Anomalous W/Z quartic couplings (QGCs) 1

Anomalous W/Z Quartic Couplings (QGCs)
Revised March 2024 by M.W. Grünewald (U. College Dublin) and A. Gurtu (CERN;TIFR Mumbay).
Quartic couplings, W W ZZ, W W Zγ, W W γγ, and ZZγγ, were studied at LEP andTevatron at energies at which the Standard Model predicts negligible contributions tomultiboson production. Thus, to parametrize limits on these couplings, an effective
theory approach is adopted which supplements the Standard Model Lagrangian withhigher dimensional operators which include quartic couplings. The LEP collaborations
chose the lowest dimensional representation of operators (dimension 6) which presumes
the SU(2)×U(1) gauge symmetry is broken by means other than the conventional Higgs
scalar doublet [1–3]. In this representation possible quartic couplings, a0 , ac , an , areexpressed in terms of the following dimension-6 operators [1,2];e2 a F µν F W~α · W~αL06 = − 16Λµν
2 0~β · W
e2 a F µα F W~αLc6 = − 16Λµβ2 c2(i)(j)eLn6 = −i 16Λ2 an ǫijk Wµα Wν

W (k)α F µν

e 0 = − e2 e~αLa F µν Feµν W~ α · W
616Λ2 0(i)(j)e n = −i e2 eLa ǫ Wµα Wν W (k)α Feµν
616Λ2 n ijk

e 0 conserves
where F, W are photon and W fields, L06 and Lc6 conserve C, P separately (L6enonly C) and generate anomalous W + W − γγ and ZZγγ couplings, Lnviolates
CP (L66violates both C and P ) and generates an anomalous W + W − Zγ coupling, and Λ is anenergy scale for new physics. For the ZZγγ coupling the CP -violating term representedby Ln
6 does not contribute. These couplings are assumed to be real and to vanish at treelevel in the Standard Model.
Within the same framework as above, a more recent description of the quartic
couplings [3] treats the anomalous parts of the W W γγ and ZZγγ couplings separately,
leading to two sets parametrized as aV0 /Λ2 and aVc /Λ2 , where V = W or Z.
With the discovery of a Higgs at the LHC in 2012, it is then useful to go to the next
higher dimensional representation (dimension 8 operators) in which the gauge symmetryis broken by the conventional Higgs scalar doublet [3,4]. There are 14 operators which
can contribute to the anomalous quartic coupling signal. Some of the operators haveanalogues in the dimension 6 scheme. The CMS collaboration, [5], have used thisparametrization, in which the connections between the two schemes are also summarized:
e2 aW
0 F F µν W +a W −µνa
8 Λ2e2 aWcFµν F µa (W +ν Wa− + W −ν Wa+ )−16 Λ2

LAQGC = −S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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2Anomalous W/Z quartic couplings (QGCs)
κW0 F Z µν W +a W −µνaΛ2e2 g 2 κW
c−Fµν Z µa (W +ν Wa− + W −ν Wa+ )2 Λ2fT,0cµν W
c µν ] × T r[Wcαβ Wc αβ ]+ 4 T r[WΛ
− e2 g 2

The energy scale of possible new physics is Λ, and g = e/sin(θW ), e being the unit
electric charge and θW the Weinberg angle. The field tensors are described in [3,4].
2W
2The two dimension 6 operators aW0 /Λ and ac /Λ are associated with the W W γγ2W2vertex. Among dimension 8 operators, κW
0 /Λ and κc /Λ are associated with the4W W Zγ vertex, whereas the parameter fT,0 /Λ contributes to both vertices. There is arelationship between these two dimension 6 parameters and the dimension 8 parameters
fM,i /Λ4 as follows [3]:22aW
0 = − 4MW fM,0 − 8MW fM,2
Λ2g 2 Λ4g ′ 2 Λ42 f2 f4MW
8MWaWM,3M,1c=−−
2224′
Λ
gΛ
Λ4g

where g ′ = e/cos(θW ) and MW is the invariant mass of the W boson. This relation
4provides a translation between limits on dimension 6 operators aW0,c and fM,j /Λ . It isfurther required [4] that fM,0 = 2fM,2 and fM,1 = 2fM,3 which suppresses contributions
to the W W Zγ vertex. The complete set of Lagrangian contributions as presented in [4]corresponds to 19 anomalous couplings in total – fS,i , i = 1, 2, fM,i , i = 0, . . . , 8 and
fT,i , i = 0, . . . , 9 – each scaled by 1/Λ4 .
Another approach to couplings is the so called K-matrix framework [7], in which theanomalous couplings can be expressed in terms of two parameters α4 and α5 , which
account for all BSM effects.
The LHC collaborations have published couplings results based on various theoreticalframeworks. It is hoped that the collaborations will agree to use at least one common setof parameters to express these limits to enable the reader to make a comparison, and toallow for a possible LHC combination.
References:
1. G. Belanger and F. Boudjema, Phys. Lett. B288, 201 (1992).
2. J.W. Stirling and A. Werthenbach, Eur. Phys. J. C14, 103 (2000);
J.W. Stirling and A. Werthenbach, Phys. Lett. B466, 369 (1999);
A. Denner et al., Eur. Phys. J. C20, 201 (2001);
G. Montagna et al., Phys. Lett. B515, 197 (2001).
3. G. Belanger et al., Eur. Phys. J. C13, 283 (2000).
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Anomalous W/Z quartic couplings (QGCs) 3
4. O.J.P. Éboli, M.C. Gonzalez-Garcia, and S.M. Lietti, Phys. Rev. D69, 095005(2004);
O.J.P. Éboli, M.C. Gonzalez-Garcia, and J.K. Mizukoshi, Phys. Rev. D77, 073005(2006).
5. S. Chatrchyan et al., Phys. Rev. D90, 032008 (2014);
S. Chatrchyan et al., Phys. Rev. Lett. 114, 051801 (2015).
6. G. Aad et al., Phys. Rev. Lett. 113, 141803 (2014).
7. A. Albateanu, W. Killian, and J. Reuter, JHEP 0811, 010 (2008).

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