- Email: [email protected]
Corrigendum to: “Complete electroweak chiral Lagrangian with a light Higgs at NLO” [Nucl. Phys. B 880 (2014) 552–573]
JID:NUPHB AID:13872 /ERR
[m1+; v1.237; Prn:5/10/2016; 14:58] P.1 (1-4)
Available online at www.sciencedirect.com
ScienceDirect 1
1
2
2
Nuclear Physics B ••• (••••) •••–•••
3
www.elsevier.com/locate/nuclphysb
4 5
Corrigendum
6 7 8 9 10 11 12
19
21
25 26 27 28
The matching of the Higgs-portal example to the chiral Lagrangian in Section 7.2 is corrected to properly account for the systematics of the relevant strong-coupling limit. The text of Section 7.2 given below supersedes the previous version. The main conclusions of Section 7.2 and the rest of the paper are not affected by this correction. The Higgs-portal model discussed here is equivalent to the Standard Model extended by a heavy scalar singlet. The chiral Lagrangian as the low-energy effective field theory of this model in the strong-coupling regime is further elaborated on in [1].
33 34 35 36 37 38 39 40 41
7.2. Higgs portal
44 45 46 47
12
14
16
18
20
22 23 24 25 26 27 28
30 31
As a specific model for a UV completion we consider the Higgs portal (see [2–5] and references therein). This model postulates the existence of a new, Standard-Model singlet scalar particle, which has allowed dimension-4 couplings to the Higgs field. This interaction modifies the scalar potential of Eq. (4) to μ2 μ2 λs λh η V = − s |φs |2 + |φs |4 − h |φh |2 + |φh |4 + |φs |2 |φh |2 , (1) 2 4 2 4 2 where φs refers to the standard scalar doublet and φh denotes the hidden scalar. Both of them acquire a vacuum expectation value, which can be written as
42 43
11
29
31 32
10
21
29 30
9
19
Editor: Hong-Jian He
20
24
8
17
Received 9 September 2016; accepted 17 September 2016
18
23
6
15
Ludwig-Maximilians-Universität München, Fakultät für Physik, Arnold Sommerfeld Center for Theoretical Physics, D-80333 München, Germany
17
22
5
13
Gerhard Buchalla, Oscar Catà, Claudius Krause
15 16
4
7
Corrigendum to: “Complete electroweak chiral Lagrangian with a light Higgs at NLO” [Nucl. Phys. B 880 (2014) 552–573]
13 14
3
32 33 34 35 36 37 38 39 40 41 42
DOI of original article: http://dx.doi.org/10.1016/j.nuclphysb.2014.01.018. E-mail address: [email protected] (G. Buchalla). http://dx.doi.org/10.1016/j.nuclphysb.2016.09.010 0550-3213/© 2014 The Author(s). Published by Elsevier B.V. All rights reserved.
43 44 45 46 47
JID:NUPHB AID:13872 /ERR
1 2 3 4 5 6
[m1+; v1.237; Prn:5/10/2016; 14:58] P.2 (1-4)
G. Buchalla et al. / Nuclear Physics B ••• (••••) •••–•••
2
vs √ = 2
λh μ2s − ημ2h , λs λh − η 2
vh √ = 2
λs μ2h − ημ2s λs λh − η 2
Expanding both scalars around their vacuum expectation value, i.e. |φi | = a potential of the form
12
λs vs2 2 λh vh2 2 η V= h + h + vs vh hs hh + O(h3i ) 4 s 4 h 2 The transformation hs H1 cos χ − sin χ = sin χ cos χ H2 hh
13
diagonalizes the mass matrix. The rotation angle χ is defined as
7 8 9 10 11
14 15
tan (2χ) =
16 17
20 21
2 M1,2
+ hi ), leads to
2ηvs vh λh vh2 − λs vs2
λh vh2 − λs vs2 1 = (λh vh2 + λs vs2 ) ∓ 4 4 cos (2χ)
24 25 26 27 28 29 30 31 32 33
6
(3)
36 37 38 39 40 41 42 43 44 45 46 47
7 8 9 10
(4)
11 12 13 14
(5)
15 16 17 18
(6)
19 20 21
The Lagrangian relevant for the two scalars then reads
22
1 1 LH = ∂μ H1 ∂ μ H1 + ∂μ H2 ∂ μ H2 − V (H1 , H2 ) 2 2 v2 2a1 2a2 b1 2 b12 b2 2 † μ + Dμ U D U 1 + H1 + H2 + 2 H1 + 2 H1 H2 + 2 H2 4 v v v v v c c 1 2 ¯ e U P− η + h.c. 1 + H1 + H2 , ¯ d U P− r + lY − v qY ¯ u U P+ r + qY v v where 1 1 V (H1 , H2 ) = M12 H12 + M22 H22 − λ1 H13 − λ2 H12 H2 − λ3 H1 H22 − λ4 H23 2 2 − z1 H14 − z2 H13 H2 − z3 H12 H22 − z4 H1 H23 − z5 H24
23 24
(7)
25 26 27 28 29 30 31
(8)
32 33
34 35
4 5
22 23
2 3
√1 (vi 2
The masses of the physical states H1 and H2 are given by
18 19
1
(2)
34
The couplings λi and zi depend on μs , μh , λs , λh and η. With the parameters of the Higgs-portal model a1 = b1 = c1 = cos χ, a2 = b2 = c2 = sin χ, b12 = 2 sin χ cos χ, (9) the theory is renormalizable and unitary. The scalar H1 is now identified with the light scalar h that was found at the LHC. H2 is assumed to be heavy such that it can be integrated out. In doing so, we take its mass M2 to be larger than all other energy scales in the model, M2 vh , vs , M1 . In this limit the couplings λs , λh and η become large. We will assume, however, that they still remain in a regime where perturbation theory is a sufficiently reliable approximation. H2 can then be integrated out at tree level by solving its equation of motion and inserting the solution into the Lagrangian (7). The H2 -part of this Lagrangian can be written as 1 LH2 = H2 (−∂ 2 − M22 )H2 + J1 H2 + J2 H22 + J3 H23 + J4 H24 2
35 36 37 38 39 40 41 42 43 44 45 46
(10)
47
JID:NUPHB AID:13872 /ERR
[m1+; v1.237; Prn:5/10/2016; 14:58] P.3 (1-4)
G. Buchalla et al. / Nuclear Physics B ••• (••••) •••–••• 1 2
where the Ji can be read off from (7) and (8). Making the dependence on M2 explicit, the Ji take the form
3
6
where is a pure polynomial in H1 ≡ h. The equation of motion for H2 reads
It can be solved order by order in powers of
10
(0)
12 13
(0)
H2
(0)
H2 =
15
19 20 21 22
∞
27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
(13)
13 14
can then be obtained in terms of etc. Inserting the solution (13) of (12) back into (10), (7), and expanding in 1/M22 , one arrives at the effective Lagrangian of the model with H2 integrated out in the limit described above. At leading order, O(1/M20), the result has the form (n) of the electroweak chiral Lagrangian in (4), with the functions FU (h), V (h) and n Yˆf (h/v)n given as infinite series in h. For example, FU (h) = 2a1
h2
h + (b1 − a1 a22 (a1 + a2 v/vh )) 2 + O(h3 ) v v 1 ), M24
(16)
OψS1 , OψS2 , OψS7 , OψS14 , OψS15 , OψS18 ,
22
24
27 28
31
(17)
32 33 34 35 36
(18)
the hermitean conjugates of the OψSi in (18), and 4-fermion operators coming from the square of the Yukawa bilinears contained in J¯1 . The 4-fermion operators that are generated have the same structure as those in the heavy-Higgs model discussed in [6], which are1
37 38 39 40 41
43
OLR1 , OLR2 , OLR3 , OLR4 , OLR8 , OLR9 , OLR10 , OLR11 , OLR12 , OLR13 , OLR17 , OLR18
(19)
and their hermitean conjugates, but they are now dressed with functions Fi (h/v).
46 47
21
30
OF Y 1 , OF Y 3 , OF Y 5 , OF Y 7 , OF Y 9 , OF Y 10 , OST 5 , OST 9 ,
45
20
29
The effective Lagrangian LN LO contains operators that modify the leading-order Lagrangian (4) as well as a subset of the next-to-leading operators of Section 4. In particular, we have OD1 , OD7 , OD11 ;
19
26
(0) (H2
≡ H0 )
2 (−∂ 2 + 2J¯2 )H0 + J¯1 + 3J¯3 H02 + 4J¯4 H03 LN LO = 2M22 (1 − 2J20 − 6J30 H0 − 12J40 H02 )
18
25
Extending the derivation to the NLO terms of O(1/M22 ) one finds Leff = LLO + LN LO + O(
17
23
(15)
42
44
15 16
(0) H2 ,
where
11 12
(14)
(1) H2
25 26
10
(l)
H2 = O(1/M22l )
rk hk
23 24
(2)
8 9
by expanding
k=2
16
18
1/M22
7
(12)
can be determined from the O(M22 ) piece of (12) as an infinite series in powers of h
14
17
(1)
4
6
H2 = H2 + H2 + H2 + . . . ,
11
2
5
(−∂ 2 − M22 + 2J2 )H2 + J1 + 3J3 H22 + 4J4 H23 = 0
8 9
(11)
Ji0
7
1
3
Ji ≡ M22 Ji0 + J¯i
4 5
3
42 43 44 45 46
1 The terms O LR2 , OLR4 , OLR11 and OLR13 had been missed in the discussion of the heavy-Higgs models in [6,7].
47
JID:NUPHB AID:13872 /ERR
1 2 3 4 5 6
[m1+; v1.237; Prn:5/10/2016; 14:58] P.4 (1-4)
G. Buchalla et al. / Nuclear Physics B ••• (••••) •••–•••
4
This discussion shows explicitly how a subset of our NLO operators is generated in the Higgs-portal scenario. After integrating out the heavy scalar H2 in the non-decoupling limit M2 vh , vs , M1 the effective theory takes the form of a chiral Lagrangian. In particular, even for Fi (h/v) → 1, it is seen that operators of canonical dimension 4 (OD1 ), 5 (OψSi ) and 6 (4-fermion terms) contribute at the same (next-to-leading) order 1/M22 . This shows that the effective Lagrangian is not simply organized in terms of canonical dimension.
7 8
11 12 13
18 19 20 21 22
4 5 6
8
10 11 12 13 14
References
16 17
3
9
We thank Alejandro Celis for collaboration on the model discussed in Sec. 7.2. This work was performed in the context of the ERC Advanced Grant project ‘FLAVOUR’ (267104) and was supported in part by the DFG grant BU 1391/2-1, the DFG cluster of excellence ‘Origin and Structure of the Universe’ and the Munich Institute for Astro- and Particle Physics (MIAPP).
14 15
2
7
Acknowledgements
9 10
1
15 16
[1] [2] [3] [4] [5] [6] [7]
G. Buchalla, O. Catà, A. Celis, C. Krause, arXiv:1608.03564 [hep-ph]. R. Schabinger, J.D. Wells, Phys. Rev. D 72 (2005) 093007, arXiv:hep-ph/0509209. B. Patt, F. Wilczek, arXiv:hep-ph/0605188. M. Bowen, Y. Cui, J.D. Wells, J. High Energy Phys. 0703 (2007) 036, arXiv:hep-ph/0701035. C. Englert, T. Plehn, D. Zerwas, P.M. Zerwas, Phys. Lett. B 703 (2011) 298, arXiv:1106.3097 [hep-ph]. G. Buchalla, O. Catà, J. High Energy Phys. 1207 (2012) 101, arXiv:1203.6510 [hep-ph]. G. Buchalla, O. Catà, C. Krause, Nucl. Phys. B 880 (2014) 552, arXiv:1307.5017 [hep-ph].
17 18 19 20 21 22
23
23
24
24
25
25
26
26
27
27
28
28
29
29
30
30
31
31
32
32
33
33
34
34
35
35
36
36
37
37
38
38
39
39
40
40
41
41
42
42
43
43
44
44
45
45
46
46
47
47
[m1+; v1.237; Prn:5/10/2016; 14:58] P.1 (1-4)
Available online at www.sciencedirect.com
ScienceDirect 1
1
2
2
Nuclear Physics B ••• (••••) •••–•••
3
www.elsevier.com/locate/nuclphysb
4 5
Corrigendum
6 7 8 9 10 11 12
19
21
25 26 27 28
The matching of the Higgs-portal example to the chiral Lagrangian in Section 7.2 is corrected to properly account for the systematics of the relevant strong-coupling limit. The text of Section 7.2 given below supersedes the previous version. The main conclusions of Section 7.2 and the rest of the paper are not affected by this correction. The Higgs-portal model discussed here is equivalent to the Standard Model extended by a heavy scalar singlet. The chiral Lagrangian as the low-energy effective field theory of this model in the strong-coupling regime is further elaborated on in [1].
33 34 35 36 37 38 39 40 41
7.2. Higgs portal
44 45 46 47
12
14
16
18
20
22 23 24 25 26 27 28
30 31
As a specific model for a UV completion we consider the Higgs portal (see [2–5] and references therein). This model postulates the existence of a new, Standard-Model singlet scalar particle, which has allowed dimension-4 couplings to the Higgs field. This interaction modifies the scalar potential of Eq. (4) to μ2 μ2 λs λh η V = − s |φs |2 + |φs |4 − h |φh |2 + |φh |4 + |φs |2 |φh |2 , (1) 2 4 2 4 2 where φs refers to the standard scalar doublet and φh denotes the hidden scalar. Both of them acquire a vacuum expectation value, which can be written as
42 43
11
29
31 32
10
21
29 30
9
19
Editor: Hong-Jian He
20
24
8
17
Received 9 September 2016; accepted 17 September 2016
18
23
6
15
Ludwig-Maximilians-Universität München, Fakultät für Physik, Arnold Sommerfeld Center for Theoretical Physics, D-80333 München, Germany
17
22
5
13
Gerhard Buchalla, Oscar Catà, Claudius Krause
15 16
4
7
Corrigendum to: “Complete electroweak chiral Lagrangian with a light Higgs at NLO” [Nucl. Phys. B 880 (2014) 552–573]
13 14
3
32 33 34 35 36 37 38 39 40 41 42
DOI of original article: http://dx.doi.org/10.1016/j.nuclphysb.2014.01.018. E-mail address: [email protected] (G. Buchalla). http://dx.doi.org/10.1016/j.nuclphysb.2016.09.010 0550-3213/© 2014 The Author(s). Published by Elsevier B.V. All rights reserved.
43 44 45 46 47
JID:NUPHB AID:13872 /ERR
1 2 3 4 5 6
[m1+; v1.237; Prn:5/10/2016; 14:58] P.2 (1-4)
G. Buchalla et al. / Nuclear Physics B ••• (••••) •••–•••
2
vs √ = 2
λh μ2s − ημ2h , λs λh − η 2
vh √ = 2
λs μ2h − ημ2s λs λh − η 2
Expanding both scalars around their vacuum expectation value, i.e. |φi | = a potential of the form
12
λs vs2 2 λh vh2 2 η V= h + h + vs vh hs hh + O(h3i ) 4 s 4 h 2 The transformation hs H1 cos χ − sin χ = sin χ cos χ H2 hh
13
diagonalizes the mass matrix. The rotation angle χ is defined as
7 8 9 10 11
14 15
tan (2χ) =
16 17
20 21
2 M1,2
+ hi ), leads to
2ηvs vh λh vh2 − λs vs2
λh vh2 − λs vs2 1 = (λh vh2 + λs vs2 ) ∓ 4 4 cos (2χ)
24 25 26 27 28 29 30 31 32 33
6
(3)
36 37 38 39 40 41 42 43 44 45 46 47
7 8 9 10
(4)
11 12 13 14
(5)
15 16 17 18
(6)
19 20 21
The Lagrangian relevant for the two scalars then reads
22
1 1 LH = ∂μ H1 ∂ μ H1 + ∂μ H2 ∂ μ H2 − V (H1 , H2 ) 2 2 v2 2a1 2a2 b1 2 b12 b2 2 † μ + Dμ U D U 1 + H1 + H2 + 2 H1 + 2 H1 H2 + 2 H2 4 v v v v v c c 1 2 ¯ e U P− η + h.c. 1 + H1 + H2 , ¯ d U P− r + lY − v qY ¯ u U P+ r + qY v v where 1 1 V (H1 , H2 ) = M12 H12 + M22 H22 − λ1 H13 − λ2 H12 H2 − λ3 H1 H22 − λ4 H23 2 2 − z1 H14 − z2 H13 H2 − z3 H12 H22 − z4 H1 H23 − z5 H24
23 24
(7)
25 26 27 28 29 30 31
(8)
32 33
34 35
4 5
22 23
2 3
√1 (vi 2
The masses of the physical states H1 and H2 are given by
18 19
1
(2)
34
The couplings λi and zi depend on μs , μh , λs , λh and η. With the parameters of the Higgs-portal model a1 = b1 = c1 = cos χ, a2 = b2 = c2 = sin χ, b12 = 2 sin χ cos χ, (9) the theory is renormalizable and unitary. The scalar H1 is now identified with the light scalar h that was found at the LHC. H2 is assumed to be heavy such that it can be integrated out. In doing so, we take its mass M2 to be larger than all other energy scales in the model, M2 vh , vs , M1 . In this limit the couplings λs , λh and η become large. We will assume, however, that they still remain in a regime where perturbation theory is a sufficiently reliable approximation. H2 can then be integrated out at tree level by solving its equation of motion and inserting the solution into the Lagrangian (7). The H2 -part of this Lagrangian can be written as 1 LH2 = H2 (−∂ 2 − M22 )H2 + J1 H2 + J2 H22 + J3 H23 + J4 H24 2
35 36 37 38 39 40 41 42 43 44 45 46
(10)
47
JID:NUPHB AID:13872 /ERR
[m1+; v1.237; Prn:5/10/2016; 14:58] P.3 (1-4)
G. Buchalla et al. / Nuclear Physics B ••• (••••) •••–••• 1 2
where the Ji can be read off from (7) and (8). Making the dependence on M2 explicit, the Ji take the form
3
6
where is a pure polynomial in H1 ≡ h. The equation of motion for H2 reads
It can be solved order by order in powers of
10
(0)
12 13
(0)
H2
(0)
H2 =
15
19 20 21 22
∞
27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
(13)
13 14
can then be obtained in terms of etc. Inserting the solution (13) of (12) back into (10), (7), and expanding in 1/M22 , one arrives at the effective Lagrangian of the model with H2 integrated out in the limit described above. At leading order, O(1/M20), the result has the form (n) of the electroweak chiral Lagrangian in (4), with the functions FU (h), V (h) and n Yˆf (h/v)n given as infinite series in h. For example, FU (h) = 2a1
h2
h + (b1 − a1 a22 (a1 + a2 v/vh )) 2 + O(h3 ) v v 1 ), M24
(16)
OψS1 , OψS2 , OψS7 , OψS14 , OψS15 , OψS18 ,
22
24
27 28
31
(17)
32 33 34 35 36
(18)
the hermitean conjugates of the OψSi in (18), and 4-fermion operators coming from the square of the Yukawa bilinears contained in J¯1 . The 4-fermion operators that are generated have the same structure as those in the heavy-Higgs model discussed in [6], which are1
37 38 39 40 41
43
OLR1 , OLR2 , OLR3 , OLR4 , OLR8 , OLR9 , OLR10 , OLR11 , OLR12 , OLR13 , OLR17 , OLR18
(19)
and their hermitean conjugates, but they are now dressed with functions Fi (h/v).
46 47
21
30
OF Y 1 , OF Y 3 , OF Y 5 , OF Y 7 , OF Y 9 , OF Y 10 , OST 5 , OST 9 ,
45
20
29
The effective Lagrangian LN LO contains operators that modify the leading-order Lagrangian (4) as well as a subset of the next-to-leading operators of Section 4. In particular, we have OD1 , OD7 , OD11 ;
19
26
(0) (H2
≡ H0 )
2 (−∂ 2 + 2J¯2 )H0 + J¯1 + 3J¯3 H02 + 4J¯4 H03 LN LO = 2M22 (1 − 2J20 − 6J30 H0 − 12J40 H02 )
18
25
Extending the derivation to the NLO terms of O(1/M22 ) one finds Leff = LLO + LN LO + O(
17
23
(15)
42
44
15 16
(0) H2 ,
where
11 12
(14)
(1) H2
25 26
10
(l)
H2 = O(1/M22l )
rk hk
23 24
(2)
8 9
by expanding
k=2
16
18
1/M22
7
(12)
can be determined from the O(M22 ) piece of (12) as an infinite series in powers of h
14
17
(1)
4
6
H2 = H2 + H2 + H2 + . . . ,
11
2
5
(−∂ 2 − M22 + 2J2 )H2 + J1 + 3J3 H22 + 4J4 H23 = 0
8 9
(11)
Ji0
7
1
3
Ji ≡ M22 Ji0 + J¯i
4 5
3
42 43 44 45 46
1 The terms O LR2 , OLR4 , OLR11 and OLR13 had been missed in the discussion of the heavy-Higgs models in [6,7].
47
JID:NUPHB AID:13872 /ERR
1 2 3 4 5 6
[m1+; v1.237; Prn:5/10/2016; 14:58] P.4 (1-4)
G. Buchalla et al. / Nuclear Physics B ••• (••••) •••–•••
4
This discussion shows explicitly how a subset of our NLO operators is generated in the Higgs-portal scenario. After integrating out the heavy scalar H2 in the non-decoupling limit M2 vh , vs , M1 the effective theory takes the form of a chiral Lagrangian. In particular, even for Fi (h/v) → 1, it is seen that operators of canonical dimension 4 (OD1 ), 5 (OψSi ) and 6 (4-fermion terms) contribute at the same (next-to-leading) order 1/M22 . This shows that the effective Lagrangian is not simply organized in terms of canonical dimension.
7 8
11 12 13
18 19 20 21 22
4 5 6
8
10 11 12 13 14
References
16 17
3
9
We thank Alejandro Celis for collaboration on the model discussed in Sec. 7.2. This work was performed in the context of the ERC Advanced Grant project ‘FLAVOUR’ (267104) and was supported in part by the DFG grant BU 1391/2-1, the DFG cluster of excellence ‘Origin and Structure of the Universe’ and the Munich Institute for Astro- and Particle Physics (MIAPP).
14 15
2
7
Acknowledgements
9 10
1
15 16
[1] [2] [3] [4] [5] [6] [7]
G. Buchalla, O. Catà, A. Celis, C. Krause, arXiv:1608.03564 [hep-ph]. R. Schabinger, J.D. Wells, Phys. Rev. D 72 (2005) 093007, arXiv:hep-ph/0509209. B. Patt, F. Wilczek, arXiv:hep-ph/0605188. M. Bowen, Y. Cui, J.D. Wells, J. High Energy Phys. 0703 (2007) 036, arXiv:hep-ph/0701035. C. Englert, T. Plehn, D. Zerwas, P.M. Zerwas, Phys. Lett. B 703 (2011) 298, arXiv:1106.3097 [hep-ph]. G. Buchalla, O. Catà, J. High Energy Phys. 1207 (2012) 101, arXiv:1203.6510 [hep-ph]. G. Buchalla, O. Catà, C. Krause, Nucl. Phys. B 880 (2014) 552, arXiv:1307.5017 [hep-ph].
17 18 19 20 21 22
23
23
24
24
25
25
26
26
27
27
28
28
29
29
30
30
31
31
32
32
33
33
34
34
35
35
36
36
37
37
38
38
39
39
40
40
41
41
42
42
43
43
44
44
45
45
46
46
47
47