DynamicalMassGenerationinWickCutkosky Model · dynamical generation of mass becomes crucial for scalars with masses much lower than the electroweak scale. This paper addresses Yukawa

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Dynamical Mass Generation in Wick Cutkosky

Model

Tajdar Mufti ∗

Lahore University of Management Sciences

Opposite Sector U, D.H.A, Lahore Cantt., 54792, Pakistan

July 2, 2020

Abstract

Studying extent of dynamical generation of mass in a quantum fieldtheory, which may have non-perturbative attributes, is an essentialstep towards complete understanding of the theory. It has historicrelevance to interactions including fermions. However, as search offurther fundamental scalars continues, inquiring into the possibility ofdynamical generation of mass becomes crucial for scalars with massesmuch lower than the electroweak scale. This paper addresses Yukawainteraction between a complex doublet field, conveniently named theHiggs, and a scalar singlet studied in terms of correlation functionsand the dynamical masses produced in the parameter space of themodel. The study is conducted using the method of Dyson Schwingerequation. The Higgs is found to be receiving negative squared masseswhile the scalar singlet field receives positive squared mass. Higgspropagators are found to be significantly more sensitive to couplingsin comparison to the scalar propagators. The vertices are relativelymore stable against the cutoff used in comparison to the propagatorsand dynamically generated masses. No indication of a critical couplingis found within the explored range of coupling values. The model isfound to have strong cutoff effects until the cutoff is raised to 100 TeV.The model suggests no sign of triviality.

1 Introduction

Importance of the standard model (SM) [1, 2] as the theory of low energyphenomenology for particle interactions can not be overstated after the dis-covery of Higgs boson [3–5] at the LHC [6–8]. The discovery also openspossibility of existence of other fundamental scalars (or pseudo-scalars) innature, particularly the ones required by various cogent theories, such as

∗[email protected], [email protected]

1

http://arxiv.org/abs/2007.00092v1

Supersymmetry [9–13], cosmic inflation [14–21], and dark matter [22–26].As in nature there exists a wide range of masses from a few orders less than1 eV to close to 200 GeVs, the possibilities of masses for further scalars is aan open question.

One of the main reasons to expect the Higgs was to render the elec-troweak gauge bosons and several other fundamental particles massive. How-ever, it is a well known fact that the lightest quarks do not direly requireHiggs mechanism as they can also generate masses via QCD interactions 1.With no information regarding the masses of further scalars, studying extentof dynamical generation of masses (DMG) [29–40] via scalar interactions invarious models naturally finds its place.

An element of ambiguity in studies related to new physics is that itis not truly known if the new physics is accessible through the approach ofperturbation. Furthermore, DMG is a non-perturbative phenomenon by def-inition. Hence, non-perturbative methods, such as that of Dyson SchwingerEquations (DSEs) [41, 42, 51–53], become necessary tool to investigate theunderlying physics.

However, as is the case with almost every non-perturbative approaches,DSEs posses a limitation that there are more unknown correlation functionsthan the equations. A commonly used approach to cope with it is to useansatz and truncations [38, 41, 42, 51, 52] which may effect the correlationfunctions. Thus, beside studying the physics, relatively simpler models thanSM offer an extremely important playground for implementing and study-ing unconventional numerical approaches. A model containing the Yukawainteraction vertex is among the highly interesting avenues to study scalarinteractions due to its relevance to particle physics phenomenology.

This paper is a continuation of studies of a variant of Wick Cutkoskymodel [45–50], and addresses the phenomenon of DMG due to a Yukawa in-teraction between Higgs field, which preserves SU(2) symmetry, and a (real)singlet scalar field using the approach of DSEs [41, 42, 51–53]. There areseveral benefits such an study offers. Firstly, the model serves as a useful av-enue to study how two mutually interacting fields with different symmetriesand under the same renormalization condition dynamically acquire massesfor various cutoff values. Secondly, understanding the features, such as exis-tence of critical coupling and phase structure, in a model containing Yukawainteraction may provide valuable information in new Physics searches. Fur-thermore, a relatively simpler interaction offers a test ground to employuncustomary approaches which could later be used in richer theories.

The model is studied for (bare) coupling values 10−2 ≤ λ ≤ 2.0 in GeVs2 and cutoff values at 40 TeV, 80 TeV, and 100 TeV. By the definition of

1There also exist various suggestions in literature, see for example [28]2Bare couplings are taken in GeV in order to keep the study in the perspective of

electroweak physics. Furthermore, no peculiarities of the Wick Cutkosky model is used.

2

DMG, both fields are massless unless they interact. The renormalizationpoint is also chosen corresponding to the physics involved.

A significant amount of studies involving scalar interactions is with inclu-sion of four point interactions in the realm of renormalizable quantum fieldtheories. It includes both the four point self interactions of Higgs or scalarfields, or the vertex formed by two scalar and two Higgs fields. However,these three type of interactions can also be formed by the three point Yukawainteraction mentioned above. Thus, in this paper only the seemingly morefundamental Yukawa interaction is considered. Inclusion of further verticesare to be reported somewhere else [54] using a different non-perturbativeapproach [43, 44].

The φ4 theory [55–57] is found to be trivial [58–62]. However, despitebeing a (complex doublet) scalar field, Higgs interactions with gauge bosonsin the Yang-Mills-Higgs theory [63, 64] is not found to render the theorytrivial. Hence, it is implicitly assumed that the model considered is not atrivial theory.

The paper encompasses study of the theory in terms of correlation func-tions and dynamically generated masses using two DSEs for the two fieldpropagators without the truncations or ansatz mentioned above. Insteadof employing the conventional approach, the DSEs for the two field prop-agators containing the propagators and a vertex are used in a numericallycontrolled environment. The only hard constraints are renormalization con-ditions, which are typical of renormalizable quantum field theories, and arestriction on the vertex to restrict violent local fluctuations. The detailsare mentioned in the next section.

2 Technical Details

The Euclidean version of the Lagrangian 3, along with the counter terms[65], is given by 4

L =(1 +A)δµν∂µh†∂νh+ (m2

h +B)h†h+ (1 + α)1

2δµν∂µφ∂νφ+

1

2(m2

s + β)φ2 + λ(1 +C

λ)φh†h

(1)

with h as the Higgs fields with SU(2) symmetry and φ a real scalar sin-glet field. A, B, C, α, and β are counter terms, and λ is the three pointinteraction coupling. The DSEs for scalar and the Higgs propagators are,

3Bare masses are shown in the Lagrangian for the sake of convenience.4A considerable part of technical details can also be found in other reports. The details

are kept in their entirety for the sake of self-sufficiency.

3

respectively, given by

S(p)−1 =(1 + α)p2 +m2

s + β+

λ(1 +C

λ)

∫d4q

(2π)4H ik(q)Γkl(q, p − q,−p)H li(q − p)

(2)

H ij(p)−1 =δij((1 +A) p2 +m2

h +B) +

2λ(1 +C

λ)

∫d4q

(2π)4S(q)Γik(−p, p − q, q)Hkj(q − p)

(3)

where Γkl(u, v, w) is the three point Yukawa interaction vertex of Higgs,Higgs bar, and scalar fields with momenta u, v, and w, respectively, Higgsand Higgs bar 5 fields have indices k and l, respectively. S(p) and H ij(p) arescalar propagator and the Higgs propagator, respectively. Setting ms = 0and mh = 0, and introducing the following definitions of B and β, respec-tively,

B = 2λ(1 +A)(1 + α)σh (4)

β = 2λ(1 +A)(1 + α)σs (5)

the dynamical squared masses for the Higgs and scalar fields, respectively,assume the following definitions 6.

m2

h,d = 2λ(1 +A)(1 + α)σh (6)

m2

s,d = 2λ(1 +A)(1 + α)σs (7)

Furthermore, defining the vertex as

Γikr = (1 +

C

λ)Γik = (1 +A)(1 + α)Γik (8)

we have the following equations 7,

S(p)−1 =(1 + α)( p2 + 2λ(1 +A)σs+

λ(1 +A)

∫d4q

(2π)4H ik(q)Γkl(q, p− q,−p)H li(q − p))

(9)

H ij(p)−1 =(1 +A)( δij( p2 + 2λ(1 + α)σh) +

2λ(1 + α)

∫d4q

(2π)4S(q)Γik(−p, p − q, q)Hkj(q − p))

(10)

5Throughout the paper, Higgs bar is referred to h†.6Since σh and σs can take any suitable value during computation, the definitions do

not impose any constraints on the dynamical squared masses.7The definitions introduced in equations 6-8 results in a multiplicative constant in

each of the equations 9 and 10 which facilitate in implementation of the renormalizationconditions.

4

The renormalization conditions [42] for the propagators are given below:

H ij(p)|p2=1 =δij

p2|p2=1 (11)

S(p)|p2=1 =1

p2|p2=1 (12)

No renormalization condition was explicitly imposed on the vertex 8. Thesymmetries in the DSEs are numerically implemented while at the same timeonly the flavor diagonal Higgs propagators are assumed to be non-vanishingin order to maintain the similarity between the Higgs propagator and itstree level structure.

As there is no DSE or ansatz used for the Yukawa vertex, three stepsare taken to ensure that the resulting correlation functions, particularly thevertex, are stable. Firstly, a condition is locally imposed on the vertex thatit never exceeds an order of magnitude relative to its neighboring momen-tum points in the 4-momentum (Euclidean) spacetime. The constraint issimilar to the well known Lipschitz condition abundantly used in literature[66–70]. The constraint is introduced to keep the vertex from fluctuatingviolently. Secondly, instead of calculating local deviations, i.e. difference be-tween the two sides of a DSE at each momentum separately, sum of squarederror is calculated, which leads to implementation of least squares method.It certainly slows the computations but it is found enormously helpful inimproving stability of the correlation functions. Furthermore, Higgs propa-gators are expanded in a polynomial form given below:

H ij(p) = δij1

c(p2 + d+ f(p))(13)

with f(p) defined as

f(p) =

N∑i=0

aip2i

N∑l=0

blp2l

(14)

where ai, bl, c, and d are parameters to be updated during a computation.Such a parameterization brings certain advantages. Firstly, the procedure ofrenormalization is significantly faster than the alternative approaches whichmay involve extrapolations. Secondly, there is a certain correspondence be-tween the self energy contribution and the expansion in equations 10 and 14,which renders the vertex a certain form. Hence, a stable vertex is attained

8It was found that introducing any renormalization condition on the vertex producesabnormal discontinuities in the vertices for very low coupling. This behavior was taken asa sign of over-constrained system.

5

whose fluctuations mostly depend upon the resolution between momentumvalues.

The computation proceeds as follows: It starts with tree level structuresfor the Higgs propagators and the vertex, and the parameters σs and σh areset to zero. Scalar propagator takes the values from its DSE, see equation 9.First, σs is calculated using Newton Raphson’s method 9 which is also usedfor updating the Higgs propagator and the vertex. The criterion for updateis that the sum of squared error (the difference between left and right handsides) in equation 10 decreases. It is followed by update of σh parameterusing the same procedure. It is followed by updates of the parameters (givenin equation 13) for the Higgs propagators in equations 9 and 10. Lastly, thevertex is updated at each momentum value while preserving the symmetryimposed by the DSEs. During updating of each of the above mentionedquantities, scalar propagator is calculated from equation 9. The parametersA and α are calculated during each update and calculation of the Higgsand scalar propagators, respectively. Hence, as a computation proceeds,σs and σh deviate from their starting value, while the correlation functionsnumerically deviate from their tree level structures. The computation endswhen either there is no further improvement in terms of the sum of squarederror or the error has reached a value below the preset minimum error 10.The minimum error is preset at 10−20.

It was found that changing the sequence of updates of the parameters orthe correlation functions does not effect the results within machine precision.It was taken as the definition of uniqueness through out the study.

Gaussian quadrature algorithm is used for numerical integration.

3 Correlation Functions

3.1 Field Propagators

Scalar and Higgs propagators are shown in figures 1 and 2, respectively. Forthe case of scalar propagators, the most interesting feature is their strong(qualitative as well as quantitative) similarity for various couplings. It pointstowards the possibility that the model favors a certain narrow range ofdynamical scalar masses which, in an extreme case, can even be a particularvalue characteristic to the model. In fact, it was the Higgs propagator whichwas expected to be stable due to the reason that Higgs mass was found to

9For the quantity f (correlation function, or parameters σh or σs), the candidate value

is developed by fn = f1 −

(f2−f1)E(f1)E(f2)−E(f1)

, where f1, f2, and fn are the current value, the

modified value, and the candidate value of the quantity, and E(f1), and E(f2) are thecurrent value, and the value of sum of squared error as a result of modifying the quantityf, respectively. If fn decreases the error, it is accepted.

10All the results presented here belong to the later case. Beyond the preset minimumerror, the sum of squared errors are taken as zero.

6

0 20 40 60 80 100 120p [TeV]

3−10

2−10

1−10

1

]-2

D(p

) [G

eVScalar propagators

2, 1001.2, 100

1, 1000.9, 100

0.4, 1000.1, 1002, 80

1.2, 801, 80

0.9, 800.4, 800.1, 80

2, 401.2, 40

1, 400.9, 400.4, 40

0.1, 40

Figure 1: Scalar propagators for different couplings λ (in GeVs) and cutoffvalues Λ (in TeVs), shown as (λ,Λ) in the legend, are plotted. Startingwith the highest value of coupling constant for a fixed cutoff, each subse-quent scalar propagator with lower coupling constant is displaced by 5 GeVmomentum in the figure.

0 20 40 60 80 100 120p [TeV]

10−10

9−10

8−10

7−10

6−10

5−10

]-2

D(p

) [G

eV

Higgs propagators

2, 1001.2, 100

1, 1000.9, 100

0.4, 1000.1, 1002, 80

1.2, 801, 80

0.9, 800.4, 800.1, 80

2, 401.2, 40

1, 400.9, 400.4, 40

0.1, 40

Figure 2: Higgs propagators for different couplings λ (in GeVs) and cutoffvalues Λ (in TeVs), shown as (λ,Λ) in the legend, are plotted. Startingwith the highest value of coupling constant for a fixed cutoff, each subse-quent Higgs propagator with lower coupling constant is displaced by 5 GeVmomentum in the figure.

less less sensitive [71]. However, the presence of cutoff effects are vivid

7

10 20 30 40 50 60 70 80 90

Scalar momentum [TeV]2040

6080

100120

140

Higgs momentum [TeV]

15−

10−

5−

0

5

10

9−10×

Ver

tex

Higgs-scalar vertex

0.02, 100

10 20 30 40 50 60 70 80 90


6080

100120

140


30−

20−

10−

0

10

20

9−10×

Ver

tex

Higgs-scalar vertex

0.5, 100

10 20 30 40 50 60 70 80 90


6080

100120

140


5−

0

5

10

15

20

25

309−

10×

Ver

tex

Higgs-scalar vertex

0.9, 100

10 20 30 40 50 60 70 80 90


6080

100120

140


0.6−

0.4−

0.2−

0

0.2

0.4

0.6

0.8

6−10×

Ver

tex

Higgs-scalar vertex

2, 100

Figure 3: Yukawa interaction vertices between Higgs and scalar fields areshown for various couplings (in GeV) with cutoff at 100 TeV.

on the propagators which is severe enough to immediately expect cutoffdependence of scalar mass. The effects become milder for higher cutoffwhich is an indication of stability of the model against cutoff of the order ofhundreds of TeV, see figure 1.

In contrast to the scalar propagators, the Higgs propagators posses dif-ferent features. First of all, they are sensitive to coupling as well as thecutoff values. For the coupling values considerably lower than 1.0 GeV,Higgs propagators are found to be suppressed. The propagators are en-hanced as the coupling rises to the vicinity of 1.0 GeV. However, signs ofsimilar suppression are observed once again for further higher coupling val-ues. This change in behavior is peculiar since it takes place in the vicinityof λ = 0.8801 GeV which is naively the fourth root of self interaction cou-pling for Higgs 0.6 GeV 11. The propagators also have signs of qualitativedifferences in the infrared region. It indicates that the deviations amongthe Higgs propagators are not solely due to the parameter A. Furthermore,as is the case for scalar propagators, there are cutoff effects though not assevere as for the case of scalar propagators, see figure 2.

3.2 Higgs-scalar vertices

The Yukawa interaction vertices, defined in equation 8, are shown in figures3, 4, and 5 for various coupling values with cutoff at 100 TeV, 80 TeV, and40 TeV, respectively 12.

11At this point, it is speculated to have implications related to the four point selfinteractions which remains to be investigated in detail.

12The presence of fluctuations is due to lesser resolution among the field momenta.

8

10 20 30 40 50 60 70


6080

100


15−

10−

5−

0

5

10

15

9−10×

Ver

tex

Higgs-scalar vertex

0.02, 80

10 20 30 40 50 60 70


6080

100


50−40−30−20−10−0

10

20

30

9−10×

Ver

tex

Higgs-scalar vertex

0.5, 80

1020 30 40 50 60

70


6080

100


10−

0

10

20

30

40

509−

10×

Ver

tex

Higgs-scalar vertex

0.9, 80

1020 30 40 50 60

70


6080

100


2−1.5−

1−0.5−

0

0.5

1

1.5

2

2.56−

10×

Ver

tex

Higgs-scalar vertex

2, 80


5 10 15 20 25 30 35


3040

50


80−

60−

40−

20−

0

20

40

60

9−10×

Ver

tex

Higgs-scalar vertex

0.02, 40

5 10 15 20 25 30 35


3040

50


0.4−

0.2−

0

0.2

0.4

6−10×

Ver

tex

Higgs-scalar vertex

0.5, 40

510 15 20 25 30

35


3040

50


0

0.1

0.2

0.3

0.4

6−10×

Ver

tex

Higgs-scalar vertex

0.9, 40

510 15 20 25 30

35


3040

50


0.6−

0.4−

0.2−

0

0.2

0.4

0.6

0.86−

10×

Ver

tex

Higgs-scalar vertex

2, 40


Firstly, the vertex is found to have qualitative dependence on the cou-pling, see figures 3 to 5. In terms of qualitative dependence, they possesa similar trend in behavior as the Higgs propagators. Starting from thecoupling value λ = 0.02 GeV the vertex changes to a different qualitativestructure as λ increases. However, beyond λ = 0.9 GeV indications of arestoration of the qualitative dependence at low coupling emerge. It impliesthat Higgs’ role is relatively more significant than that of scalar field for

9

0 0.5 1 1.5 2 [GeV]λ

1

1.5

2

2.5

3

3.5

4

4.5

5

[MeV

]s,

dm

Scalar Dynamical Masses

100

80

40

Figure 6: Dynamically generated masses of scalar field ms as a function ofYukawa coupling λ at different cutoff values (in TeVs) are shown.

dynamical mass generation.The vertex is indeed found to have cutoff dependence to some extent,

though these effects are considerably weaker than for the case of the prop-agators. It supports the speculation that, as the vertex is not stronglyinfluenced by the cutoff values, it may be the masses whose cutoff effectstranslate to the propagators, see equations 9 and 10.

Qualitative dependence on field momenta is a clear indication that thetheory is not a trivial theory and the deviations in Higgs propagators aremore than a mere multiplicative constant to the corresponding tree levelstructure due to the contribution by self energy term, see equations 10.

4 Dynamical Renormalized Masses

The dynamical masses are plotted in figures 6 and 7 for scalar and Higgsfields, respectively. An immediate observation is strong cutoff effects onmasses of both fields, particularly below 50 TeVs. For the case of DMG, therenormalized masses do not have any contribution from tree level value. Itis also the reason that the cutoff effects appear vividly in the figures 6 and7. The stability in the masses ensues as the cutoff is raised considerablyabove 50 TeVs.

One of the most interesting features in the model is negative squaredmasses produced for the Higgs field. A negative squared masses is taken asa sign of symmetry breaking. In this context, Higgs is indeed found to bedynamically breaking the symmetry.

Furthermore, magnitude of the masses of both fields (for higher cutoff

10

0 0.5 1 1.5 2 [GeV]λ

25−

20−

15−

10−

5−

0]2 [M

eV2 h,

dm

Higgs Squared Dynamical Masses

100

80

40

Figure 7: Dynamically generated squared masses of the Higgs field ms as afunction of Yukawa coupling λ at different cutoff values (in TeVs) are shown.

values) are around a few MeVs over a wide range of Yukawa coupling values(taken in GeVs). It is close to the magnitude of the two lightest quarks [3]13. However, the cutoff at which the Masses tend to stabilize is well over ahundred TeV instead of a few tens of GeVs which is a typical cutoff for QCDphysics. This remarkable difference may be attributed to the simplicity ofthe model which does not contain any but a Yukawa interaction.

There is a certain deviation in both of the masses beyond λ = 1.0 GeVfrom the vicinity of 1 MeV magnitude. For both Higgs and the scalar sin-glet fields, the magnitude of squared masses are found to be suppressed asthe cutoff is increased. Since the cutoff for the Higgs related physics is tobe pushed far above hundreds of TeVs, it is expected that these qualitativedeviations for strong coupling may further subside, hence establishing sta-bility of masses over the Yukawa coupling values. Another possibility is areversal of the behavior beyond 1.0 GeV coupling. If such a situation doestake place in the parameter space of the theory, it may be a strong sign ofa phase structure in the model, possibly similar to a previous study [63] ofa model. However, probing such a feature requires further improvement inresolution.

From the figures 6 and 7, it is clear that the model does not have thecritical value of coupling [38] above λ = 10−2 GeV. Hence, given the observedqualitative behavior of the dynamical masses, it is expected that the criticalvalue of the coupling may very well be at least a few orders smaller than10−2 GeV if there exists any for the model.

13Let us note here that the renormalization schemes used for the current investigationand the one to calculate masses of quarks [3] are not identical.

11

5 Conclusion

Despite that the phenomenon of dynamical mass generation has mostly beenrelated to interactions involving fermions, it does find strong relevance toscalar sector even when the interaction vertex in the model is as simpleas a three point Yukawa interaction. The current study is an addition ofexploration of Wick Cutkosky model using the method of DSEs. The maingoals are to estimate extent of DMG in the model for the two fields, and tostudy the correlation functions. The study also presented an opportunityto explore Higgs’ role from the perspective of symmetry breaking, thoughdynamically.

The Higgs propagators are indeed found to be developing a pole in Eu-clidean (momentum version of) spacetime as the squared renormalized massassumes negative sign over the explored coupling values. Hence, the roleof Higgs is found to be qualitatively fulfilled in the model, despite that thesame renormalization condition is imposed on the two field propagators andthat both fields achieve similar magnitudes of masses. In other words, themodel demonstrates that the symmetries and group structures may stronglyinfluence the dynamical masses even if the fields are of the same nature.

As is the case with QCD interactions, the model is found to be pro-ducing a few MeV masses which serves as another limitation on the extentof dynamical mass generation. However, an important difference is thatthe masses stabilize over most of the coupling values for cutoff above 100TeVs which is many orders of magnitude higher than that for QCD inter-actions. Hence, the model serves as a reminder of the remarkable strengthand diversity of QCD interactions which could be competed, though in arestricted sense, with a simpler Yukawa interaction only at relatively muchhigher cutoff values.

No vertex was found to be vanishing. Hence, if there is a critical couplingin the model, below which no dynamical masses are generated, it may be atthe coupling values at least few orders of magnitude less than λ = 10−2 GeV.Within the studied region of the parameter space, the study serves as yetanother addition to the already known observation that Higgs interactionwith other fields does not render the theory trivial.

6 Acknowledgments

I am deeply indebted to Dr. Shabbar Raza for a number of valuable discus-sions during this endeavor. I would also like to express my deepest gratitudeto Professor Johan Hansson for his critical advices and inspiration by someof his work.

This work was supported by Lahore University of Management Sciences,Pakistan.

12

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DynamicalMassGenerationinWickCutkosky Model · dynamical generation of mass becomes crucial for scalars with masses much lower than the electroweak scale. This paper addresses Yukawa