

2 For SU(2) U(1)B ,

3 × 3 3 A (SU(2)2U(1) ) = Tr[τ aτ b B]= B = . (A.2) 2 B 2 Q 2

( )2 ( ) For U 1 Y U 1 B ,

3 A (U(1)2 U(1) ) = Tr[YYB]=3 × 3(2Y 2 B − Y 2 B − Y 2 B ) =− . (A.3) 3 Y B Q Q u u d d 2

( )2 ( ) For U 1 BU 1 Y ,

A ( ( )2 ( ) ) = [ ]= × ( 2 − 2 − 2 ) = . 4 U 1 BU 1 Y Tr BBY 3 3 2BQYQ Bu Yu Bd Yd 0 (A.4)

( )3 For U 1 B ,

A ( ( )3 ) = [ ]= × ( 3 − 3 − 3) = . 5 U 1 B Tr BBB 3 3 2BQ Bu Bd 0 (A.5)

© Springer International Publishing AG, part of Springer Nature 2018 133 N. D. Barrie, Cosmological Implications of Quantum Anomalies, Springer Theses, https://doi.org/10.1007/978-3-319-94715-0 134 Appendix A: Baryon and Lepton Number Anomalies in the Standard Model

2 Fig. A.1 1-Loop corrections to a SU(2) U(1)B , where the loop contains only left-handed quarks, ( )2 ( ) and b U 1 Y U 1 B where the loop contains only quarks

For U(1)B ,

A6(U(1)B ) = Tr[B]=3 × 3(2BQ − Bu − Bd ) = 0, (A.6) where the factor of 3 × 3 is a result of there being three generations of quarks and three colours for each quark. The δab terms are not included in the anomalies (Fig. A.1).

A.2 Lepton Number Anomalies

When introducing right handed neutrinos into the SM the quantum anomalies for a gauged lepton number, or leptonic current, are the following, 2 For SU(3) U(1)L , ⎛ ⎞ 3   A (SU(3)2U(1) ) = Tr[λaλb L]= ⎝ L − L ⎠ = 0. (A.7) 1 L 2 i i lef t right

2 For SU(2) U(1)L ,

3 3 A (SU(2)2U(1) ) = Tr[τ aτ b L]= L = . (A.8) 2 L 2 L 2

( )2 ( ) For U 1 Y U 1 L ,

3 A (U(1)2 U(1) ) = Tr[YYL]=3(2Y 2 L − Y 2 L − Y 2 L ) =− . (A.9) 3 Y L L L e e ν ν 2 Appendix A: Baryon and Lepton Number Anomalies in the Standard Model 135

( )2 ( ) For U 1 LU 1 Y ,

A ( ( )2 ( ) ) = [ ]= ( 2 − 2 − 2 ) = . 4 U 1 LU 1 Y Tr LLY 3 2L L YL Le Ye Lν Yν 0 (A.10)

( )3 For U 1 L ,

A ( ( )3 ) = [ ]= ( 3 − 3 − 3) = . 5 U 1 L Tr LLL 3 2L L Le Lν 0 (A.11)

For U(1)L ,

A6(U(1)L ) = Tr[L]=3(2L L − Le − Lν ) = 0. (A.12)

If the right handed neutrinos are not included in the SM, A5 and A6 will be non-zero. That is, A5 = 3 and A6 = 3, where A6 is to the graviton-lepton anomaly.

A.3 Mixed Gauged Baryon and Lepton Number Anomalies

If these two gauge groups are introduced then the interactions between the leptonic and baryonic currents must also be anomaly free, ( )2 ( ) For U 1 BU 1 L ,

A( ( )2 ( ) ) = [ ]= . U 1 BU 1 L Tr BBL 0 (A.13)

( ) ( )2 For U 1 BU 1 L ,

A( ( )2 ( ) ) = [ ]= . U 1 LU 1 B Tr LLB 0 (A.14)

For U(1)BU(1)LU(1)Y ,

A(U(1)BU(1)LU(1)L ) = Tr[BLY]=0. (A.15)

These will only be non-zero if fermions such as leptoquarks are added to the SM. There are no fermions in the SM which can couple to both a leptophobic gauge boson and a leptophillic gauge boson. Some recent models have introduced leptoquarks along with gauged baryon and lepton number symmetries into the SM [1, 2]. To ensure that these mixed interactions don’t lead to new gauge anomalies, the number of types of leptoquarks and the quantum numbers they carry are such that these quantum corrections remain zero. They can also be used to cancel the gauge anomalies that are also present with these gauge bosons in combination with the SM gauge fields. 136 Appendix A: Baryon and Lepton Number Anomalies in the Standard Model

References 1. P.V. Dong, H.N. Long, A simple model of gauged lepton and baryon charges. Phys. Int. 6(1), 23–32 (2010). https://doi.org/10.3844/pisp.2015.23.32 2. M. Duerr, P.F. Perez, M.B. Wise, Gauge theory for baryon and lepton numbers with leptoquarks. Phys. Rev. Lett. 110, 231801 (2013). https://doi.org/10.1103/ PhysRevLett.110.231801 Appendix B Further Details of Chap. 3 Calculations

B.1 F+ Coefficients, Eq. (3.20)

Matching superhorizon modes with the plane waves, we obtain the following relation,

  − √ 3−k 1 (1+ )  1 2 4 k π = 4 √ −   . C1 −1 √ C2 + (B.1) (1−k )  3 k 2 4 π 2k 4

The Wronskian normalisation implies:  2 π   π 1 1 +  + 2   = . C1C2 sin 1 k C2 + (B.2)  4   1 k 2k k k 2

Solving the above conditions we find that the coefficients for the F+ modes are,      1 1 − (1+ ) 3−k − ( +3) 1+k 3−k 2 4 k  2 2 k  ( )  = √ 4 − 4 4 k , C1 + (B.3) (3 k ) πk πk 4 and     1+ 1+ 1  k   k 4 4 k 4 k C2 = √ = √ . (B.4) 2 2π k 2 2π κ

B.2 F− Coefficients, Eq. (3.21)

Similarly as above, we obtain the following relations from the matching,

  − √ 3−ik 1 (1+i )  1 2 4 k π = 4 √ −   , C4 −1 √ C3 +  (B.5) (1−ik )  3 i k 2 4 π 2k 4 © Springer International Publishing AG, part of Springer Nature 2018 137 N. D. Barrie, Cosmological Implications of Quantum Anomalies, Springer Theses, https://doi.org/10.1007/978-3-319-94715-0 138 Appendix B: Further Details of Chap. 3 Calculations and the Wronskian normalisation,

√ −   ∗ π k −  √ π k iC e 4 k 2 +| |2 + 4  4  = . C3 C4 2C3e 2πIm +  (B.6)  1 i k k 2 2

These two equations determine the coefficients for the F− modes,   2    −   1 − π k 1  3 i k  C3 = √ k e 4 −   , (B.7) 2 2kP(k) π 4 and   ⎛ ⎞ −i √    3 k  π  −  2 4 ⎝ π − k 1  3 i k  ⎠ C4 = √ 1 −   k e 4 −   , −1 ( −  ) 1 ( +  ) 3+i 4 1 i k 4 5 i k ( ) k π 4 2 2πk 2 P k 4 (B.8) where  √     / 3−i 3 4   k 2 − π k i 4 4 P(k) = √ 2πe Im    − Re  . (B.9) i k 1+i i k π 4  k 4 2 4 2 Appendix C Further Details of Chap. 5 Calculations

C.1 Dimensional Regularisation Integrals and Useful Relations

The following dimensional regularisation integrals were utilised in Chap. 5,    d N p 1 1 4π2λ2  i =− (), (C.1) (2π)N (p2 − m2)2 16π2 M2    d N p 1 1 4π2λ2  (1 + ) i = , (C.2) (2π)N (p2 − m2)3 32π2 M2 M2    d N p p p 1 4π2λ2  i μ ν = M2( − 1)g , (C.3) (2π)N (p2 − m2)2 32π2 M2 μν    d N p p p 1 4π2λ2  i μ ν =− ()g , (C.4) (2π)N (p2 − m2)3 64π2 M2 μν

   N 2 2  d p pμ pν pρ pσ 1 4π λ 2 i = M ( − 1)(gμνgρσ + gμσgνρ + gμρgνσ). (2π)N (p2 − m2)3 128π2 M2 (C.5)

Some other useful relations are,

5 Tr(γμγαγργβγ ) =−4iεμαρβ, (C.6)

© Springer International Publishing AG, part of Springer Nature 2018 139 N. D. Barrie, Cosmological Implications of Quantum Anomalies, Springer Theses, https://doi.org/10.1007/978-3-319-94715-0 140 Appendx C: Further Details of Chap. 5 Calculations

/ 5 Tr(γμ(/p + k + m)γρσαβγ (/p + m)) = 2 2 λ 4{εμραβ[m − p − (kp)]−k [εαβρλ pμ − εαβμλ pρ]}, (C.7)

/ 5 Tr(γρ(/p − k + m)γμσαβγ (/p + m)) = 2 2 λ 4{εμραβ[p − m − (kp)]−k [εαβρλ pμ − εαβμλ pρ]}. (C.8)

Upon taking  → 0 the following are obtained,

1 1 (1 + )| →  1,()| →  − γ,( − 1)| → − + γ − 1, (C.9)  0  0   0 

μν ημν η  4 − 2, (C.10)   4πλ2  4πλ2 | →  1 +  ln . (C.11) M2  0 M2